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arxiv |
Electric-field induced capillary interaction of charged particles at a polar interface
7 Feb 2014
Lionel Foret
CPMOH
Université Bordeaux 1
351 cours de la Libération33405TalenceFrance
Alois Würger
CPMOH
Université Bordeaux 1
351 cours de la Libération33405TalenceFrance
Electric-field induced capillary interaction of charged particles at a polar interface
7 Feb 2014
We study the electric-field induced capillary interaction of charged particles at a polar interface. The algebraic tails of the electrostatic pressure of each charge results in a deformation of the interface u ∼ ρ −4 . The resulting capillary interaction is repulsive and varies as ρ −6 with the particle distance. As a consequence, electric-field induced capillary forces cannot be at the origin of the secondary minimum observed recently for charged PMMA particles at on oil-water interface.PACS numbers:Electrostatic forces operating on charged particles determine to a large extent the properties of emulsions and foams, and stabilize colloidal aggregates[1,2]. Charged particles at interfaces or in thin films form 2D hexagonal crystals[3][4][5][6][7], while for bulk colloidal suspensions several 3D crystal phases have been reported[8]. At a typical lattice spacing of a few microns, the interparticle forces are of the order of picoNewton.A charge at a polar interface and its counterion cloud carry a finite dipole moment perpendicular to the interface (Cf.Fig. 1.) The resulting electrostatic force acting on two neighboring particles is repulsive and varies with the inverse fourth power of their distance[3,[9][10][11][12]. This law has been confirmed experimentally for charged polystyrene particles[6].The pair potential of PMMA particles at an oil-water interface has been measured by recording the trajectories and evaluating distance correlations. Besides the dipolar repulsion at short distances, an attractive force has been found at larger distances, and a minimum in the potential energy has been shown to occur at about 5.7 µm [7]. As a possible explanation, these authors propose that the deformation of the interface by the particles gives rise to a capillary interaction that varies logarithmically with their distance[7].Quite generally, a capillary or elastic interaction is obtained when equilibrating an external force acting on the particles with the surface tension or the bending rigidity. A variety of such models have been studied, both for spherical and anisotropic defects[13][14][15][16][17][18][19]; these works deal with forces that act on the particle only, such as gravity. A somewhat different pattern arises for forces that operate both on the particle and on the surrounding interface. For example, a polymer grafted on a membrane exerts a force at the point of attachment and an opposite entropic pressure on the interface[20].In this Letter we study the deformation of a polar interface due to a charged colloidal particle, and we derive the resulting capillary interaction. The present work is confined to the case most relevant for micron size colloidal particles, where the distance is much larger than the Debye length. After a reminder of the free energy of a deformation field, we calculate the electrostatic pressure profile exerted by a charge and the associated counterions (as shown schematically inFig. 1.)The electric-field FIG. 1: a) Pressure profileπ(r) induced on the interface by a charged particle (filled grey circle); open circles are the counterions. b) The corresponding deformation fieldû(r).arXiv:cond-mat/0310657v3 [cond-mat.soft]
induced capillary interaction is compared to experimental findings and to the case of δ-force that was proposed in [7].
We consider particles of charge Q trapped at an electrolyte-insulator interface. In the absence of forces, the interface S is flat and its energy reads γS, where γ is the surface tension. The charged particles and their counterions exert on the interface a pressure that is given by the normal component of the stress tensor and comprises entropic and electrostrictive contributions,
π(r) = k B T (n + + n − ) + T ins zz − T el zz .(1)
The first term is the entropic pressure of the excess density of positive and negative ions, n ± = n s e ∓eφ(r)/k B T − 1 , induced by N surface charges. Electroneutrality requires e dV (n + −n − ) = −N Q. The remaining terms involve the Maxwell tensor
T ij = ε E i E j − 1 2 E 2 δ ij ,(2)
evaluated at the electrolyte and insulating side of the interface. Note that both the dielectric constant ε and the normal component E z of the electric field vector are discontinuous across the interface. The corresponding jump of the normal component of the Mawell tensor describes an electrostrictive force that arises from the difference in the electric field density in the two media. As a consequence of the inhomogeneous pressure, the interface is no longer flat; the deformation u(r) increases the total surface and thus the surface energy by the amount
γ dS 1 + (∇u) 2 − 1 ,
where ∇ is the 2D gradient in the interface plane. In the following, r denotes the in-plane coordinates, and z that normal to the interface. In the case of weak deformations, |∇u| 1, we may expand the square root and retain the leading term only. Then the free energy functional reads
f [u(r)] = γ 2 dS (∇u) 2 − dSπ(r)u(r).(3)
The actual deformation is determined by minimizing this functional. Linearizing the deformation about the equilibrium value u(r) and integrating the first term by parts, one finds that the mimimum free energy f occurs for the deformation satisfying the Young-Laplace equation,
γ∇ 2 u(r) + π(r) = 0,(4)
and reads
f = − γ 2 dS (∇u) 2 .(5)
As the most severe approximation of the present paper, we assume that the total pressure π(r) is the linear superposition
π(r) = απ (r − r α ),(6)
whereπ(r) is the pressure profile of a single charge at the origin. In Debye-Hückel approximation, the electrostatic potential is a linear superposition of single-particle terms φ = α ϕ α , and the total pressure is quadratic in the potential and its derivatives, π(r) ∼ φ 2 . Besides the diagonal termsπ α ∼ ϕ 2 α , it comprises cross-terms ϕ α ϕ β that are significantly smaller and thus have been neglected in (6).
Because of the linearity of (4), a similar relation holds true for the deformation
u(r) = αû (r − r α ).
It is then sufficient to consider two particles at positions ± 1 2 ρ. Discarding terms that do not depend on their distance ρ, one finds immediatly the pair potential
∆f (ρ) = −γ d 2 r∇û r + 1 2 ρ · ∇û r − 1 2 ρ . (7)
The single-charge pressureπ(r) comprises the entropic pressure of the excess soluted ion density and the electric field energy. It is obvious from Fig. 1 that the force on the charge is directed towards the electrolyte, whereas the electrostrictive force on the surrounding interface is in the opposite direction. It is essential to note that the total force vanishes, i.e.,
dSπ (r) = 0.(8)
In view of (5) we need to calculate the gradient of the deformation field. This is achieved most easily by applying Gauss' theorem on (4),
γ ∂D dsn · ∇û = − D dSπ (r) ,
that relates the pressure integrated over the area D to the "flux" of the gradient field through the boundary ∂D. For a disk centered at the origin and polar coordinates r and θ, we have ds = rdθ and n = r/r. Using the condition (8) and the fact thatû is isotropic, the integral over the disk is replaced by that over the infinite space outside,
∇û(r) = 1 γ r r 2 ∞ r dr r π(r ).(9)
Thus we have expressed the interaction free energy ∆f (ρ) by integrals over the single-charge electrostatic pressurê π(r) that remains to be determined. Starting from the screened potential ϕ(r, z) of a single interfacial charge at the origin, we evaluate the pressure profile according to Eqs. (1) and (2). Expanding the ion densities to quadratic order, n + + n − = n s (eϕ/k B T ) 2 , and noting κ 2 = (2n s e 2 /ε el k B T ), we find the entropic pressure
k B T (n + + n − ) = ε el 2 κ 2 ϕ(r, z) 2 . The relevant component T zz = ε 2 (E 2 z −E 2 r )
of the Maxwell tensor depends on both normal and in-plane derivatives of the potential. The latter one, E r = ∂ r ϕ, being continuous at the interface, the electrostrictive force simplifies to
T ins zz − T el zz = ε el − ε ins 2 E 2 r − ε el 2 E 2 z el + ε ins 2 E 2 z ins
In Debye-Hückel approximation (DHA), the 2D Fourier transform of ϕ is known, and explicit forms of the potential at the interface can be given in the domains separated by the Bjerrum length B and the Debye screening length κ −1 . Here we confine ourselves to the case of large distances well beyond κ −1 and the particle size a, where the potential in the electrolyte reads [9][10][11][12] ϕ el (r, z) = (Qε ins /2πκ 2 ε 2 el )e −κz r −3 (r κ −1 ).
(High charge densities may result in a renormalized value for Q.) The in-plane electric field E r ∼ r −4 is negligible at large r; for the normal component one finds E z | el = ∂ z ϕ el = −κϕ el . Its value at the insulating side of the interface is obtained from the continuity condition ε ins E z | ins = ε el E z | el . Thus the first term in T ins zz − T el zz cancels the entropic pressure, the second one is negligible, and the third one giveŝ
π(r) = ε ins 2 ε el ε ins κϕ el (r, 0) 2 .(10)
Now it is straightforward to calculate the gradient of the interface deformation,
∇û(r) = 1 32π 2 ε ins Q 2 γε 2 el κ 2 r r 6 ,(11)
This remarkable result, ∇û ∼ r −5 , relies only on the asymptotic form of the potential ϕ el ∼ r −3 and the fact that the net force (8) on the interface vanishes. As a consequence, the deformation field is negative and varies asû ∼ (−1/r 4 ). Now we discuss the behavior ofπ(r) and ∇û(r) at small distances. The singularities for r → 0 are clearly unphysical and would cause the deformation free energy ∆f to diverge. The above screened potential ϕ el and electrostatic pressureπ are valid at large distances r κ −1 , a. At intermediate distances a < r < κ −1 screening is irrelevant; evaluating the stress tensor with the bare electrostatic potential ϕ el ∼ r −1 , one finds a modified power lawπ(r) ∼ r −4 [21]. For micrometer size charged colloids, however, the particle radius a provides the most relevant physical cutoff. For this reason, it will be referred to in the remainder of this paper, although one should keep in mind that the actual situation may be more complex, especially for small molecules.
Thus the power laws of bothπ(r) and ∇û(r) cease to be valid at distances smaller than a. Moreover, Eq. (8) requires a strong negative force operating on the particle at r = 0. An important conclusion can be drawn directly from the relation (8): ∇û is positive everywhere and tends towards zero at r = 0.
In order to regularize the surface integral in Eq. (7), we explicitly introduce a cut-off a and replace (11) with
∇û(r) = 1 32π 2 ε ins Q 2 γε 2 el κ 2 r (r 2 + a 2 ) 3 .(12)
Although the precise form of this cut-off function is somewhat arbitrary, it satisfies the limit ∇û = 0 for r → 0 as imposed by Eq. (9). For r a, it shows the long-range behavior that has been obtained rigorously from (8) and (10).
Inserting ∇û(r± 1 2 ρ) in (7), one finds that the capillary interaction ∆f (ρ) = −(ε ins Q 2 /32π 2 γε 2 el κ 2 ) 2 I depends on particle distance and size through the integral
I = ∞ 0 drr 2π 0 dθ r 2 − 1 4 ρ 2 r 2 + 1 4 ρ 2 + a 2 2 − r 2 ρ 2 cos 2 θ 3 .
This definite integral can be performed analytically. Since in all applications, the particle distance ρ exceeds significantly the size a, we retain only the leading term in powers of (a/ρ) and thus have
I = − 2π a 2 ρ 6 1 + O a 2 /ρ 2 .
The variation with the particle radius ∼ a −2 does not depend on the precise form of the cut-off function chosen above. Thus we obtain the capillary interaction
∆f (ρ) = 1 2 9 π 3 ε 2 ins Q 4 ε 4 el κ 4 1 γa 2 ρ 6(13)
that is repulsive and varies with the inverse sixth power of the distance. Note that ∆f depends on the particle charge, its size, and the Debye length. We compare our result for the capillary interaction with experimental findings and previous work. In Ref. [7], a minimum in the interaction potential of PMMA particles was reported to occur at a distance of 5.7 microns. As a possible explanation, these authors considered the competing electrostatic and capillary interactions. Interfacial charges are subject to the well-known repulsive potential
V = ε ins 2πε 2 el p 2 ρ 3 ,
where p is dipole moment formed by the charge and its screening cloud [10]. On the other hand, assuming an electric-field induced δ-forceπ(r) = −π 0 δ(r), one obtains an attractive capillary interaction ∆f δ (ρ) = (π 2 0 /2π) ln ρ. The sum of V and ∆f δ shows a minimum at finite distance that has been considered in Ref. [7]. Note, however, that this force does not satisfy the condition (8).
As shown schematically in Fig. 1 and discussed below (11), the electrostatic pressure profile induced by a screened surface charge comprises a long-range contribution and a δ-force, π(r) = g(r)r −6 − π 0 δ(r), with a cut-off function g that is constant for r a, vanishes at the origin, and satisfies Eq. (8). By now it should be clear that the long-range contribution to the pressure completely changes the capillary forces: While that of a pure δ-force is attractive and depends logarithmically on distance, the capillary interaction of interfacial charges is repulsive and varies as ∆f (ρ) ∼ ρ −6 . We conclude that the mimimum in the interaction potential reported in [7] cannot arise from V + ∆f .
Finally we compare the relative magnitude of the forces arising from V and ∆f . If the particle's charge Q is located at the interface, the dipole moment reads p = Q/κ, and the capillary interaction can be expressed in terms of the dipolar interaction V and the surface energy
γa 2 , ∆f (ρ) = V 2 2 7 πγa 2 .(14)
The total force −∂ ρ (V + ∆f ) is always repulsive. A change in the exponent occurs where ∂ ρ V = ∂ ρ ∆f , defining the cross-over distance
ρ * = 1 4 ε ins π 2 ε 2 el p 2 γa 2 1/3
, for ρ < ρ * , the capillary repulsion −∂ ρ ∆f ∼ ρ −7 dominates, whereas at larger distances the dipolar force −∂ ρ V ∼ ρ −4 takes over. For micron size particles at an oil-water interface with typical parameters, one finds that the cross-over distance hardly attains the micrometer range and thus is of little relevance. Quite a different situation arises if a significant fraction of the charge is located at the side of the particle immersed in the insulator.
For micron size particles, the electrostatic potential ϕ is enhanced by a factor of the order (κa) 2 [6], and the capillary interaction increases by a factor (κa) 4 . The modified dipole moment reads p ∼ Qa instead of Q/κ, resulting in a larger cross-over distance ρ * . Note, however, that at distances much shorter than ρ * , the gradient of the deformation |∇û| is no longer small, requiring to go beyond the quadratic approximation for the surface energy in (3).
In summary, we have studied the capillary interaction induced by the electric field of charged particles at an electrolyte-insulator interface. The electrostatic pressure of a single charge consists of a force acting on the particle and the opposite electrostrictive force on the interface that varies as ρ −6 with the lateral distance. The resulting deformation, u ∼ ρ −4 , induces a capillary interaction that is repulsive and obeys a power law ∆f ∼ ρ −6 . As a consequence, this capillary force cannot be at the origin of the secondary minimum observed recently for charged PMMA particles at an oil-water interface [7].
N ote added. After submission of this Letter, a comment by Megens and Aizenberg on the paper by Nikolaides et al. [7] appeared in Nature [22]. In agreement with our discussion and our Eq. (8), Megens and Aizenberg point out that there is no net force on the interface and thus invalidate the attractive logarithmic potential proposed in Ref. [7].
FIG. 2 :
2The capillary interaction arises from the superposition of the deformation fields of nearby particles.
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| {'fraction_non_alphanumeric': 0.060964766033122154, 'fraction_numerical': 0.03401133711237079, 'mean_word_length': 3.8846362649294246, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study the electric-field induced capillary interaction of charged particles at a polar interface. The algebraic tails of the electrostatic pressure of each charge results in a deformation of the interface u ∼ ρ −4 . The resulting capillary interaction is repulsive and varies as ρ −6 with the particle distance. As a consequence, electric-field induced capillary forces cannot be at the origin of the secondary minimum observed recently for charged PMMA particles at on oil-water interface.PACS numbers:Electrostatic forces operating on charged particles determine to a large extent the properties of emulsions and foams, and stabilize colloidal aggregates[1,2]. Charged particles at interfaces or in thin films form 2D hexagonal crystals[3][4][5][6][7], while for bulk colloidal suspensions several 3D crystal phases have been reported[8]. At a typical lattice spacing of a few microns, the interparticle forces are of the order of picoNewton.A charge at a polar interface and its counterion cloud carry a finite dipole moment perpendicular to the interface (Cf.Fig. 1.) The resulting electrostatic force acting on two neighboring particles is repulsive and varies with the inverse fourth power of their distance[3,[9][10][11][12]. This law has been confirmed experimentally for charged polystyrene particles[6].The pair potential of PMMA particles at an oil-water interface has been measured by recording the trajectories and evaluating distance correlations. Besides the dipolar repulsion at short distances, an attractive force has been found at larger distances, and a minimum in the potential energy has been shown to occur at about 5.7 µm [7]. As a possible explanation, these authors propose that the deformation of the interface by the particles gives rise to a capillary interaction that varies logarithmically with their distance[7].Quite generally, a capillary or elastic interaction is obtained when equilibrating an external force acting on the particles with the surface tension or the bending rigidity. A variety of such models have been studied, both for spherical and anisotropic defects[13][14][15][16][17][18][19]; these works deal with forces that act on the particle only, such as gravity. A somewhat different pattern arises for forces that operate both on the particle and on the surrounding interface. For example, a polymer grafted on a membrane exerts a force at the point of attachment and an opposite entropic pressure on the interface[20].In this Letter we study the deformation of a polar interface due to a charged colloidal particle, and we derive the resulting capillary interaction. The present work is confined to the case most relevant for micron size colloidal particles, where the distance is much larger than the Debye length. After a reminder of the free energy of a deformation field, we calculate the electrostatic pressure profile exerted by a charge and the associated counterions (as shown schematically inFig. 1.)The electric-field FIG. 1: a) Pressure profileπ(r) induced on the interface by a charged particle (filled grey circle); open circles are the counterions. b) The corresponding deformation fieldû(r).arXiv:cond-mat/0310657v3 [cond-mat.soft]', 'arxivid': 'cond-mat/0310657', 'author': ['Lionel Foret \nCPMOH\nUniversité Bordeaux 1\n351 cours de la Libération33405TalenceFrance\n', 'Alois Würger \nCPMOH\nUniversité Bordeaux 1\n351 cours de la Libération33405TalenceFrance\n'], 'authoraffiliation': ['CPMOH\nUniversité Bordeaux 1\n351 cours de la Libération33405TalenceFrance', 'CPMOH\nUniversité Bordeaux 1\n351 cours de la Libération33405TalenceFrance'], 'corpusid': 27909638, 'doi': '10.1103/physrevlett.92.058302', 'github_urls': [], 'n_tokens_mistral': 5753, 'n_tokens_neox': 4898, 'n_words': 3160, 'pdfsha': '695bf38ec5035fd38e44796344fdd449672a0bd2', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0310657v3.pdf'], 'title': ['Electric-field induced capillary interaction of charged particles at a polar interface', 'Electric-field induced capillary interaction of charged particles at a polar interface'], 'venue': []} |
arxiv |
The product formula for Lusternik-Schnirelmann category
2001
Joseph Roitberg
The product formula for Lusternik-Schnirelmann category
Algebraic & Geometric Topology A TG
12001491AMS Classification 55M30 Keywords Phantom mapMislin (localization) genusLusternik-Schnirel- mann categoryHopf invariantCuplength
If C = C φ denotes the mapping cone of an essential phantom map φ from the suspension of the Eilenberg-Mac Lane complex K = K(Z, 5) to the 4-sphere S = S 4 , we derive the following properties: (1) The LS category of the product of C with any n-sphere S n is equal to 3;(2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C . These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1-connected CW-complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If φ is such that its p-localizations are inessential for all primes p, then by the main result of [10], the pair C * = S ∨ Σ 2 K , C provides such an example in the case m = 1.
Introduction
In this sequel to [10], we record two additional curious properties of the space X described in the main Theorem of [10].
Recall that X is the mapping cone of an essential map φ from ΣK , the suspension of the Eilenberg-Mac Lane complex K = K(Z, 5), to S = S 4 , the 4-sphere. Henceforth we use the notation C = C φ for this space. Recall from [10] that φ factors uniquely as Both φ and ψ are phantom maps in the sense of [14] (see also [9, End of section 1]); they become inessential when restricted to finite subcomplexes of their respective domains. We study D = D ψ , the mapping cone of ψ , side by side with C . The space D is simpler (it is finite-dimensional) and less interesting (it does not have finite type) than C and serves as a model for the latter. The two results below apply to both spaces; the proofs in the (easier) case of D provide motivation and guidance for the proofs in the case of C .
φ = ψ • r,
The first result states, in view of the main Theorem of [10], that the spaces C, D satisfy the Ganea "conjecture" for all spheres. Thus:
Theorem 1.1 For any n > 0, cat(C × S n ) = 3 = cat(D × S n ).
As an application of Theorem 1.1, we obtain a pair of spaces in the same Mislin genus having LS categories 2 and 3. Namely, if C * (respectively D * ) denotes the mapping cone of the trivial map with domain ΣK (respectively S 6 (0) ), and if φ is a special phantom map as in [10,Example 2] (that is the p-localization of φ is inessential for every prime p), then the spaces C × S n , C * × S n are in the same Mislin genus, as are the spaces D × S n , D * × S n , and cat(C × S n ) = 3, cat(C * × S n ) = 2;
cat(D × S n ) = 3, cat(D * × S n ) = 2.
This application is only moderately interesting since we may generalize Example 2 of [10] to get an example of a pair of spaces in the same Mislin genus having LS categories m and m + 1 for any m > 0. Indeed, we need only replace K by K(Z, 4m + 1) and S by HP m , quaternionic projective m-space, and argue as in [10], applying appropriate results of Iwase [8]; details are omitted. What would be more interesting is to find a pair of spaces in the same Mislin genus with LS categories m and 2m for any m > 0. For according to a result of Cornea [1], with improvement by Stanley [11] (see also Félix-Halperin-Thomas [3]),
cat(X) ≤ 2· sup p cat(X (p) ) , (1.1)
where X is a 1-connected CW-complex of finite type and X (p) is its p-localization.
Examples of the type just mentioned would show that the inequality (1.1) is sharp for all m > 0. To that end, our original thought was to try the m-fold product spaces C m * , C m (or D m * , D m ). However, this attempt fails, as the next result shows.
Theorem 1.2 cat(C × C) = 3 = cat(D × D).
Thus C is a 1-connected CW-complex of finite type such that cat(C × C) is strictly smaller than 2·cat(C) . Examples of strict inequality in the product formula cat(A × B) ≤ cat(A) + cat(B) go back to Fox [4]. However, the two factors in Fox's example are distinct spaces: they are Moore spaces S 2 ∪ p e 3 , S 2 ∪ q e 3 with respect to distinct primes p and q . See also Ganea-Hilton [5] for a generalization of Fox's example as well as the observation that cat(K(Q, 1) × K(Q, 1)) ≤ 3 (which is strictly smaller than 2·cat(K(Q, 1)) = 4); of course, K(Q, 1), unlike C , is neither 1-connected nor of finite type. An alternate derivation of the inequality cat(K(Q, 1) × K(Q, 1)) ≤ 3 is possible along the lines of the first part of the proof below of Theorem 1.2.
A more recent example of strict inequality in the product formula, similar in flavor to Fox's example (in both situations, A ∨ B is homotopy equivalent to A × B ), is contained in Félix-Halperin-Lemaire [2]. In this example, A is a 1-connected CW-complex whose (reduced) homology groups are all finite and B is a 1-connected, rational CW-complex. The main result of [2] states that -unlike the situation when A, B are both K(Q, 1) -cat(A × B) = cat(A) + cat(B) whenever A, B are rationalizations of 1-connected CW-complexes of finite type.
The landscape changed radically when Iwase [7] discovered examples of strict inequality in the product formula in which one of the factors (but not both!) is a sphere. Systematic approaches to Ganea's "conjecture", with many more examples, have been developed since then; see [8], [11], Vandembroucq [13], Stanley [12] and Harper [6].
We remark that the examples in [6] and [12], illustrating the strict inequality cat(A × A) < 2·cat(A) for A 1-connected of finite type are quite different from those in Theorem 1.2. In each of Harper's examples, A is the mapping cone of a suitable map from a sphere to another sphere and cat(A) = 2, cat(A × A) ≤ 3. In Stanley's example, A is the mapping cone of a suitable map from a sphere to a bouquet of two spheres and cat(A) = 2 = cat(A × A). In contrast to the situation in Theorem 1.1, the examples in [6] and [12] do not satisfy the Ganea "conjecture". In fact, cat(A × S n ) = 2 for all n ≥ 2 in two of the three examples in [6] and cat(A × S n ) = 2 for all n ≥ 1 in the third example in [6] and in the example in [12].
The proofs of Theorems 1.1 and 1.2 are contained in the next section. The proof of Theorem 1.1 builds on the proof of the main Theorem of [10], bringing in more of the Hopf invariant technology of [8].
We also take the opportunity to correct a mis-statement in the last paragraph on p. 99 of [10]; I thank Jianzhong Pan for bringing the error to my attention.
As for the proof of Theorem 1.2, the inequalities
cat(C × C) ≤ 3, cat(D × D) ≤ 3
follow from classical obstruction theory arguments. Similar, but slightly more elaborate, arguments lead to the inequalities
cat(C k ) ≤ k + 1, cat(D k ) ≤ k + 1
for 3 ≤ k ≤ 6 as well, but these arguments break down for k = 7. To establish the opposite inequalities
cat(C × C) ≥ 3, cat(D × D) ≥ 3,
we exploit techniques of [8] together with cuplength arguments. In fact, we show that D × D has cuplength 3. This latter fact fails for C × C (whose cuplength is 2). To remedy this failure, we introduce a notion of refined cup products -which in the case of C ×C consists of a blend of ordinary cohomology and cohomotopy -and show that C × C has non-zero refined cup products of length 3, thereby leading to the desired inequality.
The latter approach also leads to an alternate, albeit less direct, proof of Theorem 1.1, as well as of the main Theorem of [10].
Proofs
Before giving the proof of Theorem 1.1, which closely follows the lines of the proof of the main Theorem of [10], we wish to point out that throughout the last paragraph on page 99 of [10], the space S 6 (0) should be replaced by ΣK . To see that the argument beginning in that paragraph and concluding on page 100 is valid for ΣK , it is only necessary to observe that we may identify the map
[ΣK, S 7 ] −→ [ΣK, E 2 (ΩS 4 )]
induced by the inclusion of the bottom S 7 into E 2 (ΩS 4 ) with the analogous map
[S 6 (0) , S 7 ] −→ [S 6 (0) , E 2 (ΩS 4 )], denoted in [10] by β 1 ; here E 2 (ΩS 4 ) is a space homotopy equivalent to the 2-fold join ΩS 4 * ΩS 4 . Indeed, we have a commutative diagram [S 6 (0) , S 7 ] −−−→ [S 6 (0) , E 2 (ΩS 4 )] [ΣK, S 7 ] −−−→ [ΣK, E 2 (ΩS 4 )]
where the vertical maps are induced by rationalization r : ΣK → S 6 (0) and are isomorphisms; see [10, (2.5)].
We now give the proof of Theorem 1.1.
Proof of Theorem 1.1 It is clear that
cat(A × S m ) ≤ 3, A = C or D,
for any m ≥ 1. To prove that
cat(A × S m ) = 3,
we appeal to a special case of [8, Theorem 3.9(2)] according to which it suffices to establish the following variant of (iii) on page 98 of [10]: Now the argument on page 99 of [10] prior to the last paragraph actually shows that Σ m H 1 (α) is essential for any m ≥ 0. Moreover the subsequent argument (suitably amended; see above) shows that E 2 (Ωi) has a left homotopy inverse. Since Σj m−1 has a left homotopy inverse, it follows that E 2 (Ωi) * j m−1 too has a left homotopy inverse, and (iii ′ ) is verified.
(E 2 (Ωi) * j m−1 ) • Σ m H 1 (α) (iii ′ ) is essential,
In the case A = D, it is even permissible to "ignore" the map E 2 (Ωi) * j m−1 ; indeed, that map induces a monomorphism from [S 6 (0) * S m−1 , E 2 (ΩS 4 ) * S m−1 ] to [S 6 (0) * S m−1 , E 2 (ΩD) * ΩS m ] since S 6 (0) has dimension 7. See also [8,Remark 3.10].
Remark Since any rationalization map r : ΣK → S 6 (0) is a suspension, it follows from [8, Proposition 2.11 (1)] that
H 1 (ϕ) = H 1 (ψ) • r.
We next give the proof of Theorem 1.2.
Proof of Theorem 1.2 To prove
cat(B) ≤ 3, B = C × C or D × D,(2.1)
it suffices to show that the Ganea fibration Two remarks:
P 3 (ΩB) −→ B,(2.
(1) An alternate approach to the existence of a section of (2.2) in this case, but not in the case B = C × C , is via a slight variant of [8, Theorem 5.5] -details are omitted.
(2) We do not assert that there is a unique section of (2.2) in the case B = D × D. In fact, the (vertical homotopy classes of) sections are classified by
H 15 (D × D; π 15 (E 4 (Ω(D × D))) ≈ Ext(Q, Z ⊕ · · · ⊕ Z),
which has the cardinality of the continuum.
To deal with the case B = C × C , first note that a homology decomposition of K, say we may view u 2 * as the composite
S 5 = K[5] ⊂ · · · ⊂ K[m] ⊂ · · · ⊂ K,µ • (u * ∧ u * ) • ∆ ′ : D −→ D ∧ D −→ K(4) ∧ K(4) −→ K(8),(2.7)
where µ represents a generator of H 8 (K(4) ∧ K(4)). From (2.6) and (2.7), we see that u 2 * may be represented as the composite
µ • (j ∧ j) • (e ∧ e) • ΣH 1 (ψ) • q : D −→ K(8),(2.8)
where j ∧ j is induced by a generator j of H 4 (S) while the other maps in the composite have been defined above.
Reasoning as in the proof of [10, (2.3)] shows that (e ∧ e) • ΣH 1 (ψ) is non-zero. It is then routine to argue that the composite (2.8) is itself non-zero. Thus (2.5), and with it (2.4), is proved.
Next let u • , v • be the images of u, v under the cohomology map induced by the inclusion D × S ⊂ D × D. Of course, the restriction of u • to D × * ⊂ D × S is just u * . To prove (2.3), it suffices to prove
u 2 • v • = 0. (2.9)
To that end, we apply [8, Corollary 4.1.1] to the cofibration sequence
S 10 (0) −→ (D × * ) ∪ (S × S) → D × S to infer that the reduced diagonal map ∆ ′ ∧ 1 S = q ′ • (∆ ′ × 1 S ) : D × S → (D ∧ D) × S −→ D ∧ D ∧ S,
where q ′ is the collapsing map, factors as
(i ∧ i ∧ 1 S ) • (e ∧ e ∧ 1 S ) • Σ 5 H 1 (ψ) • q ! : D × S −→ D ∧ D ∧ S(5) i ∧ i ∧ 1 S : S ∧ S ∧ S → D ∧ D ∧ S is induced by i.
From (2.10), we see that u 2
• v • may be represented as the composite
ν • (j ∧ j ∧ j) • (e ∧ e ∧ 1 S ) • Σ 5 H 1 (ψ) • q ! : D × S −→ K(12) (2.11)
where ν represents a generator of H 12 (K(4) ∧ K(4) ∧ K(4)).
Arguing as with (2.8), we readily verify that the composite (2.11) is non-zero. Thus (2.9), and with it (2.3), is proved.
Finally, we adapt the preceding argument to prove the inequality
cat(C × C) ≥ 3.
If u, v now stand for the canonical generators (up to sign) of H 4 (C × C), the analogs of (2.3) and (2.4) both fail. In fact, as the cohomology of C is indistinguishable from that of C * , the cuplength of C is precisely 1 and the cuplength of C × C is precisely 2. One might try replacing cohomology by cohomotopy since π 8 (C) = [C, While neither cohomology alone nor cohomotopy alone supports a cuplength argument necessary to establish the inequality cat(C × C) ≥ 3, the following compromise between cohomology and cohomotopy provides a way out of the dilemma. Fix a CW-model for K(m) such that the m-skeleton is S m and the N -skeleton K(m) N has only finitely many cells for all N ≥ m.
Lemma 2.1 (1) For N ≥ 9, the inclusions K(4) N ⊂ K(4) N +1 ⊂ K(4) induce a commutative diagram [S, K(4) N ] ←−−− [C, K(4) N ] [S, K(4) N +1 ] ←−−− [C, K(4) N +1 ] [S, K(4)] ←−−− [C, K(4)]
(2.13)
with each of the sets in (2.13) equivalent to Z, and each of the arrows a bijection;
(2) For N ≥ 8, the inclusion K(8) N ⊂ K(8) N +1 induces a commutative diagram [C, K(8) N ] ←−−− [Σ 2 K, K(8) N ] [C, K(8) N +1 ] ←−−− [Σ 2 K, K(8) N +1 ]
(2.14)
with each of the sets in (2.14) equivalent to Ext(Q, Z), and each of the arrows a bijection;
(3) For N ≥ 12, the inclusion K(12) N ⊂ K(12) N +1 induces a commutative diagram
[C × S, K(12) N ] ←−−− [Σ 6 K, K(12) N ] [C × S, K(12) N +1 ] ←−−− [Σ 6 K, K(12) N +1 ] (2.15)
with each of the sets in (2.15) equivalent to Ext(Q, Z), and each of the arrows a bijection.
Proof To prove the first part of (1), consider the commutative diagram with exact rows and (2.16) collapses to the upper square in (2.13). The proof of the rest of (1), and also (2) and (3), are similar. Observe that the diagram in (3) is induced by the cofibration sequence as the maps uniquely determined by µ, ν . It may be assumed that these maps are cellular. We define the refined cup square u 2 * N as the element represented by the composite µ N • (u * N ∧ u * N ) • ∆ ′ C : C −→ C ∧ C −→ K(4) N ∧ K(4) N −→ K(8) 2N +1 . We define the refined 3-fold cup product, u 2 •N v •N similarly, using ν N . Observe that u 2 * N compresses uniquely into S 8 by part (2) of Lemma 2.1. In other words, u * N is essentially a 4−dimensional cohomology class while u 2 * N is essentially an 8−dimensional cohomotopy class. Similarly, by part (3) of Lemma 2.1, u 2 •N v •N is essentially a 12−dimensional cohomotopy class.
[ΣK, K(4) N ] ←− [S, K(4) N ] ←− [C, K(4) N ] ←− [Σ 2 K, K(4) N ] [ΣK, K(
We now invoke the analog of (2.8) with D replaced by C , S 7 (0) replaced by Σ 2 K and K(m) replaced by K(m) N , N ≥ 9,to conclude, with the help of Lemma 2.1, that u 2 * N = 0, that is the refined cuplength of C is at least 2. Moreover, arguing with the appropriate analog of (2.11), we find, once again with the help of Lemma 2.1, that u 2 •N v •N = 0, that is the refined cuplength of C × C is at least 3.
The classical argument showing that cuplength is a lower bound for cat readily generalizes to show that refined cuplength is also a lower bound for cat. Hence cat(C × C) is at least 3, completing the proof of Theorem 1.2.
rationalization map and ψ : S 6 (0) −→ S. c Geometry & Topology Publications
where the left-most map is the join of the map induced by the inclusion i : S ⊂ A with the canonical map j m−1 : S m−1 → ΩS m , α = φ ( respectively ψ ) if A = C (respectively D), and H 1 (α) is the Berstein-Hilton-Hopf invariant of α as generalized by Iwase ([8, Definition 2.4]) and Stanley ([11, Definition 3.4]).
H
n (D × D; π n−1 (E 4 (Ω(D × D))) = 0 for all n. Since the obstructions to a section of (2.2) in the case B = D × D lie in these cohomology groups, it is clear that a section exists.
where each K[m] is a finite complex, naturally induces a homology decomposition of C, S = C[4] ⊂ · · · ⊂ C[m] ⊂ · · · ⊂ C,where each C[m] is likewise a finite complex. Explicitly, we take C[m] to be the mapping cone of the restriction of φ to ΣK[m − 1]. Thus C × C is the ascending union of the finite subcomplexes C[m] × C[m], about which we make two claims: (1) cat(C[m]×C[m]) = 2, m ≥ 4. In fact, the homotopy equivalence C[m] ≃ S ∨Σ 2 K[m−1], m ≥ 4, resulting from the fact that φ is phantom, implies that cat(C[m]) = 1, hence that cat(C[m] × C[m]) = 2, m ≥ 4.(2) For each m, H n (C[m] × C[m])is finite for all n > 14. This follows immediately from the fact that H n (C × C) is finite for all n > 14.It follows from(1)that for each m, the inclusion C[m] × C[m] ⊂ C × C lifts to the total space P 3 (Ω(C × C)) of the fibration (2.2). Moreover, the number of such lifts is finite since, thanks to (2), the cohomology groups H n (C[m] × C[m]; π n (E 4 (Ω(C × C))) are finite for all n. A variant of the classical argument utilized in the first paragraph of page 100 of [10] then leads to the conclusion that (2.2) admits a section in the case B = C × C . We now verify that cat(B) ≥ 3, B = C × C or D × D, beginning with the case B = D ×D. Our strategy is to detect a non-zero 3-fold cup product in the integral cohomology ring of D × D. More precisely, if u and v are the canonical generators (up to sign) of H 4 (D × D), we show that generalize the proof of (2.4) to obtain (2.3). Let u * be the restriction of u to D × * ⊂ D × D. To prove (2.4), it suffices to prove −→ S −→ D, we infer that the reduced diagonal map ∆ ′ = ∆ ′ D : D −→ D ∧ D factors as the composite (i ∧ i) • (e ∧ e) • ΣH 1 (ψ) • q : D −→ D ∧ D , (2.6) where (1) q : D → D/S = S 7 (0) is the collapsing map, (2) ΣH 1 (ψ) : S 7 (0) → ΣE 2 (ΩS) ≃ ΣΩS ∧ ΣΩS is the suspension of the Berstein-Hilton-Hopf invariant of ψ , and (3) (i ∧ i) • (e ∧ e) is induced by the evaluation map e : ΣΩS → S and the inclusion map i : S ⊂ D.Abbreviating K(m) = K(Z, m) and viewing u * as (the homotopy class of) a map u * : D −→ K(4),
) q ! : D × S → D × S/(D × * ) ∪ (S × S) = (D/S) ∧ S is the collapsing map,(2) D × S/(D × * ) ∪ (S × S) = (D/S) ∧ S = S 11 (0) ,(3)Σ 5 H 1 (ψ) : S 11 (0) → ΣΩS ∧ ΣΩS ∧ S is the 5-fold iterated suspension of H 1 (ψ), (4) e ∧ e ∧ 1 S : ΣΩS ∧ ΣΩS ∧ S → S ∧ S ∧ S is induced by e, and
S 8 ] (as also π 8 (D) = [D, S 8 ] ≈ H 8 (D)) turns out to be set-theoretically equivalent to Ext(Q, Z) as noted in Lemma 2.1 below. Unfortunately, the Puppe sequence [ΣK, S] ←− [S, S] ←− [C, S] ←− [Σ 2 K, S] (2.12) associated to the cofibration sequence ΣK → S → C gives that π 4 (C) = [C, S] = 0 since the left-most map in (2.12) sends [S, S] monomorphically to the (infinite cyclic) group generated by φ, and [Σ 2 K, S] = 0 by [14, Theorem D] (see also [9, Theorem 4.2]). In other words, the cohomology classes u, v are not compressible into S .
4) N +1 ] ←− [S, K(4) N +1 ] ←− [C, K(4) N +1 ] ←− [Σ 2 K, K(4) N +1 ] (2.16) induced by the cofibration sequence ΣK → S → C . Notice that if M ≥ 9, π 7 (K(4) M ) = 0 = π 8 (K(4) M ). Therefore, by [14, Theorem D] (see also [9, Theorem 4.2]), [ΣK, K(4) M ] = 0 = [Σ 2 K, K(4) M ]
Σ 5 K
5−→ (C × * ) ∪ (S × S) −→ C × S. Define u * N , u •N , v •N to be the unique compressions of u * , u • , v • into K(4) N ,N ≥ 9, guaranteed by part (1) of Lemma 2.1. Further, define two mapsµ N : K(4) N ∧ K(4) N −→ K(8) 2N +1 , and ν N : K(4) N ∧ K(4) N ∧ K(4) N −→ K(12) 3N +1
for n > 14; note that unless the coefficients are explicitly indicated, homology and cohomology groups are understood to be integral. Thus2)
with fiber
E 4 (ΩB) ≃ ΩB * ΩB * ΩB * ΩB,
admits a section; see [8, Theorem 2.1]. For either value of B in (2.1), we easily
verify that E 4 (ΩB) is 14-connected -more precisely, E 4 (ΩB) has the homotopy
type of a CW-complex of the form
(S 15 ∨ · · · ∨ S 15 ) ∪ cells of dimension ≥ 18 .
Next observe that
H n (D × D) = 0
Algebraic & Geometric Topology, Volume 1 (2001)
This research was supported in part by a grant from the City University of New York PSC-CUNY Research Award Program.
Lusternik-Schnirelmann-categorical sections. O Cornea, Annales Scientifiques de l'École Normale Supérieure, 4 e Série. 28O. Cornea, Lusternik-Schnirelmann-categorical sections, Annales Scientifiques de l'École Normale Supérieure, 4 e Série, 28 (1995), 689-704.
The rational LS category of products and of Poincare duality complexes. Y Félix, S Halperin, J.-M Lemaire, Topology. 37Y. Félix, S. Halperin and J.-M. Lemaire, The rational LS category of products and of Poincare duality complexes, Topology 37 (1998), 749-756.
Lusternik-Schnirelmann category of skeleta. Y Félix, S Halperin, J.-C Thomas, PreprintY. Félix, S. Halperin and J.-C. Thomas, Lusternik-Schnirelmann category of skeleta. Preprint.
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A ∞ method in Lusternik-Schnirelmann category. N Iwase, PreprintN. Iwase, A ∞ method in Lusternik-Schnirelmann category. Preprint.
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The Lusternik-Schnirelmann category of certain infinite CWcomplexes. J Roitberg, Topology. 39J. Roitberg, The Lusternik-Schnirelmann category of certain infinite CW- complexes, Topology 39 (2000), 95-101.
Spaces with Lusternik-Schnirelmann category n and cone length n + 1. D Stanley, Topology. 39D. Stanley, Spaces with Lusternik-Schnirelmann category n and cone length n + 1, Topology 39 (2000), 985-1019.
On the Lusternik-Schnirelmann category of maps. D Stanley, PreprintD. Stanley, On the Lusternik-Schnirelmann category of maps. Preprint.
Suspension des fibrations de Ganea et invariant de Hopf. L Vandembroucq, Université de LilleThèseL. Vandembroucq, Suspension des fibrations de Ganea et invariant de Hopf, Thèse, Université de Lille (1998).
On phantom maps and a theorem of H. Miller. A Zabrodsky, Israel Journal of Mathematics. 58A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel Journal of Mathematics 58 (1957), 129-143.
CUNY 695 Park Avenue. New York, NY 10021, USA EmailDepartment of Mathematics and Statistics Hunter [email protected] ReceivedDepartment of Mathematics and Statistics Hunter College, CUNY 695 Park Avenue, New York, NY 10021, USA Email: [email protected] Received: 26 October 2000 Revised: 7 May 2001
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arxiv |
Constructing local models for general measurements on bosonic Gaussian states
Michael G Jabbour
Department of Physics
Technical University of Denmark
2800Kongens LyngbyDenmark
Jonatan Bohr Brask
Department of Physics
Technical University of Denmark
2800Kongens LyngbyDenmark
Constructing local models for general measurements on bosonic Gaussian states
We derive a simple sufficient criterion for the locality of correlations obtained from given measurements on a Gaussian quantum state. The criterion is based on the construction of a local-hiddenvariable model which works by passing part of the inherent Gaussian noise of the state onto the measurements. We illustrate our result in the setting of displaced photodetection on a two-mode squeezed state. Here, our criterion exhibits the existence of a local-hidden-variable model for a range of parameters where the state is still entangled.
Introduction.-Quantum mechanics allows for correlations than are impossible classically and which can be exploited in a variety of applications. In particular, entangled quantum states are a key resource for quantum information science, enabling advantages in computing, communication, and sensing [1][2][3][4]. Furthermore, as shown by Bell [5], measurements on certain entangled states can lead to observations that violate a so-called Bell inequality and are then incompatible with local causal explanations. This phenomenon, known as nonlocality, demonstrates a profound departure from classical physics and is a cornerstone of modern understanding of quantum physics [6]. Nonlocal correlations also enable advantages for communication [7,8], and information processing at an unprecedented level of security [9][10][11].
Entanglement and nonlocality, however, are not equivalent. While entanglement is a prerequisite for nonlocality, in general only carefully chosen measurements on a given entangled state will produce nonlocal observations, and while such measurements can always be found for pure entangled states [12], there exist mixed entangled states which are local for any possible measurements [13,14]. Deciding whether given states can give rise to nonlocality is desirable both for applications and fundamentally. However, this is far from trivial. To demonstrate nonlocality, it is sufficient to find a particular set of measurements which leads to violation of a particular Bell inequality. Demonstrating that a state cannot give rise to nonlocality is much harder because there are infinitely many possible measurements and Bell inequalities. It requires the construction of local-hidden-variable (LHV) models that can reproduce the observations for any combination of measurements. Constructing such models is challenging, even for particular classes of measurements. A number of methods for constructing LHV models have nevertheless been developed [13][14][15][16][17][18][19][20], applicable to a variety of entangled states and measurements. Very often, a clear connection between the introduction of noise and the vanishing of nonlocality can be identified in these models, e.g. in [13] and [16].
While most previous work is concerned mainly with systems of finite dimension, another relevant class is that of so-called continuous-variables systems [21]. Most particularly, Gaussian bosonic states and transformations are ubiquitous in quantum theory and in experiments in e.g. optical, superconducting, and mechanical platforms. At the same time, Gaussian systems are relatively easy to model. Their entanglement properties have been extensively studied [22,23] and their nonlocality [24][25][26][27][28][29][30][31][32][33] and steering [34] have also been explored. The relation between noise and nonlocality has also been investigated [35,36]. For Gaussian measurements on Gaussian states, the resulting observations are always local, because the positive Wigner function of such states enables the construction of an LHV model for any set of Gaussian measurements (as explained in more detail below). However, little is known about the existence of LHV models for Gaussian states subject to non-Gaussian measurements.
Here, we develop a sufficient criterion for the existence of LHV models for general measurements on Gaussian states. Given a state and a candidate family of measurements, the criterion enables one to certify that they will never lead to nonlocal correlations. The idea behind our result follows the lines of Werner and Wolf's criterion for the separability of Gaussian states [22]. Furthermore, we provide an interesting interpretation in terms of the role of noise for the vanishing of nonlocality, separating the inherent quantum noise resulting from the uncertainty relations from additional classical Gaussian noise. Before presenting our main result, we review some elements of the theory of bosonic systems and nonlocality.
Bosonic systems and Bell nonlocality.-A bosonic system [21] is described by N modes, where each mode is associated with an infinite-dimensional Hilbert space and a pair of bosonic field operatorsâ k ,â † k , where k = 1, . . . , N denotes the mode. The total system Hilbert space is the tensor product over the modes. The field operators satisfy the bosonic commutation relations
[â i ,â † j ] = δ ij , [â i ,â j ] = 0, [â † i ,â † j ] = 0
. Alternatively, the system can be described using the quadrature operators {q k ,p k } N k=1 defined asq k :=â k +â † k ,p k := i(â † k −â k ) (we take = 2 throughout), which can also be arranged in the vectorr := (q 1 ,p 1 , · · · ,q N ,p N ) T . The quadratures satisfy [r k ,r l ] = 2iΩ kl , where Ω := Wigner function in phase space. If the operator is of unit trace (e.g. the density matrix ρ of a quantum state), its Wigner function integrates to unity. Two quantities of particular interest are the two first statistical moments: the mean of the quadraturesr := Tr[rρ] and the covariance matrix V with V ij := Tr[{∆r i , ∆r j }ρ]/2, where ∆r i :=r i −r i and {·, ·} is the anticommutator. Whenever ρ is a genuine quantum state, the 2N × 2N real, symmetric covariance matrix satisfies the uncertainty principle V + iΩ ≥ 0, which also implies V ≥ 0.
As already mentioned, the so-called Gaussian states [37] are ubiquitous in quantum experiments. These are the states whose Wigner function is a multivariate Gaussian distribution. As such, they are completely described by their first two statistical moments, and their Wigner function can be written as
W (r) = 1 (2π) N √ det V e − 1 2 (r−r) T V −1 (r−r) .(1)
The entanglement in a Gaussian state is determined by its covariance matrix alone. A bipartite Gaussian state with covariance matrix V AB will be separable if and only if there exists genuine covariance matrices γ A and γ B of parties A and B such that V ≥ γ A ⊕ γ B [22]. A stronger form of correlations, Bell nonlocality is defined at the level of the observed input-output distribution in an experiment with multiple observers. In particular, a bipartite experiment with observers A and B is characterised by the distribution p(ab|xy), where x, y label the choice of input (measurement setting) of A and B, respectively, and a, b label their outputs (measurement outcomes). The distribution is called nonlocal if it does not admit an LHV model, i.e. if it cannot be written as
p(ab|xy) = dλ q(λ)p(a|x, λ)p(b|y, λ),(2)
where the integral is over the (hidden) variable λ which is distributed according to a probability density q(λ) and where p(a|x, λ) and p(b|y, λ) are local response functions. Entanglement is necessary but not sufficient for the generation of nonlocal correlations [6]. In a general bipartite quantum experiment, A and B share a stateρ AB and each perform a generalised measurement with positiveoperator-valued-measure (POVM) elements Q a|x and R b|y , respectively. The corresponding probabilities are p(ab|xy) = Tr[ρ AB Q a|x ⊗ R b|y ]. If the quantum state and all the POVM elements have positive Wigner functions, p(ab|xy) is necessarily local. Indeed, ifρ AB , Q a|x and R b|y have respective Wigner functions W , Q a|x and R b|y , we have
p(ab|xy) = dr W (r) Q a|x (r A ) (4π) −N A R b|y (r B ) (4π) −N B ,(3)
with r = (r A , r B ), where r A and r B are the phase-space variables and N A and N B the number of modes of party A and B, respectively. This can be understood as an LHV model (2) with r as the hidden variable. W is normalised and is hence a probability density over r. Since a Q a|x = I, with I the identity operator, the Wigner functions fulfill a Q a|x (r A ) = (4π) −N A for all x and r A , because the Wigner function of the identity on N modes is the constant (4π) −N in our convention. Similarly for R b|y . It follows that the last two terms in (3) are probability distributions over a and b, respectively, and can be interpreted as local response functions. Hence (3) is of the form (2). An immediate consequence is that correlations obtained by Gaussian measurements on a Gaussian state will never be nonlocal.
Constructing the LHV model -We denote by Gs ,γ the multivariate Gaussian distribution with means and covariance matrix γ, and by f * g the convolution of functions f and g. We also define 0 := (0, · · · , 0) T . The following statement provides a sufficient criterion for the existence of LHV models for Gaussian states subject to specific measurements. Theorem 1. Letr be the mean and V the covariance matrix of a Gaussian stateρ AB and Q a|x and R b|y the Wigner functions of the POVM elements Q a|x and R b|y . If there exist matrices γ A ≥ 0 and γ B ≥ 0 such that
V ≥ γ A ⊕ γ B ,(4)
and
Q a|x * G 0,γ A ≥ 0 and R b|y * G 0,γ B ≥ 0,(5)
for all a, u and b, v, then the probabilities p(ab|xy) = Tr[ρ AB Q a|x ⊗ R b|y ] exhibit an LHV model.
Proof. Let ω = V − γ A ⊕ γ B ≥ 0. We have p(ab|xy) = (4π) N dr A dr B Gr ,V (r A , r B )Q a|x (r A )R b|y (r B ) = (4π) N dr A dr B (Gr ,ω * G 0,γ A ⊕γ B )(r A , r B )Q a|x (r A )R b|y (r B ) = dr A dr B Gr ,ω (r A , r B )Q a|x (r A ) (4π) −N AR b|y (r B ) (4π) −N B(6)
whereQ a|x := Q a|x * G 0,γ A ≥ 0,R b|y := R b|y * G 0,γ B ≥ 0. Since for the constant distribution c = (4π) −N A , it holds that c * G 0,γ A = c, we also have that (4π) N A aQ a|x = 1, and similarly forR b|y . Eq. (6) can therefore be interpreted as an LHV model.
It is instructive to have a closer look at the situation when the stateρ AB is separable. In that case, there exist covariance matrices γ A and γ B of quantum states (i.e., which satisfy the uncertainty principle), such that V ≥ γ A ⊕ γ B [22], so that
Q a|x (r A ) = ds A Q a|x (s A )G 0,γ A (r A − s A ) = ds A Q a|x (s A )G r A ,γ A (s A ) = (4π) −N A Tr Q a|xσA ,(7)
whereσ A is the density matrix of the Gaussian state with mean value r A and covariance matrix γ A . Sinceσ A is a genuine density matrix, we have thatQ a|x (r A ) ≥ 0 for all r A . The same reasoning can of course be made for party B. We therefore see that whenρ AB is separable, we are always provided with an LHV model whatever the measurements, as should indeed be the case.
In fact, while the Wigner functionsQ a|x andR b|y will always become positive when subject to enough noise (that is, noise coming from a separable stateρ AB ), one can push the analysis further. Consider the bivariate con-volutionQ (t) a|x := Q a|x * G 0,γ A with the choice γ A = tI 2 , for some t ≥ 0, where I 2 is the 2 × 2 identity matrix. It is well known that the distributionQ
where ∆ is the Laplacian, with initial conditionQ (0) a|x = Q a|x . In the limit of t → ∞, the distributionQ (t) a|x approaches a Gaussian, whose Wigner function is necessarily positive. The convolution Q a|x * G 0,γ A actually always makes the distribution Q a|x "less negative" as the parameter t increases. More precisely, the local minima of Q (t) a|x have nonnegative Laplacian, which implies from the heat equation (8) that their t-derivative is nonnegative, so that their values never decrease when t increases.
One can give an operational interpretation of Theorem 1 in terms of the effect that added local Gaussian noise has on nonlocality. Consider a bipartite pure Gaussian stateρ AB with covariance matrix V and suppose it can be written as V = ω + γ q A ⊕ γ q B with ω, γ q A , γ q B ≥ 0 (where q is for quantum). Suppose further that we apply local noise toρ AB in the form of classical additive Gaussian noise channels [21], i.e., local quantum convolutions in the sense of Ref. [38]. These channels are completely characterized by their action on the covariance matrix, which is of the form V → V + γ c A ⊕ γ c B with γ c A , γ c B ≥ 0 (where c is for classical). The resulting mixed Gaussian stateρ AB has covariance matrix ω+(γ q A +γ c A )⊕(γ q B +γ c B ). Now apply Theorem 1 toρ AB with the POVM elements Q a|x and R b|y . An LHV model will exist if Q a|x * G 0,γ q A +γ c A ≥ 0 and R b|y * G 0,γ q B +γ c B ≥ 0, (9) for all a, u and b, v. Eq (9) expresses the fact thatρ AB will become local with respect to the POVMs Q a|x and R b|y when the noise provided by the convolutions with the Gaussian distributions G 0,γ q A +γ c A and G 0,γ q B +γ c B is important enough. There are two contributions to the noise. The first, characterized by γ q A and γ q B , is quantum noise; that is, the uncertainty inherent to quantum mechanics coming from the fact that the pure stateρ AB is subject to the uncertainty relation. The second, characterized by γ c A and γ c B , is classical Gaussian additive noise making the state mixed. An interesting situation arises when either of γ q A + γ c A and γ q B + γ c B is not a genuine covariance matrix, so thatρ AB is still entangled, while the noise is important enough so that there exists an LHV model. We provide an example of this in the following.
An application.-For the sake of illustration, we consider a two-mode squeezed state (TMSS)ρ AB with zero mean and covariance matrix
V = νI 2 √ ν 2 − 1Z √ ν 2 − 1Z νI 2 ,(10)
where ν ≥ 1 and Z := diag(1, −1). It is entangled for ν > 1. We consider a scheme similar to that of Ref. [32] for demonstrating nonlocality with a TMSS (see Fig. 1). First, we take losses into account by applying a local pure-loss channel [21] E η of parameter η ∈ [0, 1] to each mode of the TMSS. The channel E η acts as
E η [σ] := Tr 2 U η (σ ⊗ |0 0|) U † η ,(11)
where U η is a beam-splitter unitary and |0 is the vacuum state. Since E η is Gaussian, the resulting stateρ AB = (E η ⊗ E η ) [ρ AB ] is also Gaussian with zero mean value and covariance matrix
V = [1 + η(ν − 1)]I 2 η √ ν 2 − 1Z η √ ν 2 − 1Z [1 + η(ν − 1)]I 2 .(12)
Furthermore, it can be seen to be entangled for any ν > 1 and η > 0 by evaluating the partial transpose [23,39,40]. Next, for the measurements we consider displacements followed by non-number-resolving single-photon detection (click/no-click). Ideally, this implements a projective measurement onto a coherent state. Here, we allow for some noise in the detection by modelling the POVM element corresponding to the no-click outcome as X +1 (ε, α) :
= D α [(1 − ε)(|0 0| + ε |1 1|]D † α ,
where D α is the displacement operator and |1 is the onephoton Fock state. The click outcome corresponds to FIG. 1. Sketch of a scheme for demonstrating nonlocality from a TMSS with loss. A TMSS is generated by injecting a couple of vacuua into a two-mode squeezer (TMS) of parameter ν, while losses are modeled by the interaction of each output mode of the TMS with a vacuum state through a beam splitter (BS) of transmittance η. The measurements characterized by the POVM elements Xa(ε, αx) and X b (ε, βy) are then performed on party A and B, respectively.
X −1 (ε, α) := I − X +1 (ε, α). The parameter ∈ [0, 1]
can be understood as the probability for an additional excitation to be introduced during measurement. Inputs x, y ∈ {0, 1} for A and B correspond to displacements α x and β y , respectively, and we label the outputs a, b ∈ {±1}, with −1 for click events. We take the noise strength ε to be the same for all measurements. Defining the correlators a x b y = a,b ab p(ab|xy), Eq. (2) implies the CHSH inequality [41]
S = a 0 b 0 + a 0 b 1 + a 1 b 0 − a 1 b 1 ≤ 2.(13)
This inequality can be violated for the quantum probabilities p(ab|xy) = Tr[ρ AB (X a (ε, α x ) ⊗ X b (ε, β y ))]. In particular, taking β y = −α x for y = x and optimising over real α x , we find violation for a range of values of the squeezing, loss, and noise, as shown in Fig. 2.
On the other hand, we can apply Theorem 1 to show that the correlations must be local for another parameter region. Let X a (ε, α) be the Wigner function of X a (ε, α). The quasi-distribution X +1 (ε, α) is nonnegative everywhere since ε < 1, while X −1 (ε, α) admits negative values. According to Theorem 1, the probability p(ab|xy) will satisfy Eq. (2) if there exist nonnegative matrices γ A and γ B such that V ≥ γ A ⊕ γ B and the Wigner functions X a (ε, α x ) * G 0,γ A and X b (ε, β y ) * G 0,γ B are nonnegative for all a, b. It is enough to find γ A and γ B such that X −1 (ε, α x ) * G 0,γ A ≥ 0 and X −1 (ε, β y ) * G 0,γ B ≥ 0. Now, consider the choice γ A = γ B = tI 2 with t ≥ 0. If we are to satisfy V ≥ γ A ⊕ γ B , we need t ≤ 1 + η(ν − 1 − √ ν 2 − 1). From Eq. (8), it follows that if X −1 (ε, α) * G 0,tI2 becomes nonnegative for some value of t, it will remain so for all larger t. Consequently, one can consider the highest acceptable value of t, that is t = 1 + η(ν − 1 − √ ν 2 − 1). Furthermore, by definition of the convolution, the value of t for which X −1 (ε, α) * G 0,tI2 becomes nonnegative does Comparisons of regions for which the lossy TMSSρ AB admits a LHVM for the choice of measurements {X+1(ε, α), X−1(ε, α)} (blue region, left) and for which it violates the CHSH inequality (orange region, right), for the choice ε = 0.02. Note that we limited the figures to values ν ∈ [1, 1.5] as this is where a significant violation of the CHSH inequality occurs, but the region of existence of LHVMs extends further when increasing the range of values of ν.
not depend on α, so that one can take α = 0. Now, (X −1 (ε, 0) * G 0,tI2 )(x, p)
= 1 4π − 1 + t + ε(x 2 + p 2 − 2) 2π(1 + t) 2 e −(x 2 +p 2 ) 2(1+t) .(14)
The region in the (η, ν)-plane for which the above distribution is nonnegative for ε = 0.02 and t = 1 + η(ν − 1 − √ ν 2 − 1) is plotted in Fig. 2. In this region, the entangled stateρ AB admits an LHV model for the family of measurements described above.
Conclusion.-In this work, we have developed a criterion for the existence of local-hidden-variable models for correlations resulting from general measurements on Gaussian states, by exploiting that measurementoperator Wigner functions can be made positive by passing Gaussian noise from the state to the measurement. We have illustrated the criterion for the case of noisy displacement-based measurements on a two-mode squeezed state subject to loss.
An interesting question is whether the statement of Theorem 1 is also a necessary criterion: if one cannot find two positive semidefinite matrices γ A and γ B such that Eq. (5) is satisfied for all POVM elements simultaneously, does this imply nonlocality of the distribution p(ab|xy) = Tr[ρ AB Q a|x ⊗ R b|y ]?
Acknowledgements.-We gratefully acknowledge sup-
. Any positive hermitian operator in state space can equivalently be completely described by its real-valued arXiv:2210.05474v1 [quant-ph] 11 Oct 2022
FIG. 2 .
2FIG. 2. Comparisons of regions for which the lossy TMSSρ AB admits a LHVM for the choice of measurements {X+1(ε, α), X−1(ε, α)} (blue region, left) and for which it violates the CHSH inequality (orange region, right), for the choice ε = 0.02. Note that we limited the figures to values ν ∈ [1, 1.5] as this is where a significant violation of the CHSH inequality occurs, but the region of existence of LHVMs extends further when increasing the range of values of ν.
from the Carlsberg Foundation and the Independent Research Fund Denmark 7027-00044B. * [email protected] † jonatan. [email protected] the Carlsberg Foundation and the Independent Research Fund Denmark 7027-00044B. * [email protected] † [email protected]
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| {'fraction_non_alphanumeric': 0.07521932534288892, 'fraction_numerical': 0.050722846904732485, 'mean_word_length': 4.022184300341297, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We derive a simple sufficient criterion for the locality of correlations obtained from given measurements on a Gaussian quantum state. The criterion is based on the construction of a local-hiddenvariable model which works by passing part of the inherent Gaussian noise of the state onto the measurements. We illustrate our result in the setting of displaced photodetection on a two-mode squeezed state. Here, our criterion exhibits the existence of a local-hidden-variable model for a range of parameters where the state is still entangled.', 'arxivid': '2210.05474', 'author': ['Michael G Jabbour \nDepartment of Physics\nTechnical University of Denmark\n2800Kongens LyngbyDenmark\n', 'Jonatan Bohr Brask \nDepartment of Physics\nTechnical University of Denmark\n2800Kongens LyngbyDenmark\n'], 'authoraffiliation': ['Department of Physics\nTechnical University of Denmark\n2800Kongens LyngbyDenmark', 'Department of Physics\nTechnical University of Denmark\n2800Kongens LyngbyDenmark'], 'corpusid': 252816023, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11246, 'n_tokens_neox': 9346, 'n_words': 5283, 'pdfsha': 'ee548fa8063f9e7c90fe5f0d94c6af9ef368e69e', 'pdfurls': ['https://export.arxiv.org/pdf/2210.05474v1.pdf'], 'title': ['Constructing local models for general measurements on bosonic Gaussian states', 'Constructing local models for general measurements on bosonic Gaussian states'], 'venue': []} |
arxiv |
A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization
January 3, 2023
Albert S Berahas
Miaolan Xie
Baoyu Zhou
A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization
January 3, 2023
A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed via inexact probabilistic zeroth-and first-order oracles. Under reasonable assumptions, a high-probability bound on the iteration complexity of the algorithm to approximate first-order stationarity is derived. Numerical results on standard nonlinear optimization test problems illustrate the advantages and limitations of our proposed method. †
Introduction
In this paper, we propose a step-search 1 sequential quadratic programming (SQP) algorithm for solving nonlinear equality-constrained stochastic optimization problems of the form min x∈R n f (x) s.t. c(x) = 0, (1.1) where f : R n → R and c : R n → R m are both continuously differentiable. We consider the setting in which exact function and derivative information of the objective function is unavailable, instead, only random estimates of the objective functionf (x; Ξ 0 (x)) ≈ f (x) and its first-order derivativeḡ(x; Ξ 1 (x)) ≈ ∇f (x) are available via inexact probabilistic oracles, where Ξ 0 (x) (with probability space (Ω 0 , F Ω 0 , P 0 )) and Ξ 1 (x) (with probability space (Ω 1 , F Ω 1 , P 1 )) denote the underlying randomness in the objective function and gradient estimates, respectively. On the other hand, the constraint function value c(x) and its Jacobian ∇c(x) T are assumed to be available. Such deterministically constrained stochastic optimization problems arise in multiple science and engineering applications, including but not limited to computer vision [37], multi-stage optimization [39], natural language processing [30], network optimization [9], and PDE-constrained optimization [35]. The majority of the methods proposed in the literature for solving deterministically equality-constrained stochastic optimization problems follow either projection or penalty approaches. The former type of methods (e.g., stochastic projection methods [21,[23][24][25]) require that the feasible region satisfies strict conditions, to ensure well-definedness, that are not satisfied by general nonlinear functions and thus are not readily applicable. In contrast, the latter, stochastic penalty methods [14,34], do not impose such conditions on the feasible region. These methods transform constrained problems into unconstrained problems via a constraint penalization term in the objective function and apply stochastic algorithms to solve the transformed unconstrained optimization problems. Stochastic penalty methods are easy to implement and well-studied, however, the empirical performance of such methods is sensitive to parameter choices and ill-conditioning, and is usually inferior to paradigms that treat constraints as constraints.
Recently, a class of stochastic SQP methods has been developed for solving (1.1). These methods outperform stochastic penalty methods empirically and have convergence guarantees in expectation [7,28]. In [7], the authors propose an objective-function-free stochastic SQP method with adaptive step sizes for the fully stochastic regime. In contrast, in [28], the authors propose a stochastic step search (referred to as line search in the paper [28]) SQP method for the setting in which the errors in the function and derivative approximations can be diminished. We note that several algorithm choices in the two papers [7,28], e.g., merit functions and merit parameters, are different. Several other extensions have been proposed [3,6,8,17,27,32], and very few of these works (or others in the literature) derive worst-case iteration complexity (or sample complexity) due to the difficulties that arise because of the constrained setting and the stochasticity. Notable exceptions are, [16] where the authors provide convergence rates (and complexity guarantees) for the algorithm proposed in [7], and [3,29] that provide complexity bounds for variants of the stochastic SQP methods under additional assumptions and in the setting in which the errors can be diminished. We note that, with the exception of [32], all methods mentioned above assume access to unbiased estimates of the gradients (and function values where necessary), whereas in this paper, we propose an algorithm that can handle biased function and gradient estimates.
For all aforementioned methods, the most vital ingredient is the quality and reliability of the random estimates of the objective function and its derivatives. In our setting, neither the objective function nor its derivatives are assumed to be directly accessible, only stochastic approximations of them are accessible to the algorithm in the form of inexact probabilistic zeroth-order and first-order oracles (precise definitions will be introduced in Section 2.3). Such oracles have been proposed and utilized in several works; e.g., [1,12,20,22]. Moreover, these probabilistic oracles and their variants have been proposed for directsearch methods [20,36], trust-region methods [1,10,15,19], and step-search methods [2,13,28,33]. We note that only [28] considers the setting with (equality) constraints, but iteration complexity (or sample complexity) results are not provided.
Contributions
In this paper, we design, analyze, and implement a step-search SQP (SS-SQP) method for solving nonlinear equality-constrained stochastic optimization problems where exact constraint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed. These stochastic approximations are computed via inexact probabilistic zeroth-and first-order oracles, which are similar to those in [22], with parameters controlling the accuracy and reliability of the approximations, and allowing for biased approximations. Our proposed algorithm is inspired by state-of-the-art line search SQP methods [11] in conjunction with the recent stochastic adaptive step-search framework developed in [22] for the unconstrained stochastic setting. At every iteration, the algorithm constructs a model of the reduction in the merit function that serves the dual purpose of a measure of sufficient progress (part of the step size computation) and a proxy for convergence. To mitigate the challenges that arise due to the noise in the objective function evaluations, our step-search method employs a relaxed sufficient decrease condition similar to that proposed in [4]. Under reasonable assumptions, we provide a high probability worst-case iteration complexity bound for the proposed algorithm. Specifically, we prove that with overwhelmingly high probability, our proposed algorithm generates a first-order ε-stationary iterate in O(ε −2 ) iterations, where ε is bounded away from zero and its lower bound is dictated by the noise and bias in the zeroth-and first-order oracles. The complexity bound derived matches that of the deterministic algorithm provided in [16]. There are two key differences between our paper and [16]: (i) our algorithm requires access to the objective function whereas the method in [16] is objective-function-free; and (ii) our first-order oracle provides estimates with sufficient accuracy only with some probability and can provide arbitrarily bad estimates otherwise. Finally, numerical results on standard nonlinear equality-constrained test problems [18] illustrate the efficiency and efficacy of our proposed algorithm.
Notation
Let R denote the set of real numbers, R n denote the set of n-dimensional real vectors, R m×n denote the set of m-by-n-dimensional real matrices, N denote the set of natural numbers, and S n denote the set of n-by-n-dimensional real symmetric matrices. For any a ∈ R, let R >a (R ≥a ) denote the set of real numbers strictly larger than (larger than or equal to) a. We use · to denote the 2 -norm. We use k ∈ N as the iteration counter of the algorithm, and for brevity, we use a subscript k for denoting information at the kth iterate, e.g., f k := f (x k ). All quantities with over-bars are stochastic, e.g.,f (x; Ξ 0 (x)) andḡ(x; Ξ 1 (x)) (see Section 2.3), andf (x; ξ 0 (x)) (resp.ḡ(x; ξ 1 (x))) denote realizations of f (x; Ξ 0 (x)) (resp.ḡ(x; Ξ 1 (x))).
Organization
The rest of this paper is organized as follows. The algorithmic framework is introduced in Section 2. The analysis of the algorithm is established in Section 3. We report numerical results in Section 4. Concluding remarks and future research directions are given in Section 5.
Algorithm
To solve (1.1), we design an iterative algorithm based on the SQP paradigm that generates: (i) a primal iterate sequence {x k }, (ii) a primal trial iterate sequence {x + k }, (iii) a primal search direction sequence {d k }, (iv) a dual iterate sequence {ȳ k }, (v) a step size sequence {α k }, (vi) a merit parameter sequence {τ k }, and, (vii) a trial merit parameter sequence {τ trial k }. We discuss each of these sequences in below. We make the following assumption throughout the remainder of this paper. The objective function f : R n → R is continuously differentiable and bounded below over X . The objective gradient function ∇f : R n → R n is L-Lipschitz continuous and bounded over X . The constraint function c : R n → R m (where m ≤ n) is continuously differentiable and bounded over X , and each gradient ∇c i : R n → R n is γ i -Lipschitz continuous and bounded over X for all i ∈ {1, . . . , m}. The singular values of J := ∇c T are bounded away from zero over X . Assumption 2.1 is a standard assumption in the deterministic constrained optimization literature [31]. Under Assumption 2.1, there exist constants {κ g , κ c , κ J , κ σ } ⊂ R >0 and f inf ∈ R such that for all k ∈ N,
f inf ≤ f k , ∇f k ≤ κ g , c k 1 ≤ κ c , J k ≤ κ J , and (J k J T k ) −1 ≤ κ σ .
We should note that by Assumption 2.1, linear independence constraint qualifications (LICQ) hold. Moreover, under Assumption 2.1, for all x ∈ R n , d ∈ R n and α ∈ R ≥0 it follows that
f (x + αd) ≤ f (x) + α∇f (x) T d + L 2 α 2 d 2 and c(x + αd) 1 ≤ c(x) + α∇c(x) T d 1 + Γ 2 α 2 d 2 , where Γ = m i=1 γ i . (2.1)
In this paper, we are particularly interested in finding some primal-dual iterate (x, y) ∈ R n × R m that satisfies the first-order stationarity conditions of (1.1). To this end, let L : R n × R m → R be the Lagrangian of (1.1), defined as
L(x, y) = f (x) + y T c(x),(2.2)
where y ∈ R m are the dual variables. The first-order stationarity conditions for (1.1), which are necessary by Assumption 2.1 (due to the inclusion of the LICQ), are
0 = ∇ x L(x, y) ∇ y L(x, y) = ∇f (x) + ∇c(x)y c(x) . (2.3)
In the remainder of this section we introduce the key algorithmic components: the merit function and its associated models, the search direction computation and merit parameter updating mechanism, and the inexact probabilistic zeroth-and first-order oracles. The main algorithm is Algorithm 1.
Merit function
The merit function φ :
R n × R >0 → R is defined as φ(x, τ ) := τ f (x) + c(x) 1 , (2.4)
where τ ∈ R >0 , the merit parameter, acts as a balancing parameter between the objective function and the constraint violation. Given the gradient (approximation) g ∈ R n and a search direction d ∈ R n , the model of merit function l :
R n × R >0 × R n × R n → R is defined as l(x, τ, g, d) := τ (f (x) + g T d) + c(x) + ∇c(x) T d 1 .
Given a search direction d ∈ R n that satisfies linearized feasibility, i.e., c(x) + ∇c(x) T d = 0, the reduction in the model of the merit function ∆l : R n × R >0 × R n × R n → R is defined as ∆l(x, τ, g, d) :=l(x, τ, g, 0) − l(x, τ, g, d)
= − τ g T d + c(x) 1 − c(x) + ∇c(x) T d 1 = − τ g T d + c(x) 1 .
(2.5)
We use the reduction in the model of the merit function (2.5) to monitor the progress made by our proposed algorithm. We discuss this in more detail in Section 2.2.
Algorithmic components
We now establish how to: (i) compute the primal search direction sequence {d k }, (ii) update the merit parameter sequence {τ k }, and (iii) update the primal iterate sequence {x k }. These sequences depend on the approximation of the gradient of the objective function sequence {ḡ(x k ; Ξ 1 (x k ))}. Letḡ(x k ; ξ 1 (x k )) denote the realization ofḡ(x k ; Ξ 1 (x k )).
To simplify the notation, in this subsection we drop the dependence on the randomness, e.g.,ḡ k =ḡ(x k ; ξ 1 (x k )). At each iteration k ∈ N, the primal search directiond k ∈ R n and the dual variablē y k ∈ R m are computed by solving the linear system of equations
H k J T k J k 0 d k y k = − ḡ k c k , (2.6)
where {H k } satisfies the following assumption.
Assumption 2.2. For all k ∈ N, H k ∈ S n is chosen independently fromḡ k . Moreover, there exist constants {κ H , ζ} ⊂ R >0 such that for all k ∈ N, H k ≤ κ H and u T H k u ≥ ζ u 2 for any u ∈ Null(J k ).
It is well known that under Assumptions 2.1 and 2.2, there is a unique solution (d k ,ȳ k ) to (2.6), and, thus, the vectorsd k ∈ R n andȳ k ∈ R m are well-defined [31].
Next, we present the merit parameter updating mechanism. Given constants { τ , σ} ⊂ (0, 1), for all k ∈ N, we computeτ k viā
τ k ← τ k−1 ifτ k−1 ≤τ trial k ; min (1 − τ )τ k−1 ,τ trial k otherwise, (2.7) whereτ trial k ← ∞ ifḡ T kd k + max d T k H kdk , 0 ≤ 0; (1−σ) c k 1 g T kd k +max{d T k H kdk ,0}
otherwise.
(2.8)
The merit parameter updating mechanism ensures that the sequence of merit parameter values is non-increasing. Moreover, the updating mechanism is designed to ensure that the reduction in the model of the merit function is sufficiently positive. By (2.7) and (2.8), it follows that (see Lemma 3.7)
∆l(x k ,τ k ,ḡ k ,d k ) ≥τ k max d T k H kdk , 0 + σ c k 1 . (2.9)
In the deterministic setting, the reduction in the model of the merit function is zero only at iterates that satisfy (2.3). After updating the merit parameterτ k , we evaluate ∆l(x k ,τ k ,ḡ k ,d k ), the stochastic model reduction of the merit function, and use it to check for sufficient progress. Specifically, given a step size α k , we compute a candidate iterate x + k := x k + α kdk and check whether sufficient progress can be made via the following modified sufficient decrease conditionφ
(x + k ,τ k ; ξ 0 (x + k )) ≤φ(x k ,τ k ; ξ 0 (x k )) − α k θ∆l(x k ,τ k ,ḡ k ,d k ) + 2τ k f ,(2.10)
whereφ(x + k ,τ k ; ξ 0 (x + k )) andφ(x k ,τ k ; ξ 0 (x k )) are merit function estimates, θ ∈ (0, 1) is a user-defined parameter and f is an upper bound on the expected noise in the objective function approximations. We note thatφ(x + k ,τ k ; ξ 0 (x + k )) andφ(x k ,τ k ; ξ 0 (x k )) are realizations of the zeroth-order oracle described in detail in Section 2.3. The positive term on the right-hand-side allows for a relaxation in the sufficient decrease condition, i.e., the merit function may increase after a step, and serves to correct for the noise in the merit function approximations. If (2.10) is satisfied, we accept the candidate point x + k by setting x k+1 ← x + k , and potentially increase the step size for the next iteration, i.e., α k+1 ≥ α k . If (2.10) is not satisfied, the algorithm does not accept the candidate iterate, instead, it sets x k+1 ← x k and shrinks the step size for the next iteration, i.e., α k+1 < α k . This step update rule is the centerpiece of our step-search method, and is fundamentally different from traditional line-search strategies; see [5,13,22] and the references therein. Contrary to line search methods, which compute a search direction and then look for a step size along that direction, in our approach the search direction changes in every iteration.
We conclude this section by drawing a few parallels to the unconstrained setting. First, in the unconstrained setting (with H k = I), the quantity ∆l(x k ,τ k ,ḡ k ,d k ) reduces to ḡ k 2 , which provides a sufficient descent measure and is an approximate first-order stationarity measure. In the constrained setting, the reduction in the model of the merit function will play a similar role. Second, in the unconstrained optimization setting, (2.10) recovers the sufficient decrease condition used by some noisy unconstrained optimization algorithm; see [4,Eq. (3.11)].
Probabilistic oracles
In many real-world applications exact objective function and derivative information cannot be readily computed. Instead, in lieu of these quantities, approximations are available via inexact probabilistic zeroth-and first-order oracles. These oracles produce approximations of different accuracy and reliability, and are formally introduced below.
Oracle 0 (Probabilistic zeroth-order oracle). Given x ∈ R n , the oracle computes f (x; ξ 0 (x)), a realization off (x; Ξ 0 (x)), which is a (random) estimate of the objective function value f (x), where Ξ 0 (x) denotes the underlying randomness (may depend on x) with associated probability space Ω 0 , F Ω 0 , P 0 . Let e(x; Ξ 0 (x)) := |f (x; Ξ 0 (x)) − f (x)|. For any x ∈ R n , e(x; Ξ 0 (x)) is a "one-sided" sub-exponential random variable with parameters {ν, b} ⊂ R ≥0 , whose mean is bounded by some constant f ∈ R ≥0 . Specifically, for all x ∈ R n and λ ∈ [0, 1/b],
E Ξ 0 (x) e(x; Ξ 0 (x)) ≤ f and E Ξ 0 (x) exp(λ(e(x; Ξ 0 (x)) − E e(x; Ξ 0 (x)) )) ≤ exp λ 2 ν 2 2 .
(2.11)
The stochastic approximation of the merit function value is defined asφ(x, τ ; ξ 0 (x)) = τf (x; ξ 0 (x)) + c(x) 1 .
Oracle 1 (Probabilistic first-order oracle). Given x ∈ R n and α ∈ R >0 , the oracle computesḡ(x; ξ 1 (x)), a realization ofḡ(x; Ξ 1 (x)), which is a (random) estimate of the gradient of the objective function ∇f (x), such that
P Ξ 1 (x) ḡ(x; Ξ 1 (x)) − ∇f (x) ≤ max g , κ FO α ∆l(x,τ (x; Ξ 1 (x)),ḡ(x; Ξ 1 (x)),d(x; Ξ 1 (x))) ≥ 1 − δ,
where Ξ 1 (x) denotes the underlying randomness (may depend on x) with associated probability space (Ω 1 , F Ω 1 , P 1 ), (1 − δ) ∈ ( 1 2 , 1] is the probability that the oracle produces a gradient estimate that is "sufficiently accurate" (related to the reliability of the oracle) and { g , κ FO } ⊂ R ≥0 are constants intrinsic to the oracle (related to the precision of the oracle).
In the rest of the paper, to simplify the notation we drop the dependence on x in ξ 0 (x) and ξ 1 (x). Moreover, we use ξ + k to represent ξ 0 (x + k ), the randomness in the zeroth-order oracle evaluated at the trial point x + k .
Remark 2.3.
We make a few remarks about Oracles 0 and 1:
• Oracles 0 and 1 are similar to those defined in [12,22]. For a full discussion and examples of the oracles, we refer interested readers to [22,Section 5].
• Oracle 1 is a natural generalization of the ones defined in [12,22] to the constrained setting. In particular, the right-hand-side of Oracle 1 reduces to max g , κ FO α ḡ(x; Ξ 1 ) in the unconstrained setting, and is precisely what is used in [12,22].
• The presence of g ∈ R ≥0 in the max term in Oracle 1 allows the gradient approximations to be biased; the magnitude of the bias is proportional to g .
Algorithmic framework
We are ready to introduce our stochastic step-search SQP method (SS-SQP) in Algorithm 1.
Remark 2.4. We make the following remarks about SS-SQP:
• (Step-search) Algorithm 1 is a step-search algorithm, whose main difference from traditional line-search methods is that only a single trial iterate is tested at every iteration. That is, if (2.10) is not satisfied, the step size is reduced and a new search direction and candidate iterate are computed in the next iteration. This strategy has been employed in other papers; e.g., see [5,13,22,28]. We should note that at every iteration, even if the iterate does not change, our algorithm requires new objective function and gradient estimates in the next iteration.
Algorithm 1 Adaptive Step-Search SQP (SS-SQP)
Require: initial iterate x 0 ∈ R n ; initial merit parameterτ −1 ∈ R >0 ; maximum step size α max ∈ (0, 1]; initial step size α 0 ∈ (0, α max ]; parameter f ∈ R ≥0 of the zeroth-order oracle (Oracle 0); and other constant parameters {γ, θ, σ, τ } ⊂ (0, 1) 1: for all k ∈ N do 2:
Generateḡ k =ḡ(x k ; ξ 1 k ) via Oracle 1 with α = α k ,d k =d(x k ; ξ 1 k ) as in (2.6), and τ k =τ (x k ; ξ 1 k ) as in (2.7)-(2.8) 3
:
Let x + k = x k + α kdk , and generateφ(x k ,τ k ; ξ 0 k ) andφ(x + k ,τ k ; ξ + k ) via Oracle 0 4:
if (2.10) holds then 5:
Set x k+1 ← x + k and α k+1 ← min{α max , γ −1 α k } 6: else 7: Set x k+1 ← x k and α k+1 ← γα k 8: end if 9: end for • (Modified sufficient decrease condition (2.10))
The 2τ k f term on the right-handside of (2.10) is a correction term added to compensate for the inexactness of the probabilistic zeroth-order oracle (Oracle 0). This correction provides a relaxation to the sufficient decrease requirement. In contrast to traditional sufficient decrease conditions, the modified condition (2.10) allows for a relaxation that is proportional to the noise level of Oracle 0.
• (Objective function evaluations; Line 3) The randomness associated with the evaluation of the objective function value at the candidate iterate x + k (Line 3) is not the same as that of the evaluation at the current point x k . Moreover, we note that even for unsuccessful iterations (where the iterates do not change) the objective function values are re-evaluated.
• (Objective gradient evaluations; Line 2) In order to generate an estimate of the gradient of the objective function that satisfies the conditions of Oracle 1, one can employ a procedure (a loop) similar to [38,Algorithm 2]. The idea is to refine the estimate progressively in order to generate one that satisfies the condition. Indeed, in many real-world problems, including empirical risk minimization in machine learning, one can improve the gradient approximation by progressively using a larger number of samples.
• (Maximum step size α max ) We pick α max ∈ (0, 1] mainly to simplify our analysis. That being said, the unit upper bound on α max is motivated by the deterministic constraint setting. In the deterministic setting (without any noise), the merit function decrease is upper bounded by a nonsmooth function, whose only point of nonsmothness is at α = 1, which complicates the analysis; see [7, Lemma 2.13].
Before we proceed, we define the stochastic process related to the algorithm.
Let M k denote {Ξ 0 k , Ξ + k , Ξ 1 k } with realizations {ξ 0 k , ξ + k , ξ 1 k }.
The algorithm generates a stochastic process:
{(G k , D k , T k ,φ(X k , T k ; Ξ 0 k ),φ(X + k , T k ; Ξ + k ), X k , A k )} with realizations {(ḡ k ,d k ,τ k ,φ(x k ,τ k ; ξ 0 k ),φ(x + k ,τ k ; ξ + k ), x k , α k )}, adapted to the filtration {F k : k ≥ 0}, where F k = σ(M 0 , M 1 , . . . , M k ) and σ denotes the σ-algebra. At iteration k, G k is the random gradient, D k is the random primal search direction, T k is the random merit param- eter,φ(X k , T k ; Ξ 0 k ) andφ(X + k , T k ; Ξ + k )
are the random noisy merit function evaluations at the current point and the candidate point, respectively, X k is the random iterate at iteration k and A k is the random step size. Note that G k , D k , T k are dictated by Ξ 1 k (Oracle 1) and the noisy merit function evaluations are dictated by Ξ 0 k , Ξ + k (Oracle 0).
Theoretical analysis
In this section, we analyze the behavior of Algorithm 1. For brevity, throughout this section, we assume Assumptions 2.1 and 2.2 hold and do not restate this fact in every lemma and theorem. We begin by presenting some preliminary results, definitions, and assumptions and then proceed to present a worst-case iteration complexity bound for Algorithm 1.
Preliminaries, definitions & assumptions
We first define some deterministic quantities that are used in the analysis of Algorithm 1, and which are never explicitly computed in the implementation of the algorithm. Let (d k , y k ) ∈ R n × R m be the solution of the deterministic counterpart of (2.6), i.e.,
H k J T k J k 0 d k y k = − ∇f k c k . (3.1)
The norm of the gradient of the Lagrangian (defined in (2.2)) of (1.1), which is used as a first-order stationarity measure, can be upper bounded at every primal-dual iterate (x k , y k ) as
∇f k + J T k y k c k = −H k d k −J k d k ≤ (κ H + κ J ) d k , (3.2)
where the equality is by (3.1) and the inequality follows by Assumptions 2.1 and 2.2. Thus, (3.2) implies that d k , the primal search direction, can be used as a proxy of the first-order stationary measure. The following lemma shows that the tuple (d k , y k ) is bounded for all k ∈ N.
Lemma 3.1. There exist constants {κ d , κ y } ⊂ R >0 such that d k ≤ κ d and y k ≤ κ y for all k ∈ N.
Proof. By the Cauchy-Schwarz inequality and (3.1), we have
d k y k = H k J T k J k 0 −1 ∇f k c k ≤ H k J T k J k 0 −1 ∇f k c k ,
where both terms on the right-hand side of the inequality are bounded by Assumptions 2.1 and 2.2, which concludes the proof.
Moreover, we define τ k ∈ R >0 and τ trial k ∈ R >0 , the deterministic counterparts of (2.7) and (2.8),
τ k ← τ k ifτ k ≤ τ trial k ; min (1 − τ )τ k , τ trial k otherwise, (3.3) where τ trial k ← ∞ if ∇f T k d k + max d T k H k d k , 0 ≤ 0; (1−σ) c k 1 ∇f T k d k +max{d T k H k d k ,0} otherwise. (3.4)
We emphasize again that {(τ k , τ trial k )} k∈N are introduced only for the purposes of the analysis, and in Algorithm 1 they are never computed (not even in the setting in which the true gradient is used, i.e.,ḡ k = ∇f (x k )). We also note that this definition is not the same as that in [7,16]. The difference is in the fact that in the computation of τ k , the comparison is made toτ k instead ofτ k−1 . This is important for the analysis, since this guarantees τ k ≤τ k .
We assume that the merit parameter sequence {τ k } generated in the stochastic setting is bounded away from zero (Assumption 3.2). Such an assumption has been adopted in previous literature [6-8, 16, 17]; we refer readers to [7, Section 3.2] and [16, Section 4.2] for detailed discussions. Finally, we note that we only assume that {τ k } is bounded away from zero, and never require the knowledge ofτ min in the algorithm.
Assumption 3.2. Let {τ k } be the merit parameter sequence generated by Algorithm 1. There exists a constantτ min ∈ R >0 such that for every realization of Algorithm 1,τ k ≥τ min for all k ∈ N.
Next, we state and prove a provide a useful property with regards to the deterministic merit parameter sequence {τ k } defined in (3.3). Lemma 3.3. Suppose Assumption 3.2 holds, then there exists a positive constant τ min ∈ R >0 such that for every realization of Algorithm 1, τ k ≥ τ min for all k ∈ N.
Proof. By [7, Lemma 2.16], {τ trial k } ⊂ R >0 ∪ {+∞} is always bounded away from zero. We define τ trial min ∈ R >0 such that τ trial min ≤ τ trial k for all k ∈ N. By (3.3)-(3.4)
and Assumption 3.2, one may pick τ min = min{(1 − τ )τ trial min ,τ min } to conclude the proof.
Our final assumption relates to the zeroth-order oracle (Oracle 0).
Assumption 3.4. Let E k and E + k be the errors in the objective function evaluations from Oracle 0, i.e.,
E k := f (X k ; Ξ 0 k ) − f (X k ) , and E + k := f (X + k ; Ξ + k ) − f (X + k )
. We assume that either {E k } and {E + k } are deterministically bounded by f ∈ R ≥0 , or that the summation of the errors {E k + E + k } are independent over different iterations.
Next, we introduce several definitions necessary for the analysis of Algorithm 1. Specifically, we define true/false iterations (Definition 1), successful/unsuccessful iterations (Definition 2) and large/small steps (Definition 3), and introduce three indicator variables respectively.
Definition 1. An iteration k ∈ N is true if ḡ k − ∇f k ≤ max g , κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ) and e k + e + k ≤ 2 f , (3.5) where ∆l(x k ,τ k ,ḡ k ,d k ) is defined in (2.5
) and the constants f , g and κ FO are the same ones as in Oracles 0 and 1. If (3.5) does not hold, we call the iteration a false iteration. We use the random indicator variable I k to denote if an iteration is true.
Definition 2. Given a constant θ ∈ (0, 1), letφ(x k ,τ k ; ξ k ) andφ(x + k ,τ k ; ξ + k )
be obtained by Oracle 0. If inequality (2.10) holds, then iteration k is successful, otherwise, it is an unsuccessful iteration. We use the random indicator variable Θ k to denote whether an iteration is successful.
Definition 3. For any k ∈ N, if min{α k , α k+1 } ≥α whereα is some problem-dependent positive real number (defined explicitly in Lemma 3.15), then we call the step a large step and set the indicator variable U k = 1. Otherwise, we call the step k a small step and set U k = 0.
We show that under appropriate conditions, if the step is a small step and the iteration is true, then, the iteration is guaranteed to be successful (see Lemma 3.15). The last definition is for the stopping time (T ε ∆l ) and a measure of progress ({Z k }).
Definition 4. For any realization of Algorithm 1, define T ε ∆l = min{k :
∆l(x k , τ k , ∇f k , d k ) ≤ ε ∆l }, the number of iterations required to reach a first-order εstationary iterate, where ε = Ω(ε ∆l ). We discuss the explicit relationship between ε and ε ∆l in Remark 3.5. Moreover, for all k ∈ N,
let Z k := φ(x k ,τ k ) − φ min − (τ k f inf −τ min f inf ),
where φ min is a lower bound of φ(·,τ min ) over X andτ min is defined in Assumption 3.2.
Remark 3.5.
A key ingredient of our algorithm is the stopping time T ε ∆l that is related to ∆l(x k , τ k , ∇f k , d k ). In fact, by (3.2), Assumption 3.2 and Lemma 3.9 (see below), the stopping time T ε ∆l defined in Definition 4 is the number of iterations needed to achieve a first-order ε-stationary iterate, i.e.,
max{ ∇f k + J T k y k , c k } ≤ ε, where ε = max{κ H ,1} √ κ l τ min · ε ∆l . (3.6)
We note that (3.6) is the same stationarity measure as that used in [16,Eq. (5)], and is a non-standard first-order stationary measure compared to
∇f k + J T k y k c k . That said, one can show that ∇f k + J T k y k c k ≤ 2 max{ ∇f k + J T k y k , c k } ≤ 2 max{κ H ,κ J } √ κ l τ min · ε ∆l = Ω(ε).
Throughout this paper we focus on (and provide complexity bounds for) (3.6) as it provides a stronger result for feasibility ( c k ) when ε < 1.
Main Technical Results
We build toward the main result of the paper (Theorem 3.18) through a sequence of technical lemmas. Our first lemma shows that Z k (defined in Definition 4) is always nonnegative.
Lemma 3.6. For all k ∈ N, Z k ≥ 0.
Proof. It follows from (2.4) and Definition 4 that
Z k = φ(x k ,τ k ) − φ min − (τ k f inf −τ min f inf ) = (τ k (f k − f inf ) + c k 1 ) − φ min +τ min f inf ≥ (τ min (f k − f inf ) + c k 1 ) − φ min +τ min f inf = (τ min f k + c k 1 ) − φ min = φ(x k ,τ min ) − φ min ≥ 0,
which concludes the proof.
The next lemma reveals the critical role of the merit parameter update.
Lemma 3.7. For all k ∈ N, (2.9) is satisfied. Furthermore, ifτ k =τ k−1 , then 0 <τ k ≤ (1 − τ )τ k−1 .
Proof. By Algorithm 1, we haveτ k ≤τ trial k . Moreover, by (2.5), (2.7) and (2.8), it follows that (2.9) is satisfied for all k ∈ N.
By (2.7), ifτ k =τ k−1 , thenτ k = min (1 − τ )τ k−1 ,τ trial k ≤ (1 − τ )τ k−1 . Moreover, when c k = 0, it follows from Assump- tion 2.2, (2.6) and (2.8) thatd k ∈ Null(J k ) andḡ T kd k +max{d T k H kdk , 0} =ḡ T kd k +d T k H kdk = c T kȳ k = 0, which impliesτ trial k = ∞. Therefore, we haveτ trial k > 0 for all k ∈ N. Finally, bȳ τ −1 ∈ R >0 and (2.7), we haveτ k > 0 for all k ∈ N.
The next lemma provides a useful lower bound for the reduction in the model of the merit function, ∆l(x k ,τ k ,ḡ k ,d k ), that is related to the primal search direction ( d k 2 ) and a measure of infeasibility ( c k ).
Lemma 3.8. There exists some constant κ l ∈ R >0 such that for all k ∈ N, ∆l(x k ,τ k ,ḡ k ,d k ) ≥ κ lτk ( d k 2 + c k ).
Proof. For any iteration k ∈ N, by [7, Lemma 3.4], there exists some constant κ l ∈ R >0 such that
−τ k (ḡ T kdk + 1 2 max{d T k H kdk , 0}) + c k 1 ≥ κ lτk ( d k 2 + c k 1 ).
Byτ k ∈ R >0 (from Lemma 3.7), this implies that
∆l(x k ,τ k ,ḡ k ,d k ) = −τ kḡ T kdk + c k 1 ≥ −τ k (ḡ T kdk + 1 2 max{d T k H kdk , 0}) + c k 1 ,
which concludes the proof.
Lemma 3.9. There exists some constant κ l ∈ R >0 such that for all k ∈ N,
∆l(x k , τ k , g k , d k ) ≥ κ l τ k ( d k 2 + c k ).
Proof. The proof follows the same logic as that of Lemma 3.8 with the stochastic quantities replaced by their deterministic counterparts. By [7,Lemma 3.4], the desired inequality is satisfied for the same constant κ l defined in Lemma 3.8.
The next lemma bounds the errors in the stochastic search directions and dual variables, respectively, with respect to the errors in the gradient approximations.
} ⊂ R >0 such that d k − d k ≤ ζ −1 ḡ k − ∇f k and ȳ k − y k ≤ ω 1 ḡ k − ∇f k , where ζ is defined in Assumption 2.2.
Proof. By the Cauchy-Schwarz inequality, Assumption 2.2, (3.1), and the fact that (d k − d k ) ∈ Null(J k ), it follows that
d k − d k ḡ k − ∇f k ≥ (d k − d k ) T (∇f k −ḡ k ) = (d k − d k ) T (H k (d k − d k ) + J T k (ȳ k − y k )) = (d k − d k ) T H k (d k − d k ) ≥ ζ d k − d k 2 ,
which proves that d k −d k ≤ ζ −1 ḡ k −∇f k . Next, by (3.1) and Assumption 2.1 it follows thatȳ
k − y k = −(J k J T k ) −1 J k (ḡ k − ∇f k ) + H k (d k − d k ) .
By the triangle inequality, the Cauchy-Schwarz inequality, Assumptions 2.1 and 2.2 and the fact that
d k − d k ≤ ζ −1 ḡ k − ∇f k , it follows that ȳ k − y k = (J k J T k ) −1 J k (ḡ k − ∇f k ) + H k (d k − d k ) ≤ (J k J T k ) −1 J k ( ḡ k − ∇f k + H k d k − d k ) ≤ κ σ κ J (1 + κ H ζ −1 ) ḡ k − ∇f k .
Setting ω 1 = κ σ κ J (1 + κ H ζ −1 ) concludes the proof.
The next lemma relates the inner product of the stochastic gradient and stochastic search direction to the stochastic reduction in the model of the merit function. We consider two cases that are related to the two cases in the max term of Oracle 1.
Lemma 3.11. For all k ∈ N:
• If ḡ k − ∇f k ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ), then τ k |ḡ T kdk | ≤ max{κ H ,κy} κ l + √τ k (1+κHζ −1 )κFOαk √ κ l ∆l(x k ,τ k ,ḡ k ,d k ). • If ḡ k − ∇f k ≤ g , τ k |ḡ T kdk | ≤ max{κ H ,κy}+1 κ l · ∆l(x k ,τ k ,ḡ k ,d k ) +τ k (1+κHζ −1 ) 2 4 2 g .
Proof. If ḡ k −∇f k ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ), by the triangle inequality, (2.6), Assumption 2.2, and Lemmas 3.1, 3.8 and 3.10, it follows that
τ k |ḡ T kdk | =τ k |(H kdk + J T k y k + J T k (ȳ k − y k )) Td k | ≤τ k (|d T k H kdk | + |y T k J kdk | + |(ȳ k − y k ) T J kdk |) ≤τ k (κ H d k 2 + y k c k + (ḡ k − ∇f k ) + H k (d k − d k ) d k ) ≤ max{κ H , κ y } ·τ k ( d k 2 + c k ) +τ k ( ḡ k − ∇f k + κ H d k − d k ) d k ≤ max{κ H ,κy} κ l ∆l(x k ,τ k ,ḡ k ,d k ) +τ k 1 + κ H ζ −1 ḡ k − ∇f k d k ≤ max{κ H ,κy} κ l ∆l(x k ,τ k ,ḡ k ,d k ) + √τ k (1+κHζ −1 )κFOαk √ κ l ∆l(x k ,τ k ,ḡ k ,d k ),
which completes the first part of the proof. Using similar logic, if ḡ k −∇f k ≤ g , by the triangle inequality, (2.6), Assumption 2.2, Lemmas 3.1, 3.3, 3.8, 3.10, and the fact that ab ≤ a 2 + b 2 4 holds for any {a, b} ⊂ R, it follows
thatτ k |ḡ T kdk | ≤ max{κ H ,κy} κ l ∆l(x k ,τ k ,ḡ k ,d k ) +τ k 1 + κ H ζ −1 ḡ k − ∇f k d k ≤ max{κ H ,κy} κ l ∆l(x k ,τ k ,ḡ k ,d k ) +τ k 1 + κ H ζ −1 g d k ≤ max{κ H ,κy} κ l ∆l(x k ,τ k ,ḡ k ,d k ) + √τ k (1+κHζ −1 ) √ κ l g ∆l(x k ,τ k ,ḡ k ,d k ) ≤ max{κ H ,κy}+1 κ l ∆l(x k ,τ k ,ḡ k ,d k ) +τ k (1+κHζ −1 ) 2 4 2 g ,
which completes the proof.
The next lemma provides a useful upper bounds for the errors related to the stochastic search directions (and gradients) for the same two cases as in Lemma 3.11. Lemma 3.12. For all k ∈ N:
• If ḡ k − ∇f k ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ), then |∇f T k d k −ḡ T kdk | ≤ (1+κ H ζ −1 )κ FO α k √ κ lτk + κ 2 FO α 2 k ζ ∆l(x k ,τ k ,ḡ k ,d k ) and |d T k H k d k −d T k H kdk | ≤ 2κ H ζ −1 κ FO α k √ κ lτk + κ H κ 2 FO α 2 k ζ 2 ∆l(x k ,τ k ,ḡ k ,d k ). • If ḡ k − ∇f k ≤ g , then |∇f T k d k −ḡ T kdk | ≤ (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) + ζ −1 2 g and |d T k H k d k −d T k H kdk | ≤ 2κ H ζ −1 g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) + κ H ζ −2 2 g .
Proof. We begin with ḡ k − ∇f k ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ). By the triangle and Cauchy-Schwarz inequalities, Assumption 2.1, and Lemmas 3.1, 3.8 and 3.10,
|∇f T k d k −ḡ T kdk | = |(ḡ k − ∇f k ) Td k + (∇f k −ḡ k ) T (d k − d k ) +ḡ T k (d k − d k )| = |(ḡ k − ∇f k ) Td k + (∇f k −ḡ k ) T (d k − d k ) − (H kdk + J T kȳk ) T (d k − d k )| ≤ |(ḡ k − ∇f k ) Td k | + |(∇f k −ḡ k ) T (d k − d k )| + |d T k H k (d k − d k )| + |ȳ T k J k (d k − d k )| ≤ ḡ k − ∇f k d k + ∇f k −ḡ k d k − d k + κ H d k d k − d k ≤ (1 + κ H ζ −1 ) ∆l(x k ,τ k ,ḡ k ,d k ) κ lτk ḡ k − ∇f k + ζ −1 ḡ k − ∇f k 2 ≤ (1+κ H ζ −1 )κ FO α k √ κ lτk + ζ −1 κ 2 FO α 2 k ∆l(x k ,τ k ,ḡ k ,d k ).
Additionally, under Assumption 2.2 it follows that
|d T k H k d k −d T k H kdk | = |2d T k H k (d k − d k ) − (d k − d k ) T H k (d k − d k )| ≤ 2|d T k H k (d k − d k )| + |(d k − d k ) T H k (d k − d k )| ≤ 2κ H d k d k − d k + κ H d k − d k 2 ≤ 2κ H ζ −1 ∆l(x k ,τ k ,ḡ k ,d k ) κ lτk ḡ k − ∇f k + κ H ζ −2 ḡ k − ∇f k 2 ≤ 2κ H ζ −1 κ FO α k √ κ lτk + κ H ζ −2 κ 2 FO α 2 k ∆l(x k ,τ k ,ḡ k ,d k ),
which completes the first part of the proof. If ḡ k − ∇f k ≤ g , following similar logic as the first part of the proof, by the triangle and Cauchy-Schwarz inequalities, (3.1), and Lemmas 3.1, 3.9 and 3.10,
|∇f T k d k −ḡ T kdk | = |(ḡ k − ∇f k ) T (d k − d k ) + (ḡ k − ∇f k ) T d k + ∇f T k (d k − d k )| = |(ḡ k − ∇f k ) T (d k − d k ) + (ḡ k − ∇f k ) T d k − (H k d k + J T k y k ) T (d k − d k )| ≤ |(ḡ k − ∇f k ) T (d k − d k )| + |(ḡ k − ∇f k ) T d k | + |d T k H k (d k − d k )| + |y T k J k (d k − d k )| ≤ ζ −1 ḡ k − ∇f k 2 + (1 + κ H ζ −1 ) d k ḡ k − ∇f k ≤ ζ −1 ḡ k − ∇f k 2 + 1+κ H ζ −1 √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) ḡ k − ∇f k ≤ ζ −1 2 g + (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ).
Additionally, under Assumption 2.2 it follows that
|d T k H k d k −d T k H kdk | = |(d k −d k ) T H k (d k −d k ) + 2d T k H k (d k − d k )| ≤ κ H d k −d k 2 + 2κ H d k d k −d k ≤ κ H ζ −2 ḡ k − ∇f k 2 + 2κ H √ ∆l(x k ,τ k ,∇f k ,d k ) √ κ l τ k ζ −1 ḡ k − ∇f k ≤ κ H ζ −2 2 g + 2κ H ζ −1 g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ),
which completes the proof.
The next lemma provides a bound on the merit function across an iteration.
Lemma 3.13. For all k ∈ N φ(x k + α kdk ,τ k ) − φ(x k ,τ k ) ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk (∇f k −ḡ k ) Td k +τ k L+Γ 2 α 2 k d k 2 .
Proof. By Algorithm 1, for any k ∈ N, 0 < α k ≤ α max ≤ 1. Moreover, by the triangle inequality, (2.1), (2.4) and (2.6), it follows that
φ(x k + α kdk ,τ k ) − φ(x k ,τ k ) =τ k (f (x k + α kdk ) − f k ) + ( c(x k + α kdk ) 1 − c k 1 ) ≤τ k (α k ∇f T kdk + L 2 α 2 k d k 2 ) + ( c k + α k J kdk 1 − c k 1 + Γ 2 α 2 k d k 2 ) ≤ α kτk ∇f T kdk + |1 − α k | c k 1 + α k c k + J kdk 1 − c k 1 +τ k L+Γ 2 α 2 k d k 2 = α kτk ∇f T kdk − α k c k 1 +τ k L+Γ 2 α 2 k d k 2 = α kτkḡ T kdk − α k c k 1 + α kτk (∇f k −ḡ k ) Td k +τ k L+Γ 2 α 2 k d k 2 = − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk (∇f k −ḡ k ) Td k +τ k L+Γ 2 α 2 k d k 2 ,
which completes the proof.
Due to the quality and reliability of the zeroth-and first-order oracles (Oracles 0 and 1), one can only guarantee convergence to a neighborhood of the solution. Assumption 3.14 provides a lower bound on the size of the convergence neighbourhood in terms of ε (and ε ∆l ).
Assumption 3.14. Let
ε > max g η , √ f ω 7 ω 8 · max{κ H ,1} √ κ l τ min ,
which is equivalent to ε ∆l > max g η ,
√ f ω 7 ω 8 by Remark 3.5, where 0 < η < 2(1 − θ) min 1 η 1 +η 2 , 1 η 3 +η 4
and {η 1 , η 2 , η 3 , η 4 } ⊂ R >0 are defined as
η 1 = (1−θ)(1+ τ )τ −1 1+ κ H ζ √ κ l τ min η 2 = (1 − θ) 2τ −1 1 + κ H ζ 2 (1+ τ ) 2τ −1 κ l τ min + τ + 4τ −1 1+ τ ω 2 κ l + (1−θ) 2 (1+ τ ) ζ η 3 = (1−θ)τ −1 τ −1 1+ 3κ H ζ +(1−σ)τ min 1+ κ H ζ (1−σ)τ min √ κ l τ min and η 4 = (1−θ) 2τ 2 −1 τ −1 1+ 3κ H ζ +(1−σ)τ min 1+ κ H ζ 2 (1−σ) 2 τ 3 min κ l + 4τ −1 κ l + 4(1 − θ) 2 τ 2 −1 1+ κ H ζ (1−σ)τ min ζ +τ −1 ζ
with p ∈ 1 2 , 1 , and {ω 2 , ω 3 , ω 4 , ω 5 , ω 6 , ω 7 , ω 8 } ⊂ R >0 defined as
ω 2 = max{κ H ,κy}+1 κ l , ω 3 = (1+κ H ζ −1 )κ FO √τ −1 αmax √ κ l +τ −1 κ 2 FO α 2 max ζ , ω 4 = max τ max{κ H ,κy} κ l + √τ −1( 1+κ H ζ −1 )κFOαmax √ κ l + ω 3 , τ −1 (1−σ)τ min (1+3κ H ζ −1 )κ FO √τ −1 αmax √ κ l + (1 + κ H ζ −1 )τ −1 κ 2 FO α 2 max ζ , ω 5 = (1 + τ )τ −1 η ζ + 1+κ H ζ −1 √ κ l τ min + ττ−1(1+κH ζ −1 ) 2 η 4 , ω 6 =τ 2 −1 · (1+κ H ζ −1 ) η ζ + 1+3κ H ζ −1 √ κ l τ min (1−σ)τ min +τ −1 η ζ + 1+κ H ζ −1 √ κ l τ min , ω 7 = 4τ −1 (p− 1 2 )θ max 1+ τ ω 2 1−ηω 5 , 1 1−ηω 6 , 1 + ω 3 + ω 4 , and ω 8 = max τ −1 κ l κ FO + L 2κ l + Γ 2τ min κ l 1−θ ,τ min L+Γ 2τ min κ l 1−θ−η τ −1 κ l max 1+ τ ω 2 1−ηω 5 , 1 √ 1−ηω 6 .
Assumption 3.14 involves many constants and is indeed hard to parse. We make all constants explicit in order to show the exact dependence on the convergence neighborhood. That being said, what is important is that the lower bound of ε is proportional to the bias in the gradient approximations and proportional to the square root of the noise level in the function approximations.
We are now ready to present the key lemma of this section. In Lemma 3.15, we first define (p,α, h(·)), where p ∈ 1 2 , 1 is a lower bound on the probability of a true iteration conditioned on the past (before the stopping time),α ∈ R >0 is the large step threshold, and h : R >0 → R >0 is a monotonically increasing function (in α) that bounds the potential progress made at any given iteration. Moreover, we prove five results that can be summarized as follows: (i) lower bound (proportional to f ) on the potential progress with step sizeα; (ii) conditioned on the past, the next iteration is true with probability at least p; (iii) bound the potential progress made in any true and successful iterations; (iv) true iterations with small step sizes are successful ; and, (v) bound (proportional to f ) the damage incurred at any iteration.
= 1 − δ − exp − min{ u 2 2ν 2 , u 2b } otherwise (with u = inf x∈X { f − E[E(x)]}, where E(x) = |f (x; Ξ 0 (x)) − f (x)|), •α = min 1−θ τ −1 κ l κ FO + L 2κ l + Γ 2τ min κ l , 2τ min κ l 1−θ−η τ −1 κ l max 1+ τ ω 2 1−ηω 5 , 1 √ 1−ηω 6 τ min L+Γ , • h(α) = αθε 2 ∆l min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 , 1 1+ω 3 +ω 4 .
Then, the following results hold:
(i) h(α) > 4τ −1 p− 1 2 f . (ii) P [I k = 1|F k−1 ] ≥ p with some p ∈ 1 2 + 4τ −1 f h(α) , 1 .
(iii) If iteration k is true and successful, then
Z k+1 ≤ Z k − h(α k ) + 4τ −1 f .
(iv) If α k ≤α and iteration k is true, then iteration k is also successful.
(
v) Z k+1 ≤ Z k + 2τ −1 f +τ −1 (e k + e + k ).
Proof. First, we note that: (1) due to the constants and the form, p is a valid probability, i.e., p ∈ ( 1 2 , 1], (2)α > 0 is guaranteed by the restriction on η in Assumption 3.14, and (3) h : R >0 → R >0 is a positive function that measures the potential progress made if iterations are true and successful. We proceed with this proof by showing all five statements separately.
(i) This result follows directly from the definition of h(α) and the lower bound on ε ∆l ; see Assumption 3.14.
(ii) This proof is essentially the same as that from [22, Proposition 1(ii)]. Let
J k := 1 G k − ∇f (X k ) ≤ max g , κ FO A k ∆l(X k , T k , G k , D k ) .
Clearly, by Definition 1,
P [I k = 0 | F k−1 ] = P J k = 0 or E k + E + k > 2 f | F k−1 ≤ P [J k = 0 | F k−1 ] + P E k + E + k > 2 f | F k−1 .
The first term on the right-hand-side of the inequality is bounded above by δ, by the first-order probabilistic oracle (Oracle 1). The second term is zero in the case where f is a deterministic bound on the noise. Otherwise, since E k and E + k individually satisfy the one-sided sub-exponential bound in (2.11) with parameters f and (ν, b), one can show that E k + E + k satisfies (2.11) with parameters 2 f and (2ν, 2b). Hence by the one-sided Bernstein inequality, the second term is bounded above by e − min u 2 2ν 2 , u 2b ,
with u = inf x∈X { f − E[E(x)]}.
As a result,
P [I k = 1 | F k−1 ] ≥ p
for all k, for p as defined in the statement. The range of p ∈ 1 2 + 4τ −1 f h(α) , 1 follows from the definitions of h(·) andα in the statement, together with the inequality on ε ∆l in Assumption 3.14.
(iii) Suppose iteration k is true and successful. Since iteration k is true, by Definition 1 we have
ḡ k − ∇f k ≤ max g , κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ) ,
and we consider the two cases separately. We further subdivide the analysis into the case where ∇f T k d k ≤ 0 and ∇f T k d k > 0.
Case A When ḡ k − ∇f (x k ) ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ), by Lemma 3.10, d k − d k ≤ ζ −1 ḡ k − ∇f (x k ) ≤ ζ −1 κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ).
Case A.1 If ∇f T k d k ≤ 0, by the fact thatτ k ≥ τ k , the triangle inequality, (2.5) and Lemma 3.12, it follows that
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) =τ kḡ T kdk − τ k ∇f T k d k ≤τ k (ḡ T kdk − ∇f T k d k ) ≤τ k |ḡ T kdk − ∇f T k d k | ≤τ k (1+κ H ζ −1 )κ FO α k √ κ lτk + κ 2 FO α 2 k ζ ∆l(x k ,τ k ,ḡ k ,d k ).
(3.7)
Case A.2 If ∇f T k d k > 0, by the triangle inequality, (2.5) and Lemma 3.12,
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) =τ kḡ T kdk − τ k ∇f T k d k ≤ |τ kḡ T kdk − τ k ∇f T k d k | ≤ |(τ k − τ k )∇f T k d k | +τ k |ḡ T kdk − ∇f T k d k | ≤ |(τ k − τ k )∇f T k d k | +τ k (1+κ H ζ −1 )κ FO α k √ κ lτk + κ 2 FO α 2 k ζ ∆l(x k ,τ k ,ḡ k ,d k ).
(3.8)
We proceed to bound the term |(τ k − τ k )∇f T k d k |; we consider three cases due to the merit parameter updating formulae ((2.7)-(2.8) and (3.3)-(3.4)).
Case A.2.1 If τ k =τ k , then |(τ k − τ k )∇f T k d k | = 0. Case A.2.2 If τ k = (1 − τ )τ k ,
by the triangle inequality and Lemmas 3.11 and 3.12,
|(τ k − τ k )∇f T k d k | = ττk |∇f T k d k | ≤ ττk (|ḡ T kdk | + |∇f T k d k −ḡ T kdk |) ≤ τ max{κ H ,κy} κ l + √τ k (1+κHζ −1 )κFOαk √ κ l ∆l(x k ,τ k ,ḡ k ,d k ) + ττk (1+κ H ζ −1 )κ FO α k √ κ lτk + κ 2 FO α 2 k ζ ∆l(x k ,τ k ,ḡ k ,d k ). Case A.2.3 Ifτ k > τ k = (1−σ) c k 1 ∇f T k d k +max{d T k H k d k ,0} , by (2.7)-(2.8), ∇f T k d k + max d T k H k d k , 0 > (1−σ) c k 1 τ k ≥ḡ T kdk + max d T k H kdk , 0 . (3.9)
By Lemma 3.3, we have τ k ≥ τ min for all k ∈ N. Moreover, it follows from (2.5) and Lemma 3.
9 that 0 ≤ ∆l(x k , τ k , ∇f k , d k ), which implies τ k ∇f T k d k ≤ c k 1 . Using the fact that τ k ∈ R >0 and ∇f T k d k > 0, |∇f T k d k | c k 1 = ∇f T k d k c k 1 ≤ 1 τ k . (3.10)
By Lemma 3.12, (3.9) and (3.10), it follows that
|(τ k − τ k )∇f T k d k | = τ k − (1−σ) c k 1 ∇f T k d k +max{d T k H k d k ,0} · |∇f T k d k | ≤ (∇f T k d k +max{d T k H k d k ,0})−(ḡ T kd k +max{d T k H kdk ,0}) ∇f T k d k +max{d T k H k d k ,0} ·τ k |∇f T k d k | ≤ |(∇f T k d k +max{d T k H k d k ,0})−(ḡ T kd k +max{d T k H kdk ,0})| (1−σ) c k 1 ·τ 2 k |∇f T k d k | ≤ |∇f T k d k −ḡ T kd k |+| max{d T k H k d k ,0}−max{d T k H kdk ,0}| (1−σ) c k 1 ·τ 2 k |∇f T k d k | ≤τ 2 k (1−σ)τ k · |∇f T k d k −ḡ T kdk | + | max d T k H k d k , 0 − max{d T k H kdk , 0}| ≤τ 2 k (1−σ)τ k · |∇f T k d k −ḡ T kdk | + |d T k H k d k −d T k H kdk | ≤τ 2 k (1−σ)τ min (1+3κ H ζ −1 )κ FO α k √ κ lτk + (1 + κ H ζ −1 )ζ −1 κ 2 FO α 2 k ∆l(x k ,τ k ,ḡ k ,d k ).
Combining (3.7), (3.8) and Cases A.2.1-A.2.3, it follows that
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) ≤ τ k (1+κ H ζ −1 )κ FO α k √ κ lτk + ζ −1 κ 2 FO α 2 k + max τ max{κ H ,κy} κ l + √τ k (1+κHζ −1 )κFOαk √ κ l + ττk (1+κ H ζ −1 )κ FO α k √ κ lτk + ζ −1 κ 2 FO α 2 k , τ 2 k (1−σ)τ min (1+3κ H ζ −1 )κ FO α k √ κ lτk + (1 + κ H ζ −1 )ζ −1 κ 2 FO α 2 k ∆l(x k ,τ k ,ḡ k ,d k ) ≤ (ω 3 + ω 4 ) · ∆l(x k ,τ k ,ḡ k ,d k ),
where {ω 3 , ω 4 } ⊂ R >0 are as defined in Assumption 3.14. By
{ω 3 , ω 4 } ⊂ R >0 , ∆l(x k ,τ k ,∇f k ,d k ) 1+ω 3 +ω 4 ≤ ∆l(x k ,τ k ,ḡ k ,d k ).
By the fact that iteration k is successful and Definition 2, it follows that
φ(x + k ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) ≤ −α k θ∆l(x k ,τ k ,ḡ k ,d k ) + 2τ k f ≤ −α k θ ∆l(x k ,τ k ,∇f k ,d k ) 1+ω 3 +ω 4 + 2τ −1 f .
Hence, it follows that
Z k+1 − Z k = φ(x k+1 ,τ k+1 ) − φ(x k ,τ k ) −τ k+1 f inf +τ k f inf ≤ φ(x k+1 ,τ k+1 ) −φ(x k ,τ k ; ξ k ) −τ k+1 f inf +τ k f inf +τ k e k = φ(x k+1 ,τ k+1 ) −φ(x k+1 ,τ k ; ξ + k ) +φ(x k+1 ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) −τ k+1 f inf +τ k f inf +τ k e k ≤ − α k θ ∆l(x k ,τ k ,∇f k ,d k ) 1+ω 3 +ω 4 + 2τ −1 f + (τ k+1 −τ k )(f (x k+1 ) − f inf ) +τ k (e k + e + k ) ≤ − α k θ ∆l(x k ,τ k ,∇f k ,d k ) 1+ω 3 +ω 4 + 2τ −1 f +τ k (e k + e + k ).
(3.11)
Case B When ḡ k − ∇f (x k ) ≤ g , by the condition that k < T ε ∆l and Definition 4, it follows that ∆l(x k , τ k , ∇f k , d k ) > ε ∆l > g η . By Lemma 3.10,
d k − d k ≤ ζ −1 ḡ k − ∇f k ≤ ζ −1 g < ζ −1 η ∆l(x k , τ k , ∇f k , d k ).
Case B.1 If ∇f T k d k ≤ 0, by the fact thatτ k ≥ τ k , the triangle inequality, (2.5) and
Lemma 3.12, it follows that
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) =τ kḡ T kdk − τ k ∇f T k d k ≤τ k (ḡ T kdk − ∇f T k d k ) ≤τ k |ḡ T kdk − ∇f T k d k | ≤τ k ζ −1 2 g + (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) ≤τ k ζ −1 η + 1+κ H ζ −1 √ κ l τ k η∆l(x k , τ k , ∇f k , d k ).
(3.12)
Case B.2 If ∇f T k d k > 0, by the triangle inequality, (2.5) and Lemma 3.12,
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) =τ kḡ T kdk − τ k ∇f T k d k ≤ |τ kḡ T kdk − τ k ∇f T k d k | ≤ |(τ k − τ k )∇f T k d k | +τ k |ḡ T kdk − ∇f T k d k | ≤ |(τ k − τ k )∇f T k d k | +τ k ζ −1 2 g + (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) ≤ |(τ k − τ k )∇f T k d k | +τ k ζ −1 η + 1+κ H ζ −1 √ κ l τ k η∆l(x k , τ k , ∇f k , d k ). (3.13) We proceed to bound the term |(τ k − τ k )∇f T k d k |. Case B.2.1 If τ k =τ k , then |(τ k − τ k )∇f T k d k | = 0. Case B.2.2 If τ k = (1 − τ )τ k ,
then by Lemmas 3.11 and 3.12 and Assumption 3.14,
|(τ k − τ k )∇f T k d k | = ττk |∇f T k d k | ≤ ττk |ḡ T kdk | + |∇f T k d k −ḡ T kdk | ≤ τ ω 2 ∆l(x k ,τ k ,ḡ k ,d k ) + ττk (1+κ H ζ −1 ) 2 4 2 g + ττk ζ −1 2 g + (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) ≤ τ ω 2 ∆l(x k ,τ k ,ḡ k ,d k ) + ττk η (1+κ H ζ −1 ) 2 η 4 + η ζ + 1+κ H ζ −1 √ κ l τ k ∆l(x k , τ k , ∇f k , d k ). Case B.2.3 Ifτ k > τ k = (1−σ) c k 1 ∇f T k d k +max{d T k H k d k ,0}
, following the same logic as in Case A.2.3, by Lemma 3.12, (3.9) and (3.10),
|(τ k − τ k )∇f T k d k | ≤τ 2 k (1−σ)τ k · |∇f T k d k −ḡ T kdk | + |d T k H k d k −d T k H kdk | ≤τ 2 k (1−σ)τ min · ζ −1 2 g + (1+κ H ζ −1 ) g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) +κ H ζ −2 2 g + 2κ H ζ −1 g √ κ l τ k ∆l(x k , τ k , ∇f k , d k ) ≤τ 2 k (1−σ)τ min (1 + κ H ζ −1 )ζ −1 η + 1+3κ H ζ −1 √ κ l τ k η∆l(x k , τ k , ∇f k , d k ).
Combining (
∆l(x k , τ k , ∇f k , d k ) − ∆l(x k ,τ k ,ḡ k ,d k ) ≤ max τ ω 2 ∆l(x k ,τ k ,ḡ k ,d k ) + ττk η (1+κ H ζ −1 ) 2 η 4 + η ζ + 1+κ H ζ −1 √ κ l τ k ∆l(x k , τ k , ∇f k , d k ), ητ 2 k · (1+κ H ζ −1 )ζ −1 η+ 1+3κ H ζ −1 √ κ l τ k (1−σ)τ min ∆l(x k , τ k , ∇f k , d k ) +τ k ζ −1 η + 1+κ H ζ −1 √ κ l τ k η∆l(x k , τ k , ∇f k , d k ) ≤ max τ ω 2 ∆l(x k ,τ k ,ḡ k ,d k ) + ηω 5 ∆l(x k , τ k , ∇f k , d k ), ηω 6 ∆l(x k , τ k , ∇f k , d k ) ,(3.
14) where {ω 2 , ω 5 , ω 6 } ⊂ R >0 are defined in Assumption 3.14. Thus, it follows,
∆l(x k ,τ k ,ḡ k ,d k ) ≥ min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 · ∆l(x k , τ k , ∇f k , d k ). (3.15)
By selecting η following Assumption 3.14, using the fact that iteration k is successful and Definition 2,
φ(x + k ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) ≤ − α k θ∆l(x k ,τ k ,ḡ k ,d k ) + 2τ k f ≤ − α k θ min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 · ∆l(x k , τ k , ∇f k , d k ) + 2τ −1 f .
Hence, following similar logic as in (3.11), it follows that
Z k+1 − Z k ≤ φ(x k+1 ,τ k+1 ) −φ(x k+1 ,τ k ; ξ + k ) +φ(x k+1 ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) −τ k+1 f inf +τ k f inf +τ k e k ≤ − α k θ min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 · ∆l(x k , τ k , ∇f k , d k ) + 2τ −1 f + (τ k+1 −τ k )(f (x k+1 ) − f inf ) +τ k (e k + e + k ) ≤ − α k θ min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 · ∆l(x k , τ k , ∇f k , d k ) + 2τ −1 f +τ k (e k + e + k ).
Combining the results for Case A and Case B, together with the assumption that the iteration is true, it follows that
Z k+1 − Z k ≤ − min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 , 1 1+ω 3 +ω 4 α k θ∆l(x k , τ k , ∇f k , d k ) + 2τ −1 f +τ −1 (e k + e + k ) ≤ − h(α k ) + 4τ −1 f ,
where the last inequality is from the conditions that ∆l(x k , τ k , ∇f k , d k ) > ε 2 ∆l and e k + e + k ≤ 2 f .
(iv) We first show that for any k ∈ N, if α k ≤α and iteration k is true, then
φ(x k + αd k ,τ k ) ≤ φ(x k ,τ k ) − α k θ∆l(x k ,τ k ,ḡ k ,d k ).
Since iteration k is true, by Definition 1, it follows that
ḡ k − ∇f k ≤ max g , κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ) ,
and we consider the two cases separately.
Case A When ḡ k − ∇f k ≤ κ FO α k ∆l(x k ,τ k ,ḡ k ,d k ), by
α k ≤α ≤ 1−θ τ −1 κ l κ FO + L 2κ l + Γ 2τ min κ l ,
the Cauchy-Schwarz inequality, Assumption 3.2 and Lemmas 3.8 and 3.13,
φ(x k + α kdk ,τ k ) − φ(x k ,τ k ) ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk (∇f k −ḡ k ) Td k +τ k L+Γ 2 α 2 k d k 2 ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk ∇f k −ḡ k d k +τ k L+Γ 2 α 2 k d k 2 ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + τ k κ l κ FO α 2 k ∆l(x k ,τ k ,ḡ k ,d k ) +τ k L+Γ 2τ k κ l α 2 k ∆l(x k ,τ k ,ḡ k ,d k ) ≤ − 1 − τ −1 κ l κ FO + L 2κ l + Γ 2τ min κ l α α k ∆l(x k ,τ k ,ḡ k ,d k ) ≤ − α k θ∆l(x k ,τ k ,ḡ k ,d k ).
Case B When ḡ k − ∇f k ≤ g and iteration k is true, (3.15) holds. Moreover, by the condition that k < T ε ∆l and Definition 4, it follows that
ḡ k − ∇f k ≤ g < ηε ∆l < η ∆l(x k , τ k , ∇f k , d k ).
Therefore, by
α k ≤α ≤ 2τ min κ l 1−θ−η τ −1 κ l ·max 1+ τ ω 2 1−ηω 5 , 1 √ 1−ηω 6 τ min L+Γ ,
the Cauchy-Schwarz inequality, Assumption 3.2, (3.15) and Lemmas 3.8 and 3.13,
φ(x k + α kdk ,τ k ) − φ(x k ,τ k ) ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk (∇f k −ḡ k ) Td k +τ k L+Γ 2 α 2 k d k 2 ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk ∇f k −ḡ k d k +τ k L+Γ 2 α 2 k d k 2 ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) +τ k L+Γ 2τ k κ l α 2 k ∆l(x k ,τ k ,ḡ k ,d k ) + α kτk η ∆l(x k , τ k , ∇f k , d k ) ∆l(x k ,τ k ,ḡ k ,d k ) κ lτk ≤ − α k ∆l(x k ,τ k ,ḡ k ,d k ) + α 2 kτ k L+Γ 2τ k κ l ∆l(x k ,τ k ,ḡ k ,d k ) + α k η τ k κ l max 1+ τ ω 2 1−ηω 5 , 1 √ 1−ηω 6 ∆l(x k ,τ k ,ḡ k ,d k ) ≤ − α k 1 − η τ −1 κ l max 1+ τ ω 2 1−ηω 5 , 1 √ 1−ηω 6 −τ min L+Γ 2τ min κ lα ∆l(x k ,τ k ,ḡ k ,d k ) ≤ − α k θ∆l(x k ,τ k ,ḡ k ,d k ).
Combining Cases A and B, together with the fact the iteration is true, we conclude the proof of (iv) bȳ
φ(x k + α kdk ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) ≤ −α k θ∆l(x k ,τ k ,ḡ k ,d k ) +τ k e k +τ k e + k ≤ −α k θ∆l(x k ,τ k ,ḡ k ,d k ) + 2τ k f .
(v) If iteration k is unsuccessful, then by definition Z k+1 = Z k , so the inequality holds trivially. On the other hand, if iteration k is successful, then starting with the second equation from (3.11)
Z k+1 − Z k ≤ φ(x k+1 ,τ k+1 ) −φ(x k+1 ,τ k ; ξ + k ) +φ(x k+1 ,τ k ; ξ + k ) −φ(x k ,τ k ; ξ k ) −τ k+1 f inf +τ k f inf +τ k e k ≤ − α k θ∆l(x k ,τ k ,ḡ k ,d k ) + (τ k+1 −τ k )(f (x k+1 ) − f inf ) + 2τ k f +τ k (e k + e + k ) ≤ 2τ −1 f +τ −1 (e k + e + k ).
Therefore, we conclude the proof of (v).
The next two lemmas will be used in the iteration complexity analysis that follows. Lemma 3.17. For any positive integer t and anyp ∈ 1 2 , 1 , we have
P T ε ∆l > t, t−1 k=0 I k ≥pt, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 = 0 where l = max − ln α 0 −lnα ln γ , 0 .
Proof. The proof is the same as [22,Lemma 3.5].
We are now ready to present the main theorem of the manuscript; the iteration complexity of Algorithm 1.
Theorem 3.18. Suppose Assumptions 2.1, 2.2, 3.2, 3.4 and 3.14 hold and that the conditions of Oracles 0 and 1 are satisfied. Then, for any s ≥ 0,p ∈ 1 2 +
4τ −1 f +s h(α) , p , and t ≥ R p− 1 2 − 4τ −1 f +s h(α) , P [T ε ∆l ≤ t] ≥ 1 − e − (p−p) 2 2p 2 t − e − min s 2 t 2(2τ −1 ν) 2 , st 2(2τ −1 b) ,
where R = Z 0 h(α) + max lnα−ln α 0 2 ln γ , 0 , and (p,α, h(·)) are as defined in Lemma 3.15.
Proof. By the law of total probability,
P [T ε ∆l > t] =P T ε ∆l > t, 1 t t−1 k=0 (2τ −1 f +τ −1 (E k + E + k )) > 4τ −1 f + s A + P T ε ∆l > t, 1 t t−1 k=0 (2τ −1 f +τ −1 (E k + E + k )) ≤ 4τ −1 f + s B .
First we bound P[A]. For each iteration k, since E k and E + k satisfy the one-sided subexponential bound (2.11) with parameters (ν, b), one can show thatτ −1 (E k + E + k ) satisfies (2.11) with parameters (2τ −1 ν, 2τ −1 b). Moreover, sinceτ −1 (E k + E + k ) has mean bounded by 2τ −1 f , applying (one-sided) Bernstein's inequality, for any s ≥ 0
P[A] ≤ P 1 t t−1 k=0τ −1 (E k + E + k ) > 2τ −1 f + s ≤ e − min s 2 t 2(2τ −1 ν) 2 , st 2(2τ −1 b) . Let l = max − ln α 0 −lnα ln γ , 0 . To bound P[B]
we apply the law of total probability,
P[B] = P T ε ∆l > t, 1 t t−1 k=0 (2τ −1 f +τ −1 (E k + E + k )) ≤ 4τ −1 f + s, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 B 1 + P T ε ∆l > t, 1 t t−1 k=0 (2τ −1 f +τ −1 (E k + E + k )) ≤ 4τ −1 f + s, t−1 k=0 Θ k I k U k ≥ p − 1 2 t − l 2 B 2 .
We first show that P[B 2 ] = 0. By Lemma 3.15, for any iteration k < T ε ∆l , it follows that
Z k+1 ≤ Z k −h(α)+2τ −1 f +τ −1 (E k +E + k ) ≤ Z k −h(α)+4τ −1 f if U k I k Θ k = 1, and Z k+1 ≤ Z k + 2τ −1 f +τ −1 (E k + E + k ) if U k I k Θ k = 0.
By the definition of the zeroth-order oracle (Oracle 0), E[E k ] and E[E + k ] are bounded above by f for all k. The event T ε ∆l > t implies that Z t > 0 (since Z t = 0 can only happen when T ε ∆l ≤ t by the proof of Lemma 3.6). This together with 1 t t−1
k=0 (2τ −1 f +τ −1 (E k + E + k )) ≤ 4τ −1 f + s in turn implies the event t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 . To see this, assume that t−1 k=0 Θ k I k U k ≥ p − 1 2 t − l 2 , then Z t ≤ Z 0 − p − 1 2 t − l 2 h(α) − t−1 k=0 (2τ −1 f +τ −1 (E k + E + k )) ≤ Z 0 − p − 1 2 t − l 2 h(α) + t(4τ −1 f + s) = Z 0 − p − 1 2 h(α) − (4τ −1 f + s) t + l 2 h(α) ≤ 0.
The last inequality above is due to the assumption thatp > 1 2 +
4τ −1 f +s h(α) and t ≥ R p− 1 2 − 4τ −1 f +s h(α)
. Hence, P[B 2 ] = 0.
We now bound P[B 1 ]; by Lemmas 3.16 and 3.17,
P[B 1 ] ≤ P T ε ∆l > t, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 = P T ε ∆l > t, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 , t−1 k=0 I k <pt + P T ε ∆l > t, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 , t−1 k=0 I k ≥pt ≤ P t−1 k=0 I k <pt + P T ε ∆l > t, t−1 k=0 Θ k I k U k < p − 1 2 t − l 2 , t−1 k=0 I k ≥pt ≤ e − (p−p) 2 2p 2 t + 0 = e − (p−p) 2 2p 2 t .
Combining P[A] and P[B], completes the proof.
+ 4τ −1 f +s αθωpε 2 ∆l , p and t ≥R p− 1 2 − 4τ −1 f +s αθωpε 2 ∆l , P [T ε ∆l ≤ t] ≥ 1 − e − (p−p) 2 2p 2 t − e − min s 2 t 2(2τ −1 ν) 2 , st 2(2τ −1 b) , (3.16) whereR = φ(x 0 ,τ −1 )−φ min −(τ −1 −τ min )f inf αθωpε 2 ∆l + max lnα−ln α 0 2 ln γ , 0 , equivalently, by Remark 3.5, R = max{κ 2 H ,1} κ l τ min · φ(x 0 ,τ −1 )−φ min −(τ −1 −τ min )f inf αθωpε 2 + max lnα−ln α 0 2 ln γ , 0 , ω p = min 1−ηω 5 1+ τ ω 2 , 1 − ηω 6 , 1
1+ω 3 +ω 4 , and the rest of the constants are defined in Assumption 3.14.
Remark 3.20. We make a few remarks about the main theoretical results of the paper (Theorem 3.18 and Corollary 3.19).
• (Iteration Complexity) By Definition 4 (and Remark 3.5) and Corollary 3.19, we conclude that, with overwhelmingly high probability, the iteration complexity of Algorithm 1 to generate a primal-dual iterate (x k , y k ) ∈ R n × R m that satisfies
max{ ∇f k + J T k y k , c k } ≤ ε is O(ε −2 )
. This iteration complexity is of the same order in terms of the dependence on ε as the iteration complexity that can be derived for the deterministic counterpart [16], with the additional restriction that ε is bounded away from zero (Assumption 3.14) due to the noise and bias in the oracles.
• (Almost Sure Convergence) We note that Algorithm 1 finds an ε-stationary iterate in a finite number of iterations with probability 1, i.e., P[∩ ∞ k=1 ∪ ∞ t=k (T ε ∆l > t)] = 0. This is a direct consequence of the Borel-Cantelli lemma, since it follows from (3.16) that the probability of failure events is summable, i.e.,
∞ t=1 P[T ε ∆l > t] = ∞ t=1 (1 − P[T ε ∆l ≤ t]) < ∞.
• (Unconstrained Setting) The high probability complexity bound in this paper is a generalization of the unconstrained version. In the unconstrained setting, the parameters reduce to σ = 0, ω 1 = 0, ω 2 = 1, Γ = 0, ζ = 1, κ H = 1, κ l = 1, τ = 0, andτ k = 1 for all k ∈ N. Using these values in the results of Corollary 3.19 does not exactly recover the result from the unconstrained setting [22]. That being said, the order of the results is the same in terms of the dependence on ε. The existence of the gap is due to complications that arise in the constrained setting related to the adaptivity of the merit parameter. We conclude by emphasizing again that though there is a constant difference in function h and valueα comparing to [22], our algorithm recovers the complexity bound of the deterministic variant algorithm [16].
Numerical Results
In this section, we present numerical results for our proposed algorithm on standard equality constrained nonlinear optimization problems. The goal of the numerical experiments is to investigate the efficiency and robustness of the SS-SQP algorithm across a diverse set of test problems with different levels of noise in the objective function and gradient evaluations. All experiments were conducted in MATLAB. Before we present the numerical results, we describe the test problems, implementation details, and evaluation metrics.
Test Problems
We ran the numerical experiments on a subset of the equality-constrained optimization problems from the CUTEst collection [18]. We selected the problems that satisfy the following criteria: (i) the objective function is not a constant function, (ii) the total number of variables and constraints are not larger than 10 3 , and (iii) the singular values of Jacobians of the constraints at all iterates in all runs were greater than 10 −8 . This resulted in 35 test problems of various dimensions. We considered noisy (noisy objective function and gradient evaluations) versions of the 35 CUTEst problems. Specifically, whenever an objective function or objective gradient evaluation was required, approximations,f (x; ξ) = N f (x), 2 f,N and g(x; ξ ) = N ∇f (x), 2 g,N n I , respectively, were utilized. We considered 4 different noise levels in the objective function and gradient evaluations, dictated by the constants f,N ∈ 0, 10 −4 , 10 −2 , 10 −1 and g,N ∈ 0, 10 −4 , 10 −2 , 10 −1 , respectively. Each CUTEst problem has a unique initial starting point, which was used as the starting point of all runs of all algorithms. Moreover, for each selected tuple of noise levels ( f,N , g,N ) ∈ 0, 10 −4 , 10 −2 , 10 −1 × 10 −4 , 10 −2 , 10 −1 ∪ {0} × {0}, where appropriate, we ran each problem with five different random seeds.
Implementation Details
We compared SS-SQP (Algorithm 1) to the adaptive stochastic SQP algorithm proposed in [7] (which we call AS-SQP) on the previously described noisy CUTEst problems. We set user-defined parameters for SS-SQP as follows: f = f,N , g = g,N , τ = 10 −2 ,τ −1 = σ = 0.1, γ = 0.5, θ = 10 −4 , α 0 = α max = 1, and H k = I for all k ∈ N. For AS-SQP [7] we set the parameters as follows (this parameter selection was guided by the choice of parameters in [7]):τ −1 = σ = 0.1,ξ −1 = 1, = 10 −2 , θ = 10 4 , H k = I and β k = 1 for all k ∈ N. The AS-SQP step size rule requires knowledge (or estimates) of the Lipschitz constants L and Γ. To this end, we estimated these constants using gradient differences near the initial point, and set L k = L and Γ k = Γ for all k ∈ N. We note that while the analysis of the SS-SQP algorithm requires that the condition of Oracles 1 hold, such conditions are not enforced or checked, and rather in each experiment, the algorithms were given random gradient estimates with the same, fixed, pre-specified accuracy (as described above). That being said, a clear distinction between SS-SQP and AS-SQP is the fact that the former requires function evaluations of the objective function (for the step search) whereas AS-SQP does not (AS-SQP is an objective-function-free method). We discuss this further when presenting the numerical results.
Termination Conditions and Evaluation Metrics
In all of our experiments, results are given in terms of infeasibility ( c(x k ) ∞ ) and stationarity (KKT) (max{ c(x k ) ∞ , min y∈R m ∇f (x k ) + ∇c(x k )y ∞ }) with respect to different evaluation metrics (iterations and work). We ran all algorithms with a budget of iterations (10 3 ), and only terminated a run early if an approximate stationary point was found, which we define as x * ∈ R n such that c(x * ) ∞ ≤ 10 −6 and min y∈R m ∇f (x * ) + ∇c(x * )y ∞ ≤ 10 −4 .
We present results in the form of performance profiles with respect to iterations and work (defined as the number of function and gradient evaluations), and use the convergence metric as described in [26], i.e., m(
x 0 ) − m(x) ≥ (1 − pp )(m(x 0 ) − m b ), where m(x) is either c(x) ∞ (infeasibility) or max{ c(x) ∞ , min y∈R m ∇f (x) + ∇c(x)y ∞ } (stationarity (KKT)
), x 0 is the initial iterate, and m b is the best value of the metric found by any algorithm for a given problem instance within the budget, and pp ∈ (0, 1) is the tolerance. For all experiments presented, we chose pp = 10 −3 .
Noisy Gradients, Exact Functions ( f = 0)
In our first set of experiments, we consider problems with exact objective function evaluations and noisy objective gradient evaluations and compare SS-SQP and AS-SQP. The goal of this experiment is to show the effect of noise in the gradient and the advantages of using (exact) function values. Each row in Figure 1 shows performance profiles for a different noise level in the gradient (bottom row, highest noise level) and each column shows a different evaluation metric. Starting from the noise-less benchmark case ( f = 0 and g = 0, the first row of Figure 1), it is clear that the performance of the methods in terms of both infeasibility error and KKT error is similar with a slight advantage in effectiveness (total problems that can be solved) for SS-SQP in terms of KKT error. As the noise in the gradient is increased, the gap between the performance of the two methods (in terms of all metrics) increases favoring SS-SQP. This, of course, is not surprising as SS-SQP uses additional information (exact function values). These results highlight the effect reliable function information can have on the performance of the methods.
Noisy Functions and Gradients
Here we present results with noise in both the objective function and gradient evaluations. As in Figure 1, in Figure 2 different rows show results for different noise levels in the gradient (the bottom row has the highest noise) and different columns show results for different evaluation metrics. Each performance profile has 4 lines: the AS-SQP (that is objectivefunction-free and is not affected by the noise in the function evaluations) and three variants of the SS-SQP method with different levels of noise in the objective function evaluations. One can make the following observations. First, not surprisingly, the performance of the SS-SQP method degrades as the noise in the objective function evaluations increases. Second, AS-SQP and SS-SQP are competitive and achieve similar robustness levels with respect to infeasibility errors. Third, and most interestingly, the performance of the methods depends on the relative errors of the function and gradient evaluations. In particular, when the objective function noise level is sufficiently small compared to the objective gradient bias, SS-SQP performs better. On the other hand, when the function estimations are too noisy compared to the noise level in the gradient evaluations, AS-SQP performs slightly better. These results highlight the power of objective-function-free optimization methods in the presence of noise (especially high noise in the objective function evaluations) and the value of quality (or at least relative quality) function evaluations in methods that require zeroth-order information.
Conclusion
We have proposed a step-search SQP algorithm (SS-SQP) for solving stochastic optimization problems with deterministic equality constraints, i.e., the setting in which constraint function values and derivatives are available, but only stochastic estimates of the objective function and its associated derivatives can be computed. We showed that under reasonable assumptions on the inexact probabilistic zeroth-and first-order oracles, with overwhelmingly high probability, in O(ε −2 ) iterations our algorithm can produce an iterate that satisfies the first-order ε-stationarity, which matches the iteration complexity of the deterministic counterparts of the SQP algorithm [16]. Numerical results provide strong evidence for the efficiency and efficacy of the proposed method. Some future directions include but are not limited to, (1) incorporating stochastic constraint evaluations into the algorithm design and analysis, and (2) extending the framework to the setting with inequality constraints. Both avenues above are subjects of future work as they require significant adaptations in the design, analysis, and implementation of the algorithm.
Assumption 2 . 1 .
21Let X ⊆ R n be an open convex set including iterates {x k } and trial iterates {x + k }.
Lemma 3 . 10 .
310For all k ∈ N, there exist constants {ζ, ω 1
Lemma 3. 15 .
15Suppose Assumptions 3.2, 3.4 and 3.14 hold. For all k < T ε ∆l , let • p = 1 − δ when the noise is bounded by f , and p
Lemma 3 . 16 .
316For all t ≥ 1, and anyp ∈ [0, p), The proof is the same as [22, Lemma 3.1].
Corollary 3. 19 .
19Under the conditions of Theorem 3.18, for any s ≥ 0,p ∈ 1 2
Figure 1 :
1Performance profiles for AS-SQP and SS-SQP on CUTEst collection[18] with deterministic objective function evaluations ( f = 0) and noisy objective gradient evaluations. Each column corresponds to a different evaluation metric (infeasibility and KKT errors vs. iterations and work). The noise in the objective gradient evaluations g increases from top to bottom (First row: g = 0; Second row: g = 10 −4 ; Third row: g = 10 −2 ; Fourth row: g = 10 −1 ).
Figure 2 :
2Performance profiles for AS-SQP and SS-SQP on CUTEst collection[18] with noise in both the objective function and gradient evaluations. Each column corresponds to a different evaluation metric (infeasibility and KKT vs. iterations and work). The noise in the objective gradient evaluations g increases from top to bottom (First row: g = 10 −4 ; Second row: g = 10 −2 ; Third row: g = 10 −1 ). The different variants of SS-SQP correspond to different levels of noise in the objective function evaluations.
AcknowledgmentsThis material is based upon work supported by the Office of Naval Research under award number N00014-21-1-2532. We would like to thank Professors Frank E. Curtis and Katya Scheinberg for their invaluable support and feedback.
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arxiv |
Published as a conference paper at ICLR 2023 MIND THE GAP: OFFLINE POLICY OPTIMIZATION FOR IMPERFECT REWARDS
Jianxiong Li
Tsinghua University
BeijingChina
Xiao Hu
Tsinghua University
BeijingChina
Haoran Xu
JD Technology
BeijingChina
Jingjing Liu
Tsinghua University
BeijingChina
Xianyuan Zhan [email protected]
Tsinghua University
BeijingChina
Shanghai Artificial Intelligence Laboratory
ShanghaiChina
Qing-Shan Jia
Tsinghua University
BeijingChina
Ya-Qin Zhang
Tsinghua University
BeijingChina
Published as a conference paper at ICLR 2023 MIND THE GAP: OFFLINE POLICY OPTIMIZATION FOR IMPERFECT REWARDS
Reward function is essential in reinforcement learning (RL), serving as the guiding signal to incentivize agents to solve given tasks, however, is also notoriously difficult to design. In many cases, only imperfect rewards are available, which inflicts substantial performance loss for RL agents. In this study, we propose a unified offline policy optimization approach, RGM (Reward Gap Minimization), which can smartly handle diverse types of imperfect rewards. RGM is formulated as a bi-level optimization problem: the upper layer optimizes a reward correction term that performs visitation distribution matching w.r.t. some expert data; the lower layer solves a pessimistic RL problem with the corrected rewards. By exploiting the duality of the lower layer, we derive a tractable algorithm that enables sampled-based learning without any online interactions. Comprehensive experiments demonstrate that RGM achieves superior performance to existing methods under diverse settings of imperfect rewards. Further, RGM can effectively correct wrong or inconsistent rewards against expert preference and retrieve useful information from biased rewards.
INTRODUCTION
Reward plays an imperative role in every reinforcement learning (RL) problem. It encodes the desired task behaviors, serving as a guiding signal to incentivize agents to learn and solve a given task. As widely recognized in RL studies, a desirable reward function should not only define the task the agent learns to solve, but also offers the "bread crumbs" that allow the agent to efficiently learn to solve the task (Abel et al., 2021;Singh et al., 2009;Sorg, 2011). However, due to task complexity and human cognitive biases (Hadfield-Menell et al., 2017), accurately describing a complex task using numerical rewards is often difficult or impossible (Abel et al., 2021;Li et al., 2019). In most practical settings, the rewards are typically "imperfect" and hard to be fixed through reward tuning when online interactions are costly or dangerous . Such imperfect rewards are widespread in real-world applications and can appear in forms such as partially correct rewards, sparse rewards, mismatched rewards from other tasks, and completely incorrect rewards (see Figure 1 for an intuitive illustration 1 ). These rewards either fail to incentivize agents to learn correct behaviors or cannot provide effective signals to speed up the learning process. Consequently, it is of great importance and practical value to devise a versatile method that can perform robust offline policy optimization under diverse settings of imperfect rewards.
Reward shaping (Ng et al., 1999) is the most common approach to tackling imperfect rewards, but it requires tremendous human efforts and numerous online evaluations. Another possible avenue is imitation learning (IL) (Pomerleau, 1988;Kostrikov et al., 2019) or offline inverse reinforcement learning methods (IRL) (Jarboui & Perchet, 2021), by directly imitating or deriving new rewards from expert behaviors. However, these methods heavily depend on the quantity and quality of expert demonstrations and offline datasets, which are often beyond reach in practice. Another key challenge is how to precisely measure the discrepancy between the given reward in the data and the true reward of the task. As evaluating the learned policy's behavior under a specific reward function through environment interactions becomes impossible under the offline setting, let alone revising the reward.
In this paper, we investigate the challenge of learning effective offline RL policies under imperfect rewards, when environment interactions are not possible. We first formally define the relative gap between the given and perfect rewards based on state-action visitation distribution matching (referred to as reward gap), and formulate the problem as a bi-level optimization problem. In the upper layer, the imperfect rewards are adjusted by a reward correction term, which is learned by minimizing the reward gap toward expert behaviors. In the lower layer, we solve a pessimistic RL problem to obtain the optimized policy under the corrected rewards. By exploiting Lagrangian duality of the lower-level problem, the overall optimization procedure can be tractably solved in a fully-offline manner without any online interactions. We call this approach Reward Gap Minimization (RGM). Compared to existing methods, RGM can: 1) evaluate and minimize the reward gap without any online interactions; 2) eliminate the strong dependency on human efforts and numerous expert demonstrations; and 3) handle diverse types of reward settings (e.g., perfect, partially correct, sparse, multi-task data sharing, incorrect) in a unified framework for reliable offline policy optimization.
Through extensive experiments on D4RL datasets (Fu et al., 2020), sparse reward tasks, multi-task data sharing tasks and a discrete-space navigation task, we demonstrate that RGM can achieve superior performance across diverse settings of imperfect rewards. Furthermore, we show that RGM effectively corrects wrong/inconsistent rewards against expert preference and effectively retrieves useful information from biased rewards, making it an ideal tool for practical applications where reward functions are difficult to design.
RELATED WORK ON DIFFERENT REWARD SETTINGS
We here briefly summarize relevant methodological approaches that handle different types of rewards.
Perfect rewards. Directly applying offline RL algorithms is a natural choice when rewards are assumed to be perfect for the given task (Fujimoto et al., 2019;Kumar et al., 2019;Xu et al., 2021;Fujimoto & Gu, 2021;Kostrikov et al., 2021a;b;Xu et al., 2022a;Lee et al., 2021;Bai et al., 2021;Xu et al., 2023). However, specifying perfect rewards requires deep understanding of the task and domain expertise. Even given the perfect rewards, some offline RL methods still need to shift the rewards to achieve the best performance (Kostrikov et al., 2021a;Kumar et al., 2020), which is shown to be equivalent to engineering the initialization of Q-function estimation that encourages conservative exploitation under offline learning (Sun et al., 2022).
Partially correct rewards. Reward shaping is the most common approach to handle partially correct rewards, by modifying the original reward function to incorporate task-specific domain knowledge (Dorigo & Colombetti, 1994;Randløv & Alstrøm, 1998;Ng et al., 1999;Marom & Rosman, 2018;Wu & Lin, 2018). However, these approaches follow a trial-and-error paradigm and require tremendous human efforts. Recent approaches such as population-based method (Jaderberg et al., 2019), optimal reward framework (Chentanez et al., 2004;Sorg et al., 2010;Zheng et al., 2018) and automatic reward shaping (Hu et al., 2020;Devidze et al., 2021;Marthi, 2007) can automatically shape the rewards when online interaction is allowed. However, to the best of the authors' knowledge, no reward shaping or correction mechanism exists for offline policy optimization. Researchers have to discard the given imperfect rewards and resort to other stopgaps like offline IL under offline settings.
Sparse rewards. Sparse rewards can be seen as a special case of partially correct rewards. The key challenge of offline policy optimization for sparse rewards is how to effectively back-propagate the sparse signals to stitch up suboptimal trajectories (Levine et al., 2020). Recent works (Kostrikov et al., 2021b;Kumar et al., 2020) use reward shaping to densify the sparse rewards for better performance. However, reward shaping requires online evaluation and tuning, which is not applicable in the offline setting. Currently, few mechanisms are specifically designed for offline RL to handle sparse rewards.
Imperfect rewards in multi-task data sharing. Sharing data across different tasks can potentially enhance offline RL performance on a target task by utilizing additional data from other relevant tasks. As the goals of other relevant tasks are different from that of the target task, the rewards designed for other tasks are naturally imperfect for solving the target task. Since directly sharing datasets from other tasks exacerbates the distribution shift in offline RL (Yu et al., 2021;Bai et al., 2023), prior work such as CDS (Yu et al., 2021) shares data relevant to the target task based on learned Q-values, but it requires access to the functional form of the reward for relabling. CDS+UDS directly set the shared rewards to zero without reward relabeling to reduce the bias in the shared rewards, but it cannot completely remedy the reward bias.
Completely incorrect rewards. When rewards are believed to be totally wrong or missing, researchers typically adopt offline imitation learning (IL) methods. These methods directly mimic the expert from demonstrations without the presence of a reward signal. Among these approaches, behavior cloning (BC) (Pomerleau, 1988;Florence et al., 2022) is the simplest one, but is vulnerable to covariate shift and compounding errors (Rajaraman et al., 2020). Recent works tackle this problem via distribution matching (Jarboui & Perchet, 2021;Kostrikov et al., 2019;Kim et al., 2021;Ma et al., 2022) or using a discriminator to measure the optimal level of the data and further guide policy learning (Zolna et al., 2020;Xu et al., 2022b;. These approaches all have strong requirements on the size and coverage of the expert datasets, and only try to imitate the expert rather than improve beyond the policies in data via RL based on the underlying reward of the task.
PRELIMINARIES
Markov decision process under imperfect rewards. We consider the typical Markov Decision Process (MDP) setting (Puterman, 2014), which is defined by a tuple M := (S, A, r, T, µ 0 , γ). S and A represent the state and action space, r : S × A → R is the perfect reward function, T : S × A → ∆(S) is the transition dynamics which represents the probability T (s t+1 |s t , a t ) of the transition from state s t to state s t+1 by executing action a t at timestep t. µ 0 ∈ ∆(S) is the distribution of the initial state s 0 , and γ ∈ (0, 1) is the discount factor.
The perfect reward function r(s, a) encodes the desired behaviors of the task. But in most cases, we only have access to an imperfect human-designed reward functionr(s, a), which may not align well with the target task. This leads to a biased MDP M := (S, A,r, T, µ 0 , γ) as compared to the original MDP M. To remedy the adverse effects of imperfect reward signals, existing offline policy learning studies (Zolna et al., 2020;Xu et al., 2022b;Ma et al., 2022;Kim et al., 2021;Jarboui & Perchet, 2021)
introduce additional expert demonstrations D E = s E 0 , a E 0 , s E 1 , · · · (i) N E i=0
to provide extra information on the desired policy behaviors. We follow a similar setup, but only consume very limited expert demonstrations. In our offline policy optimization setting, we are given a pre-collected
dataset D = (s 0 , a 0 ,r 0 , s 1 , · · · ) (i) N i=0
that is generated by an unknown behavior policy π β and annotated with imperfect rewardsr. We aim to learn an effective policy π r : S → ∆(A) to capture the optimized agent behavior in M rather than M using both D and a very small expert dataset D E .
Reinforcement learning. Given a MDP and the reward function r(s, a), the goal of RL is to find an optimized policy π * r to maximize the expected cumulative discount reward: π * r = arg max
πr (1 − γ)E[ ∞ t=0 γ t r (s t , a t ) |s 0 ∼ µ 0 (·), a t ∼ π r (·|s t ) , s t+1 ∼ T (·|s t , a t )]
. This optimization objective can be equivalently written into the following succinct form (Puterman, 2014;Nachum et al., 2019b) by defining the normalized discounted state-action visitation distribution d πr (s, a) (in the rest of the paper, we omit "normalized discounted state-action" for brevity unless otherwise specified):
π * r = arg max πr E (s,a)∼d πr [r(s, a)](1)d πr (s, a) = (1 − γ) ∞ t=0 γ t Pr[s t = s, a t = a|s 0 ∼ µ 0 (·), a t ∼ π r (·|s t ) , s t+1 ∼ T (·|s t , a t )]
This RL objective is not directly applicable to offline setting, as it is no longer possible to sample from d πr via online interactions, and serious distributional shift (Kumar et al., 2019) may occur without proper data-related regularization when learning from offline datasets. To tackle these problems, several recent works (Nachum et al., 2019b;Nachum & Dai, 2020;Lee et al., 2021) incorporate a regularizer into Eq. (1) to formulate a pessimistic RL framework that is solvable in the offline setting:
π * r = arg max πr E (s,a)∼d πr [r(s, a)] − αD d πr d D(2)
where d D is the visitation distribution of dataset D, D (· ·) represents some statistical discrepancy measures and α > 0 controls the strength of the regularization.
REWARD GAP MINIMIZATION
To handle diverse imperfect reward settings, three challenges have to be tackled:
1) Measure the gap between the given rewards and the underlying unknown perfect rewards; 2) Unify different reward settings and bridge the reward gap;
3) Perform offline policy optimization using an integrated framework.
Our solution to these challenges is Reward Gap Minimization (RGM). We formally define the reward gap in the perspective of visitation distribution matching and introduce a correction term to correct the problematic rewards. Then, we model RGM as a bi-level optimization problem, with the upper layer minimizing the reward gap and the lower layer solving a pessimistic RL problem. To derive a tractable algorithm, we leverage Lagrangian duality to eliminate the requirement for online samples.
DEFINITION OF REWARD GAP
As observed in recent literature, some tasks cannot be captured by a numerical Markovian reward function (Abel et al., 2021). Hence, learning an explicit proxy of the perfect reward function and comparing it to the given rewards is unlikely the best option to characterize the reward gap. In this study, we define the reward gap based on the outcome of the learned agent behavior, i.e., from the perspective of visitation distribution matching. Definition 1. (Reward gap) Given an arbitrary reward functionr(s, a) and the visitation distribution d * of the optimal policy induced from the perfect rewards r, the reward gap betweenr and r is:
D f d π * r d * (3) where D f (p q) = E z∼q f p(z) q(z)
is the f -divergence between distributions p and q, and d π * r represents the visitation distribution induced by π * r , which is derived using Eq.
(2) withr.
Note that d * is unobtainable since the perfect reward function is unknown. We can alternatively use the visitation distribution d E induced by unknown π E in expert demonstrations D E to approximate d * . Next, we discuss how to adjustr to minimize the reward gap.
BI-LEVEL OPTIMIZATION
Reward correction. In our study, we considerr(s, a) :=r(s, a) + ∆r(s, a,r), where ∆r(s, a,r) is a learnable reward correction term that is correlated with the given imperfect rewardsr in D.
The introduction of ∆r(s, a,r) enables us to exploit useful information within the partially correct rewards, while also correcting the wrong or inconsistent reward signals. We can further use it to construct a bi-level optimization formulation for RGM, where the upper-level problem optimizes the reward correction term to minimize the f -divergence between d π * r and d E , and the lower-level problem solves π * r as the optimal policy of a pessimistic RL problem with the corrected rewards:
∆r * = arg min ∆r D f d π * r d E(4)s.t. π * r = arg max πr E (s,a)∼d πr [r(s, a)] − αD f d πr d D(5)
The above bi-level optimization formulation poses several technical difficulties, stemming from the complexity of deriving d π * r from π * r , as well as the requirement of online samples from d π * r , which is impossible under the offline setting. In the following, we present reformulations for both lower and upper-level problems, which leads to a tractable form and an easy-to-implement algorithm.
Reformulation of the lower-level problem.
We first reformulate the lower-level problem by exploiting duality and the Bellman flow constraint (Puterman, 2014).
Definition 2. (Bellman flow constraint) Let T d(s) = s,ā T (s|s,ā)d(s,ā) denote the transpose (or adjoint) transition operator, the Bellman flow constraint for the visitation distribution d(s, a) is:
a d(s, a) = (1 − γ)µ 0 (s) + γT d(s), ∀s ∈ S(6)
If d(s, a) ≥ 0 satisfies the Bellman flow constraint, then d(s, a) is feasible and there is a one-to-one correspondence between d and the related policy π: i.e., d is the only visitation distribution for policy π(a|s) = d(s,a) ā d(s,ā) , while π is the only policy whose visitation distribution is d (for detailed proof see Puterman (2014)). Then, the lower level problem Eq. (5) can be re-written to a constraint maximization problem w.r.t. d in place of πr:
d π * r = arg max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D s.t. a d(s, a) = (1 − γ)µ0(s) + γT d(s), ∀s ∈ S(7)
The Lagrange dual problem of Eq. (7) is as follow:
min V (s) max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γT d(s) − a d(s, a)(8)
where V (s) are Lagrange multipliers. Note that the primal problem Eq. (7) is convex w.r.t. d, and under a mild assumption (see Assumption 1 in Appendix A.2), the Slater's condition (Boyd et al., 2004) holds, which means by strong duality, we can solve the original primal problem by solving Eq. (8). After rearranging the terms, Eq. (8) can be equivalently written as the following form (see Lemma 2 in Appendix A.2 for detailed deduction):
min V (s) max d≥0 (1 − γ)E s∼µ0 [V (s)] + E (s,a)∼d [r(s, a) + γT V (s, a) − V (s)] − αD f d d D (9)
in which T V (s, a) = s T (s |s, a)V (s ) denotes the transition operator. Next, by exploiting the Fenchel conjugate, we can further transform the minimax problem Eq. (9) into a tractable single-level unconstrained minimization problem (see Proposition 1 in Appendix A.2 for detailed derivation), which eliminates the requirement of online samples:
min V (s) (1 − γ)E s∼µ0 [V (s)] + α E (s,a)∼d D f (r (s, a) + γT V (s, a) − V (s) α )(10)
where f is the Fenchel conjugate of f . In the above formulation, the Lagrange multipliers V (s) can be equivalently perceived as some sort of state-value function, which can be learned and optimized via a parameterized neural network, similar to the treatment used in the DICE-family of RL algorithms (Nachum et al., 2019a;Nachum & Dai, 2020).
Reformulation of the upper-level problem. Using the property of Fenchel conjugate, the optimal d * and V * from the lower level problem satisfy the following nice relationship (see Proposition 2 in Appendix A.3 for details):
d π * r (s, a) d D (s, a) = f r(s, a) + γT V * (s, a) − V * (s) α(11)
Plugging the above equation into Eq. (5), we can obtain a new objective for the upper-level problem: Figure 2: Illustration of the reformulated bi-level optimization problem.
∆r * = arg min ∆r D f f r + γT V * − V * α d D d E(12)
For simplicity, we denote f r+γT V * −V * α as g.
By expanding the f -divergence, we have:
D f d D g d E = E (s,a)∼d E f d D (s, a)g(s, a) d E (s, a) = E (s,a)∼d D d E (s, a) d D (s, a) f d D (s, a) d E (s, a) g(s, a)(13)
The above objective involves computing the distribution ratio w(s, a) d E (s, a)/d D (s, a). In the tabular case, we can empirically estimate w(s, a) = (s,ā)∈D E 1(s=s,ā=a)/N E (s,ā)∈D 1(s=s,ā=a)/N . But in the continuous state-action settings, estimating the distribution ratio w using only samples from d D and d E becomes a challenge. Inspired by previous studies (Goodfellow et al., 2020;Ma et al., 2022), we instead train a discriminator h :
S × A → (0, 1) to infer if (s, a) samples are from D E or not: h * = arg min h E (s,a)∼d D [log(h(s, a))] + E (s,a)∼d E [log(1 − h(s, a))](14)
where the optimal discriminator is h et al., 2020). We can optimize the above objective to obtain the optimal h * , and further recover w(s, a) = 1/h * (s, a) − 1.
* (s, a) = d D (s,a) d D (s,a)+d E (s,a) (Goodfellow
Finally, combining all the reformulations, the final tractable form of the original bi-level optimization problem Eq. (4)-(5) is given as follows:
∆r * = arg min ∆r E (s,a)∼d D w(s, a) · f f r(s, a) + γT V * (s, a) − V * (s) α /w(s, a) s.t. V * (s) = arg min V (s) (1 − γ)Es∼µ 0 [V (s)] + α E (s,a)∼d D f r(s, a) + γT V (s, a) − V (s) α(15)
Policy extraction. With the learned reward correction term ∆r(s, a,r), we can in principle use existing offline RL algorithms to learn the policy with the corrected rewards. However, this implicates additional policy evaluation and policy improvement steps. A more elegant way is to extract the policy through weighted BC as follows, which is substantially more robust and less expensive:
π * = arg min π −E (s,a)∼d π * r [log π(a|s)] = arg min π −E (s,a)∼d D d π * r (s, a) d D (s, a) log π(a|s)(16)
where d π * r (s,a) d D (s,a) can be calculated from Eq. (11).
PRACTICAL IMPLEMENTATION
In our implementation, we use stochastic first-order two-timescale optimization technique (Borkar, 1997), which has been successfully applied in several RL algorithms (Hong et al., 2020;, to solve bi-level optimization problems. Specifically, we make the gradient update step size of the upper layer much smaller than the one of the lower layer (see Figure 2 for RGM framework. Refer to Appendix B for additional implementation details of RGM).
EXPERIMENTS
In this section, we present empirical evaluations of RGM under diverse imperfect reward settings, including partially correct rewards, completely incorrect rewards, sparse rewards, and multi-task data sharing setting on Robomimic (Mandlekar et al., 2021), D4RL-v2 (Fu et al., 2020) and a dataset of a grid-world navigation task. As D4RL MuJoCo tasks are deterministic, we use only one expert trajectory to assist the reward correction and policy learning for these tasks. The scores are from the final 10 evaluations with 5 seeds. (T), (P) and (C) mean policy optimization with true rewards, partially correct rewards and completely incorrect rewards, respectively. "-r","-m","-m-r", and "-m-e" are short for random, medium, medium-replay, and medium-expert, respectively. We obtain the results by running author-provided open-source code, and some scores are reported from TD3+BC and IQL papers. For each dataset, the top 2 scores under partially correct rewards are marked in blue.
D4RL Dataset Offline IL Offline RL RGM (T / P / C) BC DWBC SMODICE TD3+BC (T / P / C) IQL (T / P / C) CQL (T / P / C)
COMPARATIVE RESULTS
Comparisons for partially correct rewards. We train RGM and SOTA offline RL methods (TD3+BC (Fujimoto & Gu, 2021), IQL (Kostrikov et al., 2021b) and CQL (Kumar et al., 2020)) under partially correct 2 rewards and report their performances evaluated based on the perfect rewards 3 in Table 1. Table 1 shows that RGM surpasses offline RL methods under partially correct rewards 4 by a large margin and achieves similar performance to offline RL policies that are trained on perfect rewards. This shows a remarkable advantage of RGM as it can alleviate severe performance degradation when perfect rewards are unattainable and hence removes the restrictive requirements on perfect rewards, which can be particularly useful for a wide range of real-world scenarios.
Comparisons for completely incorrect rewards. When rewards are believed to be completely incorrect, one generally resorts to IL methods. We compare RGM with BC and SOTA offline IL methods (DWBC (Xu et al., 2022b) and SMODICE (Ma et al., 2022)) that can learn from mixedquality data. Only offline IL methods are considered as baselines, because other existing methods that tackle incorrect rewards can only be applied in the online settings (see Section 2 for discussions).
In our setting, we train offline IL baselines using the original D4RL dataset D, which may not cover enough expert trajectories. However, DWBC and SMODICE both build on the strong assumption that D already covers a large proportion of expert datasets, which is a rare case in real scenarios. As a result, Table 1 shows that these two methods suffer from inferior performance when the restrictive requirements on the quality and state-action space coverage of expert data are not satisfied. RGM, however, performs well when nearly no expert trajectories are contained in the offline dataset, because RGM is optimizing an RL objective that relaxes the requirements on the quality of the dataset.
To further illustrate the superiority of RGM, we compare RGM against DWBC and SMODICE under their settings by adding 100∼200 expert trajectories into D. Results show that RGM can still outperform SOTA offline IL methods by a large margin (see Table 8 in Appendix D).
Comparisons for sparse rewards. We evaluate RGM against BC and offline RL methods TD3+BC, CQL and IQL on Robomimic (Mandlekar et al., 2021) Lift and Can tasks. We also evaluate on the well-known extremely difficult AntMaze tasks. We report the average max success rate as the evaluation metric in Table 2 (See Appendix C.2 for task descriptions and experimental setups). Table 2 shows that the offline RL baselines fail miserably on AntMaze tasks 5 , as sparse rewards are hard to back-propagate through a very long horizon (≈ 1K steps), while RGM can correctly provide dense signals to guide the ant navigate to the destination. For Robomimic Lift and Can tasks, RGM again outperforms existing methods, while other methods can also achieve reasonable performance. We suspect that these offline datasets may already contain near-optimal trajectories as BC can achieve reasonable performance. Moreover, the planning horizon of both tasks are relatively short (≈ 150 steps), thus is relatively simple for offline RL to back-propagate the sparse signals. Extension to multi-task data sharing. We highlight that RGM can also perform well in the offline multi-task data sharing tasks (Yu et al., 2021), which utilize datasets from other relevant tasks to enhance the offline RL performance on a target task. Prior works either require the functional form of rewards to be known for relabeling (Yu et al., 2021) or partially correct the reward biases . In contrast, RGM systematically corrects the reward biases without reward relabelling, using just one expert trajectory from the target task. To demonstrate the efficacy of RGM compared to SOTA multi-task data sharing algorithms CDS (Yu et al., 2021) and CDS+UDS , we conduct experiments in multi-task Walker (Stand, Walk, Run, Flip) and Quadruped (Walk, Run, Roll-Fast, Jump) domains built on DeepMind Control Suite (Tassa et al., 2018). For each task, we use TD3 (Fujimoto et al., 2018) to collect three types of datasets (expert, medium, replay), and share the replay dataset of the relevant task with the medium dataset of the target task. For RGM, we only draw one expert trajectory for the discriminator training. We report the experimental results in Figure 3, which shows that RGM substantially outperforms CDS and CDS+UDS (see Appendix C.3 and D.5 for more experiment details and results).
INVESTIGATIONS ON REWARD CORRECTION
Benefits of learned rewards. We investigate the potential benefits of the learned rewards via demonstrative experiments in an 8×8 grid world environment. We observe the learned rewards in RGM enjoy three desirable properties that are unlikely to be provided in other existing methods: 1) encode long horizon information; 2) correct wrong rewards against expert preference; and 3) retrieve useful information from existing rewards, as shown in Figure 4. Specifically, Figure 4b shows that the learned rewards not only recover correct learning signals on the path of the expert, but also generalize well on regions not covered by expert data. In most locations, the agent can navigate to the destination by simply maximizing the one-step reward, meaning that the learned rewards encode long-horizon information. Moreover, Figure 4c shows that the learned rewards can avoid the dangerous fire locations by retrieving useful information provided in imperfect r, meanwhile correcting the wrong rewards against expert preference.
Offline RL with corrected rewards. The learned corrected rewardsr obtained by RGM can also be used in other offline RL approaches. To be mentioned, the corrected rewards are optimized based on the specific α in Eq. (5), hence may not be optimal to other offline RL methods. Nevertheless, Figure 5 shows that the corrected rewards can largely remedy the negative effects of the partially correct rewards and even surpass perfect rewards in some datasets. Ablations on learned rewards. Additionally, we investigate the learned rewards in highdimensional continuous control tasks by inspecting the learning process of both the reward correction term ∆r and the final learned rewardsr. Figure 6a shows that the reward correction term ∆r initially cannot distinguish expert and non-expert data well, but adapts and converges quickly. After a few training steps, ∆r can correctly reward expert data and punish non-expert data very well. We also perform ablations on the effect of diverse types of imperfect rewardsr on ∆r andr. Figure 6b shows that a perfectr is beneficial to enlarge reward differences on expert and non-expert samples, and incorrectr can be counterproductive. Nevertheless, RGM can largely correct the wrong rewards and produce reasonable learning signals. Similar effects are also observed on ∆r, as Figure 6c shows.
DISCUSSION AND CONCLUSION
In this paper, we propose RGM (Reward Gap Minimization), a unified offline policy optimization approach applicable to diverse settings of imperfect rewards. RGM is formulated as a bi-level optimization problem, which achieves reward correction and simultaneous policy learning in a fully offline paradigm. Extensive experiments and illustrative examples show that RGM can perform robust policy optimization under imperfect rewards. Several desirable properties are also identified in the corrected rewards learned by RGM. One limitation of RGM is the need for a small expert dataset, which may not be easily accessible in some applications. However, RGM relaxes the strong dependencies on online reward tuning and tedious human efforts, which renders it a powerful tool to solve many real-world problems.
ACKNOWLEDGMENTS
f (y) = sup x∈X (y T x − f (x))(17)
where the domain of the f (y) is given by:
dom f = y : sup x∈dom f y T x − f (x) < ∞(18)
If f is convex and lower semi-continuous as well, we have the duality f (x) = f (x). Furthermore, if f is also differentiable, then the maximizer x * of f (y) satisfies:
x * = f (y)
Next, we present the interchangeability principle, which plays a key role in Proposition 1. Lemma 1. (Interchangeability principle) Let ξ be a random variable on Ξ and assume for any ξ ∈ Ξ, function g(·, ξ) is a proper and upper semi-continuous concave function. Then
E ξ max u∈R g(u, ξ) = max u(·)∈G(Ξ) E ξ [g(u(ξ), ξ)](20)
where G(Ξ) = {u(·) : Ξ → R} is the entire space of functions defined on support Ξ .
Proof. Please refer to (Dai et al., 2017;Rockafellar & Wets, 2009).
A.2 PROOF OF TRACTABLE TRANSFORMATION OF THE LOWER-LEVEL PROBLEM
We start our proof from the original bi-level optimization problem Eq. (4) and Eq. (5). Using the Bellman flow constraint for Eq. (5) yields:
∆r * = arg min ∆r D f d π * r d E s.t. d π * r = arg max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D s.t.
We note that this assumption is mild since when every state is reachable from the initial state distribution, the assumption is satisfied, which is common in practice.
Slater's theorem (Boyd et al., 2004) states that strong duality holds, if the optimization problem is strictly feasible (Slater's condition holds) and the problem is convex. So under Assumption 1 with the fact that the lower level problem is convex w.r.t. d, the strong duality holds, which means that the above lower level problem can be re-written as the following form:
min V (s) max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γT d(s) − a d(s, a)(23)
Lemma 2. The minimax problem:
min V (s) max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γT d(s) − a d(s, a)(24)
can be equivalently written as:
min V (s) max d≥0 (1 − γ)Es∼µ 0 [V (s)] + E (s,a)∼d [r(s, a) + γT V (s, a) − V (s))] − αD f d d D (25)
Proof.
E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γT d(s) − a d(s, a) = E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γ s,ā T (s|s,ā)d(s,ā) − a d(s, a) = s,a d(s, a)r(s, a) − αD f d d D + (1 − γ) s µ0(s)V (s) + γ s,ā d(s,ā) s T (s|s,ā)V (s) − s,a d(s, a)V (s) = s,a d(s, a)r(s, a) − αD f d d D + (1 − γ) s µ0(s)V (s) + γ s,a d(s, a) s T (s |s, a)V (s ) − s,a d(s, a)V (s) = (1 − γ) s µ0(s)V (s) + s,a d(s, a) r(s, a) + γ s T (s |s, a)V (s ) − V (s) − αD f d d D = (1 − γ)Es∼µ 0 [V (s)] + E (s,a)∼d [r(s, a) + γT V (s, a) − V (s))] − αD f d d D (26)
Proposition 1. The minimax problem:
min V (s) max d≥0 E (s,a)∼d [r(s, a)] − αD f d d D + s V (s) (1 − γ)µ0(s) + γT d(s) − a d(s, a)(27)
shares the same optimal value as the following minimization problem:
min V (s) (1 − γ)E s∼µ0 [V (s)] + α E (s,a)∼d D f (r (s, a) + γT V (s, a) − V (s) α ) (28) where f is the Fenchel conjugate function of f with dom f = {u : u ≥ 0}
Proof. Using Lemma 2, this minimax problem can be re-written as:
min V (s) max d≥0 (1 − γ)E s∼µ0 [V (s)] + E (s,a)∼d [r(s, a) + γT V (s, a) − V (s)] − αD f d d D (29) Next, min V (s) max d≥0 (1 − γ)Es∼µ 0 [V (s)] + E (s,a)∼d [r(s, a) + γT V (s, a) − V (s)] − αD f d d D = min V (s) (1 − γ)Es∼µ 0 [V (s)] + max d≥0 E (s,a)∼d [r(s, a) + γT V (s, a) − V (s)] − αD f d d D = min V (s) (1 − γ)Es∼µ 0 [V (s)] + α max d≥0 E (s,a)∼d r(s, a) + γT V (s, a) − V (s) α − D f d d D L(30)
Published as a conference paper at ICLR 2023 L in the last step reduces to:
α max d≥0 E (s,a)∼d r(s, a) + γT V (s, a) − V (s) α − D f d d D = α max d≥0 E (s,a)∼d D d(s, a) d D (s, a) (r(s, a) + γT V (s, a) − V (s)) α − E (s,a)∼d D f d(s, a) d D (s, a) = α max d≥0 E (s,a)∼d D d(s, a) d D (s, a) (r(s, a) + γT V (s, a) − V (s)) α − f d(s, a) d D (s, a) = α E (s,a)∼d D max d(s,a)≥0 d(s, a) d D (s, a) (r(s, a) + γT V (s, a) − V (s)) α − f d(s, a) d D (s, a) = α E (s,a)∼d D max d(s,a) d D (s,a) ≥0 d(s, a) d D (s, a) y(s, a) − f d(s, a) d D (s, a) = α E (s,a)∼d D [f (y(s, a))](31)
where y(s, a) =r (s,a)+γT V (s,a)−V (s) α , the third step follows the interchangeability principle (Lemma 1) and the last step comes from the Fenchel conjugate of convex function f 6 .
Using this result, we finally yield the tractable lower-level problem Eq. (10).
A.3 PROOF OF TRACTABLE TRANSFORMATION OF THE UPPER-LEVEL PROBLEM
Proposition 2. The original upper-level problem
min ∆r D f d π * r d E(32)
can be equivalently written as:
min ∆r D f f r + γT V * − V * α d D d E(33)
where d π * r is the optimal state-action visitation distribution of Eq. (7) Proof. By the property Eq. Given V * , we have:
d π * r (s, a) d D (s, a) = f r(s, a) + γT V * (s, a) − V * (s) α(35)
Substituting this result into the original upper-level problem completes the proof.
Next, we denote f r+γT V * −V * α as g. By expanding the f -divergence, we have the upper-level objective:
D f d D g d E = E (s,a)∼d E f d D (s, a)g(s, a) d E (s, a) (36) = E (s,a)∼d D d E (s, a) d D (s, a) f d D (s, a) d E (s, a) g(s, a)(37)
= E (s,a)∼d D w(s, a)f g(s, a) w(s, a)
where the distribution ratio w(s, a) d E (s, a)/d D (s, a).
Finally, by combining proposition 1 and proposition 2, the original bi-level optimization problem Eq.
(4)-(5) is rewritten equivalently as follows:
∆r * = arg min ∆r E (s,a)∼d D w(s, a)f f r(s, a) + γT V * (s, a) − V * (s) α /w(s, a) s.t. V * (s) = arg min V (s) (1 − γ)E s∼µ0 [V (s)] + α E (s,a)∼d D f r(s, a) + γT V (s, a) − V (s) α (39) B IMPLEMENTATION DETAILS OF RGM B.1 RGM WITH KL-DIVERGENCE
In this section, we introduce the implementation details of RGM. For KL-divergence, we have f (x) = x log x and its Fenchel conjugate is f (x) = e x−1 . However, this exponential form is numerically unstable and prone to value explosion in practice. We address this issue by using the fact that the conjugate of the negative entropy function, restricted to the probability simplex, is the log-sum-exp function (Boyd et al., 2004), i.e., D ,f (y) = log E x∼q [exp y(x)]. Then, the optimization problem of RGM with KL divergence is 40), we need to calculate a log-sum-exp value in the denominator of the log(Softmax) term, where log (Softmax(Adv(∆r, V * )/α)) = Adv(∆r, V * )/α − log s,a∈S×A exp(Adv(∆r, V * )/α). In low-dimensional discrete state-action space, we can easily get this value via summing over the overall space. In high-dimensional continuous MDPs, however, it is pretty difficult to retrieve the value because it requires integration over the entire space. CQL (Kumar et al., 2020) approximates this value via importance sampling but requires additional samples from the entire state-action space. There are some other methods like Markov Chain Monte Carlo (MCMC) or Score Match (SM) (Song & Kingma, 2021) that can approximate the update gradient but bring additional computation costs and suffer from some technical issues.
min ∆r E (s,a)∼d D Softmax Adv(∆r, V * ) α log d D (s, a) d E (s, a) + log Softmax Adv(∆r, V * ) α s.t.V * = arg min V (1 − γ)E s∼µ0 [V (s)] + α log E (s,a)∼d D exp Adv(∆r, V ) α(
Fortunately, we can subtly circumvent the log-sum-exp term by optimizing the upper bound of the original upper-level problem using the following inequality (Boyd et al., 2004):
max xi∈B {x 1 , ..., x n } ≤ max{x 1 , ..., x n } ≤ log n i exp (x i )(42)
where max xi∈B {x 1 , ..., x n } is the max value in a mini-batch B which is sampled from {x 1 , ..., x n }.
For simplicity, we denote max
Upper(40) = E (s,a)∼d D Softmax Adv(∆r, V * ) α log d D (s, a) d E (s, a) + Adv(∆r, V * ) α − log s,a∈S×A exp Adv(∆r, V * ) α ≤ E (s,a)∼d D Softmax Adv(∆r, V * ) α log d D (s, a) d E (s, a) + Adv(∆r, V * ) α − max B Adv(∆r, V * ) α ∝ E (s,a)∼d D exp Adv(∆r, V * ) α log d D (s, a) d E (s, a) + Adv(∆r, V * ) α − max B Adv(∆r, V * ) α(43)
where Upper(40) denotes the upper level objective in Eq. (40).
Replacing Eq. (43) to the upper level objective in Eq. (40), we obtain the final optimization problem: When extracting the policy, we can ignore the annoying sum-exp term in the denominator of Softmax and get the following ratio, because it does not influence the direction of gradients to update the policy.
min ∆r E (s,a)∼d D exp Adv(∆r, V * ) α log d D (s, a) d E (s, a) + Adv(∆r, V * ) α − max B Adv(∆r, V * ) α s.t.V * = arg min V (1 − γ)E s∼µ0 [V (s)] + α log E (s,a)∼d D exp Adv(∆r, V * ) α(44)ψ * (s, a) = d π * r (s, a) d D (s, a) ∝ exp r + ∆r + γT V * (s, a) − V * (s) α :=ψ * (s, a)(45)
However, using Eq.(45), we can only get an unnormalized distribution ratio instead of an exact one. We resort to self-normalized importance sampling (Owen, 2013) to obtain a normalized ratio:
ψ * (s, a) =ψ * (s, a) E (s,a)∼d D [ψ * (s, a)](46)
B.2 RGM WITH X 2 -DIVERGENCE Additionally, we can also implement RGM using
X 2 -divergence. For X 2 -divergence, we have f (x) = 1 2 (x − 1) 2 with dom f = {x :
x ≥ 0} 7 and its Fenchel conjugate is f (x) = 1 2 (x + 1) 2 and f (x) = max (0, x + 1). Then, the optimization objective of RGM with X 2 divergence is
min ∆r E (s,a)∼d D d E (s, a) 2d D (s, a) max 0, Adv(∆r, V * ) α + 1 d D (s, a) d E (s, a) − 1 2 s.t V * = arg min V (1 − γ)Es∼µ 0 [V (s)] + α 2 E (s,a)∼d D Adv(∆r, V ) α 2(47)
The importance ratio used to extract the policy is:
ψ * (s, a) = d π * r (s, a) d D (s, a) = max 0,r + ∆r + γT V * (s, a) − V * (s) α + 1(48)
For RGM with KL-divergence, the upper layer contains an exponential term exp( Adv(δr,V * ) α ), which may pose numerical instability. For RGM with χ 2 divergence, f (x) = max (0, x + 1) and so the gradient vanishes when x + 1 < 0, which makes the policy learning slow or even fail. In practice, we follow the criteria from SMODICE (Ma et al., 2022) by monitoring the initial policy loss to choose the types of f -divergence.
B.3 RGM HYPERPARAMETERS AND PSEUDOCODE
For continuous MDPs with high dimensional state-action spaces, we implement RGM by parameterizing h τ , ∆r φ , V θ and π w using deep neural networks with parameter τ, φ, θ and w, respectively. We implement RGM based on a two-time scale first-order stochastic gradient update, where the reward correction term is updated much slower than the Lagrangian multiplier V . We choose the cosine annealing learning rate schedule of the reward correction term and policy network to stabilize the training process. To make the reward correction term comparable w.r.t the original imperfect rewards, we normalize the imperfect rewards to standard Gaussian distribution N (0, 1) and strict the output range of ∆r φ to [−3, 3] by Tanh function. The conclusive hyperparameters can be found in Table 3. The pseudocode of RGM with deep neural networks can be found in Algorithm 1. We run RGM on one RTX 3080Ti GPU with about 1h30min training time to apply 1M gradient steps.
We report the wall-clock training time of RGM compared with SOTA offline RL methods as well as SOTA offline IL methods that can learn from mixed quality data in Table 4. RGM is as efficient as most baselines but has an additional ability to combat the negative impacts of imperfect rewards.
C EXPERIMENTAL DETAILS
In this section, we introduce the detailed experimental setups in our paper.
C.1 D4RL EXPERIMENTS
Task Descriptions. The D4RL Fu et al. (2020) tasks we try to solve include Hopper, Halfcheetah and Walker2d. For these tasks, RL policies need to control the robots to move in the forward (right) direction by applying torques on the joints. Dataset composition. The D4RL Fu et al. (2020) datasets that we used in this paper contain 5 types of datasets: random: roll out a random policy for 1M steps. expert: roll out an expert policy that trained with SAC (Haarnoja et al., 2018) for 1M steps. medium: roll out a medium policy that achieves 1/3 the performance of the expert for 1M steps. medium-replay: replay buffer of a SAC agent that is trained to the performance of the medium policy. medium-expert: equally mixed dataset combines medium and expert data. We sample only one trajectory from the expert dataset to serve as the expert demonstration D E . The other datasets are treated as non-expert datasets D. Imperfect rewards.
We assume the original rewards in D4RL datasets are perfect, since we evaluate the policy performance based on the perfect reward function in the original gym environment during evaluation. We randomly flip the sign of 50% original rewards to construct partially correct rewards, where half rewards can provide correct learning signals while the other half cannot. We flip all signs of the original rewards to construct completely incorrect rewards.
C.2 SPARSE REWARD EXPERIMENTS
Task descriptions. The Robomimic Mandlekar et al. (2021) tasks we try to solve include Lift and Can. For the Lift task, RL policy needs to control a 7-DOF robot arm to learn to lift a cube that is randomly located at a table. For the Can task, RL policy needs to control a 7-DOF robot arm to learn to pick a can that is randomly located at a table and place it in a specific location. The AntMaze tasks we try to solve include AntMaze medium tasks, where an ant not only needs to learn to walk but also navigates from the goal to the destination in a medium-size maze. This task is extremely difficult due to the non-markovian and mixed-quality offline dataset, the stochastic property of environments, and the high dimensional state-action space (Fu et al., 2020). Lift and Can task, taking agent checkpoints that are saved regularly during training, and collecting 300 rollout trajectories from each checkpoint. We treat PH dataset as the expert dataset since the environment is stochastic, thus only one expert trajectory is difficult to capture the expert distribution. We use MG datasets as the large potentially suboptimal dataset rather than MH datasets since MH datasets are non-markovian and thus are hard to be solved by modern offline RL methods (Mandlekar et al., 2021), which is not the main challenge we try to solve. AntMaze dataset composition. The expert dataset of RGM is composed of 30 successful trajectories (which may be suboptimal) that are collected by training IQL with dense rewards. We set the original D4RL Antmaze-medium-play-v2 and Antmaze-medium-diverse-v2 datasets as non-expert datasets.
C.3 MULTI-TASK DATA SHARING EXPERIMENTS
Task descriptions. The multi-task data sharing experiments contain 2 domains with 4 tasks per domain built on DeepMind Control Suite (Tassa et al., 2018). The immediate rewards in the 8 tasks are all in the unit interval, r(s, a) ∈ [0, 1]. (a) For Walker (Stand, Walk, Run, Flip) domain, the agent needs to control a biped in a 2D vertical plane to master four different locomotion skills. The observation space is 24 dimensional, and the action space is 6 dimensional. The episode length is set to 1000. (b) For Quadruped (Walk, Run, Roll-Fast, Jump) domain, the agent needs to control a quadruped within a 3D space to master four different moving skills. The observation space is 78 dimensional, and the action space is 12 dimensional. The episode length is set to 1000. Dataset composition. We take the same rule of dataset generation and similar task settings as the work (Bai et al., 2023). For each task, we utilize TD3 (Fujimoto et al., 2018) to collect three types of datasets (expert, medium, replay). The expert dataset contains only one expert episode, the medium dataset contains 1000 episodes of interactions, and the replay dataset contains 2000 episodes of interactions. For Walker (Stand, Walk, Run, Flip) domain, the Stand task is set to the target task, and the others are relevant tasks. For Quadruped (Walk, Run, Jump, Roll-Fast) domain, the Walk task is set to the target task, and the others are relevant tasks. We conduct two-task data sharing experiments, in which we share the replay dataset of the relevant task with the medium dataset of the target task.
C.4 GRID WORLD EXPERIMENTS
Dataset composition. The offline dataset D we use in grid world experiments consists of 1000 trajectories generated by a completely random policy (Figure 11 (b)). There are two settings of imperfect rewardsr: (i)r = +10 when reaching the goal whiler = 0 anywhere else. (ii) ( Figure 11 (c))r = +10 when reaching the goal,r = −10 when encountering the fire (true fire or fake fire),r = 0 everywhere else. The expert demonstration dataset D E consists of only one expert demonstration (Figure 11 (a)).
D ADDITIONAL RESULTS
In this section, we provide additional comparative and ablation results of RGM against baseline methods.
D.1 ADDITIONAL COMPARISON TO OFFLINE IL
Recall that DWBC (Xu et al., 2022b) and SMODICE (Ma et al., 2022) all assume the offline dataset already covers a lot of expert trajectories, which is more restrictive compared to the requirement of RGM. Therefore, we further demonstrate the superiority of RGM compared to these offline IL methods by evaluating RGM under the same settings of DWBC and SMODICE. We combine the original D4RL dataset with 200 or 100 expert trajectories as the offline dataset D, see Table 7 for descriptions of the expert trajectories. The comparisons under these dataset configurations can be found in Table 8. We can observe from Table 8 that RGM still outperforms existing SOTA offline IL methods under their settings. We also implemented the discounted visitation distribution sampling in RGM. This is done by augmenting the D4RL datasets that adds the timestep of each (s, a) pair in an episode. When performing sampling in Eq.(14-16) and calculating the gradient, we sample (s, a, t) in the D4RL datasets and then multiply the gradient by γ t . Empirically, we found that the performance of the discounted visitation distribution version is not better than the sampling distribution version of RGM. Figure 12 and Table 9 show that RGM (sampling distribution) surpasses RGM (discounted visitation distribution) in most cases with lower variance, while the latter wins by a slight margin in only a few cases.
D.3 EXPERIMENTS ON NOISY PARTIALLY CORRECT REWARDS
We add i.i.d Gaussian noises with different standard deviation σ to original D4RL rewards to construct noisy imperfect rewards with different degrees of imperfection. We set σ = 1 to construct partially correct rewards and σ = 10 as largely incorrect rewards, see Table 10 for detailed results. Table 10 shows that RGM under perfect rewards slightly outperforms RGM with partially correct rewards, indicating that RGM can largely remedy the negative impacts caused by reward noises with σ = 1. Meanwhile, the highly noisy rewards (σ = 10) surely impact the performance, but its mean Table 8: Average normalized scores of RGM compared with SOTA offline IL methods that can learn from mixed quality data under their settings. The notation "-w.e" stands for the mixed dataset that combines the original D4RL dataset with some expert trajectories. The scores are taken over the final 10 evaluations with 5 random seeds. We obtain the results via ruining author-provided open-source codes. RGM achieves 7 highest scores in 12 tasks. designed for mixed-quality data (DWBC and SMODICE). It is found that RGM also enjoys a higher level of performance gains when the amount of expert data is increased.
D.5 EXPERIMENTS ON MULTI-TASK DATA SHARING
We present concrete results of the multi-task data sharing experiment. Table 14 shows the evaluated scores on multi-task data sharing, which are illustrated in Fig. 3.
D.6 ADDITIONAL LEARNING CURVES OF RGM
We present the learning curves of RGM compared with offline IL and RL baselines on D4RL datasets related to the results presented in Table 1.
D.7 ILLUSTRATIVE EXAMPLE FOR THE NON-TABULAR SCENARIOS
The results of the 8×8 grid world experiments in Section 5.2 and Appendix C.4 illustrate the potential benefits of the learned rewards in the tabular case. In this subsection, we consider a one-dimensional random walk task in the non-tabular case and provide the visualization of the learned corrected rewardsr. In this task, the state space is a straight line from [0, +3] and the agent can move at each step in the range of [-0.5, 0.5]. If the agent goes beyond the edge (s < 0 or s > +3), then we keep it at the edge (s = 0 or s = +3). The agent needs to start from state s = 0 and reach the destination located at s = 3 as fast as possible. The expert dataset D E consists of one trajectory where the expert takes action a = 0.5 at every state. The offline dataset D consists of 1000 trajectories generated by a completely random policy where the agent takes action uniformly from [-0.5, 0.5] at every state. The sparse rewardsr = +10 is set when reaching the destination whiler = 0 anywhere else. The visualization of learned rewardsr at each state-action pair is shown in Figure 14.
E DISCUSSION ON THE APPLICABILITY TO ONLINE SETTINGS
It should be noted that the proposed RGM framework can also be applied to the online setting. This can be achieved by simply setting α = 0 in Eq. (4-5), and we have the bi-level objective of the online
Since we could get online samples from d π * r in the online setting, so we don't have to eliminate d π * r . One can use the existing popular online RL algorithms to solve the lower-level problem, while leveraging the online samples from d π * r to solve the upper-level problem. Hence the online version of RGM can be perceived as a reduced and simplified version of the original RGM. The core idea of the reward correction has not been changed in the online setting, which illustrates that to some extent, our proposed RGM is a unified policy optimization method for imperfect rewards.
Published as a conference paper at ICLR 2023 , the learned rewardr gets a larger value when the action gets closer to 0.5. The expert data has contain 7 states (s = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0), but the learned rewards can still generalize well in the state space even in regions that are not covered by the expert data. Similar to the 8×8 grid world experiment, we can successfully navigate to the destination by only maximizing per-step rewardr, which means that the learned rewards also encode long-horizon information.
Figure 1 :
1Diverse settings of imperfect rewards.
Figure 3 :
3Results on multi-task data sharing tasks.
Figure 4 :
4Learned rewardsr and optimal distribution d π * r trained on two types of imperfect rewardsr. The opacity of each square represents the value of marginal state distribution d π * r (s). The opacity of the arrow shows the learned rewardr, where the darkest arrow points to the direction of the highest reward. The expert starts from , follows the path and arrow to reach the goal .r in (b) is +10 at the goal and is zero at other states.r in (c) falsely punishes the agent on and correctly punishes the RL agent on fire marks .
Figure 5 :Figure 6 :
56Performance drop of normalized returns of SOTA offline RL methods on D4RL datasets under perfect and RGM corrected rewards. The wrong rewards are the partially correct rewards as in Table 1. H: Hopper; HC: HalfCheetah; W: Walker2d.(a) Learning curve of ∆r (b) Effect ofr onr (c) Effect ofr on ∆r Experiments on learned rewards in hopper-m-r task. The superscript "¯" denotes the mean value of mini-batch samples. The subscript "E" and "O" denote the value on expert and non-expert data. In (b)(c), largē rE −rO and ∆rE − ∆rO indicate that expert and non-expert data are clearly distinguishable according to the learned rewards, and small values mean the opposite.
Assumption 1 .
1There exists at least one d such that: a d(s, a) = (1 − γ)µ 0 (s) + γT d(s), d(s) > 0, ∀s ∈ S
(19), the maximizer d(s,a) d D (s,a) * of f (y(s, a)) in Eq. (31) satisfies d(s, a) d D (s, a) * = f r(s, a) + γT V (s, a) − V (s) α (34)
Adv(∆r, V ) :=r(s, a)+γT V (s, a)−V (s) =r(s, a)+∆r(s, a,r)+γT V (s, a)−V (s) and log d D(s,a) d E(s,a) can be obtained by training a discriminator log d D(s,a) d E(s,a) = − log 1 h * − 1 using Eq. (14) in continuous MDPs. The importance ratio used to extract the policy is ψ * (s, a) = d π * r (s, a) d D (s, a) = Softmax r + ∆r + γT V * (s, a) − V * (s) α (41) B.1.1 OPTIMIZE WITHOUT SUM-EXP Note that in the upper level objective of Eq. (
{x 1 , ..., x n } as max xi∈B {x}. Substituting Eq. (42) into the upper-level problem of Eq. (40), we get the upper bound of the original upper-level optimization objective:
We practically utilize the same mini-batch B as that of SGD gradient update step to calculate maxB Adv(∆r,V * ) α . Note that the exp term in the upper-level problem is prone to value explosion in practice, we clip the exp value to (−∞, 100] like IQL (Kostrikov et al., 2021b) does to improve training stability.
Figure 7 :
7D4RL MuJoCo tasks
Figure 8 :
8Robomimic tasks
Figure 9 :
9AntMaze medium task. Robomimic dataset composition. The Robomimic Mandlekar et al. (2021) datasets that we used in this paper contain 3 types of datasets: PH (Proficient-Human): datasets are collected by a single, experienced human operator. MH (Multi-Human): datasets are collected by 6 human operators of varying proficiency. MG (Machine-Generated): datasets are collected by first training SAC on the
Figure 10 :
10Different tasks in Walker and Quadruped domain
Figure 11 :
11(a) The only one expert demonstration path, which starts from , follows the path and arrow to reach the goal . (b) The empirical distribution heatmap of offline dataset D O , which consists of trajectories generated by random policy starting from . The darker the color is, the more frequently the agent passes. (c) Illustration of imperfect rewards. Agent getsr = −10 when reaching ,r = +10 when reaching ,r = 0 everywhere else.
Figure 13 :Figure 14 :
1314Learning curves of RGM trained on D4RL datasets under imperfect rewards.(a) Empirical distribution of D (b) Visualization of learned rewardsr (a) The empirical distribution of offline dataset D in a continuous one-dimensional random walk task. Most states in the offline dataset are distributed near the starting point. (b) At each state (at each vertical line)
Table 1 :
1Average normalized scores of RGM compared with offline IL and RL baselines on D4RL datasets.
Table 2 :
2Results on sparse reward tasks.Dataset
BC
TD3+BC
CQL
IQL
RGM
Antmaze-m-p
0
0
0
0
13.7
Antmaze-m-d
0
0
0
0
3.3
Lift-MG
65.3
87.3
64.0
56.0
90.3
Can-MG
64.7
55.3
64.7
50.0
66.7
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A PROOFS
A.1 BACKGROUND
We begin by briefly introducing the Fenchel conjugate (also known as convex conjugate or Legen-
dre-Fenchel transformation):
Definition 3. (Fenchel conjugate) In a real Hilbert space X , if a function f (x) is proper, then the
Fenchel conjugate f of f at y is:
Table 3 :
3The hyperparameters of RGM with deep neural networksHyperparameter
Value
Architecture
Reward correction hidden dim
256
Reward correction layers
2
Reward correction activation function
ReLU
Discriminator hidden dim
512
Discriminator layers
3
Discriminator activation function
Tanh
V hidden dim
256
V hidden layers
2
V activation function
ReLU
Policy hidden dim
256
Policy hidden layers
2
Policy activation function
ReLU
RGM Hyperparameters
Optimizer
Adam (Kingma & Ba, 2015)
Reward correction learning rate l φ
3e-7
Reward correction learning rate schedule cosine annealing
Discriminator learning rate lτ
1e-3
V θ learning rate l θ
3e-4
Policy learning rate lw
3e-4
Policy learning rate schedule
cosine annealing
V θ gradient L2-regularization
1e-4
Discount factor
0.99
f -divergence
χ 2 for Robomimic tasks
KL for other tasks
α
4 for walker2d-medium-replay
0.5 for other D4RL tasks
0.5 for Antmaze tasks
2 for Lift and Can tasks
0.3 for Quadruped-walk + Quadruped-jump and
3 for the others in multi-task data sharing experiments
Table 4 :
4Wall-clock run time comparison of RGM and other baselinesBC
DWBC
SMODICE TD3+BC CQL
IQL
RGM(ours)
30min 2h40min 2h20min
45min
4h30min 1h30min 2h30min
Table 5 :
5Dataset compositions for D4RL ExperimentsTask
State Dim
Expert Dataset
Number of Trajectories Expert Data Size
Hopper
11
hopper-expert-v2
1
1000
Halfcheetah
17
halfcheetah-expert-v2
1
1000
Walker2d
17
walker2d-expert-v2
1
1000
Table 6 :
6Dataset compositions for Robomimic DatasetsTask State Dim Expert Dataset Expert Size Non-expert Dataset Non expert Size
Lift
19
Lift-PH
9666
Lift-MG
225K
Can
23
Can-PH
23207
Can-MG
585K
Table 7 :
7The details about the expert data that are used to construct the non-expert dataset in offline IL settings.D.2 EXPERIMENTS ON SAMPLING FROM DISCOUNTED DISTRIBUTIONSTask
State Dim
Expert Dataset
Number of Trajectories Expert Data Size
Hopper
11
hopper-expert-v2
200
193430
Halfcheetah
17
halfcheetah-expert-v2
200
199800
Walker2d
17
walker2d-expert-v2
100
99900
Table 9 :
9Normalized scores of RGM sampling from discounted distribution and undiscounted distributionDataset
RGM (Discounted) RGM (Undiscounted)
hopper-r
19.8±0.2
21.2 ±0.4
halfcheetah-r
0.2 ±0.0
0.2 ±0.0
walker2d-r
1.2±1.7
7.7 ±3.3
hopper-m
51.1±4.9
55.5 ±1.0
halfcheetah-m
40.3±1.6
40.7 ±1.4
walker2d-m
62.2±22.5
72.3 ±10.7
hopper-m-r
43.3±11.6
59.1 ±15.3
halfcheetah-m-r 34.5±4.5
37.8 ±2.6
walker2d-m-r
34.3±11.0
48.6 ±3.6
hopper-m-e
65.3±19.5
87.1 ±10.7
halfcheetah-m-e
87.3 ±7.8
81.5±0.8
walker2d-m-e
108.4±0.6
108.8 ±0.4
Mean score
45.7±7.2
52.0 ±4.2
Table 10 :
10Normalized scores of RGM on different degrees of noisy datasets. Dataset RGM(T) RGM (σ = 1) RGM (σ = 10)hopper-r
29.6
8.5
9.8
halfcheetah-r
0.2
0.3
0.2
walker2d-r
3.9
0.6
-0.1
hopper-m
56.2
52.0
47.9
halfcheetah-m
40.4
41.2
38.4
walker2d-m
73.3
71.9
72
hopper-m-r
60.3
58.0
40.0
halfcheetah-m-r
37.9
38.3
28.1
walker2d-m-r
46.3
42.5
43.8
hopper-m-e
106.1
82.0
82.8
halfcheetah-m-e
85.6
88.7
69.1
walker2d-m-e
109.2
108.2
108.5
Mean score
54.1
49.4
45.0
Table 11 :
11Normalized scores of RGM and offline IL baselines when D E contains 10 expert trajectories. Dataset DWBC (N E = 10) SMODICE (N E = 10) RGM (N E = 10)hopper-r
52.5
1.3
30.8
halfcheetah-r
-0.3
2.1
0.2
walker2d-r
96.2
0.3
6.1
hopper-m
31.1
53.8
54.5
halfcheetah-m
5.0
40.9
41.4
walker2d-m
22.4
3.3
72.9
hopper-m-r
37.4
33.2
55.5
halfcheetah-m-r
3.9
36.7
34.9
walker2d-m-r
90.7
34.7
43.1
hopper-m-e
31.2
85.0
89.2
halfcheetah-m-e
10.9
86.6
79.4
walker2d-m-e
46.3
14.1
109.0
Mean score
35.6
32.7
51.4
Table 12 :
12Normalized scores of RGM and offline IL baselines when D E contains 40 expert trajectories. Dataset DWBC (N E = 40) SMODICE (N E = 40) RGM (N E = 40)hopper-r
54.6
67.2
36.9
halfcheetah-r
8.8
14.8
18.7
walker2d-r
78.7
92.9
-0.1
hopper-m
13.5
54.2
57.0
halfcheetah-m
5.6
44.5
40.9
walker2d-m
16.9
3.5
74.3
hopper-m-r
54.3
47.2
54.3
halfcheetah-m-r
46.1
54.6
46.1
walker2d-m-r
87.8
35.7
61.2
hopper-m-e
34.5
75.9
92.4
halfcheetah-m-e
3.4
85.1
84.7
walker2d-m-e
57.2
17.1
108.6
Mean score
38.5
44.9
56.5
Table 13 :
13Normalized scores of RGM and offline IL baselines when D E contains 80 expert trajectories. Dataset DWBC (N E = 80) SMODICE (N E = 80) RGM (N E = 80)hopper-r
65.1
92.7
47.3
halfcheetah-r
2.3
48.1
40.4
walker2d-r
86.1
98.8
109.1
hopper-m
8.8
53.4
59.7
halfcheetah-m
6.7
51.2
42.5
walker2d-m
36.5
2.9
73.4
hopper-m-r
35.3
46.6
66.2
halfcheetah-m-r
36.1
59.6
54.0
walker2d-m-r
85.8
32.8
64.9
hopper-m-e
8.8
83.7
97.1
halfcheetah-m-e
12.1
87.4
83.9
walker2d-m-e
64.5
43.7
108.6
Mean score
37.5
58.4
70.6
Table 14 :
14Evaluated scores on multi-task data sharing. version of RGM: ∆r * = arg min ∆r D f d π * r d E s.t. π * r = arg max π E (s,a)∼d πr [r(s, a)]Domain
Dataset
CDS
CDS+UDS
RGM
Walker
stand-medium + walk-replay
486.1±7.2
415.3±44.8
753.3±107.6
Walker
stand-medium + run-replay
455.8±21.9
440.0±8.4
620.0±31.0
Walker
stand-medium + flip-replay
492.4±19.0
371.2±123.0
745.6±100.8
Quadruped
walk-medium + run-replay
527.6±213.0
155.7±53.0
900.0±48.6
Quadruped walk-medium + roll_fast-replay
476.2±45.1
439.1±90.7
493.2±37.3
Quadruped
walk-medium + jump-replay
533.5±168.5
521.9±236.9
490.9±119.4
Mean score
495.3±79.1
390.5±92.8
667.0±74.1
The signs of 50% D4RL rewards are flipped and hence only half rewards can give correct learning signals.3 We regard the original D4RL rewards as perfect since we evaluate the policies in terms of these rewards, which can be perceived as solving the tasks encoded in the original D4RL rewards.4 All sign of the original rewards is flipped.
Note that in IQL and CQL papers, they turn the original sparse rewards into dense rewards by applying the reward subtraction trick (minus 1 on every reward, so the reward becomes negative except at the goal).
dom f = {u : u ≥ 0} and f is convex, so f (y) = −f (0) when y ≤ f (0).
On account of the state-action visitation distribution d ≥ 0
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Deepthermal: Combustion optimization for thermal power generating units using offline reinforcement learning. Xianyuan Zhan, Haoran Xu, Yue Zhang, Xiangyu Zhu, Honglei Yin, Yu Zheng, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial IntelligenceXianyuan Zhan, Haoran Xu, Yue Zhang, Xiangyu Zhu, Honglei Yin, and Yu Zheng. Deepthermal: Combustion optimization for thermal power generating units using offline reinforcement learning. In Proceedings of the AAAI Conference on Artificial Intelligence, 2022.
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Offline learning from demonstrations and unlabeled experience. Konrad Zolna, Alexander Novikov, Ksenia Konyushkova, Caglar Gulcehre, Ziyu Wang, Yusuf Aytar, Misha Denil, Scott Nando De Freitas, Reed, arXiv:2011.13885Dataset BC DWBC SMODICE RGM. arXiv preprintOursKonrad Zolna, Alexander Novikov, Ksenia Konyushkova, Caglar Gulcehre, Ziyu Wang, Yusuf Aytar, Misha Denil, Nando de Freitas, and Scott Reed. Offline learning from demonstrations and unlabeled experience. arXiv preprint arXiv:2011.13885, 2020. Dataset BC DWBC SMODICE RGM (Ours)
Experiments on sampling from discounted and undiscounted distributions score is 45.0, which is still considerably higher than other Offline RL and IL methods under partially correct rewards with the largest mean value of 35. Figure. 125 as shown in Table 1Figure 12: Experiments on sampling from discounted and undiscounted distributions score is 45.0, which is still considerably higher than other Offline RL and IL methods under partially correct rewards with the largest mean value of 35.5 as shown in Table 1.
ABLATIONS ON THE NUMBER OF EXPERT TRAJECTORIES We add the ablations on the number of expert trajectories in D E (N E ) for RGM, SMODICE and DWBC. Table 11, 12 and 13 show that RGM achieves better performance than offline IL methods. D.4 ABLATIONS ON THE NUMBER OF EXPERT TRAJECTORIES We add the ablations on the number of expert trajectories in D E (N E ) for RGM, SMODICE and DWBC. Table 11, 12 and 13 show that RGM achieves better performance than offline IL methods
| {'fraction_non_alphanumeric': 0.06745757366231182, 'fraction_numerical': 0.034232201559979136, 'mean_word_length': 4.153341251484654, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Reward function is essential in reinforcement learning (RL), serving as the guiding signal to incentivize agents to solve given tasks, however, is also notoriously difficult to design. In many cases, only imperfect rewards are available, which inflicts substantial performance loss for RL agents. In this study, we propose a unified offline policy optimization approach, RGM (Reward Gap Minimization), which can smartly handle diverse types of imperfect rewards. RGM is formulated as a bi-level optimization problem: the upper layer optimizes a reward correction term that performs visitation distribution matching w.r.t. some expert data; the lower layer solves a pessimistic RL problem with the corrected rewards. By exploiting the duality of the lower layer, we derive a tractable algorithm that enables sampled-based learning without any online interactions. Comprehensive experiments demonstrate that RGM achieves superior performance to existing methods under diverse settings of imperfect rewards. Further, RGM can effectively correct wrong or inconsistent rewards against expert preference and retrieve useful information from biased rewards.', 'arxivid': '2302.01667', 'author': ['Jianxiong Li \nTsinghua University\nBeijingChina\n', 'Xiao Hu \nTsinghua University\nBeijingChina\n', 'Haoran Xu \nJD Technology\nBeijingChina\n', 'Jingjing Liu \nTsinghua University\nBeijingChina\n', 'Xianyuan Zhan [email protected] \nTsinghua University\nBeijingChina\n\nShanghai Artificial Intelligence Laboratory\nShanghaiChina\n', 'Qing-Shan Jia \nTsinghua University\nBeijingChina\n', 'Ya-Qin Zhang \nTsinghua University\nBeijingChina\n'], 'authoraffiliation': ['Tsinghua University\nBeijingChina', 'Tsinghua University\nBeijingChina', 'JD Technology\nBeijingChina', 'Tsinghua University\nBeijingChina', 'Tsinghua University\nBeijingChina', 'Shanghai Artificial Intelligence Laboratory\nShanghaiChina', 'Tsinghua University\nBeijingChina', 'Tsinghua University\nBeijingChina'], 'corpusid': 256598204, 'doi': '10.48550/arxiv.2302.01667', 'github_urls': [], 'n_tokens_mistral': 27440, 'n_tokens_neox': 24048, 'n_words': 13136, 'pdfsha': 'b2950bce85a14904d1e30018fa4c8d21ba0651a8', 'pdfurls': ['https://export.arxiv.org/pdf/2302.01667v1.pdf'], 'title': ['Published as a conference paper at ICLR 2023 MIND THE GAP: OFFLINE POLICY OPTIMIZATION FOR IMPERFECT REWARDS', 'Published as a conference paper at ICLR 2023 MIND THE GAP: OFFLINE POLICY OPTIMIZATION FOR IMPERFECT REWARDS'], 'venue': []} |
arxiv |
Thermal-bath effects in quantum quenches within quantum critical regimes Ettore Vicari
Francesco Tarantelli
Dipartimento di Fisica dell'Università di Pisa and INFN
Dipartimento di Fisica dell'Università di Pisa
Largo Pontecorvo 3, Largo Pontecorvo 3, II-56127, 56127Pisa, PisaItaly, Italy
Thermal-bath effects in quantum quenches within quantum critical regimes Ettore Vicari
(Dated: May 15, 2023)
We address the out-of-equilibrium dynamics arising from quantum-quench (QQ) protocols (instantaneous changes of the Hamiltonian parameters) in many-body systems within their quantum critical regime and in contact with (homogeneously coupled) thermal baths. We consider two classes of QQ protocols. In one of them the thermal bath is used to prepare the initial finite-temperature Gibbs state; then, after quenching, the thermal bath is removed and the dynamics of the system is unitary. We also address a more complex QQ protocol where the thermal bath is not removed after quenching, thus the quantum evolution is also driven by the interaction with the bath, which may be described by appropriate master equations for the density matrix of the system, where a further relevant time scale, or inverse decay rate, characterizes the system-bath coupling. Under these QQ protocols, the critical system develops out-of-equilibrium scaling behaviors, which extend those for isolated critical systems, by introducing further scaling variables proportional to the temperature of the thermal bath and the decay rate of the system-bath interactions. These out-of-equilibrium scaling behaviors are checked by analyzing QQ protocols within fermionic Kitaev wires, or equivalently quantum Ising chains, supplemented with a particular modelization of thermal bath that guarantees the asymptotic thermalization within the Lindblad master equation for the dynamics of open systems.
I. INTRODUCTION
Thanks to the recent experimental progress in the realization and control of the dynamics of quantum manybody systems, see e.g. Refs. [1,2], the out-of-equilibrium quantum dynamics of many-body systems has become an important theoretical issue. In particular, out-ofequilibrium phenomena have been addressed within the critical regimes of many-body systems at continuous quantum transitions (CQTs) [3][4][5], where collective behaviors give rise to zero-temperature singularities in the equilibrium low-energy properties of the system, and the universal critical behaviors are determined by a limited number of relevant features, such as the global symmetry, the symmetry-breaking pattern, dimensionality, etc.. Within critical regimes and in the appropriate thermodynamic or finite-size scaling (FSS) limits, one can achieve a complete characterization of the complex dynamics of many-body systems by controlling a limited number of renormalization-group (RG) perturbations. The universal scaling behaviors at CQTs extend beyond the equilibrium conditions [5]. Indeed dynamic protocols entailing out-of-equilibrium evolutions develop scaling behaviors as well, in the appropriate limits, related to the universality class of the CQT. For example, out-of-equilibrium scaling behaviors emerge when analyzing the quantum evolutions arising from a quantum quench (QQ), see e.g. Refs. [5][6][7][8][9][10][11], or from slow changes of the Hamiltonian parameters across the transition point, such as the protocols associated with the so-called quantum Kibble-Zurek problem, see e.g. Refs. [5,[12][13][14][15][16][17][18][19][20][21][22][23].
These out-of-equilibrium issues have been mostly addressed within isolated many-body systems, unitarily driven by their Hamiltonian and the Schrödinger equation. In this paper we extend such studies to investigate how the interaction with a thermal bath, coupled homogeneously to the system, affects the out-ofequilibrium dynamics of many-body systems within the critical regime of a zero-temperature quantum transition, such as that arising from a QQ or a slow crossing of the quantum critical regime.
The role of the temperature within the equilibrium critical behavior at a CQT is generally associated with one of the relevant RG perturbations at the stable fixed point of the RG flow controlling the quantum criticality [3][4][5]24]. Therefore, the quantum scaling behavior can be only observed in the zero-temperature limit. More precisely, the quantum scaling limit requires that the zero-temperature critical point is approached keeping the ratio T /∆ fixed, where ∆ is the gap at the quantum critical point, which is generally power-law suppressed. For example, in the FSS limit the gap is suppressed as ∆ ∼ L −z at the critical point, where L is the size of the system and z > 0 is the universal dynamic exponent associated with universality class of the CQT. Within the equilibrium critical regime the temperature enters the asymptotic FSS laws through a further dependence of the scaling functions on the scaling variable Ξ ≡ T L z ∼ T /∆.
The role of the temperature becomes less definite when we consider out-of-equilibrium behaviors, because the temperature of the system is an equilibrium concept. However, one may consider the effects of thermal baths in contact with the system during its out-of-equilibrium dynamics. The main feature of a thermal bath is that it eventually drives the system toward thermalization at its temperature T , in the large-time limit of the evolu-arXiv:2305.05494v2 [cond-mat.stat-mech] 12 May 2023 tion of the system in contact with the thermal bath. The thermalization process must somehow introduce a further time scale τ in the problem, characterizing the approach of the system to the thermal state when it is put in contact with the thermal bath. Such time scale is expected to play an inportant role in the out-of-equilibrium dynamics of the system in contact with the thermal bath. In this paper we investigate these issues within the simplest dynamic protocols giving rise to out-of-equilibrium behaviors, i.e. those entailing instantaneous QQs of the Hamiltonian parameters starting from equilibrium thermal conditions.
A quench protocol is generally performed by suddenly varying a parameter within a family of Hamiltonians, such asĤ
(w) =Ĥ c + wĤ p ,(1)
whereĤ c andĤ p are independent of the parameter w, and [Ĥ c ,Ĥ p ] = 0. In a standard QQ protocol for closed systems, one usually starts from the ground state |Φ 0 , w i of the HamiltonianĤ(w i ) associated with an initial value w i of the parameter w, with corresponding density matrix ρ i = |Φ 0 , w i Φ 0 , w i |. At a given time, t = 0 say, the Hamiltonian parameter is suddenly changed from w i to w = w i , and the subsequent quantum evolution is supposed to be unitarily driven by the HamiltonianĤ(w), that is |Ψ(t) = e −iĤ(w)t |Φ 0 , w i (hereafter we set = 1). Several interesting issues have been investigated within QQ dynamic protocols. They include the long-time relaxation and the consequent spreading of quantum correlations and entanglement, the statistics of the work, localization effects due to the mutual interplay of interactions and disorder, dynamical phase transitions, the dynamic scaling close to quantum transitions, effects of dissipation or of measurements due to interactions with an environment (see, e.g., Refs. [5,9,).
To focus on the out-of-equilibrium dynamics close to a quantum transition, we assume that the Hamiltonian H c in Eq. (1) is critical, thus w = w c = 0 represents a quantum critical point. We recall that the critical behavior around the CQT point w c = 0 is characterized by a diverging length scale ξ ∼ |w| −ν of the quantum critical modes, and the power-law suppression ∆ ∼ ξ −z of the gap. The out-of-equilibrium dynamics at CQTs develops scaling behaviors controlled by the universality class of the quantum transition, for example when the Hamiltonian parameters are slowly varied across the critical regime [5,21,23], and in the case of soft QQ protocols when both the initial and final values of the quenching parameters are such to maintain the system within the critical regime [5,9,59]. In particular, soft QQs require that the energy scale of the QQ [i.e. the difference of the energy Ψ(t)|Ĥ(w)|Ψ(t) of the evolving state |Ψ(t) for t > 0 and the ground state ofĤ(w)] is sufficiently small, i.e. comparable with the energy gap ∆ ∼ L −z of the spectrum at the transition point in finite-size systems.
To study the effects of a thermal bath in the out-of-equilibrium behavior arising from a QQ within the critical regime, we consider two protocols where the thermal baths are involved in different ways: (i) Within the first protocol the thermal bath is used to prepare the system in a finite-temperature Gibbs state, described by the thermal density matrix (hereafter we set the Boltzmann constant k B = 1)
ρ t (w i , T ) = n e −En(wi)/T |Φ n , w i Φ n , w i |,(2)
where |Φ n , w i are the eigenstates ofĤ(w i ). Then the quantum evolution after the quench of the Hamiltonian parameters at t = 0 is unitary and driven by the Hamilto-nianĤ(w) only, i.e., the thermal bath is removed during the quantum evolution for t > 0. Therefore, the evolution of the density matrix is driven by the equation
∂ t ρ(t) = −i[Ĥ(w), ρ(t)], ρ(t = 0) = ρ t (w i , T ). (3)
(ii) In the second protocol the starting point is the same, i.e. the Gibbs state (2), but the thermal bath is not removed after quenching. Therefore, the out-ofequilibrium quantum evolution for t > 0 is not unitary anymore, but it is also driven by the interaction with the thermal bath. Under some conditions, discussed in Refs. [5,[79][80][81][82][83][84], the nonunitary evolution arising from the thermal baths can be described by a Lindbald master equation governing the time evolution of the density matrix of the system, which can be written as
∂ t ρ = L[ρ] ≡ −i Ĥ (w), ρ + γ D T [ρ],(4)
where L is a Liouvillian superoperator, and D T is a dissipative driving whose strength is controlled by the homogeneous coupling γ, playing the role of the decay rate (inverse time scale) associated with the interactions between the system and the bath. The operator D T is assumed to be such that the Lindbald master equation (4) drives the system toward an equilibrium Gibbs state at temperature T in the large-time limit. We argue that, for both types of protocols and sufficiently small temperatures of the thermal baths, the out-of-equilibrium time evolution within the critical regime develop a nontrivial out-of-equilibrium FSS (OFSS) limit, with peculiar scaling behaviors, similar to those arising for closed systems. The effects of the thermal baths can be taken into account by appropriate extensions of the out-of-equilibrium zero-temperature scaling laws describing soft quantum QQs within the critical regime of isolated systems, already put forward by earlier works [5,9]. As a theoretical laboratory to check our extended OFSS laws, we consider the quantum Ising chain [4], or the equivalent fermionic Kitaev wire [85], supplemented with a particular modelization of the thermal bath that guarantees the asymptotic thermalization within the Lindblad formulation of the dynamics of open systems with quadratic Hamiltonians [84,86], such as the fermionic Kitaev wire.
Our analyses are developed within FSS frameworks, which generally simplify the study of the universal features of critical behaviors, with respect to studies in the thermodynamic limit. In the FSS limit the general requirement of a large length scale ξ of the critical correlations is not subject to further conditions on the system size L. It only requires that ξ ∼ L, while critical behaviors in the thermodynamic limit requires ξ L. Therefore much larger systems are necessary to probe analogous length scales ξ in the thermodynamic limit. Equilibrium and out-of-equilibrium FSS behaviors are often observed for systems of moderately large size, see e.g. Refs. [5,9,57,87,88]. Thus FSS behaviors should be more easily accessed by numerical computations and experiments where the quantum dynamics can be monitored for a limited number of particles or spins, such as experiments with quantum simulators in laboratories, e.g., by means of trapped ions [89,90], ultracold atoms [91,92], or superconducting qubits [93,94].
The paper is organized as follows. In Sec. II we present the fermionic Kitaev wire, equivalent to the quantum Ising chain, and the model of thermal bath that we use as theoretical laboratory for our study; we also outline the QQ protocols that we consider and define the observables to monitor the quantum evolution after quenching. In Sec. III we outline the out-of-equilibrium scaling scenarios that are expected to be developed under the dynamic QQ protocols considered, and support them by numerical computations for the fermionic Kitaev wires in contact with the thermalizing bath. Finally, in Sec. IV we summarize, draw our conclusions, and add some remarks on the extension of this study to the dynamic Kibble-Zurek protocols slowly crossing quantum critical regimes. The appendix reports some details on the numerical computations for the QQ protocols within fermionic Kitaev wires in contact with a thermal bath.
II. KITAEV FERMIONIC WIRES AND THERMAL BATHS
A. The fermionic Kitaev chain
We consider fermionic Kitaev wires of L sites with open boundary conditions, whose quantum unitary dynamics is driven by the Hamiltonian [85]
H K = −J L−1 x=1 ĉ † xĉx+1 +ĉ † xĉ † x+1 + h.c. − µ L x=1n x , (5)
whereĉ x is the fermionic annihilation operator associated with the site x of the chain,n x ≡ĉ † xĉx is the particle density operator. In the following we assume J as the energy scale, thus we set J = 1.
The Hamiltonian (5) can be mapped into a quantum Ising chain, by means of the Jordan-Wigner transformation, see, e.g., Ref. 4. The corresponding spin model is the quantum Ising chain with open boundary conditions, i.e.Ĥ
Is = − L−1 x=1σ (1) xσ (1) x+1 − g L x=1σ (3) x , (6) σ (k)
x being the Pauli matrices and g = −µ/2. In the following we prefer to stick with the Kitaev quantum wire, because the thermal baths and observables that we consider are best defined within the fermionic model. However, the general scaling scenarios that will emerge apply to both models.
The Kitaev model undergoes a CQT at µ = µ c = −2 (corresponding to g = g c = 1 in the quantum Ising chain), between a disordered quantum phase for µ < µ c (corresponding to g > 1) and an ordered quantum phase for |µ| < |µ c | (corresponding to |g| < 1). Thus, we define
w = µ − µ c = µ + 2,(7)
so that one can easily see the correspondence between the Kitaev Hamiltonian (5) and the generic one reported in Eq. (1), i.e.Ĥ c corresponds to the Hamiltonian (5) for µ = µ c , andĤ p = − L x=1n x . The continuous transition at w = w c belongs to the two-dimensional Ising universality class [4,5], characterized by the length-scale critical exponent ν = 1, related to the RG dimension y w = 1/ν = 1 of the Hamiltonian parameter w. This implies that, approaching the critical point, the length scale ξ of the critical quantum fluctuations diverges as ξ ∼ |w| −ν . The dynamic exponent z = 1 associated with the unitary quantum dynamics can be obtained from the power law ∆ ∼ ξ −z of the vanishing gap with increasing ξ. Moreover, the RG dimension of the fermionic operatorsĉ j andĉ † j at the CQT is y c = 1/2, and that of the particle density operatorn x is y n = 1 [4,5].
B. Modelization of the thermal bath
In our study we consider a modelization of interaction with a thermal bath within the Lindblad master equation (4), whose asymptotic large-time behavior leads to a Gibbs density matrix at a given finite temperature T . In particular, we consider the proposal developed in Ref. [84] which applies to quantum models described by quadratic Hamiltonians, such as that of the fermionic Kitaev wires. This provides a relatively simple modelization of a thermal bath leading to thermalization in the largetime limit of the corresponding Lindblad master equation for the density matrix of the system. The Kitaev Hamiltonian (5) with open boundary conditions can be diagonalized in the Nambu field space by a Bogoliubov transformation, see e.g. Refs. [84,95,96], so that we can rewrite it aŝ
H K = L k=1 ω kb † kb k ,(8)
where ω k are values of the spectrum of the Bogoliubov eigenoperatorsb k (we are neglecting an irrelevant constant term). Note that both ω k andb k depend on the Hamiltonian parameter µ. The relation between the fermionic operatorsĉ x and the Bogoliubov eigenoperatorsb k can be generally written as [84,95,96]
c x = L k=1 A xkbk + B xkb † k ,(9)
where A and B are appropriate L×L matrices depending on µ. Following Refs. [84,86], we write the dissipator D T [ρ] in the Lindblad master equation (4) in terms of the Bogoliubov eigenoperators as
D T [ρ] = k [1 − f (ω k , T )] 2b k ρb † k − {b † kb k , ρ} + k f (ω k , T ) 2b † k ρb k − {b kb † k , ρ} ,(10)
where
f (ω k , T ) = 1 + e ω k /T −1 .(11)
When using this homogeneous dissipator term, the Lindblad master equation (4) ensures the asymptotic largetime thermalization [84]. Therefore,
lim t→∞ ρ(t) = ρ t (w, T ),(12)ρ t (w, T ) = n e −En(w)/T |Φ n , w Φ n , w|,(13)
where ρ t (w, T ) is the density matrix representing the thermal state, E n (w) and |Φ n , w are the eigenvalues and eigenstates ofĤ(w). The asymptotic approach to the thermal distribution is controlled by the decay-rate parameter γ [84]. Indeed the Liouvillian gap ∆ L that controls the exponential approach to the asymptotic stationary state of the Lindblad equation is proportional to the decay rate γ, i.e.
∆ L ∼ γ.(14)
The above modelization of thermal baths provides a useful theoretical laboratory to investigate issues related to the out-of-equilibrium dynamics in the presence of thermal baths. Its derivation has been thoroughly discussed in Ref. [84]. We also mention that it has been employed in Refs. [86,97]. Some details of the computations using the Lindblad master equation (4) with the dissipator (10) are reported in the appendix.
C. Quantum-quench protocols
As already anticipated in Sec. I, we consider two protocols, differing for the absence or presence of the contact with the thermal bath during the quantum evolution after quenching, giving respectively rise to unitary or dissipative dynamics after quenching. We call them unitary and dissipative QQ protocols, respectively.
• Unitary QQ protocol: In this simplest QQ protocol the role of the thermal bath is limited to that of preparing the initial Gibbs state ρ t (w i , T ) at t = 0, reported in Eq. (2). This can be obtained by keeping the thermal bath in contact with the system for a sufficiently long time t th , i.e t th γ −1 . Then at t = 0 the Hamiltonian parameter is instantaneously quenched from w i < 0 to w ≥ 0 and the thermal bath is removed, so that the subsequent time evolution is that of a closed fermionic wire, i.e. it is unitary and only driven by the Hamiltonian of the system, cf. Eq. (3).
• Dissipative QQ protocol: The quantum evolution starts from the same initial Gibbs state ρ t (w i , T ), but the thermal bath is maintained in contact with the system after the QQ from w i < 0 to w ≥ 0, at t = 0. Therefore, the quantum evolution for t > 0 is driven by the Lindblad master equation (4) with the dissipator term (10). Note that this dynamic protocol entails a further time scale τ = γ −1 , characterizing the asymptotic exponential approach to the large-time stationary Gibbs state associated with the HamiltonianĤ(w) and temperature T .
D. Observables monitoring the time evolution
To characterize the dynamic properties of the quantum evolution after the QQ at t = 0, we consider the subtracted particle-density average
n s (t, L) = 1 L Tr ρ(t) L x=1n x − n c (L),(15)
where n c (L) is the ground-state energy density of the Kitaev wire of size L at the critical point w c = 0 (in the infinite-size limit n c = 1/2 − 1/π [95]). Note that the particle density operatorn x and the transverse spin componentσ (3) x of the quantum Ising chain (6) are trivially related, indeedσ
(3) x = 2n x .
In the definition of n s , the subtraction of n c (L) simplifies the scaling behavior of n s (t, L) within the critical regime, cancelling the leading analytical behavior [5,24]. To monitor the spatial correlations, we also consider
P (x, y, t) = Tr[ρ(t) (ĉ † xĉ † y +ĉ yĉx )],(16)C(x, y, t) = Tr[ρ(t) (ĉ † xĉy +ĉ † yĉx )].(17)
Some details on the computation of the above quantities during the time evolution of the QQ protocols are reported in the appendix.
III. OUT-OF-EQUILIBRIUM SCALING
We now discuss the out-of-equilibrium behaviors arising from the QQ protocols outlined in Sec. II C. We show that they develop OFSS behaviors where the effects of the thermal baths are taken into account by appropriate extensions of the out-of-equilibrium zerotemperature scaling laws describing soft QQs in closed systems within their critical regime, already put foward by earlier works [5,9]. 4) and (10). These curves refer to a system of size L = 60, temperature T = 2 of the thermal bath, quenching from wi = −0.01 to w = 0, and various values of the decay rate γ (the case γ = 0 corresponds to the evolution of the close system). We plot the difference ns(t, L, T ) − ns,eq(L, T ) which is expected to vanish in the large-time limit. In this figure and in the following ones, the unity that we use are such that = 1, kB = 1, and J = 1.
The OFSS behaviors that we put forward for QQ protocols considered are verified by numerical computations for the fermionic Kitaev wire up to relatively large sizes. See the appendix for details on such calculations.
As a preliminary example of out-of-equilibriun QQ behaviors that we want to address, in Fig. 1 we show some results for the quantum evolution of the subtracted particle density (15) along the dissipative protocol outlined in Sec. II C, after quenching a fermionic Kitaev wire of size L = 60, from w i = −0.01 to w = 0, in the presence of a thermal bath at a temperature T = 2, and various values of the decay rate γ. The quantum evolution turns out to have a significant dependence on the decay-rate parameter γ that characterized the interactions between the system and the thermal bath. Indeed, the curves of the substracted particle density appear to approach its equilibrium value n s,eq (w = 0, T = 2) ≈ 0.0004601... (while at t = 0 we have n s,eq (w = w i , T = 2) ≈ 0.126598...), faster and faster with increasing γ, actually exponentially as exp(−t/τ ) with τ ∼ γ −1 , conferming the role of decay rate of the parameter γ within the Lindblad master equation, cf. Eq. (14). Analogous results are obtained for other observables, such as fermionic correlation functions defined in Sec. II D. In the following we put forward an out-of-equilibrium scaling theory for these out-of-equilibrium phenomena within the quantum critical regime.
A. Zero-temperature scaling in quantum quenches
We now provide a brief summary of the out-ofequilibrium scaling theory for close systems, describing QQ protocols within the critical regime [5,9]. The initial state is the ground state associated with an initial value w i < 0, and, after the instantaneous quench at t = 0 from w i to w, the quantum evolution is driven by the Schrödinger equation.
Out-of-equilibrium scaling laws can be obtained by extending those valid at equilibrium, allowing for a time dependence essentially controlled by the time scaling variable Θ ∼ t ∆, which is obtained by assuming that the relevant time scale of the critical modes is proportional to the inverse energy difference ∆ of the lowest states. We refer to Ref. [5] for a through presentation of the scaling arguments leading to the asymptotic OFSS behaviors.
Let us consider the out-of-equilibrium evolution (after quenching) of generic observables, such as the expectation value O at time t of a local operatorÔ(x) and its fixed-time correlations G O = Ô (x)Ô(y) . The general working hypothesis underlying out-of-equilibrium FSS frameworks is that the expectation value ofÔ(x) and its correlation functions obey asymptotic homogeneous scaling laws [5], such as
O(t, x, L, w i , w) ≈ b −yo O(t/b z , x/b, L/b, b yw w i , b yw w),(18)
where b is an arbitrary (large) length scale, y o is the RG dimension of the local operatorÔ x and the RG exponents y w and z are determined by the universality class of the CQT (they are the RG dimensions of the Hamiltonian parameter w and the temperature T , respectively). Thus both the initial and final values of w, i.e. w i and w, take the same RG exponent y w , being coupled to the RG perturbationĤ p within the Hamiltonian. Note that we do not assume translation invariance, which is generally broken by the presence of boundaries, such as those arising from open boundary conditions. OFSS can be straightforwardly derived by fixing b = L in the above homogenous scaling law. Then, we expect the OFSS of the expectation value O of a generic local operatorÔ x , of its spatial averageÔ a = L −d xÔ x , and its two-point correlation function G O , develop the asymptotic OFSS behavior [5,9]
O(t, x, L, w i , w) ≈ L −yo O(Θ, X, Φ i , Φ), O a (t, L, w i , w) ≈ L −yo O a (Θ, Φ i , Φ),(19)G O (t, x 1 , x 2 , L, w i , w) ≈ L −2yo G O (Θ, X 1 , X 2 , Φ i , Φ),
where the scaling variables appearing in the scaling functions O, O a , and G O are defined as
Θ ≡ t L z , X i ≡ x i L , Φ i ≡ L yw w i , Φ ≡ L yw w. (20)
The OFSS limit is obtained in the large-L and large-t limit keeping the above scaling variables fixed. These conditions ensure that the system remains within the universal critical regime during the quantum evolution. Note that in the scaling law (20) the dynamic features are essentially encoded in the time dependence of the scaling variable Θ ∼ t ∆. The other features, in particular when w i = w, are analogous to those arising from equilibrium FSS at CQTs [5,24], where the argument Φ = L yw w of the scaling functions is controlled by the RG dimension y w of the relevant parameters w at the RG fixed point associated with the CQT. The above OFSS equations can be straightforwardly applied to the observables defined in Sec. II D, after a quench from w i to w at t = 0, keeping into account that the RG dimension of the subtracted particle density is y n = 1, and that of the fermionic operatorĉ x is y c = 1/2. Note that the dominant analytical contributions to the particle density [5,24] coming from the analytical background are canceled in the difference n s defined in Eq. (15), whose leading asymptotic behavior arises from the quantum critical modes, therefore it is analogous to that of O a in Eq. (19), with y o = y n . Analogously one can apply the OFSS in Eq. (19) to observables and correlation functions constructed with the spin operators of the quantum spin chain (6). The OFSS functions are expected to be universal with respect to the microscopic details of the model, apart from nonuniversal multiplicative rescaling and normalizations of its arguments. Within isolated fermionc Kitaev wires and quantum Ising chains, the OFSS arising from soft QQs has been verified by numerical computations for various boundary conditions, and also along their quantum first-order transition line [5,9].
The OFSS limit is expected to be approached with power-law suppressed corrections. There are various sources of scaling corrections when approaching the OFSS. Of course, they include those that are already present at equilibrium. In particular, the irrelevant RG perturbations are sources of scaling corrections for the asymptotic behavior of the free-energy density [5,99]. In the case of one-dimensional quantum systems undergoing CQTs belonging to the two-dimensional Ising universality class, the leading scaling corrections from irrelevant RG perturbations are suppressed as L −ω with ω = 2 [24,98]. However, other contributions may become more relevant [5,24,99], such as those arising from the presence of analytical backgrounds, from the presence of boundaries (which generally gives rise to O(1/L) corrections), and, in the case of correlation functions, from RG mixings of the source fields [this for example happens in the case of the correlation functions of the fermionic fieldĉ x , for which corrections are O(1/L)].
These scaling corrections have been confirmed by numerical results [5,24]. Therefore, we expect that the asymptotic OFSS of fermionic Kitaev wires and quantum Ising chains with open boundary conditions is generally approached with O(1/L) corrections.
B. OFSS along the unitary QQ protocol
For the simplest unitary protocol reported in Sec. II C, where the quantum evolution is that of the isolated fermionic wire, the request that the dynamics remains within the critical regime implies that the temperature of the initial Gibbs state must be appropriately suppressed in the large-L OFSS limit, to obtain a nontrivial out-ofequilibrium critical limit. This is analogous to what happens within the equilibrium FSS, where one introduces the scaling variable [3][4][5]
Ξ ≡ L z T,(21)
to allow for a nonzero temperature in the FSS of the observables. Therefore, like equilibrium FSS, we conjecture that the temperature of the initial Gibbs state enters the OFSS associated with the unitary QQ protocol by adding a further dependence on Ξ in the scaling functions (19).
In other words, a nontrivial asymptotic OFSS limit is expected to be realized in the large-L and large-t limits keeping also Ξ fixed, beside the scaling variables already defined in Eq. (20). Therefore, we expect that the OFSS of standard QQ protocols starting from ground states, cf. Eq. (19), changes into (22) and analogously for its spatial average O a and the correlation function G O . The numerical analysis for the fermionic Kitaev wire under the unitary protocol fully support to this OFSS, obtained by extending the QQ FSS behaviors of closed systems starting from an initial ground state. This is clearly demonstrated by the curves reported in Fig. 2, associated with the quantum evolutions of the subtracted particle density n s (t) and the fermionic correlation P (x, y, t) (the other fermionic correlation C(x, y, t) develops an analogous OFSS).
O(t, x, L, w i , w, T ) ≈ L −yo O(Θ, X, Φ i , Φ, Ξ),
C. OFSS along the dissipative QQ protocol
We now discuss the dynamics arising from the dissipative protocol outlined in Sec. II C, when the quantum evolution after quenching is described by the Lindblad master equation (4) with the thermal-like dissipator (10), to modelize the interaction with a thermal bath characterized by a temperature T (which does not change after quenching) and decay rate γ.
We expect that the temperature T of the thermal bath must be rescaled as in the case of the unitary QQ protocol, i.e. we must consider again the associated scaling variable Ξ already defined in Eq. (21). However, since the QQ moves the system out-of-equilibrium, also the decay rate γ, and corresponding time scale τ = γ −1 , associated with the interactions with the thermal bath is expected to play a relevant role to establish a corresponding nontrivial OFSS limit. This was already noted in Ref. [97] in the analysis of dynamic protocols entailing the variation of the temperature at the critical point. When keeping τ constant in the FSS limit where the scaling variable Θ = t/L z is kept fixed, in the large-L limit we have eventually that
t = Θ L z τ,(23)
which is the condition ensuring thermalization for any finite value Θ > 0. Therefore, when keeping τ fixed, the quantum evolution is not expected to develop a nontrivial OFSS limit. Indeed, in the large-L limit, the : Quantum evolution of the subtracted particle density arising from the dissipative QQ protocol, when rescaling all quantities involved in the quench protocol, except for the decay rate γ. With increasing L, the curves appear to approach the equilibrium FSS value at finite temperature (where the temperature dependence enters through the scaling variable Ξ = L z T ) faster and faster, reflecting a nonuniform convergence for any Θ > 0. The dashed line shows the equilibrium value of ns for Φ = 0 and Ξ = 1, which is asymptotically approached by the various curves.
system turns out to suddenly approach an equilibrium Gibbs state (associated with the Hamiltonian parameter w and temperature T ) with respect to the rescaled time Θ, without any further relevant evolution of the system for any Θ > 0. Therefore, if the temperature is rescaled by keeping Ξ = L z T fixed, we must recover the equilibrium FSS behavior in the presence of a thermal bath at temperature T , such as that associated with the subtracted particle density [5,24] n s,eq (w, L, T ) ≈ L −yn N (Φ, Ξ),
where Φ = L yw w, and the temperature dependence enters through the associated scaling variable Ξ = L z T . In Fig. 3 we show some equilibrium data at the critical point w = Φ = 0, versus Ξ, showing the approach to the asymptotic large-L equilibrium FSS (24). The realization of the equilibrium FSS within the QQ protocol at fixed γ is demonstrated by the plots reported in Fig. 4, which show the somewhat trivial convergence toward the equilibrium FSS for any finite Θ > 0.
The above results suggest that also the the decay rate γ of the system-bath interactions must be rescaled to observe a nontrivial OFSS limit as a function of the time scaling variable Θ, to create the conditions for a balanced competition between the critical Hamiltonian driving and the interactions with the thermal bath. As already put forward in the case of other homogeneous dissipative terms in the Lindblad equation [5,55,[100][101][102], for example associated with particle-decay or particlepumping dissipative mechanisms, a nontrivial OFSS limit is obtained by rescaling the decay rate of the dissipative term, so that the scaling variable
Γ ≡ L z γ ∼ γ/∆(25)
is kept fixed in the OFSS limit, where ∆ is the energy difference of the lowest eigenstates ofĤ(w) at the critical point w = w c = 0. Then an OFSS behavior emerges from the nontrivial competition between the critical unitary dynamics and the dissipative driving arising from the thermal bath.
In conclusion, on the basis of the above scaling arguments, the OFSS arising from the dissipative QQ protocols in the presence of a thermal bath is expected to be given by (26) and
O a (t, L, w i , w, T, γ) ≈ L −yo O a (Θ, Φ i , Φ, Ξ, Γ),G O (t, x 1 , x 2 , L, w i , w, T, γ) ≈ (27) L −2yo G O (Θ, X 1 , X 2 , Φ i , Φ, Ξ, Γ).
In the large-Γ limit the above OFFS behaviors at fixed Ξ is expected to approach the corresponding equilibrium FSS, faster and faster in terms of Θ, matching the behavior at finite γ. Moreover, we also expect that the equilibrium FSS is also approached in the large-Θ limit at fixed Γ and Ξ, independently of Γ, but faster and faster with increasing Γ.
Again, the numerical results for the particle density n s (t) and correlation functions P and C fully support the above OFSS equations, i.e. Eq. (26) for n s (t) with y o = y n = 1, and Eq. (27) for P and C with y o = y c = 1/2. Some results are reported in Fig. 5. We also stress that analogous results are expected for other observables, for example the correlation functions of the spin operator of the equivalent formulation provided by the quantum Ising chains.
IV. CONCLUSIONS
We have reported a study of the effects of thermal baths to the out-of-equilibrium dynamics of many-body systems within their quantum critical regime close to a zero-temperature CQT. In particular, we analyze the outof-equilibrium quantum evolution arising from QQs of the Hamiltonian parameters within two different protocols involving a thermal bath coupled homogeneously to the system. Within the first protocol, named unitary QQ protocol, the thermal bath is used to prepare the system at t = 0 in a finite-temperature Gibbs state, then the dynamics after quenching of the Hamiltonian parameters is assumed unitary, i.e., the thermal bath is removed during the quantum evolution for t > 0. The second protocol, named dissipative QQ protocol, starts from the same initial condition, but the thermal bath is not removed after quenching, and the quantum evolution for t > 0 is assumed to be described by the Lindblad master equation (4). The dissipative term of the Lindblad equation is supposed to simulate a thermal bath, such that the manybody system is driven to a large-time finite-temperature Gibbs state. This dissipative protocol is characterized by a further time scale τ = γ −1 , related to the decay rate of the interactions between the system and the bath.
Within OFSS frameworks, we argue that, when the thermal baths are associated with a sufficiently small temperature, their effects can be taken into account by appropriate extensions of the zero-temperature out-ofequilibrium scaling laws describing soft QQs of isolated systems within the critical regime. For the unitary QQ protocol, where the thermal bath only determines the initial Gibbs state and the evolution is unitary, a nontrivial OFFS limit is simply obtained by rescaling the temperature as T ∼ L −z , similarly to equilibrium FSS. Along the dissipative QQ protocol, where the thermal bath is not removed after quenching, the dynamics is more complicated, and the decay rate γ plays a relevant role. Indeed, in addition to the rescaling of the temperature T associated with thermal bath, one also needs to rescale γ as γ ∼ L −z to obtain a nontrivial OFSS. Otherwise, when keeping γ fixed, the dynamics converges toward the equilibrium FSS at finite temperature, which happens suddenly after quenching with respect to the time scale t c ∼ L z of the critical regime. Therefore the scaling behavior when keeping γ fixed becomes somehow trivial, reproducing the equilibrium FSS for any rescaled time Θ = L −z t > 0 in the large-L limit.
Our scaling arguments are supported by numerical results with the paradigmatic fermionic Kitaev model, or equivalently quantum Ising chain, at its CQT separating quantum disordered and ordered phases. We consider a particular modelization of the thermal bath that guarantees the asymptotic thermalization within the Lindblad formulation of the dynamics of open systems. However, we note that the scaling arguments used to arrive at the OFSS laws for critical QQs are general, and therefore we expect that the emerging out-of-equilibrium scenar-ios also apply to many-body systems at generic CQTs in contact with homogenous thermal baths, in any dimension.
We finally remark that the out-of-equilibrium scaling arguments we put forward, leading to the OFSS of QQs in the presence of a thermal bath, can be extended to other protocols giving rise to out-of-equilibrium dynamics. Another interesting class of dynamic protocols entails slow variations of the Hamiltonian parameters across the critical regime of a quantum transition, such as those associated with the quantum Kibble-Zurek (KZ) problem (see e.g. Refs. [5,[12][13][14][15][16][17][18][19][20][21][22][23]). In standard KZ protocols starting from the ground state for an initial parameter w i < 0, the out-of-equilibrium quantum evolution arises from the linear time dependence of one Hamiltonian parameter, w(t) = t/t s in Eq. (1), where t s is the time scale of the KZ protocol. Since w(t) crosses the critical point at t = 0, the system passes through the quantum critical regime, moving it away from equilibrium even in the large-t s limit, and developing a peculiar out-of-equilibrium scaling behaviors. In particular, the interplay between the size L of the system and the time scale t s of the protocol develops OFSS behaviors [5,23] when t s → ∞ and L → ∞, keeping the scaling variables Ω t ≡ t/t κ s = t/t z/(yw+z) s and Υ ≡ t s /L yw+z (thus Ω t = t/t 1/2 s and Υ = t s /L 2 for the fermionic Kitaev wire or quantum Ising chain) fixed.
KZ-like protocols can be also extended to systems interacting with a thermal bath, such as that outlined in Sec. II B, starting from a Gibbs state for an initial w i < 0 and the temperature T of the thermal bath. Then we may consider a time evolution driven by the Lindblad master equation (4), with a time-dependent Hamiltonian H[w(t)] and the dissipator term (10), where also the Bogoliubov operators are assumed to be time dependent to adapt themselves to the time dependence of w. Analogously to the OFSS of QQs in contact with thermal baths, to define a nontrivial OFSS limit in KZ protocols, we expect that both the temperature T and the decay rate γ associated with the bath must be rescaled, as T ∼ L −z and γ ∼ L −z . If only the temperature of the thermal bath is rescaled as T ∼ L −z , while γ > 0 is kept fixed, the time interval associated with a variation of Ω t in the KZ scaling limit, i.e. ∆ Ω t ∼ t κ s ∆Ω t , becomes eventually much larger than the time scale τ ∼ γ −1 of the interaction with the thermal bath. Since τ /∆ Ω t → 0 in the KZ limit, the system effectively thermalizes at each rescaled time Ω t . Therefore, in the KZ limit the quantum evolution is expected to pass through equilibrium finite-temperature states, thus effectively resulting into adiabatic evolutions reproducing the equilibrium finitetemperature FSS as a function of L yw w(t). Therefore, like dissipative QQ protocols, the observation of a nontrivial OFSS in KZ protocols requires the simultaneous rescaling of the time scale τ associated with the interaction with the thermal bath. The necessary rescaling of the decay rate γ of the dissipative term in the Lindblad master equation has been also put forward for KZ pro-tocols in the presence of other dissipative mechanisms, such as those related to particle decay or pumping [100].
The dynamics of the system in contact with the thermal bath described by the Lindblad master equation (4) with the dissipator term (10) leads to thermal states, such as those described by the density matrix reported in Eq. (13). To compute the correlation functions of the fermionic operatorsĉ x in thermal states of the Hamilto-nianĤ(w), one can use the relation with the Bogoliubov eigenoperatorsb k , cf. Eq. (9), and the thermal correlations of the Bogoliubov operatorsb k , i.e.
b † k b q ≡ Tr[ρ t (w, T )b † k b q ] = δ kq 1 + e ω k /T ,(A1)
corresponding to the standard Fermi-Dirac distribution function. Note also that the other correlations b k b q and b † k b † q vanish. Then the correlation functions of the original fermionic fieldĉ x can be straightforwadly obtained from Eq. (9).
Computations for the unitary protocol
In the unitary QQ protocol, one starts from a Gibbs state associated with the Hamiltonian parameter w i and the temperature T , then at t = 0 one instantaneously changes w i → w and removes the contact with the thermal bath. Therefore the quantum evolution is unitary, described by the Schrödinger equation (3). One may easily obtain closed equations for the evolution of the correlation functions C and P defined in Eqs. (16) and (17).
We introduce the correlations C x,y = Tr ρ(t)ĉ † xĉy , P x,y = Tr ρ(t)ĉ † xĉ † y , (A2) whose quantum evolution can be written as dC x,y dt = i C x,y+1 − C x−1,y + C x,y−1 − C x+1,y − −i P † y,x−1 − P † y,x+1 +i P x,y−1 − P x,y+1 , (A3) dP x,y dt = −i P x,y+1 + P x+1,y + P x,y−1 + P x−1,y − − 2 i µ P x,y − i δ x−1, y − δ x+1, y − − i C x,y−1 − C y,x−1 − C x,y+1 + C y,x+1 . (A4)
The initial conditions are easily obtained by the relations with the thermal correlations of the Bogoliubov operators associated with the initial Gibbs state. Then the fermionic correlation function are obtained by C(x, y, t) = 2 ReC x,y (t), P (x, y, t) = 2 ReP x,y (t). (A5)
The above differential equations are solved using the fourorder Runge-Kutta method. The particle density is obtained from the data of C x,x = Tr ρ(t)ĉ † xĉx .
Computations for the dissipative protocol
For the dissipative QQ protocol, where the thermal bath is kept in contact with the system, the evolution is driven by the Lindblad master equation (4), which can be equivalently written in terms of the time dependence of Heisenberg operatorsÔ H (t), i.e. [84,86]:
∂ tÔH (t) = i Ĥ (w),Ô H (t) + γ D T [Ô H (t)],(A6)
where
D T [Ô H (t)] = k f (ω k ) 2b † kÔ H (t)b k − Ô H (t),b kb † k + k (1 − f (ω k )) 2b kÔH (t)b † k − Ô H (t),b † kb k ,(A7)
whereb k are the Bogoliubov operators associated with the HamiltonianĤ(w). The initial state at t = 0 is the Gibbs state for the Hamiltonian parameter w i . This state corresponds to the steady state solution of the Eq. (A6) withĤ(w i ). Then, the change of the Hamiltonian parameter to w = w i leads to a change of the Bogoliubov operators diagonalizing the Hamiltonian. We call {b k } the operators which diagonal-izesĤ(w),Ĥ
(w) = L k=1 ω kb † kb k ,(A8)
where {ω k } is the Bogoliubov spectrum associated witĥ H(w). To evaluate the correlations of the Bogoliubov operatore {b k }, one can solve the Eq. (A6) for couples of operators {b k }, obtaining [84] b
† k b k = (1 − e −2γt )f (ω k ) + e −2γt b † k b k 0 , b † k b q = e i(ω k −ω q )t−2γt b † k b q 0 , b † k b † q = e i(ω k +ω q )t−2γt b † k b † q 0 , b k b q = e −i(ω k +ω q )t−2γt b k b q 0 .(A9)
The initial values b † k b q 0 of the correlations is computed on the initial Gibbs state associated with w i , and it can be obtained using the relations between {b k } to {b k }. This relation can be formally derived as follows [84]. Introducing the fermionic Nambu field C † = (ĉ † 1 , ...,ĉ † L ,ĉ 1 , ...,ĉ L ), their relations with the Bogoliubov operators B(w) † = (b † 1 , ...,b † L ,b 1 , ...,b L ) corresponding to the HamiltonianĤ K (w) are obtained by a unitary transformation, C = T(w)B(w). See e.g. Ref. [84] for more details. Therefore one can formally derive the relation between the Bogoliubov operatorsb k andb k , corresponding to the Hamiltonian parameters w i and w respectively, from the general relation B(w 2 ) = T(w 2 ) † T(w 1 )B(w 1 ).
(A10)
Finally, to compute the time-dependent observables defined in Sec. II D, one can use the relations between the fermionic correlation functions associated withĉ x and those of the Bogoliubov operatorsb k , such as
C(x, y) = L k,q=1 A * xk A yq b † k b q + B * xk B yq b k b † q +A * xk B yq b † k b † q + B * xk A yq b k b q (A11)
where A and B are the matrices entering Eq. (9).
FIG. 1 :
1The quantum evolution of the subtracted particle density ns(t), cf. Eq. (15), for the dissipative QQ protocol entailing a dissipative dynamics after the QQ at t = 0 of the Hamiltonian parameter w, describing the persistent interaction with the thermal bath, cf. Eqs. (
FIG. 2 :
2OFSS behavior of the subtracted particle density (bottom) and the fermionic correlation function P (x = L/3, y = 2L/3, t), cf. Eq.(16), arising from the unitary QQ protocol, for various lattice sizes L, at fixed Ξ = L z T = 1, Φi = L yw wi = −1 and Φ = L yw w = 0, versus the time scaling variable Θ = t/L z . These computations nicely support the OFSS behaviors reported in Eq.(22). The inset of the bottom figure shows that the approach to the OFSS limit is consistent with O(1/L) corrections. Analogous results are obtained for other values of the scaling variables.
FIG. 3 :
3Equilibrium FSS of the subtracted particle density ns,eq at the critical point w = 0, versus the rescaled temperature Ξ = L z T . With increasing L, the data show the expected convergence to the equilibrium FSS reported in Eq. (24) with yn = 1.
FIG. 5 :
5Quantum evolutions along the dissipative protocol, fully supporting the OFSS reported in Eqs.(26) and(27). We report curves for L ns (bottom), L P (x = L/3, y = 2L/3, t) (middle), andC(x = L/3, y = 2L/3, t) (top),for various values of L, at fixed Φi = −1, Φ = 0, Ξ = 1, and two values of Γ = L z γ, i.e. Γ = 1, 10 (except for the top figure where we only report data for Γ = 10 to ensure a good readability). The inset of the top figure shows that the OFSS is approached with O(1/L) corrections. Analogous results are obtained for other values of the scaling variables.
. Asymptotic thermal states
AcknowledgmentsWe thank Giulia Piccitto and Davide Rossini for interesting and useful discussions.Appendix A: Details on the computationsIn this section we provide some details of the computations for the fermionic Kitaev wire in the presence of a thermal bath.
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| {'fraction_non_alphanumeric': 0.06369361788928836, 'fraction_numerical': 0.030304189296004487, 'mean_word_length': 4.143540983606558, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 14, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We address the out-of-equilibrium dynamics arising from quantum-quench (QQ) protocols (instantaneous changes of the Hamiltonian parameters) in many-body systems within their quantum critical regime and in contact with (homogeneously coupled) thermal baths. We consider two classes of QQ protocols. In one of them the thermal bath is used to prepare the initial finite-temperature Gibbs state; then, after quenching, the thermal bath is removed and the dynamics of the system is unitary. We also address a more complex QQ protocol where the thermal bath is not removed after quenching, thus the quantum evolution is also driven by the interaction with the bath, which may be described by appropriate master equations for the density matrix of the system, where a further relevant time scale, or inverse decay rate, characterizes the system-bath coupling. Under these QQ protocols, the critical system develops out-of-equilibrium scaling behaviors, which extend those for isolated critical systems, by introducing further scaling variables proportional to the temperature of the thermal bath and the decay rate of the system-bath interactions. These out-of-equilibrium scaling behaviors are checked by analyzing QQ protocols within fermionic Kitaev wires, or equivalently quantum Ising chains, supplemented with a particular modelization of thermal bath that guarantees the asymptotic thermalization within the Lindblad master equation for the dynamics of open systems.', 'arxivid': '2305.05494', 'author': ["Francesco Tarantelli \nDipartimento di Fisica dell'Università di Pisa and INFN\nDipartimento di Fisica dell'Università di Pisa\nLargo Pontecorvo 3, Largo Pontecorvo 3, II-56127, 56127Pisa, PisaItaly, Italy\n"], 'authoraffiliation': ["Dipartimento di Fisica dell'Università di Pisa and INFN\nDipartimento di Fisica dell'Università di Pisa\nLargo Pontecorvo 3, Largo Pontecorvo 3, II-56127, 56127Pisa, PisaItaly, Italy"], 'corpusid': 258564873, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 24623, 'n_tokens_neox': 21648, 'n_words': 12788, 'pdfsha': '13a25028a4f40606de2a56db53281dbc446da785', 'pdfurls': ['https://export.arxiv.org/pdf/2305.05494v2.pdf'], 'title': ['Thermal-bath effects in quantum quenches within quantum critical regimes Ettore Vicari', 'Thermal-bath effects in quantum quenches within quantum critical regimes Ettore Vicari'], 'venue': []} |
arxiv |
ELEMENTARY PROOFS OF REPRESENTATION BY TERNARY QUADRATIC FORMS
Benjamin Rainear
Katherine Thompson
ELEMENTARY PROOFS OF REPRESENTATION BY TERNARY QUADRATIC FORMS
arXiv:2206.00589v1 [math.NT] 1 Jun 2022
Mordell in 1958 [15]gave a new proof of the three squares theorem. Those techniques were generalized by Blackwell, et al., in 2016 [1] to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries". We continue the generalization of these techniques to four additional forms.
Introduction
The theory of quadratic forms has a long and rich history. Of particular interest is the question of representation of an integer by a form. In studying universal and almost universal positive definite forms (which, in particular, concerns four or more variables), knowing which integers are represented by ternary subforms not only is key from both theoretical and computational purposes, but also is a delicate and nontrivial matter. The four-squares theorem of Lagrange appeared in 1770, and the three-squares theorem of Legendre did not appear until 1797. And yet, assuming the three-squares theorem, the proof of the four squares theorem is at most a few lines. Even much more recent results such as the 451 paper of Rouse [18], which gives conditions under which quadratic forms are guaranteed to represent all odd positive integers, makes assumptions about certain ternary subforms-conditional on the Generalized Riemann Hypothesis; in considering the 24888 escalators Rouse used knowledge of ternary subforms to handle 9812 of these cases. Among these was a form where, if instead one took a more standard analytic approach and considered its corresponding theta series, would have required looking at a space of modular forms where the cuspidal subspace was 2604 dimensional.
This project is heavily influenced by recent work of the second author in Blackwell et al. [1]. Many of the results in that paper were not new; however, it was the technique that was unique. Concentrating mainly on the Ramanujan-Dickson ternaries (which, in particular, were ternary forms of determinant at most 10), the authors showed which positive were represented by certain positive definite ternary forms. Proving what fails to be represented by a quadratic form is typically simple and straightforward; it is proving that m ∈ N is represented that is challenging. The authors began with a generic quadratic form of determinant D. They then showed a series of congruence conditions simultaneously held which guaranteed that the form in question represented a particular m ∈ N, and also represented values that inequivalent forms of the same determinant D failed to represent.
As noted in the abstract, this paper begins by considering ternary forms not handled in [1]; indeed, the forms considered here are not as famous as the Ramanujan-Dickson forms and the following representation results do not seem to appear anywhere else in the literature. More crucially, we consider determinants D much higher than those considered before, thus forcing many more candidate forms to be simultaneously eliminated. Last, we note that in [1] those excepted values by the other candidates happened to lie in the same congruence class, a congruence class which in turn had little to do directly with the determinant of the form. That is not the case here. The excepted values of the other candidates of determinant D are of the form Dk, where k is a quadratic nonresidue modulo D. While in [1] it was designed so that x = 1, y = z = 0 would be a vector so that Q(x, y, z) produces an excepted value, that is not possible here because of what specifically is not represented by the other forms of determinant D. Therefore, the arguments for specific vector evaluation and subsequent elimination of other candidate forms are much more intricate.
Our main results are: Theorem 1. A positive integer m is represented by 2x 2 +2xy+2xz+2y 2 +2yz+3z 2 if and only if m 4 k (8ℓ+1).
Theorem 2.
A positive integer m is represented by x 2 + 2y 2 + 2yz + 6z 2 if and only if m 4 k (8ℓ + 5).
Theorem 3. A positive integer m is represented by x 2 + 3y 2 + 2yz + 5z 2 if and only if m 4 k (16ℓ + 2). Theorem 4. A positive integer m is represented by 2x 2 + 2xy + 3y 2 + 2yz + 5z 2 if and only if m 4 k (8ℓ + 1).
The remainder of the paper is organized as follows: after a brief but more detailed background section, we proceed to the proofs of the theorems in order. As these proofs are constructive, we end with concrete examples.
Background
For general references on the theory of quadratic forms, we refer the reader to [3] and [13].
A n-ary quadratic form over Z is a polynomial Q : Z n → Z given by
Q( x) = 1≤i≤ j≤n a i j x i x j ∈ Z[x 1 , ..., x n ].
We say a quadratic form is positive definite if Q( x) ≥ 0 for all x ∈ Z n and if Q( x) = 0 if and only if x = 0.
To each quadratic form there is an associated symmetric matrix A Q ∈ M n 1 2 Z whose entries c i j are given by
c i j = a i j , i = j a i j /2, i j.
With this, we note that x T A Q x = Q( x). When A Q ∈ M n (Z) we say that Q is classically integral. We say that the determinant det(Q) = det(A Q ). Last, we say that two forms Q and Q ′ are equivalent if there exists M ∈ S L n (Z) such that M t A Q M = A Q ′ .
From now on, when we use the word "form" we mean "ternary, positive definite, classically integral quadratic form."
In [1] the main idea was as follows: to show that some square-free m ∈ N is represented by a form R of determinant D, suppose mR(x, y, z) = (Ax + By + mz) 2 + ax 2 + 2hxy + by 2 .
If one can show all of the following conditions hold for some integers A, B, h, b, a:
• ab − h 2 = Dm; • A 2 + a ≡ B 2 + b ≡ 2AB + 2h ≡ 0 (mod m); • −Dm a = 1 = −a p
where p|m is prime;
• R( x) = k where k ∈ N is not represented by forms Q ′ of determinant D with Q ′ not equivalent to Q, then R must be equivalent to Q. We continue to use this same basic approach. However, in [1] simplicities were made that can no longer be afforded. Namely, they took b ≡ B ≡ h ≡ 0 (mod m), and they made A 2 +a m = k (therefore, x = (1, 0, 0)). Here, we do not (necessarily) have b ≡ B ≡ h ≡ 0 (mod m), and moreover, we have (with one exception) x = (1, 1, 0).
We note that the choice of b ≡ B ≡ h ≡ 0 (mod m) in [1] was to make 2AB + 2h ≡ 0 (mod m) immediate. One key realization here is that once A and B have been determined so that A 2 + a ≡ B 2 + b ≡ 0 (mod m), even if A, B 0 (mod m), one still can ensure 2AB + 2h ≡ 0 (mod m) and in fact that AB + h ≡ 0 (mod m). For a prime p|m:
(AB + h) 2 ≡ A 2 B 2 + 2ABh + h 2 (mod p) (AB + h) 2 ≡ h 2 + 2ABh + h 2 (mod p) (AB + h) 2 − 2h(h + AB) ≡ 0 (mod p) (AB + h)(AB − h) ≡ 0 (mod p).
And so, AB + h ≡ 0 (mod p) can be ensured for all p|m, which by the Chinese Remainder Theorem guarantees AB + h ≡ 0 (mod m) has a solution.
3. Proof of Theorem 1
In this section, we provide a proof of Theorem 1, noting that there are three forms of determinant 7: Q 1 : 2x 2 + 2xy + 2xz + 2y 2 + 2yz + 3z 2 , Q 2 : x 2 + y 2 + 7z 2 , Q 3 :
x 2 + 2y 2 + 4z 2 + 2yz. Lemma 1. For any m ≡ 1 (mod 8), Q 1 does not represent m.
Proof. This is a simple exercise left to the reader.
Lemma 2. If m is odd, 4m is represented by Q 1 if and only if m is.
Proof. One direction is trivial. So suppose 4m is represented, where m is odd. Note that then 4m ≡ 4 (mod 8). By a computer search, one can determine that this forces all of x, y, z to be even. Substituting
x = 2X, y = 2Y, z = 2Z we have 4m = 4(2X 2 ) + 4(2XY) + 4(2XZ) + 4(2Y 2 ) + 4(2YZ) + 4(3Z 2 )
and dividing through, we see m is represented by Q 1 .
Lemma 3. If m = 4 k (8ℓ + 1) for integers k, ℓ, then m is not represented by Q 1 .
Proof. This follows immediately from the previous two lemmas.
Lemma 4. If m = 7n where n ≡ 3, 5, 6 (mod 7), then m is not represented by Q 2 .
Proof. Suppose that Q 2 represents 7n for some n ∈ Z. Then necessarily x ≡ y ≡ 0 (mod 7), and substituting x = 7X and y = 7Y we see
49X 2 + 49Y 2 + 7z 2 = 7n 7X 2 + 7Y 2 + z 2 = n.
This implies that n is a quadratic residue modulo 7.
Lemma 5. If m = 7n where n ≡ 3, 5, 6 (mod 7) then m is not represented by Q 3 .
Proof. A computer search shows that for Q 3 to represent any number divisible by 7, x ≡ 0 (mod 7). Moreover, modulo 7, (y, z) ∈ {±(1, 5), ±(2, 3), ±(3, 1)}. Consider the first case, writing x = 7X, y = 7Y + 1 and z = 7Z + 5:
49X 2 + 2(7Y + 1) 2 + 4(7Z + 5) 2 + 2(7Y + 1)(7Z + 5) = 7n 7X 2 + 14Y 2 + 14Y + 14YZ + 28Z 2 + 42Z + 16 = n.
This implies that n ≡ 2 (mod 7). Similarly, substituting y = 7Y + 2, z = 7Z + 3 we have
49X 2 + 2(7Y + 2) 2 + 4(7Z + 3) 2 + 2(7Y + 2)(7Z + 3) = 7n 7X 2 + 14Y 2 + 14Y + 14YZ + 28Z 2 + 28Z + 8 = n,
which means n ≡ 1 (mod 7). Last, with y = 7Y + 3, z = 7Z + 1 we have
49X 2 + 2(7Y + 3) 2 + 4(7Z + 1) 2 + 2(7Y + 3)(7Z + 1) = 7n 7X 2 + 14Y 2 + 14YZ + 28Z 2 + 14Z + 4 = n,
and n ≡ 4 (mod 7). Now we suppose m 1 (mod 8) is squarefree. We will show m is represented by Q 1 . We proceed by cases.
In the interest of space, and so as not to belabor the reader with repetition, we make note of when cases become identical to those completed in more detail.
(Case 1) Suppose m ≡ 3 (mod 4). We choose a ≡ 1 (mod 4) and a ≡ 3 (mod 49) a prime such that ( −a p ) = 1 for all primes p|m. This guarantees
−7m a = −1 a 7 a m a = 1 = −a m .
Considering the equation ab
−h 2 = 7m, we see that modulo 7, (b, h) ∈ {(0, 0), (3, ±3), (5, ±1), (6, ±2)}.
Switching h with −h as necessary, we can safely assume modulo 7, (b, h) ∈ {(0, 0), (3,4), (5, 1), (6, 2)}. This automatically guarantees there is a solution to
(A + B) 2 + a + b + 2h ≡ 0 (mod 7)
which means that Q will represent multiples of seven. Moreover, for all but the case where (b, h) = (3,4) there are solutions to each of (A + B) 2 + a + b + 2h ≡ 0, 7, 14, 21, 28, 35, 42 (mod 49)
This will guarantee that when x ≡ y ≡ 1 (mod 49) and z = 0, Q will represent 7n where n is a quadratic nonresidue modulo 7 (setting the equation to 21, 35, 42 (mod 42) when m is a quadratic residue (mod 7) and to 7, 14, 28 when m is a quadratic nonresidue (mod 7) ). In the case where (b, h) = (3, 4) there is a solution to which means when x ≡ 2 (mod 49), y ≡ 1 (mod 49) and z = 0 then Q will represent 7n where n is a quadratic nonresidue modulo 7 (with similar restrictions based on m being a quadratic residue (mod 7) or not.). Regardless, in each case A and B are predetermined (mod m), such that
4A 2 + 4AB + B 2 + 4a + b + 4h ≡ 2 2 (A 2 + a) + 2(2AB + 2h) + B 2 + b ≡ 0,A 2 + a ≡ B 2 + b ≡ 2(AB + h) ≡ 0 (mod m). (Case 2) Suppose m ≡ 6 (mod 8). Then m = 2m ′ where m ′ ≡ 3 (mod 4). Choose a ≡ 1 (mod 8) and a ≡ 3 (mod 49) to be prime, where additionally ( −a p ) = 1 for all primes p|m ′ . This guarantees −7m a = −1 a 7 a 2 a m ′ a = 1 = −a m .
The rest of this case is identical to (Case 1).
(Case 3) Suppose m ≡ 5 (mod 8). Let a = 2a ′ where a ′ is a prime satisfying a ′ ≡ 26 (mod 49), a ′ ≡ 1 (mod 4) and where for all p|m, ( −2a ′ p ) = 1. This guarantees −7m a ′ = −1 a ′ 7 a ′ m a ′ = 1 = −a m .
Moreover, if a ′ ≡ 26 (mod 49), 2a ′ ≡ 3 (mod 49). That means the rest of this case will reduce to (Case 1). The rest of this case is identical to (Case 1).
Proof of Theorem 2
There are three forms of determinant 11: Q 1 : x 2 + 2y 2 + 2yz + 6z 2 , Q 2 : x 2 + y 2 + 11z 2 , and Q 3 :
x 2 + 3y 2 + 2yz + 4z 2 .
Lemma 6. If m ≡ 5 (mod 8), then Q 1 does not represent m.
Proof. This is a simple proof by exhaustion and is left to the reader. Proof. One direction is trivial. Suppose 4m is represented by Q 1 . Looking (mod 2) this implies x is even.
Looking then (mod 4), we see that y and z cannot both be odd; however, any one of them even forces the other to be even. Writing x = 2X, y = 2Y, z = 2Z and dividing through by 4 gives the result.
Lemma 8. If m = 4 k (8ℓ + 5) then m is not represented by Q 1 .
Proof. This follows immediately from the previous lemmas.
Lemma 9. If m = 11n where n is a quadratic nonresidue modulo 11, then m is not represented by Q 2 .
Proof. Without loss of generality, suppose m is squarefree. Suppose Q 2 represents m = 11n. This immediately forces x ≡ y ≡ 0 (mod 11). Substituting x = 11X and y = 11Y this means 121X 2 + 121Y 2 + 11z 2 = 11n 11X 2 + 11Y 2 + z 2 = n which means that n is a quadratic residue modulo 11.
Lemma 10. If m = 11n where n is a quadratic nonresidue modulo 11, then m is not represented by Q 3 .
Proof. For Q 3 to represent any multiple of 11, we must have x ≡ 0 (mod 11). Assuming that m is squarefree, this moreover, modulo 11, means (y, z) ∈ {± (1,8), ±(2, 5), ±(3, 2), ±(4, 10), ±(5, 7)}. Again, we proceed by cases. Writing first y = 11Y + 1 and z = 11Z + 8 gives
121X 2 + 3(11Y + 1) 2 + 4(11Z + 8) 2 + 2(11Y + 1)(11Z + 8) = 11n 11X 2 + 33Y 2 + 44Z 2 + 22Y + 66Z + 22YZ + 25 = n
which means that n ≡ 3 (mod 11). Next if y = 11Y + 2, z = 11Z + 5 we have 121X 2 + 3(11Y + 2) 2 + 4(11Z + 5) 2 + 2(11Y + 2)(11Z + 5) = 11n 11X 2 + 33Y 2 + 44Z 2 + 22Y + 44Z + 22YZ + 12 = n which means n ≡ 1 (mod 11). Supposing next y = 11Y + 3 and z = 11Z + 2 we see 121X 2 + 3(11Y + 3) 2 + 4(11Z + 2) 2 + 2(11Y + 3)(11Z + 2) = 11n 11X 2 + 33Y 2 + 44Z 2 + 22Y + 22Z + 22YZ + 5 = n and again n is a quadratic residue modulo 11. When y = 11Y + 4 and z = 11Z + 10 we have 121X 2 + 3(11Y + 4) 2 + 4(11Z + 10) 2 + 2(11Y + 4)(11Z + 10) = 11n 11X 2 + 33Y 2 + 44Z 2 + 44Y + 88Z + 22YZ + 48 = n which makes n ≡ 4 (mod 11). Last, if y = 11Y + 5 and z = 11Z + 7: 121X 2 + 3(11Y + 5) 2 + 4(11Z + 7) 2 + 2(11Y + 5)(11Z + 7) = 11n 11X 2 + 33Y 2 + 44Z 2 + 44Y + 66Z + 22YZ + 31 = n and n ≡ 9 (mod 11).
Next, suppose that m 5 (mod 8) is squarefree. This gives the following cases:
Proof of Theorem 3
There are three forms of determinant 14: Q 1 : x 2 + 3y 2 + 2yz + 5z 2 , Q 2 : x 2 + y 2 + 14z 2 , Q 3 : x 2 + 2y 2 + 7z 2 .
Lemma 11. If m ≡ 2 (mod 16) then m is not represented by Q 1 .
Proof. We leave the proof to the reader.
Lemma 12. Let m be even. Then m is represented by Q 1 if and only if 4m is represented by Q 1 .
Proof. For the nontrivial direction, if m is even, then 4m ≡ 0 (mod 8), which forces x, y, z all even.
Lemma 13. If m = 4 k (16ℓ + 2), then m is not represented by Q 1 .
Proof. This follows from the previous lemmas.
Lemma 14. If m = 7n where n is a quadratic nonresidue modulo 7, then m is not represented by Q 2 .
Proof. Suppose m = 7n is squarefree and is represented by Q 2 . This forces x ≡ y ≡ 0 (mod 7). Substituting x = 7X and y = 7Y and simplifying gives 7X 2 + 7Y 2 + 2z 2 = n which means that n is a quadratic residue modulo 7.
Lemma 15. If m = 7n where n is a quadratic nonresidue modulo 7, then m is not represented by Q 3 .
Proof. Suppose m = 7n is squarefree and is represented by Q 2 . This forces x ≡ y ≡ 0 (mod 7). Substituting x = 7X and y = 7Y and simplifying gives 7X 2 + 14Y 2 + z 2 = n which means that n is a quadratic residue modulo 7.
We now proceed to show that if m 2 (mod 16) is squarefree, then m is represented by Q 1 . Again, we proceed by caess. The proof now follows (Case 1) (Case 4) Let m ≡ 10 (mod 16). Then m = 2m ′ where m ′ ≡ 5, 13 (mod 16). We note that the total number of primes p|m which are congruent to either 5 or 7 (mod 8) is odd. With that, we take a = 2a ′ where a ′ is prime, satisfying a ′ ≡ 1 (mod 8), a ′ ≡ 26 (mod 49), and −2a ′ p = 1 for all p|m ′ . This gives
−14m a ′ = −1 a ′ 2 a ′ 2 7 a ′ m ′ a ′ = 1.
Moreover, we note that 2a ′ ≡ 3 (mod 49), which means next considering 2a ′ b − h 2 = 14m we are reduced to earlier cases.
Proof of Theorem 4
We begin by noting there are five forms of determinant 23: Q 1 : 2x 2 +2xy+3y 2 +2yz+5z 2 , Q 2 : x 2 +y 2 +23z 2 , Q 3 : x 2 + 2y 2 + 2yz + 12z 2 , Q 4 : x 2 + 3y 2 + 2yz + 8z 2 , and Q 5 :
x 2 + 4y z + 2yz + 6z 2 .
Lemma 16. If m ≡ 1 (mod 8) then Q 1 does not represent m.
Proof. Left to reader.
Lemma 17. Let m ∈ N be odd. Then Q 1 represents m if and only if Q 1 represents 4m.
Proof. One direction is trivial, so suppose Q 1 represents 4m where m is odd. Then 4m ≡ 4 (mod 8) and a computer search will verify that in this case each of x, y, z must be even in order for Q 1 (x, y, z) ≡ 4 (mod 8).
Lemma 18. If m = 23n where n is a quadratic nonresidue modulo 23, then m is not represented by Q 2 .
Proof. Considering x 2 + y 2 + 23z 2 ≡ 0 (mod 23) immediately yields x ≡ y ≡ 0 (mod 23). Substituting x = 23X, y = 23Y gives (23X) 2 + (23Y) 2 + 23z 2 = 23n 23X 2 + 23Y 2 + z 2 = n, which means n is a quadratic residue (mod 23).
Lemma 19. If m = 23n where n is a quadratic nonresidue modulo 23, then m is not represented by Q 3 .
Proof. Setting Q 3 (x, y, z) ≡ 0 (mod 23) immediately gives x ≡ 0 (mod 23). There are additional constraints on y and z modulo 23. These cases behave similarly to those in previous sections, and so in the interest of space, we simply provide a summary of the data.
(y (mod 23), z (mod 23)) n (mod 23)
(±1, ±21) 2 (±2, ±19) 8 (±3, ±17) 18 (±4, ±15) 9 (±5, ±13) 4 (±6, ±11) 3 (±7, ±9) 6 (±8, ±7) 13 (±9, ±5) 1 (±10, ±3) 16 (±11, ±1) 12
In each case, n is a quadratic residue (mod 23), which completes the proof.
Lemma 20. If m = 23n where n is a quadratic nonresidue modulo 23, then m is not represented by Q 4 .
Proof. Setting Q 4 (x, y, z) ≡ 0 (mod 23) immediately gives x ≡ 0 (mod 23). There are additional constraints on y and z modulo 23. These cases behave similarly to those in previous sections, and so in the interest of space, we simply provide a summary of the data.
(y (mod 23), z (mod 23)) n (mod 23) (±1, ±20)
3 (±2, ±17) 12 (±3, ±14) 4 (±4, ±11) 2 (±5, ±8) 6 (±6, ±5) 16 (±7, ±2) 9 (±8, ±22) 8 (±9, ±19) 13 (±10, ±16) 1 (±11, ±13) 18
In each case, n is a quadratic residue (mod 23), which completes the proof.
Lemma 21. If m = 23n where n is a quadratic nonresidue modulo 23, then m is not represented by Q 5 .
Proof. Setting Q 5 (x, y, z) ≡ 0 (mod 23) immediately gives x ≡ 0 (mod 23). There are additional constraints on y and z modulo 23. These cases behave similarly to those in previous sections, and so in the interest of space, we simply provide a summary of the data.
(y (mod 23), z (mod 23)) n (mod 23)
(±1, ±19) 4 (±2, ±15) 16 (±3, ±11) 13 (±4, ±7) 18 (±5, ±3) 8 (±6, ±22) 6 (±7, ±18) 12 (±8, ±14) 3 (±9, ±10) 2 (±10, ±6) 9 (±11, ±2) 1
In each case, n is a quadratic residue (mod 23), which completes the proof. Now suppose m 1 (mod 8) is squarefree. We will show that m is represented by Q 1 with the following cases: Considering the equation ab−h 2 = 23m and replacing h with −h as necessary we see that modulo 23, (b, h) ∈ {(0, 0), (5,5), (7,9), (10,2), (11,3), (14,1), (15,11), (17,4), (19,7), (20, 10), (21, 6), (22,8)}. This automatically guarantees there is a solution to
((A + B) 2 + a + b + 2h ≡ 0 (mod 23)
which means that Q will represent multiples of 23. Moreover, for each case there are solutions to each of (A + B) 2 + a + b + 2h ≡ 23k for k = 0, 1, 2, ..., 22. This will guarantee that when x ≡ y ≡ 1 (mod 529) and z = 0 that Q will represent 23n where n is a quadratic nonresidue modulo 23 (with different congruence conditions necessary when m is or is not a quadratic residue (mod 23)). But also, noting that a ≡ 2 · 267 ≡ 5 (mod 529) we are able to mimic previous cases at this point. And as the conditions (mod 23) on a are the same as in (Case 1), the rest of the proof follows similarly.
Examples
We end this paper with hopefully helpful if not entertaining to the reader concrete examples of choices of A, B, a, b, h as outlined in the proofs of the theorems.
Example. To show that m = 51 = 3 · 17 is represented by 2x 2 + 2xy + 2xz + 2y 2 + 2yz + 3z 2 , we consider (Case 1) of the proof of Theorem 1.
We choose a prime a ≡ 1 (mod 4) and a ≡ 3 (mod 49), and (without loss of generality) a ≡ 2 (mod 3) and a ≡ 1 (mod 17). The smallest prime satisfying all of these conditions is a = 4217. Then solving 4217b − h 2 = 7 · 51 for b and h, we see we can take b = 1613 and h = 2608. Note this is not the "smallest" solution with respect to b > 0; however, here b ≡ 3 (mod 7) and h ≡ 4 (mod 7 Q(x, y, z) = 48291x 2 + 64544xy + 3136xz + 21567y 2 + 2096yz + 51z 2 .
Last, we note that Q(2, 1, 0) = 7 · 49117, where as 49117 ≡ 5 (mod 7) means Q represents 7n where n is a quadratic nonresidue modulo 7. We conclude that Q is equivalent to 2x 2 + 2xy + 2xz + 2y 2 + 2yz + 3z 2 .
Example. To show that m = 67 is represented by x 2 + 2y 2 + 2yz + 6z 2 we refer to (Case 3) of the proof of Theorem 2.
We choose a prime a ≡ 1 (mod 4) and a ≡ 2 (mod 121) and noting ( −2 67 ) = 1, we also take a ≡ 2 (mod 67). The smallest prime satisfying all of these conditions is a = 170249. Then solving 170249b − h 2 = 11 · 67 for b and h, we see we can take b = 4413 and h = −27410. The requirements of A 2 + a ≡ B 2 + b ≡ 2AB + 2h ≡ 0 (mod 67), give A ≡ 20 (mod 67) and B ≡ 64 (mod 67), or A ≡ 47 (mod 67) and B ≡ 3 (mod 67). We choose the former. Solving (A + B) 2 + a + b + 2h ≡ 22 (mod 121) gives, among many options, A ≡ 60 (mod 121) and B ≡ 0 (mod 121). We then take A = 4174 and B = 3146. This then means Q(x, y, z) = 262575x 2 + 391164xy + 8348xz + 147787y 2 + 6292yz + 67z 2 .
We note that Q(1, 1, 0) = 801526 = 2 · 11 · 36433, and ( 2·36433 11 ) = −1. Example. To show that m = 26 = 2 · 13 is represented by x 2 + 3y 2 + 2yz + 5z 2 , we consider (Case 4) of Theorem 3.
We choose a prime a ′ ≡ 1 (mod 8) and a ′ ≡ 26 (mod 49) and a ′ ≡ 2 (mod 13). The smallest such prime is a ′ = 27809. Set a = 2a ′ . Considering next the equation ab − h 2 = 14 · 26, we get b = 8440 and h = 21666 as a possible solution. Solving A 2 + a ≡ 0 (mod 26) and B 2 + b ≡ 0 (mod 26) gives A ∈ {10, 16} and B ∈ {6, 20}. Taking into account we must have 2AB + 2h ≡ 0 (mod 26) we see the pairs (A, B) are (10,20) and (16, 6). Because 26 is a quadratic residue mod 7, we next solve for A, B (mod 49) such that (A + B) 2 + a + b + 2h ≡ 7 (mod 35). This gives 98 pairs. One such pair is A ≡ 1 (mod 49) and B ≡ 43 (mod 49). Using the Chinese Remainder Theorem, we take A = 1128 ad B = −6. This yields Q(x, y, z) = 51077x 2 + 1146xy + 2256xz + 326y 2 − 12yz + 26.
Last, we note that when x = y = 1 and z = 0, Q(x, y, z) = 7 · 7507, and 7507 ≡ 3 (mod 7).
Lemma 7 .
7A natural number m ∈ N is represented by Q 1 if and only if 4m is.
(
Case 1) m ≡ 1 (mod 8). We set a = 2a 1 , where a 1 is an prime satisfying a 1 ≡ 1 (mod 4) and a 1 ≡ 1 (mod 121) and ( −a p ) = 1, for all primes p|m. with −h as necessary, we can assume (b, h) (mod 11) + B) 2 = a + b + 2h ≡ 0 (mod 11) which means that Q will represent multiples of 11. Moreover, for each pair (b, h) there are solutions to(A + B) 2 + a + b + 2h ≡ 11k(mod 121)for k = 0, 1, 2, ..., 10. This will guarantee that when x ≡ y ≡ 1 (mod 121) and z = 0, Q will represent 11k where k is a quadratic nonresidue modulo 11 (setting the equation to 11, 33, 44, 55, 99 when m is a quadratic residue (mod 11) and to 22, 55, 66, 77, 88, 110 when m is a quadratic nonresidue (mod 11)). (Case 2) m ≡ 2 (mod 8). We start with writing m = 2ℓ, where ℓ ≡ 1 (mod 4). a to be a prime satisfying a≡ 5 (mod 8), a ≡ 2 (mod 121), and ( −a p ) = 1 for all odd p|m. The rest of the proof then follows (Case 1). (Case 3) m ≡ 3 (mod 4). Here we choose a prime a ≡ 1 (mod 4), a ≡ 2 (mod 121) and ( −a p ) = 1 for all primes p|m. rest of the proof mimics (Case 1). (Case 4) m ≡ 6 (mod 8). We write m = 2ℓ, where ℓ ≡ 3 (mod 4). Here we choose a prime satisfying a ≡ 1 (mod 8), a ≡ 2 (mod 121), and ( −a p ) = 1 for all primes p|m. Then −11m a = −a m = 1 and the rest of the proof follows like the others.
(Case 1 ))
1Suppose m ≡ 1 (mod 4). Choose a prime a such that ( −a p ) = 1 for all p|m, where additionally a ≡ 5 (mod 8) and a ≡ 3 (mod 49). equation to consider now is ab − h 2 = 14m, this now behaves identically to the proof of Theorem 1 (Case 1). (Case 2) Suppose m ≡ 3 (mod 4). Choose a prime a such that ( −a p ) = 1 for all p|m, where additionally a ≡ 1 (mod 8) and a ≡ 3 (mod 49). Let m ≡ 6, 14 (mod 16). Then m = 2m ′ where m ′ ≡ 3, 7, 11, 15 (mod 16). Let a ≡ 1 (mod 8) and a ≡ 3 (mod 49), and ( −a p ) = 1 for all p|m ′ . This is enough to guarantee that
Case 1 )
1Let m ≡ 3 (mod 4). Let a be a prime satisfying a ≡ 1 (mod 4), a ≡ 5 (mod 529), and ( −a p ) = 1 for all primes p|m. This will ensure
(Case 2 )
2Let m ≡ 6 (mod 8), so m = 2m ′ where m ′ ≡ 3 (mod 4). Let a be a prime satisfying a ≡ 1 (mod 8), a ≡ 5 (mod 529), and ( −a p ) = 1 for all primes p|m ′ . the conditions (mod 23) on a are the same as in (Case 1), the rest of the proof follows similarly. (Case 3) Let m ≡ 5 (mod 8). We note that the total number of primes p|m which are congruent to either 5 or 7 (mod 8) is odd. With that, we write a = 2a ′ where a ′ is a prime satisfying a ≡ 1 (mod 4), a ≡ 267 (mod 529) and ( −a p ) = 1 for all p|m.
(Case 4 )
4Let m ≡ 2 (mod 8). Let m ≡ 2 (mod 8), so m = 2m ′ where m ′ ≡ 1 (mod 4). Let a be a prime satisfying a ≡ 5 (mod 8), a ≡ 5 (mod 529), and ( −a p ) = 1 for all primes p|m ′ .
(Case 4) Suppose m ≡ 2 (mod 8). Then m = 2m ′ where m ′ ≡ 1, 5 (mod 8). Let a ≡ 5 (mod 8) and a ≡ 3 (mod 49) be prime with ( −a p ) = 1 for all primes p|m ′ . This guarantees−7m
a
=
−1
a
7
a
2
a
m ′
a
= 1.
). Noting now that A 2 + a ≡ 0 (mod 51) means modulo 51, A ∈ {4, 13, 38, 47}. Similarly with B 2 + b ≡ 0 (mod 51) we see B ∈ {11, 23, 28, 40}. Noting, however, that we must also have 2AB + 2h ≡ 0 (mod 51) we seethe possible pairs of (A, B) modulo 51 are (A, B) ∈ {(4, 11), (13, 23), (38, 28), (47, 40)}. Accounting for 4A 2 + 4AB + B 2 + 4a + b + 4h ≡ 2 2 (A 2 + a) + 2(2AB + 2h) + B 2 + b ≡ 21 (mod 49) gives 98 choices for (A, B) (mod 49). Among these choices is A ≡ 0 (mod 49) and B ≡ 19 (mod 49). Selecting from our (mod 51) conditions A ≡ 38 (mod 51) and B ≡ 28 (mod 51) and using the Chinese Remainder Theorem gives minimum positive values of A = 1568 and B = 1048. This then means
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The Sage Developers. SageMath, the Sage Mathematics Software System. United States Naval Academy, Annapolis, MD921402SageMath, the Sage Mathematics Software System (Version 9.3), The Sage Developers, 2015, http://www.sagemath.org B.Rainear, Department of Mathematics, United States Naval Academy, Annapolis, MD., 21402
. E-Mail Address, B Rainear, [email protected] address, B. Rainear: [email protected]
K Thompson, Department of Mathematics. United States Naval Academy, Annapolis, MD21402K. Thompson, Department of Mathematics, United States Naval Academy, Annapolis, MD., 21402
. E-Mail Address, K Thompson, [email protected] address, K. Thompson: [email protected]
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arxiv |
DUAL PAIRS OF OPERATORS, HARMONIC ANALYSIS OF SINGULAR NON-ATOMIC MEASURES AND KREIN-FELLER DIFFUSION
Palle E T Jorgensen
James Tian
DUAL PAIRS OF OPERATORS, HARMONIC ANALYSIS OF SINGULAR NON-ATOMIC MEASURES AND KREIN-FELLER DIFFUSION
We show that a Krein-Feller operator is naturally associated to a fixed measure µ, assumed positive, σ-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, L 2 (µ) and L 2 (λ), where λ denotes Lebesgue measure. An associated operator pair consists of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. This then yields a rigorous analysis of the corresponding µ-Krein-Feller operator as a closable quadratic form. As an application, for a given measure µ, including the case of fractal measures, we compute the associated diffusion, semigroup, Dirichlet forms, and µ-generalized heat equation.2000 Mathematics Subject Classification. Primary: 47B32, 47B25, 47E05, 46N20. Secondary: 46E22, 46N30, 46N50, 60G15.
Introduction
Recently there have been several advances to an harmonic analysis of Krein-Feller operators for classes of singular measures. (Intuitively, a Krein-Feller operator is an analogue of a Laplacian for classical domains, as they arise in diffusion problems and in potential theory.)
In fact there are recent papers which cover the theory from the point of fractal analysis, see e.g., [Fre08,Fre03,Iy89,LOSS20, ARCG + 20, QS13]; as well as applications to physics and to signal processing, e.g., the papers [AN06,Fle96,Zag87,Iy85,Kas12,Wat98,Fuj87]. Our present approach to Krein-Feller operators is motivated by both of these new trends; but our approach is based on a new duality. It combines a new transform theory based on the theory of reproducing kernel Hilbert spaces (RKHSs), and a new technology introduced here, based on pairs of unbounded densely defined operators, each one contained in the adjoint of the other.
A word about the terminology "Krein-Feller operator" (details are cited inside the paper): Mark Krein has pioneered a number of powerful Hilbert space-based tools which have found numerous applications, and the present problem is a case in point. Krein's operator theory (cited below) forms the foundation in our approach to problems for unbounded operators with dense domain in Hilbert space. Sections 2 and 4 below will elaborate on this. William Feller, in the name "Krein-Feller operator" refers to the role of the KF-operator in the study of diffusion. Indeed, W. Feller was one of the pioneers in our understanding of diffusion, diffusion-semigroups, and their analysis. Hence later authors have adopted the name "Krein-Feller operators" for the associated semigroup generators. There are interesting connections to inverse problems, and prediction theory, see [DM76]. A nice presentation of this, and early work of Krein and Feller, is [DM76,chapter 5]. We shall include additional details on this point in Section 3 below.
Starting with a fixed positive non-atomic Borel measure µ (with support contained in R), then, informally, the associated Krein-Feller operator (denoted K F = K (µ)
F ) is K F = d dµ d dx .
The meaning of "d/dµ" will be made precise. If x > 0, set g µ (x) = µ ([0, x]), i.e., the cumulative distribution. For ψ ∈ C 1 , we have d dµ (ψ • g µ ) = ψ • g µ . A key step in our consideration is a rigorous study of K F as an unbounded (symmetric) operator in L 2 (µ).
Organization: We begin in Section 2 with the framework for our dual pair analysis. This is presented in the rather general setting of pairs of Hilbert spaces, and associated pairs of densely defined (unbounded) operators. Particular choices of dual pairs of operators are then applied to a rigorous analysis of Krein-Feller operators in Section 3. The framework for these considerations is a fixed measure µ, assumed positive, sigma-finite, and non-atomic. Hence, our starting point is a specified and fixed measure µ (generally singular, e.g., a Cantor measure). The two Hilbert spaces for the corresponding dual pair of unbounded operators will then be L 2 (µ) and L 2 (λ), where λ denotes Lebesgue measure, or its restriction to a chosen interval. The rest of Section 3 will deal with an analysis of the associated diffusion, Markov process, semigroup, and a corresponding µ-generalized heat equation. Section 6 studies Stieltjes measures df globally. For this purpose, we introduce a Hilbert space H class of "sigma functions" as a Hilbert space of certain equivalence classes. Starting with a fixed Stieltjes measure df , we then identify its pairwise mutually singular components with corresponding orthogonal "pieces" in the Hilbert space H class .
In a general framework, these settings correspond to suitably specified Dirichlet forms; the subject of Section 4. An application to iterated function system (IFS) measures is also included in Section 4. Our analysis of specific applications relies on several new tools; one in particular derives from consideration of associated reproducing kernel Hilbert spaces (RKHSs), and Gaussian fields. This is covered in Section 5.
Dual pairs of operators in Hilbert space
The notion of dual pairs we shall need here is in Definition 2.6 below. But the operators in question will act between two Hilbert spaces to be specified. Hence, we shall first need to recall some properties of unbounded operators with specified dense domains; especially the precise definition (Def 2.1) of the adjoints of such operators. With this accomplished, the dual pair definition (Def 2.6) for a pair of densely defined operators amounts to the assertion that each operator in the pair be contained in the dual of the other (Lemma 2.7). We shall need this in our analysis of classes of Krein-Feller operators introduced in Section 3 below. For a given measure µ and associated Krein-Feller operator K F , we shall then identify a dual pair which provides a factorization of this Krein-Feller operator K F . Of course, K F is an unbounded operator, symmetric and semibounded; so our dual pair factorization will present us with a canonical selfadjoint extension, see Lemma 2.7. Background references for this include [CP68,DM72,DS88,Fel54,Kat95,Sch12].
Let H 1 and H 2 be complex Hilbert spaces. If H 1 T − − → H 2 represents a linear operator from H 1 into H 2 , we shall denote dom (T ) = {ϕ ∈ H 1 | T ϕ is well-defined} , the range of T . The closure of ran (T ) will be denoted ran (T ).
Definition 2.1. Let T : H 1 → H 2 be a densely defined operator, and let
dom(T * ) = h 2 ∈ H 2 | ∃C = C h2 < ∞, s.t. | h 2 , T ϕ 2 | ≤ C ϕ 1 holds for ∀ϕ ∈ dom (T ) . (2.3)
By Riesz' theorem, there is a unique η ∈ H 1 for which
η, ϕ 1 = h 2 , T ϕ 2 , h 2 ∈ dom(T * ), ϕ ∈ dom (T ) , (2.4)
and the adjoint operator is defined as T * h 2 = η. See the diagram below:
H 1 T ' ' H 2 T * g g Definition 2.2. The graph of T : H 1 → H 2 is G T := ϕ T ϕ | ϕ ∈ dom (T ) ⊂ H 1 ⊕ H 2 , (2.5)
where H 1 ⊕ H 2 is a Hilbert space under the natural inner product
ϕ 1 ϕ 2 , ψ 1 ψ 2 := ϕ 1 , ψ 1 H1 + ϕ 2 , ψ 2 H2 . (2.6) Definition 2.3. Let T : H 1 → H 2 be a linear operator. (1) T is closed if G T is closed in H 1 ⊕ H 2 .
(2) T is closable if G T is the graph of an operator.
(3) If (2) holds, the operator corresponding to G T , denoted T , is called the closure, i.e.,
G T = G T . (2.7)
We shall need the following two results for unbounded operators, see e.g., [DS88,Sch12,Rud91]. To clarify notation, and for the benefit of the reader, we have included them below in the form they are needed.
Theorem 2.4. Let T : H 1 → H 2 be a densely defined operator. Then
(1) T * is closed;
(
2) T is closable ⇐⇒ dom (T * ) is dense; (3) T is closable =⇒ (T ) * = T * .
Theorem 2.5 (von Neumann, polar decomposition/factorization, [DS88]). Let H i , i = 1, 2, be two Hilbert spaces, and let T be a closed operator from H 1 into H 2 having dense domain in H 1 ; then T * T is selfadjoint in H 1 , T T * is selfadjoint in H 2 , both with dense domains. Moreover, there is a partial isometry J :
H 1 → H 2 such that T = J (T * T ) 1 2 = (T T * ) 1 2 J (2.8)
holds on dom (T ). (Equation (2.8) is called the polar decomposition of T .)
Definition 2.6 (symmetric pair). For i = 1, 2, let H i be two Hilbert spaces, and suppose D i ⊂ H i are given dense subspaces.
We say that a pair of operators (S, T ) forms a symmetric pair if dom (T ) = D 1 , and dom (S) = D 2 ; and moreover, T u, v H2 = u, Sv H1 (2.9) holds for ∀u ∈ D 1 , ∀v ∈ D 2 . See the diagram below: [JP16]). Let (S, T ) be the pair of operators specified in (2.9). Then, we have T ⊂ S * , S ⊂ T * (2.10) (containment of graphs.) Moreover, the two operators S * S and T * T are selfadjoint.
H 1 T & & H 2 S f f Lemma 2.7 (Dual Pair
It is immediate from (2.10) that both S and T are closable.
Definition 2.8. We say that a symmetric pair is maximal if T = S * , and S = T * .
With the starting point, a fixed positive non-atomic Borel measure µ with support on an interval, we now show how the operator theoretic framework of dual pairs (Definition 2.6) offers an explicit setting for the study of spectral theory of the associated Krein-Feller operator. In particular, we show in the subsequent sections how key features of our dual pair framework from the discussion above serves to yield explicit new results for the Krein-Feller operator, for example Theorem 3.1, Lemma 3.24, Corollary 3.38.
Krein-Feller operators, and their properties
For a given measure µ we shall offer several tools in our analysis of the associated Krein-Feller operator K F . One will make use of an appropriate dual pair of operators (see Sec 2, and Theorem 3.1 and Corollary 3.38 below). The other is more direct; it is sketched in the present section. In Theorem 3.22, we present the inverse of K F as an explicit integral operator. This will be especially useful in our analysis of the spectrum of diverse selfadjoint extensions of K F . Background references for this include [AJ12, AJL11, AL08, Hid80, IM74]. For basics on Stieltjes measures, fractals, and transformation rules for measures, readers may wish to consult [BP17,DM72,Hut81,Kol83,Nel67,Roh52,Rud91,SZ09,DJ14].
Perhaps it is appropriate to add a comment on the role of W. Feller, in the name "Krein-Feller operator." Feller was one of the pioneers in the study of diffusion, diffusion-semigroups, and their analysis. Hence later authors have adopted the name "Krein-Feller operators" for the associated semigroup generators. We shall elaborate this point in the next section. A list of references which covers this viewpoint is long, but it includes the following, [Fel54,FM56,Yos68].
Terminology convention. Fixing a measure µ as specified, then formally, the notation ∇ µ (see (3.8)) and T µ stand for the same operation, but in the theorem below, we are referring to a specific pair of Hilbert spaces, and the notation T µ is used to stress this point. The conclusion of Theorem 3.1 is that the Krein-Feller operator K F then has a symmetric dual-pair realization in the sense of Definition 2.6.
If f is a function on R (or defined on a subinterval), assumed to be locally of bounded variation, then we shall denote by df the corresponding Stieltjes measure. (Recall df is defined first on intervals (x, y] by df ((x, y]) := f (y) − f (x), and then extended to the Borel σ-algebra B by the usual σ-algebra-completion procedure.) If µ is a fixed positive Borel measure, we then consider the corresponding Radon-Nikodym derivative, denoted
f (µ) = ∇ (µ) f = df /dµ. (3.1) It is determined by, f (y) − f (x) = y x ∇ (µ) f dµ; (3.2) abbreviated ∇ (µ) f dµ = df .
The the Krein-Feller operator K F is defined as
K F = d dµ d dx = ∇ µ d dx . (3.3)
In what follows, we denote by J the unit interval [0, 1].
Theorem 3.1 (A symmetric pair for ∇ µ ). If ϕ ∈ C ∞ c (J) then − ϕ f dx = J ϕ (T µ f ) dµ; (3.4)
so we obtain the dual pair of operators:
L 2 (µ) Tµ ' ' L 2 (µ) D=− d dx g g
Here,
L 2 (µ) := dom (T µ ) L 2 (λ)
, (3.5) i.e., the L 2 (λ)-closure of dom (T µ ); see (3.7).
Proof. One checks that
ϕ df = − ϕ f dx (3.6)
holds for all ϕ ∈ C ∞ c (J), using integration by parts. Details: Let f and ϕ be as specified, ϕ ∈ C 1 c (J), f locally bounded variation s.t. f (µ) = T µ f ∈ L 2 loc (µ) is well defined. For the integral J ϕ (T µ f ) dµ, we therefore get the following approximation via choices of partitions in the interval J:
x 0 < x 1 < · · · : J ϕ (T µ f ) dµ i xi+1 xi ϕ (T µ f ) dµ i ϕ (x i ) xi+1 xi f (µ) dµ = (3.2) i
3.1. Realization of T µ as a skew-symmetric operator with dense domain in L 2 (µ). Fix a non-atomic measure µ on [0, 1]. Let
D 1 := f : f (x) = f (0) + x 0 f (µ) dµ, f (µ) ∈ L 2 (µ) , for all x .
(3.7)
Then D 1 ⊂ L 2 (µ) ∩ C ([0, 1]). Define ∇ µ f = f (µ) , ∀f ∈ D 1 . (3.8)
In the lemma below we express eq (3.8) for the operator ∇ µ (acting on functions f ) in terms of associated Stieltjes measures. This point is summarized best in eq (3.11) in Lemma 3.2 below, where df then denotes the Stieltjes measure corresponding to some function f . In the sequel we shall reserve the notation df for Stieltjes measure, (not to be confused with notions of differential.) Recall that for the Stieltjes measure df to make sense, the function f must be assumed to be locally of bounded variation.
Lemma 3.2. Let f be a function on R, assumed to be locally of bounded variation, so that the Stieltjes measure df is well defined. Let µ be a positive measure defined on the Borel σ-algebra B, and assume that df µ, (3.9)
i.e., that the implication (3.10) below holds:
µ (B) = 0 =⇒ df (B) = 0. (3.10) Let f (µ) be the corresponding Radon-Nikodym derivative (also denoted f (µ) = ∇ µ f ), then df = f (µ) dµ. (3.11)
Proof. The assertion in (3.11) amounts to the identity
df (B) = B f (µ) dµ, (3.12)
for all B ∈ B. But since f is locally of bounded variation, (3.12) follows from the corresponding assumption for intervals, i.e., B = [x, y] for all x < y; so
f (y) − f (x) = y x f (µ) dµ.
(3.13) Condition (3.13) in turn is equivalent to the definition of f (µ) = ∇ µ f given in (3.7) above.
Remark 3.3. Let f be a locally bounded variation function, and let µ be a positive Borel measure. Suppose that the two measures df and µ are mutually singular; we then set ∇ µ f = 0. See eq (3.14) below for justification.
Remark 3.4. We can decompose the restriction df µ in the definition of ∇ µ f as follows:
(a) Let f and µ be as stated, and pass to the Jordan-decomposition of the signed measure df (as a Stieltjes measure). Then
df = (∇ µ f ) dµ + (df ) s (3.14)
where the term (df ) s in (3.14) is mutually singular w.r.t. µ.
(b) In section 6, we shall consider a more detailed and global analysis of (3.14) for a given Stieltjes measure df . Indeed, when df is given, then the second term on the RHS in (3.14) will typically contain contributions from other measures ν, mutually singular, and each ν relatively singular w.r.t. µ.
Lemma 3.5. For all f, g ∈ D 1 , it holds that
∇ µ (f g) = f ∇ µ g + (∇ µ f ) g, Leibnitz' rule (3.15)
and
(f g) (1) − (f g) (0) = ∇ µ f, g L 2 (µ) + f, ∇ µ g L 2 (µ) . (3.16)
Proof. In our considerations below we make use of (3.7), and the definition (3.8) for the new "µ-derivative." And we further make use of basic facts for the corresponding Stieltjes measures; in particular the integration by parts formula for Stieltjes measures.
Details: If f, g ∈ D 1 (see (3.7)), then
x 0 f ∇ µ g dµ = x 0 f dg = f g x 0 − x 0 g df = f g x 0 − x 0 g∇ µ f dµ.
That is,
f (x) g (x) − f (0) g (0) = x 0 (f ∇ µ g + g∇ µ f ) dµ, so that ∇ µ (f g) = f ∇ µ g + (∇ µ f ) g,
and (3.16) also follows.
The proof above relies on key facts for Stieltjes integrals which might perhaps not be widely known. For the benefit of readers, we have therefore included the following alternative proof:
Second proof of Lemma 3.5. Let f, g ∈ D 1 as above, then
f (1) g (1) = f (0) + 1 0 ∇ µ f dµ g (0) + 1 0 ∇ µ gdµ = f (0) g (0) + f (0) 1 0 ∇ µ gdµ + g (0) 1 0 ∇ µ f dµ + 1 0 ∇ µ f dµ 1 0 ∇ µ gdµ , where 1 0 ∇ µ f dµ 1 0 ∇ µ gdµ = 1 0 1 0 (∇ µ f ) (s) (∇ µ g) (t) µ (ds) µ (dt) = 1 0 t 0 (∇ µ f ) (s) µ (ds) + 1 t (∇ µ f ) (s) µ (ds) (∇ µ g) (t) µ (dt) = 1 0 [f (t) − f (0)] (∇ µ g) (t) µ (dt) + 1 0 1 t (∇ µ f ) (s) µ (ds) (∇ µ g) (t) µ (dt) = 1 0 [f (t) − f (0)] (∇ µ g) (t) µ (dt) + 1 0 (∇ µ f ) (s) (g (s) − g (0)) µ (ds) .
Thus,
(f g) (1) − (f g) (0) = 1 0 f ∇ µ gdµ + 1 0 g∇ µ f dµ which is (3.16).
Remark 3.6. (a) In a C * -algebraic framework, operators with dense domain and satisfying a general Leibnitz rule of the form (3.15) occur under the name "unbounded derivations." They arise in a wider applied context, beyond that of fractal analysis, and have been extensively studied. They play an important role in dynamics, see e.g., [BR87]. (b) A non-atomic measure µ is fixed, and we assume that µ is supported in the unit interval [0, 1]. We now turn to the corresponding boundary-value problem for the operator ∇ µ , see (3.20). Its operator theory will be identified relative to the Hilbert space L 2 (µ).
To emphasize choice of Hilbert space, we shall use the terminology T µ for the operator, and then add subscripts to indicate domains. The theory of von Neumann (see [DS88]) of selfadjoint extensions of symmetric operators will be used. Only, for convenience, we shall use the equivalent formulation in the form where we consider instead skew-adjoint extensions of a fixed (minimal) skew-symmetric operator with dense domain.
Definition 3.7. Set T µ,0 : L 2 (µ) → L 2 (µ) by T µ,0 = ∇ µ D0 (3.17) where D 0 = C c (J) ∩ D 1 . (3.18)
Theorem 3.8. The operator T µ,0 from (3.17)-(3.18) is skew-symmetric, densely defined in the complex Hilbert space L 2 (µ). Moreover, T µ,0 has deficiency indices (1, 1), and the corresponding skew-adjoint extensions are specified by
dom (T µ,α ) = {f ∈ D 1 : f (1) = αf (0)} , |α| = 1, (3.19)
and
T µ,α = ∇ µ dom(Tµ,α) . (3.20)
Proof. From the identity
∇ µ f, g L 2 (µ) + f, ∇ µ g L 2 (µ) = f g (1) − f g (0) ,
which is valid for all f, g ∈ D 1 , it follows that the right-hand side vanishes if and only if f, g are in dom (T µ,α ), see (3.19).
For more details on extensions of skew-symmetric operators, we refer to [DS88,Sch12]. We now turn to the unitary one-parameter groups which are generated by the skew-adjoint extension operators above, from Theorem 3.8, eq. (3.20).
Consider the two unitary one parameter groups with the respective skew adjoint generators, and periodic boundary condition f (0) = f (1).
L 2 ([0, 1] , λ) ψ U λ (t) − −−− → ψ ([· + t] F ) ∈ L 2 ([0, 1] , λ) (3.21)
where [·] F denotes the fractional part of a real number. Let µ be a non-atomic Borel measure on [0, 1], and let
g (x) = µ ([0, x]) . (3.22) 0 1 2 n -1 n n + 1 Figure 3.1. [·] F : R → [0, 1] ψ ([· + t] F ) Wµ / / ψ ([g (·) + t] F ) ψ U λ (t) 1 1 Wµ 0 0 ψ • g Uµ(t) D D (3.23) We have U µ (t) (ψ • g) (·) = ψ ([g (·) + t] F ) (3.24) U µ (t) W µ = W µ U λ (t) ,
and summarized in the diagram below:
U λ (t) ψ ∈ L 2 (λ) Wµ / / U µ (t) f ∈ L 2 (µ) ψ ∈ L 2 (λ) U λ (t) O O Wµ / / f ∈ L 2 (µ) Uµ(t) O O
Lemma 3.9. Fix µ and g, and set W µ ψ = ψ • g. Then TFAE:
(1) W µ U λ (t) = U µ (t) W µ : L 2 (λ) → L 2 (µ) (2) U µ (t) = W µ U λ (t) W * µ : L 2 (µ) → L 2 (µ) (3) U λ (t) = W * µ U µ (t) W µ : L 2 (λ) → L 2 (λ)
Starting with µ, specified as before, we then set g = g µ , g (x) := µ ([0, x]), the "cumulative distribution". It follows that then the operator W µ , given by W µ ψ := ψ • g, will be an isometric isomorphism of L 2 (λ) onto L 2 (µ), with adjoint W * µ : L 2 (µ) → L 2 (λ), given by (3.28). We add that a detailed analysis of this operator W µ , and its applications, will be undertaken below. Corollary 3.10 (Time-change). We have
(U µ (t) f ) (x) = W µ U λ (t) W * µ f (x) (3.25) = W * µ f ([g (x) + t] F ) , (3.26) where W µ ψ (·) = ψ • g, (3.27) and W * µ f (y) = g −1 ({y}) f dρ y , (3.28) and so W * µ f ([g (x) + t] F ) = g −1 ({[g(x)+t] F }) f dρ [g(x)+t] F . (3.29)
In (3.28), dρ y denote conditional measures. Since µ • g −1 = λ we have conditional measures {ρ y } y∈J subject to the partition g −1 ({y}) y∈J . By the disintegration theorem, applied to µ, we therefore get the representation:
µ (·) = J ρ y (·) dy.
( Remark 3.11 (The middle-third Cantor measure).
(a) For a detailed account of harmonic analyses on fractals, see [Jor18,HJW19b]; as well as the papers cited there. Below we discuss the special case when µ = µ 3 is assumed to be the middle-third Cantor measure. In details, if the pair σ 0 , σ 1 denotes σ 0 (x) = x 3 , σ 1 (x) = x+2 3 , x ∈ R; then µ 3 is the corresponding normalized IFS-measure fixed by the Cantor-measure µ solving equation (3.31) arises as a special case of a wider class of self-similar measures, also called IFS-measures. In the general case of IFS-measures, the corresponding equation is (4.1). It is defined from a finite system of endomorphisms σ i (called function systems), and the corresponding iterated function system (IFS)measure µ will then arise as a Markov chain-average (Figure 4.1) of its corresponding σ i transforms. In this general case, the solution µ can be found, see Theorem 4.2: Every IFS-measure µ allows a representation (4.7), defined as a pull-back of an infinite-product measure (4.4). (c) Note that the function g defined this way (see (3.22)) will have the following properties:
µ 3 = 1 2 µ 3 • σ −1 0 + µ 3 • σ −1 1 ,(3.
It is monotone (increasing, or more precisely non-decreasing). Moreover, by standard measure theory, it follows that the initial measure µ will then agree with the corresponding Stieltjes measure (here denoted dg), i.e., we have µ = dg. Hence it is possible to define branches of an inverse to the function g, i.e., g −1 defined a.e., w.r.t. µ. Intuitively we think of the function g as a time-change, see (3.26). (d) In the case when µ is the standard middle third Cantor measure, then the corresponding function g is illustrated in Figure 3.2 (often called "the Devil's staircase.") At each iteration in the construction of g, we take it to be constant on the omitted middle-third intervals.
In more details, let σ 0 , σ 1 : [0, 1] → [0, 1] be given by
σ 0 (x) = x 3 , σ 1 (x) = x + 2 3 . (3.32)
Then, for the Cantor set C 3 , we have
σ 0 (C 3 )∪σ 1 (C 3 ) = C 3 .
Using bit representations, we get
g −1 ∞ n=1 b n 2 n = ∞ n=1 2b n 3 n , b n ∈ {0, 1} .
To see this, recall that every x ∈ C 3 has the following representation:
x = ∞ n=1 a n 3 n (a n ∈ {0, 2})
= a 1 3 + ∞ n=2 a n 3 n = a 1 3 + 1 3 a 2 3 + a 3 3 2 + · · · shift of bits .
(e) Let µ (= µ 3 ) be the Cantor measure of (3.31) above. We then get the following transformation of pairs of Borel measures on the unit-interval J: µ • g −1 = λ 1 , where λ 1 is Lebesgue measure on J.
Note, the defining properties of the Cantor measure µ, supp (µ) = C 3 , and the Lebesgue measure λ are as follows:
Cantor ϕ dµ = 1 2 ϕ • σ 0 dµ + ϕ • σ 1 dµ , ∀ϕ ∈ C; see (3.31)-(3.32) above; Lebesgue ϕ dλ = 1 2 ϕ x 2 dλ (x) + ϕ x + 1 2 dλ (x) , ∀ϕ ∈ C.
3.2. A symmetric pair of operators for L 2 (µ) and L 2 (ν). Let µ, ν be two non-atomic measures on J = [0, 1].
Lemma 3.12. For f ∈ L 2 (ν) ∩ dom (T µ ) and g ∈ L 2 (µ) ∩ dom (T ν ), we have (f g) (x) − (f g) (0) = x 0 (∇ µ f ) gdµ + x 0 f (∇ ν g) dν.
Proof. A direct computation:
x 0 (∇ µ f ) gdµ = x 0 gdf = gf x 0 − x 0 f dg = gf x 0 − x 0 f (∇ ν g) dν ⇓ (f g) (x) − (f g) (0) = x 0 (∇ µ f ) gdµ + x 0 f (∇ ν g) dν d (f g) = (∇ µ f ) gdµ + f (∇ ν g) dν.
See also the proof of Lemma 3.5, and that of Theorem 3.1.
Corollary 3.13. Given a pair of non-atomic measures µ and ν as described; with Lemma 3.12 and the arguments in sect 2, and 3.1, we then arrive at the following dual-pair realization for the associated operators:
dom (T µ ) ∩ L 2 (ν) L 2 (ν) Tµ * * dom (T ν ) ∩ L 2 (µ) L 2 (µ) −Tν j j (3.33) T µ ⊂ −T * ν , −T ν ⊂ T * µ . (3.34)
Remark 3.14. Let K F = d dµ d dx be as before. Note it has a quadratic form representation as follows:
ϕ, K F ψ L 2 (µ) = − ϕ , ψ L 2 (λ) (3.35) where ϕ = dϕ dx , ψ = dψ dx .
This follows from the dual pair d/dx, d/dµ in Theorem 3.1. Details:
ϕ d dµ d dx ψ dµ = − d dx ϕ d dx ψdλ = − ϕ ψ dλ. (3.36)
Then we get the selfadjoint operator
T µ T * µ = V T * µ T µ V * (3.37)
where V is a partial isometry, and T µ T * µ has dense domain in L 2 (µ). (See Theorem 2.5 and Definition 2.6) So T µ T * µ is a selfadjoint operator extension for the quadratic form QF (K F ), where
QF (K F ) (ϕ, ψ) = − ϕ ψ dλ. (3.38) Similarly, in (3.33) we get QF (K F ) ⊂ T µ T * µ . (3.39)
The notion of "extension" of a closable positive quadratic form Q is made precise in, for example [Kat95]. It is a quadratic form-version of the analogous extension of Friedrichs (see e.g., [DS88]). If K is such a selfadjoint extension operator for Q, then the requirement is that the domain of Q be contained in the domain of the square root of K. 2)). Let µ = µ 3 , the middle 1/3 Cantor measure,
µ 3 = 1 2 µ 3 • σ −1 0 + µ 3 • σ −1 1 , (3.40)
supported on the Cantor set
C 3 = [0, 1] \ {middle intervals} , (3.41) so that λ (C 3 ) = 0. Let g 3 (x) = µ 3 ([0, x]). Set K (3) F = d dµ3 d dx . Then K (3) F (ψ • g 3 ) = d dµ (ψ • g) d dx g 3 = 0 (3.42)
since d dx g 3 = 0 in the sense of distribution, i.e, g 3 = 0 a.e. λ. 3.3. The L 2 (µ)-boundary value problem. We now turn to a detailed harmonic analysis of the skew-adjoint extension operators introduced in Theorem 3.8. Recall that, when α is fixed (on the complex circle), then the corresponding skew-adjoint extension operator generates a unitary one-parameter group (depending on α) of operators U (t) acting in L 2 (µ). The harmonic analysis of this unitary one-parameter group was presented in detail above in Lemma 3.9. The following result offers a complete spectral picture.
Lemma 3.16. Set v x (y) := µ ([0, y ∧ x]) . (3.43)
Then we have
T µ v x = dv x dµ = χ [0,x] . (3.44)
Moreover, for any F ∈ C 1 , we get
T µ (F (v x )) = F (v x ) χ [0,x] .
(3.45)
In particular,
T µ e ivx = ie ivx χ [0,x] . (3.46) Proof. Note that v x (y) = y 0 χ [0,x] (s) dµ (s) = µ ([0, y ∧ x]), which is (3.44). Now, if F ∈ C 1 then d (F (v x )) = F (v x ) dv x . That is, F (v x (y)) − F (v x (0)) = y 0 F (v x ) dv x = y 0 F (v x ) χ [0,x] dµ,
so that (3.45) holds, and (3.46) follows from this.
We now turn to the detailed spectral expansion for the indexed system of skew-adjoint operators T µ,θ discussed in Theorem 3.8.
Theorem 3.17. Let T µ,θ (α = e iθ ) be as above. In particular, elements in dom (T µ,θ ) satisfy the boundary condition
f (1) = e iθ f (0) , θ ∈ R.
(3.47) Then T µ,θ has the following spectral representation:
− iT µ,θ = n∈Z λ n |ϕ n ϕ n | (3.48) where λ n = θ + 2nπ µ (J) , ϕ n (x) = 1 µ (J) e iλnµ([0,x]) , n ∈ Z, (3.49)
and {ϕ n } is an ONB in L 2 (µ). And the associated unitary one-parameter group U (t) = e tT µ,θ is given by e tT µ,θ = n∈Z e itλn |ϕ n ϕ n | .
(3.50)
Proof. With Lemma 3.16 we justify the eigenvalue/eigenfunction assertions in (3.49) in the Theorem. Note that Lemma 3.18. The adjoint operator of T µ,0 is given by T * µ,0 = −∇ µ , defined on D 1 . Proof. For any f ∈ D 0 and g ∈ D 1 , one has
T µ,θ f = iλf ⇐⇒ f (x) − f (0) = iλ x 0 f dµ.1 0 (T µ,0 f ) gdµ + 1 0 f ∇ µ gdµ = 0 and so T * µ,0 ⊂ −∇ µ D1 . Conversely, for all g ∈ dom T * µ,0 ⊂ L 2 (µ), let h = T * µ,0 g then 1 0 (T µ,0 f ) gdµ = 1 0 f hdµ, ∀f ∈ D 0 . Set H (x) = H (0) + x 0 hdµ ∈ D 1 , then 1 0 (T µ,0 f ) gdµ = − 1 0 f ∇ µ Hdµ = − 1 0 (T µ,0 f ) Hdµ, ∀f ∈ D 0 .
It follows that g = −H in L 2 (µ).
Even though our present focus is on the case when µ is assumed singular w.r.t. Lebesgue measure, λ, for the sake of illustration, the next two results, Lemma 3.19 and Corollary 3.20, cover the other extreme, i.e., when µ λ holds. Then
∇ µ f (x) = M −1 f (x) . Proof. Indeed, if f ∈ D 1 then f (x) − f (0) = x 0 f (x) dx = x 0 f M −1 (x) M (x) dx = x 0 ∇ µ f (x) dµ (x) .
Corollary 3.20. Let M (x) = dµ/dx be as above. The deficiency subspaces of T µ,0 are determined as follows:
D ± (T µ,0 ) := g : g M −1 (x) = ±g = span exp ± M (x) dx .
In particular, this implies that T µ,0 has deficiency indices (1, 1).
Proof. One checks that
∇ µ g = ±g g M −1 = ±g (ln g) = ±M
and the conclusion follows.
Definition 3.21. Set ∆ µ = d dµ d dx = ∇ µ d dx defined on dom (∆ µ ) := {f | f ∈ D 1 } = c + t 0 f (0) + x 0 ∇ µ f dµ dx | ∇ µ f ∈ L 2 (µ) . Set ∆ µ,0 = ∆ µ Cc∩dom(∆µ) .
Then ∆ µ,0 has two particular selfadjoint extensions that correspond to Dirichlet and Neumann boundary conditions. Theorem 3.22. Fix µ. Let K F denote a corresponding selfadjoint realization of ∆ µ . For every g ∈ L 2 (µ), set
f (t) := t 0 x 0 gdµ dx.
(3.52)
Then
(K F f ) (t) = (f ) µ = g, (3.53)
and the eigenvalue problem may be stated as
g (t) = λ t 0 gdµ. (3.54)
Proof. We have
(K F f ) (t) = λf g = λf g (t) = λ t 0 x 0 gdµ dx.
And so g (t) = λ t 0 gdµ.
Corollary 3.23. Assume µ dx, and dµ/dx = M > 0. Let
dW (µ) t = M −1/2 (x) dB t ,
where B t is standard Brownian motion. Then,
u (t, x) := E x f W (µ) t satisfies ∂u ∂t = 1 2 K F u.
Proof. An application of Ito's lemma (see e.g., [Hid80]) gives
df (W t ) = f (W t ) M −1/2 dB t + 1 2 f (W t ) M −1 dt and d dt E (f (W t )) t=0 = 1 2 M −1 (x) f (x) = 1 2 K F f (x) ,
where K F f = M −1 f , by Lemma 3.19.
3.4. Krein-Feller diffusion. We now turn to a detailed discussion of diffusion, and its connection to the Krein-Feller operator. As before, our starting point is a fixed measure µ supported on the real line, and we let K F be the corresponding Krein-Feller operator. The measure µ under consideration is assumed to be defined on the Borel-sigma algebra, and further assumed positive, sigma-finite, and non-atomic. The measure µ may be singular, and the main novelty in our analysis of the diffusion semigroup is for the singular case; including the case of IFSmeasures. We first introduce the centered Gaussian process W (µ) having µ as its quadratic variation. We then note that Ito's lemma applies to W (µ) , see (3.94). We further study the Markov semigroup ((3.79) and Lemma 3.29) corresponding to W (µ) . In our two main results Lemma 3.24 and Theorem 3.30 below, we identify the selfadjoint extension of K F from section 2 as the infinitesimal generator for this diffusion semigroup, Lemma 3.24. In Theorem 3.30 we introduce a time-change in our characterization of the diffusion semigroup.
Below we show that every positive non-atomic Borel measure (see Lemma 3.2) gives rise to a naturally associated dual pair of operators, as per Definition 2.6. The dual pair is made precise in (3.4), and the figure in Theorem 3.1.
Fix a measure space (J, B, µ) with J = [0, ∞). We introduce the corresponding positive definite kernel: Here we shall need the following: Let ϕ ∈ C 2 , and consider the corresponding diffusion semigroup (depending on µ):
K µ (x, y) = µ ([0, x ∧ y]) . (3.55) Note that if µ = λ = dx = the Lebesgue measure then K λ (x, y) = x ∧ y,(3.u (t, x) := E ϕ W (µ) t | W (µ) 0 = x (3.59)
where E in (3.59) refers to the expectation E (·) = Ω (·) dP for the probability space associated to (3.55). Then E(· · · | W (µ) 0 = x) in (3.59) refers to conditioning with all paths ω s.t. ω (0) = x. Here we also use the representation
W (µ) t (ω) = ω (t) , ∀ω ∈ Ω (3.60) for the Gaussian process {W (µ) t }.
Since the Gaussian process W (µ) has independent increments, it follows that eq. (3.59) defines a semigroup (see e.g., [IM74,Phi61b,Kas12,KS16]), and we shall call it the Markov semigroup. Its properties and its infinitesimal generator will be identified in Lemma 3.24 below, and in the subsequent discussion.
The Gaussian process from (3.58) and (3.60) is often called a generalized Brownian motion, or a Gaussian field. The associated Ito-integral is also used in the proof of Lemma 3.24 below.
We now introduce the Krein-Feller operator
K F := ∂ 2 ∂µ∂x
(3.61) (see above.) In our discussion of (3.59), we shall consider K F as acting in the x-variable.
Lemma 3.24. The diffusion (3.59) is generated by the following generalized heat equation:
∂u ∂t = 1 2 K F u, u (0, x) = ϕ (x) (3.62)
where ϕ is a fixed continuous function.
Remark 3.25. The special case of (3.62) corresponding to µ = λ = Lebesgue measure is . Let K F be as in (3.61), then
∂u ∂t = 1 2 ∂ 2 u ∂x 2 , u (0, x) = ϕ (x) .K F ⊆ T µ T * µ ,(3.65)
both are operators in L 2 (µ). But T µ T * µ is selfadjoint by general theory (see Section 2), and restriction of symmetric is symmetric. Proof. Since L 2 (µ) ⊂ L 1 (µ) the function y −→ y 0 ϕ (s) µ (ds) is continuous, and so x −→ (Aϕ) (x) is C 1 (one time differentiable with (Aϕ) ∈ C (J).) Hence, for the LHS of (3.67) we have
K F A : ϕ −→ ∇ µ d dx Aϕ = ∇ µ x 0 ϕ (s) µ (ds) = ϕ (x) ,
which is the desired conclusion (3.67).
Corollary 3.27. Let the measure µ be specified as above, and set Proof. Without loss of generality we may work with b = 1 and real Hilbert space. An easy application of Schwarz to L 2 (µ) shows that A : L 2 (µ) → L 2 (µ) is bounded. We have
Aϕ, ψ L 2 (µ) = 1 0 x 0 (x − s) ϕ (s) ψ (x) µ (ds) µ (dx) . (3.69)
The asserted symmetry follows from this, or equivalently, Figure 3.3. To see that the operator A of (3.66) is compact as an operator in L 2 (µ), we make use of (3.69) and (3.70) as follows: We recall the fact the compact operators are the norm-closure of finite rank operators. We then create such norm-limits of finite rank operators with the use of the kernel (3.68), and a choice of a filter of partitions P with disjoint Borel subsets of the support of µ. For each such partition P , we form rank-one operators from the corresponding indicator functions from pairs of sets B, B in P , and we then form the associated span of the rank-one operators |χ B χ B | by evaluation of (3.68) with sample points chosen from the partition sets. (We use Dirac's terminology |· ·| for rank-one operators.) The Borel sets B, making up partitions, are chosen with µ(B) finite, so the corresponding indicator functions are in L 2 (µ). For a fixed partition, we then form pairs of such indicator functions, and the corresponding rank-one operators. Since µ is chosen non-atomic, the partition-limit refinements can be constructed such that the limit of the corresponding numbers µ(B) is zero. 0
(Aϕ) (x) = 1 0 a (x, s) ϕ (s) µ (ds) (3.70) where a (x, s) = χ [0,x] (s) (x − s), see
x 1 The following result gives a spectral decomposition for the Krein-Feller operator K F .
Corollary 3.28. Let J = [0, 1], and µ be a non-atomic positive Borel measure on J. Set
g (x) = µ ([0, x]). Consider K F = d dµ d
dx as a selfadjoint operator in L 2 (µ) with Neumann boundary condition, i.e.,
dom (K F ) = ψ (x) = ψ (0) + x 0 y 0 f dµ dx : f ∈ L 2 (µ) , ψ (0) = ψ (1) = 0 . Let V : L 2 (µ) → L 2 (µ) be the integral operator defined as V f (x) = 1 0 H (x, t) f (t) dt (3.71) where H (x, t) = (g (x) − 1) g (t) t ≤ x, (g (t) − 1) g (x) x ≥ t. (3.72)
Then V is compact and selfadjoint, and we have the following eigenvalue correspondence:
c ∈ sp (K F ) ⇐⇒ c −1 ∈ sp (V ) . (3.73)
Proof. Selfadjointness of V follows from (3.72), and the argument for compactness is the same as that of Corollary 3.27.
Let ψ (x) = ψ (0) + x 0 t 0 f dµ dt ∈ dom (K F ), where f ∈ L 2 (µ), so that K F ψ (x) = f (x). Assume that K F ψ = cψ,
for some constant c < 0. Then,
K F ψ = cψ f (x) = c x 0 t 0 f dµ dt ⇓ f (x) = c x 0 f dµ.
The last line can be written as follows:
Set g (x) := µ ([0, x]), then f (x) = c x 0 f dµ = c x 0 f dg = c f (x) g (x) − x 0 g (t) f (t) dt = c f (0) + x 0 f (s) ds g (x) − x 0 g (t) f (t) dt = cf (0) g (x) + c x 0 (g (x) − g (t)) f (t) dt.
(3.74)
Since f = cψ , the boundary condition ψ (1) = 0 and (3.74) imply that
cf (0) + c 1 0 (1 − g (t)) f (t) dt = 0. (3.75)
Substitute (3.75) into (3.74), then
f (x) = c x 0 (g (x) − g (t)) f (t) dt + cf (0) g (x) = c x 0 (g (x) − g (t)) f (t) dt − g (x) 1 0 (1 − g (t)) f (t) dt = c x 0 (g (x) − 1) g (t) f (t) dt + 1 x g (x) (g (t) − 1) f (t) dt = c 1 0 H (x, t) f (t) dt
with H as defined in (3.72), and the assertion (3.73) follows.
3.5. Path-space and Markov transition. It is also of general interest to relate K F directly to the generator of the diffusion semigroup. Notation: (Ω, F , P), Ω path space, F cylinder σ-algebra, P probability measure,
K µ (A ∩ B) = µ (A ∩ B), W (µ) t (ω) = ω (t) , ∀ω ∈ Ω. (3.76) E (·) = Ω · · · dP (x ∈ J) (3.77) E x (·) = Ωx · · · dP, ω ∈ Ω x = {ω, ω (0) = x} = ω, W (µ) 0 ω = x .
(3.78)
In the discussion below we omit µ in W (µ) to simplify notation.
(S t ϕ) (x) := E x ϕ • W (µ) t .
(3.79)
We showed that K F is the generator of S t in (3.79). It is known that S t is a semigroup (t ∈ R + ) also S 0 = I, so S s S t = S s+t , ∀s, t ∈ R + .
For semigroups and generators in the Hilbert space framework, see e.g., [Phi11,CP68,Phi61a], and for diffusion semigroups, we refer to e.g., [Phi61b,Won21,KS16]. Also see [Kni81, EL93, ch 3].
Lemma 3.29. S t is selfadjoint in L 2 (µ) ∀t ∈ R + , so
(S t ϕ) (x) ψ (x) µ (dx) = ϕ (x) (S t ψ) (x) µ (dx) , ∀ϕ, ψ ∈ L 2 (µ) , (3.80) also |S t ϕ| 2 dµ ≤ |ϕ| 2 dµ. (3.81)
Proof. The proof of the properties (3.80)-(3.81) is contained in the literature of diffusion semigroups. But the following proof sketch for (3.80) is new: Fix t > 0 and any 0 ≤ s ≤ t, then
d ds (S t−s ϕ) (x) (S s ψ) (x) µ (dx) ≡ 0 (3.82) so s → (S t−s ϕ) (x) (S s ψ) (x) µ (dx)
is constant value at s = 0 = value at s = t, and (3.80) follows. Proof of (3.82).
LHS (3.82) = d ds S t−s ϕ, S s ψ L 2 (µ) = − K F S t−s ϕ, S s ψ L 2 (µ) + S t−s ϕ, K F S s ψ L 2 (µ) = 0 since K F is symmetric w.r.t. L 2 (µ).
3.6. The conditioning W (µ) 0 = x. For our considerations in (3.78) and (3.79) we used the notation E x and Ω x with reference to conditioning paths ω which "start" at x, so ω (0) = x. The justification is as follows. We have selected the sample space Ω to be [0,∞) R (Cartesian product), and functions ω : R ≥0 → R (infinitely many "paths".) (It is known that the continuous functions will have full measure relative to (Ω, C , P) where C = the usual cylinder σ-algebra of subsets of Ω.)
Here Now consider the Radon-Nikodym derivative:
dP • π −1 0 dµ = E x , (3.87)
or equivalently, for all random variables F on (Ω, C ) we have:
E (F ) = R E x (F ) µ (dx) .
(3.88) 3.7. The µ-heat equation. We assume a fixed non atomic Borel measure µ supported in an interval J = [0, α] where α may be finite or infinite. We shall denote by W (µ) the corresponding generalized Brownian motion, i.e., determined by:
W (µ) is Gaussian, real-valued E(W (µ) ) = 0, E(W (µ) A W (µ) B ) = µ (A ∩ B) (3.89)
for all Borel sets A, B ⊂ J.
For every continuous function ϕ on R, we consider ϕ(W (µ)
A ) = ϕ • W (µ) A . If A = [s, t] we make a choice of W (µ) t such that W (µ) [s,t] = W (µ) t − W (µ) s , and we set S (µ) t ϕ (x) = E x ϕ • W (µ) t , t ∈ R + .
(3.90)
Since, by (3.89), the process W (µ) has independent increments, it follows that S (µ) t is a Markov semigroup.
When ϕ is given, we set
u (t, x) = S (µ) t ϕ (x) = E x ϕ W (µ) t (3.91)
where the conditional expectation E x corresponds to W (µ) 0 = x. Then the probability space Ω consists of continuous ω, and
W (µ) t (ω) = ω (t) , 0 ≤ t ≤ ∞.
(3.92)
We then get the boundary condition u (0, x) = ϕ (x) directly from (3.91).
We shall further consider the operator d/dµ acting in the t-variable. For convenience we shall write ∇ (µ) t . We now turn to the corresponding diffusion equation:
Theorem 3.30. Let µ, W (µ) , S (µ)
t , and u (t, x) be as specified. We then have
∇ (µ) t u = 1 2 ∂ 2 ∂x 2 u.
(3.93)
Proof. Without loss of generality we may assume ϕ ∈ C 2 . Then by Ito's lemma, we get
dϕ W (µ) t = ϕ W (µ) t dW (µ) t + 1 2 ϕ W (µ) t dµ (3.94)
where ϕ = (d/dx) 2 ϕ. For the derivation of (3.94), we refer to the cited papers (e.g., [IM74, KS16, EL93, AJL17]), we also use the familiar quadratic variation formula
QV = dW (µ) t 2 = µ (dt) .
(3.95) Note that (3.95) holds since µ was assumed non-atomic. Further note that (3.94) refers to Ito-differentials. In general (3.94) is equivalent to the corresponding integral formula version:
ϕ W (µ) t − ϕ W (µ) 0 = t 0 ϕ W (µ) s dW (µ) s Ito-integral + 1 2 t 0 ϕ W (µ) s µ (ds) . (3.96)
Now apply the expectation E x to both sides in (3.95) we arrive at
E x ϕ W (µ) t − ϕ (x) = 1 2 t 0 E x ϕ W (µ) s µ (ds) , (3.97)
or equivalently:
S (µ) t ϕ (x) − ϕ (x) = 1 2 t 0 ∂ 2 ∂x 2 u (s, x) µ (ds) . (3.98)
From the definition of the operator T (µ)
(in t) = ∇ (µ) t (see Lemma 3.2), we therefore get ∇ (µ) t u = 1 2 ∂ 2 ∂x 2 u, (3.99)
which is the desired conclusion (3.93) in the Theorem.
Proposition 3.31. Consider the heat equation
∂ ∂t u (t, x) = K F u (t, x) , (t, x) ∈ R + × [0, 1] , (3.100) with K F = ∂ ∂µ ∂
∂x given a selfadjoint realization in L 2 ([0, 1] , µ). Then the corresponding solution to (3.100) has the following form:
u (t, x) = ∞ 1 e −tλn k n (x) ,
where λ n are the eigenvalues of K F and k n are the corresponding eigenfunctions.
Proof. The argument is based on separation of the two variables t and x, and use of spectral data; but now with reference to K F and ∇ µ .
For details about choices of selfadjoint realizations of K F , see Corollary 3.28, as well as Remark 3.32 below.
Details as follows: Set
u (t, x) = h (t) k (x) . (3.101)
Substituting (3.101) into (3.100) leads to
h (t) k (x) = h (t) ∇ µ x k (x) , so that h h (t) = ∇ µ x k (x) k (x) = const = −λ.
Thus, h (t) = conste −λt , and λ is specified by
− λk (x) = ∇ µ x k (x) . (3.102)
Note, the eigenvalue problem (3.102) is equivalent to (1) Dirichlet boundary. (For a related discussion, see also Corollary 3.28 above.) Dirichlet conditions:
h (t) x 0 k (y) µ (dy) = h (t) (k (x) − k (0)) .f (0) = f (1) = 0. Specifically, dom (K F ) = f ∈ L 2 (µ) : f (x) = x 0 f (0) + y 0 ϕ (s) µ (ds) dy, f (1) = 0, ϕ ∈ L 2 (µ) .
In this case, one has
K −1 F ϕ (x) = 1 0 K Dirichlet (x, s) ϕ (s) µ (ds) , where K Dirichlet (x, s) = (x − 1) s s ≤ x, x (s − 1) s ≥ x.
(2) f (0) = f (1) = 0. That is,
dom (K F ) = f ∈ L 2 (µ) : f (x) = x 0 f (0) + y 0 ϕ (s) µ (ds) dy, f (1) = 0, ϕ ∈ L 2 (µ) .
Then,
K −1 F ϕ (x) = − 1 0 min (x, s) ϕ (s) µ (ds) .
Remark 3.33. Our analysis of W (µ) and the associated semigroup is related to what is often referred to as "change of time;" see e.g., [BNS15] and (3.26) in Lemma 3.9.
Remark 3.34. Dym and McKean developed a version of Krein-Feller operators in a context of what they call "strings", see e.g., [DM72,DM76] and also [Man68]. In principle, there is the following dictionary: string = positive measure µ on a finite interval. Reasoning: every positive measure on an interval is a Stieltjes measure by a monotone function, say F . In Dym & McKean, the monotone function F measures the accumulation of mass as you move forward on the string, and µ = dF as a Stieltjes measure. However, Dym & McKean do not seem to distinguish their analysis for the dichotomy: µ singular or not. Recall, for the 1/3 Cantor measure, µ = dF where F is the Devil's staircase function; see Figure 3.2. The case when µ Lebesgue is covered in many other places, e.g., books and papers by Edward Nelson, e.g., [Nel92,Nel73,Nel67,Nel64].
Remark 3.35 (Summary of extension theory for unbounded operators). There is a theory in the case of unbounded operators in Hilbert space, see e.g., [CP68,Phi61b,Phi11].
Here, we emphasize the correspondence between skew-adjoint operators, and generators of unitary one-parameter groups. In the case when skew-adjoint operators arise as operator extensions, then they are specified by partial isometries. On the other hand, dissipative operators correspond to generators of contraction semigroups; and dissipative extensions are specified by partial contractions.
Generators of unitary one-parameter groups are maximal skew-symmetric extensions. Examples of maximal skew-symmetric extensions that might not be skew-adjoint will be when one of the indices is 0, so the cases (0, m) or (n, 0). Generators of contraction semigroups are maximal dissipative extensions (for details, see [DS88].) One can have semigroup generators for the cases (n, 0). But there are other semigroup generators.
Consider the operator (d/dx) 2 in L 2 ([0, 1]).
E x (ϕ (W t )) = (S t ϕ) (x)
where W t is the Brownian motion (and W (µ) for the general case).
Two cases:
(1) all sample paths ω, ω (0) = x (2) restricted sample paths; e.g., ω (t) ∈ [0, 1] or ω (t) ∈ [−1, 1].
Operators and generalized Dirichlet forms.
Lemma 3.37. Let µ be a σ-finite positive measure as before, (J, B) with Borel σ-algebra B.
Here J = [0, 1) or J = [0, ∞). Then the following are equivalent:
(
1) f (y) − f (x) = y x T µ f dµ; (2) J ϕ df = J ϕT µ f dµ, ∀ϕ ∈ C ∞ c (J); (3) df µ and df dµ = T µ f . If H (K µ ) is the RKHS of the p.d. kernel K µ (A, B) := µ (A ∩ B), then in (3) we have df H (Kµ) = T µ f L 2 (µ) .
(3.104)
Proof. In our previous paper (see e.g., [JT17,JT19a]) we studied the RKHS H (K µ ) as a Hilbert space of measures ρ such that ρ µ and dρ dµ ∈ L 2 (µ), ρ H (Kµ) = dρ/dµ L 2 (µ) ; and so we apply this result to the current setting, with ρ = df as a Stieltjes measure. We will come back to this point in Section 5.
Details:
(1)⇔(2). We have
ϕdf ≈ i ϕ (x i ) (f (x i+1 ) − f (x i )) (3.105) = by (1) i ϕ (x i ) xi+1 xi T µ f dµ ≈ J ϕ (T µ f ) dµ (standard integral approximation) .
(2)⇒(3). Rewrite (2) with ϕ = χ B , B ∈ B, then
B df = B (T µ f ) dµ.T µ : L 2 (µ) −→ L 2 (µ) ; and D = − d dx : L 2 (µ) −→ L 2 (µ) , where L 2 (µ) := dom (T µ ) L 2 (λ)
, and λ = d/dx = the usual Lebesgue measure restricted to the fixed interval J. Recall, µ is assumed supported in J.
We therefore obtain the following two selfadjoint operators:
T µ T * µ : L 2 (µ) −→ L 2 (µ) ; (3.107)
and
T * µ T µ : L 2 (µ) −→ L 2 (µ) (3.108)
with corresponding Dirichlet forms:
ϕ, T µ T * µ ϕ L 2 (µ) = J |ϕ | 2 dµ; (3.109) and f, T * µ T µ f L 2 (µ) = J |f (µ) | 2 dx. (3.110)
For a given non-atomic measure µ, we shall refer to the quadratic form (3.109) as the Dirichlet form induced from µ. It further follows from (3.107) that this Dirichlet form has a selfadjoint, semibounded (A ≥ 0) realization, say A, where A is a selfadjoint extension of our Krein-Feller operator K F . Here, initially K F is considered as a symmetric operator with dense domain in L 2 (µ). We shall further show that, when the diffusion semigroup is realized in L 2 (µ), then its infinitesimal generator is -A. It is known from general theory that the Dirichlet form determines the diffusion semigroup; and vice versa. See, e.g., [Kni81,Fel54,FM56,Fuj87], and also Section 3.4.
Applications to IFS measures
The present section deals with applications of Krein-Feller operators in the setting of IFS measures. For the operator theoretic framework, see Section 3, especially Corollary 3.38. It is subdivided into two subsections. The first subsection introduces a general class of
i : X → X. Let {p i } N i=1 , p i > 0, N i=1 p i = 1 be fixed.
A Borel measure µ on X is said to be an iterate function system (IFS) measure w.r.t. the data iff (Def) the following identity holds:
µ = N i=1 p i µ • σ −1 i (4.1)
on the Borel σ-algebra B X .
We now turn to an explicit realization of the IFS-measure from e.q. (4.1).
Theorem 4.2. Let X, N , {σ i } N i=1 , {p i } N i=1
be as above. Consider the infinite product
Ω N := N0 {1, 2, · · · , N } , (4.2)
and suppose, for all ω ∈ Ω N , the intersection below is a singleton, i.e.,
∞ n=1 ω|n={i1,··· ,in} σ in σ in−1 · · · σ i1 (X) = {x (ω)} ; (4.3)
then there is an associated IFS measure µ constructed from the infinite product
π := p × p × p × · · · on Ω N . (4.4)
Because of assumption (4.3), we get a well-defined X-valued random variable W (for the probability space (Ω N , π)), and the IFS-measure µ from (4.1) is then the distribution of W . See Proof. With (4.3), we define the random variable W (IF S) (ω) := x (ω), ω ∈ Ω N . If (i 1 , · · · , i n ) ∈ n 1 {1, · · · , N }, then the measure π is specified on cylinder sets as follows:
π ([i 1 , · · · , i n ]) = p i1 p i2 · · · p in . (4.5)
The measure π is then defined on Ω N via Kolomogorov's consistency extension theorem, see [Hid80,Kol83,Kol77]. Let W : Ω N −→ X (4.6) be the random variable specified by the condition in (4.3), and set µ := π • W −1 .
(4.7)
One checks that µ will then satisfy the IFS condition in (4.1).
Corollary 4.3. In this corollary we fix a system {σ i } N i=1 of endomorphisms, and we consider the IFS measure µ as it depends on the choice of probability weights p = (p i ) N i=1 , p i = 1. Set µ (p) = the solution to (4.1); (see also (4.6)). Then if p = q (i.e., ∃i such that p i = q i ) then the two measures µ (p) and µ (q) are mutually singular.
Proof. The result is immediate from (4.7) and Kakutani's dichotomy theorem for infinite product measures, applied to (4.4). By Kakutani [Kak43], the two measures × N p and × N q are mutually singular; and by (4.7), so are the two IFS measures µ (p) and µ (q) .
IFS measures supported on compact intervals.
Here we take
σ i (x) := λ i x + b i (4.8) where 0 < λ i < 1, b i ∈ R, x ∈ R, 1 ≤ i ≤ N . Fix {p i } N i=1
as above. Then the corresponding IFS measure µ (see (4.4)) will satisfy
N i=1 p i f (λ i x + b i ) µ (dx) = f (x) µ (dx) (4.9)
for all bounded continuous functions f on R. One checks that µ will then be supported on a compact interval J ⊂ R. (1)
σ 1 (x) = x 2 σ 2 (x) = x+1 2 J = [0, 1], µ = restricted Lebesgue measure.
(2) σ 1 (x) = Remark 4.5. There is an important difference between the cases (2) and (3) above. Naturally they have different geometries, different Hausdorff dimension, and they are mutually singular. They are both IFS measures, but the most striking difference is their respective harmonic analysis. For the middle fourth Cantor measure µ 4 in (3), the corresponding L 2 (µ 4 ) admits an orthogonal Fourier series expansion; while the middle third Cantor measure µ 3 in (2) does not. Even more striking is the fact that L 2 (µ 3 ) does not admit three orthogonal Fourier exponentials. For this subject, and related, readers are referred to [JP98b,JP98a,Jor18].
Corollary 4.6. Consider a measure µ specified as in (4.9) above, so it includes the cases (1)-(3) in Example 4.4. Then for the Dirichlet form (3.109) in Corollary 3.38 we have
J |ϕ | 2 dµ = N i=1 p i λ 2 i d dx (ϕ (λ i x + b i )) 2 µ (dx) .
(4.10)
Cumulative functions g µ for general IFS measures µ. Let µ be an IFS measure as in Theorem 4.2, and let
g µ (x) := µ ([0, x]) .
The scale 3 Cantor measure µ 3 with g µ3 is discussed in Remark 3.11 and Example 4.4, see also Consider more general IFS measures on [0, 1], i.e., the unique solutions µ to:
µ = p 1 µ • σ −1 1 + p 2 µ • σ −1 2 . Define f 0 (x) = xχ [0,1] (x) + χ (1,∞) (x) ,
and f n (x) := p 1 f n−1 σ −1 1 (x) + p 2 f n−1 σ −1 2 (x) , n ∈ N.
(4.11)
Then lim n→∞ f n (x) = g µ (x) , ∀x ∈ [0, 1] .
(4.12)
Example 4.7. Let p = (p i ) 2 i=1 = 1 3 , 2
3 . An illustration of g µ (·) for the two cases below is in Figure 4.3.
(1) σ 1 (x) = x 2 , σ 2 (x) = x+1 2 ; (2) σ 1 (x) = x 3 , σ 2 (x) = x+2 3 .
For case (1), we have
(a i ) ∈ N {0, 1} , ∞ 1 a i 2 i ≤ x µ − − → lim m→∞ 2 3 m 1 2 i1 + 1 2 i2 + · · · + 1 2 im = g µ (x) .(b) σ 1 (x) = x 3 , σ 2 (x) = x+2 3 . Figure 4.3. g µ (x) with p = 1 3 , 2 3 .
For the related IFS-measures and their cumulative distributions discussed earlier, see Figure 3.2 (Remark 3.11, scale-3 Cantor), Example 4.4, Figure 4.2 (scale-4 Cantor). For these cases, the fair-coin measure is used. And by contrast, Figure 4.3 illustrates a choice of biased Markov chain-weights.
In particular, it follows from Corollary 4.3 above that the measure µ in Example 4.7 (1), see Figure 4.3a, is mutually singular with respect to Lebesgue measure λ. They are mutually singular despite the fact that both measures, µ and λ, on the unit-interval, arise from the same pair of maps, {σ i } by IFS-recursive iteration.
Special case (intervals) vs general measure spaces
Observation: For general measure spaces (X, B, µ), in an earlier paper, we established a canonical isometry T µ of the RKHS (K µ ) onto L 2 (µ).
In the special case of X = an interval, and B = the Borel sigma-algebra, µ a singular nonatomic measure, we also have an operator T µ and it is a special case of the T µ we introduced in our earlier paper on RKHS theory. Background references for this include [JPT16, Zag87, Nel67, Nel73, Nel92, AJL17, Jor18, HJW19b].
Remark 5.1 (Distinction between first order and second order operators). Our KF-Laplacian (second order) has selfadjoint extensions, for example T * T . As we study the KF-Laplacian with minimal domain, we study its selfadjoint extensions.
Let (X, B, µ) be a σ-finite measure space. We then consider the p.d. kernel on B f in × B f in , defined as
K µ (A, B) := µ (A ∩ B) , A, B ∈ B f in (5.1)
with H (K µ ) being the associated RKHS.
Theorem 5.2. We have
H (K µ ) = F : F σ-finite measure on (X, B) s.t. (5.2) F µ, T µ F := dF/dµ, F H (K) = dF/dµ L 2 (µ) . (5.3)
Proof. We also included proof details for the conclusions (5.2)-(5.3) when K = K µ is specified as in (5.1).
Recall that µ (· ∩ A) ∈ H (K µ ) . (5.4)
So if F is a signed measure on (X, B) and F ∈ H (K µ ), then we assign the inner product
F, µ (· ∩ A) H (Kµ) = F (A) , (5.5)
using the reproducing property of H (K µ ). From (5.5), F is a function on B, and we showed that from the axioms of the RKHS H (K µ ) that F (·) will be σ-additive, so a signed measure.
Specifically, if B = ∪ i B i , B i ∈ B, B i ∩B j = ∅ for i = j, one has F (B) = i F (B i ) . (5.6)
But we also derive the axioms for H (K µ ) as follows:
F µ dF dµ = Radon Nikodym derivative (5.7)
In particular if A ∈ B f in is fixed, then
µ (· ∩ A) µ (5.8) and dµ (· ∩ A) dµ = χ A (·) . (5.9)
To see (5.9), note that
B χ A (·) dµ = µ (A ∩ B) .
Moreover, for all F, G ∈ H (K µ ), we have
F, G H (Kµ) = X dF dµ dG dµ dµ.
(5.10)
The formula (5.10) offers another way to verify (5.5). Indeed,
LHS (5.5) = by (5.10) X dF dµ (·) dµ (· ∩ A) dµ dµ = by (5.9) X dF dµ (·) χ A (·) dµ = A dF dµ dµ = F (A) = RHS (5.5) .
Conclusion. Fix (X, B, µ). Recall: H (K µ ) RKHS of K µ , consisting of signed measures F s.t. F µ and dF/dµ ∈ L 2 (µ).
F ∈ H (K µ ) Tµ ( ( L 2 (µ) T * µ h h T µ F = dF/dµ ∈ L 2 (µ) T * µ T µ = I H (Kµ) , and T µ T * µ = I L 2 (µ) T * µ ψ (A) = A ψdµ, ∀A ∈ B If F is
A Hilbert space of equivalence classes
In the earlier literature, authors typically only focus their analysis on a fixed positive Borel measure µ. This µ might be compared to Lebesgue measure λ. When Stieltjes measures df are considered, it will then be relative to just this one measure µ. So when discussing ∇ (µ) f = f (µ) , then consideration of the equation df = f (µ) dµ is really only picking out one component of df .
Recall that the family of Stieltjes measures df account for all Borel measures. And, in general, a Stieltjes measure will contain other non-zero components.
The focus in section 6 is the following: When we apply the Jordan decomposition to a fixed Stieltjes measure df , then the part of df that is singular w.r.t. λ may contain multiple components, chosen in such a way that each of these components is mutually singular w.r.t. the others. The emphasis below is this: We introduce a Hilbert space H class of "sigma functions". Starting with a Stieltjes measure df , we may identify its mutually singular components with orthogonal "pieces" in the Hilbert space H class .
We shall consider pairs (f, µ) where f is a locally-bounded variation function, and µ is a positive non-atomic Borel measure. Following [Nel69], one checks that the ∼ as specified below will be an equivalence relation on pairs:
(f 1 , µ 1 ) ∼ (f 2 , µ 2 ) iff One further checks that orthogonality in the inner product in H class happens precisely for classes with measures µ 1 and µ 2 which are mutually singular.
Theorem 6.1. If f is given, locally of bounded variation. In addition, assume that the sum in (6.4) is finite, so the Stieltjes measure df is in H class . It follows that H class induces a Hilbert norm as follows:
f 2 H class = µ df dµ 2 dµ (6.4)
where the sum on the RHS in (6.4) is over all µ s.t. df | supp(µ) µ| supp(µ) , and distinct terms in the sum correspond to mutually singular measures µ.
Proof. The result is immediate from the discussion above, and [Nel69, ch 6], commutative multiplicity theory. When the function f is fixed, one checks from the definition of the equivalence relation (6.1)-(6.2), and an easy calculation, that each of the individual terms on the RHS in (6.4) in the sum-expression only depends on the equivalence class determined by the measures µ entering into the summation. (Note that the Hilbert space H class of equivalence classes is also called the Hilbert space of sigma-functions.)
Remark 6.2. Nelson's sigma Hilbert space [Nel69] serves as a tool allowing us to make precise the formal assertion for Stieltjes measures:
df = µ ∇ (µ) f dµ (6.5)
where the measures µ in (6.5) are specified as in (6.4). Hence (6.5) is justified when the function f (in (6.5)) yields a finite sum for the RHS in (6.4).
For the Stieltjes measure df , we therefore get the following evaluation formula: For all B ∈ B 1 (= the Borel σ-algebra), we have
df (B) = µ∈M+(df ) B∩esssup(µ) ∇ (µ) f dµ, (6.6)
where "esssup" refers to essential support.
The following example illustrates that there are choices of functions f for which the sum might be infinite.
of T , and ran (T ) = {T ϕ | ϕ ∈ dom (T )} , (2.2)
Figure 3 .
32. g µ3 (·) the middle third Cantor measure as a Stieltjes measure. See Remark 3.11.
3
3the Cantor set. The cumulative distribution function of µ 3 is the function g (x) := µ 3 ([0, x]). It is sketched inFigure
to verify that f (x) = e iλµ([0,x]) satisfies (3.51). Indeed,x 0 e iλµ([0,s]) dµ (s) = x 0 e iλg(s) dg (s) = 1 iλ e iλg(x) − 1 , where g (x) = µ ([0, x]) as before.
Lemma 3. 19 .
19Assume µ dx, where dx = the Lebesgue measure on [0, 1], and let dµ/dx = M (x).
=
56)the usual covariance function for Brownian motion.Starting with µ (assumed non-atomic) and the RKHS H (µ), we arrive at a generalized Brownian motion W K µ (x, y) , ∀x, y ∈ [0, ∞).(3.58)A detailed description of {W (µ)x } and its Ito-calculus is contained in many relevant papers, and books; see e.g., [JT20, JT19b, AJL17, AJ15, AJ12, AJL11, IM74, IM63, HØUZ10, JT21].
.
We shall refer to the literature, e.g., [JT20, JT19b, AJL17, AJ15, JT21]. Suffice it to say that the Ito-lemma for the Gaussian process {W (µ) t } is a key tool; together with the following fact for the quadratic variation: Fix µ, σ-finite and non-atomic. Let λ = dx = Lebesgue measure,
Theorem 3. 26 .
26Consider a fixed positive non-atomic Borel measure µ on J = [0, b], b < ∞;and define an operator A on L 2 (µ) = L 2 (J, µ) as follows:For ϕ ∈ L 2 (µ) ∩ C,
ϕ
(s) µ (ds) dy (see (3.66)), which defines a compact integral operator in L 2 (µ).Then A = A µ is bounded and compact with triangular integral kernela (x, s) = χ [0,x] (s) (x − s) .(3.68)
Figure 3 . 3 .
33The kernel a (x, s) = χ [0,x] (s) (x − s) = [x − s] + = max (0, x − s).
Ω x := {ω ∈ Ω : ω (0) = x} , and (3.83) P denotes the probability measure on (Ω, C ), such that Borel sets A, B ∈ C .From the construction the projection π 0 : Ω → Ω, π 0 (ω) = ω (0),
( 3 . 103 )
3103Remark 3.32. Solutions to (3.102) depend on choices of boundary conditions, i.e., selfadjoint realizations of the Krein-Feller operator. Two examples are included below:
•
Two particular selfadjoint extensions:-Neumann: f (0) = f (1) = 0 -Dirichlet: f (0) = f (1) = 0 • Maximal dissipative extension (diffusion semigroups)
Figure 3
3Remark 3.36 (Diffusion paths for Brownian motion, and for generalized Brownian motion). Diffusion paths:
(B) = 0 ⇒ df (B) = 0, and df µ with df /dµ = T µ f which is (3).(3)⇒(2) is clear.
Corollary 3 . 38 .
338From Theorem 3.1 we now get the following dual pair (with dense domains)
iterated function system (IFS) measures; while the second specializes to IFS measures with support contained in finite intervals, and with the Krein-Feller operators. Background references for this include [Hut81, Jor18, JP98a, JP98b, Roh49, Roh52, Roh64a, LOSS20, ARCG + 20, QS13, AJL11]. 4.1. Iterated function system measures. The theory of iterated function system (IFS) measures is extensive. IFS measures arise in diverse applications, geometric analysis, fractal harmonic analysis, chaotic dynamics and more. Here we wish to cite the following papers of most direct relevance for our current discussion, [JT19a, HJW19b, BJ19, HJW19a, BJ18, JT18, JPT16, JP16, BF21, MP21, WLLZ20, AJL11, Jor18].
Definition 4 . 1 .
41Let (X, B X ) be a measurable space, N ∈ N, and let {σ i } N i=1 be a system of continuous endomorphisms σ
Figure 4 . 1 .
41
Figure 4
4Figure 4.1
Example 4. 4 (
4Three IFS measures, Lebesgue measure and two Cantor measures). Let N = 2 and {p i } = 1 2 , 1 2 .
Figure 3 . 2 .Figure 4 .
324The scale 4 Cantor measure µ 4 with g µ4 from Example 4.4 is shown inFigure 4.2 below. Note these two cases both have equal weights p = (p i ) 2. g µ4 (·) the scale 4 Cantor measure as a Stieltjes measure.
represented as a signed Stieltjes measure,F = df , then ∇ (µ) f = T µ F . RKHS; different settings for K µ (A, B) = µ (A ∩ B) (X, B) general, µ (A ∩ B) special measures on (X, B) X ⊂ R, so [0, 1] or [0, ∞] etc. A = [0, x], B = [0, y], K µ (A, B) = µ ([0, x ∧ y]) Special case µ = λ = dx = Lebesgue measure, K λ = x ∧ y GENERAL F ∈ H (K µ ) with σ-finite measure s.t. F µ, T µ F = dF dµ X ⊂ R, F = df where f is a bounded variation function on R; Stieltjes measure df µ, T µ F = dF dµ
a) X = R, B = [0, x], A = [x, y] (b) (X, B, µ) general measure spaceFigure 5.1. µ (· ∩ A)Stieltjes measuresIt follows from Theorem 5.2, that elementsF in H (K µ ) are signed measures on (X, B) s.t. F µ. Set T µ F := dF dµ , then T µ : H (K µ ) ≈ − − → L 2 (µ) (5.11) is an isometric isomophism, i.e., dF/dµ L 2 (µ) = F H (µ) .(5.12)In the special case of (X, B) = (R, B), or J an interval, e.g., J = [0, 1] or J = [0, ∞], the general conclusions (5.11)-(5.12) simplify as follows.All signed measures on (R, B) have the form F = df for a bounded variation function f on (X, B), i.e., ϕdF = ϕdf (5.13) as a Stieltjes measure, where df ([x, y]) = f (y) − f (x) (5.14) for intervals and extend to all B ∈ B. Moreover the operator T µ f = f (µ) from before then agree (5.11)-(5.12), i.e.,T µ f = df dµ (5.15) since df ∈ H (K µ ) ⇐⇒ df µ, (5.16)so that the Radon-Nikodym derivative df dµ in (5.15) is well defined and (see (5.14))f (y) − f (x) = X (T µ f ) dµ, (5.17)which is the form we used before when T µ f = f (µ) is the Stieltjes measure, B ∈ B, and B a σ-algebra. So we have T µ f := f (µ) , and (5.18) is the standard extension of df defined first on intervals df ([x, y]) = f (y) − f (x), and then extended to Borel sets df (B). Note that every µ is a Stieltjes measure f (x) := µ ([0, x]), so of the form µ = df .
Example 6 . 3 .
63Consider J = [0, 1]. Let µ n = Lebesgue measure restricted to 1 2 n+1 , 1 2 n , n ∈ N. Let f (x) = √ x over [0, 1], so that f (s)
Table 5 .1
5
i=1 = {1/2, 1/2}.
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. E T (palle, Jorgensen, address: [email protected] F. Tian) Mathematical Reviews. 4164Department of Mathematics, The University of Iowa, Iowa CityIA 52242-1419, U.S.A. Email address: [email protected](Palle E.T. Jorgensen) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A. Email address: [email protected] (James F. Tian) Mathematical Reviews, 416 4th Street Ann Arbor, MI 48103-4816, U.S.A. Email address: [email protected]
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arxiv |
Antiferromagnetism, spin splitting, and spin-orbit interaction in MnTe
Suman Rooj
Department of Physics
Indian Institute of Science Education and Research Bhopal
462066Bhauri, BhopalIndia
Jayita Chakraborty
Department of Physics
Indian Institute of Science Education and Research Bhopal
462066Bhauri, BhopalIndia
Nirmal Ganguli
Department of Physics
Indian Institute of Science Education and Research Bhopal
462066Bhauri, BhopalIndia
Antiferromagnetism, spin splitting, and spin-orbit interaction in MnTe
(Dated: 20 January 2023)
Hexagonal MnTe emerges as a critical component in designing magnetic quantum heterostructures, calling for a detailed study. After finding a suitable combination of exchange-correlation functional and corrections, our study within ab initio density functional theory uncovers an insulating state with a preferred antiferromagnetic order. We compute the exchange interaction strengths to estimate the antiferromagnetic ordering temperature via Monte Carlo calculations. Our calculations and symmetry analysis reveal a large spin splitting in the system due to the antiferromagnetic order without considering spin-orbit interaction, except in the kx-ky plane. Critically examining the band dispersion and spin textures obtained from our calculations and comparing them with an insightful symmetry analysis and analytical model, we confirm a combined Rashba-Dresselhaus interaction in the kx-ky plane, around the K point of the system. Finally, we find ferroelectricity in the system for a higher energy magnetic configuration. Our results and insights would help design heterostructures of MnTe for technological applications. *
I. INTRODUCTION
Besides exhibiting interesting physical properties, magnetic quantum materials and heterostructures hold immense promise for future technology. A combination of topologically nontrivial properties with magnetism, Rashba-like spin-orbit interaction, spin Hall effect, and strong electronic correlation may lead to numerous possibilities for designing and operating devices [1,2]. Antiferromagnetic spintronics, where antiferromagnetic materials with substantial spin-orbit interaction can be used for superfast computational processing and nonvolatile memory by manipulating the spin-textures, may be envisaged as one such possible direction [3]. The idea of antiferromagnetic spintronics bases itself on the realization of externally manipulable antiferromagnetic spin textures via spin-orbit torque. Rashba-like spin-orbit interaction or spin Hall effect may help realize such tunable spinorbit torque, storing and processing information via spin textures at a THz frequency and tiny power consumption. Recently, spin splitting due to antiferromagnetic order in the absence of combined time reversal, spatial inversion symmetry, and combined spin reversal, translation symmetry has emerged as a promising research direction [4,5]. However, the search for suitable materials for the purpose is still on, inspiring researchers to design new heterostructures with the necessary ingredients [6]. In this context, MnTe has been recognized as an interesting antiferromagnetic band insulator and a component for designing suitable heterostructures [7][8][9][10][11][12][13][14]. Experiments revealed room-temperature antiferromagnetism along with anisotropic magnetoresistance and spin-flop transitions in bulk MnTe [7,8]. Strong magnetic anisotropy has been reported in Lidoped MnTe [9]. MnTe|InP heterostructure exhibits interfacial conduction along with ferromagnetism [10], while thin films of antiferromagnetic MnTe reportedly host planar Hall effect [11]. Several possible heterostructures combining MnTe have been envisaged, with a few already synthesized [12][13][14]. Bi 2 Te 3 |MnTe bilayers have been proposed to realize charge-magnon conversion, useful in antiferromagnetic magnonics to improve the performance of magnon transistors and magnon torque memories [12]. Ferromagnetic|antiferromagnetic|ferromagnetic heterostructures of Cr 2 Te 3 |MnTe|Cr 2 Te 3 has been synthesized to host high coercivity and exchange bias [13]. Pournaghavi et al. [14] proposed Chern insulator and axion insulator phases in MnTe|Bi 2 (Se,Te) 3 |MnTe heterostructures based on density functional theory (DFT) and tight-binding model calculations. A similar compound, GeTe, has been reported from DFT calculations to host a strong Rashba interaction along with ferroelectricity in the bulk form, where the Te atoms play a major role in realizing the spin-orbit interaction [15]. Based on DFT calculations, bulk MnTe has recently been shown to host a spin splitting even without considering spin-orbit interaction, a feature that may offer more robust application potential than the spin splitting due to spin-orbit interaction [16].
The above discussion reveals the usefulness of MnTe for quantum technologies. It highlights the importance of thoroughly understanding the physical properties, particularly the electronic structure, magnetic properties, and spin-orbit interaction, so that heterostructures involving MnTe can be rationally designed to host the desired properties. Therefore, using first principles density functional theory within a judicially chosen combination of exchange-correlation functional and corrections, we study the interesting physical properties of MnTe in its hexagonal bulk structure. After carefully understanding the electronic structure, preferred magnetic configuration, and magnetic interactions, we critically examine the spin splitting without and with spin-orbit interaction using DFT calculations, symmetry analysis, and an analytical model. Subsequently, we explore the possibility of ferroelectricity in the system. The remainder of the
II. CRYSTAL STRUCTURE AND METHOD
Bulk MnTe crystallizes in a hexagonal NiAs structure with space group P 6 3 /mmc, Mn and Te occupying 2a and 2c Wyckoff positions, respectively. Fig. 1(a) depicts a unit cell of MnTe, illustrating face-sharing MnTe 6 octahedra. The total energy, electronic structure, magnetic properties, spin-orbit interaction, and ferroelectricity calculations presented here are performed within density functional theory, as implemented in the vasp code [17,18]. A plane wave basis set with 500 eV kinetic energy cutoff is employed to expand the wavefunctions within the projector augmented wave (PAW) method [19]. We systematically tested various exchangecorrelation functionals combined with appropriate corrections including local density approximation (LDA) [20,21], generalized gradient approximation (GGA) [22], a meta-GGA functional viz. SCAN [23], Hubbard-U correction [24], and van der Waals correction rVV10 [25] to find which combination of functional and corrections best suite our purpose. The Brillouin zone integration is performed within corrected tetrahedron method [26] using a Γ-centered k-point mesh of 9 × 9 × 6 and 5 × 5 × 4 for the unit cell and a 2 × 2 × 2 supercell, respectively. A U eff = U − J = 3 eV is used for the description of Mn-3d states with reasonably strong Coulomb correlation. The atomic positions and lattice vectors are optimized to minimize the Hellman-Feynman force on each atom to a threshold of 10 −2 eVÅ −1 and the stress on the simulation cell, respectively. Table I compares the predictive capability of different exchange-correlation functionals and corrections, revealing spin-polarized SCAN + rVV10 + Hubbard-U and spin-polarized GGA + Hubbard-U func- tionals to best predict the lattice constants. Since a combination of SCAN and rVV10 has been demonstrated to outperform other functionals in layered materials [25], we continue our further calculations within the framework of SCAN + rVV10 + Hubbard-U . Whenever appropriate, spin-orbit interaction is considered in our calculations. The Berry-phase method is employed to calculate the ferroelectric properties of the system [28,29].
III. RESULTS AND DISCUSSIONS
After choosing an appropriate combination of exchange-correlation functional and corrections, we systematically investigate the electronic structure, magnetic properties, spin-orbit interaction, and ferroelectric properties of MnTe.
A. Electronic Structure and magnetic properties
In a nominal sense of ionic compound, Mn and Te atoms in MnTe are expected in 2+ and 2− oxidation states, respectively. We start with defining the possible magnetic configurations and identifying the exchange paths in the system. Besides the ferromagnetic (FM) configuration, where magnetic moments from all Mn atoms align along the same direction, several antiferromagnetic (AFM) configurations may be envisaged depending on the exchange interactions, leading to no net magnetic moment of the system. The arrangement of magnetic moments in AFM1, AFM2, AFM3, AFM4, and AFM5 are illustrated in Fig. 1(b),(c),(d),(e),(f), respectively. Table II shows the relative energies for all the magnetic configurations, suggesting AFM4 to have the lowest energy. The magnetic moments of Mn ions in AFM4 configuration are parallel to each other in the hexagonal basal plane, whereas antiparallel along the cdirection. Unless stated otherwise, our results discussed subsequently corresponds to the lowest energy AFM4 configuration. The density of states (DoS) for both spins and band dispersion for one spin of the system in AFM4 configuration, shown in Fig. 2(a) and Fig. 2(b), respectively, suggest a band gap of ∼1.16 eV, which is in reasonable agreement with 1.27 eV gap reported from experiment [31]. We attribute our choice of SCAN + rVV10 + Hubbard-U framework to this fair agreement. The projected DoS (see Fig. 2(a)) indicates that while the valence band comprises hybridized Mn-3d and Te-5p orbitals, the conduction band predominantly consists of only Mn-3d orbitals. The band dispersion (see Fig. 2(b)) reveals an indirect band gap with the valence band maximum at A-point and the conduction band minimum at K-point.
a b c (a) (b) (c) (d) (e) (f) AFM1 AFM2 AFM3 AFM4
A projected magnetic moment of 4.54 µ B is observed at each Mn, suggesting a high-spin configuration of Mn 2+ ion. We note that the projected magnetic moment is often slightly less than the actual magnetic moment when a plane wave basis set is used for expanding the wave-
4 3 2 1 0 1 2 3 DoS (eV -1 cell -1 ) -4 -2 0 2 4 E -E F (eV) Total Mn-3d Te-5p Γ K M Γ A L H A Spin A Spin B (a) (b) FIG. 2.
Spin-polarized density of states along with Mn-3d and Te-5p orbital-projected DoS for AFM4 configuration is displayed in (a), while (b) represents the band dispersion from both spins marked as spin A and spin B. [8]. The band dispersion shows a perfect overlap of the bands of both spins along the chosen high-symmetry directions.
Exchange interaction and ordering temperature
The magnetic exchange interaction strengths J in the spin Hamiltonian H = − ij J ij S i S j with S and i, j representing spins and site indices, respectively, can be evaluated along different directions from the relative energies of the various magnetic configurations listed in Table II using a method vividly described in ref. [32,33]. The important exchange interaction strengths along the paths marked as J 1 , J 2 , and J 3 marked in Fig. 1(b) are estimated and tabulated in Table III. While the Mn ions connected via J 1 path can have direct exchange interaction, J 2 and J 3 paths allow superexchange via one and two Te-ions, respectively, explaining the different orders of their magnitude. The negative signs of the prominent J-values suggest an antiferromagnetic ground state, consistent with our findings.
A magnetic ordering temperature, known as the Néel temperature for antiferromagnetic systems, may be estimated using the exchange interaction strengths given in Table III. We compute the magnetic specific heat per spin C given as [34]
C = 1 N ∂U ∂T = ε 2 − ε 2 N k B T 2 ,(1)
where ε, N , k B , and T represent the energy of each magnetic configuration, number of spins, Boltzmann constant, and temperature, respectively. U (T ) = ε is the average internal energy, ... representing thermal average at a given temperature. We calculate the energy of magnetic configurations as
ε = −J 1 ij S i S j − J 3 ij S i S j ,(2)
where i, j run over the atoms satisfying the distance of the corresponding exchange path J 1 , or J 3 . Owing to its small value, we ignore J 2 for our calculations but retain J 3 due to twice as many neighbors at that distance compared to J 2 , making it significant. A large projected magnetic moment at the Mn site indicates the high spin configuration of Mn 2+ ions, justifying our approach. Considering three different lattice sizes 15×15×15, 18×18×18, and 20 × 20 × 20 within periodic boundary condition, we evolved the system in imaginary time using Metropolis Monte Carlo simulation algorithm and Boltzmann distribution function to calculate ε 2 and ε at different temperatures. A spin S i is randomly selected from the lattice, and the change in energy ∆ε upon flipping the spin is computed from Eq. (2). If the corresponding Boltzmann weight exp(−∆ε/k B T ) is greater than a uniform random number r ∈ (0, 1), the spin-flip is accepted. Subsequently, we move on to the next time step by randomly choosing another spin S i [35,36]. The averaging is performed over 10 5 imaginary time steps after bringing the system to thermal equilibrium in 10 7 imaginary time steps for a given temperature. Our results, displayed in Fig. 3 for lattice sizes 15 × 15 × 15, 18 × 18 × 18, and 20 × 20 × 20 lattice sizes within periodic boundary condition, indicate an ordering temperature of ∼240 K with negligible variation for different lattice sizes. We note that experiments reported a magnetic ordering temperature around 310 K [8], indicating a minor underestimate from our calculations, possibly due to errors in the calculated J-values.
Spin splitting
After confirming the magnetic ground state and estimating the magnetic exchange interactions, hence the ordering temperature, we analyze the spin splitting in the system driven by antiferromagnetic exchange interaction [16]. While the band dispersion shown in Fig. 2
(b) along Γ → K → M → Γ → A → L → H → A direction
reveals no spin splitting, the band dispersion plotted alonḡ L/2 → Γ → L/2 direction, displayed in Fig. 4(a) without considering spin-orbit interaction reveals pronounced spin splitting, in agreement with Ref. [16], calling for a careful analysis of the spin splitting. The magnetic system in AFM4 configuration does not preserve the combined time-reversal and spatial inversion symmetry; a combined spin rotation (reversal) and translation symmetry is also broken, except for some local symmetries in the k x -k y plane, owing to the parallel arrangement of spins in the ab plane and antiparallel arrangement along the cdirection. Thus, the spin degeneracy remains protected in the k x -k y plane [4], as seen from Fig. 4(b) and Fig. 4(c), exhibiting the 3D band dispersion as a function of (k x , k y ) and the isoenergetic contours for E − E F = −0.6 eV in the k x -k y plane, respectively, for the pair of bands within a small energy range [−0.6, −0.44] eV relative to the Fermi level, intersecting a horizontal dotted (red) line at E − E F = −0.6 eV in Fig. 4(a). However, since the local symmetry does not hold in other planes, the 3D band dispersion as a function of (k y , k z ) and the corresponding isoenergetic contours for E − E F = −0.6 eV in the k y -k z plane, shown in Fig. 4(d) and Fig. 4(e), respectively, for the same pair of bands exhibit a pronounced spin splitting without considering spin-orbit interaction. Similarly, the isoenergetic contours in the k xk z plane displayed in Fig. 4(f) also exhibit large spin splitting, owing to the absence of the local symmetry in the plane. We note that the spin splitting without spin-orbit interaction discussed here may not be useful for spintronic applications in a heterostructure involving (0001)-terminated MnTe unless the magnetic configuration drastically changes upon forming the heterostructure since the Brillouin zone would effectively collapse to a k x -k y plane in such a heterostructure. However, if realized, a spin splitting in the k x -k y plane due to spin-orbit interaction may be useful for spintronic applications in a heterostructure involving (0001)-terminated MnTe, encouraging us to investigate the implications of spin-orbit interaction in the system under consideration.
B. Spin-orbit interaction
Upon considering spin-orbit interaction in our DFT calculations, we find substantial magneto-crystalline anisotropy and splitting of bands in the momentum space due to Rashba-like interaction along with a projected orbital moment of 0.011 µ B at each Mn site. Below we discuss our results in detail.
Magneto-crystalline anisotropy
Spin-orbit interaction often leads to anisotropy in spin quantization along different crystallographic directions, particularly for non-cubic crystals. We compute the relative energies for quantizing the spin along four crystallographic directions viz. (1000), (0100), (0001), and (1120) for hexagonal MnTe; and tabulate the results in Table IV. Our results reveal (1120) direction in abplane to be the most preferred spin quantization direction among the ones considered, consistent with a previous report of magneto-crystalline anisotropy in hexagonal MnTe [8], resulting in a collinear magnetic arrangement (AFM4 configuration), leaving the magnetic space group unchanged.
Rashba-like interaction
In order to understand the electronic structure of MnTe upon considering spin-orbit interaction, we plot the band dispersion along the high-symmetry lines of a hexagonal Brillouin zone and density of states, as shown in Fig. 5(a) and Fig. 5(b), respectively. We note that considering spin-orbit interaction has reduced the calculated band gap to ∼0.9 eV. Some bands below and above the Fermi level exhibit a Rashba-like spin splitting in the momentum space; we take a closer look at the bottom of the conduction band having a predominant Mn-3d character, as featured in the inset of Fig. 5(b).
To better understand the spin-orbit interaction in the lowest pair of conduction bands, we plot the energy dispersion as functions of (k x , k y ), as displayed in Fig. 6(a). A cross-section of the bands in an appropriate part of the Brillouin zone, as displayed in Fig. 6(b), reveals Rashbalike splitting of the pair of bands in momentum space. Fig. 6(c) exhibits a pair of isoenergetic contours for the same pair of bands at E − E F = 1.20 eV along with projected spin texture, as obtained over an energy range [0.88, 1.25] relative to the Fermi level from our DFT calculations [37]. We find the projected spins arranging themselves like a persistent spin helix (PSH) [38], a characteristic of combined Rashba and Dresselhaus spin-orbit interactions of similar strengths. The idea of Rashba-Dresselhaus interaction in bulk MnTe intrigues us as the structure corresponds to a centrosymmetric space group P 6 3 /mmc. Dresselhaus and Rashba interactions require bulk inversion asymmetry (BIA) and structure inversion asymmetry (SIA) with a microscopic electric field, respectively. The asymmetry may arise from the point groups corresponding to some atomic sites instead of the space group [39][40][41][42]. In the present case, Mn and Te ions occupy 2a and 2c Wyckoff positions corresponding to point groups D 3d and D 3h , respectively. While the point group D 3d corresponding to Mn-sites preserves centrosymmetry, the point group D 3h corresponding to Te-sites, although nonpolar, lacks centrosymmetry. The bulk inversion asymmetry thus introduced is expected to lead to only a Dresselhaus (D-2) interaction [39]. Additionally, the microscopic arrangement of alternating +2| − 2 charged layers along the (0001) direction due to the arrangement of Mn 2+ and Te 2− ions leads to a microscopic electric field along the same direction [43], resulting in Rashba interaction. Hence, finding a combined Rashba-Dresselhaus spin-orbit interaction in the system is possible. Further, the crys-tal point group symmetry coincides with the wave vector point group symmetry D 3h at the high-symmetry K point [44]. We identify a magnetic space group Cmcm for our system [45], a type I magnetic space group according to Ref. [46] and spin splitting prototype 4A according to Ref. [5], where spin splitting is expected even without spin-orbit interaction. However, owing to a collinear magnetic arrangement, the local symmetry discussed earlier for the k x -k y plane ensures that the spin expectation value S n ( k) for the n th band satisfies the condition S n (− k) = − S n ( k) in the k x -k y plane around the K point, as verified from the spin texture displayed in Fig. 6(c). Hence, the Rashba-Dresselhaus terms would take their conventional form in the Hamiltonian for the k x -k y plane around the K point [41].
To examine the hypothesis of Rashba-Dresselhaus interactions against our DFT results, considering the above-mentioned symmetry conditions, we construct an analytical model of combined Rashba-Dresselhaus spinorbit interaction in the k x -k y plane around the K point, as described by the Hamiltonian H RD (in the units ofh) [6]
H RD = H 0 + H R + H D , where(3)H 0 = − 1 2m * ∂ 2 ∂x 2 + ∂ 2 ∂y 2 and H R = α(k y σ x − k x σ y ); H D = β(k y σ y − k x σ x ),
with (σ x , σ y ) and m * representing the Pauli matrices and the effective mass, respectively. Considering the Rashba and Dresselhaus terms as perturbations to the free electron-like Hamiltonian H 0 , we find the energy eigenvalues
ε ± RD (k x , k y ) = k 2 x + k 2 y 2m * ± (α 2 + β 2 )(k 2 x + k 2 y ) − 4αβk x k y = k 2 2m * ± k α 2 + β 2 − 2αβ sin 2φ,(4)
where we denote (k , φ) as the polar coordinates in the k x -k y plane. Setting the expressions for eigenvalues in Eq. (4) to a suitable constant energy will give us the equations for isoenergetic contours in k x -k y plane. The eigenstates for the Hamiltonian may be expressed as
|± RD = 1 √ 2 (±ζ RD | ↑ + | ↓ ) , where(5)ζ RD = i α 2 + β 2 − 2αβ sin 2φ α exp(iφ) − iβ exp(−iφ)
, ζ * RD ζ RD = 1.
The projected spin components S x and S y may be
S x + RD = 1 2 RD +|σ x |+ RD = 1 4 (ζ RD + ζ * RD ) = 1 2 α sin φ − β cos φ α 2 + β 2 − 2αβ sin 2φ = − 1 2 RD −|σ x |− RD = − S x − RD , S y + RD = 1 2 RD +|σ y |+ RD = 1 4 (ζ RD − ζ * RD ) = 1 2 β sin φ − α cos φ α 2 + β 2 − 2αβ sin 2φ = − 1 2 RD −|σ y |− RD = − S y − RD S z + RD = 1 2 RD +|σ z |+ RD = 1 4 (ζ * RD ζ RD − 1) = 0 = 1 2 RD −|σ z |− RD = S z − RD .(6)
C. Magnetostriction and ferroelectricity
Finally, we explore the possibility of ferroelectric polarization in hexagonal bulk MnTe system with and without spin-orbit interaction. Our results reveal that upon optimizing, the atomic positions of the lowest energy AFM4 magnetic configuration do not change from the Wyckoff 2a and 2c positions. As discussed earlier, the P 6 3 /mmc space group preserved overall centrosymmetry and none of the D 3d and D 3h point groups associated with Mn and Te sites, respectively, exhibit any polar nature. Hence, as expected, our calculations reveal no ferroelectric polarization for bulk MnTe in the AFM4 configuration. However, we observe a pronounced magnetostrictive effect for AFM1 configuration with the Mn magnetic moments arranged in ↑↑↓↓ order along the c direction (see Fig. 1(b)) [47]. Upon relaxation, the distance between Mn atoms along the c-direction with parallel (antiparallel) spin ori-entation increases (decreases) by ∼0.025Å. Such structural changes due to magnetostriction induce a substantial electric polarization of 537 µC m −2 (535 µC m −2 ) (without) considering spin-orbit interaction. Te-5s lone electron pair may contribute to the electric polarization, as discussed in Ref. [48]. However, since the electric polarization does not correspond to the lowest energy magnetic configuration, we cannot call it a multiferroic material.
IV. CONCLUSION
To conclude, we studied bulk hexagonal MnTe within first-principles density functional theory to understand its physical properties, including electronic structure, magnetism, spin-orbit interaction, and ferroelectricity. We begin by examining different exchange-correlation functionals and corrections to find an optimum combination for our calculations. After converging on an appropriate combination of exchange-correlation functional and corrections, we study the system's electronic structure and magnetic properties. Our results reveal an insulating nature of MnTe in its bulk form with a collinear antiferromagnetic order. We estimate the exchange interaction parameters along various exchange paths by mapping the interactions to a spin Hamiltonian. Subsequently, we evaluate the antiferromagnetic ordering temperature based on the exchange interaction strengths using the Metropolis Monte Carlo algorithm to be ∼240 K. We find a large spin splitting in the lowest energy antiferromagnetic configuration of the system even without considering spin-orbit interaction, except in the k x -k y plane where the spin degeneracy remains protected by local symmetry, as long as spin-orbit interaction is not considered. Our systematic and critical examination of spin-orbit interaction via usual band dispersion, 3D band dispersion, isoenergetic contours, and projected spin directions obtained from DFT results compared with an insightful analytical model unequivocally confirms a combined Rashba-Dresselhaus interaction in the k x -k y plane, around the K point of the system. We attribute the Rashba-Dresselhaus interaction in a structure with centrosymmetric space group P 6 3 /mmc to a non-centrosymmetric, nonpolar point group corresponding to the Te site and a microscopic electric field along the (0001) direction owing to alternating +2| − 2 charged layers. A heterostructure involving (0001)-terminated MnTe may host spin splitting only of Rashba-Dresselhaus type, owing effectively to the accessibility of only k x -k y plane in the reciprocal space for such a heterostructure, provided the magnetic arrangement remains unaltered. If the magnetic arrangement realized in a possible heterostructure is such that the local symmetry protecting antiferromagnetic spin degeneracy is no longer preserved, an antiferromagnetic spin splitting combined with a Rashba-Dresselhaus spin splitting may be realized, with the former dominating over the latter. Finally, we explore the possibility of ferroelectricity in the system. Although the lowest energy antiferromagnetic configuration reveals no electric polarization, another antiferromagnetic configuration with ↑↑↓↓ arrangement of magnetic moments along the c-direction exhibits a pronounced magnetostrictive effect, resulting in substantial electric polarization. Although the bulk material may not have a stable AFM1 configuration to host ferroelectricity, the feature may be present in a possible heterostructure if the relevant magnetic configuration gets stabilized due to the synthesis conditions and/or stress. Further, lattice distortion in the process of formation of a possible heterostructure may stabilize a ferroelectric phase in MnTe. Our studies help understand the electronic structure, magnetism, spin splitting, and spin-orbit interaction in bulk hexagonal MnTe, paving the way for its application in potential spintronic devices. ACKNOWLEDGMENTS SR acknowledges CSIR, India, for a research fellowship through grant number 09/1020(0157)/2019-EMR-I. NG acknowledges financial support from SERB, India, through grant number CRG/2021/005320. The use of high-performance computing facilities at IISER Bhopal and PARAM Seva within the framework of the National Supercomputing Mission, India is gratefully acknowledged.
FIG. 1 .
1The face sharing MnTe6 octahedra are illustrated in the depiction of a unit cell in (a), while The magnetic exchange paths and the possible antiferromagnetic configurations AFM1, AFM2, AFM3, AFM4, and AFM5 are depicted in (b), (c), (d), (e), and (f), respectively. The illustrations have been prepared using vesta software[30].
FIG. 3 .
3The magnetic specific heat with temperature for lattice sizes (a) 15 × 15 × 15, (b) 18 × 18 × 18, and (c) 20 × 20 × 20 are shown here.
FIG. 4 .
4Panel (a) shows the spin-polarized MnTe bands alongL/2 → Γ → L/2 direction. We identify a pair of such spin-split bands within a small energy range [−0.6, −0.44] eV relative to the Fermi level, intersecting a horizontal dotted (red) line at E − EF = −0.6 eV, and show their dispersion as functions of (kx, ky) with kz = 0 in (b). The corresponding isoenergetic contours in the kz = 0 plane are shown in (c) for E − EF = −0.6 eV. Panel (d) shows the dispersion of the same pair of bands as functions of (ky, kz) with kx = 0. Panels (e) and (f) exhibit the isoenergetic contours for E − EF = −0.6 eV in the kx = 0 and ky = 0 planes, respectively.
FIG. 5 .
5The band structure and density of states of MnTe considering spin-orbit interaction are shown in (a) and (b), respectively. The inset of (b) enlarges the bottom of the conduction band at the high-symmetry point K to reveal Rashbalike splittings of the bands in momentum space.
FIG. 6 .
6The energy dispersion as functions of (kx, ky) (3D bands) for the lowest two conduction bands is shown in panel (a). Panel (b) exhibits a cross-section of the 3D bands near the high-symmetry point K. Isoenergetic contours for the same bands at E − EF = 1.20 eV along with projected spins over an energy range [0.88, 1.25] eV relative to the Fermi level from DFT calculations are displayed in (c). Panels (d), (e), and (f) represent the 3D bands, a cross-section of the 3D bands, and isoenergetic contours with projected spins, respectively, obtained from our combined Rashba-Dresselhaus interaction model. evaluated using the eigenstates in Eq. (5) as
Fig. 6 (
6d) andFig. 6(e) illustrate the eigenvalues obtained in Eq. (4), whileFig. 6(f) depict the corresponding isoenergetic contours for a suitable energy and the projected spin directions obtained from Eq.(6). We note the remarkable match between our DFT results and a similar depiction from the analytical model considered here, suggesting that our model essentially describes the spin-orbit interaction observed in the system. The model reasonably fits the DFT results for m * = 0.003m e , m e denoting the mass of an electron, α = 0.39 eVÅ, and β = −0.30 eVÅ, thus conclusively characterizing the nature of spin-orbit interaction in the system. The remarkably small value of the effective mass m * indicates a super-light electron-like band under consideration.
TABLE I .
ILattice constants of MnTe obtained from the experiment and calculations using different exchange-correlation functionals and corrections are tabulated here.article is organized as follows: The crystal structure and our calculation methodologies are described in Sec. II. The results of our calculations are thoroughly discussed in Sec. III. Finally, we summarize the work in Sec. IV.Experiment and different functionals
a = b (Å) c (Å)
Experiment [27]
4.15
6.71
Spin-unpolarized LDA
4.002
4.802
Spin-unpolarized LDA + U
4.021
4.849
Spin-polarized LDA
3.795
5.828
Spin-polarized LDA + U
4.067
6.436
Spin-unpolarized GGA
4.097
4.899
Spin-unpolarized GGA + U
4.128
5.020
Spin-polarized GGA
4.102
6.444
Spin-polarized GGA + U
4.193
6.704
Spin-unpolarized SCAN + rVV10
4.041
4.861
Spin-unpolarized SCAN + rVV10 + U
4.101
5.105
Spin-polarized SCAN + rVV10
4.079
6.493
Spin-polarized SCAN + rVV10 + U
4.165
6.683
TABLE II .
IIThe relative energies for different magnetic configurations are tabulated here.Magnetic configuration
Relative energy (meV)
FM
938.87
AFM1
379.46
AFM2
26.69
AFM3
400.44
AFM4
0.0
AFM5
769.39
TABLE III .
IIIThe exchange interaction parameters (Js) are tabulated here.Exchange path Distance (Å) Exchange strength (meV)
J1
3.342
−3.79
J2
4.165
−0.02
J3
5.34
−0.04
functions. Neutron diffraction experiments revealed a
magnetic moment of 4.7−5 µ B per Mn atom
TABLE IV .
IVThe relative energies of spin quantization along different crystallographic directions considering spin-orbit interaction are tabulated here.Spin quantization axis
Energy (meV/formula unit)
(1000)
0.19
(0100)
3.23
(0001)
0.075
(1120)
0
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| {'fraction_non_alphanumeric': 0.061827518747962176, 'fraction_numerical': 0.04835751548744702, 'mean_word_length': 4.379631659723745, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Hexagonal MnTe emerges as a critical component in designing magnetic quantum heterostructures, calling for a detailed study. After finding a suitable combination of exchange-correlation functional and corrections, our study within ab initio density functional theory uncovers an insulating state with a preferred antiferromagnetic order. We compute the exchange interaction strengths to estimate the antiferromagnetic ordering temperature via Monte Carlo calculations. Our calculations and symmetry analysis reveal a large spin splitting in the system due to the antiferromagnetic order without considering spin-orbit interaction, except in the kx-ky plane. Critically examining the band dispersion and spin textures obtained from our calculations and comparing them with an insightful symmetry analysis and analytical model, we confirm a combined Rashba-Dresselhaus interaction in the kx-ky plane, around the K point of the system. Finally, we find ferroelectricity in the system for a higher energy magnetic configuration. Our results and insights would help design heterostructures of MnTe for technological applications. *', 'arxivid': '2301.07985', 'author': ['Suman Rooj \nDepartment of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia\n', 'Jayita Chakraborty \nDepartment of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia\n', 'Nirmal Ganguli \nDepartment of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia\n'], 'authoraffiliation': ['Department of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia', 'Department of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia', 'Department of Physics\nIndian Institute of Science Education and Research Bhopal\n462066Bhauri, BhopalIndia'], 'corpusid': 256000041, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16270, 'n_tokens_neox': 13306, 'n_words': 7382, 'pdfsha': '7cab06459defb0cff10b4d53dac9d21b5710bc8a', 'pdfurls': ['https://export.arxiv.org/pdf/2301.07985v1.pdf'], 'title': ['Antiferromagnetism, spin splitting, and spin-orbit interaction in MnTe', 'Antiferromagnetism, spin splitting, and spin-orbit interaction in MnTe'], 'venue': []} |
arxiv |
Entanglement-enhanced dual-comb spectroscopy
Haowei Shi
Ming Hsieh Department of Electrical and Computer Engineering
University of Southern California
90089Los AngelesCaliforniaUSA
Zaijun Chen
Ming Hsieh Department of Electrical and Computer Engineering
University of Southern California
90089Los AngelesCaliforniaUSA
Scott E Fraser
Translational Imaging Center
University of Southern California
90089Los AngelesCaliforniaUSA
Mengjie Yu
Ming Hsieh Department of Electrical and Computer Engineering
University of Southern California
90089Los AngelesCaliforniaUSA
Zheshen Zhang
Department of Electrical Engineering and Computer Science
University of Michigan
48109Ann ArborMIUSA
Quntao Zhuang
Ming Hsieh Department of Electrical and Computer Engineering
University of Southern California
90089Los AngelesCaliforniaUSA
Department of Physics and Astronomy
University of Southern California
90089Los AngelesCaliforniaUSA
Entanglement-enhanced dual-comb spectroscopy
(Dated: April 27, 2023)
Dual-comb interferometry harnesses the interference of two laser frequency combs to provide unprecedented capability in spectroscopy applications. In the past decade, the state-of-the-art systems have reached a point where the signal-to-noise ratio per unit acquisition time is fundamentally limited by shot noise from vacuum fluctuations. To address the issue, we propose an entanglementenhanced dual-comb spectroscopy protocol that leverages quantum resources to significantly improve the signal-to-noise ratio performance. To analyze the performance of real systems, we develop a quantum model of dual-comb spectroscopy that takes practical noises into consideration. Based on this model, we propose quantum combs with side-band entanglement around each comb lines to suppress the shot noise in heterodyne detection. Our results show significant quantum advantages in the uW to mW power range, making this technique particularly attractive for biological and chemical sensing applications. Furthermore, the quantum comb can be engineered using nonlinear optics and promises near-term experimentation.
I. INTRODUCTION
Dual-comb interferometry, a frequency-comb-based precision measurement technique harnessing the interference of two laser frequency combs of slightly different repetition rates in a static device, has emerged to provide unprecedented capability in various applications including spectroscopy [1][2][3], hyperspectral imaging [4,5], and light detection and ranging (LiDAR) [6][7][8][9][10]. In combination with on-chip frequency comb generators, dual-comb technique has been demonstrated with various platforms, including quantum cascade lasers [11,12], micro-resonatorbased soliton combs [13,14], electro-optic micro-rings [15] and on-chip semiconductor lasers [16].
In terms of spectroscopy, dual-comb interferometry has unique advantages of (1) rapid effective data acquisitions without mechanical moving parts [17][18][19]; (2) broad spectral coverage over the large span of a comb generator [20][21][22][23]; (3) spectral resolution reaching the comb line spacing of sub-picometers [21,24]; (4) frequency scale calibrated with the accuracy of an atomic clock [25]; (5) feasibility to on-chip integration [13,14], whose high repetition rates supports rapid measurements. For linear spectroscopy, it acquires thousands of molecular transitions simultaneously [20][21][22], providing rich spectral information for quantitative concentration analysis on a sample [26][27][28]. For nonlinear spectroscopy, the high peak power of the comb sources has been utilized for coherent anti-stokes Raman spectroscopy [17] and four-wave-mixing multidimensional spectroscopy [29] on bio-chemicals, with the potential to improve the measurement speed by several orders of magnitude. * [email protected] Similar to other broadband spectroscopic techniques, the sensitivity in a dual-comb measurement is inversely proportional to the optical bandwidth as the laser power in photo-detection is constrained due to detector nonlinearity or sample damage [2,30]. In this regard, the product of signal-to-noise ratio (SNR) per unit acquisition time and the number of resolved spectral elements is taken as the figure of merit for a dual-comb system [2]. As long-term coherent averaging [21,24,31,32] improves the sensitivity at the cost of acquisition time [20][21][22][23], these techniques hinder real-time sensing and fail to improve the figure of merit. In the power range of state-of-the-art biological and chemical sensing systems, such a figure of merit is fundamentally limited by the shot noise.
To go beyond the shot noise limit, quantum resources such as squeezing and entanglement are necessary. For example, the Laser Interferometer Gravitational-wave Observatory (LIGO) [33][34][35] and the Haloscope At Yale Sensitive To Axion CDM (HAYSTAC) dark matter search [36] inject squeezed light to suppress the shot noise. Photonic radar [37] and optomechanical force sensing [38] adopted entanglement with the distributed sensing paradigm [39]. In terms of spectroscopy, amplitudesqueezing has been demonstrated in nonlinear spectroscopy [40] and entangled two-mode squeezed vacuum has been shown to benefit linear absorption spectroscopy [41]. However, none of these advantages directly applies to dual-comb spectroscopy, as its essential component of heterodyne detection presumably precludes the use of squeezing and entanglement.
In this work, we develop a quantum description of dualcomb spectroscopy and then propose an entanglementenhanced scheme that utilizes quantum combs of light to gain sensitivity enhancement in dual-comb spectroscopy. We first provide a complete quantum model for dual-arXiv:2304.01516v2 [quant-ph] 26 Apr 2023 comb spectroscopy, which recovers the SNR results of Ref. [30] in the case of classical source. Furthermore, the quantum model allows us to design a quantum comb composed of pair-wise entanglement around each strong comb line to improve the SNR drastically. The quantum advantage is robust against loss and phase misalignment. We also provide an experimental design to engineer the quantum comb with off-the-shelf components.
II. RESULTS
A. Overview of the protocol Dual-comb spectroscopy employs the interference between the signal comb (shown in red) and the local comb (shown in blue), as shown in Fig. 1(a). The protocol's strength stems from the selection of signal and local combs with slightly different frequency spacings, f r and f r + ∆f r , respectively, as illustrated in Fig. 1(b). The signal comb interrogates the sample and undergoes loss and phase shift that can be modeled as a bosonic quantum channel, with frequency-dependent transmissivity κ(f ) and phase-shift α(f ). Meanwhile, the local comb serves as a local oscillator (LO) for the final heterodyne measurement. After mixing the LO and return at a balanced beamsplitter, information about the sample can be extracted from the photocurrent differ-
enceN (t) =ĉ † + (t)ĉ + (t) −ĉ † − (t)ĉ − (t)
, obtained from the photocurrent measurement on both output portsĉ ± (t). While we utilize quantum operator language to describe the measurement to prepare our analyses for quantum combs, our analyses also recover the results obtained from semi-classical analyses for classical combs [30].
In the classical protocol, both combs are classical-the quantum state of each comb line is in a coherent state, obeying the shot-noise-limited standard quantum limit (SQL).We propose to engineer quantum combs to further improve the performance of dual-comb spectroscopy. To begin with, we consider the case of signal comb being quantum engineered. To suppress noise below the SQL, squeezing is commonly adopted in quantum sensing protocols. However, in the case of dual-comb acquisitions, squeezing a single mode alone is inadequate, as heterodyne measurement is necessary to read out quadratures across the entire spectrum. To surpass the SQL, entanglement between different frequency modes is required so that joint quadratures are squeezed. Thus, we propose entangling the side-bands of the signal comb around each local comb line, as indicated in Fig. 1(b) by the dashed lines. Such an entangled comb with squeezing gain G allows both measured quadratures in heterodyne to be squeezed, resulting in fundamental noise a factor of 1/G below the SQL Furthermore, as we will detail in later part of the paper, in general, the LO can be engineered to be similarly entangled, which can further improve the performance, especially when the power of the local comb is similar as or lower than that of the signal comb. Note (a) Conceptual schematic for entanglementenhanced dual comb spectroscopy. The teeth share intermodal entanglement within the signal comb (red beam). The tissue is depicted in a cartoon fashion, while the real tissue slice is almost transparent in practice. (b) Schematic of the quantum comb. Each pair of signal modes beating with the same LO comb tooth (purple line) for the same intermediate frequency is entangled, indicated by a black dashed line connecting a pair of purple circles. (c) Practical SNR involving NEP-type and RIN-type noises, plotted versus signal power (analog to Fig. 2 of Ref. [30]), normalized to unit acquisition time T = 1s. We assume an ideal detector with unity efficiency, and zero loss and noise κ ≈ 1, η = 1. In (c), both signal and LO are entangled with equal gain G, which increases from 0dB (coherent-state) to 30dB in steps of 10dB, plotted in color from blue to magenta. The NEP/RIN-dictated SNR is presented by green-dot-dashed/black-dot-dashed line, along with the shot noise (SN) limit in blue-dashed. N = 10 5 , λ = 1µm, RIN= −170dBc/Hz [42], NEP= 5 × 10 −13 W/Hz 1/2 (NEP= 4.5 × 10 −15 W/Hz 1/2 is actually achievable, e.g. by Thorlabs FGA01FC-InGaAs Photodiode), PLO/PS = 5. Inset: Quantum advantage in SNR (in decibel unit) versus various values of RIN for total signal power PS=10mW (blue) and PS=10µW (red) at G=20dB. The LO-signal power ratio γ ≡ PLO/PS = 5 is fixed-although this figure shows the case of both signal and LO entangled, as we show later, only the signal needs to be entangled under large γ. that quantum engineering almost preserves the power of the comb, as the joint-squeezing power is negligible compared to the mean field in dual-comb systems.
To analyze the performance of the proposed quantum dual-comb spectroscopy protocol, we model the signal and all noise involved in the protocol systematically. In this overview, the case of bright LO is considered, where the power P LO is much larger than the signal comb power P S , while the full analysis is presented later. The information about the sample is derived from the amplitude decay and phase-shift of the signal comb, e iα √ κP S , which is subject to contamination from various sources of noise. For low-loss (κ ∼ 1) and low-noise scenarios, typical for thin sample slices and signal wavelength 5um at room-temperature, the fundamental noise stemming from fluctuation properties of the light field, ∼ 1/2G for the quantum comb. In terms of estimating the transmissivity √ κ, it contributes an inverse-law noise term O(1/P S ). In practice, the detector noises characterized by noise equivalent power (NEP) and the relative intensity noise (RIN) in laser sources also mix in, which dominate the estimation error at the low and high comb power region respectively, as analyzed for the classical protocol in Ref. [30]. In addition to the inverse-law term due to the fundamental noise, the RIN adds a constant noise term O(1) independent of P S , denoted as RIN-type noise, and the NEP adds an inverse-square-law noise term O(1/P 2 S ) to the estimation, denoted as NEP-type noise. We ignore detector dynamical range noise, as it can be resolved by engineering the detector array [30] and it has similar effects as RIN-type noise.
With all noises into consideration, we consider the realistic performance of the quantum dual-comb spectroscopy system. As we detail in later part of the paper, the signal power P S is constrained within the range of uW-mW (10 −6 W−10 −3 W) to avoid damage of tissues. We plot the signal-to-noise ratio (SNR) per second versus the total signal power under practical experimental settings in Fig. 1(c). In this uW-mW region, the performance of state-of-the-art classical systems (blue solid) is limited the shot noise (SN) limit (blue dashed). While at low/high signal power limit, the SNR converges to the NEP/RIN-dictated limit (green dot-dashed/black dot-dashed). With the quantum comb, we see that a practical entangled source of 10dB gain (G = 10, purple solid) yields a quantum advantage up to 4.9dB over the coherent-state source (blue solid). As the gain increases (from blue to magenta), the quantum advantage improves further until it saturates subject to the limits dictated by NEP-type noise alone (greed dot-dashed) and the RIN-type noise alone (black dot-dashed). In the scenario of Fig. 1(c), we observe that the ultimate limit of quantum advantage can go up to 13.4dB at P S ≈ 0.1mW, which is of great interest to bio-sensing applications. Additionally, we provide predictions at the power levels when such saturation happens (dots on the blue-magenta curves), as we detail in Methods. Our analyses show that for a state-of-art dual-comb system to en-joy quantum advantage, RIN-type of noise is often the major constraint: in the inset of Fig. 1, we show that for 10mW signal power, to enjoy a significant quantum advantage it requires RIN-type noise to be smaller than ∼ −170dBc/Hz, challenging but still possible [43]; For lower power of 10uW, the requirement is less stringent, RIN= −150dBc/Hz is readily achievable [30,42].
B. Linear absorption spectroscopy
In a linear absorption spectroscopy sensing process, one is interested in the input-output relation of light for a range of frequencies f . The pattern of the output light reveals information about the composition of the sample under study. The mathematical model for the inputoutput relation involves a thermal-loss phase-shift channel, which has frequency-dependent transmissivity κ(f ) and phase-shift α(f ). Given an input light mode described by the annihilation operatorâ S [44], the output field annihilation operator is given by the linear relation
a R = √ κe iαâ S + √ 1 − κâ E .(1)
The channel attenuates the mean of input signal modê a S by √ κ, shift the phase by α, and mixes in the environment modeâ E with mean thermal photon number given by the Bose-Einstein distribution
E(f ) = 1/[exp(hf /k B T B ) − 1]
, with h being the reduced Planck constant, k B the Boltzmann constant and T B being the sample environment temperature. Although the thermal noise E(f ) 1 is negligible at the frequency of interest, we will keep it in our analyses to tackle the general case.
Although our results work for the simultaneous estimation of phase-shift and transmissivity, as enabled by dual-comb technique, we consider two special scenarios to simplify the SNR analyses. In the first scenario, we are concerned with only the transmissivity κ(f ), while the phase-shift is negligible due to phase cancellation via sending both combs to the sample. In the second scenario, the absorption is almost zero (κ(f ) 1), while the phase-shift α(f ) (despite also being small) provides the major information. For example, imaging the subtle changes in phase contrast on the order of 0.1 milli-radians allows the study of neural activities at the single neuron level [45,46]. Overall, the absorption or phase-shift can be very weak due to the low concentration of sample, as is the case in atmospheric sensing and human breath analysis [28] at parts per billion and in radiocarbon detection at few parts per quadrillion [47].
C. Quantum model of dual comb spectroscopy
Now we formulate the quantum theory for dual comb spectroscopy of N frequency components. In the rotating frame of the carrier frequency ν 0 (ν 0 f r ), the signal Figure 2. The frequency arrangement of the comb modes. A red(blue) peak Bn(An) in LO(signal) results a quadrature noise from red(blue) mode pair in signal(LO), which are to be entangled to improve the SNR. Around each peak, there are N such pairs. Here we explicitly label the pairs at the edges.
� +1, � +1,− � , + Δ 2 Δ � ,− peaks 2 Δ � +1, +1+ � , − � , + � +1, +1− 2 Δ 2 Δ peaks … … … … … … … …
comb is represented by the field operator
A(t) = 1 √ T â(t) + N n=1 A n e i2πn(fr+∆fr)t ,(2)
while the local oscillator (LO) comb is represented bŷ
B(t) = 1 √ T b (t) + N n=1 B n e i2πnfrt .(3)
Here T is the acquisition time, and f r > 1/T to avoid aliasing. The sum in each comb consists of the strong mean fields of a frequency comb source at discrete frequencies. The light power is mainly contributed by these mean fields. Specifically, the power is P S = hν 0 N n=1 |A n | 2 /T for the signal, and P LO = hν 0 N n=1 |B n | 2 /T for the LO, while the additional power due to squeezing is negligible in this paper.
The quantum-operator term in each comb describes the noisê
z(t) = N n=1 N δ=−Nẑ n,δ e i2π[nfr+δ∆fr]t(4)
whereẑ ∈ {â,b}, and we have quantized the frequency modes of field into the field annihilation operators, satisfying the commutation relation [â n,δ ,â † n,δ ] = [b n,δ ,b † n,δ ] = 1 and all the other commutators are zero. Here we have included all frequency modes relevant to the heterodyne measurement. In a classical strategy, the noise property of all modes is vacuum-limited, but in this work, we propose to engineer the noise property via squeezing and entanglement.
Field propagation through the sample can be formulated by a bosonic quantum channel, as shown in Eq. (1). We are interested in the transmissivity κ(f ) and phase α(f ) induced by the sample. Note that the non-ideal LO storage also induces a channel of transmissivity η(f ) and phase-shift β(f ). We assume that the transmissivity and phase-shift spectra are smooth enough such that their values at sidebands of each comb line are identical. For example, κ(nf r + δ∆f r ) = κ n for all sideband frequencies −N ≤ δ ≤ N . We define α n and η n , β n similarly. Our formalism can be easily generalized to rapidly-varying spectra, while the formula will turn much lengthier. After travelling through the sample, channel output fieldŝ A (t) for the sample return andB (t) for the LO can be decomposed in the same form as Eqs. (2) and (3). According to Eq. (1), the input-output relation yields A n → √ κ n A n e iαn and B n → √ η n B n e iβn for mean fields,
andâ n,δ = √ κ n e iαnâ n,δ + √ 1 − κ nên,δ , b n,δ = √ η n e iβnb n,δ + 1 − η nfn,δ ,(5)
for noise modes, whereê n,δ 's andf n,δ 's are environmental noise modes of thermal photon number E n ≡ E(nf r ).
At the receiver, the two combs are combined by a balanced beamsplitter, yielding two output combs,ĉ
± (t) = [Â (t) ±B (t)]/ √ 2.
Then the photon counts of the two output combs are measured, and subtracted from each
otherN (t) =ĉ † + (t)ĉ + (t) −ĉ † − (t)ĉ − (t) =Â † (t)B (t) + B † (t)Â (t).
Taking into account that ∆f r f r , we can filter out the direct current (DC) term and the fastoscillating terms at |f | ∆f r . The resulting alternative current (AC) N AC (t) is a random variable with mean
N AC (t) = 1 T N n=1 √ κ n η n e i(αn−βn) A n B n e i2πn∆frt + c.c. ,(6)
where c.c. represents the complex conjugate. One can perform a finite-time-T Fourier transform to obtain the spectrum
N AC (m∆f r ) = √ κ m η m A m B m e i(αm−βm) , 1 ≤ m ≤ N,(7)
from which we can extract the information about the transmissivities and phase-shifts across the entire N -line spectrum.
To evaluate the fluctuation of the readout, now we consider the contribution to N AC (m∆f r ) from noise modeŝ a,b. As the amplitudes A n , B n 1, the noise in
N AC (m∆f r ) iŝ Σ AC (m∆f r ) N n=1 √ η n κ n B nXn,m + √ η n κ n A nQn,m + η n (1 − κ n )B nX (e) n,m + (1 − η n )κ n A nQ (f ) n,m .(8)
Here we have adopted the nomenclature widely used in quantum optics [48,49] that defines the joint quadrature
operatorŝ X n,m ≡ 1 √ 2 â n,m e i(αn−βn) +â † n,−m e −i(αn−βn) , Q n,m ≡ 1 √ 2 b n,n+m e −i(αn−βn) +b † n,n−m e i(αn−βn) ,(9)
for the signal (â n,m ,â n,−m beat with the strong mean field B n at frequency nf r ) and for the LO (b n,n+m ,b n,n−m beat with A n at frequency n(f r + ∆f r )) respectively. Simiarly, we define the quadratureŝ X (e) ,Q (f ) for the environment modesê andf in Eq. (5). Note that these quadratures, along withΣ AC , are usually not Hermitian (real-valued) observables, thus their variances are defined as varX ≡ X †X for any non-Hermitian complex operatorX.
Dual-comb spectroscopy aims to estimate the transmissivity κ n , phase-shift α n or both simultaneously, for all 1 ≤ n ≤ N frequencies of the sample from the photocurrent of Eq. (7). We define the amplitude SNR at each comb line as
SNR = |N AC (m∆f r )|/ var [N AC (m∆f r )] .(10)
The noise, which is defined in Eq. (8), collects the beating modes near all N comb lines. As shown in Methods, it is a good indicator for the minimum mean square error of either the transmissivity or the phase-shift estimation task. Furthermore, a neat figure of merit is the overall quality factor N · SNR, which eliminates the dependence on total line number N .
To evaluate the SNR, we make use of the independence between modes around different comb lines and evaluate the variance from Eq.
where the thermal noise N n = η n B 2 n (1 − κ n )(2E n + 1) + κ n A 2 n (1 − η n )(2E n + 1) is determined by the sample, the LO storage and the environment temperature; the complex quadrature noises varX n,m and varQ n,m are determined by the quantum state of the signal comb source and local comb source.
D. SNR with entangled quantum comb
From the definition of quadratures in Eq. (9), we see that the noise varX n,m can be suppressed by entangling the modesâ n,±m in a two-mode squeezed vacuum state (see Methods). By such means, the joint quadraturê X n,m is squeezed with suppressed variance (see Meth-
ods) varX n,m = 1 2G − G 2 − 1 cos (2α n − 2β n ) + (G 2 + 1) ,(12)
where squeezing gain G ≥ 1. When phases are perfectly matched as α n − β n = 0, varX n,m is minimized to 1/G. Similarly, we can squeeze the joint quadratureQ n,m of the local comb by gain G LO . When G = G LO = 1, the variance reduces to the classical dual-comb spectroscopy. In this case the variance of complex operator varX n,m is twice of the SQL 1/2, because it is defined as a sum of variances of its real and imaginary parts.
To model the full SNR of the dual-comb spectroscopy system, we involve device and source imperfections. For simplicity, we assume the N comb lines are generated symmetric (A n = A n and B n = B n ). In Methods, we derive the full formula of the SNR at intermediate
frequency m∆f r SNR −2 = N 2 T a NEP 1 P 2 S + a quad P S + a RIN ,(13)
where T is the acquisition time, P S is the total signal power, the NEP-type noise coefficient a NEP ≡ NEP 2 /η m κ m γ, the RIN-type noise coefficient a RIN ≡ RIN/2, and the quadrature noise coefficient
a quad ≡ hν 0 N N n=1 varX n,m + 1 γ varQ n,m + N n κ n η n .(14)
Here γ ≡ P LO /P S is the LO-to-signal power ratio, hν 0 is the energy per photon. Note the quadrature noises can be suppressed by the entanglement (joint quadrature squeezing) in Eq. (12). The proposed SNR quantifies the performance of both the transmissivity estimation and phase estimation scenarios.
The SNR of Eq. (13) versus total signal power P S has been evaluated in Fig. 1(c) for the case of G = G LO > 1, which highlights the quantum advantages from entanglement. We have taken the case where the phase mismatch α n − β n 1/G are all small. This is the case when one estimates the transmissivity with good phase locking, or estimates small phase-shift caused by weak samplesphase-shift as small as 0.1 milli-radians allows the study of neural activities at the single neuron level [45,46].
Our formula can be regarded as a quantum version of Eq. (2) of Ref. [30]. We note that our quantum model yields a SNR-γ relation different from the semiclassical model in Ref. [30]. Specifically, for a fixed signal comb power P S , we find that the optimum is at γ → ∞ in our quantum model, while the optimum is finite in the semiclassical model. This is because when γ is large, the RIN-type noise increases proportional to γ in the semiclassical model, while the RIN-type noise remains constant in our quantum model. Our result on the RINtype noise agrees with Ref. [50].
E. Strategies of applying quantum combs
In the above analyses, we have allowed both the signal comb and local comb to be quantum engineered. In general, having quantum entangled combs in both arms might be not necessary and also experimentally challenging. Here we address different scenarios of applying the quantum combs.
From Eq. (11), we see that the noises from signal and LO are indeed amplified by the mean field of the other, i.e. the signal noise is amplified by the LO mean and vice versa. Hence, squeezing merely the signal is sufficient to yield significant advantage when P LO dominates, while squeezing merely the LO is sufficient when P S dominates. Now we evaluate the quantum advantages of various entanglement (or joint squeezing) strategies over the classical coherent-state source in terms of amplitude SNR. In Fig. 3, row 1 shows the ideal advantage without practical noises (note that the ideal advantage depends on the relative power ratio γ = P LO /P S between the signal and LO only, not the absolute magnitudes of power). Row 2 and row 3, operating under signal power P S = 10mW and P S = 10µW respectively, show the practical advantage with NEP-type and RIN-type noise involved. Along the horizontal axis, all contours are centered at the signal power P LO = P S , thus at the left half the signal dominates while at the right half the LO dominates.
In row 1, we verify our results of fundamental limits for the ideal advantages. In Fig. 3(a), only the signal is squeezed, we see that the advantage peaks at the LOdictated region (right half); in Fig. 3(b), only the LO is squeezed, the advantage now peaks at the signal-dictated region (left half); finally in Fig. 3(c), both the signal and LO are squeezed, here we enjoy both the advantageous areas in the two squeezing strategies above. It is noteworthy that when LO is squeezed, the quantum advantage survives even when κ → 0 as shown at the bottom-left corners of subplots (b)(c), which is useful when the sample is lossy and LO power is limited.
In row 2 and row 3, we consider the effect of practical noises. In Fig. 3(d)-(f), the signal power P S = 10mW is relatively large. In this scenario, the advantage is mainly constrained by the RIN-type noise which is more significant for large P LO . We find that the advantages at the LO-dictated region (right half) of all subplots (d)(e)(f) are significantly undermined, which is especially noticeable in subplot (d). On the other hand, in Fig. 3(g)-(i), the signal power P S = 10µW is extremely small. In this scenario, the advantageous region is mainly affected by the NEP-type noise which is more significant for small P LO . As expected, the advantages at the left half region of subplots (g)(h)(i) are undermined, which is especially noticeable in subplot (e). In subplot (f) or (i) where both the signal and LO are squeezed, the patterns in the two squeezing strategies shown in subplot (d)(e) or (g)(h) occur simultaneously.
F. Performance under total power constraints
To begin with, we consider the power dependence of the SNR on P LO , P S . Here we explore the scenario where the LO is sent along with the signal, thus η m = κ and the total power exposure P S + P LO is to be constrained. Fig. 4(a)(b) shows that the total power P S + P LO contour and the SNR contour are tangent at P LO = P S . This explains that in some applications one tends to use comparable LO and signal rather than very strong LO to save the total power consumption. Now consider the quantum advantage. Note that only the fundamental noise is suppressed by the quantum engineering, we expect to maximize these ratios σ 2 quad /σ 2 NEP and σ 2 quad /σ 2 RIN to see a significant quantum advantage. In Methods, we show that NEP-type noise σ 2 NEP ∼ 1/(P LO P S ), fundamental noise σ 2 quad ∼ (P S + P LO )/P S P LO , RIN-type noise σ 2 RIN ∼ 1. The ratio σ 2 quad /σ 2 NEP is proportional to the total power P S + P LO , while σ 2 quad /σ 2 RIN is maximized at P S → 0 or P LO → 0.
Total signal power (W) For the NEP-dictated scenario, i.e. the total power is small, the quantum advantage grows with the absolute SNR as total power increases, while it does not depend on γ = P LO /P S . In this case one can let γ → 0 or ∞ to make signal or LO dominate so that squeezing on the other is no longer needed, as discussed previously. For the RIN-dictated scenario, i.e. the total power is large, we note that the quantum advantage decreases with the total power and it is minimized at P S = P LO given a fixed total power, which is opposite to the absolute SNR case. This is not a preferred scenario for quantum advantage. Fig. 4(c)(d) verifies that the total power contour and the quantum advantage contour almost overlap in the region of small total power; they are again tangent at P S = P LO in the region of large total power, while the gradient direction of the quantum advantage contour is reversed. Comparing subplot (c) of lossless sample κ = 1 and subplot (d) of lossy sample κ = 0.5, we see that the quantum advantage degrades significantly when the sample is lossy now that both signal and LO suffer such loss, while a 1dB advantage still survives.
Total LO power (W) (a) (b) (c) (d)
G. Performance limits in biological applications
An important application of the proposed entanglement-enhanced dual-comb spectroscopy system is in sensing fragile bio-tissues. In bio-sensing, the power of signal light shining on the sample is typically between 10−100 uW to avoid kill, bleach or perturb the specimen under analysis [51][52][53]. For example, Ref. [40] showed an extreme case of power at ∼ 10mW which causes severe sample damage. In another extreme case retina sensing, the safety standard permits much lower power [54]. The maximum permissible radiant power is a function of the exposure duration, wavelength and visual angles. In long exposure time limit, the maximum power can be as low as
1uW (see Fig. 2 of Ref. [54]). In other scenarios, the power can be higher, e.g. ∼ 10 − 200uW is adopted in some studies [55,56]. To gain better SNR, therefore one cannot simply increase the power, but rather consider suppressing the quantum-limited noise such as in our proposal.
In bio-sensing, a major limitation of the applicable frequency region comes from water absorption. The transmissivity spectrum of water absorption can be derived from the Lambert absorption coefficient spectrum α(f ) [57][58][59] via κ(f ) = exp{−α(f )L}, where L is the sample depth. For the optical domain of wavelength λ < 1µm, the absorption is weak: α 10 −4 /µm. We take a typical sample depth of L = 15um and evaluate the transmissivity in Fig. 5(a) assuming the sample absorption is majorly dominated by water, with absorption coefficients taken from Refs. [57][58][59]. We see that the absorption is substantial starting around λ ∼ 2um. In wavelength below 2um, the thermal noise described by the Bose-Einstein distribution E(f ) 1 is negligible at room temperature. In this ≤ 5um frequency of interest, E ≤ 10 −4 and is negligible at room temperature of 300K. Even at 10um, E ∼ 0.008 is still small. With the absorption and noise in hand, we can evaluate the quantum advantage in absence of any NEP-type or RIN-type noise from Eq. (14). To simplify the evaluation, we assume uniform absorption across all comb lines to get a sense of the quantum advantage. In general, the quantum advantage will be an average across a frequency region analysed here. In Fig. 5(b), we find the advantage indeed appreciable below about 2um and increases with the gain G, while above 2um water absorption starts to limit the possible advantage. Note that the quantum advantage monotonically increases with G, we expect the contour lines rise to higher G when transmissivity dips. As expected, we see that the contour lines are almost the reverse of the transmissivity spectrum in Fig. 5(a).
III. DISCUSSION
Before closing, we discuss about experimental generation of the quantum comb. One can send a local frequency comb source to a periodically-poled LN (PPLN) waveguide to generate a comb source at the second harmonic frequencies, which is then used to pump another PPLN waveguide for two modes squeezing around each of the local comb line via spontaneous parametric down- conversion (SPDC), during which the original local frequency comb serves as the seed. By appropriately controlling the phase between the two frequency combs, one can operate in the parametric de-amplification regime to generate amplitude squeezed state at each comb line. The phase matching bandwidth of the SPDC needs to be less than half of the comb line spacing [60]. The sideband entangled outputs are then combined with the sample frequency comb at a highly transmissive beamsplitter, forming the quantum frequency comb source in the proposed dual-comb configuration. We believe that the integrated photonic platform is well suited to achieve the stringent phase-matching bandwidth at each comb line over a broad optical bandwidth since both the group velocity dispersion and group velocity can be tailored via engineering of the nanophotonic waveguide dimension. In addition, both the classical local and sample comb sources can be generated on chip via Kerr nonlinearity with a large comb line spacing of over 100 GHz, which increases the higher bound of the phase matching bandwidth. We envision the proposed scheme can be fully integrated on thin film LN platform thanks to its large second order and Kerr nonlinearity as well as capability of achieving quasi-phase matching via electrical poling [61]. Note that the proposed quantum comb in this work is different from Ref. [62]: there the comb lines are themselves pair-wise entangled to serve as resource for quantum computation; while our proposed quantum comb has side-band of each comb line pair-wise entangled to benefit dual-comb spectroscopy sensing precision.
In this work, we proposed a entanglement-enhanced dual-comb spectroscopy protocol, where both the signal comb and local comb can be quantum engineered. When local comb is stronger than the signal comb, the protocol promises signal-to-noise ratio advantages in detecting low-loss samples such as thin slice of bio-tissues and molecular gas. When the local comb is weak com-pared with signal comb, quantum engineering of the local comb provides signal-to-noise ratio advantages regardless of the sample loss, making the quantum advantage robust against experimental imperfections.
Dual-comb interferometry is evolving into one of the most powerful tools for broadband laser spectroscopy, ranging and imaging, and our work extends its advantages beyond the standard quantum limit. Such a potential will enable new comb-based spectroscopy and metrology with unprecedented precision and sensitivity. The boost in SNR will directly lead to orders of magnitude improvement in measurement speeds for real-time sensing and bio-imaging.
Appendix A: Two-mode squeezed vacuum
In this paper, we are interested in the two-mode squeezed vacuum (TMSV) state, the continuous-variable version of Einstein-Podolsky-Rosen (EPR) state [63]. Consider two modesâ 1 andâ 2 , with real and imaginary quadrature operatorsq j ≡ (â j +â † j )/
√ 2,p j ≡ (â j −â † j )/ √ 2i, j = 1, 2.
The entanglement between the two modes is described by the joint squeezing on modeŝ a + = (â 1 +â 2 )/ √ 2 andâ − = (â 1 −â 2 )/ √ 2, such that the variances of joint quadraturesq + ,p − are suppressed to e −2r /2 ≡ 1/2G. On the other hand, the variances ofq − ,p + are amplified to e 2r /2 ≡ G/2. In this case, when r → ∞ it has the ideal EPR correlation ofp 1 =p 2 ,
q 1 = −q 2 .
Take the phase-matched case (α n = β n ) of Eq. (9) and indexâ n,m →â 1 andâ n,−m →â 2 , the operator X n,m ≡ 1 √ 2 â 1 +â † 2 =q + + ip − . Therefore, the TMSV state enables varX n,m to be suppressed to e −2r ≡ 1/G. This is the keystone of the SNR improvement proposed in this paper. For the general case, X n,m = cos(α n − β n )q + − sin(α n − β n )p + + i cos(α n − β n )p − + i sin(α n − β n )q − , (A1) from Eq. (9). Then, one can obtain the variance in Eq. (12).
The quadrature fluctuations of Eq. (8) are sums of contributions from N comb lines varq AC (m∆f r ) = N n=1 N n 2 + η n κ n B 2 n var ReX n,m + A 2 n var ReQ n,m , varp AC (m∆f r ) = N n=1 N n 2 + η n κ n B 2 n var ImX n,m + A 2 n var ImQ n,m .
(B2) The sum of the two noise gives Eq. (11). From Eq. (A1) for TMSV state, we can see that ReX n,m , ImX n,m , ReQ n,m , ImQ n,m are mutually independent when α n = β n .
We begin with the estimation of κ n 's, assuming perfect phase matching, α n = β n . In this case, two-mode squeezing has var ReX n,m = var ImX n,m = 1 2 varX n,m , var ReQ n,m = var ImQ n,m = 1 2 varQ n,m , thus varq AC (m∆f r ) = varp AC (m∆f r ) = 1 2 var N AC (m∆f r ) .
(B3) Also, note thatq AC andp AC commute, and indeed they are mutually independent Gaussian variables [which can be verified from Eq. (A1)]. Thus, we can define the distribution of the readouts q, p forq AC (m∆f r ),p AC (m∆f r ) as P q (q) · P p (p). Then the minimum mean square error (MMSE) for unknown parameters κ n is given by the Cramér-Rao lower bound of Gaussian distribution:
−1 ≡ ˆd q ∂ log P q (q) ∂ √ κ m 2 P q (q) −1 + ˆd p ∂ log P p (p) ∂ √ κ m 2 P p (p) −1 = dN AC (m∆fr) d √ κm 2 varq AC (m∆f r ) + 2 · dvarqAC(m∆fr) d √ κm 2 2[varq AC (m∆f r )] 2 dN AC (m∆fr) d √ κm 2 varq AC (m∆f r ) = η m B 2 m A 2 m varq AC (m∆f r ) .(B4)
In the last equality, we have assumed that the modulation on the readout variance is negligible, which is true due to squeezing power much lower than the comb power. Thus, in comparison with Eq. (10),
[MMSE κ m ] −1 = 2/κ m · SNR 2 .(B5)
Now we estimate the phase mismatch θ m ≡ α m − β m .
Note that dN AC (m∆fr) dθm 2 = 2 · κηA 2 B 2 . Similar to the transmissivity estimation above, we assume a TMSV input state such that varq AC (m∆f r ) = varp AC (m∆f r ) and the independence betweenq AC andp AC still holds.
Similarly, the Cramér-Rao lower bound gives
−1 = η m κ m B 2 m A 2 m varq AC (m∆f r ) (B6)
Note that varq AC (m∆f r ) = 1 2 var N AC (m∆f r ). Thus, in comparison with Eq. (10),
[MMSE θ m ] −1 = 2 · SNR 2 .
(B7)
Appendix C: Full formula of estimation error
Here we derive Eq. (13), including the inverse-squarelaw, the inverse-law, and constant noise terms with respect to source power P S . We denote the total power of the signal or the LO as P S or P LO , and define their ratio γ ≡ P LO /P S . We can identify
T P S = hν 0 N m=1 |A m | 2 , T P LO = hν 0 N m=1 |B m | 2 ,
where hν 0 is the energy per photon. To simplify the formulas we assume symmetric comb lines A m = A, B m = B for any 1 ≤ m ≤ N . To connect to the SNR of Eq. (10), we normalize each noise by the power of mean field Eq. (7),
N AC (m∆f r ) 2 = η m κ m A 2 B 2 = η m κ m · (P S T /N hν 0 ) · (P LO T /N hν 0 ).
The inverse-square-law noise comes from the detector noise. It is modelled as a fluctuation of constant noise equivalent power (NEP) on the photon current readout, including dark current, Johnson noise, amplifier noise figure, etc.. Note that NEP-type noise is circular symmetric in the phase space, it affects both real and imaginary parts of N AC . Since we defined var N AC as the sum of the two quadrature noises, the physical NEP-type noise is 2NEP 2 ∆f in power. In photon number, the noise is 2(NEP · T /N hν 0 ) 2 ∆f . Here the bandwidth is defined as ∆f = 1/2T for single-sided NEP spectral density. According to Eq. (10), the detector noise results in the normalized NEP-type noise at intermediate frequency m∆f r
σ 2 NEP = 2 (NEP · T /N hν 0 ) 2 ∆f η m κ m (P LO T /N hν 0 )(P S T /N hν 0 ) = N 2 T NEP 2 η m κ m γP 2 S . (C1)
Here NEP is has the unit of W/Hz 1/2 . Now we see that the NEP-type noise-power relation is
σ 2 NEP ∝ 1 PS·PLO , which is an inverse-square-law term ∼ O( 1 P 2 S ) when γ is fixed.
The inverse-law noise comes from the intrinsic quantum noise Eq. (8). For the case where phase noise is negligible, the quadrature fluctuation leads to the normalized quadrature noise
σ 2 quad = varΣ AC η m κ m B 2 (P S T /N hν 0 ) = N 2 T c γ 4hν 0 P S ,(C2)
where
c γ ≡ varΣ AC 4η m κ m B 2 · N(C3)
with varΣ AC defined in Eq. (11). For vacuum input G = 1, G LO = 1, unit transmissivities η n = κ n = 1 and zero noiseẼ n = 0 for any 1 ≤ n ≤ N , we have c γ = 1 4 (1 + 1 γ ). In this case, varΣ AC ∝ P S + P LO , we see that the fundamental noise-power relation is σ 2 quad ∝ PS+PLO PS·PLO , which is an inverse-law term ∼ O(1/P S ) when γ is fixed.
The constant noise results from the relative intensity noise (RIN). Consider the power of each comb tooth, hν 0 A 2 /T for signal and hν 0 B 2 /T for LO. RIN is modelled as a single-sided white noise var (hν 0 A 2 /T ) = (RIN∆f )P 2 S , var (hν 0 B 2 /T ) = (RIN∆f )P 2 LO on the power of field amplitudes A, B, generated from additive amplified spontaneous emission (ASE) from the laser or from any subsequent optical amplification. Similar to NEP-type noise, the bandwidth is defined as ∆f = 1/2T for single-sided RIN spectral density. RIN-type noise is also circular symmetric in the phase space, which affects both real and imaginary parts of N AC and yields a factor of 2 in the complex-observable variance var N AC . Consequentially, the physical noise is
2η m κ m [var (A) · B 2 + A 2 · var (B)] = 2η m κ m [ var (A 2 ) 4P S · P LO + P S · var (B 2 ) 4P LO ] = 2η m κ m T hν 0 2 · RIN∆f · P 2 S ( 1 4 · γ + γ 2 4γ ) = η m κ m T hν 0 2 · RIN∆f · γP 2 S (C4)
Here RIN is in unit 1/Hz, and in the first equality we have used var (A 2 ) = ( ∂ ∂A A 2 ) 2 var (A) = 4A 2 var (A), var (B 2 ) = 4B 2 var (B), assuming A 2 var A and B 2 var B. It is valid to consider fluctuations on amplitudes A, B instead, because that the RIN physically comes from the amplified spontaneous emission, which is modelled as a Gaussian noise on the field quadratures in quantum optics. The amplitude fluctuations result in the normalized RIN-type noise σ 2 RIN = η m κ m RIN∆f · γ(P S T /hν 0 ) 2 η m κ m (P LO T /N hν 0 )(P S T /N hν 0 ) = N 2 T 2c γ 2 RIN .
(C5) Here the coefficient c γ 2 ≡ 1/4. We immediately see that the RIN-type noise-power relation is σ 2 RIN ∼ O(1), which does not depend on the signal or LO power.
Overall, we can write the full formula of the SNR Eq. (10) at intermediate frequency m∆f r as SNR −2 = σ 2 NEP + σ 2 quad + σ 2 RIN , which gives Eq. (13) in main text. To recover the classical results in Ref. [30], we further assume that the frequency spectra of all parameters are almost uniform, such that κ m ≈ κ, η m ≈ η for any 1 ≤ m ≤ N . When η, κ → 1 and classical source is used (G = G LO = 1), our result recovers the formula of σ H [Eq. (2)] in Ref. [30] mostly [H(f ) is the transfer function of electrical field, equivalent to κ(f )]. Note that in the quantum model the shot noise term a shot is instead formulated as the quadrature fluctuation a quad , while the quadrature fluctuation incidentally gives a similar result c γ = 1 4 (1+ 1 γ ), up to an extra 1/2 factor. Also, c γ 2 = 1 4 · 2γ 2γ is different from (1 + γ 2 )/2γ in Eq. (2) of Ref. [30], which is the result of our derivation of the physical RIN-type noise Eq. (C4) in the balanced detection. The independence of c γ 2 and thereby of σ 2 RIN in our Eq. (13) agrees with the recent results of Eqs. (58) (59) in Ref. [50] assuming white noise spectrum. Meanwhile, our derivation recovers the result c unbal
γ 2 = 1+γ 2 2γ
in Ref. [30] for the unbalanced detection case where the physical noise is ∼ var (A 2 ) + var (B 2 ) instead. To summarize, our result when applied to the classical dual-comb SNR can be obtained by letting c γ = 1 4 (1 + 1 γ ), b = 1, c γ 2 = 1/4 in Eq. (2) of Ref. [30].
We note that our quantum model yields an SNRgamma relation different from the semiclassical model in Ref. [30]. For example, at a fixed P S there is a finite optimum value of γ to maximize the SNR in the semiclassical model while the optimum is at γ → ∞ in our quantum model. This is because when γ is large, the RIN-type noise increases with γ in the semiclassical model, while RIN-type noise remains a constant in our derivation which agrees with Ref. [50].
Finally, we address the saturation in the SNR with respect to the squeezing gain G, due to NEP or RIN noise. For simplicity, we consider κ m ≈ κ, η m ≈ η for any 1 ≤ m ≤ N . The saturation of SNR begins when the squeezing gain G is large enough such that NEP-type or RIN-type noise overwhelms the fundamental noise. By solving σ 2 NEP > σ 2 quad or σ 2 RIN > σ 2 quad , we derive the saturation threshold as P NEP S,sat = G · NEP 2 /{hν 0 (γ[G(1−κ)+κ]+κ)} for the NEP-type and P RIN S,sat = hν 0 (γ(G(1 − κ) + κ) + κ)/{Gγκ · RIN} for the RIN-type. The above thresholds are indicated in Fig. 1 as the dots on the SNR curves, showing a good agreement with numerical results.
Figure 1 .
1Figure 1. (a) Conceptual schematic for entanglementenhanced dual comb spectroscopy. The teeth share intermodal entanglement within the signal comb (red beam). The tissue is depicted in a cartoon fashion, while the real tissue slice is almost transparent in practice. (b) Schematic of the quantum comb. Each pair of signal modes beating with the same LO comb tooth (purple line) for the same intermediate frequency is entangled, indicated by a black dashed line connecting a pair of purple circles. (c) Practical SNR involving NEP-type and RIN-type noises, plotted versus signal power (analog to Fig. 2 of Ref. [30]), normalized to unit acquisition time T = 1s. We assume an ideal detector with unity efficiency, and zero loss and noise κ ≈ 1, η = 1. In (c), both signal and LO are entangled with equal gain G, which increases from 0dB (coherent-state) to 30dB in steps of 10dB, plotted in color from blue to magenta. The NEP/RIN-dictated SNR is presented by green-dot-dashed/black-dot-dashed line, along with the shot noise (SN) limit in blue-dashed. N = 10 5 , λ = 1µm, RIN= −170dBc/Hz [42], NEP= 5 × 10 −13 W/Hz 1/2 (NEP= 4.5 × 10 −15 W/Hz 1/2 is actually achievable, e.g. by Thorlabs FGA01FC-InGaAs Photodiode), PLO/PS = 5. Inset: Quantum advantage in SNR (in decibel unit) versus various values of RIN for total signal power PS=10mW (blue) and PS=10µW (red) at G=20dB. The LO-signal power ratio γ ≡ PLO/PS = 5 is fixed-although this figure shows the case of both signal and LO entangled, as we show later, only the signal needs to be entangled under large γ.
n κ n B 2 n varX n,m + A 2 n varQ n,m ,
Figure 3 .
3The quantum advantage (in decibel unit) in amplitude SNR over the coherent-state source at the squeezing gain of 10 dB, versus various LO-signal power ratio γ ≡ PLO/PS and sample transmissivity κ. The absorption spectrum is assumed to be slowly-varying as κm ≈ κ for any m. Rows: without practical noise; with both NEP-type and RIN-type noises, total signal power PS = 10mW; with both NEP-type and RIN-type noises, PS = 10µW. Columns: signal squeezed only; local comb (LO) squeezed only; both signal and LO squeezed. We assume ideal detector with unity efficiency and ideal LO link ηm = 1. N = 10 5 , T = 1s, λ = 1µm, NEP= 5 × 10 −13 W/Hz 1/2 , RIN= −170dBc/Hz.
Figure 4 .
4The dependence of (a)(b) the amplitude SNR using the squeezed source and (c)(d) its quantum advantage over the coherent-state source, both in decibel unit, on various power constraints PLO, PS. The absorption spectrum is assumed to be slowly-varying as κm ≈ κ for any m. Both signal and LO are squeezed at the squeezing gain of 10dB. (a)(c) κ = 1; (b)(d) κ = 0.5. The black dashed lines are contours of total power PLO + PS. We assume ideal detector with unity efficiency, while the LO is sent along with the signal so that ηm = κ. N = 10 5 , T = 1s, λ = 1µm, NEP= 5 × 10 −13 W/Hz 1/2 , RIN= −170dBc/Hz.
Figure 5 .
5(a) The absorption spectrum of pure water κ(λ) at room temperature 295K[57][58][59], evaluated for path length L = 15µm. (b) The fundamental limit of quantum advantage in amplitude SNR (in decibel unit) enforced by water absorption. The spectrum data subplot (a) is unknown to the observer. An additive thermal Gaussian noise at room temperature is mixed in. PLO/PS = 5.
Appendix B: Justification of the SNR definitionFirst, we construct the estimator for κ n and show that the estimation error is connected to our definition of SNR in Eq.(10). Consider the real quadraturesq AC (m∆f r ) ≡ Re N AC (m∆f r ) + ReΣ AC (m∆f r ), p AC (m∆f r ) ≡ Im N AC (m∆f r ) + ImΣ AC (m∆f r ) of the complex heterodyne readout N AC [see Eq. (7) and Eq.(8)]. Their means are given by Eq.(7)as
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| {'fraction_non_alphanumeric': 0.06995329822419691, 'fraction_numerical': 0.03449401677857113, 'mean_word_length': 3.971981085468276, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Dual-comb interferometry harnesses the interference of two laser frequency combs to provide unprecedented capability in spectroscopy applications. In the past decade, the state-of-the-art systems have reached a point where the signal-to-noise ratio per unit acquisition time is fundamentally limited by shot noise from vacuum fluctuations. To address the issue, we propose an entanglementenhanced dual-comb spectroscopy protocol that leverages quantum resources to significantly improve the signal-to-noise ratio performance. To analyze the performance of real systems, we develop a quantum model of dual-comb spectroscopy that takes practical noises into consideration. Based on this model, we propose quantum combs with side-band entanglement around each comb lines to suppress the shot noise in heterodyne detection. Our results show significant quantum advantages in the uW to mW power range, making this technique particularly attractive for biological and chemical sensing applications. Furthermore, the quantum comb can be engineered using nonlinear optics and promises near-term experimentation.', 'arxivid': '2304.01516', 'author': ['Haowei Shi \nMing Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n', 'Zaijun Chen \nMing Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n', 'Scott E Fraser \nTranslational Imaging Center\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n', 'Mengjie Yu \nMing Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n', 'Zheshen Zhang \nDepartment of Electrical Engineering and Computer Science\nUniversity of Michigan\n48109Ann ArborMIUSA\n', 'Quntao Zhuang \nMing Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n\nDepartment of Physics and Astronomy\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA\n'], 'authoraffiliation': ['Ming Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA', 'Ming Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA', 'Translational Imaging Center\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA', 'Ming Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA', 'Department of Electrical Engineering and Computer Science\nUniversity of Michigan\n48109Ann ArborMIUSA', 'Ming Hsieh Department of Electrical and Computer Engineering\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA', 'Department of Physics and Astronomy\nUniversity of Southern California\n90089Los AngelesCaliforniaUSA'], 'corpusid': 257921833, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23625, 'n_tokens_neox': 20200, 'n_words': 11784, 'pdfsha': '31b7bb6c84b0228b22d5243be45cea5a7c307490', 'pdfurls': ['https://export.arxiv.org/pdf/2304.01516v2.pdf'], 'title': ['Entanglement-enhanced dual-comb spectroscopy', 'Entanglement-enhanced dual-comb spectroscopy'], 'venue': []} |
arxiv |
Exchange-Driven Intermixing of Bulk and Topological Surface State by Chiral Excitons in Bi2Se3
Bowen Hou
Department of Mechanical Engineering and Material Sciences
Yale University
06511New HavenCTUSA
Dan Wang
Department of Mechanical Engineering and Material Sciences
Yale University
06511New HavenCTUSA
Bradford A Barker
Department of Physics
University of California
95343MercedCAUSA
Diana Y Qiu [email protected]
Department of Mechanical Engineering and Material Sciences
Yale University
06511New HavenCTUSA
Exchange-Driven Intermixing of Bulk and Topological Surface State by Chiral Excitons in Bi2Se3
1
Topological surface states (TSS) in the prototypical topological insulator (TI) Bi2Se3 are frequently characterized using optical probes, but electron-hole interactions and their effect on surface localization and optical response of the TSS remain unexplored. Here, we use ab initio calculations to understand excitonic effects in the bulk and surface of Bi2Se3. We identify multiple series of chiral excitons that exhibit both bulk and TSS character, due to exchange-driven mixing. Our results address fundamental questions about the degree to which electron-hole interactions can relax the topological protection of surface states and dipole selection rules for circularly polarized light in TIs by elucidating the complex intermixture of bulk and surface states excited in optical measurements and their coupling to light. In the prototypical topological insulator (TI) Bi2Se3, strong spin-orbit coupling (SOC) drives the inversion of bulk bands near the Fermi level, resulting in the appearance of a pair of topological surface states (TSS) with spin-momentum locking[1-13]. These TSS have the potential to be exploited for the generation of topologically-protected spin currents[14], leading to the promise of faster, low-powered spintronics. Due to the high residual conductivity of the bulk[15-20], light-based probes, such as the circular photogalvanic effect, are frequently used to probe the surface states[15,[21][22][23][24][25][26]. In such probes, changes to the photocurrent direction with the helicity of light are interpreted as evidence of photocurrent generation from the TSS. However, the frequency of the probe light is typically 1.5 eV or larger, leading to the excitation of a complex manifold of
transitions between numerous bulk and surface states, which can be challenging to disentangle without the help of microscopic theory. For instance, it can be difficult to differentiate photocurrent generated from the TSS near the Fermi level (SS1) and a higher-energy unoccupied topological surface state (SS2), with recent time-resolved angle-resolved photoemission spectroscopy (tr-ARPES) indicating that previous photocurrent measurements likely involve excitations of both states [16]. Furthermore, recent experiments have identified long-lived chiral excitons around 2.4 eV, which are speculated to arise from transitions between a low-energy occupied Rashba surface state (RSS) and SS2 [27].
Despite the important role of optical probes in the characterization of TSS, electron-hole interactions leading to the formation of excitons have largely been neglected, with analyses of optical spectra relying on selection rules derived from an independent particle picture. However, recent discovery of long-lived chiral surface excitons [27] raises the question of how excitonic effects can alter the character of the surface excitations. In this work, we utilize the state-of-theart ab initio GW plus Bethe Salpeter equation (GW-BSE) method to investigate the quasiparticle (QP) band structure and optical absorption spectrum including electron-hole interactions for both bulk and quasi-two-dimensional slabs of Bi2Se3. Our calculations reveal multiple previously unidentified excitons with chiral optical selection rules (chiral excitons) in an energy window from 0-2.82 eV, including bright exciton states with p-like symmetry arising from topologically nontrivial bands [28][29][30]. Remarkably, these chiral excitons are composed of electrons and holes that combine the character of both bulk and topologically-protected surface states, whose intermixing is driven by the Coulomb exchange interaction. Our calculations address fundamental questions about the degree to which electron-hole interactions change the surface localization of opticallyexcited TSS, relax the dipole selection rules for circularly polarized light compared to the independent particle picture, and elucidate the complex intermixture of bulk and surface states excited in optical measurements, in good agreement with recent experiments [27,31].
In this Letter, we perform ab initio mean-field calculations of bulk, one quintuple layer (QL), 3QL, and 5QL Bi2Se3 using density functional theory (DFT) in the local density approximation (LDA) [32][33][34] as implemented in Quantum ESPRESSO [35]. GW and GW-BSE calculations, employing a fully-relativistic spinor formalism [36], as implemented in BERKELEYGW [37][38][39] are performed on top of the DFT mean field. Computational details can be found in the Supplemental Information (SI). Fig. 1(a) shows the bandstructures of bulk Bi2Se3.
The top of the valence band at the LDA level has a "camelback" feature, which becomes parabolic at the GW level, consistent with k•p models [1,44] and previous GW calculations [36,42,[44][45][46][47][48][49][50][51][52].
After the many-body electron-electron interactions are included at the GW level, the indirect LDA bandgap of 0.32 eV is renormalized into a direct QP gap of 0.31 eV at the Γ point, which agrees well with previous experimental and theoretical results [42,48]. We note that there is a kink in the one-shot G0W0 bandstructure along the Γ to Z direction, which becomes smooth when the QP wavefunction is updated self consistently (see SI Fig. S6) [36,46]. [53]. At the DFT level, the surface state is gapped by 0.53 eV in the 1QL system and becomes nearly metallic in the 3 QL system with a gap 0.017 eV. At the GW level, in 1 QL, the bandgap increases to 1.27 eV, which is 0.95 eV larger than the GW gap of bulk Bi2Se3. 3QL and 5QL Bi2Se3 have bandgaps of 0.17 eV and 0.02 eV respectively at the GW level, which agrees well with previous ARPES measurements of 0.15 eV and 0.04 eV [53] and improves on previous GW calculations, where SOC is included perturbatively [44]. Both SS2 and RSS remain ungapped in the 3QL and 5QL slabs at both the DFT and GW levels.
Next, we use the GW-BSE approach to investigate the optical properties of 1QL and 3QL
Bi2Se3. Fig. 2 shows the absorption spectra for 1QL and 3QL Bi2Se3, with and without electron- Next, we turn to the identification of the experimentally observed chiral excitons [27]. Here, we focus on the 3QL Bi2Se3, which is the thickest slab size that is computationally tractable for our GW-BSE calculations. We need to solve the BSE with at least 18 valence and 14 conduction bands to include all bulk and surface transitions in the energy range of the RSS to SS2 transition.
Minima of the surface states occupy only a small part of the Brillouin zone near Γ and require very high k-point sampling to resolve, so we retain only a patch within a radius of 0.028 a0 -1 of the Γ point (grey shadow shown in Fig. 1 (c)) [55][56][57]. Solving the BSE Hamiltonian in this smaller patch allows us to increase the k-point sampling to 300×300×1, resolving the surface excitons, which are highly localized in reciprocal space. Since excitons with right and left-circularly polarized selection rules are degenerate, we illustrate their different behavior by calculating the intensity of the instantaneous emission (i.e. without considering thermalization) of light with polarization ′ following excitation by light of polarization as I σσ′ ∝ | < 0|̂|S >< S|̂′|0 > | 2 where S denotes the quantum number of exciton and ̂ is the momentum operator. We note, however, that in actual photoemission processes, excitons with a large admixture of bulk states will have a shorter non-radiative lifetime and thus be less apparent in photoemission spectra. This is because CX8 is composed of 30% RSS-to-(CBM+3) transitions, which have circularly polarized dipole selection rules (see Fig. S3(c)) and 68% non-circularly polarized (VBM-2)-to-(CBM+3) transitions (see Fig. S3(b)). The admixture of the latter significantly relaxes the circular dichroism. The highest-energy chiral exciton peak we identify, CX10, which is located at 2.78 eV and mainly consists of the RSS-to-SS2 transitions, preserves a high degree of circular polarization.
From the features of CX8 and CX10, we can speculate that the experimentally observed broad circularly-polarized photoemission peak around 2.4 eV in previous work [27] may be due to both CX8 and CX10.
Finally, to elucidate the effects of the mixing of TSS and non-protected states, we plot the modulus squared of the electronic part of the exciton wavefunction for CX3 and CX10, as shown in Fig. 4 (c-d), when the hole is fixed at a position maximizes the electron amplitude. For comparison, two important individual electronic states of SS1' and RSS at the Γ point are plotted in Fig. 4 (a-b) (Other electronic and excitonic states may be found in the SI). For SS1', 18% of the electron wavefunction resides on the middle QL. For CX1 (Fig. S5 (c)), which is composed of SS1to-SS1' transitions, the electron wavefunction on the middle QLs is slightly reduced to 14%, reflecting enhanced localization due to the exciton binding energy compared to the independentparticle picture. For CX3, however, which consists primarily of (VBM-2)-to-SS1' transitions, 32%
of the electronic part of the exciton wavefunction extends into the central QL, showing that the excitonic intermixing of the bulk state and surface state results in reduced localization at the surface. In comparison, for CX8 ( Fig. S5 (d)), where the electron comes primarily from bulk states, 91% of the electron wavefunction is on the central QL. Finally, Fig. S5 (a) and Fig. 4(b) show that 97% and 92% of the independent particle wavefunctions are localized on the surface QLs for SS2' and RSS. Similarly to CX1, the exciton CX10 that mainly comes from the transitions between SS2'
and RSS exhibits enhanced localization, with 99% of the electron wavefunction residing on the surface QLs. Thus, electron-hole interactions between two topologically or non-topologically protected surface states can enhance surface localization, while electron-hole interactions between the surface and bulk state reduce the surface localization.
Finally, though our full GW-BSE calculations focus on 3QL Bi2Se3, the mechanism of exchange-driven intermixing of bulk and surface states is generalizable to Bi2Se3 of arbitrary thickness and topological surface states in general. Fig. 4(e-g) shows the evolution of the electronhole exchange matrix elements ⟨ | | ′ ′ ′ ⟩. The matrix elements reveal that the exchange interaction allows for scattering between surface and bulk states at all thicknesses from 3QL to 6QL. The direct Coulomb interaction, which does not allow for significant scattering, is shown in the SI.
In summary, we have performed GW-BSE calculations of the QP bandstructure and optical spectra of bulk, 1QL, 3QL, and 5QL Bi2Se3, identified multiple series of novel excitonic features arising from transitions between an admixture of topologically protected surface states, topologically trivial surface states and bulk states, and studied their evolution with layer thickness.
We characterize the lowest energy bound excitons in 1QL and 3QL Bi2Se3 and find that the bright excitons are doubly-degenerate with an angular momentum of = ±1, consistent with p-like hydrogen atom states. The exciton binding energy decreases with increasing layer number, but exciton effects remain significant in 3QL Bi2Se3 for resonant states above the QP bandedge, where they are dominated by the electron-hole exchange interaction, which allows for significant mixing of bulk and surface states. We reveal the existence of ten chiral excitons, which arise from surfacestate-to-surface-state or surface-state-to-bulk-state transitions, suggesting that thin films of Bi2Se3 may be an interesting platform for exploring inter-exciton processes involving topological surface states. Furthermore, our calculations suggest that the previously observed broad chiral PL peak can be ascribed to the combined effects of two chiral excitons located at 2.48 and 2.78 eV [27].
These results provide a comprehensive understanding of the chiral excitons of Bi2Se3, paving the way for engineering chiral excitons for spin optoelectronics.
Reference
Fig. 1
1(b-d) shows the bandstructures for 1QL, 3QL, and 5QL slabs of Bi2Se3. In finite slabs of Bi2Se3, interaction between surface states at the top and bottom surface open a bandgap at the Γ point in the Dirac cone crossing the Fermi level (SS1)
hole interactions. Two prominent excitonic peaks, located at 0.8 eV and 1.0 eV, which we label peaks A and B, respectively, appear in the absorption spectrum of 1QL Bi2Se3. These excitons are strongly bound, with binding energies of 0.46 eV and 0.26 eV, respectively. Peak A consists of contributions from two degenerate bright excitons that are nearly degenerate with a lower energy dark exciton (see SI Fig. S2 for full spectrum), all of which are composed of electron and hole states occupying the doubly degenerate highest valence and lowest conduction bands. We expand each exciton in the electron-hole basis as | ⟩ = ∑ | ⟩ , where is the k-space envelope of the wavefunction for the exciton state S, composed of electron-hole pairs | ⟩. For 1QL Bi2Se3, we plot the phase winding of for the two bright excitons in peak A (Fig. 2 (b-c)) along with the phase winding of the dipole matrix element < |±| > between the highest valence and lowest conduction bands for right (̂+) and left-hand (̂−) circularly polarized light (Fig. 2 (d-e)). We find that the lowest energy bright excitons are p-like states with orbital angular momentum m=+1 and m=-1, while the dipole matrix elements have a winding number of l-=+1(l+=-1 ) for left(right)-circularly polarized light. The bright p-like exciton states in conjunction with the non-trivial topology of the underlying independent-particle bands is consistent with the generalized selection rule = − ∓ [28-30,54]. The B exciton peak is composed primarily of holes residing in the second-highest valence band and electrons in the lowest conduction band.Excitonic effects in 3QL Bi2Se3 are considerably smaller(Fig. 2 (f)) due to its smaller bandgap. The first peak in the absorption spectrum, labelled C, is nearly identical regardless of whether electron-hole interactions are included. We find that the binding energy of the first bright exciton is 10 meV. Since GW significantly increases the bandgap, we perform one self-consistent update of the screened Coulomb interaction W utilizing the GW bandgap of 0.17 eV. After the self-consistent update, the binding energy of peak C increases to 20 meV (see SIFig. S7). Despite the small binding energy, the optical selection rules of the bound excitons in 3QL Bi2Se3 follow the same pattern as in 1QL Bi2Se3: p-like exciton states with orbital angular momentum m=-1 and m=+1 are bright due to the l-=+1 and l+=-1 phase winding of the dipole matrix element between the valence and conduction bands composed of gapped chiral fermions (Fig. 2 (g-j)). At higher energies, resonant excitonic effects become more prominent, with considerable deviation between the spectra with and without electron-hole interactions. Intriguingly, asFig.4(e-g) shows, excitonic effects above the bandgap are dominated by the repulsive electron-hole exchange interaction. Unlike the direct Coulomb interaction, which scales with the overlap of the electron wavefunctions, the exchange scales with the overlap of the electron and hole states, thus allowing for mixing of electron states of different character, facilitating the intermixing of bulk and surface states.
Fig. 3 (
3a) depicts IRL and IRR. We identify ten excitonic peaks with circular-polarizationpreserving emission, labelled CX1-CX10, in the energy window from 0 to 2.82 eV. Of these, CX1, CX3, CX8, and CX10 exhibit the greatest circular dichroism and are the focus of the following discussions (see SI for other states). The figure insets show the k-space amplitude of the excitons contributing to each peak, where all identified excitons are well-localized within the patch around Γ. To elucidate the character of these excitons, we break down each exciton state into its component transitions.Fig. 3(b) shows the contributions of each occupied and unoccupied band to a given exciton state, weighted by the degree of circular emission ( − )[58]. The first circularly polarized emission peak, CX1, which is located at 0.17 eV corresponds to the lowest energy excitons in peak C, discussed previously.Among the ten identified excitons, CX1 is the only one that is composed entirely of TSS.The other higher energy peaks are resonant states composed of a combination of surface states with chirality-dependent optical transitions and a continuum of lower-energy bulk states. CX3, located at 1.1 eV, primarily arises from transitions between the third highest degenerate set of bulk valence bands (VBM-2) and the surface state SS1' (here, we use prime to identify the upper half of the gapped Dirac cone). Compared to CX1 and CX3, CX8 exhibits reduced circular dichroism.
This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Early Career Award No. DE-SC0021965. D.Y.Q. acknowledges support by a 2021 Packard Fellowship for Science and Engineering from the David and Lucile Packard Foundation. Development of the BerkeleyGW code was supported by Center for Computational Study of Excited-State Phenomena in Energy Materials (C2SEPEM) at the Lawrence Berkeley National Laboratory, funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE-C02-05CH11231. The calculations used resources of the National Energy Research Scientific Computing (NERSC), a DOE Office of Science User Facility operated under contract no. DE-AC02-05CH11231; the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562; and the Texas Advanced Computing Center (TACC) at The University of Texas at Austin. FIG.1 Electronic bandstructure at the G0W0 (solid red curve) and LDA (dashed blue curve) levels of (a) bulk, (b) 1QL, (c) 3QL, (d) 5QL Bi2Se3. The grey shadow in (c) shows the extent of a Γcentered patch with a radius of 0.028 0 −1 . FIG.2 Calculated absorption spectra of (a) 1QL and (f) 3QL Bi2Se3 with(without) electron-hole interactions included at the GW-BSE(GW-RPA) level in red(blue). The gray vertical line indicates the G0W0 quasiparticle bandgap. (b-c) The phase winding of the exciton envelope wavefunction, , of the degenerate bright excitons contributing to peak A of 1QL Bi2Se3 and (g-h) degenerate bright excitons contributing to peak C of 3QL Bi2Se3. The angle of the arrows indicates the phase, and the length of each arrow indicates the amplitude. The index m denotes the winding number around the Γ point. (d-e) The winding of the optical interband dipole transition matrix element between the highest valence band and the lowest conduction band for 1QL and (i-j) 3QL Bi2Se3. The index l-(+) denotes the winding number of left(right) circularly polarized light respectively. A maximally smooth local gauge is imposed following Refs. [59,60]. The exciton wavefunction and optical interband transitions are plotted in a patch centered around the Γ point in the BZ, marked by grey area in Fig. 1(c). FIG.3 (a) Instantaneous emission intensity of left-hand (IRL, dark blue) or right-hand (IRR, orange curve) polarized light after excitation by right-hand polarized light from all exciton states in a patch of radius of 0.028 0 −1 around Γ in the BZ. The inset shows the amplitudes of the envelope wavefunction of CX1, CX3, CX8, and CX10 in the patch in reciprocal space. (b) Contribution of each band to each exciton state. The bands are labeled as either bulk-like bands relative to the valence band maximum (VBM) and conduction band minimum (CBM) or as surface states SS1, SS2, and RSS, with the occupied band contributions in blue and unoccupied band contributions in red. Each bulk band is doubly degenerate, while surface states sum contributions from all the occupied or unoccupied surface states. The size of each dot is proportional to ( a-b) The modulus of the electronic wavefunction in 3QL Bi2Se3 for the surface states SS1' and RSS at the Γ point. (c-d) The modulus of the exciton wavefunction for CX3 and CX10, when the hole (yellow sphere) is fixed. The hole position is set at the location that maximizes the wavefunction amplitude. The integral of the modulus squared of each wavefunction over the x-y plane is shown as a side panel in (a-d). The percentage label indicates the contribution to the wavefunction from the central QL. For clarity, the Bi and Se atoms are not explicitly shown. The bonds are depicted as purple lines in the vicinity of Bi and green lines in the vicinity of Se. (e-g) Electron-hole exchange matrix elements ⟨ | | ′ ′ ′ ⟩ for = ′ = Γ and = ′ = SS1 for (e) 3QL, (f) 5QL and (g) 6QL slabs of Bi2Se3.
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| {'fraction_non_alphanumeric': 0.06549300400709453, 'fraction_numerical': 0.05271628456940156, 'mean_word_length': 4.2055052145665925, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Topological surface states (TSS) in the prototypical topological insulator (TI) Bi2Se3 are frequently characterized using optical probes, but electron-hole interactions and their effect on surface localization and optical response of the TSS remain unexplored. Here, we use ab initio calculations to understand excitonic effects in the bulk and surface of Bi2Se3. We identify multiple series of chiral excitons that exhibit both bulk and TSS character, due to exchange-driven mixing. Our results address fundamental questions about the degree to which electron-hole interactions can relax the topological protection of surface states and dipole selection rules for circularly polarized light in TIs by elucidating the complex intermixture of bulk and surface states excited in optical measurements and their coupling to light. In the prototypical topological insulator (TI) Bi2Se3, strong spin-orbit coupling (SOC) drives the inversion of bulk bands near the Fermi level, resulting in the appearance of a pair of topological surface states (TSS) with spin-momentum locking[1-13]. These TSS have the potential to be exploited for the generation of topologically-protected spin currents[14], leading to the promise of faster, low-powered spintronics. Due to the high residual conductivity of the bulk[15-20], light-based probes, such as the circular photogalvanic effect, are frequently used to probe the surface states[15,[21][22][23][24][25][26]. In such probes, changes to the photocurrent direction with the helicity of light are interpreted as evidence of photocurrent generation from the TSS. However, the frequency of the probe light is typically 1.5 eV or larger, leading to the excitation of a complex manifold of', 'arxivid': '2302.11056', 'author': ['Bowen Hou \nDepartment of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA\n', 'Dan Wang \nDepartment of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA\n', 'Bradford A Barker \nDepartment of Physics\nUniversity of California\n95343MercedCAUSA\n', 'Diana Y Qiu [email protected] \nDepartment of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA\n'], 'authoraffiliation': ['Department of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA', 'Department of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA', 'Department of Physics\nUniversity of California\n95343MercedCAUSA', 'Department of Mechanical Engineering and Material Sciences\nYale University\n06511New HavenCTUSA'], 'corpusid': 257078682, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10092, 'n_tokens_neox': 8412, 'n_words': 4811, 'pdfsha': '92d89b39a88c2469f321031f46e75e7d803f85c6', 'pdfurls': ['https://export.arxiv.org/pdf/2302.11056v1.pdf'], 'title': ['Exchange-Driven Intermixing of Bulk and Topological Surface State by Chiral Excitons in Bi2Se3', 'Exchange-Driven Intermixing of Bulk and Topological Surface State by Chiral Excitons in Bi2Se3'], 'venue': []} |
arxiv |
On Five-dimensional Superspaces
20 Feb 2006 July, 2005 Revised version: January, 2006
Sergei M Kuzenko [email protected]@physics.umd.edu
School of Physics M013
The University of Western Australia
35 Stirling Highway, Crawley W.A. 6009Australia
‡
IIIWilliam D Linch
Center for String and Particle Theory Department of Physics
University of Maryland College Park
20742-4111MDUSA
On Five-dimensional Superspaces
20 Feb 2006 July, 2005 Revised version: January, 2006arXiv:hep-th/0507176v5 3 Address after September 1, 2005: C. N. Yang Institute for Theoretical Physics and the Department of Mathematics, State University of New York, Stony Brook, NY 11794.
Recent one-loop calculations of certain supergravity-mediated quantum corrections in supersymmetric brane-world models employ either the component formulation (hep-th/0305184) or the superfield formalism with only half of the bulk supersymmetry manifestly realized (hep-th/0305169 and hep-th/0411216). There are reasons to expect, however, that 5D supergraphs provide a more efficient setup to deal with these and more involved (in particular, higher-loop) calculations. As a first step toward elaborating such supergraph techniques, we develop in this letter a manifestly supersymmetric formulation for 5D globally supersymmetric theories with eight supercharges. Simple rules are given to reduce 5D superspace actions to a hybrid form which keeps manifest only the 4D, N = 1 Poincaré supersymmetry. (Previously, such hybrid actions were carefully worked out by rewriting the component actions in terms of simple superfields). To demonstrate the power of this formalism for model building applications, two families of off-shell supersymmetric nonlinear sigma-models in five dimensions are presented (including those with cotangent bundles of Kähler manifolds as target spaces). We elaborate, trying to make our presentation maximally clear and self-contained, on the techniques of 5D harmonic and projective superspaces used at some stages in this letter.
Introduction
Supersymmetric field theories in dimensions higher than the four accessible in our everyday experiences have been contemplated for many years now. Besides being forced on us by our current understanding of superstring theory, it has also proven to be of possible phenomenological importance in which discipline such theories are known as supersymmetric "brane-world" models [1,2,3,4]. Currently, this type of theory is being used by various groups in attempts to implement supersymmetry breaking in a manner consistent with the stringent bounds coming from flavor changing neutral currents. One particular application uses a supersymmetric gravitational theory in a five-dimensional spacetime of which the "extra" spacial dimension is a compact interval of some length 1 ℓ Z 2 Figure 1: The orbifold construction. [4,5]. At each end of the interval is a copy of 4D Minkowski spacetime (the branes). Brane world models consist of postulating that the standard model of particle physics is localized on one of these branes (the "visible" or "infra-red" brane) while other fields propagate in the interior ("bulk") of the 5D spacetime or on the other "hidden" (or "ultraviolet") brane (see figure 2). In the particular model under consideration, supersymmetry is broken on the hidden brane by allowing the F -(or D-term [6]) of some field localized on that brane to acquire a vacuum expectation value. This is then communicated to our brane by supergraviton loops in the bulk which induce the breakdown of supersymmetry on the standard model brane. The final result is a four-dimensional effective action for the standard model fields on the visible brane. 1 An elegant way of constructing such a space is to start with a 5D Minkowski space and toroidally compactify one of the spacial directions, producing a space of topology R 4 × S 1 . One subsequently defines a non-free action of Z 2 on the circle which has two antipodal fixed points (i.e. a reflection through a "diameter") and mods the circle by this action. Since the action was not free, the quotient S 1 /Z 2 is not a smooth manifold (figure 1). Nevertheless it is a manifold-with-boundary diffeomorphic to the closed finite interval [0, 1]. The "orbifold" R 4 × (S 1 /Z 2 ) has as boundary two hyperplanes (the "orbifold fixed planes"), each isometric to a 4D Minkowski space, at the fixed points of the Z 2 action. Figure 2: The 5D brane-world scenario of the gravitational mediation of supersymmetry breaking in the standard model. The F -term of a chiral field X on the hidden brane acquires a vacuum excitation value. This is then communicated to the standard model (visible) brane by gravitational messengers propagating through the bulk.
Needless to say, calculations such as that of gravitational loop corrections are difficult to perform in components (although it was done in [7] at one loop). On the other hand, it is commonly believed that the development of a full-fledged 5D superspace formulation has the annoying drawback that the result, which is desired to be a four-dimensional effective action, is given in a complicated form. More exactly, the output of such an effort is manifestly supersymmetric in five dimensions and must be dimensionally reduced in the final stages of the calculation. For this reason a "hybrid" formalism was developed for supergravity in five dimensions which keeps manifest only 4D, N = 1 super-Poincaré invariance [8]. This hybrid is given in terms of supergravity prepotentials which allows one to apply the powerful supergraph techniques necessary for perturbative quantum calculations. Indeed, in [9] it was used to compute, in a more econmical way than in the component approach of [7], the leading gravity loop contribution to supersymmetry breakdown described above in a very simple way.
Although the formalism was successfully extended to allow a "warping" of the extra dimension and the gravity-mediation scenario investigated in this background [10], it has a major drawback which arises as follows. This approach is essentially a superfield Noether procedure in which one starts with a linearized supergravity action, and then tries to reconstruct interaction terms, order by order, by consistently deforming the gauge transformations, etc. Usually the Noether procedure can be completed if it requires a finite number of iterations, as is the case with polynomial actions. But superfield supergravity is a highly nonlinear theory in terms of its prepotentials (see [11,12] for reviews). As a result, the limitations of this hybrid approach are called into question. More importantly, it turns out to be difficult to discover the rules governing the coupling of this theory to other matter fields in the bulk. In the end, we are forced to turn to the known (full-fledged) off-shell formulations for 5D simple supergravity, 2 with or without supersymmetric matter, in the hope of deducing a useful superfield formulation.
Off-shell 5D simple supergravity was sketched in superspace, a quarter century ago, by Breitenlohner and Kabelschacht [13] and Howe [14] (building on a related work [15]). More recently, it was carefully elaborated by Zucker [16] at the component level, and finally perfected in [17,18] within the superconformal tensor calculus.
Using the results of the 5D superconformal tensor calculus for supergravity-matter systems, one can develope a hybrid N = 1 superspace formalism by fitting the component multiplets into superfields. Such a program has been carried out in [19]. Although useful for tree-level phenomenological applications, we believe this approach is not the optimum (economical) formulation for doing supergraph loop calculations. The point is that the superconformal tensor calculus usually corresponds to a Wess-Zumino gauge in superfield supergravity. But such gauge conditions are impractical as far as supergraph calculations are concerned.
When comparing the superconformal tensor calculus for 5D simple supergravity [17,18] with that for 4D, N = 2 and 6D, N = (1, 0) supergravities (see [17,18] for the relevant references), it is simply staggering how similar these formulations are, modulo some fine details. From the point of view of a superspace practitioner, the reason for this similarity is that the three versions of superconformal calculus are generated from (correspond to a Wess-Zumino gauge for) a harmonic superspace formulation for the corresponding supergravity theory, and such harmonic superspace formulations 3 look almost identical in the space-time dimensions 4, 5 and 6, again modulo fine details. For example, independent of the space-time dimension, the Yang-Mills supermultiplet is described by (formally) the same gauge superfield V ++ , with the same gauge freedom δV ++ = −D ++ λ, and with the same Wess-Zumino-type gauge 2 We prefer to use the term "5D simple supergravity," since in the literature 5D simple supersymmetry is called sometimes N = 1 and sometimes N = 2, depending upon taste and background. 3 The harmonic superspace formulation for 4D, N = 2 supergravity is reviewed in the book [21]. For the case of 6D, N = (1, 0) supergravity, such a formulation was constructed in [22], and it can be used to derive a relevant formulation for 5D simple supergravity by dimensional reduction.
i V ++ = θ + Γ m θ + A m (x) + θ + Γ 5 θ + A 5 (x) + θ + Γ 6 θ + A 6 (x) + O(θ 3 ) ,
where θ + is a four-component anticommuting spinor variable, and m = 0, 1, 2, 3.
The concept of harmonic superspace was originally developed for 4D, N = 2 supersymmetric theories including supergravity [20], and by now it has become a textbook subject 4 [21]. Actually it can be argued that harmonic superspace is a natural framework for all supersymmetric theories with eight supercharges, both at the classical and quantum levels. In the case of four space-time dimensions, probably the main objection to this approach was the issue that theories in harmonic superspace are often difficult to reduce to N = 1 superfields (the kind of reduction which brane-world practitioners often need). But this objection has been lifted since the advent and subsequent perfection of 4D, N = 2 projective superspace [23,24] which allows a nice reduction to N = 1 superfields and which appears to be a truncated version of the harmonic superspace [25].
What is the difference between harmonic superspace and projective superspace? In five space-time dimensions (to be concrete), they make use of the same supermanifold R 5|8 × S 2 , with R 5|8 the conventional 5D simple superspace. In harmonic superspace, one deals with so-called Grassmann analytic (also known as twised chiral) superfields that are chosen to be smooth tensor fields on S 2 . In projective superspace, one also deals with Grassmann analytic superfields that are holomorphic functions on an open subset of S 2 . It is clear that the harmonic superspace setting is more general. Actually, many results originally obtained in projective superspace can be reproduced from harmonic superspace by applying special truncation procedures [25]. The remarkable features of projective superspace are that (i) the projective supermultiplets are easily represented as a direct sum of standard 4D, N = 1 superfields; (ii) this approach provides simple rules to construct low-energy effective actions that are easily expressed in terms of 4D, N = 1 superfields. Of course, one could wonder why both harmonic and projective superspaces should be introduced? The answer is that, in many respects, they are complementary to each other. (This is analogous to the relation between the theorems of existence of solutions for differential equations and concrete techniques to solve such equations.)
To avoid technicalities, in this paper we do not consider 5D superfield supergravity at all, and concentrate only on developing a 5D simple superspace approach to globally supersymmetric gauge theories. One of our main objectives is to demonstrate that 5D superspace may be useful, even in the context of 4D effective theories with an extra dimension. Here we develop manifestly 5D supersymmetric techniques which, on the one hand, allow us to construct many of the 5D supersymmetric models originally developed within the "hybrid" formulation. One the other hand, these techniques make it possible to construct very interesting supersymmetric nonlinear sigma-models whose construction is practically beyond the scope of the "hybrid" formulation. Examples of such 5D supersymmetric sigma-models are constructed for the first time below. We therefore believe that the paper should be of some interest to both superspace experts and newcomers.
It is worth saying a few words about the global structure of this paper. We are aiming at (i) elaborating 5D off-shell matter supermultiplets and their superfield descriptions; (ii) developing various universal procedures to construct manifestly 5D supersymmetric action functionals, and then applying them to specific supermultiplets; (iii) elaborating on techniques to reduce such super-actions to 4D, N = 1 superfields. New elements of 5D superfield formalism are introduced only if they are essential for further consideration. For example, the Yang-Mills off-shell supermultiplet can be realized in 5D conventional superspace in terms of constrained superfields. In order to solve the constraints, however, one has to introduce the concept of harmonic superspace. This paper is organized as follows: In section 2 we describe, building on earlier work [15,26], the 5D Yang-Mills supermultiplet and its salient properties, both in the conventional and harmonic superspaces. We also describe several off-shell realizations for the 5D hypermultiplet. In section 3 we present two procedures to construct 5D manifestly supersymmetric actions for multiplets with and without intrinsic central charge, and give several examples. Section 4 is devoted to 5D supersymmetric Chern-Simons theories. Their harmonic superspace actions are given in a new form, as compared with [26], which allows a simple reduction to the projective superspace. We also uncover the 5D origin for the superfield constraints describing the so-called 4D, N = 2 nonlinear vector-tensor multplet. In section 5, some of the results developed in the previous sections are reduced to a "hybrid" formulation which keeps manifest only 4D, N = 1 super Poincaré symmetry. Section 6 introduces 5D simple projective superspace and projective multiplets. Here we also present two families of 5D off-shell supersymmetric nonlinear sigma-models which are formulated, respectively, in terms of a (i) 5D tensor multiplet; (ii) 5D polar mutiplet. Section 7 deals with the vector multiplet in projective superspace. A brief conclusion is given in section 8. This paper also includes three technical appendices. Appendix A contains our 5D notation and conventions, inspired by those in [27,28], as well as some important identities. Appendix B is devoted to a review of the well-known one-to-one correspondence between smooth tensor fields on S 2 = SU(2)/U(1) and smooth scalar functions over SU(2) with definite U(1) charges. Finally, in appendix C we briefly demonstrate, mainly following [25], how to derive the projective superspace action (6.14) from the harmonic superspace action (3.2).
5D Supersymmetric Matter
Vector multiplet in conventional superspace
To describe a Yang-Mills supermultiplet in 5D simple superspace R 5|8 parametrized by coordinates z = (xâ, θα i ) we introduce gauge-covariant derivatives 5
DÂ = (Dâ, D î α ) = DÂ + i VÂ(z) , [DÂ , DB} = TÂBĈ DĈ + CÂB ∆ + i FÂB ,DÂ → e iτ (z) DÂ e −iτ (z) , τ † = τ , [∆ , τ ] = 0 , (2.2)
with the gauge parameter τ (z) being arbitrary modulo the reality condition imposed. The gauge-covariant derivatives are required to obey some constraints [15] such that
{D î α , D ĵ β } = −2i ε ij (Γĉ)αβ Dĉ + εαβ (∆ + i W) , D î α , ∆ = 0 , Dâ , D ĵ β = i (Γâ)βγ D ĵ γ W , Dâ , Db = − 1 4 (Σâb)αβ D î α Dβ i W = i Fâb , (2.3)
with the matrices Γâ and Σâb defined in Appendix A. Here the field strength W is hermitian, W † = W, and obeys the Bianchi identity (see e.g. [26])
D (î α D j) β W = 1 4 εαβ Dγ (i D j) γ W ,(2.4)
and therefore D (î
α D ĵ β D k) γ W = 0 . (2.5)
The independent component fields contained in W are:
ϕ = W|| , i Ψ iα = D î α W|| , −4i Fαβ = D i (α Dβ )i W|| , −4i X ij = Dα (i D j) α W|| . (2.6)
Here and in what follows, U|| denotes the θ-independent component of a superfield U(x, θ).
It is worth noting that Fâb = Fâb|| .
Vector multiplet in harmonic superspace
The most elegant way to solve the constraints encoded in the algebra (2.3) is to use the concept of harmonic superspace originally developed for 4D, N = 2 supersymmetric theories [20,21] (related ideas appeared in [29]). In this approach, the conventional superspace R 5|8 is embedded into R 5|8 × S 2 , where the two-sphere S 2 = SU(2)/U(1) is parametrized by so-called harmonic u i − and u i + , that is group elements
(u i − , u i + ) ∈ SU(2) , u + i = ε ij u +j , (u +i ) * = u − i , u +i u − i = 1 . (2.8)
As is well-known, tensor fields over S 2 are in a one-to-one correspondence with functions over SU(2) possessing definite harmonic U(1) charge (see [25] for a review). A function
Ψ (p) (u) is said to have harmonic U(1) charge p if Ψ (p) (e iα u + , e −iα u − ) = e ipα Ψ (p) (u + , u − ) , |e iα | = 1 . (2.9)
Such functions, extended to the whole harmonic superspace R 5|8 × S 2 , that is Ψ (p) (z, u), are called harmonic superfields. Introducing the harmonic derivatives [20]
D ++ = u +i ∂ ∂u −i , D −− = u −i ∂ ∂u +i , D 0 = u +i ∂ ∂u +i − u −i ∂ ∂u −i , [D 0 , D ±± ] = ±2D ±± , [D ++ , D −− ] = D 0 ,(2.10)
one can see that D 0 is the operator of harmonic U(1) charge, D 0 Ψ (p) (z, u) = p Ψ (p) (z, u). Defining 11) one observes that the operators D A possess the same transformation law (2.2) as DÂ.
D A ≡ (DÂ, D ++ , D −− , D 0 ) , D ±± = D ±± , D 0 = D 0 ,(2.
Introduce a new basis for the spinor covariant derivatives:
D + α = D î α u + i and D − α = D î α u − i . Then, eq. (2.3) leads to {D + α , D + β } = 0 , D ++ , D + α = 0 , {D + α , D − β } = 2i Dαβ + εαβ (∆ + i W) , (2.12) D ++ , D − α = D + α , D −− , D + α = D − α .
In harmonic superspace, the integrability condition {D + α , D + β } = 0 is solved by
D + α = e −iΩ D + α e iΩ ,(2.13)
for some Lie-algebra valued harmonic superfield Ω = Ω(z, u) of vanishing harmonic U(1) charge, D 0 Ω = 0. This superfield is called the bridge. The bridge possesses a richer gauge freedom than the original τ -group (2.2) e iΩ(z,u) → e iλ(z,u) e iΩ(z,u) e −iτ (z) ,
D + α λ = 0 , [∆ , λ] = 0 . (2.
14)
The λ-and τ -transformations generate, respectively, the so-called λ-and τ -groups.
One can now define covariantly analytic superfields constrained by
D + α Φ (p) = 0 . (2.15) Here Φ (p) (z, u) carries U(1)-charge p, D 0 Φ (p) = pΦ (p)
, and can be represented as follows
Φ (p) = e −iΩ φ (p) , D + α φ (p) = 0 ,(2.16)
with φ (p) (ζ) being an analytic superfield -that is, a field over the so-called analytic subspace of the harmonic superspace parametrized by
ζ ≡ {xâ, θ +α , u + i , u − j } , (2.17) where xâ = xâ + i (Γâ)βγ θ +β θ −γ , θ ± α = θ î α u ± i . (2.18)
In particular, the gauge parameter λ in (2.14) is an unconstrained analytic superfield of vanishing harmonic U(1) charge, D 0 λ = 0. It is clear that the superfields Φ (p) and φ (p) describe the same matter multiplet but in different frames (or, equivalently, representations), and they transform under the τ -and λ-gauge groups, respectively.
Φ (p) (z, u) → e iτ (z) Φ (p) (z, u) , φ (p) (z, u) → e iλ(z,u) φ (p) (z, u) . (2.19)
By construction, the analytic subspace (2.17) is closed under the supersymmetry transformations. Unlike the chiral subspace, it is real with respect to the generalized conjugation (often called the smile-conjugation)˘ [20] defined to be the composition of the complex conjugation (Hermitian conjugation in the case of Lie-algebra-valued superfields) with the operation ⋆ acting on the harmonics only
(u + i ) ⋆ = u − i , (u − i ) ⋆ = −u + i ⇒ (u ± i ) ⋆⋆ = −u ± i , (2.20) hence (u +i )˘= −u + i (u − i )˘= u −i . (2.21)
The analytic superfields of even U(1) charge may therefore be chosen to be real. In particular, the bridge Ω and the gauge parameter λ are real.
The covariant derivatives in the λ-frame are obtained from those in the τ -frame, eq. (2.11), by applying the transformation
D A → e iΩ D A e −iΩ .
(2.22)
Then, the gauge transformation of the covariant derivatives becomes
D A → e iλ(ζ) D A e −iλ(ζ) ,λ = λ , [∆ , λ] = 0 . (2.23)
In the λ-frame, the spinor covariant derivatives D + α coincide with the flat ones, D + α = D + α , while the harmonic covariant derivatives acquire connections,
D ±± = D ±± + i V ±± .
(2.24)
The real connection V ++ is seen to be an analytic superfield, D + α V ++ = 0, of harmonic U(1) charge plus two, D 0 V ++ = 2V ++ . The other harmonic connection V −− turns out to be uniquely determined in terms of V ++ using the zero-curvature condition
[D ++ , D −− ] = D 0 ⇐⇒ D ++ V −− − D −− V ++ + i [V ++ , V −− ] = 0 , (2.25)
as demonstrated in [30]. The result is
V −− (z, u) = ∞ n=1 (−i) n+1 du 1 . . . du n V ++ (z, u 1 ) V ++ (z, u 2 ) · · · V ++ (z, u n ) (u + u + 1 )(u + 1 u + 2 ) . . . (u + n u + )
, (2.26)
with (u + 1 u + 2 ) = u +i 1 u + 2 i , and the harmonic distributions on the right of (2.26) defined, e.g., in [21]. Integration over the group manifold SU(2) is normalized according to [20]
du 1 = 1 , du u + (i 1 · · · u + in u − j 1 · · · u − jm) = 0 , n + m > 0 . (2.27)
As far as the connections V − α and Vâ are concerned, they can be expressed in terms of V −− with the aid of the (anti-)commutation relations
D −− , D + α = D − α , {D + α , D − β } = 2i Dαβ + εαβ (∆ + i W λ ) . (2.28)
In particular, one obtains
W λ = i 8 (D + ) 2 V −− , (D + ) 2 = D +α D + α ,(2.29)
where W λ stands for the field strength in the λ-frame. Therefore, V ++ is the single unconstrained analytic prepotential of the theory. With the aid of (2.25) one can obtain the following useful expression
W = i 8 du (D − ) 2 V ++ + O (V ++ ) 2 . (2.30)
In the Abelian case, only the first term on the right survives. In what follows, we do not distinguish between W and W λ .
With the notation (D + ) 2 = D +α D + α , the Bianchi identity (2.4) takes the form
D + α D + β W = 1 4 εαβ (D + ) 2 W ⇒ D + α D + β D + γ W = 0 . (2.31)
Using the Bianchi identity (2.31), one can readily construct a covariantly analytic
descendant of W −i G ++ = D +α W D + α W + 1 4 {W , (D + ) 2 W} , D + α G ++ = D ++ G ++ = 0 . (2.32)
Vector multiplet in components
The gauge freedom
−δV ++ = D ++ λ = D ++ λ + i [V ++ , λ] (2.33)
can be used to choose a Wess-Zumino gauge of the form
V ++ (x, θ + , u) = i (θ + ) 2 ϕ(x) − i θ + Γmθ + Am(x) + 4(θ + ) 2 θ +α u − i Ψ iα (x) − 3 2 (θ + ) 2 (θ + ) 2 u − i u − j X ij (x) , (2.34) where (θ + ) 2 = θ +α θ + α , θ + Γmθ + = θ +α (Γm)αβ θ +β = −(Γm)αβ θ +α θ +β . (2.35)
In this gauge, the expression (2.26) simplifies considerably
V −− (z, u) = du 1 V ++ (z, u 1 ) (u + u + 1 ) 2 + i 2 du 1 du 2 [V ++ (z, u 1 ) , V ++ (z, u 2 )] (u + u + 1 )(u + 1 u + 2 )(u + 2 u + ) + terms of third-and fourth-order in V ++ . (2.36)
Here the explicit form of the cubic and quartic terms is not relevant for our consideration. One of the important properties of the Wess-Zumino gauge is
Dm|| ≡ ∂m + i Vm|| = ∂m + i Am(x) . (2.37)
The component fields of W and V ++ can be related to each other using the identity 38) and the analyticity of V ++ . (The latter property implies, for instance,
F + 2 = (u + 1 u + 2 ) F − 1 − (u − 1 u + 2 ) F + 1 , F ± = F i u ± i ,(2.D + V ++ (z, u 1 ) = −(u + u + 1 ) D − 1 V ++ (z, u 1 ).) Thus one gets W|| = i 8 (D + ) 2 V −− || = i 8 du 1 (D − 1 ) 2 V ++ (z, u 1 )|| = ϕ(x) , D + α W|| = − i 8 du 1 (u + u + 1 ) D − 1α (D − 1 ) 2 V ++ (z, u 1 )|| = i Ψ iα (x) u + i , (2.39) (D + ) 2 W|| = i 8 du 1 (u + u + 1 ) 2 (D − 1 ) 2 (D − 1 ) 2 V ++ (z, u 1 )|| = −4i X ij (x) u + i u + j .
Finally, eq. (2.37) implies that the component field Fαβ = F (αβ) in (2.6) is (the bispinor form of) the gauge-covariant field strength Fmn generated by the gauge field Am.
Fayet-Sohnius hypermultiplet
Following the four-dimensional N = 2 supersymmetric construction due to Fayet and Sohnius [31,32], an off-shell hypermultiplet with intrinsic central charge, which is coupled to the Yang-Mills supermultiplet, can be described by a superfield q i (z) and its conjugatē
q i (z),q i = (q i ) † , subject to the constraint D (î α q j) = 0 .
(2.40)
Introducing q + (z, u) = q i (z) u + i andq + (z, u) = −q i (z) u + i , the constraint (2.
40) can be rewritten in the form
D + α q + = D + αq + = 0 , D ++ q + = D ++q+ = 0 . (2.41)
Thus q + is a constrained analytic superfield. Using the algebra of gauge-covariant derivatives, the constraints can be shown to imply 6
⌢ q + = 0 , (2.42) ⌢ = DâDâ + (D +α W) D − α − 1 4 (D +α D + α W) D −− + 1 8 [D +α , D − α ] W + (∆ + i W) 2 .
6 By analogy with the four-dimensional case [33], the operator ⌢ can be called the covariant analytic d'Alembertian. Given a covariantly analytic superfield Φ (q) , the identity
⌢ Φ (q) = − 1 64 (D + ) 2 (D + ) 2 (D −− ) 2 Φ (q)
holds, and therefore ⌢ preserves analyticity.
Therefore, the requirement of a constant central charge, ∆ q + = m q + , with m a constant mass parameter, is equivalent to an equation of motion for the hypermultiplet.
Independent component fields of q i (z) can be chosen as
C i = q i || , λα = i √ 8 D î α q i || , F i = ∆q i || . (2.43)
All other components can be related to these and their derivatives. For example,
(D − ) 2 q + = 8i ∆q − − 8W q − , D − α ∆q + = DαβD + β q − + i W D + α q − + 2i (D + α W) q − , (D − ) 2 ∆q + = −8i DâDâq − + 8W ∆q − + 8i W 2 q − + 4i (D −α W) D + α q − +2i (D −α D + α W) q − − 2i (D −α D − α W) q + . (2.44)
Off-shell hypermultiplets without central charge
One of the main virtues of the harmonic superspace approach [20] is that it makes possible an off-shell formulation for a charged hypermultiplet (transforming in an arbitrary representation of the gauge group) without central charge. Such a q + -hypermultiplet is described by an unconstrained analytic superfield q + (z, u) and its conjugateq + (z, u),
D + α q + = 0 , ∆ q + = 0 . (2.45)
In this approach, the requirement that q + be a holomorphic spinor field over the twosphere, D ++ q + = 0, is equivalent to an equation of motion. 7 The harmonic dependence of the q + -hypermultiplet is non-trivial. One can represent q + (z, u) by a convergent Fourier series of the form (B.9) with p = 1. The corresponding Fourier coefficients q i 1 ···i 2n+1 (z), where n = 0, 1, . . . , obey some constraints that follow from the analyticity condition in (2.45).
Given a hypermultiplet that transforms in a real representation of the gauge group, it can be described by a real anaytic superfied ω(z, u),
D + α ω = 0 , ∆ ω = 0 ,(2.46)
called the ω-hypermultiplet [20]. The gauge parameter λ in (2.23) is of this superfield type. It is then clear that the ω-hypermultiplet can be used, for instance, to formulate a gauge-invariant Stückelberg description for massive vector multiplets.
Supersymmetric Actions
In the case of vanishing central charge, ∆ = 0, it is easy to construct manifestly supersymmetric actions within the harmonic superspace approach [20]. Given a scalar harmonic superfield L(z, u) and a scalar analytic superfield L (+4) (ζ), supersymmetric actions are:
S H = d 5 x d 8 θ du L = d 5 x du (D − ) 4 (D + ) 4 L , (3.1) S A = dζ (−4) L (+4) = d 5 x du (D − ) 4 L (+4) , D + α L (+4) = 0 , (3.2) where (D ± ) 4 = − 1 32 (D ± ) 2 (D ± ) 2 . (3.3)
As follows from (3.1), any integral over the full superspace can be reduced to an integral over the analytic subspace,
d 5 x d 8 θ du L = dζ (−4) L (+4) , L (+4) = (D + ) 4 L . (3.4)
The massless q + -hypermultiplet action [20] is
S = − dζ (−4)q+ D ++ q + . (3.5)
This action also describes a massive hypermutliplet if one assumes that (i) the gauge group is G × U(1), and (ii) the U(1) gauge field V ++ 0 possesses a constant field strength W 0 = const, |W 0 | = m, see [34] for more details. 8 Similarly to the chiral scalar in 4D, N = 1 supersymmerty, couplings for the q +hypermultiplet are easy to construct. For example, one can consider a Lagrangian of the form
L (+4) = −q + D ++ q + + λ (q + q + ) 2 +q + σ 1 (D + ) 2 W + iσ 2 G ++ q + ,(3.6)
with the quartic self-coupling first introduced in [20]. Consistent couplings for the Fayet-Sohnius hypermultiplet are much more restrictive, as a result of a non-vanishing intrinsic central charge.
Four-derivative vector multiplet action
As another example of supersymemtric action, we consider four-derivative couplings that may occur in low-energy effective actions for a U(1) vector multiplet.
S four−deriv = dζ (−4) G ++ κ 1 G ++ + i κ 2 (D + ) 2 W + d 5 x d 8 θ H(W) ,(3.7)
with κ 1,2 coupling constants, the analytic superfield G ++ given by (2.32), and H(W) an arbitrary function. The third term on the right is a natural generalization of the fourderivative terms in 4D, N = 2 supersymmetry first introduced in [36].
Multiplets with intrinsic central charge
For 5D off-shell supermultiplets with ∆ = 0, the construction of supersymmetric actions is based on somewhat different ideas developed in [32,37] for the case of 4D, N = 2 supersymmetric theories.
When ∆ = 0, there exists one more useful representation (in addition to the τ -frame and λ-frame) for the covariant derivatives
D A → ∇ A = e i(Ω+Σ) D A e −i(Ω+Σ) ≡ D A + i V A , Σ = −θ −α θ + α ∆ . (3.8)
For the operators D A = e iΣ D A e −iΣ one obtains
∇ + α = D + α = ∂ ∂θ −α , (3.9) D ++ = D ++ + i (θ + ) 2 ∆ , D ++ = u +i ∂ ∂u −i + i (Γâ)βγ θ +β θ +γ ∂ ∂xâ + θ +α ∂ ∂θ −α , where (θ + ) 2 = θ +α θ + α . (3.10)
As is seen, in this frame the spinor gauge-covariant derivative ∇ + α coincides with partial derivatives with respect to θ −α , while the analyticity-preserving gauge-covariant derivative ∇ ++ = D ++ + i V ++ acquires a term proportional to the central charge.
In accordance with [37], the supersymmetric action involves a special gauge-invariant analytic superfield L ++
D + α L ++ = 0 , D ++ L ++ = 0 . (3.11)
The action is
S = i dζ (−4) (θ + ) 2 L ++ . (3.12)
Although S involves naked Grassmann variables, it turns out to be supersymmetric,due to the constraints imposed on L ++ . Its invariance under the supersymmetry transformations can be proved in complete analogy with the four-dimensional case [37]. The action (3.12) possesses another nice representation obtained in Appendix C, eq. (C.13).
One can transform L ++ to the τ -frame in which
L ++ (z, u) = e −iΣ L ++ = L ij (z) u + i u + j . (3.13)
This gauge-invariant superfield obeys the constrains
D + α L ++ = 0 , D ++ L ++ = 0 . (3.14)
Doing the Grassmann and harmonic integrals in (3.12) gives
S = i 12 d 5 xD ij L ij ,D ij = Dα (i D j) α , (3.15)
where we have replaced, for convenience, ordinary spinor covariant derivatives by gaugecovariant ones (this obviously does not change the action, for L ij is gauge invariant).
In four space-time dimensions, the super-action (3.15) was postulated by Sohnius [32] several years before the discovery of harmonic superspace. It is quite remarkable that only within the harmonic superspace approach, this super-action can be represented as a superspace integral having a transparent physical interpretation. To wit, the factor i(θ + ) 2 in (3.12) can be identified with a vacuum expectation value V ++ ∆ of the central-charge gauge superfield V ++ ∆ (compare with (2.39)). With such an interpretation, the superaction admits simple generalizations to the cases of (i) rigid supersymmetric theories with gauged central charge [38], and (ii) supergravity-matter systems [39].
The super-action (3.12), and its equivalent form (3.15), can be used for superymmetric theories without central charge; an example will be given below. It is only the constraints (3.14) which are relevant in the above construction.
Fayet-Sohnius hypermultiplet
An example of a theory with non-vanishing central charge is provided by the Fayet-Sohnius hypermultiplet. The Fayet-Sohnius hypermultiplet coupled to a Yang-Mills supermultiplet is described by the Lagrangian [32,37] L ++ FS = 1 2q
+ ←→ ∆ q + − i mq + q + ,(3.16)
with m the hypermultiplet mass.
To compute the component action that follows from (3.16), one should use the definitions (2.6) and (2.43) for the component fields of W and q i , respectively.
S FS = d 5 x − D aCk D a C k − iλ Dλ +F k F k + mλλ +λϕλ − i 2C k X k ℓ C ℓ − 1 2C k ϕ 2 C k − i mF k C k − 1 √ 8λ Ψ k C k + iF k ϕ C k + c.c. . (3.17)
Vector multiplet
The Yang-Mills supermultiplet is described by the Lagrangian 9
L ++ YM = 1 4 tr G ++ , ∆ L ++ YM = 0 ,(3.18)
with G ++ given in (2.32). The corresponding equation of motion can be shown to be
(D + ) 2 W = 0 ⇔ D + α D + β W = 0 . (3.19)
It follows from this that
DâDâ W = 1 2 {Dα i W , D î α W} . (3.20)
In the Abelian case, eq. (3.19) reduces to
D + α D + β W = 0 ⇒ ∂â∂â W = 0 . (3.21)
From the point of view of 4D, N = 2 supersymmetry, this can be recognized as the off-shell superfield constraints [40,37,41] describing the so-called linear vector-tensor multiplet discovered by Sohnius, Stelle and West [42] and re-vitalized fifteen years later in the context of superstring compactifications [43].
The Yang-Mills action with components defined by (2.34) is
S YM = d 5 x tr − 1 4 FâbFâb − 1 2 DâϕDâϕ + 1 4 X ij X ij + i 2 Ψ k DΨ k − 1 2 Ψ k [ϕ, Ψ k ] . (3.22) 9
In terms of the analytic prepotential V ++ , the super Yang-Mills action is non-polynomial [30].
Chern-Simons Couplings
Consider two vector multiplets: (i) a U(1) vector multiplet V ++ ∆ ; (ii) a Yang-Mills vector multilpet V ++ YM . They can be coupled to each other, in a gauge-invariant way, using the interaction
S int = dζ (−4) V ++ ∆ tr G ++ YM ,(4.1)
where G ++ YM corresponds to the non-Abelian multiplet and is defined as in eq. (2.32). Invariance of S int under the U(1) gauge transformations
δV ++ = −D ++ λ , D + α λ = 0 ,(4.2)
follows from the constraints (2.32) to which G ++ YM is subject. Let us assume that the physical scalar field in V ++ ∆ possesses a non-vanishing expectation value (such a situation occurs, for instance, when V ++ ∆ is the vector multiplet gauging the central charge symmetry). In accordance with [34], this condition is expressed as W ∆ (z) = µ = 0; then, there exists a gauge fixing such that
V ++ ∆ (ζ, u) = i µ (θ + ) 2 , µ = const . (4.3)
Now, combining the interaction (4.1) with the gauge-invariant kinetic terms for V ++ ∆ and V ++ YM , the complete action becomes
S = dζ (−4) V ++ ∆ g −2 ∆ G ++ ∆ + g −2 YM tr G ++ YM ,(4.4)
with g ∆ and g YM coupling constants. A different form for this action was given in [26]. The theory (4.4) is superconformal at the classical level. It would be interesting to compute, for instance, perturbative quantum corrections.
Let us consider the special case of a single Abelian gauge field, V ++ ∆ = V ++ YM ≡ V ++ . The equations of motion for the corresponding Chern-Simons theory,
S CS = 1 12g 2 dζ (−4) V ++ G ++ ,(4.5)
can be shown to be
−i G ++ = D +α W D + α W + 1 2 W (D + ) 2 W = 0 . (4.6)
Using the Bianchi identity (2.31), one can rewrite this in the form
D + α D + β W = − 1 2 εαβ D +γ W D + γ W W . (4.7)
From the point of view of 4D, N = 2 supersymmetry, this can be recognized as the off-shell superfield constraint describing the so-called nonlinear vector-tensor multiplet 10 [44,38]. Resorting to the two-component spinor notation, eq. (4.7) leads to
D + αD + . α W = 0 , D +α D + α W = − 1 W D +α W D + α W −D + . α WD + . α W . (4.8)
In the case of the dynamical system (4.4), the equation of motion for the Abelian gauge field is 1
κ G ++ ∆ = tr G ++ YM ,(4.9)
with κ a coupling constant. With properly defined dimensional reduction 5D → 4D, this can be recognized as the superfield constraint describing the Chern-Simons coupling of a nonlinear vector-tensor to an external N = 2 Yang-Mills supermultiplet [38].
The super Chern-Simons actions can be readily reduced to components in the Wess-Zumino gauge (2.34) for the Abelian gauge field V ++ . If L ++ (z, u) = L ij (z) u + i u + j is a real analytic superfield of the type (3.14), then
S = dζ (−4) V ++ L ++ (4.10) = d 5 x X ij L ij + i 12 ϕD ij L ij + i 12 Aâ (D i ΓâD j )L ij − 2 3 Ψ iα D ĵ α L ij .
The Abelian supersymmetric Chern-Simons theory (4.5) leads to the following component action:
S CS = 1 2g 2 d 5 x 1 3 ǫâbĉdê AâFbĉFdê − 1 2 ϕFâbFâb − ϕ∂âϕ∂âϕ + 1 2 ϕX ij X ij − i 2 Fâb(Ψ k ΣâbΨ k ) + i ϕ(Ψ k ∂Ψ k ) − i 2 X ij (Ψ i Ψ j ) .
5D Supermultiplets in Reduced Superspace
Some of the results described in the previous sections can easily be reduced to a "hybrid" formulation which keeps manifest only 4D, N = 1 super Poincaré symmetry. As the 5D superfields depend on two sets of 4D anticommuting Majorana spinors, (θ α 1 ,θ 1 . α ) and (θ α 2 ,θ 2 . α ), such a hybrid formulation is equivalent to integrating out, say, the second set and keeping intact the first set of variables
θ α = θ α 1 ,θ . α =θ 1 . α .
(5.1) 10 The nonlinear vector-tensor multiplet was discovered in [45].
In this approach, one deals with reduced (or N = 1) superfields U|, D 2 α U|,D . α defined in the obvious way:
U| = U(x, θ α i ,θ i . α ) θ 2 =θ 2 =0 , D α = D 1 α θ 2 =θ 2 =0 ,D . α =D . α 1 θ 2 =θ 2 =0 . (5.2)
Our consideration below naturally reproduces many of the 5D supersymmetric models originally derived in the hybrid formulation [46].
Vector multiplet
Let us introduce reduced gauge covariant derivatives
D α ,D . α , D a , D 5 = D 1 α , D . α 1 , D a , D 5 . (5.3)
As follows from (2.3), their algebra is
{D α , D β } = {D . α ,D . β } = 0 , {D α ,D . β } = −2i D α . β , [D α , D β . β ] = −2iε αβW . β , [D . α , D β . β ] = −2iε. α . β W β , [D α , D 5 + F ] = 0 , [D . α , D 5 − F ] = 0 , (5.4) where F = W| , W α = D 2 α W| . (5.5)
It can be seen that the field strengths F , W α andW .
α are the only independent N = 1 descendants of W.
The strengths F and W α obey some constraints which follow from the Bianchi identities (2.4) and (2.5). Consider first the constraint (2.4) with two derivatives of W. Taking the part with (i, j,α,β) = (1, 1, α, . α) gives the "N = 1 chirality" of W ᾱ D .
α W α = 0 . (5.6)
Taking instead the part with (i, j,α,β) = (1, 2, α, β) gives the familiar Bianchi identity
D α W α −D . αW . α = 0 . (5.7)
Next, the (i, j,α,β) = (1, 1, α, β) and (i, j,α,β) = (1, 2, α, .
α) parts, respectively, givē
D 2 . γD . γ 2 W| = −D 2 F , D 2 αD 2 . β W| = D αD . α F . (5.8)
The latter identities support the statement that F , W α andW .
α are the only independent N = 1 descendants of W. Finally, decomposing the second constraint (2.5) with (i, j, k) = (2, 2, 1) and (α,β,γ) = (α,β, γ) gives
− 1 4D 2 D α F + D 5 W α − [F , W α ] = 0 . (5.9)
In accordance with (3.18), the super Yang-Mills action is
S YM = i 12 d 5 xD ij L ij YM , L ij YM = i 4 tr Dα i W D ĵ α W + 1 4 {W ,D ij W} . (5.10)
Its reduced form can be shown to be
S YM = tr d 5 x 1 4 d 2 θ W α W α + 1 4 d 2θW . αW . α + d 4 θ F 2 . (5.11)
Here the Grassmann measures d 2 θ and d 4 θ are part of the chiral and the full superspace measures, respectively, in 4D, N = 1 supersymmetric field theory.
It is instructive to solve the constraints encoded in (5.4). A general solution to the equations {D
α , D β } = [D α , D 5 + F ] = 0 is D α = e −Ξ D α e Ξ , D 5 + F = e −Ξ ∂ 5 + Φ † e Ξ , D α Φ † = 0 ,(5.D 5 − F = e Ξ † ∂ 5 − Φ e −Ξ † ,D . α Φ = 0 . (5.13)
The prepotentials introduced possess the following gauge transformations
e Ξ † → e iτ (z) e Ξ † e −iλ(z) , Φ → e iλ(z) Φ − ∂ 5 e −iλ(z) · 1 ,D . α λ = 0 . (5.14)
Here the λ-gauge group occurs as a result of solving the constraints in terms of the unconstrained prepotentials.
By analogy with the 4D N = 1 super Yang-Mills case, one can introduce a chiral representation defined by applying a complex gauge transformation with τ = −Ξ † . This gives
D α = e −V D α e V ,D . α =D . α , D 5 + F = e −V ∂ 5 + Φ † e V D 5 − F = ∂ 5 − Φ ,(5.15)
where e V = e Ω e Ω † , V † = V .
Here the real Lie-algebra valued superfield V is the standard N = 1 super Yang-Mills prepotential. For F we obtain
2 F = Φ + e −V Φ † e V + e −V (∂ 5 e V ) .(5.16)
We have thus reproduced the results obtained by Hebecker within the hybrid approach [46].
Fayet-Sohnius hypermultiplet
The Fayet-Sohnius hypermultiplet q i generates two independent N = 1 superfields transforming in the same representation of the gauge group,
Q † = q 1 | , Q = q 2 | ,(5.17)
and obeying the constraints
D αQ † = 0 ,D . α Q = 0 . (5.18)
These constraints follow from (2.40). Thus Q andQ † are covariantly chiral and antichiral, respectively. The central charge transformation of these superfields is:
i ∆Q = 1 4D 2Q † + (F − D 5 )Q , i ∆Q † = 1 4 D 2 Q + (F + D 5 )Q † . (5.19)
In accordance with (3.16), the action for the Fayet-Sohnius hypermultiplet is
S FS = i 12 d 5 xD ij L ij FS , L ij FS = − 1 2q (i ↔ ∆ q j) − i mq (i q j) . (5.20)
It can be shown to reduce to the following N = 1 action
S FS = d 5 x d 4 θ (Q † Q +QQ † ) + d 2 θQ(F − D 5 + m)Q + c.c. . (5.21)
As follows from (5.4), the operator D 5 − F preserves chirality.
Projective Superspace and Dimensional Reduction
Throughout this section, we consider only 5D supermultiplets without central charge, ∆ = 0. However, many results below can be extended to include the case ∆ = 0.
Doubly punctured harmonic superspace
Let ψ (p) (z, u) be a harmonic superfield of non-negative U(1) charge p. Here we will be interested in solutions to the equation
D ++ ψ (p) = 0 ⇒ D ++ D + α ψ (p) = 0 , p ≥ 0 . (6.1)
If ψ (p) (z, u) is globally defined and smooth over R 5|8 × S 2 , it possesses a convergent Fourier series of the form (B.9). If ψ (p) (z, u) is further constrained to obey the equation (6.1), then its general form becomes
ψ (p) (z, u) = ψ i 1 ···ip (z) u + i 1 · · · u + ip . (6.2)
Therefore, such a globally defined harmonic superfield possesses finitely many component fields, and this can thought of as a consequence of the Riemann-Roch theorem [47] specified to the case of S 2 . A more interesting situation occurs if one allows ψ (p) (z, u) to have a few singularities on S 2 .
For further consideration, it is useful to cover S 2 by two charts and introduce local complex coordinates in each chart, as defined in Appendix B. In the north chart (parametrized by the complex variable w and its conjugatew) we can represent ψ (p) (z, u) as follows
ψ (p) (z, u) = (u +1 ) p ψ(z, w,w) . (6.3)
If ψ (p) (z, u) is globally defined over R 5|8 × S 2 , then ψ(z, w,w) ≡ ψ (p) N (z, w,w) is given as in eq. (B.10). It is a simple exercise to check that
D ++ ψ (p) (z, u) = (u +1 ) p+2 (1 + ww) 2 ∂w ψ(z, w,w) , p ≥ 0 ,(6.4)
and therefore
D ++ ψ (p) = 0 , p ≥ 0 ⇔ ∂w ψ = 0 . (6.5)
Assuming that ψ (p) (z, u) may possess singularities only at the north and south poles of S 2 , we then conclude that
ψ(z, w) = +∞ n=−∞ ψ n (z) w n . (6.6)
Now, consider an analytic superfield φ (p) obeying the constraint (6.1).
D + α φ (p) = 0 , D ++ φ (p) = 0 , p ≥ 0 . (6.7)
We assume that φ (p) (z, u) is non-singular outside the north and south poles of S 2 . Then, representing φ (p) (z, u) = (u +1 ) p φ(z, w,w) and defining
D + α = −u +1 ∇α(w) , ∇α(w) = −D î α w i , w i = (−w, 1) , (6.8)
eq. (6.7) is solved as
∇α(w) φ(z, w) = 0 , φ(z, w) = +∞ n=−∞ φ n (z) w n . (6.9)
These relations define a projective superfield, following the four-dimensional terminology [24]. Since the supersymmetry transformations act simply as the identity transformation on S 2 , the above consideration clearly defines supermultiplets. Such supermultiplets turn out to be most suited for dimensional reduction.
The projective analogue of the smile-conjugation (2.21) is
φ(z, w) = +∞ n=−∞ (−1) nφ −n (z) w n , ∇α(w)φ(z, w) = 0 . (6.10)
Ifφ(z, w) = φ(z, w), the projective superfield is called real. The projective conjugation (6.10) can be derived from the smile-conjugation (2.21), see [25] for details.
There are several types of projective superfields [24]. A real projective superfield of the form (7.11) is called a tropical multiplet. A real projective superfield of the form
φ(z, w) = +n −n φ n (z) w n ,φ = φ , n ∈ Z (6.11)
is called a real O(2n) multiplet. 11 A projective superfield Υ(z, w) of the form (6.27) is called an arctic multiplet, and its conjugate,Υ(z, w), an antarctic multiplet. The Υ(z, w) andΥ(z, w) constitute a polar multiplet. More general projective superfields occur if one multiplies any of the considered superfields by w n , with n an integer.
At this stage, it is useful to break the manifest 5D Lorentz invariance by switching from the four-component spinor notation to the two-component one. Representing
∇α(w) = ∇ α (w) ∇ . α (w) , ∇ α (w) ≡ wD 1 α − D 2 α ,∇ . α (w) ≡D . α 1 + wD .
α 2 , (6.12) 11 One can also introduce complex O(2n+1) multiplets [24].
the constraints (6.10) can be rewritten in the component form If the series in (6.9) is bounded from below (above), then eq. (6.13) implies that the two lowest (highest) components in φ(w)| are constrained N = 1 superfields. For example, in the case of the arctic multiplet, eq. (6.27), the leading components Φ = Υ 0 | and Γ = Υ 1 | obey the constraints (6.28).
Given a real projective superfield L(z, w), one can construct a supersymmetric invari-
ant S = 1 2π i C dw w d 5 x d 4 θ L(w) ≡ 1 2π i C dw w S(w) , (6.14)
with C a contour around the origin (in what follows, such a contour is always assumed). For S(w) there are several equivalent forms:
S(w) = 1 16 d 5 x D 2D2 L(z, w) = 1 16 d 5 x (D 1 ) 2 (D 1 ) 2 L(z, w) (6.15)
assuming only that total space-time derivatives do not contribute. The invariance of S(w) under arbitrary SUSY transformations is easy to demonstrate. Defining
D 4 = 1 16 (D 1 ) 2 (D 1 ) 2 , (6.16)
one can argue as follows:
δS(w) = i d 5 x ε α i Q i α +ε i . αQ . α i D 4 L(z, w) = − d 5 x ε α i D i α +ε i . αD . α i D 4 L(z, w) = − d 5 x ε α 2 D 2 α +ε 2 . αD . α 2 D 4 L(z, w) = − d 5 x D 4 ε α 2 D 2 α +ε 2 . αD . α 2 L(z, w) = − d 5 x D 4 ε α 2 D 1 α w −ε 2 . αD . α 1 w −1 L(z, w) = 0 , (6.17)
with Q i α andQ . α i the supersymmetry generators.
Tensor multiplet and nonlinear sigma-models
The tensor multiplet (also called O(2) multiplet) is described by a constrained real analytic superfield Ξ ++ :
D + α Ξ ++ = 0 , D ++ Ξ ++ = 0 . (6.18)
The corresponding projective superfield Ξ(z, w) is defined by Ξ ++ (z, u) = i u +1 u +2 Ξ(z, w). Without distinguishing between Ξ(z, w) and Ξ(z, w)|, we have
Ξ(w) = Φ + w G − w 2Φ ,Ḡ = G , (6.19)
where the component superfields obey the constraints
D . α Φ = 0 , − 1 4D 2 G = ∂ 5 Φ . (6.20)
Here we consider a 5D generalization of the 4D, N = 2 supersymmetric nonlinear sigma-model 12 studied in [48] and related to the so-called c-map [49]. Let F be a holomorphic function of n variables. Associated with this function is the following supersymmetric action
S = − d 5 x d 4 θ 1 2π i dw w F (Ξ I (w)) w 2 + c.c. . (6.21) Since F (Ξ I (w)) = F Φ I + w G I − w 2ΦI = F (Φ) + w F I (Φ)G I − w 2 F I (Φ)Φ I − 1 2 F IJ (Φ) G I G J + O(w 3 ) ,
the contour integral is trivial to do. The action is equivalent to
S[Φ,Φ, G] = d 5 x d 4 θ K(Φ,Φ) − 1 2 g IJ (Φ,Φ) G I G J , (6.22) where K(Φ,Φ) =Φ I F I (Φ) + Φ IF I (Φ) , g IJ (Φ,Φ) = F IJ (Φ) +F IJ (Φ) . (6.23)
The Kähler potential K(Φ,Φ) generates the so-called rigid special Kähler geometry [50].
Let us work out a dual formulation for the theory (6.22). Introduce a first-order action
S[Φ,Φ, G] + d 5 x d 2 θ Ψ I ∂ 5 Φ I + 1 4D 2 G I + c.c. = S[Φ,Φ, G] − d 5 x d 4 θ (Ψ I +Ψ I ) G I + d 2 θ Ψ I ∂ 5 Φ I + c.c.
, (6.24) where the superfield G I is now real unconstrained, while Ψ I is chiral,D .
α Ψ I = 0. In this action we can integrate out G I using the corresponding equations of motion. This gives
S[Φ,Φ, Ψ,Ψ] = d 5 x d 4 θ H(Φ,Φ, Ψ,Ψ) + d 2 θ Ψ I ∂ 5 Φ I + c.c.
, (6.25) where
H(Φ,Φ, Ψ,Ψ) = K(Φ,Φ) + 1 2 g IJ (Φ,Φ)(Ψ I +Ψ I )(Ψ J +Ψ J ) . (6.26)
The potential H(Φ,Φ, Ψ,Ψ) is the Kähler potential of a hyper Kähler manifold. By construction, this potential is generated by another Kähler potential, K(Φ,Φ), which is associated with the holomorphic function F (Φ) defining the rigid special Kähler geometry [50]. The correspondence K(Φ,Φ) → H(Φ,Φ, Ψ,Ψ) is called the rigid c-map [49].
Polar hypermultiplet and nonlinear sigma-models
According to [24], the polar hypermultiplet is generated by projective superfields
Υ(z, w) = ∞ n=0 Υ n (z) w n ,Υ(z, w) = ∞ n=0 (−1) nῩ n (z) 1 w n . (6.27)
The projective superfields Υ andΥ are called arctic and antarctic [24], respectively. The constraints (6.13) imply that the leading components Φ = Υ 0 | and Γ = Υ 1 | are constrained
D . α Φ = 0 , − 1 4D 2 Γ = ∂ 5 Φ . (6.28)
The other components of Υ(w) are complex unconstrained superfields, and they appear to be non-dynamical (auxiliary) in models with at most two space-time derivatives at the component level.
Here we consider a 5D generalization of the 4D, N = 2 supersymmetric nonlinear sigma-model studied in [51]. It is described by the action for the model (6.29). A holomorphic reparametrization A I → f I A of the Kähler manifold has the following counterparts
S[Υ,Υ] = d 5 x d 4 θ 1 2πi dw w K Υ(w),Υ(w) .Φ I → f I Φ , Υ I (w) → f I Υ(w) (6.33)
in the 4D and 5D cases, respectively. Therefore, the physical superfields of the 5D theory
Υ I (w) w=0 = Φ I , dΥ I (w) dw w=0 = Γ I ,(6.Υ I n = ∞ p=o G I J 1 ...J n+pL1 ...Lp (Φ,Φ) Γ J 1 . . . Γ J n+pΓL1 . . .ΓL p , n ≥ 2 . (6.36)
Upon elimination of the auxiliary superfields, 13 the action (6.29) takes the form
S tb [Φ,Φ, Γ,Γ] = d 5 x d 4 θ K Φ,Φ − g IJ Φ,Φ Γ IΓJ + ∞ p=2 R I 1 ···IpJ 1 ···Jp Φ,Φ Γ I 1 . . . Γ IpΓJ 1 . . .ΓJ p ,(6.37)
where the tensors R I 1 ···IpJ 1 ···Jp are functions of the Riemann curvature R IJKL Φ,Φ and its covariant derivatives. Each term in the action contains equal powers of Γ andΓ, since the original model (6.29) is invariant under rigid U(1) transformations For the theory with action S tb [Φ,Φ, Γ,Γ], we can develop a dual formulation involving only chiral superfields and their conjugates as the dynamical variables. Consider the first-order action It is instructive to consider a free hypermultiplet described by the Kähler potential
Υ(w) → Υ(e iα w) ⇐⇒ Υ n (z) → e inα Υ n (z) .(6.S tb [Φ,Φ, Γ,Γ] − d 5 x d 2 θ Ψ I ∂ 5 Φ I + 1 4D 2 Γ I + c.c. = S tb [Φ,Φ, Γ,Γ] + d 5 x d 4 θ Ψ I Γ I − d 2 θ Ψ I ∂ 5 Φ I + c.c. ,(6.K free (A,Ā) =Ā A. Then S free [Υ,Υ] = d 5 x d 4 θ ∞ n=0 (−1) nῩ n (z)Υ n = d 5 x d 4 θ Φ Φ −Γ Γ + . . . (6.40)
Here the dots stand for the auxiliary superfields' contribution. Now, eliminating the auxiliary superfields and dualizing Γ into a chiral scalar, one obtains the action for the free Fayet-Sohnius hypermultiplet, equation (5.21).
Vector Multiplet in Projective Superspace
In the Abelian case, the gauge transformation (2.33) simplifies
δV ++ = −D ++ λ , D + α λ = 0 ,λ = λ . (7.1)
The field strength (2.30) also simplifies
W = i 8 du (D − ) 2 V ++ . (7.2)
It is easy to see that W is gauge invariant.
The gauge freedom (7.1) can be used to choose the supersymmetric Lorentz gauge [20] D ++ V ++ = 0 . In other words, in this gauge V ++ becomes a real O(2) multiplet,
V ++ = i u +1 u +2 V (z, w) , V (z, w) = 1 w ϕ(z) + V (z) − wφ(z) . (7.4)
Since W is gauge invariant, for its evaluation one can use any potential V ++ from the same gauge orbit, in particular the one obeying the gauge condition (7.3). This Lorentz gauge is particularly useful for our consideration. Using the relation (C.6) and noting that |u +1 | 2 = (1 + ww) −1 , we can rewrite W in the form
W = 1 2 du P(w) V (z, w) . (7.5)
This can be further transformed to
W = 1 4πi dw w P(w) V (z, w) . (7.6)
Indeed, the consideration in Appendix C justifies the following identity
lim R→∞ lim ǫ→0 du φ R,ǫ (u) = 1 2πi dw w φ(w) ,(7.7)
with the regularization φ R,ǫ (u) = φ R,ǫ (w,w) of a function φ(w) holomorphic on C * defined according to (C.2). Since the integrand on the right of (7.5) is, by construction, a smooth scalar field on S 2 , we obvoiusly have
du P(w) V (z, w) = lim R→∞ lim ǫ→0 du P(w) V R,ǫ (z, u) . (7.8)
The representation (7.6) allows one to obtain a new formulation for the vector multiplet. Let Λ(z, w) be an arctic multiplet
Λ(z, w) = ∞ n=0 Λ n (z) w n , ∇α(w)Λ(z, w) = 0 ,(7.9)
andΛ(z, w) its smile-conjugate. It then immediately follows that
dw w P(w) Λ(z, w) = dw w P(w)Λ(z, w) = 0 . (7.10)
Now, introduce a real tropical multiplet V (z, w),
V (z, w) = +∞ n=−∞ V n (z) w n , ∇α(w)V (z, w) = 0 ,V n = (−1) n V −n ,(7.11)
possessing the gauge freedom δV (z, w) = i Λ (z, w) − Λ(z, w) . (7.12) With such gauge transformations, eq. (7.6) defines a gauge invariant field strength. Next, in accordance with the superfield structure of the tropical and arctic multiplets, the gauge freedom can be used to turn V (z, w) into a real O(2) multiplet, i.e. to bring V (z, w) to the form (7.4). We thus arrive at the projective superspace formulation 14 for the vector multiplet [24]. Now, we are in a position to evaluate the N = 1 field strengths (5.5) in terms of the prepotentials V n . It follows from (C.6) that
F = W| = 1 4πi dw w P(w) V (w) = 1 2 Φ +Φ + ∂ 5 V , (7.13) where we have defined Φ = 1 4D 2 V 1 | , V = V 0 | . (7.
14)
The spinor field strength W α is given by
W α (z) = D 2 α W| = 1 4πi dw w [D 2 α , P(w)] + P(w)D 2 α V (w) . (7.15)
However, as [D 2 α , P(w)] = w ∂ 5 D 1 α and given that for any projective superfield φ(w) we have D 2 α φ(w) = wD 1 α φ(w), this expression reduces to
W α = 1 4πi dw w 1 4D 2 D α V (w) = 1 8D 2 D α V . (7.16)
It can be seen that the gauge transformation (7.12) acts on the superfields in (7.14) as follows:
δV = i (Λ − Λ) , δΦ = i ∂ 5 Λ Λ = Λ 1 | . (7.17)
The approach presented in this section can be applied to reformulate the supersymmetric Chern-Simons theory (4.5) in projective superspace, and the possibility for this is based on the following observation. Let L ++ be a linear multiplet, that is a real analytic superfield obeying the constraint D ++ L ++ = 0. Then, the functional dζ (−4) V ++ L ++ is invariant under the gauge transformations (7.1). We can further represent L ++ = (iu +1 u +2 )L(z, w), where L(z, w) is a real O(2) multiplet. Then the functional
− 1 2π i d 5 x d 4 θ dw w V (w) L(w)
is invariant under the gauge transformations (7.12).
In the case of Chern-Simons theory (4.5), the role of L ++ is played by the gaugeinvariant superfield (12g 2 ) −1 G ++ , with G ++ defined in (2.32). With the real O(2) multiplet G(z, w) introduced by
G ++ = (iu +1 u +2 ) G(z, w) , G(w) = − 1 w Ψ + K + wΨ ,(7.18)
the Chern-Simons theory (4.5) is equivalently described by the action
12g 2 S CS = − 1 2π i d 5 x d 4 θ dw w V (w) G(w) ≡ 12g 2 d 5 x L CS . (7.19)
Direct evaluation of Ψ and K gives
Ψ = −W α W α + 1 2D 2 (F 2 ) , K = −F D α W α − 2(D α F )W α + c.c. + 2∂ 5 (F 2 ) . (7.20)
These results lead to
12g 2 L CS = d 2 θ Φ W α W α + d 4 θ V [F D α W α + 2(D α F )W α ] + c.c. + 4 d 4 θ F 3 . (7.21)
Here we have chosen to present the answer in the form (potential) × (fieldstrength) × (fieldstength) analogously to the standard representation of the bosonic Chern-Simons action. 15 The structure of the superspace action obtained is the following. The first and second line of (7.21) are separately invariant under the gauge transformation (7.17) up to surface terms as is easily seen. The relative factor of 4 is fixed by five-dimensional Lorentz invariance. This could be derived either from the component projection or, less painfully, by checking the five-dimensional mass-shell condition on the super-fieldstrengths using 15 The result presented here was given previously in the first reference of [46]. In comparing the results one should keep in mind that terms such as d 4 θ V 1 2 (Φ +Φ)D α W α + D α ΦW α + c.c. can be rewritten to look like d 2 θ ΦW α W α + c.c., thereby changing the appearance of the action. their equations of motion together with their Bianchi identities. Finally, under the shift Φ → Φ + 1, the action shifts by S CS → S CS + S YM + surface term, where S YM is the 5D Yang-Mills action (5.11) with the proper normalization.
For completeness, we also present here projective superspace extensions of the vector multiplet mass term and the Fayet-Iliopoulos term (their harmonic superspace form is given in [20]). The vector multiplet mass term is
−m 2 dζ (−4) (V ++ ) 2 −→ m 2 2π i d 5 x d 4 θ dw w V 2 (w) . (7.22)
The gauge invariant Fayet-Iliopoulos term is
dζ (−4) c ++ V ++ −→ − 1 2π i d 5 x d 4 θ dw w c(w)V (w) ,(7.23)
where c ++ = c ij u + i u + j , with a constant real iso-vector c ij . Defining c ++ = iu +1 u +2 c(w), with c(w) = w −1ξ C + ξ R − wξ C , the FI action then reduces to
ξ R d 5 x d 4 θ V + 2Re ξ C d 5 x d 2 θ Φ . (7.24)
So far the considerations in this section have been restricted to the Abelian case. It is necessary to mention that the projective superspace approach [24] can be generalized to provide an elegant description of 5D super Yang-Mills theories, which is very similar to the well-known description of 4D, N = 1 supersymmetric theories. In particular, the Yang-Mills supermultiplet is described by a real Lie-algebra-valued tropical superfield V (z, w) with the gauge transformation e V (w) → e iΛ(w) e V (w) e −iΛ(w) , (7.25) which is the non-linear generalization of the Abelian gauge transformation (7.12). The hypermultiplet sector is described by an arctic superfield Υ(z, w) and its conjugate, with the gauge transformation Υ(w) → e iΛ(w) Υ(w) .
Conclusion
In the present paper we have developed the manifestly supersymmetric approach to five-dimensional globally supersymmetric gauge theories. It is quite satisfying that 5D superspace techniques provide a universal setting to formulate all such theories in a compact, transparent and elegant form, similarly to the four-dimensional N = 1 and N = 2 theories. We believe that these techniques are not only elegant but, more importantly, are useful. In particular, these techniques may be useful for model building in the context of supersymmetric brane-world scenarios. The two examples of supersymmetric nonlinear sigma-models, which were constructed in section 6, clearly demonstrate the power of the 5D superspace approach.
Five-dimensional super Yang-Mills theories possess interesting properties at the quantum level [55]. Further insight into their quantum mechanical structure may be obtained by carrying out explicit supergraph calculations. Supersymmetric Chern-Simons theories (4.4) are truly interesting in this respect.
Note Added: After this paper was posted to the hep-th archive, we were informed of a related interesting work on 6D, N = (1, 0) supersymmetric field theories [57].
Acknowledgements:
SMK is grateful to Jim Gates and the Center for String and Particle Theory at the University of Maryland, where this project was conceived, for hospitality. The work of SMK is supported in part by the Australian Research Council. The work of WDL is supported by the University of Maryland Center for String and Particle Theory.
A 5D Notation and Conventions
Our 5D notation and conventions are very similar to those introduced in [28].
The 5D gamma-matrices Γm = (Γ m , Γ 5 ), with m = 0, 1, 2, 3, defined by
{Γm , Γn} = −2ηmn 1 , (Γm) † = Γ 0 Γm Γ 0 (A.1)
are chosen in accordance with [27,12] (
Γ m )αβ = 0 (σ m ) α . β (σ m ) . αβ 0 , (Γ 5 )αβ = −i δ α β 0 0 i δ . α . β , (A.2)
such that Γ 0 Γ 1 Γ 2 Γ 3 Γ 5 = 1. The charge conjugation matrix, C = (εαβ), and its inverse,
C −1 = C † = (εαβ) are defined by C Γm C −1 = (Γm) T , εαβ = ε αβ 0 0 −ε. α . β , εαβ = ε αβ 0 0 −ε . α . β . (A.3)
The antisymmetric matrices εαβ and εαβ are used to raise and lower the four-component spinor indices.
A Dirac spinor, Ψ = (Ψα), and its Dirac conjugate, Ψ = (Ψα) = Ψ † Γ 0 , look like
Ψα = ψ ᾱ φ . α ,Ψα = (φ α ,ψ . α ) . (A.4)
One can now combineΨα = (φ α ,ψ . α ) and Ψα = εαβΨβ = (ψ α , −φ . α ) into a SU(2) doublet,
Ψα i = (Ψ α i , −Ψ . αi ) , (Ψ α i ) * =Ψ . αi , i = 1, 2 , (A.5)
with Ψ α 1 = φ α and Ψ α 2 = ψ α . It is understood that the SU (2) indices are raised and lowered by ε ij and ε ij , ε 12 = ε 21 = 1, in the standard fashion: Ψα i = ε ij Ψα j . The Dirac spinor Ψ i = (Ψ iα ) satisfies the pseudo-Majorana conditionΨ i T = C Ψ i . This will be concisely represented as (Ψ iα ) * = Ψα i . The two equivalent descriptions Vm ↔ Vαβ and and Fmn ↔ Fαβ are explicitly described as follows: with εαβγδ the completely antisymmetric fourth-rank tensor.
Vαβ = Vm (Γm)αβ , Vm = − 1 4 (Γm)
Complex conjugation gives
(εαβ) * = −εαβ , (Vαβ) * = Vαβ , (Fαβ) * = Fαβ , (A.11)
provided Vm and Fmn are real.
The conventional 5D simple superspace R 5|8 is parametrized by coordinates z = (xâ, θα i ). Then, a hypersurface x 5 = const in R 5|8 can be identified with the 4D, N = 2 superspace R 4|8 parametrized by
z A = (x a , θ α i ,θ i . α ) , (θ α i ) * =θ . αi . (A.12)
The Grassmann coordinates of R 5|8 and R 4|8 are related to each other as follows:
θα i = (θ α i , −θ . αi ) , θ î α = θ i ᾱ θ .D î α = ∂ ∂θα i − i (Γb)αβ θβ i ∂b − i θ î α ∆ . (A.16)
One can relate the operators
D i ≡ (D î α ) = D i ᾱ D . αi ,D i ≡ (Dα i ) = (D α i , −D .D i α = ∂ ∂θ α i + i (σ b ) αβθ . βi ∂ b − i θ i α (∆ + i ∂ 5 ) , D . αi = − ∂ ∂θ . αi − i θ β i (σ b ) β . α ∂ b − iθ . αi (∆ − i ∂ 5 ) . (A.18)
These operators obey the anti-commutation relations
{D i α , D j β } = −2i ε ij ε αβ (∆ + i ∂ 5 ) , {D . αi ,D . βj } = 2i ε ij ε. α . β (∆ − i ∂ 5 ) , {D i α ,D . βj } = −2i δ i j (σ c ) α . β ∂ c , (A.19)
which correspond to the 4D, N = 2 supersymmetry algebra with a complex central charge (see also [38]).
In terms of the operators (A.17), the operation of complex conjugation acts as follows
(D i F ) † Γ 0 = −(−1) ǫ(F )D i F * , (A.20)
with F an arbitrary superfield and ǫ(F ) its Grassmann parity. This can be concisely represented as
(D î α F ) * = −(−1) ǫ(F ) Dα i F * . (A.21)
B Tensor Fields on the Two-Sphere
In this appendix we recall, following [25], the well-known one-to-one correspondence between smooth tensor fields on S 2 = SU(2)/U(1) and smooth scalar functions over SU (2) with definite U(1) charges. The two-sphere is obtained from SU(2) by factorization with respect to the equivalence relation
u +i ∼ e iϕ u +i ϕ ∈ R . (B.1)
We start by introducing two open charts forming an atlas on SU(2) which, upon identificationon (B.1), leads to a useful atlas on S 2 . The north patch is defined by
u +1 = 0 , (B.2)
and here we can represent
u +i = u +1 w i , w i = (1, u +2 /u +1 ) = (1, w) , u − i = u +1w i ,w i = (1,w) , |u +1 | 2 = (1 + ww) −1 . (B.3)
The south patch is defined by u +2 = 0 , (B.4) and here we have u +i = u +2 y i , y i = (u +1 /u +2 , 1) = (y, 1) , The variables w and y are seen to be local complex coordinates on S 2 considered as the Riemann sphere, S 2 = C ∪ {∞}; the north chart U N = C is parametrized by w and the south patch U S = C * ∪ {∞} is parametrized by y.
u − i = u +2ȳ
Along with w i andw i , we often use their counterparts with lower (upper) indices
w i = ε ij w j = (−w, 1) ,w i = ε ijw j = (w, −1) , w i = −w i , (B.8)
and similar for y i andȳ i .
Let Ψ (p) (u) be a smooth function on SU(2) with U(1)-charge p chosen, for definiteness, to be non-negative, p ≥ 0. Such a function possesses a convergent Fourier series of the form Ψ (p) (u) = ∞ n=0 Ψ (i 1 ···i n+p j 1 ···jn) u + i 1 · · · u + i n+p u − j 1 · · · u − jn , p ≥ 0 . (B.9)
In the north patch we can write Ψ (p) (u) = (u +1 ) p Ψ Ψ (i 1 ···i n+p j 1 ···jn) w i 1 · · · w i n+pw j 1 · · ·w jn (1 + ww) n .
(B.10)
In the south patch we have Ψ (p) (u) = (u +2 ) p Ψ Ψ (i 1 ···i n+p j 1 ···jn) y i 1 · · · y i n+pȳ j 1 · · ·ȳ jn (1 + yȳ) n .
(B.11)
Finally, in the overlap of the two charts Ψ and thus defines a smooth tensor field on S 2 .
C Projective Superspace Action
In this appendix we briefly demonstrate, following [25], how to derive the projective superspace action (6.14) from the harmonic superspace action (3.2). More details can be found in [25].
Consider an arbitrary projective superfield φ(z, w), eq. (6.10), which is allowed to be singular only at w = 0 and w = ∞ (i.e. φ(z, w) is holomorphic on the doubly punctured sphere S 2 \{N ∪S}). It is possible to promote φ(z, w) to a smooth analytic superfield over S 2 by smearing (regularizing) its singularities with functions used in the construction of the partition of unity in differential geometry.
Consider a smooth cut-off function F R,ǫ (x) sketched in figure 3. This function extrapolates smoothly from unit magnitude to zero in a small region between R, with is assumed to be large number, and R+ǫ where ǫ is small. The derivative of this function localizes whatever it multiplies to this region and is normalized so that in the limit lim ǫ→0 F ′ R,ǫ (x) = −δ(x − R) (C.1) as a distribution. Now, we can regularize the projective superfield φ(z, w) as follows:
φ(z, w) −→ φ R,ǫ (z, w,w) = F R,ǫ (|w| −1 ) φ(z, w) F R,ǫ (|w|) , (C.2) and the result is a a smooth neutral analytic superfield over the harmonic superspsace. If φ(z, w) is regular at w = 0 or w = ∞, then the factor F R,ǫ (|w| −1 ) or F R,ǫ (|w|) on the right of (C.2) can be removed.
The above procedure can also be used to generate charged analytic superfields. For instance, if Λ(z, w) is a real projective superfield,Λ = Λ, then the following superfields L ++ R,ǫ (z, u) = iu +1 u +2 F R,ǫ (|w| −1 ) L(z, w) F R,ǫ (|w|) ≡ iu +1 u +2 L R,ǫ (z, w,w) , (C.3) L (+4) R,ǫ (z, u) = (u +1 u +2 ) 2 F R,ǫ (|w| −1 ) L(z, w) F R,ǫ (|w|) ≡ (u +1 u +2 ) 2 L R,ǫ (z, w,w) (C. 4) are real analytic superfields of charge +2 and +4, respectively. One can use L (+4) R,ǫ (z, u) in the role of Lagrangian in (3.2). In the final stages we will remove the regulator by taking first ǫ → 0 and then R → ∞.
As is seen from (3.2) and (3.3), the analytic action involves a square of (D − ) 2 , and therefore we sould express the operators (D − ) 2 in local coordinates. What actually we need here is this operator acting on analytic or projective superfields Φ such that where we have defined the projective differential operator
P(w) = 1 4w (D 1 ) 2 + ∂ 5 − w 4 (D 1 ) 2 . (C.6)
It is worth pointing out that eq. (C.5) also holds in the presence of a non-vanishing central charge ∆. Using the analyticity of Φ again, it is easy to show that (D − ) 4 Φ = (u +1 ) 4 (1 +ww) 4 w 2 D 4 Φ + total derivatives , (C.7)
with the D 4 operator defined by (6.16). The latter operator determines the projective superspace measure, see eq. (6.15). Finally making use of the identity du = d 2 w π(1 + ww) 2 , (C. 8) one obtains (note |u +1 | 2 = (1 + ww) −1 )
dζ (−4) L (+4)
R,ǫ (z, u) = 1 π d 5 x d 2 w (1 + ww) 2 D 4 L R,ǫ (z, w,w) .
(C.9)
Representing here 1 (1 + ww) 2 = − This is exactly the projective action.
The formalism developed in this appendix can be applied to obtain a nice representation for the supersymmetric action (3.12) which is equivalent to Representing L ++ = iu +1 u +2 L(z, w) and using eq. (C.5), we obtain S = d 5 x du P(w) L(z, w) . (C.12)
Finally, making use of (7.7) gives S = 1 8πi d 5 x dw w 1 wD 2 − wD 2 L(z, w) . (C.13)
As an example of the usefulness of such a form, we can consider the super Yang-Mills Lagrangian (3.18). A trivial contour integration in (C.13) then immediately reproduces the action for this theory in reduced superspace (5.11).
= (∂â, D î α ) the flat covariant derivatives obeying the anti-commutation relations (A.15), ∆ the central charge, and V the gauge connection taking its values in the Lie algebra of the gauge group. The connection is chosen to be inert under the central charge transformations, [∆ , VÂ] = 0. The operators D possess the following gauge transformation law
all are usually independent) and 4D, N = 1 spinor covariant derivatives D α andD .
12) for some Lie-algebra-valued prepotentials Ξ and Φ † , of which Ξ is complex unconstrainedand Φ † antichiral. Similarly, the constraints {D . α ,D . β } = [D . α , D 5 − F ] = 0 are solved bȳ D . α = e Ξ †D . α e −Ξ † ,
the covariant derivatives of 4D, N = 2 central charge superspace, with x 5 being the central charge variable. The relations (6.13) imply that the dependence of the component superfields φ n on θ α 2 andθ 2 . α is uniquely determined in terms of their dependence on θ α 1 andθ 1 .α . In other words, the projective superfields depend effectively on half the Grassmann variables which can be choosen to be the spinor coordinates of 4D, N = 1 superspace (5.1). In other words, it is sufficient to work with reduced superfields φ(w)| and 4D, N = 1 spinor covariant derivatives D α andD . α defined in (5.2).
supersymmetric sigma-model respects all the geometric features of its 4D, N = 1 predecessor[52] S[Φ,Φ] = d 4 x d 4 θ K(Φ,Φ) ,(6.30)where K(A,Ā) is the Kähler potential of some manifold M. The Kähler invariance of (6.30) K(Φ,Φ) −→ K(Φ,Φ) + Λ(Φ) +Λ(Φ)
regarded, respectively, as a coordinate of the Kähler manifold and a tangent vector at point Φ of the same manifold. That is why the variables (Φ I , Γ J ) parametrize the tangent bundle T M of the Kähler manifold M. The auxiliary superfields Υ 2 , Υ 3 , . . . , and their conjugates, can be eliminated with the aid of the corresponding algebraic equations of motion dw w n−1 ∂K(Υ,Υ) ∂Υ I = 0 , n ≥ 2 . (6.35)Their elimination can be carried out using the ansatz[53]
39) where the tangent vector Γ I is now complex unconstrained, while the one-form Ψ I is chiral,D .α Ψ I = 0. Upon elimination of Γ andΓ, with the aid of their equations of motion, the action turns into S cb [Φ,Φ, Ψ,Ψ]. Its target space is the cotangent bundle T * M of the Kähler manifold M.
x d 4 θΥ(w) e V (w) Υ(w) . (7.27)
definition Σmn = −Σnm = − 1 4 [Γm, Γn], the matrices {1, Γm, Σmn} form a basis in the space of 4 × 4 matrices. The matrices εαβ and (Γm)αβ are antisymmetric, εαβ (Γm)αβ = 0, while the matrices (Σmn)αβ are symmetric. Given a 5-vector Vm and an antisymmetric tensor Fmn = −Fnm, we can equivalently represent them as the bi-spinors V = Vm Γm and F = 1 2 Fmn Σmn with the following symmetry properties Vαβ = −Vβα , εαβ Vαβ = 0 , Fαβ = Fβα . (A.7)
5 as a central charge variable, one can view R 5|8 as a 4D, N = 2 central charge superspace (see below). The flat covariant derivatives D = (∂â, D î α ) obey the algebra {D î α , D ĵ β } = −2i ε ij (Γĉ)αβ ∂ĉ + εαβ ∆ , [D î α , ∂b] = [D î α , ∆] = 0 , (A.14) or equivalently [D , DB} = TÂBĈ DĈ + CÂB ∆ , (A.15) with ∆ the central charge. The spinor covariant derivatives are
4D, N = 2 covariant derivatives D A = (∂ a , D i α ,D .
Figure 3 :
3The function F R,ǫ (x) smoothy interpolates from 1 to 0 in the region of width ǫ starting at R. It's derivative is a bump function with support [R, R + ǫ] and unit area.
∇ α (w)Φ =∇ . α (w)Φ = 0, with operators ∇ α and∇ . α (w) defined in (6.12). The analyticity allows us to move all D 2 α andD .α 2 derivatives onto Φ and rewrite them in terms of D 1 α and D . α 1 . When this is done, we find in local coordinates for an analytic Φ (D − ) 2 Φ = −4 (u +1 ) 2 (1 +ww) 2 w P(w)Φ , (C.5)
x du (D − ) 2 L ++ , D + α L ++ = 0 , D ++ L ++ = 0 . (C.11)
αβ Vαβ , These results can be easily checked using the identities (see e.g.[26]): εαβγδ = εαβ εγδ + εαγ εδβ + εαδ εβγ ,Fαβ =
1
2
Fmn(Σmn)αβ ,
Fmn = (Σmn)αβ Fαβ .
(A.8)
εαγ εβδ − εαδ εβγ = −
1
2
(Γm)αβ (Γm)γδ +
1
2
εαβ εγδ ,
(A.9)
and therefore
εαβγδ =
1
2
(Γm)αβ (Γm)γδ +
1
2
εαβ εγδ ,
(A.10)
The book[21] contains a list of relevant publications in the context of harmonic superspace.
Our notation and conventions are collected in Appendix A.
The equation of motion for the massless Fayet-Sohnius hypermultiplet, which is characterised by the kinematic constraint D ++ q + = 0, can be shown to be ∆ q + = 0, if the dynamics is generated by the Lagrangian (3.16) with m = 0.
A different approach to formulate massive hypermultiplets was proposed in[35].
The construction given in[48] has recently been reviewed and extended in[58].
An alternative procedure to deduce the projective superspace formulation for the 4D, N = 2 vector multiplet from the corresponding harmonic superspace formulation can be found in[25].
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| {'fraction_non_alphanumeric': 0.09718147249790808, 'fraction_numerical': 0.04493120505025897, 'mean_word_length': 3.4637132996075835, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 135, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Recent one-loop calculations of certain supergravity-mediated quantum corrections in supersymmetric brane-world models employ either the component formulation (hep-th/0305184) or the superfield formalism with only half of the bulk supersymmetry manifestly realized (hep-th/0305169 and hep-th/0411216). There are reasons to expect, however, that 5D supergraphs provide a more efficient setup to deal with these and more involved (in particular, higher-loop) calculations. As a first step toward elaborating such supergraph techniques, we develop in this letter a manifestly supersymmetric formulation for 5D globally supersymmetric theories with eight supercharges. Simple rules are given to reduce 5D superspace actions to a hybrid form which keeps manifest only the 4D, N = 1 Poincaré supersymmetry. (Previously, such hybrid actions were carefully worked out by rewriting the component actions in terms of simple superfields). To demonstrate the power of this formalism for model building applications, two families of off-shell supersymmetric nonlinear sigma-models in five dimensions are presented (including those with cotangent bundles of Kähler manifolds as target spaces). We elaborate, trying to make our presentation maximally clear and self-contained, on the techniques of 5D harmonic and projective superspaces used at some stages in this letter.', 'arxivid': 'hep-th/0507176', 'author': ['Sergei M Kuzenko [email protected]@physics.umd.edu \nSchool of Physics M013\nThe University of Western Australia\n35 Stirling Highway, Crawley W.A. 6009Australia\n', '‡ ', 'IIIWilliam D Linch \nCenter for String and Particle Theory Department of Physics\nUniversity of Maryland College Park\n20742-4111MDUSA\n'], 'authoraffiliation': ['School of Physics M013\nThe University of Western Australia\n35 Stirling Highway, Crawley W.A. 6009Australia', 'Center for String and Particle Theory Department of Physics\nUniversity of Maryland College Park\n20742-4111MDUSA'], 'corpusid': 14478009, 'doi': '10.1088/1126-6708/2006/02/038', 'github_urls': [], 'n_tokens_mistral': 36061, 'n_tokens_neox': 31224, 'n_words': 16664, 'pdfsha': '0c360027a37f8b7f2829c1e4712d4e91ccc74929', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/0507176v5.pdf'], 'title': ['On Five-dimensional Superspaces', 'On Five-dimensional Superspaces'], 'venue': []} |
arxiv |
Manifestation of classical bifurcation in the spectrum of the integrable quantum dimer
30 Aug 1998
S Aubry
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
S Flach
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
K Kladko
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
E Olbrich
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
H =
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
)
Max-Planck-Institute for Physics of Complex Systems
Laboratoire Léon Brillouin
CEN Saclay
Bayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany
Manifestation of classical bifurcation in the spectrum of the integrable quantum dimer
30 Aug 1998(March 24, 2022)
We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model. 03.20.+i, 03.65.Sq
The problem of correspondence between classical and quantum-mechanical properties of nonlinear systems is currently an object of wide interest [1]. One interesting topic concerns Hamiltonian systems with a given symmetry (e.g. some permutational symmetry), where classical trajectories exist which are not invariant under the corresponding symmetry operation. This topic appears in analyzing selective bond excitation in chemistry and in the quantization of discrete breathers [2].
We consider an integrable system with two degrees of freedom (TDF), whose classical version exhibits exactly one bifurcation (of periodic orbits) and separatrix manifold. This manifold cuts the phase space into three parts -one with invariant trajectories, and two with noninvariant trajectories, where the corresponding symmetry is the permutational one. By varying a single parameter it is possible to 'switch' between these phase space parts by crossing the separatrix. It appears natural to expect in the quantum case a drastic change in the splittings of energy levels (which should be zero in the classical limit for the noninvariant phase space parts). However the splittings are nonzero for any given value of the control parameter. The only way to avoid contradiction between the classical and quantum cases is to assume that the quantum level splittings tend to a step-like function (of e.g. the level pair number) in the classical limit. The step should occur at the position of the classical separatrix. This problem can be also coined dynamical tunneling through a separatrix. There exist studies of the influence of classical chaos on dynamical tunneling [3], however we are not aware of any systematic study in the absence of chaos.
We are able to trace the splittings of the level pairs using quantum perturbation methods. We consider the quasiclassical regime and show that the step indeed occurs. Therefore we are able to extract information about the classical separatrix and bifurcation. Further we show, that the quantum density of states (the second integral of motion is fixed) exhibits a sharp maximum at the separatrix energy. By calculating the corresponding classical quantity (with the help of Weyl's formula) we find that this singularity appears due to the integration over a part of the separatrix manifold which includes a hyperbolic isolated orbit.
Let us consider the integrable dimer model with Hamiltonian [4] H = 1 2
P 2 1 + P 2 2 + X 2 1 + X 2 2 + 1 8 (P 2 1 + X 2 1 ) 2 + (P 2 2 + X 2 2 ) 2 + C 2 (X 1 X 2 + P 1 P 2 ) .(1)
Here P 1,2 , X 1,2 are canonically conjugated momenta and positions of two degrees of freedom. System (1) is integrable, because the classical Poisson bracket of
B = P 2 1 + P 2 2 + X 2 1 + X 2 2(2)
with H vanishes. Further (1) is invariant under permutation of indices.
With Ψ = 1/ √ 2(X + iP ) (1) becomes H = Ψ * 1 Ψ 1 + Ψ * 2 Ψ 2 + 1 2 (Ψ * 1 Ψ 1 ) 2 + (Ψ * 2 Ψ 2 ) 2 + C (Ψ * 1 Ψ 2 + Ψ * 2 Ψ 1 ) .(3)
The equations of motion becomeΨ 1,2 = i∂H/∂Ψ * 1,2 . Isolated periodic orbits (IPO) satisfy the relation gradH || gradB. Let us parametrize the phase space of (3) with Ψ 1,2 = A 1,2 e iφ1,2 , A 1,2 ≥ 0. It follows A 1,2 time independent and φ 1 = φ 2 + ∆ with ∆ = 0, π andφ 1,2 = ω time independent. Solving the algebraic equations for the amplitudes of the IPO's we obtain
I : A 2 1,2 = 1 2 B , ∆ = 0 , ω = 1 + C + 1 2 B ,(4)II : A 2 1,2 = 1 2 B , ∆ = π , ω = 1 − C + 1 2 B ,(5)
III :
A 2 1 = 1 2 B 1 ± 1 − 4C 2 /B 2 , ∆ = 0 , ω = 1 + B .(6)
IPO III corresponds to two elliptic solutions which break the permutational symmetry. IPO III exist for B ≥ B b with B b = 2C and occur through a bifurcation from IPO I [4]. The corresponding separatrix manifold is uniquely defined by the energy of IPO I at a given value of B ≥ B b . This manifold separates three regions in phase space -two with symmetry broken solutions, each one containing one of the IPO's III, and one with symmetry conserving solutions containing the elliptic IPO II. The separatrix manifold itself contains the hyperbolic IPO I. For B ≤ B b only two IPO's exist -IPO I and II, with both of them being of elliptic character. Remarkably there exist no other IPO's, and the mentioned bifurcation and separatrix manifold are the only ones present in the classical phase space of (1) [4].
To conclude the analysis of the classical part, we calculate the energy properties of the different phase space parts separated by the separatrix manifold. First it is straightforward to show that the IPO's (4)-(6) correspond to maxima, minima or saddle points of the energy in the allowed energy interval for a given value of B, with no other extrema or saddle points present [4]. It follows
E 1 = H(IPO I) = B + 1 4 B 2 + CB ,(7)E 2 = H(IPO II) = B + 1 4 B 2 − CB ,(8)E 3 = H(IPO III) = B + 1 2 B 2 + C 2 . (9) For B < B b we have E 1 > E 2 (IPO I -maximum, IPO II -minimum). For B ≥ B b it follows E 3 > E 1 > E 2 (IPO III -maxima, IPO I -saddle, IPO II -minimum). If B < B b , then all trajectories are symmetry conserving. If B ≥ B b , then trajectories with energies E 1 < E ≤ E 3 are symmetry breaking, and trajectories with E 2 ≤ E ≤ E 1 are symmetry conserving.
The quantum eigenvalue problem can be properly analyzed in second quantization, which amounts to replacing the complex functions Ψ, Ψ * in (3) with the boson annihilation and creation operators a, a + with standard commutation relations (to enforce invariance under exchange Ψ ⇔ Ψ * the substitution has to be done after rewriting ΨΨ * = 1/2(ΨΨ * + Ψ * Ψ)):
H = 5 4 + 3 2 a + 1 a 1 + a + 2 a 2 + 1 2 (a + 1 a 1 ) 2 + (a + 2 a 2 ) 2 + C a + 1 a 2 + a + 2 a 1 .(10)H nm = 5 4 + 3 2 b + 1 2 n 2 + (b − n) 2 n = m C n(b + 1 − n) n = m + 1 C (n + 1)(b − n) n = m − 1 0 else(11)
and n, m = 0, 1, 2, ..., b. Notice that the matrix H nm is a symmetric band matrix. The additional symmetry H nm = H (b−n),(b−m) is a consequence of the permutational symmetry of H. For C = 0 the matrix H nm is diagonal, with the property that each eigenvalue is doubly degenerated (with exception of the state |b/2) for even values of b). The classical phase space contains only symmetry broken trajectories, with the exception of IPO II and the separatrix with IPO I (in fact in this limit the separatrix manifold is nothing but a resonant torus containing both IPO's I and II). So with the exception of the separatrix manifold, all tori break permutational symmetry and come in two groups separated by the separatrix. Then quantizing each group will lead to pairs of degenerated eigenvalues -one from each group. There is a clear correspondence to the spectrum of the diagonal (C = 0) matrix H nm . The eigenvalues H 00 = H bb correspond to the quantized IPO's III. With increasing n the eigenvalues H nn = H (b−n),(b−n) correspond to quantized tori further away from the IPO III. Finally the states with n = b/2 for even b or n = (b − 1)/2 for odd b are tori most close to the separatrix. Switching the side diagonals on by increasing C will lead to a splitting of all pairs of eigenvalues. In the case of small values of b these splittings have no correspondence to the classical system properties. However in the limit of large b we enter the semiclassical regime, and due to the integrability of the system eigenfunctions should correspond to tori in the classical phase space which satisfy the Einstein-Brillouin-Keller quantization rules [1].
Increasing C from zero will lead to a splitting ∆E n of the eigenvalue doublets of C = 0. In other words we find pairs of eigenvalues, which are related to each other through the symmetry of their eigenvectors and (for small enough C) through the small value of the splitting. Let us calculate the splittings in leading perturbation order. This is done by applying standard perturbation theory to each of the states |n) and |(b − n)) and calculating the perturbed eigenvectors until the matrix element of the two perturbed eigenvectors with H does not vanish. Due to the band structure of our matrix the final result has the following form [6]:
∆E n = 2 b−n−1 i=n H i,(i+1) b−n−1 i=n+1 (H nn − H ii ) −1 .(12)
For even b withñ = n − b/2 and (11) it follows
∆E n = 2C 2|ñ| ( b 2 + |ñ|)! (2|ñ| − 1)! 2 ( b 2 − |ñ|)! .(13)
For odd b withñ = n − b/2 + 1/2sgn(n − b/2) and (11) we find
∆E n = 2C 2|ñ|−1 ( b−1 2 + |ñ|)! (2|ñ| − 2)! 2 ( b+1 2 − |ñ|)! .(14)
The integerñ counts the pairs of equal diagonal elements of (11) from the center of H nm towards the corners (b even: |ñ| = 0, 1, 2, ..., b/2 and b odd: |ñ| = 1, 2, ..., (b + 1)/2). Note that for the corner states the obtained expression for the splitting is identical with the results in [5]. Let us define |ñ| = αb/2 with 0 < α < 1. For fixed α application of Stierling's formula to (13),(14) yields
∆E n ≈ b πe 1 + α 1 − α 1/2 γ αb , γ = eC √ 1 − α 2 2α(αb − 1) 1 + α 1 − α 1/(2α) .(15)
For large αb the expression (15) should be close to zero if γ < 1 and its inverse should be close to zero if γ > 1. So the perturbation result predicts a step-like change in the splitting values for γ = 1 in the limit of large αb. The considered asymptotic limit corresponds to the classical limit of (10). Thus we expect that the splittings of the eigenvalue pairs which correspond to symmetry broken classical tori should vanish in this limit. Consequently the condition γ = 1 predicts the position of the classical separatrix with respect to the variable α. Now we calculate the eigenvalue spectrum of (10) numerically 1 (for b = 20 this was done in [8]). In Fig.1 we show the eigenvalues (grouped with respect to their eigenfunctions being symmetric or antisymmetric with respect to permutation) as a function ofñ for b = 600 and C = 50. The classical model has symmetry broken trajectories, and a separatrix with energy E sep = E 1 = 120600. For the quantum problem we find an inflection point in the eigenvalue spectrum of each subgroup at precisely this energy (ñ ≈ 150). Sinceñ(E) is the integrated density of states, its derivative with respect to E gives the density of states ρ(E), which hereby exhibits a peak at the separatrix energy of the classical system (inset in Fig.1). Using Weyl's formula we can calculate its classical counterpart [1]
ρ cl (E, b) = d 2 P d 2 Xδ (E − H(P, X)) δ (b − B(P, X)) .(16)
This integral can be rewritten as ρ cl (E, b) = 1/(|∇H||∇B|sinΘ)dS, where the integration is done over the surface of constant H and B and Θ is the angle between the two gradients. The denominator vanishes on IPOs. Expanding the denominator in a Taylor series in the neighbourhood of an IPO it follows, that for elliptic IPOs no singularity develops (because the torus surface vanishes) whereas for hyperbolic IPOs (i.e. on the separatrix) a logarithmic singularity appears. By parametrizing the classical phase space using A 1,2 and φ 1,2 the expression (16) can be reduced to a single integral:
ρ cl (E, b) = 1 π dy C 2 b 2 − 4C 2 y 2 − (E − b − b 2 /4 − y 2 ) 2 .(17)
The integration has to be done over all values of y where the expression under the root is nonnegative. This integral shows up with a singularity at the classical separatrix energy. The numerical integration is compared in the inset in Fig.1 with the quantum density of states. We find excellent agreement. In the inset in Fig.2 the splittings are shown with respect toñ. The splittings become anomalously small in the region of classical symmetry broken solutions, which is bounded again by the separatrix energy. In Fig.2 we compare the numerically obtained splittings with the perturbation theory result (b = 150, C = 10). Even though the true splittings become as small as 10 −100 compared to the averaged spacings, the perturbation theory reproduces at the best the order of magnitude, but fails by e.g. 50% in the absolute value. Consequently we note that higher order terms in the perturbation theory are important even when the true splittings are anomalously small. Still there is useful information in the perturbation result as shown in (15). In Fig.3 we show the classical separatrix energy E 1 for different values of C (b = 600) and compare it to the peak energy in the quantum density of states and to the condition γ = 1 (which gives us a certain α, which in turn yields a givenñ and through the numerically obtained quantum eigenvalue spectrum a corresponding energy). First we note the remarkable agreement between the classical curve and the exact quantum counterpart. But even the perturbation theory gives values which deviate by only 6% from the exact result. So while the perturbation theory fails in reproducing the absolute values of the splittings, it still contains the information about a classical separatrix with good precision.
Finally we can easily trace the classical bifurcation by considering the dependence of the largest eigenvalue of the quantum spectrum as a function of C: E max = f (C). According to the classical system this function is given by (7) for C > b/2 and by (9) for C < b/2. Differentiating this function twice with respect to C should thus yield a step function with the step located at C = b/2. In the inset of Fig.3 d 2 f /dC 2 is shown for b = 600. The step at C = 300 is nicely observed.
Note thath = 1 here, so the eigenvalues b of B = a + 1 a 1 + a + 2 a 2 are integer numbers. Since B commutes with H we can diagonalize the Hamiltonian in the basis of eigenfunctions of B. Each value of b spans a subspace of dimension (b + 1) in the space of eigenfunctions. These eigenfunctions are products of the number states |n > of each degree of freedom and can be characterized by a symbol |n, m > where we have n bosons on site 1 and m bosons on site 2. For a given value b it follows m = b − n. So we can actually label each state by just one number n: |n, (b − n) >≡ |n). Consequently the eigenvalue problem at fixed b amounts to diagonalizing the matrix H nm with
This was done using standard Fortran routines with double precision. When splittings had to be calculated with values below 10 −16 we used Mathematica routines, where the precision can be of any value[7].
AcknowledgementsWe thank L. Bernstein, C. Eilbeck, H. Kantz, B. Mehlig, K. Müller and A. Scott for valuable discussions.
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Algebraic Approach to Simple Quantum Systems. B G Adams, SpringerNew YorkB. G. Adams. Algebraic Approach to Simple Quantum Systems. Springer New York, 1994.
. S Wolfram Mathematica, Addison Wesley RedwoodS. Wolfram. Mathematica. Addison Wesley Redwood, 1991.
FIGURE CAPTIONS Fig.1: Eigenvalues of the symmetric eigenstates (solid line) and antisymmetric eigenstates (dashed line) versus quantum numberñ for b = 600 and C = 50. Inset; Density of states for the eigenvalue spectrum from above (solid line) versus energy. The dashed line is the classical prediction using Weyl's formula. Fig.2: Eigenvalue splittings versus quantum numberñ for b = 150 and C = 10 (calculated with precision 512). L J Bernstein, Physica D. 68174Solid lineexact diagonalization, dashed line -perturbation theory result. Note that even forñ ≈ 80 the ratio of both values is of the order of 0.5. Inset: Eigenvalue splittings versusñ for b = 600 and C = 50 (compare Fig.1) from exact diagonalization. Splittings are of the order of average spacing forñ < 150 and collapse to zero forñ > 150. Fig.3: Separatrix energy versus C for b = 600 for the classical system. solid line). The thick long-dashed line is the position of the maximum in the quantum density of states. the thin dashed-dotted line is the perturbation theory prediction (γ = 1L. J. Bernstein. Physica D, 68:174, 1993. FIGURE CAPTIONS Fig.1: Eigenvalues of the symmetric eigenstates (solid line) and antisymmetric eigenstates (dashed line) versus quantum numberñ for b = 600 and C = 50. Inset; Density of states for the eigenvalue spectrum from above (solid line) versus energy. The dashed line is the classical prediction using Weyl's formula. Fig.2: Eigenvalue splittings versus quantum numberñ for b = 150 and C = 10 (calculated with precision 512). Solid line - exact diagonalization, dashed line -perturbation theory result. Note that even forñ ≈ 80 the ratio of both values is of the order of 0.5. Inset: Eigenvalue splittings versusñ for b = 600 and C = 50 (compare Fig.1) from exact diagonalization. Splittings are of the order of average spacing forñ < 150 and collapse to zero forñ > 150. Fig.3: Separatrix energy versus C for b = 600 for the classical system (solid line). The thick long-dashed line is the position of the maximum in the quantum density of states. the thin dashed-dotted line is the perturbation theory prediction (γ = 1).
Inset: Second derivative of the C-dependence of the maximum eigenvalue of the quantum spectrum for b = 600 versus C. The classical prediciton is a step function with values 2,0 and step position C = 300. Inset: Second derivative of the C-dependence of the maximum eigenvalue of the quantum spectrum for b = 600 versus C. The classical prediciton is a step function with values 2,0 and step position C = 300.
| {'fraction_non_alphanumeric': 0.05793860540760317, 'fraction_numerical': 0.033085993088026025, 'mean_word_length': 3.9303432723628164, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 35, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model. 03.20.+i, 03.65.Sq', 'arxivid': 'cond-mat/9808332', 'author': ['S Aubry \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n', 'S Flach \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n', 'K Kladko \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n', 'E Olbrich \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n', 'H = \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n', ') \nMax-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany\n'], 'authoraffiliation': ['Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany', 'Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany', 'Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany', 'Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany', 'Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany', 'Max-Planck-Institute for Physics of Complex Systems\nLaboratoire Léon Brillouin\nCEN Saclay\nBayreuther Str. 40 H.1691191, D-01187Gif-sur-Yvette, DresdenFrance, Germany'], 'corpusid': 16632816, 'doi': '10.1103/physrevlett.76.1607', 'github_urls': [], 'n_tokens_mistral': 6159, 'n_tokens_neox': 5220, 'n_words': 3391, 'pdfsha': '1399941459fd94f5d6a55a408f6ef2949d2a3b1b', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/9808332v1.pdf'], 'title': ['Manifestation of classical bifurcation in the spectrum of the integrable quantum dimer', 'Manifestation of classical bifurcation in the spectrum of the integrable quantum dimer'], 'venue': []} |
arxiv |
Magnetic impurity coupled to interacting conduction electrons Typeset using REVT E X 1
arXiv:cond-mat/9510062v1 12 Oct 1995
Tom Schork
Max-Planck-Institut für Physik komplexer Systeme
Bayreuther Str. 40 Haus 16D-01187DresdenGermany
Magnetic impurity coupled to interacting conduction electrons Typeset using REVT E X 1
arXiv:cond-mat/9510062v1 12 Oct 1995(February 21, 2022)
We consider a magnetic impurity which interacts by hybridization with a system of weakly correlated electrons and determine the energy of the ground state by means of an 1/N f expansion. The correlations among the conduction electrons are described by a Hubbard Hamiltonian and are treated to lowest order in the interaction strength. We find that their effect on the Kondo temperature, T K , in the Kondo limit is twofold: First, the position of the impurity level is shifted due to the reduction of charge fluctuations, which reduces T K . Secondly, the bare Kondo exchange coupling is enhanced as spin fluctuations are enlarged. In total, T K increases. Both corrections require intermediate states beyond the standard Varma-Yafet ansatz. This shows that the Hubbard interaction does not just provide quasiparticles, which hybridize with the impurity, but also renormalizes the Kondo coupling. 75.20.Hr, 75.30.Hx, 71.28.+d 71.27.+a Typeset using REVT E X Recently, heavy-fermion behavior has been observed in the electron-doped cupratewith a large Sommerfeld coefficient γ ≃ 4J/(mole Nd · K 2 ). In the same temperature regime, the spin susceptibility is found to be independent of the temperature and the Sommerfeld-Wilson ratio is of order unity. These are characteristic features of heavy-fermion excitations. 2However, the characteristic low energy scale of the order of 1 K which is associated with this behavior cannot be explained by applying the usual theory of the Kondo effect which assumes that the conduction carriers behave as free particles. 3 This is not too surprising because undoped Nd 2 CuO 4 is an antiferromagnetic charge-transfer insulator instead of a metal, 4,5 despite of one hole per unit cell. Upon doping the Nd ions are therefore coupling to a system of strongly correlated electrons 6,7 rather than to weakly or uncorrelated ones.Hamiltonian and Scaling. In order to explain this new type of heavy-fermion behavior, it has therefore been proposed to include the correlations among the conduction electrons by including an on-site repulsion. 3,8 Thus, the total Hamiltonian
I. INTRODUCTION
H = H c + H f + H cf (1.1) goes beyond that of the single-site Anderson impurity model. 9 H c is a Hubbard Hamiltonian describing the conduction electrons
H c = H t + H U H t = k,σ ǫ(k)c † kσ c kσ H U =Ũ 2N s kk ′ q,σ =σ ′ : c † k+δσ c kσ c † k ′ −δσ ′ c k ′ σ ′ : .
(1.2) c † kσ creates an electron with spin σ and momentum k, N s is the number of lattice sites. The non-interacting dispersion is given by ǫ(k). : · · · : denotes normal ordering with respect to the Fermi sea |FS where all states below the Fermi momentum, k F , are occupied. The magnetic impurity is assumed to contain one orbital (e.g., 4f ), which is either empty or singly occupied. Double occupancies are excluded because of the strong repulsion of electrons in that orbital. The energy of the f -orbital is then given by
H f = ǫ f σf † σfσ ,(1.3)
wheref † σ = |σ 0| are Hubbard operators forbidding a double occupancy of the impurity site and ǫ f < 0. The two subsystems are coupled by a local hybridization
H cf =Ṽ √ N s k,σ f † σ c kσ + c † kσfσ .
( . . N f , which could be viewed as an SU(N f ) generalization of the original model which has SU(2) symmetry. We thereby create an artificial model, which no longer corresponds to the physical situation of an N f -fold degenerate impurity hybridizing with a correlated s-band. The advantage in doing so is, however, that a controlled approximation becomes possible for this model, namely an expansion in 1/N f . 12
In taking the limit N f → ∞, we keep the density of conduction electrons per spin constant, so that the kinetic energy increases ∝ N f . To have a proper limit N f → ∞, the hybridization coupling constantṼ has to be scaled according toṼ = V / N f . 11 As regards the Hubbard interaction, we setŨ /2 = U/(2N f ) as was suggested in Ref. 13. With this scaling, the correction to the ground-state energy of the Hubbard model (1.2) is of order N 0 f , both in second-order perturbation theory in U and when summing the diagrams of the random-phase approximation (RPA), which is one order less than the U = 0 energy.
A straightforward variational ansatz. In case the conduction electrons are uncorrelated (U = 0) Varma and Yafet 14 proposed the following variational ansatz for the ground-state wave function of (1.1)
|Ψ 0 = 1 + qσ α qf † σ c qσ |FS .
(1.5)
|FS denotes the filled Fermi sea with empty f -level. Via H cf , it couples to the states f † σ c qσ |FS . Each of them describes a singlet formed between the f level and the free electron state with momentum q, where q is restricted to occupied states (|q| ≤ k F ). Minimizing Ψ 0 |H −E S |Ψ 0 with respect to α q yields an approximate ground-state energy E S . Compared to the energy E M of the multiplet f † σ |FS , this collective singlet formation gives rise to a gain in kinetic energy. With this energy gain a characteristic temperature scale T K , the Kondo temperature, is associated. In the Kondo limit (|ε| ≪ D ≪ |ǫ f |, 2D = band width) one finds 14 (in units of the band width)
T K = n exp ǫ f − µ ρV 2 . (1.6)
Here we assumed a constant density of states ρ = 1/(2D) of the conduction electrons. µ is the chemical potential of the conduction electrons and n = (D + µ)ρ denotes the filling per spin. Subsequently, it has been shown that in an expansion in the inverse degeneracy of the magnetic impurity, the ansatz (1.5) yields the ground-state energy to order (1/N f ) 0 . 15
In the case U = 0, it is, therefore, tempting to generalize the ansatz (1.5) by replacing the non-interacting ground state |FS by the (unknown) one of the Hubbard model, |g . 16 The expectation values with respect to |g which arise in a variational calculation are given by the moments of the spectral function of the Hubbard model, which can be taken from, e.g., applying the projection technique. 16 If we assume Fermi-liquid behavior for the Hubbard model, the result of this generalized ansatz is obvious. We introduce quasiparticlesc † via
c † qσ = √ Zc † qσ + . . .
where Z denotes their renormalization factor. These quasiparticles hybridize with the impurity site rather than bare electrons, the effective hybridization being, however, renormalized by √ Z. Therefore, we expect a Kondo temperature
T K ∝ exp ǫ f ρ QP ZV 2 , (1.7)
where ρ QP is the quasiparticle density of states at the chemical potential. Noting that ρ = Zρ QP is the many-particle density of states we see that the correlations enter only via ρ. In particular, T K is not modified for small U. This is in contrast to Ref. 17 where it has been shown for the Kondo model by a mean-field decoupling that due to polarization effects the Kondo temperature increases, even to lowest order in U. In the strongly correlated case Eq. (1.7) cannot be correct as well since the Kondo exchange coupling should be V 2 /U rather than V 2 /ǫ f . 8
To clarify the quality of the variational approach, we will restrict ourselves to the weakly correlated case in this paper and perform a 1/N f expansion to lowest order in U. The theoretical framework, the Brioullin-Wigner perturbation series, is introduced in the next section. In particular, we will show that to order 1/N f additional contribution arise from the Hubbard interaction in the singlet channel which do not occur in the multiplet channel (Sec. III) and conclude that these contributions modify the Kondo temperature. They are estimated in Sec. IV and the results are discussed in Sec. V.
II. BRIOULLIN-WIGNER PERTURBATION THEORY
The The resulting equation corresponds to the lowest pole of the propagator of the occupied f -state. The important energy for the low-temperature thermodynamics is given by the energy difference of singlet and multiplet ground state, E M − µ − E S , which is related to the Kondo temperature. 21 Note that the multiplet has one electron more than the singlet in our definition.
E S = FS|H 1 ∞ n=0 Q E S −H H 1 n |FS . (2.1) Here, Q = 1 − |FS FS| andH = L t + H f , where the Liouvillean L t is defined by L t A = [H t , A] − . Equation (2.1) is
Renormalization of the bare propagators. Already to order (1/N f ) 0 the bare empty flevel (single wiggle line, 1/z) has to be renormalized (double wiggle line, G 0 (z)). This renormalization arises from a partial summation in Eq. (2.1) shown in Fig. 1:
G 0 (z) = 1 z + 1 z I (0) (z)G 0 (z) = 1 z − I (0) (z)
.
(2.
2)
The self-energy I (0) (z) (see also Fig. 2a) evaluates to
I (0) (z) = V 2 N s q 1 z + ǫ q − ǫ f . (2.
3)
The propagator of the occupied f -state, G 1 (z), is not renormalized to this order
G 1 (z) = 1 z − ǫ f .I (0) (z) = V 2 N s q G 1 (z + ǫ q ) . (3.1)
There are no diagrams ∝ U to this order. To order 1/N f we first find the diagram shown in Fig. 2b:
I (1) (z) = V 4 N f N 2 s qQ [G 1 (z + ǫ q )] 2 G 0 (z + ǫ q − ǫ Q ) . (3.2)
As mentioned previously, applying H U does not change the order of a diagram to lowest order in 1/N f . Therefore, I (1) can be regarded as parent diagram in which we insert vertices of the interaction, H U . Thereby we restrict ourselves to first order in U, i.e., we apply H U only once in the series (2.1). We then find the diagrams shown in Fig. 2c. As the diagrams are time ordered, I
A differs from I
I (1) A (z) = − 2UV 4 N f N 3 s qrr ′ R G 1 (z + ǫ q )G 1 (z + ǫ r + ǫ r ′ − ǫ R )G 0 (z + ǫ r − ǫ R )G 1 (z + ǫ r ) δ r ′ −q,R−r I (1) B (z) = UV 4 N f N 3 s qrQR G 1 (z + ǫ r )G 0 (z + ǫ r − ǫ R )G 0 (z + ǫ q − ǫ Q )G 1 (z + ǫ q ) δ q−Q,r−R I (1) C (z) = UV 6 N f N 4 s qrr ′ QR G 1 (z + ǫ r )G 0 (z + ǫ r − ǫ R )G 1 (z + ǫ r + ǫ r ′ − ǫ R ) × G 1 (z + ǫ q + ǫ r ′ − ǫ Q )G 0 (z + ǫ q − ǫ Q )G 1 (z + ǫ q ) δ q−Q,r−R I (1) D (z) = 2UV 4 N f N 3 s qrQR G 0 (z + ǫ q + ǫ r − ǫ Q − ǫ R )G 1 (z + ǫ q + ǫ r − ǫ R ) × G 0 (z + ǫ r − ǫ R )G 1 (z + ǫ r ) δ Q−q,r−R I (1) E (z) = 2UV 6 N f N 4 s qrr ′ QR G 1 (z + ǫ r ′ )G 1 (z + ǫ q + ǫ r + ǫ r ′ − ǫ Q − ǫ R )G 0 (z + ǫ q + ǫ r − ǫ Q − ǫ R ) × G 1 (z + ǫ q + ǫ r − ǫ Q )G 0 (z + ǫ q − ǫ Q )G 1 (z + ǫ q ) δ Q−q,r−R . (3.3)
According to Eq. (2.1), the ground-state energy (relative to E FS ) is given by the smallest solution of
E S = I (0) (E S ) + I (1) (E S ) + E i=A I (1) i (E S ) . (3.4)
There is no contribution ∝ UV 0 in this expression for the ground-state energy since we
E S = E (0) S + 1 N f E(1)S + o(1/N f ) 2 ,(0)S = I (0) (E (0) S ) (3.5) E (1) S = I (1) (E (0) S ) + E i=A I (1) i (E (0) S ) 1 − ∂I (0) (E (0) S )/∂E S .
Diagrams for the multiplet. We now turn to the ground-state energy of a multiplet state.
To order (1/N f ) 0 it is given by E M = ǫ f (relative to E FS ). There is only one diagram contributing to order 1/N f , which is shown in Fig. 3. It is
J (1) (z) = V 2 N f N s Q R 0 (z − ǫ Q ) ,(3.6)
and we find therefore
E M = ǫ f + J (1) (ǫ f ) + o(1/N f ) 2 . (3.7)
Kondo temperature. We associate the Kondo temperature, T K , with the difference between singlet and multiplet ground-state energy 21 (in units of the band width)
T K = (E M − µ − E S )ρ . (3.8)
With this definition we find from Eqs. (3.5) and (3.7) to order (1/N f ) 0
T (0) K = (ǫ f − µ)ρ − I (0) (ǫ f − µ − T (0) K /ρ) . (3.9)
Assuming a constant density of states we have
I (0) (z) = ρV 2 µ −D dǫ 1 z + ǫ − ǫ f = ρV 2 log (ǫ f − µ) − z (D + µ) + (ǫ f − µ) − z ,(3.10)
and hence
T (0) K = (ǫ f − µ)ρ − (ρV ) 2 log T (0) K n + T (0) K ,(3.11)
where n = (D + µ)ρ denotes the filling per spin. This is solved for small
J K = −V 2 /(ǫ f − µ) by T (0) K = n exp − 1 ρJ K , (3.12)
cf. Eq. (1.6). To order 1/N f , we find from Eqs. (3.5), (3.7), and (3.8)
T (1) K = ρ J (1) (ǫ f ) − I (1) (E (0) S ) + E i=A I (1) i (E (0) S ) 1 − ∂I (0) (E (0) S )/∂E S .
(3.13)
Connection to variational approach. To find the result (3.13) variationally the following states, which occur as intermediate states in the diagrams, have to be included in the trial state for the singlet ground statê
f † σ c qσ ; c † Qσ c qσ ; c † Qσ c qσf † σ ′ c q ′ σ ′ |FS (3.14) c † Qσ c qσ c † Q ′ σ ′ c q ′ σ ′ ; c † Qσ c qσ c † Q ′ σ ′ c q ′ σ ′f † σ ′′ c q ′′ σ ′′ |FS .
IV. ESTIMATING THE KONDO TEMPERATURE
In this section, we estimate the effect of the diagrams ∝ 1/N f on the Kondo temperature.
We scale the energies by ρ and study the dependence on T K ≪ x ≪ n), we may safely approximate
i(x) = − 1 ρV 2 I (0) (E (0) S − x/ρ) = log x + n + T (0) K x + T (0) K g 1 (x) = − 1 ρ G 1 (E (0) S − (x/ρ − µ)) = 1 T (0) K + x g 0 (x) = −ρV 2 G 0 (E (0) S − x/ρ) = ρ 2 V 2 x + T (0) K − ρ(ǫ f − µ) − ρ 2 V 2 i(x)g 0 (x) ∼ ρJ K (4.2)
for small T
K . The validity of this replacement for the whole x range in the diagrams has been checked numerically.
Diagrams of order U 0 . We begin with the contributions ∝ U 0 in Eq. (3.13)
I (1) (E (0) S ) = − ρV 2 N f n 0 du 1−n 0 dx g 2 1 (u)g 0 (u + x) (4.3) J (1) (ǫ f ) = − 1 ρN f 1−n 0 dx g 0 (x − T (0) K ) , (4.4)
where we again assumed a constant density of states. Inserting (4.2) we find for the multiplet energy
J (1) (ǫ f ) = − J K N f (1 − n) , (4.5)
where the corrections are of higher order in 1/(ǫ f − µ)ρ. For the singlet energy we use the same approximation for g 0 to obtain
I (1) (E (0) S ) = (ρV ) 2 J K N f (1 − n) 1 n + T (0) K − 1 T (0) K . (4.6)
Together with the denominator in Eq. (3.13)
1 − ∂I (0) (E (0) S ) ∂E S = 1 − (ρV ) 2 1 n + T (0) K − 1 T (0) K (4.7)
and neglecting the 1, we find that both contributions ∝ U 0 cancel. (Loosely speaking, these terms describe the energy gain due to hybridization with unoccupied states which is the same for multiplet and singlet state.)
Diagrams of order U. We continue with the estimation of the diagrams ∝ U. The numerical evaluation of the sums I
A , . . . , I
E is difficult because of the δ-functions, which ensure momentum conservation in the Hubbard interaction. Since we are interested only in the qualitative behavior, we may neglect them. This implies that the interaction U acts only at the lattice site 0 with which the impurity hybridizes and corresponds to taking the limit of infinite dimensions. 22 Then the sums ∝ U in Eq. (3.3) read
I (1) A (E (0) S ) = 2UV 2 ρ 2 N f log T (0) K n + T (0) K n 0 du 1−n 0 dx g 1 (u) g 0 (u + x) i(u + x) I (1) B (E (0) S ) = U N f n 0 du 1−n 0 dx g 1 (u) g 0 (u + x) 2 I (1) C (E (0) S ) = − UV 2 ρ 2 N f n 0 du dv 1−n 0 dx dy g 1 (u) g 0 (u + x) g 1 (v) g 0 (v + y) × i(u + x) − i(v + y) u + x − (v + y) I (1) D (E (0) S ) = 2U N f n 0 du dv 1−n 0 dx dy g 1 (u) g 0 (u + x) g 1 (u + v + x) g 0 (u + v + x + y) I (1) E (E (0) S ) = − 2UV 2 ρ 2 N f n 0 du dv 1−n 0 dx dy g 1 (u) g 0 (u + x) g 1 (u + v + x) × g 0 (u + v + x + y) i(u + v + x + y) − i(0) u + v + x + y . (4.8)
For the integrals I
I (1) A (E (0) S ) = UV 2 ρ 2 C(n) N f log T (0) K n I (1) B (E (0) S ) = U N f (1 − n) 2 (4.9) I (1) D (E (0) S ) = − UC(n) N f (1 − n) 1 log T (0) K → 0 with C(n) = −2 [n log n + (1 − n) log(1 − n)] > 0. The integral I(1)
E simplifies as one integration can be performed:
I (1) E (E (0) S ) = − 2UV 2 ρ 2 N f n 0 du dv 1−n 0 dx g 1 (u) 1 i(u + x) − i(0) ×g 1 (u + v + x) log u + v + x + 1 − n u + v + x .
(4.10)
By numerical evaluation, one finds that I B in Eq. (3.13), which corresponds to considering only the states (3.14) in a variational calculation, we obtain in the limit of small T
(0) K T (1) K = − T (0) K N f ρUC(n) log T (0) K n + U ρV 2 (1 − n) 2 = T (0) K N f U J K C(n) − U ρV 2 (1 − n) 2 .
(4.11)
The first contribution to T
T K (Ũ ) T K (0) = exp α ρJ K (1 + α) (4.12)
with α = (3/2)ρŨ log 2. In contrast to our result (4.11), the increase of the Kondo temperature seems to depend exponentially onŨ. Note, however, that in our treatment the interaction had to be scaled by 1/N f . ScalingŨ in Eq. In particular, we assumed that the correlations are weak and calculated their effect on the Kondo temperature to lowest order in 1/N f . We found two competing effects: The first contribution is related to charge fluctuations. Because the energy of the virtual state in the spin exchange process increases, the Kondo temperature is reduced. This corresponds effectively to a shift of the position of the f -level. A similar effect has been found in Ref. 8.
The second contribution is related to the enhancement of spin fluctuations of the conduction electrons. The Kondo exchange coupling is effectively enhanced and the Kondo temperature increases. In the Kondo limit the second contribution dominates the first one, so that we find in total an increase of the Kondo temperature for small U.
To our opinion more interesting is that corrections to the Kondo temperature occur already to lowest order in the Hubbard interaction U. To obtain them in a variational approach, trial states are needed which in the uncorrelated case yield corrections to the ground-state energy which are of order 1/N f (and higher). Thus, our result cannot be obtained by an ansatz of the Varma-Yafet type (1.5). This shows that the effect of the Hubbard correlations is more intricate than just to provide quasiparticles with a modified density of states at the Fermi surface which hybridize with the f -orbital as it was described by Varma and Yafet for the uncorrelated case, U = 0.
If we wish to proceed to higher order in U, we note that to order 1/N f only RPA-type diagrams contribute since each U vertex carries a factor 1/N f , which has to be compensated by a spin summation, i.e., a closed loop of conduction electrons (in fact, the ground-state energy of the Hubbard model (1.2) to order 1/N f is given by summing the diagrams of RPA type and neglecting the σ = σ ′ constraint). Only few more intermediate states will occur. This is, however, an artifact of our scaling of the Hubbard interaction and one expects that in a realistic model (without the restrictive scaling of H U and finite N f ) intermediate states with more and more excited electron-hole pairs contribute, the number of which increases with increasing order of U. Therefore, it seems questionable that a systematic 1/N f treatment grasps the correct physics for realistic models of interacting conduction electrons in the limit of strong correlations.
twofold degeneracy of the f -orbital (σ = 1, 2), the model defined in Eqs. (1.1-1.4) corresponds 3 to the situation found in Nd 2−x Ce x CuO 4 since the crystal-field ground state of Nd is a doublet. 10 In order to perform a systematic expansion we set σ = 1 . . . N f and consider large N f . This generalization deserves some comment: If the conduction electrons are uncorrelated (Ũ = 0), this corresponds to treating an N f -fold degenerate impurity which hybridizes with an s-wave conduction band. This is seen by expanding the conduction electron states in partial waves about the impurtity site and, assuming a spherically symmetric hybridization, only conduction electrons with the same total angular momentum are coupled to the impurity while the others play a passive role and can be dropped. 11 Due to the interactions among the conduction electrons this change of basis does not simplify the Hamiltonian (1.1). Nevertheless, we will consider the Hamiltonian in Eqs. (1.1-1.4) for σ = 1 .
equivalent to the zero temperature limit of the equation for the lowest lying pole of the empty f -state propagator that appears in the partition function (see, e.g., Ref. 19). Diagrams. The individual terms of the series (2.1) can be visualized by diagrams: In H 1 , each H cf changes the occupation of the impurity level from 0 (wiggled line) to 1 (dashed line) destroying an conduction electron (solid line), since no double occupancy is allowed (and vice versa). This vertex carries a factor V / N f N s . The impurity line changing always between occupied and unoccupied f -level constitutes the backbone of a diagram. H 1 contains H U as well. The vertex H U has two incoming and two outgoing conduction electron lines. It yields a factor U/(N f N s ) and a δ-function ensuring momentum conservation. Taking the expectation value with respect to |FS we connect the conduction electron lines in all possible ways. The resolvent Q/(ε −H) yields the energy of the intermediate states and, because of the Liouvillean L t only the energy difference with respect to the filled Fermi sea enters. Conduction electron lines pointing to the right correspond to particle-like excitations (with a momentum denoted by a capital letter, |Q| > k F ) while those pointing to the left are hole-like (denoted by |q| ≤ k F ). Without Hubbard interaction H U these rules correspond to the standard ones 11,19 for the self energy of the propagator of the empty state in the partition function at zero temperature. For an expansion in 1/N f we note that each closed loop of fermions yields a summation over spin and, hence, a factor N f , whereas each V -vertex is ∝ 1/ N f . To lowest order in 1/N f , the application of H U does not change the order of a diagram: If we connect two conduction electron lines of a closed loop by the 4-point interaction H U we create two loops with the only restriction σ = σ ′ , which is of higher order in 1/N f . Multiplet energy. Similarly to Eq. (2.1), we obtain the energy E M for the multiplet ground state by taking the expectation values with respect to the multiplet state f † σ |FS .
III. GROUND-STATE ENERGIES TO ORDER 1/N F Diagrams for the singlet energy. To order (1/N f ) 0 only the diagram shown inFig. 2aoccurs. It was already evaluated in Eq.(2.3)
B
, etc. Also, applying H U over a doubly wiggled line deserves some comment (see, e.g., I(1) B ): Such a diagram would not be unambiguous since it is not clear whether H U acts while the f -level is empty or occupied, when we expand the renormalized empty f -propagator as in Fig. 1. For that reason we define that H U acts while the (bare) f -level is empty. The other case yields a different diagram (here, I (1) C ). The contributions of the diagrams of Fig. 2b are given by
introduced the Hubbard interaction in normal ordered form in Eq. (1.2) and restricted to first order in U. Hence the Hubbard interaction enters only via the hybridization V in the ground-state energy. Expanding Eq. (3.4) in 1/N f we obtain
coefficients are determined up to first order in U and to leading order in 1/N f . 16 In the free case (U = 0) the first state corresponds to the ansatz of Varma and Yafet, cf. Eq. (1.5), which gives the result correctly to order (1/N f ) 0 . The next two yield the 1/N f corrections while the last two are of order (1/N f ) 2 .
the small quantity. The transformed propagators read
K
and n. The empty-state propagator, g 0 (x) diverges ∝ T(0) K /x for x → 0. [This corresponds to the spin fluctuation peak at z = E (0) S in G 0 (z).
D
it numerically proves sufficient to replace g(x) as in Eq. (4.2) in the limit T
positive and, therefore, enhances the Kondo temperature. Since it depends on U/J K it is related to spin degrees of freedom. A similar contribution was found in Ref. 17 and it was attributed to the enhancement of spin fluctuations that result from the reduction of charge degrees of freedom when turning on U. The second contribution in Eq. (4.11) depends on ρV 2 rather than J K . It is related with charge degrees of freedom. A similar effect has been found in Ref. 8 and has been interpreted as the increase in energy of the virtual state in the spin-exchange process because in the virtual state a conduction site is doubly occupied. It decreases the Kondo temperature. However, in the limit ρ|ǫ f −µ| ≫ 1 that we considered throughout, the first term dominates: Overall we find an increase of the Kondo temperature. This interpretation can be put onto more solid grounds by the following observation: If we would not scale the Hubbard interaction among the conduction electrons by 1/N f , the corrections due to U, I(1)i (z), would be of the same order as I (0) . Then the integral I
B
(z), which remains constant as z → 0, would effectively shift the position of the f -level to ǫ * f , whereas I(1) A (z) ∼ log z would renormalize the exchange coupling V 2 /ǫ * f [cf. Eqs. (3.9) and (3.11)]. However, without scaling there would be contributions of higher order in U which diverge as N f → ∞. Comparison to previous results. In Ref. 17, a Kondo model (N f = 2) with correlated conduction electrons has been investigated to lowest order in the interaction strengthŨ by a mean-field decoupling of the Kondo-exchange interaction. The following increase of the Kondo temperature has been found
( 4 .
412) and expanding in 1/N f yields the first term of our result (4.11) (with a factor log 2 for N f = 2 at half filling instead of 3/4 log 2).The second term of Eq. (4.11), which describes the effective shift of the f -level, cannot be found in Ref.17 because there the Kondo model has been investigated, where the charge degrees of freedom of the impurity have already been projected out. Therefore, J K in(4.12)is an effective coupling constant which depends on U.8 Without scaling the Hubbard interaction, all integrals would be of order (1/N f ) 0 . As discussed above, the position of the f -level and the Kondo coupling constant are modified, and these corrections occur in the exponent as in Eq. (4.12).V. CONCLUSIONThe aim of this paper was to investigate the influence of correlations among the conduction electrons on the Kondo effect. Lead by the situation prevailing in Nd 2−x Ce x CuO 4 , we proposed a model with a twofold degenerate impurity (N f = 2) which hybridizes with a correlated s-band and straightforwardly generalized it to arbitrary N f . As discussed in the introduction, this generalization does not correspond to the physical situation of an N f -fold degerate impurity hybridizing with a correlated s-band for N f > 2, in contrast to the uncorrelated case. Although artificial, we saw that this model allows for systematically studying the effects of the correlations on those diagrams which are usually considered in the uncorrelated case.
FIGURESFIG. 1 . 0 FIG. 2 .
102Renormalization of the empty state propagator to order (1/N f ) Diagrams for the singlet ground-state energy. a. Order U 0 and (1/N f ) 0 , I (0) (z). b. Order U 0 and (1/N f ) 1 , I (1) (z). c. Order U 1 and (1/N f ) 1 , I (1) i (z) (i = A, . . . , E) FIG. 3. Diagram for the multiplet ground-state energy, J (1) (z) I
1/N f -expansion for the ground-state energy can be derived with the help of Brioullin-Wigner perturbation theory:15,18,19 We decompose H from Eq. (1.1) into H 0 + H 1 and choose H 1 = H U + H cf as perturbation. In order to obtain the singlet ground-state energy we take as unperturbed ground state the filled Fermi sea |FS . The energy E S of the ground state of H (relative to the energy of |FS ) is given by20
ACKNOWLEDGMENTSThe author has benefited from numerous discussions with Professor P. Fulde, Dr. G. Khaliullin, and Dipl.-Phys. K. Fischer.
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A similar problem occurs for the Anderson lattice model. See the discussion in. Phys. Rep. 143277A similar problem occurs for the Anderson lattice model. See the discussion in, e.g., G. Czycholl, Phys. Rep. 143, 277 (1986).
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| {'fraction_non_alphanumeric': 0.07893771060621226, 'fraction_numerical': 0.03386381076302005, 'mean_word_length': 3.275281700185423, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 35, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We consider a magnetic impurity which interacts by hybridization with a system of weakly correlated electrons and determine the energy of the ground state by means of an 1/N f expansion. The correlations among the conduction electrons are described by a Hubbard Hamiltonian and are treated to lowest order in the interaction strength. We find that their effect on the Kondo temperature, T K , in the Kondo limit is twofold: First, the position of the impurity level is shifted due to the reduction of charge fluctuations, which reduces T K . Secondly, the bare Kondo exchange coupling is enhanced as spin fluctuations are enlarged. In total, T K increases. Both corrections require intermediate states beyond the standard Varma-Yafet ansatz. This shows that the Hubbard interaction does not just provide quasiparticles, which hybridize with the impurity, but also renormalizes the Kondo coupling. 75.20.Hr, 75.30.Hx, 71.28.+d 71.27.+a Typeset using REVT E X Recently, heavy-fermion behavior has been observed in the electron-doped cupratewith a large Sommerfeld coefficient γ ≃ 4J/(mole Nd · K 2 ). In the same temperature regime, the spin susceptibility is found to be independent of the temperature and the Sommerfeld-Wilson ratio is of order unity. These are characteristic features of heavy-fermion excitations. 2However, the characteristic low energy scale of the order of 1 K which is associated with this behavior cannot be explained by applying the usual theory of the Kondo effect which assumes that the conduction carriers behave as free particles. 3 This is not too surprising because undoped Nd 2 CuO 4 is an antiferromagnetic charge-transfer insulator instead of a metal, 4,5 despite of one hole per unit cell. Upon doping the Nd ions are therefore coupling to a system of strongly correlated electrons 6,7 rather than to weakly or uncorrelated ones.Hamiltonian and Scaling. In order to explain this new type of heavy-fermion behavior, it has therefore been proposed to include the correlations among the conduction electrons by including an on-site repulsion. 3,8 Thus, the total Hamiltonian', 'arxivid': 'cond-mat/9510062', 'author': ['Tom Schork \nMax-Planck-Institut für Physik komplexer Systeme\nBayreuther Str. 40 Haus 16D-01187DresdenGermany\n'], 'authoraffiliation': ['Max-Planck-Institut für Physik komplexer Systeme\nBayreuther Str. 40 Haus 16D-01187DresdenGermany'], 'corpusid': 16939564, 'doi': '10.1103/physrevb.53.5626', 'github_urls': [], 'n_tokens_mistral': 10437, 'n_tokens_neox': 9319, 'n_words': 5802, 'pdfsha': '9c69a815df42a12b8b65e92c84fd795ae3da4517', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/9510062v1.pdf'], 'title': ['Magnetic impurity coupled to interacting conduction electrons Typeset using REVT E X 1', 'Magnetic impurity coupled to interacting conduction electrons Typeset using REVT E X 1'], 'venue': []} |
arxiv |
Extensions of mixed Hodge modules and Picard-Fuchs equations
Aug 2018
Pedro L Del Angel
CONACYT CIMAT
R Cimat
CONACYT CIMAT
José J Hernández
CONACYT CIMAT
C
CONACYT CIMAT
Extensions of mixed Hodge modules and Picard-Fuchs equations
Aug 2018
The normal function associated to algebraic cycles in higher Chow groups defines a differential equation. This Picard-Fuchs equation defines an extension of D-modules as well as an extension of local systems. In this paper, we show that both extensions define the same extension of mixed Hodge modules determined by the normal function.
Introduction
When we study a family of higher algebraic cycles, i.e. elements in CH r (X, m), for X a smooth projective variety, we can associate a differential equation to the family, given by the Picard-Fuchs operator on the normal function of a cycle. It turns out that the non vanishing of this equation is related to the problem of finding indecomposable cycles, see [3]. Thus, the properties of the differential equation can be studied in view of its relation with finding non trivial cycles. In [4], they study its rationality and field of definition of its coefficients. In this paper we show how to relate the extension of D-modules that arises from the differential equation with the extension that comes from the normal function where the Picard-Fuchs operator is evaluated, using the Abel-Jacobi map and the theory of mixed Hodge modules.
Mixed Hodge modules provide a generalization of classical Hodge theory. They can be thought as perverse sheaves with a mixed Hodge structure. The category of mixed Hodge modules was defined by M. Saito in [12], [11]. In this work we are mainly interested in the Abel Jacobi map associated to higher Chow cycles. It turns out that this map can be factored through an extension in the category of mixed Hodge modules, thus providing an interpretation of normal functions as sections of mixed Hodge modules.
When dealing with admissible smooth families and higher algebraic cycles defined over a whole family, one has the Abel Jacobi map on the total space but also on the fibres, so one can think on both, the extension of MHM associated to the total space as well as the corresponding extensions associated to the fibres. In the later case, the normal functions can be thought of as sections on some smooth family of intermediate jacobians and they are related to nonhomogeneous Picard-Fuchs equations.
Extensions of Mixed Hodge Modules
Let X be a smooth variety over C. Given any D X -module M, via the de Rham complex, we have the de Rham functor from D X -modules to complexes. Moreover, we can say the following: Theorem 2.1 (Riemann-Hilbert correspondence). Let X be a smooth variety over C. Then the de Rham functor M → DR(M) induces an equivalence of categories between the category of holonomic D X -modules with regular singularities and the category of perverse sheaves Perv(C X ).
Let PervW(C X ) be the category of weight filtered perverse sheaves in Perv(C X ) and MFW rh (X) be the category of regular holonomic D X -modules with a good filtration together with a weight filtration. There is a natural functor from MFW rh (X) to PervW(C X ). It takes the pair (M, W M ) to the pair (DR(M), DR(W M )). If MFW rh (X; Q) is the fibre product PervW(Q X )× PervW(CX ) MFW rh (X), its objects are of the form (K • Q , W Q , M, F, W, α) where K • Q is a perverse sheaf with weight filtration W Q , M is a holonomic D X -module with regular singularities and weight filtration W , F is a good filtration on M and α is an isomorphism of filtered objects, i.e. α(K
• Q ⊗C, W Q ⊗C) ≃ (DR(M), DR(W M ))
, where DR is the de Rham functor.
Theorem 2.2 (M. Saito). For any smooth variety X over C there exists an abelian category MHM(X) that is a full subcategory of MFW rh (X; Q). MHM(X) is called the category of mixed Hodge modules.
The category of mixed Hodge modules contains a semi-simple full subcategory of modules of pure weight. These are called polarizable Hodge modules. In the bounded derived category D b (MHM(X)) all expected operations are defined: f * , f * , f ! , f ! , D, etc. MHM(Spec(C)) is isomorphic to the category of graded polarizable mixed Hodge structures. For k a subfield of C, the category MHM(X) of mixed Hodge modules of a smooth variety X over k is well defined.
Let r ∈ Z. The D X -module O X has a good filtration defined by
Gr F r (O X ) = O X . Since DR(O X ) is quasi-isomorphic to C X [dimX], we get a mixed Hodge module Q X (r)[dimX] given by (Q X (r)[dimX], W Q , O X , F, W, α), where Gr W Q dimX−2r (Q X (r)[dimX]) = Q X (r)[dimX]
and W and α have the only possible interpretation. This is called the constant (or Tate) mixed Hodge module. We usually use the shifted module
Q X (r) = Q X (r)[dimX][−dimX].
The following proposition and its corollary are essential to construct the maps we are looking for.
Proposition 2.3 ([13]
). Let f : X → Y be a proper morphism of smooth varieties. If M ∈ MHM(X) is pure, we have a non canonical isomorphism
f * M ≃ i H i f * M [−i] in D b MHM(Y ).
Corollary 2.4. Let f : X → Y be a proper smooth morphism of quasiprojective smooth varieties over k ⊂ C. Then there is a Leray spectral sequence
E p,q 2 = Ext p MHM(Y ) (Q Y (0), R q f * M ) ⇒ Ext p+q MHM(X) (Q X (0), M ),
which degenerates at E 2 , for any pure M ∈ MHM(X).
Proof. The existence of the Leray filtration on Ext p+q
MHM(X) (Q X (0), M ) such that E p,q 2 = Ext p MHM(Y ) (Q Y (0), R q f * M )
is clear. Then we use the decomposition in proposition (2.3) and apply Deligne's criterion to conclude that the Leray spectral sequence associated to the Leray filtration degenerates at E 2 .
This spectral sequence induces a canonical Leray filtration on Ext p+q MHM(X) (Q X (0), M ). We also have the following short exact sequence: Proposition 2.5. Let Y be a smooth quasiprojective variety over k ⊂ C, M ∈ MHM(Y ) and g : Y → Spec(k) be the natural morphism. Then there exists a short exact sequence
0 → Ext 1 MHM(Spec(k)) (Q Spec(k) (0), R q−1 g * M ) → Ext q MHM(Y ) (Q Y (0), M ) → hom MHM(Spec(k)) (Q Spec(k) (0), R q g * M ) → 0
Proof. By Corollary (2.4) there is the Leray spectral sequence
E p,q 2 = Ext p MHM(Spec(k)) (Q Spec(k) (0), R q g * M ) ⇒ Ext p+q MHM(Y ) (Q Y (0), M ), which degenerates at E 2 . Then, if L is the Leray filtration on Ext p+q MHM(Y ) , E p,q 2 = Gr p L Ext p+q MHM(Y ) (Q Y (0), M ).
On the other hand, MHM(Spec(k)) is a subcategory of the category of mixed Hodge structures (see next section), and we know that Ext ℓ MHS = 0 for all ℓ ≥ 2, and similarly for MHM(Spec(k)). Then
L 1 Ext q MHM(Y ) (Q Y (0), M ) = Gr 1 L Ext q MHM(Y ) (Q Y (0), M ) = Ext 1 MHM(Spec(k)) (Q Spec(k) (0), R q−1 g * M ). Since hom MHM(Spec(k)) (Q Spec(k) (0), R q g * M ) = Gr 0 L Ext q MHM(Y ) (Q Y (0), M ), we get that hom MHM(Spec(k)) (Q Spec(k) (0), R q g * M ) is the quotient of Ext q MHM(Y ) (Q Y (0), M ) and Ext 1 MHM(Spec(k)) (Q Spec(k) (0), R q−1 g * M )
, and this implies that we have the short exact sequence required.
Maps for Higher Chow Groups
Let X be a smooth variety over k ⊂ C. Lets denote the r-th higher Chow group of X tensored with Q by CH r (X, m; Q). There is a cycle class map from this group to an extension of mixed Hodge modules. It was constructed by M. Saito (see [1] as well).
c r,m : CH r (X, m; Q) → Ext 2r−m MHM(X C ) (Q X C (0), Q X C (r)).
If we consider the natural morphism g : X C → Spec(C), we can use lemma (2.5) to get a short exact sequence
0 → Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 g * Q X C (r)) → Ext 2r−m MHM(X C ) (Q X C (0), Q X C (r)) → hom MHM(Spec(C)) (Q Spec(C) (0), R 2r−m g * Q X C (r)) → 0
for X a smooth quasiprojective variety over k ⊂ C. Since MHM(Spec(C)) is isomorphic to the category of graded polarizable mixed Hodge structures, we have the isomorphisms
Q Spec(C) (0) ≃ Q(0), R 2r−m−1 g * Q X C (r) ≃ H q−1 (X, Q(r)), R 2r−m g * Q X C (r) ≃ H q (X, Q(r)).
where H • (X, Q(r)) means H • (X an C , Q(r)), a notation we will continue to use in this paper. Then we can rewrite the short exact sequence as
0 → Ext 1 MHS (Q(0), H 2r−m−1 (X, Q(r))) → Ext 2r−m MHM(X C ) (Q X C (0), Q X C (r)) → hom MHS (Q(0), H 2r−m (X, Q(r))) → 0 (1)
Using the morphism of Theorem (3.1),
c r,m : CH r (X, m; Q) → Ext 2r−m MHM(X C ) (Q X C (0), Q X C (r)),
we have a map
CH r (X, m; Q) → hom MHS (Q(0), H 2r−m (X, Q(r))).(2)
We denote by [ξ] the image of ξ ∈ CH r (X, m; Q) in hom MHS (Q(0), H 2r−m (X, Q(r))) and if this image is zero (this is always the case when X is complex projective and m > 0) then ξ maps to an element of Ext 1 MHS (Q(0), H 2r−m−1 (X, Q(r))) as we can see from (1). We call this map AJ. The kernel of the (2) is denoted by CH r hom (X, m; Q). is called the Abel-Jacobi map.
This definition of AJ(ξ) coincides with the definition of higher Abel-Jacobi maps defined by integration of currents when X is smooth projective over C; it is proved by Kerr, Lewis and Muller-Stach in [7], see also [8].
The extension associated to a family
Let X be a smooth projective variety over C, g : X → C the natural morphism. Then, in the previous section we constructed a map CH r hom (X, m; Q)
AJ + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ / / Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 g * Q X (r)) = Ext 1 MHS (Q(0), H 2r−m−1 (X, Q(r))).(3)
Thus, to any ξ ∈ CH r hom (X, m; Q) we can associate an extension E ∈ Ext 1 MHS (Q(0), H 2r−m−1 (X, Q(r))).
Since we are working in the category of mixed Hodge modules, we should note that these extension groups are defined as extensions in the categorical sense. For our interests, we would like to have a different point of view. More precisely, that E is an extension in the sense of Yoneda is a consequence of the following theorem of Verdier, [15] (see also Gelfand-Manin [5] or Peters-Steenbrink [10] ): Therefore E is an extension (more precisely an equivalence class of extensions) of mixed Hodge structures of the form:
0 → H 2r−m−1 (X, Q(r)) → E → Q(0) → 0.
As we can see in (3), E is automatically a mixed hodge module over Spec(C). Now, let's work in the relative case, i.e. let's consider a family of smooth proyective varieties over C,
X t / / X f Spec(C) t / / S(4)
where f is smooth and proper with fibre X t = f −1 (t), for t ∈ S and X , S are defined over k ⊂ C algebraically closed. In the family each fibre is of the form X t = X × S Spec(C), where k(t) = C and we have natural morphisms Spec(C) → S, X t → Spec(C).
From Corollary (2.4), for f : X → Y a proper smooth morphism of quasiprojective smooth varieties over k ⊂ C there is a Leray spectral sequence
E p,q 2 = Ext p MHM(Y ) (Q Y (0), R q f * M ) ⇒ Ext p+q MHM(X) (Q X (0), M ),
which degenerates at E 2 , for any pure M ∈ MHM(X). In particular we have a Leray filtration L on Ext. For f : X → S a family of varieties over C, then we have a map
L 1 Ext 2r−m MHM(X ) (Q X (0), Q X (r)) → Gr 1 L Ext 2r−m MHM(X ) (Q X (0), Q X (r)) = Ext 1 MHM(S) (Q S (0), R 2r−m−1 f * Q X (r)).
where Gr 1 L = L 1 /L 2 . L also induces a filtration F L on CH r (X , m; Q) given by where c r,m is the cycle class map from Theorem (3.1). Using the map above and the cycle class map we get
F 1 L CH r (X , m; Q) → Ext 1 MHM(S) (Q S (0), R 2r−m−1 f * Q X (r)).
Since the Leray spectral sequence is functorial, then from the diagram 4 and the pullback on X t → X we get the diagram
F 1 L CH r (X , m; Q) / / F 1 L CH r (Xt, m; Q) Ext 1 MHM(S) (QS(0), R 2r−m−1 f * QX (r)) / / Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 f * QX t (r))(5)
We can relate this construction to the previous maps in the following:
Proof. We have an injective map
Gr 0 FL CH r (X t , m; Q) ֒→ Gr 0 L Ext 2r−m MHM(Xt) (Q Xt (0), Q Xt (r)). But Gr 0 L Ext 2r−m MHM(Xt) (Q Xt (0), Q Xt (r)) = Ext 0 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m f * Q Xt (r))
and
Ext 0 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m f * Q Xt (r)) = hom MHS (Q(0), H 2r−m (X t , Q(r)))
Recall from the morphism Spec(C) → S that we have a morphism MHM(S) →MHM(Spec(C)) and MHM(Spec(C)) is the category MHS. That
Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 f * Q Xt (r)) = Ext 1 MHS (Q(0), H 2r−m−1 (X t , Q(r)))
follows from this. Since X t is projective and because we are working in the category of mixed Hodge structures, for m ≥ 1 by weight reasons:
hom MHS (Q(0), H 2r−m (X t , Q(r))) = 0. Therefore Gr 0 L Ext 2r−m MHM(Xt) (Q Xt (0), Q Xt (r)) = 0 so L 0 = L 1 and F 0 L = F 1 L .
This shows that F 1 L CH r (X t , m; Q) = CH r (X t , m; Q). Also, from (1), for X t projective the map (2) is always zero and therefore CH r (X t , m; Q) = CH r hom (X t , m; Q).
In the proof above F 0 L = F 1 L and CH r (X t , m; Q) = CH r hom (X t , m; Q) for any smooth projective variety when m ≥ 1. So, we have a diagram CH r (X , m; Q)
/ / CH r hom (X t , m; Q) AJ Ext 1 MHM(S) (Q S (0), R 2r−m−1 f * Q X (r)) / / Ext 1 MHS (Q(0), H 2r−m−1 (X t , Q(r))).
If we choose a cycle ξ ∈ CH r (X , m; Q) and ξ t = ξ| Xt we get maps
CH r hom (X t , m; Q) → Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 f * Q Xt (r))
and every ξ t ∈ CH r hom (X t , m; Q) maps to an extension E t of mixed Hodge modules. Moreover, the morphism Spec C → S induces the bottom map in the diagram
Ext 1 MHM(S) (Q S (0), R 2r−m−1 f * Q X (r)) → Ext 1 MHM(Spec(C)) (Q Spec(C) (0), R 2r−m−1 f * Q Xt (r)).
In summary, we have the following:
Theorem 4.3.
To any smooth projective family f : X → S over C and cycle ξ ∈ CH r (X , m; Q), m ≥ 1, we can associate a mixed Hodge module E that is an extension
0 → R 2r−m−1 f * Q X (r) → E → Q S (0) → 0
of mixed Hodge modules, that restricts to the extensions given by the Abel-Jacobi map in the fibres.
The extension of mixed Hodge modules that captures the family is, precisely, in Ext 1 MHM(S) (Q S (0), R 2r−m−1 f * Q X (r)). Let MF rh (Y ) denote the category of regular holonomic D Y -modules with a good filtration, where Y is a smooth variety over a closed field k of characteristic 0. We can map any mixed Hodge module to its corresponding filtered regular holonomic D Y -module. In particular, for f :
X → S, Q S (0) maps to O S and R 2r−m−1 f * Q X (r) to R 2r−m−1 f * Ω • X /S (r) = H 2r−m−1 DR (X /S)(r)
.
Ω • X /S denotes the complex of relative differentials of X over S and H q DR (X /S) is the q-th relative de Rham cohomology sheaf of X over S that by definition is
H q DR (X /S) := R q f * Ω • X /S
for any X and S smooth quasiprojective varieties over k and f : X → S a proper smooth morphism. Then we have a natural map The extension E (we denote the correponding D S -module also by E), can be seen as an extension of D S -modules
0 → H 2r−m−1 DR (X /S)(r) → E → O S → 0.(6)
Extensions and normal functions
Let π : X → S be a flat morphism of relative dimension k between proper, irreducible smooth varieties defined over C. Assume dim (S) = 1 and let X ⊂ X , S ⊂ S be open dense subsets such that π := π| X : X → S is smooth. If H is the local system over S associated to the primitive part of R k π * Q and H = H ⊗ O S , then the Gauss-Manin connection ∇ : H → H ⊗ Ω 1 S induces a differential equation
D P F f = 0(7)
called the Picard-Fuchs equation of the periods of the family X → S.
Let {ω 1 , . . . , ω n } be a local base of solutions of 7, g be an holomorphic function on S, h be a solution of the differential equation
D P F h = g(8)
in a neighborhood U of a point t 0 ∈ S and γ be a closed path on S with starting point t 0 . If we extend h along γ, we will get back another solution of 8, which then looks like
γ(h) = h + i a i ω i
Let h 1 be another solution of 8 in U, then h 1 − h will be a solution of 7 and therefore they differ by a linear combination i c i ω i . If the action of the monodromy π 1 (S, t 0 ) on the periods is given by ρ, then γ will act on the periods by M = ρ(γ) and we can extend h 1 along γ as well, getting
γ(h 1 ) = h + i a i ω i + i c i ρ(γ)(ω i ) = h 1 + i a i ω i + i c i (M − I)ω i
so that we can associate to 8 a well defined class (see [14], part II section 3)
α = [γ → a γ ] ∈ H 1 (π 1 (S, t 0 ), H 0 ) = H 1 (S, H),
where H 0 is the fiber of H at t 0 and a γ = (a 1 , . . . , a n ). If ξ ∈ CH r (X , m; Q) and ξ t = ξ| Xt , the Abel-Jacobi map (3.2) let us define:
Definition 5.1. The normal function ν ξ : S → J associated to ξ is given by
ν ξ (t) = AJ(ξ t ),
where J fits in the short exact sequence
0 → R 2r−m−1 π * Z → R 2r−m−1 π * C ⊗ O S /F r (R 2r−m−1 π * C ⊗ O S ) → J → 0
and J t = Ext 1 MHS (Q(0), H 2r−m−1 (X t , Q(r))). If k = 2r − m − 1, let ν ξ be a lifting of ν ξ to R k π * C ⊗ O S /F r (R k π * C ⊗ O S ) and let us assume that ν ξ is not a solution of the differential equation 7 and the family X → S is admisible (i.e. the monodromy representation at the fibers at infinity is irreducible and unipotent), then g(t) := D P F ν ξ (t) defines a non zero holomorphic function which does not depend on the choice of the lifting (see [4], thm. 1.1 and 3.2) and we can associate to it (and therefore to the normal function ν ξ ) a non zero class α ξ in H 1 (S, H). Moreover, in this case the solutions of the equation D P F ν ξ (t) = g(t) are also solutions of the homogeneous equation
( d dt − d log g)D P F h(t) = 0(9)
which is the homogeneous equation associated to a local system E. Observe that the solutions of the equation D P F h(t) = 0 are solutions of 9 as well, therefore E is an extension of H of the form
0 → H → E → Q(−r) → 0(10)
and we claim that this extension is determined by the class α ξ . Indeed, with the notation above, if the action of γ on {w 1 , . . . , w n } (a flat basis of H = H⊗O S ) is given by M = ρ(γ), then the action of γ on {w 1 , . . . , w n , ν ξ } (a flat basis of E = E ⊗ O S ) is given by M a γ 0 1
i.e., it is encoded in α ξ so that we can recover the extension 10 from it. After tensoring 10 with O S we get the extension of D S modules (see 6) 0 → H 2r−m−1 DR (X /S) → E → O S (−r) → 0 associated to the local systems H, E and g · Q(−r). Thanks to theorem (4.3), we can think of the normal function ν ξ as an extension of mixed Hodge modules whose underlying extension of D S modules is precisely the extension 6. This reflects the fact that to a MHM(S) one can associate both, a D S -module and a perverse sheaf and so to an extension of MHM(S) should correspond both, an extension of D S -modules as well as an extension or perverse sheafs on S and both are determined by the image of ν ξ on the corresponding extension groups.
Proposition 5.2. The extension of D S -modules associated to equation 9 coincides with the one given by ν ξ on Ext 1 MHM(S) (Q S (0), R 2r−m−1 π * (X , Q(r))), whereas the extension of local systems given by α ξ coincides with the one given by ν ξ on Ext 1 Perv(S) (Q S (0), R 2r−m−1 π * (X , Q(r))).
Moreover, Brosnan et. al (see [2]) have shown that the class α ξ is actually a Tate class of weight 0 in IH 1 (S, H).
Theorem 3.1 (M.Saito). Let X be a smooth variety over k ⊂ C. Then there exists a cycle map
Definition 3. 2 .
2The map AJ : CH r hom (X, m; Q) → Ext 1 MHS (Q(0), H 2r−m−1 (X, Q(r))).
Proposition 4. 1 .
1If A is an abelian category then for any A and B objects in A, Ext n (A, B) is the same as the Yoneda extension group.
F j L
jCH r (X , m; Q) := c −1 r,m (L j Ext 2r−m MHM(X) (Q X (0), Q X (r))),
Proposition 4. 2 .
2The right vertical map in (5) is the Abel-Jacobi map for m ≥ 1.
S) (QS(0), R 2r−m−1 f * QX (r)) → Ext 1 MF rh (S) (OS, H 2r−m−1 DR (X /S)(r)).
Motives and algebraic de Rham cohomology. M Asakura, The arithmetic and geometry of algebraic cycles, Proceedings of the CRM summer school. 24M. Asakura. Motives and algebraic de Rham cohomology. In The arithmetic and geometry of algebraic cycles, Proceedings of the CRM summer school, volume 24 of CRM Proceedings and Lecture Notes, pages 133-155, 2000.
Singularities of admissible normal functions. P Brosnan, H Fang, Z Nie, G Pearlstein, Invent. math. 177P. Brosnan, H. Fang, Z. Nie, and G. Pearlstein. Singularities of admissible normal functions. Invent. math., 177:599-629, 2009.
The transcendental part of the regulator map for K1 on a mirror family of K3 surfaces. P L Angel, S Müller-Stach, Duke Math. J. 1123P. L. del Angel and S. Müller-Stach. The transcendental part of the reg- ulator map for K1 on a mirror family of K3 surfaces. Duke Math. J., 112(3):581-598, 2002.
Differential equations associated to families of algebraic cycles. P L Angel, S Müller-Stach, Ann. Inst. Fourier (Grenoble). 586P. L. del Angel and S. Müller-Stach. Differential equations associated to families of algebraic cycles. Ann. Inst. Fourier (Grenoble), 58(6):2075- 2085, 2008.
Methods of homological algebra. S Gelfand, Y Manin, SpringerS. Gelfand and Y. Manin. Methods of homological algebra. Springer, 2003.
Filtrations on higher chow groups and arithmetic normal functions. J Hernández, University of AlbertaPhD thesisJ. Hernández. Filtrations on higher chow groups and arithmetic normal functions. PhD thesis, University of Alberta, 2012.
The Abel-Jacobi map for higher Chow groups. M Kerr, J Lewis, S Müller-Stach, Compositio Mathematica. 1422M. Kerr, J. Lewis, and S. Müller-Stach. The Abel-Jacobi map for higher Chow groups. Compositio Mathematica, 142(2):374-396, 2006.
M Kerr, James D Lewis, The Abel-Jacobi map for higher Chow groups, II. Inventiones Mathematicae. 170M. Kerr and James D. Lewis. The Abel-Jacobi map for higher Chow groups, II. Inventiones Mathematicae, 170(2):355-420, 2007.
Algebraic cycles and Mumford-Griffiths invariants. J Lewis, S Saito, American Journal of Mathematics. 1296J. Lewis and S. Saito. Algebraic cycles and Mumford-Griffiths invariants. American Journal of Mathematics, 129(6):1449-1499, 2007.
Mixed Hodge structures. C Peters, J Steenbrink, SpringerC. Peters and J. Steenbrink. Mixed Hodge structures. Springer, 2008.
Modules de Hodge polarisables. M Saito, Publ. RIMS, Kyoto Univ. 24M. Saito. Modules de Hodge polarisables. Publ. RIMS, Kyoto Univ., 24:849-995, 1988.
Mixed Hodge modules. M Saito, Publ. RIMS, Kyoto Univ. 26M. Saito. Mixed Hodge modules. Publ. RIMS, Kyoto Univ., 26:221-333, 1990.
On the formalism of mixed sheaves. M Saito, PreprintM. Saito. On the formalism of mixed sheaves. Preprint, 2006.
Automorphic Forms and the Picard Number of an Elliptic Surface. P Stiller, ViewegP. Stiller. Automorphic Forms and the Picard Number of an Elliptic Sur- face. Vieweg, 1984.
Des catégories dérivées des catégories abéliennes. J.-L Verdier, Astérisque. 239J.-L. Verdier. Des catégories dérivées des catégories abéliennes. Astérisque, 239, 1996.
An introduction to homological algebra. Charles A Weibel, Cambridge University PressCharles A. Weibel. An introduction to homological algebra. Cambridge University Press, 1994.
| {'fraction_non_alphanumeric': 0.09593998234774934, 'fraction_numerical': 0.025463371579876434, 'mean_word_length': 3.287795648060549, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The normal function associated to algebraic cycles in higher Chow groups defines a differential equation. This Picard-Fuchs equation defines an extension of D-modules as well as an extension of local systems. In this paper, we show that both extensions define the same extension of mixed Hodge modules determined by the normal function.', 'arxivid': '1808.02549', 'author': ['Pedro L Del Angel \nCONACYT CIMAT\n\n', 'R Cimat \nCONACYT CIMAT\n\n', 'José J Hernández \nCONACYT CIMAT\n\n', 'C \nCONACYT CIMAT\n\n'], 'authoraffiliation': ['CONACYT CIMAT\n', 'CONACYT CIMAT\n', 'CONACYT CIMAT\n', 'CONACYT CIMAT\n'], 'corpusid': 52995454, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8518, 'n_tokens_neox': 7606, 'n_words': 4527, 'pdfsha': '7cf58e6b4545ece70920649b873f812d640a39d8', 'pdfurls': ['https://arxiv.org/pdf/1808.02549v1.pdf'], 'title': ['Extensions of mixed Hodge modules and Picard-Fuchs equations', 'Extensions of mixed Hodge modules and Picard-Fuchs equations'], 'venue': []} |
arxiv |
Null Singularities in Colliding Waves
arXiv:gr-qc/0103092v1 26 Mar 2001 March 24, 2022
Ozay Gurtug [email protected]
Department of Physics
Eastern Mediterranean University G.Magusa
Mersin 10 -TurkeyNorth Cyprus
Mustafa Halilsoy
Department of Physics
Eastern Mediterranean University G.Magusa
Mersin 10 -TurkeyNorth Cyprus
Null Singularities in Colliding Waves
arXiv:gr-qc/0103092v1 26 Mar 2001 March 24, 2022
Colliding Einstein -Maxwell -Scalar fields need not necessarily doomed to become in a spacelike singularity. Examples are given in which null singularities emerge as intermediate stages between a spacelike singularity and a regular horizon.
Coupling scalar fields to a static charged black hole (BH) [1,2] (with the respective Reissner-Nordstrom (RN) and Newman-Janis-Winicour (NJW) limits) converts its horizons into null singularities is a known fact. This fact was rediscovered recently in a related context involving the perturbation of a charged BH by ingoing pulses of scalar fields [3,4]. It has been shown that the inner horizon of a RN BH is unstable against such perturbations and transforms it into a null singularity. This is an interesting development since it would mean that a BH upon being slightly perturbed will not find it so simple to act as a gateway to 'other worlds' . With this example in mind and the similarity in the dynamics of collapse of a scalar field and colliding Einstein-Maxwell-Scalar (EMS) fields motivates us to explore analogous singularities in EMS fields. Ori has already discussed null singularities in plane symmetric spacetimes [5]. This must be important because as Tipler [6] has shown the tidal distortion experienced by an infalling object is small as it hits such a singularity.
In this Letter we consider two concrete cases in the field of colliding plane waves (CPW). Our first case is the collision of two electromagnetic (em) shock waves, known as the Bell-Szekeres (BS) [7] solution that yields a quasiregular 'singularity' ( this we consider equivalent to a horizon). In our second example we consider the horizon forming CPW found by Chandrasekhar and Xanthopoulos (CX) [8] which is isometric to the region in between the horizons of the Kerr-Newman BH. We show that by choosing appropriate scalar fields both the quasiregular 'singularity' of the BS spacetime and the horizon of the CX metric transform into null singularities.
Our first line element is the linearly polarized metric
ds 2 = ∆ 1−A Z 2 dτ 2 ∆ − dσ 2 δ − δdx 2 − ∆ A Z −2 dy 2(1)
where the notation goes as
τ + σ = 2u √ 1 − v 2 τ − σ = 2v √ 1 − u 2 ∆ = 1 − τ 2 δ = 1 − σ 2 2Z = a(1 + τ ) A + b(1 − τ ) A(2)
in which (a, b) and A are constants and (u, v) are the null coordinates. We choose 0 < A < 1 to represent the scalar charge while (a, b) stand for the em parameters. The scalar field is
φ(τ ) = 1 2 √ 1 − A 2 ln 1 + τ 1 − τ(3)
which implies that for A = 0 there would be a background scalar field already and the singularity is spacelike. As we increase A toward unity the scalar field diminishes and the singularity of the spacetime transforms to the removable quasiregular 'singularity'. The singularity at τ = 1, however, becomes null for 0 < A < 1. Since the null coordinates (u, v) are to be multiplied by the step functions θ(u) and θ(v) apt for the collision problem the u-dependent incoming pulse energy density is
4πT uu = Φ (0) 22 = θ(u) 4Z 2 (1 − u 2 ) 2 (1 − A 2 ) b 2 (1 − u) 2A + a 2 (1 + u) 2A +2ab(1 + A 2 )(1 − u 2 ) A (4) where 2Z = a(1 + u) A + b(1 −u) A .
2 is found to be
√ 1 − u 2 √ 1 − v 2 Ψ (0) 2 = 1 − A ∆ + A 4Z 2 a 2 (1 + τ ) 2A−1 + b 2 (1 − τ ) 2A−1 +2ab(1 − 2A)∆ A−1(5)
in which the τ = 1 singularity is manifest. The remaining scalars Ψ As a second example we consider the CX metric
ds 2 = X dτ 2 ∆ − dσ 2 δ − ∆δ X Y dy 2 − Y X (dx − q 2 dy) 2(6)
where
X = 1 α 2 (1 − αpτ ) 2 + α 2 q 2 σ 2 Y = 1 − p 2 τ 2 − q 2 σ 2 q 2 = − qδ pα 2 1 + α 2 − 2αpτ 1 − p 2 τ 2 − q 2 σ 2(7)
in which the constant parameters α, p and q must satisfy
0 < α ≤ 1 p 2 + q 2 = 1(8)
This metric admits a horizon instead of a spacelike singularity at τ = 1. We add now a scalar field φ satisfying ✷φ = 0, or equivalently (∆φ τ ) τ = (δφ σ ) σ as follows [9]: By shifting the metric function X → Xe −Γ we couple the scalar field consistently with Einstein-Maxwell's fields where Γ is determined from the line integral
Γ = 2 φ 2 u U u du + 2 φ 2 v U v dv(9)
in which e −U = √ ∆δ. Choosing a simple class of scalar field such as φ(τ ) =
k 2 ln 1+τ 1−τ , with k = constant results in e −Γ = 1−τ 2 τ 2 −σ 2 k 2
. With the addition of this scalar field we can see from the energy momentum scalar T α α and T µν T µν , which are divergent that τ = 1 is singular. Furthermore, the fact that as τ → 1 the metric function g τ τ → 0 for the case k 2 < 1 implies that it is a null singularity. For k 2 ≥ 1, however, it retains the spacelike character which is standard to CPW. Letting q = 0 and using the transformation
t = mαx, y = φ, τ = m − r √ m 2 − e 2 , σ = cos θ(10)
with mα = √ m 2 − e 2 we obtain ds 2 = 1 − 2m r + e 2 r 2 dt 2 − e −Γ 1 − 2m r + e 2 r 2 dr 2 −r 2 e −Γ dθ 2 + sin 2 θdφ 2 (11) which is a spherically non-symmetric extension of the RN metric with a null singular horizon. Our method generates infinitly many metrics with null ( or directionally null, depending on the choice of the scalar field) singular horizons that may find application in BH's.
For A = 1 (and a = b ) this metric reduces to the well-known BS solution. The scale invariant Weyl scalar Ψ
similar structure but these latter two have in addition an impulsive component. For A = 1 ( and a = b ) all conformal curvatures vanish for u > 0, v > 0 and the singularity τ = 1 becomes removable. Thus the null singularity arises as an intermediate stage in between a spacelike quasiregular singularity and a horizon.
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. O Gurtug, M Halilsoy, gr-qc 0006038ReportO.Gurtug and M.Halilsoy, Report No: gr-qc 0006038.
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arxiv |
An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification
Ankur Pal
Indian Institute of Science Education and Research Kolkata
741246MohanpurNadia, West BengalIndia
Abhishek Shukla
IMOMEC division
IMEC
Wetenschapspark 1B-3590DiepenbeekBelgium
Institute for Materials Research (IMO)
Hasselt University
Wetenschapspark 1B-3590DiepenbeekBelgium
Anirban Pathak
Department of Physics
Materials Science & Engineering Jaypee Institute of Information Technology
Uttar Pradesh
Sector-62A-10, 201309NoidaIndia
An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification
An algorithm for image processing is proposed. The proposed algorithm, which can be viewed as a quantumclassical hybrid algorithm, can transform a low-resolution bitonal image of a character from the set of alphanumeric characters (A-Z, 0-9) into a high-resolution image. The quantum part of the proposed algorithm fruitfully utilizes a variant of Grover's search algorithm, known as the fixed point search algorithm. Further, the quantum part of the algorithm is simulated using CQASM and the advantage of the algorithm is established through the complexity analysis. Additional analysis has also revealed that this scheme for optical character recognition (OCR) leads to high confidence value and generally works in a more efficient manner compared to the existing classical, quantum, and hybrid algorithms for a similar task.
I. INTRODUCTION
Ever since Richard P. Feynman's 1982 [1] paper on the viability of using quantum resources (computers) to simulate quantum mechanical systems appeared, researchers have proposed various ways in which we can harness the power of quantum mechanics to solve problems that are difficult to be solved using classical computers in a reasonable time. Interestingly, many of the problems that quantum computers can solve faster than their classical counterparts are apparently unrelated to quantum mechanics, or even physics. For example, the Bernstein-Vazirani algorithm [2] and the Deutsch-Jozsa algorithm [3] were solutions to problems that were unrelated to quantum mechanics. The problems that they solved were of proof of concept nature, having very little practical uses. Interestingly, there exists a set of quantum algorithms having important practical applications. These algorithms also perform relevant computational tasks faster than their classical counterparts. An excellent example is Shor's algorithm [4], which has significant practical applications in the area of cryptanalysis. It essentially provides an efficient scheme for the factorization of the odd bi-primes, a problem not directly related to Physics. Here, we specifically stress on this aspect as we plan to propose an algorithm that will use quantum resources to solve a problem related to optical character recognition (OCR) which is also apparently unrelated to quantum mechanics. This just illustrates the large domain of applicability of quantum computing and does not indicate that quantum computing is not useful in solving problems closely connected to physics. Relevance of quantum algorithms in solving problems related to quantum physics is demonstrated through simulation of many-body Fermi systems on a universal quantum computer [5] and more recently through the realization of a variational quantum eigenvalue solver using photonic quantum processors [6,7].
With time, it's realized that quantum computers are far more versatile than originally expected, and several new applications of quantum computing have been proposed. One such application is Quantum digital image processing (QDIP) which has recently been developed as a field that uses the power of quantum computation to solve a specific class of problems i.e., image processing-related problems which have applications in daily life. To understand its importance, we first need to note that a digital image is basically a digital object containing various visual information of an object of interest distributed over the space discretely and represented by a matrix. The smallest addressable element of the matrix is known as a pixel. The information about the color shade at every pixel is described by the pixel value (usually it's a number between 0-255 as an 8-bit/1-byte register is used for describing 256 shades of a bi-tonal image. Higher number of shades would require relatively large bit string, but that's not our concern at the moment). The full information of the shade of particular color is usually described by the parameters: hue, saturation, and luminosity [8]. As long as any processing operation is linear and reversible and the related information can be encoded in the quantum systems, the traditional (classical) processing can be replaced with the unitary operation and a quantum algorithm can be used for obtaining the computational advantage [9]. As in this procedure, the basic process of image processing not only involves quantum resources, but also a quantum algorithm, it's referred to as QDIP.
Before we describe our quantum-classical hybrid protocol for image processing, it would be apt to note that in the recent past a set of schemes for representing the image information using quantum systems (here termed as quantum digital image representation) have been de-veloped and several applications of those have been proposed (for a review see [10]). For example, the qubit lattice model is used to map the spatial information about the image into probability amplitudes of qubit states [11]. Further, in Ref. [12], a scheme is proposed that can not only encode the shade and color information, but also the spatial information i.e., pixel index. Similarly, quite a few quantum algorithms for edge detection have been proposed, like QSobel [13], Laplacian and Sobel algorithm based scheme proposed in [14], and improved Sobel operator based algorithm of Ma et al. [15]. Following an independent path, quantum support vector machine is deveopled and is used for the grouping of images for classification requiring feature map [16]. Furthermore, a convolutional neural network has shown very good performance for exploiting the correlation information when the convolutional layer and pooling layer are replaced by the quantum layer [17]. Schemes for QDIP usually improve efficiency and results of processing classically stored digital images, using protocols allowing us to interpret the measurement outcomes of qubits to represent classical information of the processed image. QDIP has many facets and a few algorithms for QDIP have already been designed for a set of specific image-processing tasks. Specifically, we may mention that algorithms have been designed for quantum image scrambling [18][19][20][21], geometrical transformation of quantum image [22,23], quantum image scaling [24], quantum image encryption and decryption [25], quantum image steganography [26], watermarking of quantum image [27,28], quantum audio [29], quantum movie [30], quantum image segmentation [31], and quantum image matching [32]. Moreover, Tseng and Huang [33] has proposed an edge detection scheme that has the same performance as the Sobel method, while the Sobel method was proved better than the corresponding classical methods for detecting noisy images [34]. In this particular work, we plan to propose a QDIP scheme that will be useful in image processing tasks of a specific type. Specifically, the quantum part of our hybrid algorithm would perfrom the preprocessing tasks of the overall OCR process for lowresolution printed text images. The motivation behind the designing of QDIP scheme for OCR is manifold. Firstly, OCR is commonly used in our daily life. Secondly, OCR-related studies form one of the most active fields in machine learning (ML), quantum ML (QML), and quantum-assisted ML (QAML). The recent success of ML, QML and QAML algorithms in performing various tasks in an efficient manner hint at the possibility of designing an efficient hybrid algorithm for image processing related to OCR where quantum resources and the concepts QML and QAML can be used in a beneficial manner.
Nowadays, with the increase in computational power, we can handle datasets of enormous sizes, and that allows OCR software to recognize not only printed char-acter text, but also handwritten texts [35]. However, efficiency and accuracy are still a concern and there still remains scope for improvement as far as the results for the aforementioned case of using OCR to identify printed text in a particular font are concerned. While training an OCR, we train OCR, for a particular type of font set. Every character belonging to that front set has a definite pixel matrix, we train OCR first to find which pixel matrix corresponds to that particular character. This ensures that we only need to provide a well-labeled dataset for each character and then rely on ML or QML for theidentification of the character. So, when we input an image into an OCR system (image acquisition), it can check which character it resembles the most [36]. This is a simplistic way of understanding how OCRs work, but on a very basic level, it interprets the information conveyed in images by the values representing each pixel. Typically, the overall process of OCR involves pre-processing consisting of normalization, denoising, and skew correction, followed by segmentation, feature extraction, and identification [37]. These processes help in increasing the accuracy and efficiency of OCR algorithms. Keeping the importance of pre-processing in mind, in the following section we will describe the important prepossessing techniques that should be used before implementing our algorithm which will be described in the subsequent sections.
The rest of the paper is organized as follows. In Sec. II, we briefly introduce the pre-processing techniques used in OCR algorithms as those will be required for our algorithm, too. Subsequently, in Sec. III, we formulate the problem, introduce the quantum systems for encoding the relevant information, and describe the flow of the overall algorithm. In Sec. IV, we describe the use of the classical part of the algorithm for pre-processing and for identification of the missing information and hence the right character. In Sec. V, we describe the quantum part of the algorithm which is based on a fixed point search algorithm. In Sec. VI, we present the confidence value obtained using our algorithm and analyze the results ending in a conclusion that our algorithm correctly and efficiently identifies the alphanumeric characters.
II. PRE-PROCESSING TECHNIQUES
One important step of OCR is pre-processing. This step is important because the input image itself may not be an ideal input for the OCR algorithm. Pre-processing is done to enhance the performance of OCRs. Some of the most commonly used pre-processing methods are :
• Image thresholding: Image thresholding is the process of removing the background, for a bitonal image. It is done by setting the pixel value in a pixel matrix to 0 (1) if it is below (above) the threshold. It reduces the amount of data by removing irrelevant details from an image (see example images for local thresholding illustrated in [38]). This method is useful when the image has been taken under irregular lighting. After removing the background, the pixel matrix (the matrix of pixel values) is normalized.
• Skew correction : It is the process of aligning the images. An example of it can be seen in Figure 1 of Ref. [39], which shows histograms of a document for two different skew angles. The histogram is created by summing up all the black pixels in the horizontal direction. For a non-zero skew angle, the number of pixels for a row number is distributed in an immiscible manner, while for a zero skew angle the histogram is regular.
• Denoising : The main sources of noise in digital images are due to irregular or low light acquired by capturing device, sensor temperature effects, and sometimes due to the error in the mathematical model. The main methods used for denoising are mean filters, median filters, and periodic noise filtering (see text images in [40] and methods for denoising them). These denoising methods are part of preprocessing process as these methods help us to clean an image before the implementation of the main algorithm for image processing.
• Segmentation It separates each character present in the text. It involves many steps like inverting the image to make the background black and foreground white to reduce space, conversion to RGB format to introduce colors other than black and white, and segmenting the word image determined by the sum of the pixel value of the foreground image to be 1 or 0 and ultimately removing the oversegmentation. Examples of all the steps are shown in Figure 3 of Ref. [41].
These pre-processing methods solve particular problems that arise in specific situations. For example, in the case of automatic number-plate recognition, one of the most common problems that arise is the low resolution of the image, usually because of the poor quality of the camera or because of the large distance of the number plate from the camera. For such cases, the use of pre-processing techniques to improve results for number-plate recognition from the low resolution images is an area of active research in classical computing. In automatic number plate recognition from low-quality videos, piece-wise gamma correction was suggested to ensure illumination invariance [42]. It was also suggested in Ref. [42] that a scheme for character segmentation based on information about character placement on the license number plate will be different worldwide. As part of pre-processing, we can also use the resolution enhancement technique to get accurate results in cases where we get no results at all, or worse, there is the chance of misidentification of characters. In the following, we propose a hybrid (quantum and classical) algorithm for the identification of distorted alphanumeric characters. The prerequisite for our algorithm is the High-resolution image of all alphanumeric characters for the Font type of interest. The pixel information is encoded in the quantum system and the quantum algorithm is used for finding pixel values that are near to the reference images. The classical OCR is then used to identify the right character by calculating the confidence value obtained from OCR. In particular, the fixed point search algorithm is used for finding the missing pixel values with the help of high-resolution reference images of all characters. So, every low-resolution image will now have the same number of images as the number of reference images with updated pixel values, for which confidence values using OCR are calculated.
III. DISTORTED ALPHANUMERIC CHARACTER IDENTIFICATION
The smallest unit of accessible information in digital objects (an image or an alphanumeric text object) is known as a pixel. The resolution of an image is quantified by the number of pixels per inch (PPI), the greater the PPI the better is the resolution of the image. Every pixel contains information about bit depth, dynamic range, file size, compression, and metadata, which vary with the file format. The bit string corresponding to bit depth is used to store the information about the color and its shades as briefly described below. Usually, the bit depth for the color images/text varies from 8 to 24 bits. Let's start with a grayscale image. For a grayscale image, a single bit is sufficient to encode black and white as 0 and 1, and the scale of grayness is defined by the pixel value (a value between 0 and 1) would signify the particular shade of a grayscale image, in order to be able to represent different shades of a grayscale image multiple bits are required. For example, an 8-bit string can be used to store 2 8 shades of a single color, while for a color image represented under RGB primary color scheme total of 24 bits are required, 8 bits for each color. Thus, a 24-bit string can represent 2 8 3 ∼ 16.7 million colors. Though in this manuscript we just focus on finding missing information in a grayscale image of an alphanumeric character using a quantum bit (qubit) to store pixel value information. The quantum advantage in storing comes from the fact that a quantum bit can be prepared in infinitely many possible probability distributions and hence probability amplitudes can be used to store information about the shades i.e., pixel value. So one qubit is sufficient as a quantum resource that can store the necessary information for our algorithm. Moreover, the use of a qubit may lead to a quantum advantage. Another advantage of our algorithm is it does not require a big dataset for training and is computationally less expensive unlike algorithms based on neural networks, e.g., in a recent work entitled, "A new image enhancement and super-resolution technique for license plate recognition" [43], a deep learning architecture has been used, which requires training on massive datasets and use of multiple loss functions, making the algorithm computationally expensive. So, we are only interested in the relative brightness of the pixel; say, 0 represents a black pixel, and 1 represents white. More specifically, when a digital camera assigns a pixel to an edge, it just takes the average value of all the values that lie within the confines of that pixel, as can be seen in Fig. 1. Here, the lower box will have a value closer to 1, and the upper box will have a lower value. In a grayscale image, the only information we have for that block is the average value. Hence, if there are very few pixels, much information needed by OCRs to identify the characters get lost. Therefore, we must find a way to retrieve that information from its average value only. This is where the classical part of our hybrid algorithm comes in. In the following section, we will first describe the classical part of the proposed hybrid algorithm. Subsequently, we will describe the quantum part of the algorithm in the next to next section.
IV. CLASSICAL PART OF THE ALGORITHM
The critical problem to solve is to find a way to retrieve the missing information needed to convert a lowresolution image to a high-resolution image. This can be made possible for character recognition of printed text, particularly if the typeface of the text is known because the missing information is restricted to a finite set (character set). To begin with, we can assume one by one, for each element of that set that it holds the missing information and try to process our image under that assumption. We can then assess which assumption is correct by assessing which of the processed image is best. Specif-ically, we compare the low-resolution image of the alphanumeric character whose resolution we want to increase with the high-resolution images of all alphanumeric characters of the same font through confidence value obtained by OCR. The alphanumeric character for which the confidence values (which is the measure of how confident are we about the character recognized by the OCR process) of the low-resolution image is closest to the confidence value of the high-resolution image is the identified character. The confidence value is defined differently for different OCRs. To be specific, let us assume that we are dealing with an English alphanumeric character set (0-9, A-Z). Suppose that we get a low-resolution image of a character that our OCR cannot identify. If we assume that the missing information to convert that low-resolution image into a high-resolution image is in the character 0, we will get the processed image such that, if we run it through OCR, the confidence value for 0 will indicate how good our assumption is. We can similarly proceed with all the characters in the character set and choose the one with the highest confidence value.
FIG. 2. (Color online) In this figure, the conventional pre-processing methods include the applicable pre-processing methods mentioned earlier. After segmentation, we can treat each segment as a separate image. This image of a single character can then go through further processing using the information about an assumed character from the character set. After this process, we input this image into the OCR, from which we get the confidence value for the assumed character. We can store this value in a list. Then we repeat the process with the next character from the character set. The character which corresponds to the highest confidence value will be the answer. In this figure, the dotted lines represent the flow of information and the solid lines represent the sequence.
A representation of this process can be seen in Fig. 2. Now, we will discuss the process through which we convert a low-resolution image into a high-resolution image using the information from the assumed character in quantum part of the algorithm.
V. QUANTUM PART OF THE ALGORITHM
A qubit is the smallest unit of quantum information, which can be realized using a quantum system. A general two level quantum state |ψ is a superposition of two eigenstates in the computational basis say, |0 and |1 , where |ψ = a|0 + b|1 . Quantum algorithms for QDIP are composed of quantum operations on quantum systems designed for achieving enhancement in performance of the digital image processing. The benefits of such a quantum algorithm are multitudinous. For example, classically, if we represent a pixel in grayscale, we need 8 bits, assuming that we want 256 shades of gray (including black and white). However, we can represent one pixel in grayscale by using just a single qubit. Additionally, we can assign any value between 0 and 1 to a qubit, making it possible to represent infinite shades of gray. This is not advantageous if we have to access the pixel directly since the probabilistic nature of quantum mechanics or more precisely the collapse on measurement postulate of quantum mechanics implies that we have to prepare multiple qubits and measure them. We will still get a computational advantage. In Fig. 2, it is mentioned that after getting a cleaner low-resolution image, we will process it using the information about the assumed character, which is the quantum part in our hybrid algorithm. In particular, we propose to use amplitude amplification as described in Grover algorithm [44]. Consider that There is an unsorted database containing N items (say, N alphanumeric characters) out of which just one item (character) satisfies a given condition and that need to be retrieved. The most efficient classical algorithm for this is to examine the items one by one requires about N 2 steps. While Grover algorithm takes √ N number of steps. We can use this algorithm for amplitude amplification of arbitrary input and search states, but that will not lead to the desired result because of the reasons described below. Ideally, we would require the difference in amplitudes of the final state and that of the search state of the low-resolution image to be directly proportional to the difference between the amplitudes of the final state and the search state of the high-resolution image. This would mean that if the low−resolution image has a pixel that corresponds to the average value of n pixels of the high-resolution image we can take new n pixels with the value that equals that of the lowresolution image, and subsequently, perform amplitude amplification on each of them with search state being that of the corresponding n pixels of the high−resolution image. However, this is not possible if we use the generalized versions of the operations described in Grover's original paper [44]. We can understand the problem by understanding Grover's search algorithm by visualizing what each operator does. For this, we can have a look at the example shown in Fig. 3. To generalize this, consider the search state as |w = |10 , and consider the initial state as |s = 1 2 (|00 + |01 + |10 + |11 ), where the oracle can be generalized as O = I − 2|w w|, and the diffuser as D = 2|s s|. We can now write |s = 1 2 |w + √ 3 2 (|00 +|01 +|11 ), and the O and D operators are defined such that the result spans the space defined by |w and |s , where |s = 1 √ 3 (|00 + |01 + |11 ), in this particular case. If we define θ 2 = tan -1 s|w s |s , we can see that the entire operation is merely a rotation by an angle θ. The representation of these transformations can be seen in Fig. 4. This, however, is problematic for us as rotation in the Grover algorithm can miss the target state. In our low-resolution image, the value of the pixel can be anything between 0 and 1. If we now consider the case of a single qubit, the initial arbitrary state |ψ can be written as a linear combination of search state |w and its orthonormal state |s . We can observe that for θ ∈ { π 3 , π, 5 π 3 }, we will get the same state as search state |w (ignoring the global phase). It is important to remember that although we are representing the initial states and the states after subsequent operations in the basis of search state |w and its orthonormal state |s , we measure in the computational basis. This means that for states for which θ ∈ { π 3 , π, 5 π 3 }, we can get an equal probability of obtaining |0 state if it is exactly in the middle of the final state and the search state |w . An example of this can be seen in the lower trace of Fig. 4. All of this will result in Fig. 5. This is not suitable for our needs as what we require is a perfect match for the case where the initial state and the target states are the same (|s = |w ), and a monotonic increase in the difference between the probability of measuring the |0 state for the final state and the probability of measuring the |0 state for the target state |w . This is why Grover's fixed-point quantum search algorithm is most suitable for our case [45]. After preparing an equal superposition of the states |00 , |01 , |10 , and |11 , the oracle reflection is applied to the state, resulting in rotation of π of the search state. Finally, we apply the diffuser operation to get the search state. Fig. 3. Here, if we take the angle made by the initial state vector with the x-axis to be θ 2 , the oracle can be thought of as a reflection about the x-axis, and the diffuser can be thought of as reflection about the initial state vector, resulting in the angle of the final state vector with the x-axis being 3 θ 2 . When combined the result is the same as rotation by an angle of θ. Taking the example of a target state such that it makes an angle of 22.5 o with |0 , for the initial state which makes an angle of 45 o with the xaxis, the final state that we will get will not be the same as target state, however, it will also make an angle of 22.5 o with |0 . Therefore, the probability of getting the state |0 on the measurement will be the same for both.
A. Fixed-point quantum search algorithm
The previously discussed algorithm can be generalized further by defining the initial state as |ψ = U |0 , oracle as O φ = I − (1 − e iφ )|t t|, and the diffuser as D ψ = U (I − (1 − e iφ )|0 0|)U † . For ψ = φ = π, we get the original algorithm [45]. Let ||U ts || 2 = | t|s | 2 = 1 − , where is the deviation of the initial state |s from the target state |w . If we now apply the operators D ψ O φ on |ψ , the component of the final state along |s will be e i 2 (ψ−φ) − 4 sin( ψ 2 ) sin φ 2 |U ts | 2 , which in the asymptotic limit |U ts | → 1, is minimized by taking ψ = φ = π 3 [46]. On applying the operators D π 3 O π 3 on the initial state |s , we get the following superposition :
U |s [e i π 3 + ||U ts || 2 (e i π 3 − 1) 2 ] + |t U ts (e i π 3 − 1). (1)
On taking the square of the inner product of the above superposition with the orthonormal to the target state |s , we get the following :
(1 − ||U ts || 2 )||[e i π 3 + ||U ts || 2 (e i π 3 − 1)2]|| 2 = 3 . (2)
We can also use this algorithm recursively by taking the final state after one iteration as the initial state of the next iteration, such that for m th iteration :
U m = U m−1 [I − (1 − e iψ )|0 0|]U † m−1 O φ U m−1 ,(3)
where, U 0 |0 = U |0 = |s . This is the ideal result because if the pixels from the high-resolution image are taken as the target states and the pixels from the low-resolution images as the initial states, then the final image should resemble the highresolution image if the high-resolution image is of the correct character.
FIG. 5. (Color online)
Here, the x-axis is the probability of measuring the |0 state for the initial state |s and the y-axis is the probability of measuring the |0 state for the target state |w . The color map represents the difference between the probability of measuring the |0 state for the final state and the probability of measuring the |0 state for the target state |w . We can see that for states at 45 o line (|s = |w ), we get perfect matches as is expected since these states will have θ = π when represented in |w and |s state. Then, we can see towards the top-left and bottom-right edges of the figure, we have two curves, which also have perfect matches. These two curves represent the states for which θ = π 3 and θ = 5 π 3 , respectively. There are also two curves that connect the curve for which θ = π to the curves for which θ = π 3 and θ = 5 π 3 . For these two curves, θ ∈ ( π 3 , 5 π 3 ). These are the cases for which even though the amplitude for the final state and the target state |w is different, the probability of measuring the |0 state will be the same, since they are at an equal angle from |0 (See lower trace of Fig. 4). We can also see that for the case where |w = |0 or |w = |1 , the difference between the probability of measuring the |0 state for the final state and the probability of measuring the |0 state for the target state |w is 1 for two values. One of the cases is trivial as the initial state |s is orthogonal to the target state |w (top-left corner and the bottom-right corner). However, for second case, the value of θ = 2 π 3 , hence the final state will be orthogonal to the target state |w .
VI. RESULTS AND ANALYSIS
Firstly, a low-resolution image of each alphanumeric character was created along with a database of highresolution images. These high-resolution images had 4 times the number of pixels than the low-resolution images. All the images were in grayscale with 8-bit representing a pixel resulting in a total of 256 shades of gray. Low-resolution images were then upscaled to match the resolution of high-resolution images by repeating each pixel 4 times. Then one by one, each of the high-resolution images was assumed to be the correct one, taking its pixels as the target state and the corresponding pixels of the up-scaled image as the initial state. Then following two types of fixed-point search algorithms are performed on them separately.
• One-qubit operation: Performing one iteration of the fixed-point algorithm on a single qubit representing the individual pixel of the upscaled image. This method is inefficient since, for a lowresolution image with a p number of pixels, we will need to perform 4p number of operations. However, this results in the final image that is closer to the high-resolution image.
• Four-qubit operation : Performing one iteration of the fixed-point algorithm on four qubits representing a 2x2 block of pixels of the upscaled image. This method is more efficient since, for a low-resolution image with a p number of pixels, we will need to perform p number of operations. However, this results in the final image that is closer to the upscaled image.
These operations were performed using CQASM simulator [47], with 256 shots for each measurement. For this particular scenario, where we are increasing the number of pixels in length and width of the image by a factor of 2, we get 4 pixels in the high-resolution image for the corresponding 1 pixel in the low-resolution image. To generalize this, if we increase the number of pixels in the length and width of the image by a factor of n, we will get n 2 times the original number of pixels to preserve the aspect ratio. The 4-qubit operation can be generalized as n 2 -qubit operation. If the low-resolution image has a p-number of pixels, we can also perform p×n 2qubit operation multiple times recursively to get the desired result. However, this is not possible in the noisy intermediate-scale quantum (NISQ) era of quantum computing. This is why two types of operations, namely, the one-qubit and the n 2 -qubit operations are used to represent the two different numbers of recursions. For the one-qubit operation, the number of times the fixed-point search will be used is p × n 2 , whereas, for the n 2 -qubit operation, it will be used p-times. After storing the final image for each corresponding high-resolution image, they were input into the MATLAB OCR to retrieve the confidence value for the corresponding character. The character which yields the highest confidence value is assumed to be the correct character. For a classical computer, let m be the number of bits being used to represent a pixel. If we want to scale up the low-resolution image by a factor of n 2 and the n 2 -qubit operation can be used once or it can be used recursively then we can compare the time complexity required to apply this transformation once. To apply the same transformation, classical computer will take O(m(2n 2 ) 3 (n 2 2n 2 + log(m))) [48]. For the proposed classical-quantum hybrid algorithm, we will need to input the unitary matrices for fixed-point Search. This will require O(2 m ) time since multiple measurements will need to be made to assign the pixel value in the processed image. In most cases, m = 8, so the proposed algorithm is mostly faster than the classical counterpart, particularly for large values of n. This algorithm will be most efficient if the fixed-point algorithm is performed recursively on the entire image. For a low-resolution image with p number of pixels, the complexity for one recursion will be O(p2 m ) [49] for the quantum computer. For classical computation, the complexity will be O(m(2 pn 2 )) 3 (pn 2 2 pn 2 + log(m)). This means that with each recursion, the quantum algorithm will be much faster. Fig. 5, here too, the x-axis is the probability of measuring the |0 state for the initial state |s and the y-axis is the probability of measuring the |0 state for the target state |w . The color map represents the difference between the probability of measuring the |0 state for the final state and the probability of measuring the |0 state for the target state |w . We can see that as we move away from the 45 o line, the difference between the final state and the target state |t increases monotonically.
In Table I, three characters ('0', '7', 'O') shown by X in the second column. These are the characters which are misidentified by OCR. Interestingly these characters can be identified after processing using our algorithm, both by the 1-qubit algorithm and 4-qubit algorithm. Our 1qubit algorithm misidentifies character 8 but the 4-qubit algorithm correctly identifies the character. Interestingly, for character 6, algorithms (both 1-qubit and 4-qubit) are TABLE I. The table shows the confidence value for all alphanumeric characters, for the low-resolution image, and after the application of the hybrid algorithm. The first column shows all alphanumeric characters, the second column shows the confidence value of the low-resolution images, and the third column shows the confidence value after the application of our 1-qubit hybrid algorithm. The fourth column shows the confidence value after the application of our 4-qubit hybrid algorithm found to misidentify the character while OCR correctly identifies '6' in the low-resolution image. For 6, the confidence value of E in the high-resolution image gives a very high confidence value. This bias for character 'E' for returning high confidence value is evident in Table II and in III. The Table III shows that the average confidence value of '0' when high-resolution image of '0' was taken as the target image for rest of the characters is considerably lower than that of 'E'. This can either be due to a biased OCR, or it can be that the shape of 'E' matches closely with the shape of characters in rest of the alphabet. The latter theory can be tested if we consider the confidence value of the rest of the characters when their high-resolution image was taken as the target state and low-resolution image of '0' and 'E' were processed. If the shape of the letter 'E' really matches closely with the shape of the rest of the characters, the OCR should return a high confidence value on average. However, it can be seen from Table IV that the average confidence value for '0' is higher. Therefore, it seems that this algorithm's results can be improved by training and using an OCR that is specially trained on images of printed individual characters. It is also important to note that the typefaces used for printing number plates is usually known. This means we can further train our OCR for that particular typeface which should provide additional improvements. This is discussed more elaborately in the last paragraph of VII. However, there is an easier way to find out the results by this algorithm in an ideal case. It can be done by using the high-resolution image themselves as reference images. By doing this, we can create a very simple model for pixel-by-pixel image matching. Since the images are in grayscale, we can calculate the following:
f (x, y) = 1 − σ i |x i − y i | 2 N(4)
Here, x and y represent the two images with N -number of pixels each and x i and y i represent the value of the i th pixel of respective images. The calculated value f (x, y) will be referred to as the 'Match Percent' between the two images. It is important to note that this is a very simplistic model which can only measure the relative similarity between two corresponding pixels of an image. It can not replace OCR completely because there may be noise, which can lower the match percent for the correct character or there may be a case of non-feature matching, which may increase the match percent of incorrect characters. Character' column represents the character in the high-resolution image taken as the target image and the '0' and 'E' columns represent the confidence value of the corresponding characters when a low-resolution image of '0' and 'E' is processed respectively.
VII. CONCLUSION
We have devised a hybrid algorithm for the identification of the missing information in a low-resolution image of the alphanumeric character. Our algorithm could identify the missing information in the low-resolution image using the high-resolution images of the alphanumeric character set. The confidence value for all known low-resolution alphanumeric characters is shown in the second column of TABLE IV. 1-qubit operation: The 'Character' column represents the character in the low-resolution image. The 'Match Percent' column gives the calculated value of the match percent for the processed image when the high-resolution image (target image) contains the same character as in the low-resolution image. The 'Mean' column gives the mean match percent for all the characters in the alphabet. The 'Mean Deviation' columns give the mean of the magnitude of absolute deviation from the mean. phanumeric characters. It is clearly visible from the confidence values shown in the third column that the algorithm is able to identify the known low-resolution image of the alphanumeric character correctly. The confidence value for the 4-qubit version is shown in the last column of Table I. One can clearly see the increased confidence value after the application of the 4-qubit hybrid algorithm. Though this scheme is restricted to typeset characters. It should also be possible to identify the missing information in handwritten characters by combining the deep learning algorithms with our algorithm.
Though this work has mostly studied the case of English alphanumeric character set for a particular typeface, it can be extended to other natural languages and to dif-ferent typefaces. However, it is important to note that natural languages have not evolved to take into account how well an OCR can be trained to identify its characters. This poses a very interesting question about the fundamental lower limit of visual information needed to tell each character in alphabet apart. Ultimately, an image is just information that needs to be processed and the amount of information depends on the resolution of the image. There must be a limit to the number of pixels, below which information can't be sufficient as to identify the correct character. This limit will be different for different natural languages and for different typefaces. This algorithm can only work in a case when there is sufficient information, so it will be important to know where the limit is, beyond which we can't trust this algorithm. Thus, we can also find the best typeface to use to print something in a practical situation where there is a possibility of involving OCR algorithm to identify a lowresolution character. Another thing that is important is that the OCR we are relying on to give the confidence value must have few properties :
• The OCR must be trained on the data-set of printed text, particularly on the typeface used to print such texts.
• It should not use Language Techniques, Dictionary Look-ups, or any other contextual editing or correction technique. For special cases, it may use these, but not when used for something like automatic number-plate recognition.
• It should perform pre-processing techniques before resolution enhancement, not after. To train such an OCR would require a lot of resources like huge data-sets of text printed in same typeface and a suitable training algorithm.
In this algorithm, the number of shots chosen (number of times measurement is done) has been taken as simply 2m, this is because it is the least number of measurements required to get the value required to be assigned by m number of classical bits. Although, this has produced desirable results, because of the probabilistic nature of measurement, it may not be enough to ensure accuracy for larger values of scale up factor n. This value will also depend on decoherence or another form of noise. Therefore, some form of quantum error correction may also be needed. Lastly, the ideal number of recursions required will need evaluation. The fixed-point algorithm will almost always converge to the target state with each recursion. This means that if an excessive number of recursions are used, only the pixels which are either completely black or completely white in the low-resolution image will remain different. Other than that pixel, the processed image will be the same as the high-resolution image taken from the database.
FIG. 1 .
1(Color online) Representation of assignment of pixel values when a digital camera captures an image.
FIG. 3 .
3(Color online) This is an example of Grover's search algorithm for a two-qubit system. Here, the search state is |10 .
FIG. 4 .
4(Color online) These three sub-figures represent the same state in the corresponding sub-figures of
FIG. 6 .
6(Color online) Similar to
A
Pal and A Pathak acknowledge support from the QUEST scheme of the Interdisciplinary Cyber-Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme-1/2019/14 (Q80). A Pal also thanks, Indian Institute of Science Education and Research, Kolkata for the support. A Shukla thanks to Interuniversity microelectronics centre, Belgium and University of Hasselt, Diepenbeek, Belgium for the financial support and computational facilities. All authors thank Kishore Thapliyal for his interest and useful technical feedback on this work. A Pal thanks Chiranjib Mitra for his interest in this work.
The fourth column shows the confidence value after the application of our 4-qubit hybrid algorithmTABLE III. The 'Character Confidence Value (0) Confidence Value (E)
0
0.92174
0.859433
1
0.029358
0.764754
2
0.029358
0.800612
3
0.030201
0.789907
4
0.719694
0.715601
5
0.075472
0.872448
6
0.069231
0.918335
7
0.058823
0.826405
8
0.84667
0.896808
9
0.056582
0.809741
A
0.039724
0.80018
B
0.072253
0.800814
C
0.110769
0.939481
D
0.792613
0.770146
E
0.9351
0.924665
F
0.9497
0.938809
G
0.726062
0.941417
H
0.035011
0.954693
I
0.039724
0.921477
J
0.039724
0.771872
K
0.048764
0.79392
L
0
0.953842
M
0.047059
0.806853
N
0.058823
0.811308
O
0.716013
0.811292
P
0.082569
0.864928
Q
0.62479
0.716127
R
0.056582
0.809852
S
0.069231
0.810731
T
0.026311
0.772327
U
0.047864
0.928568
V
0.722633
0.746693
W
0.64834
0.660381
X
0
0.752046
Y
0.037257
0.774703
Z
0.03121
0.881578
Mean
0.224609
0.8276320
TABLE II. The table shows the confidence value for all alphanumeric characters, for the low-resolution image, and after the
application of the hybrid algorithm. The first column shows all alphanumeric characters, the second column shows the confidence
value of the low-resolution images, and the third column shows the confidence value after the application of our 1-qubit hybrid
algorithm. Character Confidence Value (0) Confidence Value (E)
0
0.921740
0.075472
1
0.0487640
0.063717
2
0.752476
0
3
0.768959
0.660494
4
0.583875
0
5
0.817715
0.774370
6
0.842950
0.829795
7
0.066390
0
8
0.750651
0
9
0.812266
0.066390
A
0
0
B
0.863754
0.881197
C
0.032000
0
D
0.814780
0.105186
E
0.859433
0.924665
F
0.052499
0.100000
G
0.707981
0.664239
H
0.760396
0.790886
I
0.802360
0.042440
J
0.123288
0.056582
K
0.714194
0.739672
L
0.850833
0.0722533
M
0.712009
0.25623
N
0.647434
0
O
0.706801
0.024324
P
0.765699
0.706166
Q
0.675175
0.591882
R
0.775897
0.68769
S
0.755298
0.726142
T
0.116788
0.024961
U
0.770081
0
V
0.738190
0.598832
W
0.622550
0.540786
X
0.650041
0
Y
0.032000
0.030201
Z
0.788975
0
Mean
0.602896
0.306516
Table I .
IIn the third column, we chose the maximum confidence value out of all the values corresponding to high-resolution images, for all known al-Character Match Percent
Mean
Mean Deviation
0
0.9848823529 0.924672658 0.02420893851
1
0.9968333333 0.8929891068 0.02881094166
2
0.992127451 0.9070144336 0.02710763132
3
0.9916666667 0.9080716231 0.03007439179
4
0.9917647059 0.8939926471 0.02426116558
5
0.9910588235 0.9190220588 0.02455718954
6
0.9880098039 0.9202140523 0.02480591866
7
0.9912254902 0.9054283769 0.02753622004
8
0.9848627451 0.9244586057 0.02539584846
9
0.9859901961 0.9152706972 0.02427874607
A
0.9524411765 0.8895054466 0.02356463326
B
0.9922941176 0.9123374183 0.02813784495
C
0.9896764706 0.9185114379 0.02348202614
D
0.9918431373 0.8994360022 0.0297573832
E
0.9954901961 0.9153924292 0.02665601549
F
0.9953333333 0.9167393791 0.02400290487
G
0.9881960784 0.901665305 0.02562212539
H
0.9873137255 0.9031555011 0.02600571895
I
0.9971470588 0.9090705338 0.02148872852
J
0.9965
0.8803474946 0.03281148632
K
0.9907058824 0.906995915 0.01916344408
L
0.9973235294 0.9120277778 0.02217102397
M
0.9815
0.8892592593 0.02196514161
N
0.9786666667 0.9055234205 0.02486504478
O
0.9865
0.9012184096 0.02780694747
P
0.9924803922 0.9121176471 0.0215664488
Q
0.9828431373 0.8884471678 0.02911374365
R
0.9899313725 0.9105966776 0.02211729908
S
0.9872941176 0.9196775599 0.02380440571
T
0.9967745098 0.8862086057 0.03107764464
U
0.9417058824 0.907370915 0.02780455701
V
0.988
0.9003551198 0.01648517308
W
0.9768627451 0.876333878 0.01377795933
X
0.9875784314 0.8996160131 0.02496205519
Y
0.9922058824 0.898956427 0.0262456427
Z
0.9909705882 0.9096955338 0.02583717623
Mean 0.9873888889 0.9050470982 0.02503693239
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Currently, the best-known matrix multiplication algorithm has been shown to have the complexity O(n 2.3728596 ) [51] for matrix multiplication of two n × n matrices. Currently, the best-known matrix multipli- cation algorithm has been shown to have the complexity O(n 2.3728596 ) [51] for matrix multiplication of two n × n matrices.
Here, the additional p has been added to preserve the calculated mean of "Match Percent. in ResultsHere, the additional p has been added to preserve the cal- culated mean of "Match Percent", in Results.
A refined laser method and faster matrix multiplication. J Alman, V V Williams, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)SIAM, 2021J. Alman and V. V. Williams, A refined laser method and faster matrix multiplication, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) (SIAM, 2021) pp. 522-539.
Gaussian elimination is not optimal. V Strassen, Numer. Math. 13354V. Strassen et al., Gaussian elimination is not optimal, Nu- mer. Math. 13, 354 (1969).
| {'fraction_non_alphanumeric': 0.04622024672271073, 'fraction_numerical': 0.05475569298160853, 'mean_word_length': 4.194930143060319, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 1, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "An algorithm for image processing is proposed. The proposed algorithm, which can be viewed as a quantumclassical hybrid algorithm, can transform a low-resolution bitonal image of a character from the set of alphanumeric characters (A-Z, 0-9) into a high-resolution image. The quantum part of the proposed algorithm fruitfully utilizes a variant of Grover's search algorithm, known as the fixed point search algorithm. Further, the quantum part of the algorithm is simulated using CQASM and the advantage of the algorithm is established through the complexity analysis. Additional analysis has also revealed that this scheme for optical character recognition (OCR) leads to high confidence value and generally works in a more efficient manner compared to the existing classical, quantum, and hybrid algorithms for a similar task.", 'arxivid': '2212.12861', 'author': ['Ankur Pal \nIndian Institute of Science Education and Research Kolkata\n741246MohanpurNadia, West BengalIndia\n', 'Abhishek Shukla \nIMOMEC division\nIMEC\nWetenschapspark 1B-3590DiepenbeekBelgium\n\nInstitute for Materials Research (IMO)\nHasselt University\nWetenschapspark 1B-3590DiepenbeekBelgium\n', 'Anirban Pathak \nDepartment of Physics\nMaterials Science & Engineering Jaypee Institute of Information Technology\nUttar Pradesh\nSector-62A-10, 201309NoidaIndia\n', 'Ankur Pal \nIndian Institute of Science Education and Research Kolkata\n741246MohanpurNadia, West BengalIndia\n', 'Abhishek Shukla \nIMOMEC division\nIMEC\nWetenschapspark 1B-3590DiepenbeekBelgium\n\nInstitute for Materials Research (IMO)\nHasselt University\nWetenschapspark 1B-3590DiepenbeekBelgium\n', 'Anirban Pathak \nDepartment of Physics\nMaterials Science & Engineering Jaypee Institute of Information Technology\nUttar Pradesh\nSector-62A-10, 201309NoidaIndia\n'], 'authoraffiliation': ['Indian Institute of Science Education and Research Kolkata\n741246MohanpurNadia, West BengalIndia', 'IMOMEC division\nIMEC\nWetenschapspark 1B-3590DiepenbeekBelgium', 'Institute for Materials Research (IMO)\nHasselt University\nWetenschapspark 1B-3590DiepenbeekBelgium', 'Department of Physics\nMaterials Science & Engineering Jaypee Institute of Information Technology\nUttar Pradesh\nSector-62A-10, 201309NoidaIndia', 'Indian Institute of Science Education and Research Kolkata\n741246MohanpurNadia, West BengalIndia', 'IMOMEC division\nIMEC\nWetenschapspark 1B-3590DiepenbeekBelgium', 'Institute for Materials Research (IMO)\nHasselt University\nWetenschapspark 1B-3590DiepenbeekBelgium', 'Department of Physics\nMaterials Science & Engineering Jaypee Institute of Information Technology\nUttar Pradesh\nSector-62A-10, 201309NoidaIndia'], 'corpusid': 255124958, 'doi': '10.48550/arxiv.2212.12861', 'github_urls': [], 'n_tokens_mistral': 18557, 'n_tokens_neox': 15768, 'n_words': 9964, 'pdfsha': 'aa7bf42122b5c443ebabbc64e7d5364e336d4241', 'pdfurls': ['https://export.arxiv.org/pdf/2212.12861v1.pdf'], 'title': ['An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification', 'An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification', 'An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification', 'An efficient quantum-classical hybrid algorithm for distorted alphanumeric character identification'], 'venue': []} |
arxiv |
ON THE POWER OF CHOICE FOR BOOLEAN FUNCTIONS *
27 Sep 2021
Nicolas Fraiman
Lyuben Lichev
Dieter Mitsche
ON THE POWER OF CHOICE FOR BOOLEAN FUNCTIONS *
27 Sep 2021Boolean functionpower of choicethresholdhitting probabilityrelevant variableAchlioptas processrandomized algorithm AMS subject classifications 94D1006E3068W2060G9968Q8768R01
In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices r ∈ N and a sequence of increasing functions (fn) n≥1 such that, for every n ≥ 1, fn : {0, 1} n → {0, 1}. Given n bits which are all initially equal to 0, at each step r 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions under which an acceleration by a factor of r(1 + o(1)) is possible, and underline the wide applicability of our results by giving examples from the fields of Boolean functions and graph theory. ).1 At the last r − 1 steps, all 0-bits are proposed.
1. Introduction. The "power of two choices" was introduced by Azar, Broder, Karlin and Upfal [2] in the context of load balancing. They showed that, when randomly allocating n balls into n bins, a dramatic decrease in the maximum load is achieved by sequentially selecting the less full bin among two random options. Many variations on this basic model have been analyzed. Berenbrink, Czumaj, Steger and Vöcking [4] studied the case when a much larger number of balls is placed. Kenthapadi and Panigrahy [11] restricted the options by placing balls in an endpoint of a random edge from a graph. More recently, Redlich [16] studied the case where you want to "unbalance" and select the fullest bin.
A classical and well-studied setting is the Erdős-Rényi graph process where the edges of the complete graph K n arrive one by one according to a uniform random permutation. The power of choice in this context was introduced by Achlioptas: he was interested in the question of delaying certain monotone graph properties with respect to the original process if at each step, r ≥ 2 edges instead of one are proposed and an agent may choose the one they need more for their purposes (we call this variation the r-choice process) 1 . In two related papers Bohman and Frieze [6] and Spencer and Wormald [20] studied the problem of delaying the appearance of a giant component by the r-choice process. Krivelevich, Loh and Sudakov [12] studied rules to avoid small subgraphs. Achlioptas, D'Souza and Spencer [1] claimed that certain rules could make the giant transition discontinuous but Riordan and Wernke [17] proved that was not the case. A more restrictive version where the agent's decisions cannot depend on the previous history and only one vertex from the random edges is revealed was studied by Beveridge, Bohman, Frieze, and Pikhurko [5]. A similar restrictive model is the so called semi-random graph process, where one vertex is chosen randomly and the agent can choose the second vertex arbitrarily, see the paper of Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojaković [3]. When the goal is to expedite rather than delay certain properties, Krivelevich and Spöhel [14] proved general upper and lower bounds on the threshold to create a copy of some fixed graph H in the r-choice process. Recently, the question of acceleration of the appearance of a Hamilton Cycle or a Perfect Matching was treated by Krivelevich, Lubetzky and Sudakov [13] who proved that there exist strategies that accelerate both properties by a factor of r+o (1). Furthermore, outside of the graph setup, Sinclair and Vilenchik [19] turned particular attention to delaying the satisfiability of the random 2-SAT formula, and Perkins [15] considered a k-SAT version of the problem.
In their seminal work, Erdős and Rényi [9] showed that many interesting graph properties exhibit sharp thresholds, that is, the probability that a random graph with n vertices and m edges has the property increases from values very close to 0 to values close to 1 in a very small interval around a certain critical value of the number of edges m (often called a critical window). Later, Bollobás and Thomason [8] proved the existence of threshold functions for all monotone graph properties. A more careful analysis of the size of the critical window was performed by Friedgut and Kalai [10].
Their arguments generalize in a straightforward way to thresholds of monotone Boolean functions. More precisely, for any n ≥ 1, consider the hypercube {0, 1} n with the probability measure µ p (x 1 , . . . , x n ) = p k (1 − p) n−k where k = x 1 + · · · + x n . Let (A n ) n≥1 be a sequence of monotone sets such that, for every n ≥ 1, A n ⊆ {0, 1} n and A n is invariant under a transitive permutation group of {1, 2, . . . , n}. If µ p (A n ) > ǫ, then Bollobás and Thomason [8] showed that there is c(ε) > 0 such that µ q (A n ) > 1−ǫ for q = c(ǫ)p. This result was improved by Friedgut and Kalai [10] to µ q (A n ) > 1 − ǫ for q = p + c log(1/2ǫ)/ log n, where c is an absolute constant. We say that a function is a sharp threshold function for the sequence of monotone subsets (A n ) n≥1 if, for every ε > 0, the probability p n such that µ pn (A n ) = ε and the probability q n such that µ qn (A n ) = 1 − ε satisfy p n = (1 + o(1))q n . Then, the threshold function is given only up to a (1 + o(1)) factor by both (p n ) n≥1 and (q n ) n≥1 for any fixed ε > 0. Equivalently, the hitting time of the event A n by the process that turns from 0 to 1 the n given bits one by one in an order, chosen uniformly at random, is of order (1 + o(1))p n asymptotically almost surely. Sharp thresholds appear in various systems in combinatorics, computer science and statistical physics (where they are more widely known as phase transitions).
Motivated by all these questions, we embark in the study of the power of choice for Boolean functions. Our goal is to characterize Boolean functions whose thresholds can be maximally accelerated. More precisely, we study the r-choice process for Boolean functions where at each step an agent is presented with r zero coordinates and selects one to flip (here and below, r ≥ 1 is a fixed positive integer). Our objective is to understand which monotone Boolean functions can be accelerated by a factor of r by the r-choice process (as we shall see in a bit, the factor r is optimal). For that purpose, we compare the hitting probabilities for the function to reach the value 1 under two increasing random walks on the hypercube.
The paper is organized as follows. In Section 2 we introduce the model of interest, state the assumptions and present the main results of the paper, which are then proved in Section 3. Section 4 contains concrete applications of our results.
2. Statements of results. We use the following standard asymptotic notation: for two sequences of functions (a n ) n≥1 and (b n ) n≥1 we say that a n = O(b n ) if there exists C > 0 and n 0 ∈ N such that, for all n ≥ n 0 , |a n | ≤ C|b n |; a n = Ω(b n ) if b n = O(A n ); a n = Θ(b n ) if a n = O(b n ) and b n = Ω(a n ); a n = o(b n ) or equivalently a n ≪ b n if lim n→∞ |an| |bn| = 0; and a n = ω(b n ) if b n = o(a n ). In case the limit is taken with respect to a different variable k, we use the notation o k (b n ), Ω k (b n ), etc. to point this out. We also say that a sequence of events (E n ) n≥1 holds a.a.s. (or asymptotically almost surely), if lim n→∞ P(E n ) = 1.
Fix any n ∈ N. A Boolean function f maps elements from the hypercube {0, 1} n to {0, 1}. We denote the vectors in {0, 1} n by lower case letters in bold such as u, v, w, etc. For a vector x, we denote by |x| the number of coordinates of x. We denote by 0 the all zeroes vector and by 1 the all ones vector.
We will see the hypercube as a partially ordered set equipped with the order relation ≤ defined by x ≤ y if x i ≤ y i for every i ∈ [n]. At the same time, construct an oriented edge between every pair of vectors x, y ∈ {0, 1} n such that x ≤ y, and x and y differ in exactly one coordinate -this allows us in turn to see the hypercube as a directed graph.
A Boolean function is monotone if x ≤ y implies f (x) ≤ f (y).
f : {0, 1} n → {0, 1} and R(f ) = {i 1 , . . . , i m }, thenf : {0, 1} m → {0, 1} is defined as f (x) = f (y) where y ij = x j for j = 1, . . . , m, and for every i ∈ [n] \ {i 1 , . . . , i m }, y i is an arbitrary bit.
We will be interested in two random walks on the (directed) hypercube {0, 1} n . The simple random walk (X t ) n t=0 starts at X 0 = 0 and evolves by choosing a directed edge uniformly at random and moves in its direction at each step. In the r-choice walk (Y t ) n t=0 starting from Y 0 = 0, an agent is presented with r zero bits chosen uniformly at random, selects one of them and moves in its direction (in the end when there are fewer than r possible edges, we assume that all zero bits are proposed). Formally, for every integer t ∈ [0, n − r], let Z t be the set of zero coordinates in Y t , and let C t be the random subset of Z t of size r, presented to the agent at step t. Then, the agent selects c t ∈ C t according to some policy and updates the set of zero coordinates Z t+1 = Z t \ {c t }. Given a monotone Boolean function f , we will study the hitting times of the preimage f −1 (1) ⊂ {0, 1} n by the two processes (X t ) n t=0 and (Y t ) n t=0 (at this moment we say that the function f is activated ). Definition 2.3. The solo and the r-choice thresholds are given by
T 1 (f ) = min t : P(f (X t ) = 1) ≥ 1/2 , T r (f ) = min t : P(f (Y t ) = 1) ≥ 1/2 .
In this paper, when we are talking about a sequence of Boolean functions (f n ) n≥1 , we will always assume that f n : {0, 1} n → {0, 1} is monotone unless explicitly mentioned otherwise. The main question we consider is if one may asymptotically accelerate by a factor r the threshold values for the r-choice process (unless explicitly stated otherwise, all asymptotics refer to the regime n → +∞).
Definition 2.4. A sequence of functions (f n ) n≥1 is fast if T r (f n ) = (1 + o(1)) T 1 (f n ) r .
A sequence is slow if it is not fast.
Notice that the constant r is best possible: indeed, define the r-complete process to be the process, in which one changes all r uniformly chosen remaining zeros to 1 at the same time. This process performs only a 1/r-fraction of the time steps of the single choice process, and is at least as fast as the r-choice process.
We need one more definition that allows us to formalize the concept that relevant sets of variables might change over the process. For every n ≥ 1, the sequence of functions (f s n ) s≥0 is defined conditionally on the sequence of updated bits (b s ) s≥1 as follows. Order the first s bits in increasing order
b i1 < · · · < b is . For every integer s ∈ [0, n] and a vector v ∈ {0, 1} n−s , define v ↑ s = (v 1 , . . . , v bi 1 −1 , 1, v bi 1 , . . . , v bi 2 −2 , 1, v bi 2 −1 , . . . , v bi s −s , 1, v bi s −s+1 , . . . , v n−s ). Define f s n : v ∈ {0, 1} n−s → f n (v ↑ s ) ∈ {0, 1}. In particular, f 0 n = f n . Observe that (|R(f s n )|) n s=0
is a non-increasing sequence since for any fixed integer s ∈ [0, n − 1], if a position i is not in the set R(f s n ), then it remains outside the set R(f s+1 n ) as well. We now present our main results. Throughout we fix an integer r ≥ 2. We first state two sufficient conditions for a sequence (f n ) n≥1 to be slow.
Theorem 2.5. If there is ε > 0 such that T 1 (f n ) ≥ εn for every n ≥ 1. Then, there exists a constant C = C(r) > 0 such that, for every n ≥ 1, T r (f n ) ≥ Cn + T 1 (f n )/r. Corollary 2.6. If |R(f n )| = ω(1), and there is δ > 0 such that T 1 (f n ) ≥ δ|R(f n )| for every n ≥ 1. Then, there exists a constant C = C(r) > 0 such that, for every n ≥ 1, T r (f n ) ≥ Cn + T 1 (f n )/r.
Now, we state two sufficient conditions for a sequence (f n ) n≥1 to be fast.
Theorem 2.7. If 1 ≪ T 1 (f n ) ≪ |R(f n )| ≪ n, then T r (f n ) = (1 + o(1))T 1 (f n )/r.
Corollary 2.8. Suppose that a.a.s. for every ε > 0 there is s = s(n) such that:
1. s ≤ εT 1 (f n ), 2. |R(f s n )| ≤ εn, 3. 1 ε ≤ T 1 (f s n ) ≤ ε|R(f s n )|. Then, T r (f n ) = (1 + o(1))T 1 (f n )/r.
In the following sections, we often omit upper and lower integer parts when rounding does not matter in the corresponding computation.
3. Proofs of the main results. We split this section into two parts with the results characterizing slow and fast sequences respectively.
Slow sequences.
We present the proofs of Theorem 2.5 and Corollary 2.6.
Lemma 3.1. Fix any ε ∈ (0, 1) and c ∈ (0, (1 − ε) r ). Then, in εn steps of the r-choice process there are at least cn elements that have been proposed at least twice a.a.s.
Proof. The probability that a given element i ∈ [n] has never been proposed by the r-choice process up to step εn is given by
0≤i≤εn−1 0≤j≤r−1 1 − 1 n − i − j = (1 + o(1)) 0≤i≤εn−1 1 − r n − i = (1 + o(1)) exp −r log n n − εn = (1 + o(1)) (1 − ε) r .
Also, the probability that two different elements have both not been proposed after εn steps is
0≤i≤εn−1 0≤j≤r−1 1 − 2 n − i − j =(1 + o(1)) 0≤i≤εn−1 1 − 2r n − i =(1 + o(1)) exp −2r log n n − εn =(1 + o(1)) (1 − ε) 2r =((1 + o(1)) (1 − ε) r ) 2 .
We conclude by a direct application of the second moment method that the number of vertices not yet proposed during any of the first εn steps, is a.a.s. at least cn, which proves the proposition.
Proof of Theorem 2.5. We argue by contradiction. In this case there is an increasing sequence (n k ) k≥1 such that rT r (f n k ) = (1 + o k (1))T 1 (f n k ). Since T r (f n ) ≥ T 1 (f n )/r ≥ εn/r for every n ≥ 1, by Lemma 3.1 there is c > 0 such that a.a.s. at least cn elements have been proposed at least twice by the r-choice process until step T r (f n ). Hence, for every n ≥ 1, the number of all elements that have been proposed at least once up to time T r (f n ) in the r-choice process is a.a.s. at most rT r (f n ) − cn. Thus, for all k ≥ 1, the number of bits proposed by the r-choice process up to step T r (f n k ) (out of all n k bits) is at most T 1 (f n k ) − cn k + o k (n k ) a.a.s. and, conditionally on their number, these are chosen uniformly at random. Therefore, the probability that f n k is activated by the above set of elements is less than 1/2 for every large enough k, which is in contradiction with our assumption.
Proof of Corollary 2.6. Fix n ≥ 1. If |R(f n )| ≥ n/8, then T 1 (f n ) ≥ T 1 (f n ) ≥ δ|R(f n )| ≥ δn/8, and the lemma follows in this case. Suppose that |R(f n )| < n/8. We prove that during the first δn/8 steps of the 1-choice process, at most δ|R(f n )|/2 elements from R(f n ) have been selected a.a.s. Indeed, for every positive integer t ≤ δn/8, the 1-choice process selects an element from R(f n ) with probability at most
|R(f n )| n − t + 1 ≤ |R(f n )| (1 − δ/8) n ≤ 2|R(f n )| n .
Since any step is made independently of all previous steps conditionally on the set of already selected bits, the number of elements in R(f n ) selected after the first δn/8 steps is stochastically dominated by a binomial random variable Bin(δn/8, 2|R(f n )|/n). Thus, since |R(f n )| = ω(1), by Chernoff's bound a.a.s. there are at most δ|R(f n )|/2 elements of R(f n ) selected after the first δn/8 steps. Hence, since by assumption T 1 (f n ) ≥ δ|R(f n )|,f n (and therefore f n as well) is activated with probability less than 1/2 after the first δn/8 steps, which proves the hypothesis of Theorem 2.5, and the corollary follows.
Fast sequences.
We present the proofs of Theorem 2.7 and Corollary 2.8.
Lemma 3.2. Fix an integer r ≥ 1 and a sequence of monotone Boolean functions (f n ) n≥1 satisfying 1 ≪ T 1 (f n ) ≪ |R(f n )| ≪ n. Then, for every δ > 0,
(1 − δ)T 1 (f n )n r|R(f n )| ≤ T r (f n ) ≤ (1 + δ)T 1 (f n )n r|R(f n )| .
Proof. First, we prove the lower bound. Define
k − = k − (n) = (1 − δ)T 1 (f n )n r|R(f n )| ,
and let (Z i ) i≥1 be an infinite sequence of independent Bernoulli random variables with parameter p x = |R(fn)| n−rk − . Let A r (t, f n ) be the number of activated bits in R(f n ) after t steps of the r-complete process and A 1 (t, f n ) be the same quantity for the 1-choice process. We have
P A r (k − , f n ) ≥ 1 − δ 2 T 1 (f n ) ≤ P A 1 (rk − , f n ) ≥ 1 − δ 2 T 1 (f n ) ≤ P rk − i=1 Z i ≥ 1 − δ 2 T 1 (f n ) = exp(−Ω δ (T 1 (f n )) = o(1),
where the penultimate equality follows from Chernoff's bound and uses that rk − p x = (1 − δ + o(1))T 1 (f n ), which follows from our assumption that T 1 (f n ) = o(|R(f n )|). Thus, a.a.s. there are at most (1 − δ/2)T 1 (f n ) bits in R(f n ) selected during the first rk − steps by the 1-choice process, so
rk − ≤ T 1 (f n ) ≤ rT r (f n ),
which proves the lower bound.
For the upper bound, define
k + = k + (n) = (1 + δ)T 1 (f n )n r|R(f n )| .
We prove that, out of the first k + steps, there are a.a.s. at least (1 + δ/2)T 1 (f n ) steps such that at least one element in R(f n ) is proposed. Denote by T the hitting time of the above event. Also, recall that C 1 , C 2 , . . . , C k + are the sets of size r of elements, proposed during the first k + steps of the r-choice process. Now, let (Y i ) i≥1 be an infinite sequence of Bernoulli random variables with parameter
p y = 1 − 1 − |R(f n )| − (1 + δ)T 1 (f n ) n r .
Note that p y bounds from below the probability that C t contains an element of R(f n ) for every t ≤ T , and since |R(f n )| = o(n), p y = (1 + o(1)) r|R(fn)|−(1+δ)T1(fn) n . Thus,
P T ≥ k + ≤ P |{t ≤ k + : C t ∩ R(f n ) = ∅}| < (1 + δ/2)T 1 (f n ) ≤ P k + i=1 Y i < (1 + δ/2)T 1 (f n ) = exp(−Ω δ (T 1 (f n ))) = o(1),
where the penultimate equality follows from Chernoff's bound and uses that k + p y = (1 + δ + o(1))T 1 (f n ), which follows from our assumption that T 1 (f n ) = o(|R(f n )|). We conclude that after k + steps in the r-choice process, at least (1 + δ/2)T 1 (f n ) elements of R(f n ) have been selected. Moreover, if at every step t ≤ k + we impose on the agent to select an element from C t ∩R(f n ) uniformly at random if |C t ∩R(f n )| ≥ 2, the set of selected elements in R(f n ) after k + steps is uniform conditionally on its size. This proves the upper bound.
Proof of Theorem 2.7. By Lemma 3.2 applied with r = 1 we have that for every δ > 0,
(1 − δ)T 1 (f n )n |R(f n )| ≤ T 1 (f n ) ≤ (1 + δ)T 1 (f n )n |R(f n )|
and for every δ > 0 and r ≥ 2, once again by Lemma 3.2,
(1 − δ)T 1 (f n )n r|R(f n )| ≤ T r (f n ) ≤ (1 + δ)T 1 (f n )n r|R(f n )| .
We deduce that for every δ > 0 and r ≥ 2,
T 1 (f n ) r ≤ (1 + δ)T 1 (f n )n r|R(f n )| = 1 + δ 1 − δ (1 − δ)T 1 (f n )n r|R(f n )| ≤ 1 + δ 1 − δ T r (f n )
and
T r (f n ) ≤ (1 + δ)T 1 (f n )n r|R(f n )| = 1 + δ 1 − δ (1 − δ)T 1 (f n )n r|R(f n )| ≤ 1 + δ 1 − δ T 1 (f n ) r .
Since the above two chains of inequalities hold for every δ > 0, this proves the theorem. Instead of presenting the (rather direct) computation in this particular case, we choose to explain the logic behind the phenomenon. At any step in the process, there is a positive probability (which is 1 − (1 − ε) r − rε(1 − ε) r−1 + o(1)) that two or more of the randomly proposed r elements are in J. Since one may select only one element at a time, one may roughly think that "one possibility of selecting an element in R(f n ) is missed" on the above event. Since the number of steps is a.a.s. Θ(log n), in a constant proportion of all steps (which is 1 − (1 − ε) r − rε(1 − ε) r−1 + o(1)) at least one element of R(f n ) is "missed" a.a.s., which causes a delay in the r-choice process.
Remark 3.4. In general, the hypothesis T 1 (f n ) = ω(1) cannot be spared either. If there is a constant M > 0 such that for infinitely many n ∈ N one has T 1 (f n ) ≤ M , then clearly there cannot be acceleration by a factor of r + o(1) for any r > M .
Proof of Corollary 2.8. Fix a sequence of positive real numbers (ε k ) k≥1 that tends to zero. By assumption, a.a.s., for every k ≥ 1 there is a sequence of positive integers (γ k,n ) n≥1 such that, for every k ≥ 1 and every large enough n, with probability at least 1 − ε k the sequence of functions (f γ k,n n ) n≥1 satisfies:
(i) γ k,n ≤ ε k T 1 (f γ k,n n ) = o k (T 1 (f γ k,n n )), (ii) |R(f γ k,n n )| ≤ ε k n = o k (n), (iii) T 1 (f γ k,n n ) ≤ ε k |R(f γ k,n n )| = o k (|R(f γ k,n n )|) and T 1 (f γ k,n n ) = ω k (1).
Thus, a.a.s. one may find a sequence (k(n)) n≥1 satisfying k(n) = ω(1) such that, for every large enough n ≥ 1, the sequence (γ n ) n≥1 = (γ k(n),n ) n≥1 satisfies
1. γ n ≤ ε k(n) T 1 (f γn n ) = o(T 1 (f γn n )), 2. |R(f γn n )| ≤ ε k(n) n = o(n), 3. T 1 (f γn n ) ≤ ε k(n) |R(f γn n )| = o(|R(f γn n )|)
and T 1 (f γn n ) = ω(1). By using conditions (2) and (3), a direct application of Theorem 2.7 for the sequence of Boolean functions (f γn n ) n≥1 shows that T r (f γn n ) = (1 + o(1))T 1 (f γn n )/r a.a.s. On the other hand, E[T 1 (f n ) − T 1 (f γn n )] = γ n (note that since the γ n bits are chosen uniformly at random, the expected number of additional rounds needed to obtain probability at least 1/2 for f n to evaluate to 1 is T 1 (f n ) − γ n ), so since γ n = o(T 1 (f γn n )), one may conclude by Markov's inequality for T 1 (f n ) − T 1 (f γn n ) that T 1 (f n ) = (1 + o(1))T 1 (f γn n ) a.a.s., and similarly T r (f n ) = (1 + o(1))T r (f γn n ) a.a.s., which concludes the proof of the corollary. Fix also a positive constant c = c(M ) satisfying M log((1 − c) −1 ) ≤ 1/2. Under the above assumption for every sufficiently large n ≥ 1 we deduce that T 1 (f n ) ≥ cn: indeed, the probability not to encounter any element of R(f n ) during the first cn steps of the 1-choice process is bounded from below by
cn−1 i=0 1 − |R(f n )| n − i ≥ cn−1 i=0 1 − M n − i = exp −M log 1 1 − c + o(1) ≥ exp − 1 2 + o(1) ,
which is larger than 1/2 for every large enough n. We conclude by Theorem 2.5 that there is C = C(r, (f n ) n≥1 ) > 0 such that, for every sufficiently large n,
T r (f n ) ≥ Cn + T 1 (f n ) r ,
and hence sequences of M -juntas are slow for any M ∈ N.
Recursive Majority.
Consider two positive integer sequences (k n ) n≥1 and (t n ) n≥1 such that, for every n ≥ 1, k n is odd and k n ≥ 3, and (k tn n ) n≥1 is an increasing sequence that tends to infinity as n → +∞. Fix n ∈ N and denote k = k n , t = t n and N = k t . Now, define the sets (S j i ) j∈{0,...,t},i∈k t−j where, for every j ∈ {0, . . . , t} and Proof. Note that if x ∈ {0, 1} N satisfies f N (x) = 1, then f N (1 − x) = 0, which shows the first statement. For the second statement, we will need the following theorem, which is a special case of a more general result that one may trace back to Sperner [21], see also [18].
i ∈ k t−j , S j i = {ik j − (k j − 1), . . . , ik j }. Note that, for every j ∈ [t] and i ∈ [k t−j ], S j−1 ki−(k−1) · ∪ . . . · ∪ S j−1 ki . Now, for i ∈ [k t ],P(f N (X) = 1 | ||X|| 1 = s) ≤ P(f N (X) = 1 | ||X|| 1 = ⌊N/2⌋) = |A| N ⌊N/2⌋ −1 < |A| + |∂A| 2 N ⌊N/2⌋ −1 ≤ P(f N (X) = 1 | ⌊N/2⌋ ≤ ||X|| 1 ≥ ⌈N/2⌉) = 1 2 ,
which finishes the proof of the second statement.
By Lemma 4.1 we conclude that for every n ≥ 1 one has T 1 (f N ) ≥ N/2, so by Theorem 2.5 we deduce that there exists C = C((f N )) such that, for every n ≥ 1,
f N = f N (n) satisfies T r (f N ) ≥ CN + T 1 (f N ) r ,
and hence recursive majorities is slow.
4.3.
Tribes. Let (s n ) n≥1 be a sequence of positive integers such that, for every n ∈ N, s n ∈ [1, n]. For every n ∈ N, write n = s n t n + r n , where r n ∈ {0, . . . , s n − 1} is the remainder of the division of n by s n . Then, for every n ∈ N, given s n , a tribe partition of [n] is a t n -tuple of sets (S 1 , S 2 . . . , S tn ) such that S 1 · ∪ S 2 · ∪ · · · · ∪ S tn = n and for every i ∈ [t n ], |S i | ∈ {s n , s n + 1}. For every n ∈ N, a tribe function of tribe size s n associated to the tribe partition (S 1 , S 2 . . . , S tn ) is a function f n : x ∈ {0, 1} n → 1 ∃1≤i≤tn,all bits in positions Si in x are 1 .
Lemma 4.3. Fix any δ > 0 and a sequence of tribe functions (f n ) n≥1 of tribe sizes (s n ) n≥1 satisfying that for all n ∈ N, s n ≥ δ log n. Then, there is a constant
C = C(δ, (f n )) > 0 such that T r (f n ) ≥ Cn + T 1 (f n )/r.
Proof. Fix p = exp(−1/δ). We first show that a.a.s. f n is not activated if every bit is put to 1 with probability p independently of all other bits. Indeed, the probability of the above event is bounded from below by Moreover, by Chernoff's inequality a.a.s. at least pn/2 bits are put to 1. We conclude that T 1 (f n ) ≥ pn/2 for every large enough n, which allows us to conclude by Theorem 2.5.
4.4.
Connectivity and k-connectivity. For every n ≥ 1, consider an ordering I n of the set of pairs of vertices of K n . Let g n be a function from {0, 1} ( n 2 ) to the set of graphs on n vertices such that, for every v ∈ {0, 1} ( n 2 ) , the i-th pair of vertices of I n is an edge in g n (v) if v i = 1, and is not an edge if v i = 0. Define
f n : v ∈ {0, 1} ( n 2 ) → 1 gn(v) is connected .
Clearly, for every n ≥ 1, the function f n is monotone and all n 2 vertex pairs of K n belong to R(f n ) (note that any set of n 2 − 1 edges does not decide if a graph is connected or not in general). It is well known that for the binomial random graph G(n, p) connectivity undergoes a sharp threshold at p = (1 + o(1)) log n/n, coinciding with the moment when the last isolated vertex becomes incident to an edge. We now show that the sequence (f n ) n≥1 fixed above is accelerated by a factor of r + o(1) in the r-choice process (note that this also holds for the threshold of disappearance of the last isolated vertex):
Lemma 4.4.
The sequence (f n ) n≥1 defined above is fast.
Proof. Fix any r ∈ N. Consider the following strategy for the r-choice process: at each of the first s = n log log n steps, select an arbitrary edge among the r proposed ones. Then, at any step, select an edge that contains at most one vertex in the largest connected component if possible, and select an arbitrary edge otherwise. It is well known (see e.g. [7]) that, after s steps, a.a.s. the graph consists of a giant component that contains all but o(n)∩ω(n 1/2 ) of all n vertices as well as ω(n 1/2 ) isolated vertices. Hence, after s steps, a.a.s. only o(n 2 ) ∩ ω(n 3/2 ) of the remaining 0-bits may change the connectivity (namely the bits corresponding to the edges incident to at least one vertex outside the giant component). We condition on this event. Suppose that the first s activated bits have indices i 1 < i 2 < · · · < i s (which is a uniform random set of s out of all n 2 bits). Denote by f s n the (random) restriction of f n over the set of vectors in {0, 1} ( n 2 ) such that each of the bits with indices i 1 < i 2 < · · · < i s is turned to 1. Hence |R(f s n )| = o(n 2 ) ∩ ω(n 3/2 ). Since T 1 (f s n ) ≤ T 1 (f n ) = Θ(n log n) (for the sharp threshold for connectivity ensuring the last equality, see again [7]) and T 1 (f s n ) ≥ n 1/2 /2 = ω(1) by our conditioning, we have 1 ≪ T 1 (f s n ) ≪ |R(f s n )| ≪ n 2 , so by Theorem 2.7 T r (f s n ) = (1 + o (1))T 1 (f s n )/r. Moreover, before the conditioning we have E[T 1 (f n )− T 1 (f s n )] = s = o(n log n) = o(T 1 (f n )), and the same holds for T r (f n )− T r (f s n ). By Markov's inequality we conclude that both T 1 (f n ) = (1 + o (1))T 1 (f s n ) and T r (f n ) = (1 + o(1))T r (f s n ) a.a.s., so T r (f n ) = (1 + o (1))T 1 (f n )/r, which proves the lemma.
Remark 4.5. For any k ≥ 2, a graph is said to be k-connected if the deletion of any k − 1 vertices leaves a connected graph. Also, the k-core of a graph G is the largest subgraph of G with minimum degree k. It is well-known that a sharp threshold for k-connectivity occurs at p = (log n + (k − 1) log log n)/n (see Theorem 7.7 of [7]) as well as the fact that after n log log n steps of the 1-choice process the k-core of the resulting random graph contains n + o(n) vertices and is k-connected a.a.s. (see again [7]). Hence, a straightforward modification of the proof of Lemma 4.4 shows that, for every r ≥ 2, k-connectivity is accelerated by a factor of r(1 + o(1)) by the r-choice process.
Remark 4.6. The appearance of both Perfect matching and Hamilton cycle on n vertices fall into the category of monotone functions f n satisfying |R(f n )| = n 2 for which the r-choice process therefore gives a r(1+o(1))-factor acceleration, see [13] for Hamilton cycle (and as they remark in Section 5, Point 4, their result also applies to Perfect matching). Unfortunately our results do not lead to a significant simplification of their argument.
Definition 2. 1 .
1A variable i is relevant for f if there exist inputs x, y ∈ {0, 1} n which differ only in coordinate i and f (x) = f (y); in this case we also say that f depends on the i-th variable. The relevant set of f , denoted by R(f ), is the set of variables relevant for f .
Definition 2 . 2 .
22The relevant contraction of a Boolean function f , denoted bỹ f , is the function obtained by restricting f to its relevant set. In other words, if
Remark 3. 3 .
3The hypothesis |R(f n )| = o(n) in the first point of the theorem cannot be spared. We show this by a counterexample. Fix ε ∈ (0, 1] and let J = [⌊εn⌋]. Let f n be activated when ⌊log n⌋ of the elements in J are activated, that is, ...<ir <...<i ⌊log n⌋ ≤⌊εn⌋ 1 ∀j∈[⌊log n⌋],vi j =1 ∧ 1.
4 .
4Applications. In this section we give several examples of application of Theorems 2.5 and 2.7 and the corresponding corollaries. 4.1. Juntas. A Boolean function f is an M -junta if |R(f )| ≤ M . Fix a sequence (f n ) n≥1 of monotone Boolean functions such that there is M ∈ N satisfying max n∈N |R(f n )| ≤ M.
we say that the set S 0 i is activated if the bit i is turned from 0 to 1, and for every j∈ [t] and i ∈ [k t−j ], S j i is activated if at least k+1 2 of the sets S j−1 ki−(k−1) , . . . , S j−1 ki are activated. Define f N : x ∈ {0, 1} N → 1 S t 1is activated by x , that is, if only the 1-bits of x have been activated, then S t 1 is activated as well. The following lemma shows that, for any s < N/2, evaluating f N at a uniformly chosen vector conditioned to have exactly s 1-bits yields 1 with probability strictly smaller than 1/2.
Lemma 4 . 1 .
41Fix a random variable X distributed uniformly over {0, 1} N . Then, P(f N (X) = 1) ≤ 1/2. Moreover, P(f N (X) = 1 | ||X|| 1 = s) < 1/2 for s < N/2.
Theorem 4 . 2 (
42Special case of the local LYM inequality). Fix A ⊆ [N ] ⌊N/2⌋ and denote ∂A = S ∈ [N ] ⌈N/2⌉ ∃S ′ ∈ A, S ′ ⊆ S . Then, |A| ≤ |∂A|. Moreover, equality holds if and only if A = ∅ or A = [N ] ⌊N/2⌋ . Denote by A the set of vectors x ∈ {0, 1} N containing ⌊N/2⌋ 1-bits and satisfying f N (x) = 1. Since A is neither empty nor contains all vectors with exactly ⌊N/2⌋ 1-bits, by Theorem 4.2 |A| < |∂A|. On the other hand, by the same symmetry considerations as above, the number of vectors x ∈ {0, 1} N with either ⌊N/2⌋ or ⌈N/2⌉ 1-bits such that f N (x) = 1 is N ⌊N/2⌋ . We conclude that
( 1
1− p sn ) tn = exp(−(1 + o(1))p sn n/s n ) ≥ exp(−(1 + o(1))/s n ) = 1 + o(1).
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arxiv |
Bose-Einstein condensation on hyperbolic spaces
14 Feb 2022 February 12, 2022
Marius Lemm
Department of Mathematics
University of Tübingen
Auf der Morgenstelle 1072076TübingenGermany
Oliver Siebert
Department of Mathematics
University of Tübingen
Auf der Morgenstelle 1072076TübingenGermany
Bose-Einstein condensation on hyperbolic spaces
14 Feb 2022 February 12, 2022
A well-known conjecture in mathematical physics asserts that the interacting Bose gas exhibits Bose-Einstein condensation (BEC) in the thermodynamic limit.We consider the Bose gas on certain hyperbolic spaces. In this setting, one obtains a short proof of BEC in the infinite-volume limit from the existence of a volumeindependent spectral gap of the Laplacian.Main result: BEC in the infinite-volume limit on hyperbolic spaceIn this paper, we study the Bose gas in a hyperbolic geometry. The intriguing features of hyperbolic geometry have long inspired mathematicians and physicists alike. The first uses of hyperbolic geometry in condensed matter theory were to our knowledge based on the AdS-CFT correspondence principle [Wit98; Mal99] which has deepened our understanding of quantum entanglement and may hold the key to quantum error correction.More recently, experimentalists have been able to physically construct hyperbolic structures in the laboratory by confining particles to discrete hyperbolic lattices in circuit QED [Hu+19; KFH19; Kol+20; Alt+21; SMR21] and by using topoelectric circuits [Len+21]. These setups can serve, among other things, as tabletop simulators of quantum gravity. These recent experimental advances spawned a new interdisciplinary subfield of theoretical physics that blends condensed-matter physics with quantum information science and
Introduction
The Bose gas plays a central role in quantum many-body physics. The key effect it displays is Bose-Einstein condensation (BEC), macroscopic occupation of a single-particle quantum state. BEC is immensely useful for technological applications because it amplifies microsopic quantum effects to macroscopic scales. The original theoretical prediction of BEC was made for an ideal (i.e., non-interacting) Bose gas by Bose and Einstein [Bos24;Ein24]. A more realistic description of the Bose gas involves interactions between particles, resulting in a full quantum many-body problem.
To describe N bosons confined to an Euclidean torus T L = (−L/2, L/2) 3 and interacting via a repulsive two-particle interaction V ≥ 0, one uses the Hamiltonian
H N = N i=1 (−∆ x i ) + 1≤i<j≤N V (x i − x j ).
(1.1) A famous conjecture, mathematically formulated by Lieb in 1998 [Lie98] but probably several decades older, states that the Hamiltonian H N exhibits Bose-Einstein condensation in its ground state in the thermodynamic limit. More precisely, Lieb's conjecture asserts that there exists a constant c > 0 such that
T L T L γ(x, y)dx dy ? ≥ cL 3 (1.2)
where γ(x, y) = T N−1 L Ψ 0 (x, x)Ψ 0 (y, x)dx denotes the 1-particle correlation function of the unique ground state Ψ 0 ≥ 0 of H N . Indeed, (1.2) captures the macroscopic occupation of a single one-particle state. Note for example that (1.2) is satisfied for the non-interacting Bose gas with V ≡ 0.
Despite the central role that the Bose gas plays in mathematical physics, this conjecture remains open. Existing proofs of BEC in the Euclidean thermodynamic limit require reflection positivity [Aiz+04] or other special positivity properties [Kom21]. Nonetheless, spectacular progress has been achieved on the mathematics of the Bose gas in the past twenty years. For instance, the famous Lee-Huang-Yang formula for the subleading energy correction in the dilute limit ρa 3 → 0 has been rigorously derived [YY09; FS20; FS21], which resolved a longstanding open problem. Many works have contributed to precise understanding of BEC and the ground state energy in the Gross-Pitaevskii scaling regime; see e.g., [Dys57; LY98; LSY01a; LS02; LS06; Lie+09; NRS16; Boc+18; Boc+19; Boc+20; DS20; Nam+21 ;Hai21]. For further background and references, see [Lie+09] and the recent survey [Rou21]. general relativity [Boe+20; MR21; IAM21; Ste+21; Zha+21]. In particular, it was shown that continuum limits of some of these hyperbolic quantum lattice gases produce suitable hyperbolic continuum models [Boe+20]. Hyperbolic Bose gases were studied, e.g., in [CV93; Kir15;Zhu+21] for example.
To summarize, it can be said that hyperbolic geometry has emerged as a viable, if still exotic, theater of condensed-matter physics in general and Bose gases in particular. Nonetheless, quantum many-body physics in hyperbolic space is mathematically considerably less explored than its Euclidean counterpart.
The main result of this paper can be summarized as follows.
Main result: Interacting Bose gases on two-and three-dimensional hyperbolic manifolds rigorously display Bose-Einstein condensation in the infinite-volume limit.
One can thus say that the hyperbolic analog of the conjecture formulated by Lieb holds true. The change of the geometry from Euclidean to hyperbolic is instrumental to proving our results. The key spectral feature in our appropriately chosen hyperbolic setting is the existence of a volume-independent spectral gap for the Laplacian. It will not come as a surprise to experts that the large spectral gap removes many of the central difficulties present in the Euclidean case and leads to a rather short proof of BEC. Of course, one still needs to account for the change to hyperbolic geometry in the analytical arguments, in particular in the relevant two-particle scattering problem, but altogether this can be handled rather straightforwardly. Apart from the size of the spectral gap, the change in geometry then mostly surfaces in a different notion of scattering length a that is described in the appendix.
We now describe the models of hyperbolic manifolds that are used in this paper and summarize the result on BEC for these models. The precise setup is discussed in more detail in Section 2.
Model 1: Quotients by congruence subgroups. We consider hyperbolic manifolds of the form
X L = H d /Γ(L), L ≥ 2
where d ∈ {2, 3} and Γ(L) is a group of isometries (congruence subgroup). The {X L } L≥2 form a family of non-compact hyperbolic manifolds with finite volume increasing to infinity, vol(X L ) → ∞, L → ∞.
Since they are generated by quotienting the whole space with respect to isometries, the X L 's can be regarded as natural hyperbolic analogs of Euclidean tori. For d = 2, the X L are known as modular surfaces.
Let us now formulate the result on BEC. Let ρ = N vol(X L ) be the particle density. We fix a potential V ≥ 0 with supp V ⊆ B R 0 (0) for some finite range R 0 > 0. We write a for a hyperbolic analog of the scattering length; see Appendix A for its precise definition. Then we introduce the auxiliary parameter
Y = ρ ln 1 tanh(a/2) , for d = 2, ρ tanh a, for d = 3.
We prove that
lim Y →0 lim N,L→∞ ρ= N vol(X L ) ψ 0 , γψ 0 = 1, where ψ 0 = ½ X L vol(X L ) , (1.3)
where inner product is taken with respect to L 2 (X L ).
Comparing with (1.2), we see that (1.3) indeed proves BEC in the infinite-volume limit for any sufficiently small Y . Crucially, "sufficiently small" does not depend on the system size L since the Y -limit is taken after the infinite-volume limit in (1.3). In fact, (1.3) proves that the occupation of the ψ 0 -state converges to 1 as Y → 0. If all the particles belong to a single state up to subleading errors, one speaks of complete condensation.
The precise statements for d = 2 and d = 3 are given in Corollaries 2.7 and 2.11 below. They show that the requirement that Y is "sufficiently small" can be made explicit in terms of the other system parameters. Moreover, the same condition on Y being sufficiently small also applies to finite systems of the same density. The infinite-volume limit in (1.3) is added only for emphasis and the actual result is more general.
Model 2: Random compact hyperbolic surfaces. There is a natural probability measure on compact hyperbolic manifolds of fixed volume called the Weil-Petersson measure P WP g . Let M g denote the set of compact hyperbolic surfaces with genus g up to isometry. By the Gauss-Bonnet theorem, the volume of any hyperbolic surface X ∈ M g equals 2π(2g − 2). By taking g → ∞, we obtain compact hyperbolic manifolds whose volumes go to infinity.
Let ε > 0. We prove that there exists a family of measurable subsets (events) A g ⊂ M g such that lim
ρ ln 1 tanh a →0 lim N,g→∞: ρ= N 2π(2g−2) P WP g (A g ) = 1 (1.4)
and for every hyperbolic manifold X ∈ A g , we have
ψ 0 , γ N ψ 0 ≥ 1 − ε, where ψ 0 = ½ X vol(X) .
This proves that BEC occurs for random compact hyperbolic manifolds with probability going to 1 in the infinite-volume limit. For the precise statement, see Corollary 2.15. These results rely on two main estimates for general hyperbolic Bose gases, Proposition 2.2 and Theorem 2.3. The key common feature of the hyperbolic models described above is that the spectral gap of the corresponding Laplacian (Laplace-Beltrami operator) is bounded from below independently of the volume by deep results of Selberg [Sel65] and Mirzakhani [Mir13]. Quantitative improvements followed in [Sar83; LPS87; EGM90; BS91; Clo03; KS03], respectively [WX21; LW21].
Comparison to the Gross-Pitaevskii regime
As mentioned above, a commonly studied scaling limit in Euclidean setting is the Gross-Pitaevskii (GP) regime which is suitable for describing dilute Bose gases with strong short-ranged interaction. It is characterized by linking the length scale L of the torus to the particle density ρ = N L 3 and the scattering length a ∈ R (which captures the essential features of strength and range of the potential V for two-boson scattering) via
L = C 1 √ ρa (1.5)
Combined with the dilute limit ρa 3 → 0 this defines the GP scaling limit. This is in contrast to the thermodynamic limit (which is the subject of the open conjecture in the Euclidean setting and which we consider here in the hyperbolic setting), where one can take N, L → ∞ independently for fixed values of ρ and a. The investigation of the ground state energy asymptotics and BEC in the GP regime is a success story of mathematical physics. Landmark works in this direction include Dyson's study [Dys57], the various works of Lieb, Seiringer, and Yngvason [LY98; LSY01a; LS02; LS06] (see also [Lie+09]) and more recent advances [Boc+18; Boc+19; Boc+20; Nam+21]
Recent approaches rigorously implement a heuristic description of excitations above the condensate due to Bogoliubov [Bog47] through localization in Fock space, higher-order Bogoliubov transformations, and other technical innovations. The occurrence of BEC can be pushed to length scales larger than the GP scale (1.5) by further exploiting the close link between energy estimates and BEC on different spatial scales [Lie+09; Fou21; ABS21]. For instance, it was shown in [Fou21] that BEC occurs up to length scales
L = C (ρa 3 ) −δ √ ρa , 0 < δ < 1 4 .
and the upper bound on δ could be further improved somewhat by using methods from [FS20] as described in [Fou21].
Despite these advances, the conjectured occurrence of BEC in the thermodynamic limit (i.e. for L arbitrarily large) has remained open. A key reason for the failure of these "energy methods" beyond certain length scales is the fact that the spectral gap of the Laplacian on the torus T L = (−L/2, L/2) 3 vanishes as L −2 for L → ∞.
Our modest observation here is that energy methods are much more powerful in certain hyperbolic spaces because the change in geometry implies that the spectral gap of the Laplacian can be bounded independently of volume. (It is worth pointing out here that there are also other plenty of other apparently-natural hyperbolic settings where the spectral gap decreases with volume, e.g., balls with Neumann boundary conditions of increasing radii [Cha84, Theorem 5], so Models 1 and 2 considered above have to be chosen carefully.) At any rate, as a consequence of the spectral gap in Models 1 and 2, the proof of the main result is quite short and does not require recent advances on rigorous implementation of Bogoliubov's heuristic. This means that the argument provides no new insight on the Euclidean case.
The result raises some potentially interesting questions for further study in the hyperbolic setting.
(i) Our result on BEC is proved without identifying even the leading order of the energy asymptotics in the dilute limit. The spatial localization that commonly appears in energetic lower bounds is technically more challenging in the hyperbolic world because the local spectral gap of the Laplacian can be smaller than the global gap depending on the choice of boundary conditions. We leave it as an open problem to identify the leading asymptotics of the ground state energy in the hyperbolic setting.
(ii) A more precise analysis of the energy asymptotics would presumably be linked to a hyperbolic rendition of Bogoliubov theory [Bog47; Boc+18; Boc+19; Boc+20; BS20; FS20; FS21; Hai21]. This should reveal finer information about excitations above the condensate. It is conceivable that, as in the case of BEC considered here, the price for studying a slightly more complicated geometry is made up by its favorable spectral properties.
(iii) Another natural question concerns the fate of the BEC in the infinite-volume limit at positive temperature. The analogous problem has been resolved in the Euclidean setting, see e.g. the recent work [DS20] and references therein. This could be of practical relevance in case one finds a significantly larger critical temperature in the hyperbolic setting, keeping in mind that it is now possible to set up quantum gases in hyperbolic structures in the laboratory [Hu+19; KFH19; Kol+20; Alt+21; SMR21; Len+21].
(iv) Similarly to Point (iii), one could consider a magnetic field in a hyperbolic geometry analogously to [Sei02; LS06; NRS16] and others who proved BEC in the presence of magnetic fields in the Euclidean GP setting.
Structure of the paper
This paper is organized as follows.
• In Section 2, we state the main abstract results, the upper and lower bounds Theorem 2.3 and Proposition 2.2. Afterwards, we apply them to the concrete infinitevolume limits of hyperbolic manifolds described above to derive Corollaries 2.7, 2.11 and 2.15.
• In Section 3, we prove the upper bound Theorem 2.3. This follows an argument going back to [Dys57] in the modern form of [LY01;Lie+09]. The change in geometry leads to a few changes and, similarly to the Euclidean case, we obtain an upper bound on the effective 2-particle problem by estimating integrals over the fundamental domain by integrals over the universal cover (for us, this is H d ), cf (3.11).
• In Section 4, we use the spectral gap of the Laplacian and the non-negativity of the potential to prove the lower bound, Proposition 2.2.
• In Appendix A we generalize the scattering length to the hyperbolic setting, defining it via the radius of a hardcore potential, cf. Theorem A.2. As in the Euclidean case, we particularly use integration by parts and the inequality (
A.3) to estimate I(f R ), J(f R ) and K(f R ).
The results are analogous to the Euclidean setting and a key role is played by the harmonic function in the hyperbolic setting (A.2).
Models and Main Results
General facts about hyperbolic Bose gases
For any d ≥ 2 let H d denote the d-dimensional hyperbolic space. In d = 2 we will work in the upper-half plane model
H 2 = {z 1 + iz 2 : z 1 ∈ R, z 2 > 0} ⊆ C equipped with the Riemannian metric ds 2 = 1 z 2 2 (dz 2 1 + dz 2 2 ).
For dimensions d ≥ 3 it will be more convenient to work in the hyperboloid model
H d := {z ∈ R d+1 : z 0 > 0 and q d (z) = 1}, q d (z) := z 2 0 − z 2 1 − . . . − z 2 d , (2.1)
equipped with the pullback of the standard Lorentzian metric on R d+1 .
Remark 2.1. Throughout this paper we always use x for elements of the hyperbolic manifolds and z for elements of their universal cover H d .
Let X be a d-dimensional hyperbolic manifold, that is, a complete Riemannian manifold of constant curvature −1. Equivalently, X = H d /Γ, where Γ is a discrete subgroup of Iso(H d ) -the group of isometries of H d . We assume that X has finite volume. Denote by −∆ ≥ 0 the standard Laplace-Beltrami operator acting on L 2 (X). Furthermore, let V ∈ L ∞ (R + ) be a function with compact support and let
R 0 > 0 such that supp V ⊆ [0, R 0 ].
For N ∈ N particles consider the Hilbert space of N bosonic particles
H N := P + N L 2 (X ×N ),
P + N being the symmetrization operator in the N components. In this space we define the Hamiltonian for the Bose gas on X by
H N := −µ N i=1 ∆ i + 1≤i<j≤N V (d(x i ,x j )), (2.2) with domain D(H N ) := P + N D( N i=1 ∆ i ) = P + N H 2 (X ×n ), where ∆ i denotes the operator acting as ∆ on the i-th component, d : X × X → [0, ∞) the distance function on X, and d(x i ,x j ) the multiplication operator by the function X n ∋ (x 1 , . . . , x n ) → d(x i , x j ).
Note that H N is self-adjoint on D(H N ) as V is assumed to be essentially bounded. Furthermore, H N has a unique normalized ground state Ψ 0 ∈ H N with corresponding ground state energy E N . This can be deduced from the strict positivity of corresponding semigroup (cf. [RS78, Theorem XIII.44]), which in turn follows from V ≥ 0 and the fact that the semigroup associated to the Laplace-Beltrami on connected manifolds is positivityimproving (proven in [Dod83], see also [KLW21,p.139]).
The one-particle density matrix γ as a bounded operator on L 2 (X) is given by the integral kernel
γ(x, x ′ ) := X ×(N−1) Ψ 0 (x, x)Ψ 0 (x ′ , x)dx. (2.3)
Furthermore, let ψ 0 := vol(X) −1/2 1 1 X be the ground state of −∆ on X, in other words, the normalized constant function on X.
In order to establish BEC in the sense of (1.3) we use the following abstract lower bound for manifolds X where the Laplacian has a gap. The proof can be found in Section 4.
Proposition 2.2 (Lower bound)
Let X = H d /Γ where Γ is a discrete subgroup of Iso(H d ) such that vol(X) < ∞. Assume there exists Ξ > 0 such that −∆(Id − |ψ 0 ψ 0 |) ≥ Ξ. Then we have ψ 0 , γψ 0 ≥ 1 − E N NΞ .
Hence, in order to obtain a concrete lower bound, we need now an upper bound for E N /N. This will be now given in terms of a diluteness parameter defined as
Y := ρ ln((tanh(a/2)) −1 ) : d = 2, ρ tanh a : d = 3, (2.4)
where a is the 'hyperbolic scattering length' which depends only on the potential V and is defined in (A.2). In fact, we can make E N /N arbitrarily small if Y is small enough. To this end, for any ε > 0 let
Y 0 (ε) := min 3 2ε 3µ +1−1 16π , (8π(R 0 + 1) 2 ) −1 : d = 2, min 3 2ε 3µ +1−1 16πe 2R 0 , (8e 2R 0 (R 0 + 1) 2 ) −1 : d = 3.
(2.5)
Then we obtain the following (see Section 3 for the proof). Given that the diluteness parameter Y (defined as in (2.4)) satisfies Y ≤ (8π
(R 0 + 1) 2 ) −1 in d = 2 or Y ≤ (8e 2R 0 (R 0 + 1) 2 ) −1 in d = 3, we have E N N ≤ 16πµY (1 + 8π 3 Y ) : d = 2, 16πµe 2R 0 Y 1 + 8 3 πe 2R 0 Y : d = 3.
(2.6)
In particular, we have E N N ≤ ε for all Y ≤ Y 0 (ε). We now apply the combination of Proposition 2.2 and Theorem 2.3 to two classes of hyperbolic manifolds which are known to have spectral gaps. In order to be able to obtain a thermodynamic limit, our goal is to find a sequence of manifolds of growing volume tending to infinity with a uniform spectral gap. The first one comprises special non-compact hyperbolic manifolds of finite volume.
Modular surfaces
The case of d = 2 dimensions where one considers so-called modular surfaces, cf. [Sar03], is the most well-studied and most thoroughly understood one. The special linear group PSL 2 (R) := SL 2 (R)/{± Id} acts on H 2 ⊂ C via Möbius transformations
a b c d z := az + b cz + d ,
and is isomorphic to the group of orientation-preserving isometries of H 2 . The modular surfaces arise by considering the action of a discrete subgroup of PSL 2 (R), the modular group PSL 2 (Z) := SL 2 (Z)/{± Id} and its congruence subgroups. The latter are defined as those subgroups, which contain one of the principal congruence subgroups given by
Γ(L) := {A ∈ SL 2 (Z) : A = Id mod L}, L ∈ N,
where mod L is to be understood as taken in each entry of the matrices. We can then define for each L ∈ N a hyperbolic surface by 3 .
X L := H 2 /Γ(L),
In particular this shows that vol(X L ) → ∞, as L → ∞. Next, we need a uniform spectral gap. First, one can show that there is a gap of 1 4 for the continuous spectrum of any hyperbolic surface. It remains to find a similar bound for the lowest non-zero eigenvalue. Selberg conjectured that one actually has the same lower bound 1/4 [Sel14]. Although this remains an open problem, there are several proofs for slightly weaker bounds, being sufficient for our application. The to the authors' best knowledge best one is given in the following theorem.
Theorem 2.5 ([KS03])
Let Γ be a congruence subgroup of PSL 2 (Z) and X = H 2 /Γ. For the smallest non-zero eigenvalue λ 1 (X) of the Laplacian on X one has
λ 1 (X) ≥ 1 4 − 7 64 2 = 975 4096 .
Remark 2.6. Selberg already proved the bound λ 1 (X) ≥ 3 16 in [Sel65]. For a discussion of further bounds we refer the reader to [Sar03]. Now, combining Proposition 2.2 and Theorem 2.3 with Theorem 2.5 in the setting of modular surfaces yields the following first application.
Corollary 2.7
Let R 0 > 0. For all ε > 0 and all potentials V with supp V ⊆ R 0 , scattering length a and all N, L ∈ N satisfying
ρ ln((tanh a) −1 ) = N vol(X L ) ln((tanh a) −1 ) < Y 0 975 4096 ε
where Y 0 (·) is defined in (2.5), we have
ψ 0 , γψ 0 ≥ 1 − ε.
Remark 2.8. The statement of Corollary 2.7 implies the double limit statement in (1.3). However, it is stronger than (1.3) in two ways: (a) it is quantitative and (b) the occurrence of BEC only requires an assumption on ρ and a, so it also holds for any finite number of particles N and volume vol(X L ) corresponding to the same density. While we focused on the infinite-volume limit in the introduction for emphasis, the result also applies to finite systems.
Quotients of hyperbolic 3-space by congruence subgroups
The gap of modular surfaces can be generalized to higher dimensions. Here, it is more convenient to work in the hyperboloid model, see (2.1). For a unit ring R let SO d,1 (R) be the group of R-valued matrices with determinant one which leave q d invariant. The group of orientation-preserving isometries in H d is then given by SO 0 d,1 (R), which is defined as the connected component of the identity matrix in SO d,1 (R).
Remark 2.9. In Section 2.2 we used that SO 0 2,1 (R) ∼ = PSL 2 (R). Now we can consider principal congruence subgroups as follows, see also [EGM90;BS91]. Let SO 0 d,1 (Z) := SO 0 d,1 (R) ∩ SO d,1 (Z). Then the principal congruence subgroups can be defined as
Γ d (L) := {A ∈ SO 0 d,1 (Z) : A = Id mod L}, L ∈ N.
In particular, note that Γ d (1) = SO 0 d,1 (Z). A congruence subgroup is then a subgroup of SO 0 d,1 (Z) which contains Γ d (L) for some L.
Analogously to the 2-dimensional case we then define X L := H d /Γ d (L) and obtain hyperbolic manifolds of finite volume. Again,
vol(X L ) = [SO 0 d,1 (Z) : Γ d (L)] vol(X 1 ),
which equally tends to infinity as L → ∞. Finally, we need a variant of Theorem 2.5, i.e., the existence of a gap, for higher dimensions. For d = 3 this was proven by Sarnak [Sar83]. In [EGM90] and [LPS87] it was first generalized to arbitrary dimension. Other versions for more general algebraic groups can be found in [BS91] and [Clo03].
Theorem 2.10
Let d ≥ 3, Γ be a congruence subgroup of SO 0 d,1 (Z) and X = H d /Γ. For the smallest non-zero eigenvalue λ 1 (X) of the Laplacian on X one has
λ 1 (X) ≥ 2d − 3 4 .
Then, applying this theorem in combination with Theorem 2.3 and Proposition 2.2 once more for the case d = 3 yields the following.
Corollary 2.11
Let d = 3 and R 0 > 0. For all ε > 0 and all potentials V with supp V ⊆ [0, R 0 ], scattering length a and all N, L ∈ N satisfying
ρ ln tanh a = N vol(X L ) ln tanh a < Y 0 3 4 ε
where Y 0 (·) is defined in (2.5), we have ψ 0 , γψ 0 ≥ 1 − ε.
Random compact hyperbolic surfaces
Another possibility is to consider compact hyperbolic manifolds. An analogy of Selberg's conjecture in this case is only known in a probabilistic sense and leads to the theory of compact random hyperbolic surfaces as developed by Mirzakhani. Recent surveys for this topic can be found in [Wri20; Mon21]. A compact hyperbolic surface is given by H 2 /Γ, where Γ ⊂ PSL 2 (R) is a discrete and co-compact subgroup. For g ∈ N let M g := compact hyperbolic surfaces of genus g / isometries , the so-called moduli space, which can be also represented as a quotient of the Teichmüller space by some group action [Mir13, Section 2]. On M g one can construct a probability measure P WP g originating from a natural symplectic form on M g , the so-called Weil-Petersson form [Wri20, 2.8]. By the Gauss-Bonnet theorem the volume of any X ∈ M g equals 2π(2g − 2) and therefore we can consider g → ∞ for an infinite volume limit.
In this limit an analog of Selberg's 1/4 conjecture was formulated in [Wri20]:
P WP g X ∈ M g : λ 1 (X) ≥ 1 4 − α ? → g→∞ 1 for all α > 0. (2.7)
As in the deterministic case, this remains an open problem but several weaker results have been established as well. The currently best one is the following. Let α > 0 and ξ < 1. Then there exists g 0 ∈ N such that all g ≥ g 0 there is a measurable set A g with
P WP g (A g ) ≥ ξ
such that for all X ∈ A g , R 0 > 0, ε > 0, and all potentials V with supp V ⊆ R 0 , scattering length a and and N ∈ N satisfying
ρ ln((tanh a) −1 ) = N 2π(2g − 2) ln((tanh a) −1 ) < Y 0 3 16 − α ε where Y 0 (·) is defined in (2.5), we have ψ 0 , γψ 0 ≥ 1 − ε.
Proof. For given α > 0 let
A g := X ∈ M g : λ 1 (X) ≥ 3 16 − α .
Then we use Theorem 2.12 and find for ξ < 1 a g 0 such that P WP g (A g ) ≥ ξ for all g ≥ g 0 . Now, for X ∈ A g and under the given assumptions we have
λ 1 (X) ≥ 3 16 − α ≥ E N εN (2.8)
by Theorem 2.3. Thus, by Proposition 2.2
ψ 0 , γψ 0 ≥ 1 − E N Nλ 1 (X) (2.8) ≥ 1 − ε.
Remark 2.16. The two properties (a) and (b) described in Remark 2.8 also apply to Corollary 2.15.
Upper Bound
In this part we give the proof of Theorem 2.3. First, we show an abstract form of an upper bound, which is in complete analogy with [Lie+09, Section 2.1], cf. also [LY01, (2.7)]. For a function f on [0, ∞) we define a trial function Ψ ∈ L 2 (H N ) by
Ψ(x 1 , . . . , x N ) := N i=2 F i (x 1 , . . . , x i ), (3.1) where F i (x 1 , . . . , x i ) := f (t i (x 1 , . . . , x i−1 )), t i (x 1 , . . . , x i ) := min{d(x i , x j ) : j = 1, . . . , i − 1}.
Proposition 3.1
For any non-decreasing function f on [0, ∞) let Ψ given by (3.1). Let ρ := N/ vol(X).
Then we have Ψ, H N Ψ Ψ 2 ≤ N (1 − ρI(f )) 2 ρJ(f ) + 2 3 µ(ρK(f )) 2 ,
given that the integrals
I(f ) := H d (1 − f (d(o, z)) 2 )dz J(f ) := H d µf ′ (d(o, z)) 2 + 1 2 V (d(o, z)) |f (d(o, z))| 2 dz, K(f ) := H d f (d(o, z))f ′ (d(o, z))dz,
for any o ∈ H d chosen as an origin, are finite and ρI(f ) < 1.
Proof. The proof is analogous to the Euclidean case [Lie+09] with some modifications for the hyperbolic setting. For a function Φ : X ×N → C let ∇ k denote the gradient on the manifold X with respect to the k-th component, that is, for each
x = (x 1 , . . . , x N ) ∈ X ×N , we get an element ∇ k Φ(x) ∈ T x k X if Φ is smooth enough around x.
We write ·, · TxX : T x X × T x X → R for the Riemannian metric of X at the point x, and · TxX for the corresponding norm on T x X. For notational convenience we will mostly drop the argument x.
By the chain rule we get for almost all x ∈ X ×N
∇ k Ψ = i≥k Ψ F i f ′ (t i )∇ k t i = i≥k Ψ F i f ′ (t i )∇ k d(x i , x i * ),
where x i * denotes the nearest neighbor among the points x 1 , . . . , x i−1 . Therefore,
N k=1 ∇ k Ψ 2 Tx k X = N i=1 |Ψ| 2 F 2 i f ′ (t i ) 2 i k=1 ∇ k d(x i , x i * ) 2 Tx k X + 2 N k=1 j>i≥k |Ψ| 2 F i F j f ′ (t i )f ′ (t j ) ∇ k d(x i , x i * ), ∇ k d(x j , x j * ) Tx k X ,
where we use that there is a unique nearest neighbor almost everywhere and that we have to sum over ordered pairs. Since ∇ x d(x, y) TxX ≤ 1 almost everywhere, observe that ∇ k d(x i , x i * ) Tx k X ≤ ǫ ik and k ǫ ik ≤ 2 for almost every x, where
ǫ ik := 1 : i = k or t i = d(x i , x k ), 0 : else.
Thus, we arrive at
Ψ, H N Ψ Ψ 2 ≤ 2µ N i=1 |Ψ| 2 F −2 i f ′ (t i ) 2 |Ψ| 2 + i<j |Ψ| 2 V (d(x i , x j )) |Ψ| 2 (3.2) + 2µ N k=1 j>i≥k |ǫ ik ǫ jk | |Ψ| 2 F i F j f ′ (t i )f ′ (t j ) |Ψ| 2 . (3.3)
Now, for j < i we define F i,j in the same way as F i with the only difference that we omit the point x j in the consideration of the nearest neighbors. Likewise, we define F i,jk by omitting x j and x k . Then F i,j does not depend on x j and F i,jk does not depend on x j and x k . Furthermore, since f is monotonically increasing and 0 ≤ f ≤ 1, we have
F 2 j+1 · · · F 2 i−1 F 2 i+1 · · · F 2 N ≤ F 2 j+1,j · · · F 2 i−1,j F 2 i+1,ij · · · F 2 N,ij , (3.4) F 2 j · · · F 2 N ≥ 1 − N k=1, =i,j (1 − f (d(x j , x k )) 2 ) F 2 j+1,j · · · F 2 i−1,j × 1 − N k=1, =i (1 − f (d(x i , x k )) 2 ) F 2 i+1,ij · · · F 2 N,ij .
(3.5) Furthermore, we trivially find
f ′ (t i ) 2 η i 2 ≤ i−1 j=1 f ′ (d(x i , x j )) 2 [∇ i d(x i , x j )] 2 , (3.6) F i ≤ f (d(x i , x j )). (3.7)
Now, the numerator of the right-hand side in (3.2) can be estimated from above using (3.4) together with (3.6) and (3.7),
N i=1 |Ψ| 2 F −2 i f ′ (t i ) 2 + j<i |Ψ| 2 V (d(x i , x j )) (3.8) ≤ 2µ j<i F 2 1 . . . F 2 j−1 F 2 j+1,j · · · F 2 i−1,j F 2 i+1,ij · · · F 2 N,ij dx 1,...,N,ij (3.9) × 2µf ′ (d(x i , x j )) 2 + f (d(x i , x j )) 2 V (d(x i , x j )) dx i dx j ,(3.10)
where dx 1,...,N,ij denotes the integration over all x 1 , . . . , x N except x i and x j . For the denominator we use (3.5) and obtain similarly
Ψ 2 ≥ F 2 1 · · · F 2 j−1 F 2 j+1,j · · · F 2 i−1,j F 2 i+1,ij · · · F 2 N,ij × vol(X) − N k=1, =i,j X (1 − f (d(x j , x k )) 2 )dx j × vol(X) − N k=1, =i X (1 − f (d(x i , x k )) 2 )dx i dx 1,...,N,ij .
Now, we use that
X=H d /Γ g(d(x, x 0 ))dx ≤ H d g(d(o, z))dz (3.11)
for any positive function g defined on H d and all x 0 ∈ X, o ∈ H d . This yields
(3.10) ≤ X H d 2µf ′ (d(o, x j )) 2 + f (d(o, x j )) 2 V (d(o, x j )) dzdx j = 2 vol(X)J(f ),
and for all k,
X (1 − f (d(x j , x k )) 2 )dx j ≤ H d (1 − f (d(o, z)) 2 )dz = I(f ).
The integral over dx 1,...,N,ij cancels in the numerator and denominator and we obtain
2µ N i=1 |Ψ| 2 F −2 i f ′ (t i ) 2 |Ψ| 2 + i<j |Ψ| 2 V (d(x i , x j )) |Ψ| 2 ≤ N(N − 1) 2 2 vol(X)J(f ) (vol(X) − (N − 1)I(f )) 2 .
Next, we estimate the non-diagonal term (3.3), cf. [LSY01b]. We get the same cancellations in the numerator and denominator up to the term
2µ N k=1 j>i≥k X×X |ǫ ik ǫ jk | f (t i )f (t j )f ′ (t i )f ′ (t j )dx i dx j ≤ 4µ N k=1 j>i>k X×X f (d(x i , x k ))f (d(x j , x k ))f ′ (d(x i , x k ))f ′ (d(x j , x k ))dx i dx j = 2 3 µN(N − 1)(N − 2)K(f ) 2 .
This shows the desired bounds.
Our choice for f in definition of the F i (3.1) will be f R , R > 0, given as in (A.2). Then we have J(f R ) = E R , which is explicitly computed in Theorem A.2. Therefore, it remains to find explicit bounds for I(f R ) and K(f R ), which is the content of the following two lemmas.
Lemma 3.2
For all R > R 0 ,
I(f R ) ≤ 2π ln tanh(R/2) tanh(a/2) (R 2 − a 2 ) : d = 2, 4π tanh a tanh R−tanh a tanh R(R 2 − a 2 ) : d = 3.
Proof. Using hyperbolic polar coordinates, we get
I(f R ) = vol(S d−1 ) R 0 (1 − f R (r) 2 ) sinh d−1 rdr ≤ vol(S d−1 ) a 0 sinh d−1 rdr + vol(S d−1 ) R a 1 − f ∞ (r) 2 f ∞ (R) 2 sinh d−1 rdr = vol(S d−1 ) R 0 sinh d−1 rdr − vol(S d−1 ) f ∞ (R) 2 R a f ∞ (r) 2 sinh d−1 rdr.
Let u(r) := r 0 sinh d−1 (r ′ )dr ′ . With integration by parts the second term can be expressed as
R a f ∞ (r) 2 sinh d−1 rdr = [f ∞ (r) 2 u(r)] R a − R a 2f ∞ (r)f ′ ∞ (r)u(r)dr = f ∞ (R) 2 R 0 sinh d−1 rdr − R a 2f ∞ (r)f ′ ∞ (r)u(r)dr. Thus, using that u(r) ≤ r sinh d−1 r and f ′ ∞ (r) sinh d−1 r = C d (a) is independent of r (cf. Remark A.3), I(f R ) ≤ 2 vol(S d−1 ) f ∞ (R) 2 R a f ∞ (r)f ′ ∞ (r)u(r)dr ≤ 2C d (a) vol(S d−1 ) f ∞ (R) 2 R a f ∞ (r) ≤f∞(R) rdr ≤ C d (a) vol(S d−1 ) f ∞ (R) (R 2 − a 2 ).
Note that we have C 2 (a) = 1 and C 3 (a) = tanh a.
Lemma 3.3
For all R > R 0 ,
K(f R ) ≤ 2πR ln tanh(R/2) tanh(a/2) : d = 2, 4π tanh aR 1− tanh a tanh R : d = 3.
Proof. Using f ′ R (r)f R (r) = 1 2 (f R (r) 2 ) ′ , partial integration and (A.3), we obtain
K(f R ) = vol(S d−1 ) 2 R 0 (f R (r) 2 ) ′ sinh d−1 rdr = vol(S d−1 ) 2 f R (R) 2 sinh d−1 (R) − vol(S d−1 ) 2 R 0 f R (r) 2 (sinh d−1 r) ′ dr ≤ vol(S d−1 ) 2 f R (R) 2 sinh d−1 (R) − vol(S d−1 ) 2f ∞ (R) 2 R a f ∞ (r) 2 (sinh d−1 r) ′ dr = vol(S d−1 ) f ∞ (R) 2 R a f ∞ (r)f ′ ∞ (r) sinh d−1 rdr.
Using again that f ′ ∞ (r) sinh d−1 r = C d (a) is independent of r, we conclude
K(f R ) ≤ C d (a) vol(S d−1 ) f ∞ (R) 2 R a f ∞ (r)dr ≤ C d (a) vol(S d−1 ) f ∞ (R) .
Plugging in vol(S 1 ) = 2π, vol(S 2 ) = 4π, C 2 (a) = 1, C 3 (a) = tanh a, and f ∞ (R) = ln tanh(R/2) tanh(a/2) for d = 2, f ∞ (R) = 1 − tanh a tanh R for d = 3 yields the claimed estimates.
Remark 3.4. One can write the estimates from Lemmas 3.2 and 3.3 in a dimensionindependent way as
I(f R ) ≤ f ′ ∞ (R) sinh d−1 R vol(S d−1 ) f ∞ (R) (R 2 − a 2 )
,
K(f R ) ≤ f ′ ∞ (R) sinh d−1 R vol(S d−1 ) f ∞ (R) ,
as it can be seen in the proofs.
E N N ≤ 4πρµ tanh a tanh R 1 − 4πρ tanh a (R 2 −a 2 ) tanh R tanh R−tanh a 2 (tanh R − tanh a) 1 + 8 3 πρ tanh a tanh R tanh R − tanh a ,
provided that 4πρ tanh a (R 2 −a 2 ) tanh R tanh R−tanh a < 1.
Proof. Plugging in the concrete upper bounds of Lemma 3.2, Lemma 3.3 (in the form of Remark 3.4) and Theorem A.2 in Proposition 3.1 yields
Ψ, H N Ψ N Ψ 2 ≤ ρC d (a)µ vol(S d−1 ) 1 − ρC d (a) vol(S d−1 ) f∞(R) (R 2 − a 2 ) 2 f ∞ (R) 1 + 2 3 ρ C d (a) vol(S d−1 ) f ∞ (R) .
By using the values for C d (a) and f ∞ (R) for d = 2 and d = 3 one obtains the claimed upper bounds.
Proof of Theorem 2.3. Choose R := max{R 0 , a + 1}, which is eligible in Proposition 3.5. Then we find, using a ≤ R 0 that R 2 − a 2 ≤ (R 0 + 1) 2 . Furthermore, for d = 2 we have ρ ln tanh(R/2) tanh(a/2) = ρ ln(tanh(a/2) −1 ) 1 − ln tanh(R/2) ln tanh(a/2)
≤ 1 1 − ln tanh((a+1)/2) ln tanh(a/2) Y ≤ 1 1 − e −1 Y ≤ 2Y, and for d = 3, tanh R tanh R − tanh a ≤ tanh(a + 1) tanh(a + 1) − tanh a ≤ e 2a ≤ e 2R 0 .
Using these estimates, the upper bounds of Proposition 3.5 simplify as follows:
E N N ≤ 4πµY (1−4π(R 0 +1) 2 Y ) 2 (1 + 8π 3 Y ) : d = 2, 4πµe 2R 0 Y (1−4πe 2R 0 (R 0 +1) 2 Y ) 2 1 + 8π 3 e 2R 0 Y : d = 3.
If we assume 4π(R 0 + 1) 2 Y ≤ 1 2 and 4πe 2R 0 (R 0 + 1) 2 Y ≤ 1 2 , respectively, we get (2.6). For the choice of Y 0 (ε) note that the inequality aY (1 + bY ) ≤ c has the solution
Y ≤ 4bc/a + 1 − 1 2b
for Y ≥ 0.
Lower Bound
In this section we prove the lower bound (Proposition 2.2). Recall that Ψ 0 ∈ L 2 (X ×N ) is the ground state of the operator H N (2.2) and ψ 0 the ground state of −∆ on X. Furthermore, the one-particle density matrix γ was defined in (2.3).
Proof of Proposition 2.2. Since V ≥ 0, we have
tr(−∆γ) = Ψ 0 , − 1 N N i=1 ∆ i Ψ 0 ≤ E N N .
Let
p n φ n , −∆P m φ n ≥ Ξ n p n |ψ 0 ψ 0 | ⊥ φ n 2 .
Thus,
ψ 0 , γψ 0 = n p n | ψ 0 , φ n | 2 = n p n 1 − |ψ 0 ψ 0 | ⊥ φ n 2 = 1 − n p n |ψ 0 ψ 0 | ⊥ φ n 2 ≥ 1 − E N NΞ .
A. Variational principle
In this part we show existence and uniqueness of the ground state for the key two-particle scattering problem in the hyperbolic setting. This will be used in the choice of the N-particle test functions in the upper bound in Section 3. We also define a 'hyperbolic scattering length' a. As in the Euclidean case it will correspond to the radius of a hardcore potential with the same scattering behavior. The arguments follow closely those in [LY01;Lie+09]. Let V : [0, ∞) → [0, ∞) be a measurable function with essential compact support and let R 0 > 0 such that supp V ⊆ [0, R 0 ]. Let o ∈ H d be fixed. For R > R 0 and φ ∈ H 1 (B R (o)), we define the functional
E R (φ) := B R (o)⊆H d |∇φ(z)| 2 + 1 2 V (d(o, z)) |φ(z)| 2 dz.
Remark A.1. With this functional we can describe two-particle energies on a d-dimensional hyperbolic manifold X, cf. the proof of Proposition 3.1. for some number a > 0. The energy corresponding to φ is given by Finally, we have that f R is non-decreasing and
E R := E R (φ) = µf ′ (R) sinh d−1 R vol(S d−1 ) f ∞ (R) = f R (r) ≥ f ∞ (r) f ∞ (R) for all r ≥ a. (A.3)
Remark A.3. (a) Notice that we indeed defined a in such a way that f ∞ (a) = 0, i.e., if V is a hardcore potential with radius R 0 , then a = R 0 .
(b) As f is an indefinite integral of (sinh d−1 ) −1 , the quantity f ′ (R) sinh d−1 R only depends on a (or V ) but not on R. Therefore, we also write spherical average of φ 2 . By the generalized Jensen inequality for probability measures, we obtain ∇ φ 2 ≤ ∇φ 2 and thus also E R ( φ) ≤ E R (φ) because the potential is assumed to be spherically symmetric as well.
Existence of a minimizer: As E R is bounded from below, there exists a minimizing sequence of spherically symmetric (φ n ) in H 1 (B R (o)) with φ n (z) = 1 for a.e. z with d(o, z) = 1 and all n. Define φ n ∈ H 1 (H d ) by φ n (z) := φ n (z) for z ∈ B R (o) and φ n (z) = h (d(o, z)) for some h ∈ C ∞ (R + ) with h(r) = 1 for r < R + 1 and h(r) = 0 for r > 2R + 1. As sup n φ n H 1 (H d ) < ∞ (and because H 1 (H d ) is reflexive, cf. [Heb96,
Proposition 2.4]), one can find a subsequence ( φ n k ) in H 1 (H d ) which converges weakly in H 1 (H d ) to some φ ∈ H 1 (H d ), which is rotationally symmetric. We then have that (φ n k ) also converges weakly to φ := φ| B R (o) ∈ H 1 (B R (o)). One gets φ(z) = 1 for a.e. z with d(o, z) = 1 because the radial part is continuous outside of the origin, and φ(z) = 1 for d(o, z) ∈ (R, R+1). By equivalence of lower semicontinuity and weak lower semicontinuity for convex functions, we obtain lim k→∞ E R (φ n k ) ≥ E R (φ) and therefore, φ is a minimizer. The Euler-Lagrange equation (A.1) follows by considering d dδ | δ=0 E R (φ + δψ) = 0 for all infinitely differentiable functions ψ which vanish for all z with d(o, z) ≥ R. Furthermore, (A.1) can be written down for the radial part f R on (0, R) given by f R (d(o, z)) := φ(z), which is a linear ODE with boundary values f R (R) = 1, f ′ R (R) = 0. Thus, it has a unique solution.
For R 0 < d(o, z) < R we infer from (A.1) that −∆φ = 0. As the Laplace-Beltrami operator on H d is given in hyperbolic polar coordinates by ∆ = sinh(r) 1−d ∂ r sinh(t) d−1 ∂ r + sinh(r) −2 ∆ Σ , we find that ∂ r sinh(t) d−1 ∂ r f R (r) = 0. The corresponding solutions for d = 2 and d = 3 are given by (A.2).
For the energy we use partial integration and f ′ ∞ (r) = 1 sinh d−1 r . Thus we get
E R = vol(S d−1 ) µ sinh d−1 rf R (r)∂ r f R (r) R 0 + R 0 −µ 1 sinh d−1 (r) ∂ r (sinh d−1 (r)∂ r f R (r)) + 1 2 V (r)f R (r) f R (r) sinh d−1 (r)dr = µ vol(S d−1 ) f ∞ (R) 2 sinh d−1 rf ∞ (r)∂ r f ∞ (r) R 0 + vol(S d−1 ) R 0 −µ∆ r f R (r) + 1 2 V (r)f R (r) =0 f R (r) sinh d−1 (r)dr = µ vol(S d−1 ) f ∞ (R) sinh d−1 (R)f ′ ∞ (R).
The last statement (A.3) follows in the same way as in [Lie+09, Lemma C.2] from the Hopf maximum principle.
Theorem 2. 3 (
3Abstract upper bound) Let X = H d /Γ where Γ is a discrete subgroup of Iso(H d ) such that vol(X) < ∞. Let V be a potential supported in [0, R 0 ] with hyperbolic scattering length a, and set E N = inf σ(H N ).
L 3 p
3A fundamental domain for X 1 is given by (cf. [DS05, Lemma 2.3.1]) F 1 = {z ∈ H 2 : Re z ≤ 1/2, |z| ≥ 1}, and from that one can compute directly that vol(X 1 ) = π 3 . Furthermore, as [SL 2 (Z) : Γ(L)] = L 3 p prime, p|L 1 − 1 p 2 , see [DS05, Exercise 1.2.3(b)], we can infer that vol(X L ) = [SL 2 (Z) : Γ(L)] vol(X 1 ) = prime
Proposition 2.4 ([Sel14; Sar03])Let X be a Riemannian surface, that is, X = H 2 /Γ, where Γ is any discrete subgroup of the group of isometries of H 2 . Then the continuous spectrum of the Laplacian on X equals [1/4, ∞).
Proposition 3. 5
5Let R ≥ R 0 and R > a. In d = 2 we have for all ρ and a 2 − a 2 ) < 1 and in d = 3 we have for all ρ and a
P m := 1 1 [0,m] (−∆) be the spectral projection of −∆ to all values smaller than m, which makes −∆P m bounded. By dominated convergence using that tr(−∆P m γ) ≤ E N /N we see that tr((−∆ + ∆P m )γ) → 0, m → ∞. Now, by the spectral theorem we can write γ = n p n ·, φ n φ n , n p n = 1, with (φ n ) being an orthonormal basis of L 2 (X)
Theorem A. 2 For
2In the class of functions φ ∈ H 1 (B R (o)) with φ(z) = 1 for a.e. z with d(o, z) = 1 there exists a unique minimizer φ R of E R . It is spherically symmetric, non-R 0 < r < R, we have φ(z) = f R (d(o, z)) with f R (r) := f ∞ (r) f ∞ (R)
Remark 2.14. In other settings of random hyperbolic manifolds, namely for conformally compact infinite area hyperbolic surfaces[MN21] and for finite area non-compact hyperbolic surfaces[HM21] the lower bound 3 16 in Theorem 2.12 could actually be improved toTheorem 2.12 ([WX21; LW21])
We have for all α > 0
lim
g→∞
P WP
g
X ∈ M g : λ 1 (X) ≥
3
16
− α = 1.
Remark 2.13. This improves a famous result by Mirzakhani [Mir13, Theorem 4.8], where
she showed the same with constant 1
4
ln 2
2π+ln 2
2 ≈ 0.02 instead of 3
16 .
1
4 , see also references therein.
Corollary 2.15
AcknowledgmentsThe authors thank Christian Brennecke, Matthew de Courcy-Ireland, Andreas Deuchert, Søren Fournais, and Christian Hainzl for useful comments.Proof. First, we show that we can restrict to non-negative and spherically symmetric functions as minimizers. Let f, g real-valued functions on R. Then we find (cf. [LL01,for all points where they are differentiable and f 2 + g 2 > 0. Thus, we have a.e.Furthermore,dz, is convex. For φ ∈ H 1 (B R (o)) let φ be the spherically symmetric function given by the square root of the
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arxiv |
The Total Matching Polytope of Complete Bipartite Graphs
1 Mar 2023
Yuri Faenza
IEOR Department
Columbia University
Luca Ferrarini
Dipartimento di Matematica "F. Casorati"
Università di Pavia
The Total Matching Polytope of Complete Bipartite Graphs
1 Mar 2023Integer ProgrammingCombinatorial OptimizationTotal MatchingPolyhedral CombinatoricsComplete Bipartite GraphsExtended Formulations
The total matching polytope generalizes the stable set polytope and the matching polytope. In this paper, we first propose new facet-defining inequalities for the total matching polytope. We then give an exponential-sized, non-redundant description in the original space and a compact description in an extended space of the total matching polytope of complete bipartite graphs.ν T (G) can be computed in polynomial time for trees and it is NP-complete already for bipartite and planar graphs. The authors in[11]propose the first polyhedral study of the total matching problem, deriving facet-defining inequalities for the polytope associated to it, defined as follows.Given a total matching T , the corresponding characteristic vector is:with x corresponding to the vertex variables and y to the edge variables. Hence, the total matching polytope of a graph G = (V, E) is defined as: P T (G) := conv{χ[T ] ⊆ R |V |+|E| : T is a total matching of G}.Contributions and organization of the paper. The goal of this paper is to prove new results on the facial structure of P T (G), and highlight connections between P T (G) and the classical theory of polyhedral combinatorics. In Section 2, we introduce basic tools and employ them to show that the natural linear relaxation of P T (G) gives a complete description when G is a tree. In Section 3, we propose two new classes of inequalities for P T (G), dubbed balanced and non-balanced
Introduction
Let G = (V, E) be a simple, loopless and undirected graph, and let D = V ∪ E be the set of its elements, i.e., edges and vertices. Elements d, d ′ ∈ D are said to be adjacent if d and d ′ are adjacent vertices, or incident edges, or d is an edge incident to a vertex d ′ . If d, d ′ ∈ D are not adjacent, they are independent. A stable set is a set of pairwise independent vertices, while a matching is a set of pairwise independent edges. A total matching is a subset T ⊆ D whose elements are pairwise independent. Hence stable sets and matching are total matchings, but the converse may not be true. For instance, the complete graph with 3 nodes has a total matching of size 2, while all its matchings and stable sets have size at most 1. The total matching problem asks for a total matching of maximum size.
Define ν T (G) := max{|T | : T is a total matching}, ν(G) := max{|M | : M is a matching} and α(G) := max{|S| : S is a stable set}. ν(G), α(G) have been extensively studied in the literature, both algorithmically and in terms of the corresponding polytopes, see, e.g., [4,6,7,8,9,10,12,15,19,20,22]. Despite the fact that it generalizes those illustrious special cases, and its connection with Vizing's total coloring conjecture (see, e.g., [11] for details), the total matching problem is less studied in the operations research literature. In particular, significant results have been obtained only for structured graphs, as cycles, paths, full binary trees, hypercubes, and complete graphs [14].
The first works on the total matching problem appeared in [1,18]. In [16], Manlove reports that lifted biclique inequalities, and show that the former are always facet-defining, while the latter are facet-defining when G is a bipartite graph. We also address complexity issues of the associated separation problem. In Section 4, we give an extended formulation for P T (G) when G is a complete bipartite graph. This formulation is based on Balas' technique to describe the convex hull of the union of polytopes [2] and on results on perfect graphs [3,4]. Using projection tools from [5] and through a careful analysis of the projection rays, we then project the extended formulation and show that P T (G) is described by the (exponential-sized, non-redundant) family of inequalities containing balanced and non-balanced lifted bicliques, and the inequalities from the natural linear relaxation of P T (G).
Notation. Given a graph G = (V, E), let n = |V | and m = |E|. For v ∈ V , we denote by δ(v) the set of edges incident to v. For a subset of vertices U ⊆ V , let G[U ] be the subgraph induced by U on G. A graph is chordal if every cycle of length greater or equal than four has a chord, that is, there is an edge connecting two non consecutive vertices of the cycle. Given a graph G, we let V (G) (resp., E(G)) be its set of vertices (resp., edges). A biclique K r,s is a complete bipartite graph, where the bipartition of V (K r,s ) is given by (A, B) with A = {v 1 , . . . , v r } and B = {w 1 , . . . , w s }.
It is balanced if r = s, non-balanced otherwise.
Preliminaries
We start with a natural linear relaxation of P T (G). Proposition 1. Let G(V, E) be a graph. The following is a linear relaxation for P T (G):
x v + e∈δ(v) y e ≤ 1 ∀v ∈ V (1) x v + x w + y e ≤ 1 ∀e = {v, w} ∈ E (2) x v , y e ≥ 0 ∀v ∈ V, ∀e ∈ E.(3)
(1)-(3) are called, node, edge, and nonnegativity inequalities, respectively. In [11], the authors prove that they are facet-defining for P T (G), for any graph G. The following definition introduces a useful tool to study total matching problems.
Definition 1. Given a graph G, the total graph T (G) of G is a graph whose vertex set is the set of vertices and edges of G, and where two vertices of T (G) are adjacent if and only if their corresponding elements are adjacent in G.
The stable set polytope STAB(G) of a graph G is the convex hull of the characteristic vectors of stable sets of G. The following fact has been observed in [11].
Proposition 2. Let G be a graph and T (G) its total graph. Then, P T (G) = STAB(T (G)).
Together with a result from [24], Proposition 2 already allows us to characterize P T (G) when G is a tree. Proof: Consider the total graph T (G) of a tree G. In [24,Theorem 5], it is proved that a connected graph is a tree if and only if its total graph is chordal. Hence, we have
P T (G) = STAB(T (G)) = x ∈ R |V (T (G))| ≥0 : v∈K x v ≤ 1, ∀ clique K of T (G) ,
where the first equality follows by Proposition 2 and the second since chordal graphs are perfect (see, e.g., [23]). Let K be a maximal clique in T (G). Since G has no cycles, the preimage of K in G is either a node and its neighborhood, or an edge and its endpoints. Hence, a maximal clique inequality corresponds to an inequality of the type (1) - (2). We deduce that (1) -(3) give a complete description of P T (G). To observe that the description is non-redundant, recall that [11] showed that inequalites (1) -(3) define facets.
Theorem 1 gives an alternative, polyhedral proof of the fact that a maximum total matching in a tree can be found in polynomial time [16], and in fact shows that even a total matching of maximum weight (with weights defined over the elements of the tree) can be found in polynomial time.
New families of facet-defining inequalities
Balanced biclique inequalities
The facet-defining inequality (2) can be seen as induced by a balanced biclique K 1,1 . We next derive a generalization of these inequalities, and we show that they are facet-defining for any graph. Lemma 1. Let G be a graph and K r,r be an induced balanced biclique of G with r ≥ 2. Then, the balanced biclique inequality:
v∈V (Kr,r)
x v + e∈E(Kr,r) y e ≤ r(4)
is facet-defining for P T (G).
Proof: Let V (K r,r ) = A∪B. The validity of the inequality follows from ν T (K r,r ) = r [14]. LetF = {z ∈ P T (G) : π T z = π 0 } be the face of P T (G) defined by (4), and let F = {z ∈ P T (G) : λ T z = λ 0 } be a face of P T (G) such thatF ⊆ F . We prove that there exists a = 0 such that (λ, λ 0 ) = a(π, π 0 ).
Let e = {v, w} ∈ E[K r,r ] with v ∈ A, w ∈ B, and define the total matchings T v := (A\{v})∪{e}
and T w := (B \ {w}) ∪ {e}. Since |T v | = |T w | = r and χ[A], χ[B] ∈F ⊆ F , we have χ[T v ], χ[T w ] ∈ F ⊆ F . Hence, λ T χ[A] = λ T χ[T v ] = λ T χ[T w ] = λ 0 . We deduce therefore that λ v = λ w = λ e . Since e ∈ E[K r,r ] arbitrarily, we deduce λ v = λ w = λ e ∀v ∈ A, w ∈ B, e ∈ E[K r,r ]. Now, consider an element d / ∈ (A ∪ B ∪ E[K r,r ]) of G. Let M be a perfect matching of K r,r .
Note that at least one of T 1 := A ∪ {d} and T 2 := B ∪ {d}, and T 3 := M ∪ {d} is a total matching. We assume that T 1 is a total matching, the other cases following analogously.
χ[A], χ[T 1 ] ∈F ⊆ F
implies that λ d = 0. This completes the proof, since we have proved that (λ, λ 0 ) = a(π, π 0 ).
Separation of balanced biclique inequalities
We next address the problem of separating balanced biclique inequalities of fixed cardinality.
The following problem is NP-Complete [21].
Name: Weighted Edge Biclique Decision Problem (WEBDP).
Input: A complete bipartite graph G with edge weights u ∈ Z E , a number k ∈ N.
Decide: If there exists a subgraph of G that is a biclique with vertex partition (A, B) such that
e∈A×B u(e) ≥ k.
The previous result implies that the following problem is NP-Complete.
Name: Weighted Edge Biclique Decision Problem of fixed Cardinality (WEBDPC).
Input: A complete bipartite graph G(V, E) with edge weights u ∈ N E , a number k ∈ N and q ≤ |V |. Decide: If there exists a subgraph of G that is a biclique with vertex partition (A, B), |A| = |B| = q, such that e∈A×B u(e) ≥ k.
Indeed, WEBDPC is clearly in NP. Suppose we want to solve WEBDP on input G, u, k. Let u ′ ∈ {N ∪ {0}} E defined as u ′ e = u e + u ∞ for e ∈ E(G). For q = 1, . . . , |V (G)| solve an instance 4 of WEBDPC with input G, u ′ , k + q 2 u ∞ , q. Suppose any of those is a yes-instance. Then G has a biclique with vertex partition (A, B), |A| = |B| = q, that satisfies
e∈A×B u ′ (e) ≥ k + q 2 u ∞ ⇔ ✘ ✘ ✘ ✘ q 2 u ∞ + e∈A×B u(e) ≥ k + ✘ ✘ ✘ ✘ q 2 u ∞ .
Hence, G, u, k is a yes-instance for WEBDP if and only if one of the |V (G)| instances of WEBDPC defined above is a yes-instance, concluding the proof.
We next show that NP-completeness of WEDPC implies NP-completeness of the following.
Name: Separation Problem for Balanced Biclique Inequalities of a Given Size in complete bipartite graphs (SPBBIGS)
Input: A complete bipartite graph G(A ′ ∪ B ′ , E), a point (x * , y * ) ∈ Q |V |+|E| + , r ∈ N, r ≤ |A ′ |.
Decide: If there exists a violated biclique inequality of P T (G) of G with r nodes on each side of the partition, that is,
there exist A ⊆ A ′ , B ⊆ B ′ , |A| = |B| = |r| such that: v∈A∪B x * v + e∈A×B y * e > r.(5)
Clearly SPBBIGS belongs to NP. Suppose we have an algorithm for SPBBIGS, we show an algorithm for WEBDPC. Let G, u, k, q be an input to WEBDPC, where G has vertex bipartition (A ′ , B ′ ).
Define y * e = u e q k−1 for each edge e of G, x * = 0, and run the algorithm for SPBBIGS on G, x * , y * , r = q. Suppose there exists an inequality of the form (5) separating (x * , y * ). Then for some A ⊆ A ′ ,
B ⊆ B ′ , we have q k − 1 e∈A×B u e = e∈A×B y * e = v∈A∪B x * v + e∈A×B y * e > r = q ⇔ e∈A×B u e > k − 1 ⇔ e∈A×B u e ≥ k,
thus giving a positive answer to WEBDPC. Similarly, if no inequality of the form (5) is violated, we
have e∈A×B u e < k, for all A ⊆ A ′ , B ⊆ B ′ , |A| = |B| = r = q.
Non-balanced bicliques inequalities
Consider a general non-balanced biclique K r,s , with s > r ≥ 2, of a graph G. By mimicking (4), it is natural to ask whether v∈V (Kr,s)
x v + e∈E(Kr,s) y e ≤ s(6)
defines a facet. This inequality is indeed valid but not facet-defining. We next give a strengthening of (6) and show that it defines a facet when G is a bipartite graph.
Proposition 3. Let K r,s with s > r ≥ 2 be a non-balanced biclique. Then, for all v ∈ A, the non-balanced lifted biclique inequality:
v∈A α v x v + w∈B x w + e∈E(Kr,s) y e ≤ s(7)
where
α v = s − (r − 1) if v = v, 1 otherwise, is facet-defining for P T (K r,s ). Proof: W.l.o.g., assume v = v 1 .
We show an appropriate lifting of the (clearly) valid inequality (7), thus showing that (7) is valid. We perform a sequential lifting [17,25] of the coefficients of {x v } v∈A following the ordering 1, 2, . . . , r of subscripts of nodes in A. We have:
w∈B x w + e∈E(Kr,s) y e ≤ s of P ′ := P T (K r,s ) ∩ {x v = 0, v ∈ A} givesα v 1 := s − max w∈W x w + e∈E(Kr,s) y e : x v 1 = 1, z ∈ N, x j = 0, j ∈ A \ {v 1 } , where N is the set of vertices of P T (K r,s ). α v 1 = s − (r − 1)
, since the characteristic vector of any matching of K r,s not covering v 1 is a feasible solution for the integer program above, while any total matching T with v 1 ∈ T does not contain any vertex from W or edge incident to v 1 . Now, we
claim that α v = 1 for v ∈ A \ {v 1 }. We have α v 2 := s − max w∈W x w + e∈E(Kr,s) y e + (s − r + 1)x v 1 : x v 2 = 1, z ∈ N, x j = 0, j ∈ V \ {v 1 , v 2 } and it is easy to see that α v 2 is achieved by setting (x, y) = χ[M ∪ {v 1 }], where M is a matching of size r − 2 in G[R ∪ S \ {v 1 , v 2 }].
Thus α v 2 = 1. Iteratively, at the step i of the sequence we have:
α vi := s−max w∈B x w + e∈E(Kr,s) y e +(s−r+1)x v1 + i−1 ℓ=2 x v ℓ : x vi = 1, z ∈ N, x j = 0, j ∈ A\{v 1 , . . . , v i−1 } .
Repeating the argument above we obtain α v i = 1 and conclude that (7) is valid for P T (G). Now letF be the face of P T (G) defined by (7) and
let F = {z ∈ P T (G) : λ T z = λ 0 } be a face of P T (G)
such thatF ⊆ F . By repeating the same argument as in the proof of Lemma 1 and recalling that
s > r ≥ 2, we deduce that λ v = λ w = λ e for all v ∈ A \ {v 1 }, w ∈ W, e ∈ E[K r,s ]. Moreover,F ⊆ F implies λ T χ[A] = λ T χ[B], which in turns implies λ v 1 + r i=2 λ v i = s j=1 λ w j . Thus, λ v 1 = (s − r + 1)λ v 2 .
The following proposition shows that, in a bipartite graph G, the lifting procedure exposed generates facet-defining inequalities.
Lemma 2. Let G(V, E) be a bipartite graph and K r,s a subgraph of G. The non-balanced lifted biclique inequalities (7) are facet-defining for P T (G).
Proof: Let V (K r,s ) = A ∪ B and F be the face induced by the non-balanced lifted biclique inequality associated to K r,s . By Proposition 3, we have a set S of |V (K r,s )| + |E(K r,s )| affinely independent points that lie in F whose support is contained in the elements of K r,s . For each
element d of G with d / ∈ Q := A ∪ B ∪ E[K r,r ], we give a total matching M d such that χ[M d ] ∈ F , and S ∪ {M d } d∈(V ∪E)\Q is linearly independent. Let d be an element of G with d / ∈ Q. If d is not adjacent to A, let M d = A ∪ {d}. Else, d is not adjacent to B since G is bipartite, and we let M d = B ∪ {d}. Clearly, M d ∈ F
, and the matrix having as columns vectors from S ∪ d∈(V ∪E)\Q M d has the following form:
M = M 1 M 2 0 χ[{d}] d∈(V ∪E)\Q ,
where the first set of rows is indexed over elements of K r,s , M 1 is the collection of vectors from S restricted to nodes of K r,s , and M 2 is an appropriate matrix. Since M 1 has full rank and the bottom right submatrix of M is the identity matrix, M has full rank, and the thesis follows.
The Total Matching Polytope of Complete Bipartite Graphs
In this section, we give a complete and non-redundant description of P T (G) when G is a complete bipartite graph. Our argument is as follows. In Section 4.1, we give a (simple) algorithm for solving the maximum weighted total matching problem on a complete bipartite graph G. In Section 4.2,
we use this algorithm, results on perfect graphs, and Balas' classical theorem on the convex hull of the union of polytopes to give a compact extended formulation Q for P T (G). Then, in Section 4.3,
we study the projection cone associated to Q to deduce the following.
Theorem 2. Let G be a complete bipartite graph. A complete and non-redundant description of P T (G) is given by the basic inequalities (1) -(3) and, for each balanced (resp., non-balanced) complete bipartite subgraph of G, the balanced biclique inequality (4) (resp., non-balanced lifted biclique inequality (7)).
Throughout the section, fix a complete bipartite graph
G = K r,s , with V (G) := V = A ∪ B, A = {v 1 , . . . , v r }, and B = {w 1 , . . . , w s }.
Algorithm
A total matching T of K r,s satisfies at least one of T ∩ A = ∅ and T ∩ B = ∅. For U ∈ {A, B}, let T (K r,s ) \ U be the subgraph of the total graph T (K r,s ) of K r,s obtained by removing nodes corresponding to elements of U and the edges incident to them.
We can solve the maximum weighted total matching problem on G by solving the maximum weighted stable set on T (K r,s ) \ U for U ∈ {R, S}, and selecting the solution of maximum weight.
The next lemma shows that such graphs have a special structure.
7
Lemma 3. Let U ∈ {A, B}. The graph T (K r,s ) \ U is perfect.
Proof: Suppose w.l.o.g. that U = B. We denote by q(i, j) (resp., v ′ i ), the vertex or T (K r,s ) associated to the edge e = {v i , w j } (resp., vertex v i ∈ A) of the original graph K r,s . We prove that neither T (K r,s ) \ B nor T (K r,s ) \ B contain an odd cycle with 5 or more nodes. The statement then follows from the well-known characterization of perfect graphs [3].
We start with T (K r,s ) \ B. By construction, for i = 1, . . . , r, every vertex v ′ i lies in exactly one inclusionwise maximal clique (corresponding to edges adjacent to v i in K r,s ) and it is not adjacent to any node outside this clique. Thus, no odd cycle with at least 5 nodes contains a vertex v ′ i . Hence, an odd cycle C contains only vertices of the kind q(i, j). We call i (resp. j) the first (resp. second) entry of the vertex q(i, j). Note that no three consecutive vertices of C can share the same first or second entry; on the other hand, two consecutive vertices of C must share the first or the second entry. Hence, if we let C = {q 0 , q 1 , . . . , q k−1 }, we can assume w.l.o.g. that, for ℓ odd, q ℓ shares the first entry with q ℓ+1 and the second entry with q ℓ−1 (indices are taken modulo k). However, this contradicts k being odd.
We now focus on
T (K r,s ) \ B. Let C = {q 0 , . . . , q k−1 }, k ≥ 5 be an odd cycle in T (K r,s ) \ B. First observe that V (C) ∩ A = ∅. Indeed, suppose by contradiction that v ′ i ∈ V (C) ∩ A, and let w.l.o.g. v ′ i = q 0 . Then q ⌈ k 2 ⌉ = q(i, j) and q ⌊ k 2 ⌋ = q(i, ℓ) for some indices j, ℓ. Then q ⌈ k 2 ⌉ and q ⌊ k 2 ⌋ are not adjacent in T (K r,s ) \ B, a contradiction. Hence, V (C) ∩ A = ∅, and let w.l.o.g. q 0 = q(1, 1).
First assume that k = 5. Since q 2 , q 3 are not adjacent to q 0 but they are adjacent to each other, we can assume w.l.o.g. that q 2 = q(1, 2), q 3 = q(2, 1). Since q 1 is adjacent to q 0 but not to q 3 , we must have q 1 = q(2, t), with t = 1. Since q 1 is adjacent to q 2 , t = 2. Symmetrically, q 4 = q(p, 2) with p = 1, 2. On the other hand q 1 and q 4 are not adjacent, hence they must share one of their two entries. Hence either t = 2 or p = 2, a contradiction. Now assume k ≥ 7. Similarly to above, w.l.o.g. q ⌊ k 2 ⌋ = q(1, 2), q ⌈ k 2 ⌉ = q(2, 1). Since q ⌊ k 2 ⌋−1 is not adjacent to v 0 or q ⌈ k 2 ⌉ , we must have q ⌊ k 2 ⌋−1 = q(t, 1) for t = 1, 2. Symmetrically, q ⌈ k 2 ⌉+1 = q(1, p) for p = 1, 2. Again, using the fact that q ⌊ k 2 ⌋−1 and q ⌈ k 2 ⌉+1 are not adjacent, we deduce t = 1 or p = 1, a contradiction.
Note that, if we consider T (K r,s ) instead of T (K r,s ) \ U for U ∈ {A, B}, the graph is no longer perfect. For instance, it can be easily checked that the total graph T (K 2,2 ) contains an odd hole.
Lemma 3 allows us to use classical semidefinite techniques [13] to solve the maximum weighted stable set problem on T (K r,s ) \ U for U ∈ {A, B}. However, in our case we do not need to employ semidefinite programming, because of the following.
Extended formulation
Define the two polytopes P A := {z ∈ P T (K r,s ) : z w = 0 for w ∈ B}, P B := {z ∈ P T (K r,s ) : z v = 0 for v ∈ A}.
Using Lemma 3 and the description of the stable set polytope of perfect graphs [4], we can describe P A (and analogously P B ) completely using cliques inequalities:
P A = {z ≥ 0 : u∈K z u ≤ 1 for K clique of V (T (K r,s ) \ B)},
and Observation 1 implies that this description has linear size. We deduce the following. Corollary 1.
P A = (x, y) ∈ R |A|+|E| ≥0 :
x
v + e∈δ(v) y e ≤ 1, ∀v ∈ A; e∈δ(w) y e ≤ 1, ∀w ∈ B , P B = (x, y) ∈ R |B|+|E| ≥0 : x w + e∈δ(w) y e ≤ 1, ∀w ∈ B; e∈δ(v) y e ≤ 1, ∀v ∈ A .
Following the discussion from Section 4.1, we can write P := P T (G) = conv(P A ∪ P B ). Balas showed that the convex hull of the union of two polytopes has an extended formulation that can be described in terms of the original formulations of the polytopes [2]. When applied to P A , P B defined as above, Balas' result gives the extended formulation for P from Corollary 2, where, for later usage, we also report certain dual multipliers.
Corollary 2. The following is an extended formulation for P :
Q := (x, y, λ 1 , y 1 e ) ∈ R |V |+|E|+1+|E| :
x v + e∈δ(v) y 1 e − λ 1 ≤ 0, ∀v ∈ A [u 1 v ] e∈δ(w) y 1 e − λ 1 ≤ 0, ∀w ∈ B [u 1 w ] x w + e∈δ(w) (y e − y 1 e ) + λ 1 ≤ 1, ∀w ∈ B [u 2 w ] e∈δ(v) (y e − y 1 e ) + λ 1 ≤ 1, ∀v ∈ A [u 2 v ] − y 1 e ≤ 0, ∀e ∈ E [u 1 e ] −x v ≤ 0, ∀v ∈ V −y e + y 1 e ≤ 0, ∀e ∈ E [u 2 e ] − λ 1 ≤ 0, [u λ 1 ] λ 1 ≤ 1 [u λ 2 ] .
Projection
In order to project the extended formulation defined in the previous section to the space P lives in, we study the associated projection cone.
Theorem 3. [5, Theorem 2.1] Let
Q = {(x, z) ∈ R n ≥0 × R p : Ax + Bz ≤ d} where A, B
have m rows, and define its projection cone C P := {u ∈ R m : uB = 0, u ≥ 0}. The projection of Q onto the x-space is proj x (Q) = {x ∈ R n ≥0 : uAx ≤ ud, for all extreme rays u of C P }.
We next deduce a description of P in terms of the projection cone from Theorem 3.
Lemma 4. P = {(x, y) ∈ R n+m ≥0 : v∈A u 1 v x v + w∈B u 2 w x w + e={v,w}∈E min j=1,2 (u j v + u j w )y e ≤ max j=1,2 w∈V u j w , ∀u ∈ Y }, where Y is the set of vectors u ∈ R 2(|A|+|B|) ≥0
that satisfy 2(|A| + |B|) − 1 linearly independent constraints from the set
u v = 0 for v ∈ V u 1 v + u 1 w = u 2 v + u 2 w for v ∈ A, w ∈ B. (8) v∈V u 1 v = v∈V u 2 v .
Proof: By specializing the description of C P from Theorem 3 to the submatrix of the constraint matrix in Corollary 2 corresponding to variables to be projected out, we obtain:
C P = u : u 1 v + u 1 w − u 1 e = u 2 v + u 2 w − u 2 e , ∀e = {v, w} ∈ E (9) v∈V u 1 v + u λ 1 = v∈V u 2 v + u λ 2 ,(10)u ≥ 0 .(11)
Using Theorem 3, all valid inequalities in the description of P have the following form:
v∈A u 1 v x v + w∈B u 2 w x w + e={v,w}∈E (u 2 v + u 2 w − u 2 e )y e ≤ w∈V u 2 w + u λ 2(12)
where u is an extreme ray of C P .
We next claim that, in (12), we can assume w.l.o.g. that, for e ∈ E, at least one of u 1 e , u 2 e is equal to 0. Indeed, since those variables are nonnegative, if they are both strictly positive we can decrease both by min{u 1 e , u 2 e } > 0 and obtain a stronger inequality (12). Similarly, at least one of u λ 1 = 0, u λ 2 = 0 holds. Hence, using (9) and (10), we can rewrite (12) as
v∈A u 1 v x v + w∈B u 2 w x w + e={v,w}∈E min j=1,2 (u j v + u j w )y e ≤ max j=1,2 w∈V u j w .
Let u be an extreme ray of C P . We first claim that the vector obtained from u by projecting out {u 1 e , u 2 e } e∈E , u λ 1 , u λ 2 is a nonnegative vector that satisfies 2(|A| + |B|) − 1 linearly independent 10 constraints from (8). By construction, u satisfies at equality a set S of 2(|A|+|B|)+2|E|+1 linearly independent constraints from (9)- (11). By basic linear algebra, any set of linearly independent constraints from (9)-(11) tight at u can be enlarged to a linearly independent set of inequalities tight at u of maximum cardinality. Hence we can assume w.l.o.g. that S contains the following set S ′ of linearly independent constraints. For e ∈ E, if u 2 e = u 1 e = 0, then constraints u 2 e = 0, u 1 e = 0 belong to S ′ . Else, from what argued above, we have u j e > 0 and u 3−j e = 0 for some j ∈ {1, 2}, and we let S ′ contain u j e = 0 and
u 1 v + u 1 w − u 1 e = u 2 v + u 2 w − u 2 e .
Similarly, either u λ 1 = 0 and u λ 2 = 0 are both valid and we let them belong to S ′ , or we let the one of them that is valid and
v∈V u 1 v + u λ 1 = v∈V u 2 v + u λ 2 belong to S ′ .
It is easy to see that constraints in S ′ are linearly independent, hence S \ S ′ is a set of 2(|A| + |B|) − 1 linearly independent constraints.
Note that an inequality (9) belongs to S \ S ′ only if both the variables u 1 e and u 2 e appearing in its support are set to 0. In particular u satisfies
u 1 v + u 1 w = u 2 v + u 2 w . Similarly, constraint (10) belongs to S \ S ′ only if v∈V u 1 v = v∈V u 2 v .
Hence, u satisfies at equality 2(|A| + |B|) − 1 linearly independent constraints from (8), and the claim follows.
Conversely, any nonnegative vector in the components {u 1 v , u 2 v } v∈V that satisfies any set of constraints from (8) can be extended to a vector of C P by appropriately adding components u 1 e , u 2 e , u λ 1 , u λ 2 , concluding the proof. We call fundamental a vector u ∈ Y whose associated inequality (as in Lemma 4) defines a facet of P different from (1)-(3), (4), (7). A set S of 2(|A| + |B|) − 1 linearly independent constraints from (8) is said to support u if u satisfies all constraints in S. For a set S supporting a u ∈ Y , we let G(S) be the graph that contains all vertices of K r,s , colors a vertex v blue (resp. red) if u 1 v = 0 (resp., u 2 v = 0) belongs to S, and contains edge vw if u 1 v + u 1 w = u 2 v + u 2 w belongs to S. Note that a node can be colored both blue and red in G(S) -we call such nodes bicolored. A node that is colored with exactly one of red and blue is monochromatic. A connected component of G(S) is non-trivial if it contains at least two nodes.
Let S be the set of constraints supporting a fundamental vector u. S is called canonical if, among all sets supporting u, G(S) maximizes the number of colored nodes (with each bicolored node counting twice) and, subject to the previous condition, maximizes the number of edges.
1 v = v∈V u 2 v . Suppose first that v∈V u 1 v = v∈V u 2 v is not contained in S.
Then there exists a node v ∈ V that is monochromatic, while all the other nodes of G(S) are bicolored. Assume w.l.o.g. that v is colored red. If u 1 v = 0, then u is the zero vector, which is clearly not fundamental. Hence, u 1 v > 0. Then the inequality corresponding to u is dominated by the edge inequality corresponding to vw for some w in the neighborhood of v, a contradiction.
Hence, assume that v∈V u 1 v = v∈V u 2 v belongs to S. Since u is not the zero vector and S is canonical, there must be nodes
v, v ′ (possibly v = v ′ ) with u 1 v = u 2 v ′ > 0, while nodes from V \ {v, v ′ } are bicolored.
If v, v ′ are on opposite sides of the vertex bipartition, then we can replace v∈V u 1 v = v∈V u 2 v with the constraint corresponding to edge vv ′ , showing that S is not canonical. Hence, assume that v, v ′ are on the same side of the bipartition, and asumme w.l.o.g. that v, v ′ ∈ A.
Then the inequality corresponding to u is again dominated by the edge inequality corresponding to vw for some w ∈ B, a contradiction. ⋄
Claim 3. Let u 1 v = 0 (resp. u 2 v = 0) for some v ∈ V .
Then v is colored blue (resp. red).
Proof of claim: Suppose w.l.o.g. that u 1 v = 0 but v is not colored blue. By Claim 2, G(S) has at least one edge. If u 1 v = 0 is linearly independent from constraints in S, then we can replace some edge constraint in S with u 1 v = 0, contradicting the canonicity of S. Hence u 1 v = 0 can be generated by constraints in S. Note that a set of equations generating u 1 v = 0 must contain either an edge constraint incident to v or v∈V u 1 v = v∈V u 2 v (or both), since those are the only other constraints whose support contains u 1 v . Hence, we can replace the one that appears in S with u 1 v = 0, contradicting the canonicity of S. ⋄ 1. We first show that v, w, cannot be colored with the same color. Suppose w.l.o.g. both v, w are colored blue. In particular, u 1 v = u 1 w = 0. Since vw is an edge of G(S), we have u 2 v + u 2 w = 0, which by nonnegativity of u implies u 2 v = u 2 w = 0. By Claim 3, u 1 v = 0, u 2 v = 0, u 1 w = 0, u 2 w = 0 belong to S. By hypothesis, u 1 w + u 1 w = u 2 w + u 2 w also belongs to S, contradicting the fact that S is linearly independent.
We conclude the proof by showing that both v, w are colored. Suppose by contradiction that this is not the case. Let C be the connected component of v, w in G(S). By Claim 1, C contains k ∈ N nodes and k − 1 edges. S therefore contains at least 2k − 1 constraints S ′ whose support intersects the variables associated to nodes of C. Suppose first v∈V u 1 v = v∈V u 2 v does not belong 12 to S. Then S ′ is contained in set of the k − 1 edge constraints, and the nonnegativity constraints associated to nodes of C. Since a node of C is not colored, there must be some node of C that is bicolored, call it v ′ . Then u 1 v ′ = u 2 v ′ = 0. Let w ′ be a node adjacent to v ′ in C. Since we showed above that two adjacent nodes cannot be colored with the same color, w ′ is not colored. Since v ′ w ′ ∈ E(G(S)), we obtain u 1 w ′ = u 2 w ′ . If u 1 w ′ = u 2 w ′ = 0, using Claim 2 we deduce that w ′ is colored, a contradiction. Hence, u 1 w ′ = u 2 w ′ > 0. Let u ′ be obtained from u by setting (u ′ ) 1 w ′ = 0. Observe that u ′ ∈ Y and that the inequality associated to u is dominated by a conic combination of the inequality associated to u ′ and the node inequality (1) associated to w ′ , a contradiction.
Now suppose v∈V u 1 v = v∈V u 2 v belongs to S, and recall that v is uncolored. If u 1 v = 0 or u 2 v = 0, we contradict Claim 3. Hence, u 1 v , u 2 v > 0. Let α = min{u 1 v , u 2 v },
and let u ′ be the vector obtained from u by decreasing both u 1 v and u 2 v by α. u ′ ∈ Y , and the inequality associated to u is a conic combination of the node inequality (1) associated to v and the inequality associated to u ′ , a contradiction. 3. By applying an argument similar to part 2. above, we deduce that, for each connected component C of G(S) with k nodes, there are at most 2k − 1 constraints from S whose support is contained on the set of variables corresponding to nodes from C. There is one more constraint in S that involves variables associated to nodes in C, and this is v∈V u 1 v = v∈V u 2 v . Hence, we distinguish the following cases: a) There are exactly two connected components in G(S), all its nodes are monochromatic, and all nodes from I are bicolored. b) There is exactly one connected component in G(S), all its nodes are monochromatic, and all nodes from I are bicolored, except one that is monochromatic.
We conclude the proof by showing that a) cannot happen. Indeed, let C α and C β be the two connected components. Since all nodes of each connected component are monochromatic (with nodes on the opposite sides of the same connected component having different colors) by part 1, all non-zero variables associated to vertices of C α (resp. C β ) have the same value α (resp. β).
Let A α , B α (resp. A β , B β ) be the two sides of the bipartition of component C α (resp. C β ). We assume without loss of generality that α = 1 ≤ β and |A β | ≤ |B β |. We also assume that nodes of A α ∪ A β are red and all nodes B α ∪ B β are blue, the other cases following analogously. Note that
v∈V u 1 v = v∈V u 2 v then implies |A α | + β|A β | = |B α | + β|B β |,(13)
which, together with |A β | ≤ |B β |, implies |A α | ≥ |B α |.
The inequality associated to u is as follows
v∈Aα∪Bα x v + v∈A β ∪B β βx v + vw∈(Aα∪A β )×(Bα∪B β ) y vw ≤ |A α | + β|A β |.(14)
If β = 1, then |A α | + |A β | = |B α | + |B β | and (14) is exactly the balanced biclique inequality associated to the subgraph of K r,s with bipartition (A α ∪ B α , A β ∪ B β ), a contradiction. So assume β > 1, let F β be the face induced by the inequality (14), and consider the non-balanced lifted biclique
inequality with bipartition (A α ∪ B α , A β ∪ B β ) and coefficient θ = |A α |+ |A β |− (|B α |+ |B β |)+ 1 ≥ 2 associated to a vertex w ∈ B β : v∈Aα∪Bα x v + v∈A β ∪(B β \{w}) x v + θx w + vw∈(Aα∪A β )×(Bα∪B β ) y vw ≤ |A α | + |A β |.(15)
Let F θ be the face induced by (15). We claim that F β ⊆ F θ , thus concluding the proof. Let z be an integer vector in F β . It suffices to show z ∈ F θ . We first claim that z is the incidence vector of a total matching T containing A β or B β . Indeed, suppose by contradiction this is not true.
Assume first T ∩ B β = ∅, T ∩ A β = ∅. Then the left-hand side of (14) is upper bounded by one of the following:
|B α | + |B β | (if T ∩ A α = ∅), |A α | + |A β | (if T ∩ B α = ∅)
. Both terms are strictly smaller than |A α | + β|A β |, a contradiction. Hence, let T ∩ A β = ∅. Then T ∩ (B α ∪ B β ) = ∅. If
T contains an edge incident to A β , then we can replace the edge with the node of A β it is incident to, strictly increasing the left-hand side of (14). Hence, either T ∩ A β = A β and we are done, or T can be enlarged by adding an element of A β , a contradiction. A similar argument shows that, if
T ∩ B β = ∅, we must have T ∩ B β = B β .
Suppose then that T ⊇ A β . Then T ∩ B β = ∅. Then by removing the variables corresponding to A β ∪ B β from the left-hand side of (14) and (15) and the sum of the coefficients of the variables corresponding to A β from the right-hand sides we obtain the same inequalities. Hence, z ∈ F θ .
Suppose therefore that T ⊇ B β . Then T ∩ A β = ∅ and by changing the left-and the right-hand sides similarly to above, we obtain that the left-hand sides of (14) and (15) coincide, the right-hand side of (14) is |A α | + β(|A β | − |B β |), and using (13) and the definition of θ, the right-hand side of (15) is |A α | + |A β | − (|B β | + |A α | + |A β | − (|B α | + |B β |)) = |B α | = |A α | + β(|A β | − |B β |).
We deduce again that z ∈ F θ , concluding the proof.
We can now complete the proof of Theorem 2. Since inequalities (1) - (3) and (4), (7) are facet-defining for P T (G), it suffices to show that no other inequality (other than positive scaling of those) defines a facet of P T (G). It suffices to consider inequalities associated to u as in Theorem 3, for u fundamental, the set S supporting u being canonical, and the smallest non-zero entry of u being 1. Thanks to Lemma 5, we have a good understanding of how such u, S look like.
Assume first v∈V u 1 v = v∈V u 2 v does not belong to S. Then G(S) has exactly one non-trivial connected component C (whose nodes are all monochromatic) and all other nodes are bicolored. are colored with opposite colors. W.l.o.g. assume that |A ′ | < |B ′ |, and suppose that all nodes in A ′ (resp., B ′ ) are colored red (resp., blue), the other cases following analogously. Similarly to the proof of Lemma 5, part 3, we deduce that all non-zero components of u indexed by nodes of C have the same value. We may assume that v ∈ A and v is colored red, for otherwise, by using v∈V u 1 v = v∈V u 2 v we obtain u = 0, a contradiction. Then, by a simple computation we derive that u 1 v = (|B ′ | − |A ′ |). The inequality associated to u is therefore
v∈A ′ x v + (|B ′ | − |A ′ |)x v + w∈B ′ x w + e={v,w},v∈A ′ ,w∈B ′ y e ≤ |B ′ |.
Theorem 1 .
1Let G be a tree. Then a complete and non-redundant description of P T (G) is given by (1) -(3).
Observation 1 .
1Let U ∈ {A, B}. The cliques of T (K r,s ) \ U correspond in G either to a node in {A, B} \ {U } and the edges incident to it, or to edges incident to a node in U . In particular, T (K r,s ) \ U has O(r + s) maximal cliques.
Lemma 5 .
5Let u be fundamental, S a canonical set supporting it, and I the isolated nodes of G(S).1. For each edge vw of G(S), v and w are monochromatic and colored with opposite colors. 2. If v∈V u 1 v = v∈V u 2 v does not belong to S, there is exactly one connected component C in G(S), all nodes of C are monochromatic, and all nodes from I are bicolored. 3. If v∈V u 1 v = v∈V u 2 v belongs to S, there is exactly one connected component C in G(S), all nodes of C are monochromatic, and all nodes from I are bicolored, except one that is monochromatic. Proof: Claim 1. G(S) does not have cycles. Proof of claim: Since K r,s is bipartite, any cycle C in G(S) must be even. Alternatively summing and subtracting the equalities corresponding to edges of C, we obtain 0 = 0, contradicting linear independence. ⋄ Claim 2. G(S) contains at least one edge. Proof of claim: Suppose the thesis does not hold. Then the only constraints in S are of the form u v = 0 and, possibly, v∈V u
2 .
2We know by part 1 that G has a non-trivial connected component C, and that all its nodes are monochromatic. Let k be the number of nodes of C. By Claim 1, C has k − 1 edges, and by Claim 2, the total number of colors used in nodes from C is exactly k.Since v∈V u 1 v = v∈V u 2 v does not belong to S, there are at most 2k − 1 constraints from S supported over some of the 2k variables indexed over nodes of C. No other non-trivial connected component of G(S) can then exist: since S contains 2(|V |) − 1 − (2k − 1) = 2(|V | − k) constraints in addition to those supported by variables indexed by nodes of C, every node not in C must be isolated and bicolored.
Moreover, nodes of C from opposite sides of the bipartition are colored with different colors. Similarly to the proof of Lemma 5, part 3, we deduce that all non-zero components of u have the same value. Hence, for some j ∈ {1, 2}, we haveu j v = u all v ∈ A ′ := A ∩ V (C), w ∈ B ′ := B ∩ V (C). If j = 2, the inequality associated to u is e={v,w},v∈A ′ ,w∈B ′ y e ≤ max{|A ′ |, |B ′ |}, while if j = 1 the inequality we obtain is v∈A ′ x v + w∈B ′ x w + e={v,w},v∈A ′ ,w∈B ′ y e ≤ max{|A ′ |, |B ′ |}.Both those inequalities coincide or are dominated by the balanced and non-balanced biclique inequalities associated to the pair A ′ , B ′ . Now, assume that v∈V u 1 v = v∈V u 2 v belongs to S. Then G(S) has exactly one singleton monochromatic node v, plus exactly one nontrivial connected component C induced by all other monochromatic nodes. Let A ′ := A ∩ V (C) and B ′ := B ∩ V (C). Again, nodes from A ′ and B ′
which is exactly the non-balanced lifted biclique inequality induced on G[(A ′ ∪ v) ∪ B ′ ] with the coefficient of the vertex v equal to |B ′ | − |A ′ |. This concludes the proof.
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Strengthened clique-family inequalities for the stable set polytope. A Letchford, P Ventura, Operations Research Letters. 49Letchford, A. and Ventura, P. (2021). Strengthened clique-family inequalities for the stable set polytope. Operations Research Letters, 49:586-589.
On the algorithmic complexity of twelve covering and independence parameters of graphs. D F Manlove, Discrete Applied Mathematics. 911-3Manlove, D. F. (1999). On the algorithmic complexity of twelve covering and independence parameters of graphs. Discrete Applied Mathematics, 91(1-3):155-175.
A polyhedral approach to edge coloring. G L Nemhauser, S Park, Operations Research Letters. 106Nemhauser, G. L. and Park, S. (1991). A polyhedral approach to edge coloring. Operations Research Letters, 10(6):315-322.
Generalizations of graphical parameters. E A Nordhaus, Proceedings of the International Conference on the Theory and Applications of Graphs. the International Conference on the Theory and Applications of Graphs642Nordhaus, E. A. (1976). Generalizations of graphical parameters. Proceedings of the International Conference on the Theory and Applications of Graphs, 642:420-425.
Clique family inequalities for the stable set polytope of quasi-line graphs. G Oriolo, Discrete Applied Mathematics. 132Oriolo, G. (2003). Clique family inequalities for the stable set polytope of quasi-line graphs. Discrete Applied Mathematics, 132:185-201.
On the facial structure of the set packing polyhedra. M Padberg, Mathematical Programming. 5Padberg, M. (1973). On the facial structure of the set packing polyhedra. Mathematical Programming, 5:199-215.
Maximum weighted edge biclique problem on bipartite graphs. A Pandey, G Sharma, N Jain, Algorithms and Discrete Applied Mathematics: 6th International Conference. Hyderabad, IndiaSpringer2020Pandey, A., Sharma, G., and Jain, N. (2020). Maximum weighted edge biclique problem on bipartite graphs. In Algorithms and Discrete Applied Mathematics: 6th International Conference, CALDAM 2020, Hyderabad, India, February 13-15, 2020, Proceedings, pages 116-128. Springer.
A branch-and-cut algorithm for the maximum cardinality stable set problem. F Rossi, S Smriglio, Operations Research Letters. 282Rossi, F. and Smriglio, S. (2001). A branch-and-cut algorithm for the maximum cardinality stable set problem. Operations Research Letters, 28(2):63-74.
Combinatorial optimization: polyhedra and efficiency. A Schrijver, Springer Science & Business Media24Schrijver, A. (2003). Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media.
Edge dominating sets in graphs. M Yannakakis, F Gavril, SIAM Journal on Applied Mathematics. 181Yannakakis, M. and Gavril, F. (1980). Edge dominating sets in graphs. SIAM Journal on Applied Mathematics, 18(1):364-372.
Lifting the facets of zero-one polytopes. E Zemel, Mathematical Programming. Zemel, E. (1978). Lifting the facets of zero-one polytopes. Mathematical Programming, pages 268-277.
| {'fraction_non_alphanumeric': 0.08867585380467345, 'fraction_numerical': 0.02312318309921665, 'mean_word_length': 3.2593572778827977, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 66, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The total matching polytope generalizes the stable set polytope and the matching polytope. In this paper, we first propose new facet-defining inequalities for the total matching polytope. We then give an exponential-sized, non-redundant description in the original space and a compact description in an extended space of the total matching polytope of complete bipartite graphs.ν T (G) can be computed in polynomial time for trees and it is NP-complete already for bipartite and planar graphs. The authors in[11]propose the first polyhedral study of the total matching problem, deriving facet-defining inequalities for the polytope associated to it, defined as follows.Given a total matching T , the corresponding characteristic vector is:with x corresponding to the vertex variables and y to the edge variables. Hence, the total matching polytope of a graph G = (V, E) is defined as: P T (G) := conv{χ[T ] ⊆ R |V |+|E| : T is a total matching of G}.Contributions and organization of the paper. The goal of this paper is to prove new results on the facial structure of P T (G), and highlight connections between P T (G) and the classical theory of polyhedral combinatorics. In Section 2, we introduce basic tools and employ them to show that the natural linear relaxation of P T (G) gives a complete description when G is a tree. In Section 3, we propose two new classes of inequalities for P T (G), dubbed balanced and non-balanced', 'arxivid': '2303.00328', 'author': ['Yuri Faenza \nIEOR Department\nColumbia University\n\n', 'Luca Ferrarini \nDipartimento di Matematica "F. Casorati"\nUniversità di Pavia\n\n'], 'authoraffiliation': ['IEOR Department\nColumbia University\n', 'Dipartimento di Matematica "F. Casorati"\nUniversità di Pavia\n'], 'corpusid': 257254948, 'doi': '10.48550/arxiv.2303.00328', 'github_urls': [], 'n_tokens_mistral': 16042, 'n_tokens_neox': 14372, 'n_words': 8865, 'pdfsha': '8d2aefb606bb83f608981f6ca2a762486c840646', 'pdfurls': ['https://export.arxiv.org/pdf/2303.00328v1.pdf'], 'title': ['The Total Matching Polytope of Complete Bipartite Graphs', 'The Total Matching Polytope of Complete Bipartite Graphs'], 'venue': []} |
arxiv |
Theoretical Aspects of Single-Spin Asymmetries Studies
S M Troshin
Institute for High Energy Physics
142284Protvino, Moscow RegionRussia
N E Tyurin
Institute for High Energy Physics
142284Protvino, Moscow RegionRussia
Theoretical Aspects of Single-Spin Asymmetries Studies
arXiv:hep-ph/9410202v1 3 Oct 1994
We consider theoretical background for experimental measurements of single-spin asymmetries. We stress the non-perturbative QCD aspects of observed asymmetries in hadronic reactions.
The very important direction in spin studies is connected with the longstanding problem of one-spin transverse asymmetries observed in violent hadron reactions [1], [2]. It is well known fact that the experimental data manifest significant one-spin transverse asymmetries.
For example, the behavior of analyzing power in hadronic scattering is rather surprising. Indeed, we could expect significant spin effects in soft reactions where the chiral SU(3) L × SU(3) R symmetry of QCD Lagrangian is spontaneously broken down to SU(3) V and therefore, there is no ground for helicity conservation. However, the observed analyzing power in the region of low transferred momenta is small and decreases with energy like an inverse power of energy.
On the other side, contrary to our QCD expectations analyzing power increases with transverse momentum when we trying to explore the region of short distances. In this kinematical region we should observe helicity conservation due to chiral invariance of QCD Lagrangian. Hadron helicity conservation in hard processes is a general principle of perturbative QCD. Violation of this principle have been observed in elastic pp-scattering, in two-body hadronic decays of J/ψ and there are also indications for such violation in the measurements of Pauli form factor F 2 (Q 2 ).
It is evident now that new ideas and experimental data are urgently needed to study dynamics of the spin effects.
We consider possible dynamical mechanism of spin effects in elastic scattereing. In Ref. [3] we used the notions of effective chiral quark model for the description of elastic scattering at small and large angles. Different aspects of hadron dynamics were accounted in the framework of effective Lagrangian presented as a sum of three terms:
L = L χ + L I + L C .(1)
L χ is the term responsible for the spontaneous chiral symmetry breaking:
L χ =ψ(i∂ µ γ µ −m)ψ + L 4 + L 6 .(2)
L 4 is the NJL four-fermion interaction, L 6 is the U A (1)-breaking 6-quark interaction. L χ is responsible for providing constituent quark masses and for the structure of constituent quark which includes valence quark and cloud of quark-antiquark pairs [4]. L I describes the interaction of constituent quarks and L C -their confinement. These parts of effective interaction were taken into account at phenomenological level. In such a model quarks appear as quasiparticles and have a complex structure. Besides its mass (consider u-quark as an example)
m u = m 0 u − g 4 uu − g 6 d d ss(3)
the constituent quark has a finite size. We assume that the strong interaction radius of q-quark r q is determined by its mass: r q = ξ/m q . The common feature of the chiral models is the representation of a baryon as an inner core carring the baryonic charge and an outer condensate surrounding this core [5]. Following this picture it is natural to represent a hadron consisting of the inner region where valence quarks are located and the outer region filled with quark condensate [3]. Such a picture for the hadron structure implies that overlapping and interaction of peripheral condensates at hadron collision occurs at the first stage. In the overlapping region the condensates interact and as a result the massive quarks appear. Being released the part of hadron energy carried by the peripheral condensates goes for the generation of massive quarks. In another words nonlinear field couplings transform kinetic energy into internal energy of dressed quarks (see the arguments for this mechanism in [6] and references therein for the earlier works). Of course, the number of such quarks fluctuates. The average number of quarks should be proportional to convolution of the condensate distributions D H c of colliding hadrons:
N(s, b) ∝ N(s) · D A c ⊗ D B c ,(4)
where the function N(s) is determined by the thermodynamics of transformation of kinetic energy of interacting condenstates to the internal energy of massive quarks. To estimate the N(s) it is feasible to assume that it is determined by the maximal possible energy dependence
N(s) ≃ κ (1 − x q ) √ s m q ,(5)
where x q is the average fraction of energy carried by valence quarks, m q is the mass of constituent quark. In the model [3] valence quarks located in the central part of a hadron are supposed to scatter in a quasi-independent way by the produced massive quarks at given impact parameter and by the other valence quarks. The averaged scattering amplitude of valence quark then may be represented in the form
f q (s, b) = [N(s, b) + N − 1] V q (b) ,(6)
where N = N 1 + N 2 is the total number of valence quarks in colliding hadrons, and V q (b) is the averaged amplitude of single quark-quark scattering [3].
In this approach elastic scattering amplitude satisfies unitarity equation since it is constructed as a solution of the following equation [7] F = U + iUDF (7) which is presented here in operator form. This relation allows one to satisfy unitarity provided the inequality ImU(s, b) ≥ 0 is fulfilled. The function U(s, b) (generalized reaction matrix) [7] -the basic dynamical quantity of this approach -is chosen as a product of the averaged quark amplitudes
U(s, b) = N q=1 f q (s, b)(8)
in accordance with assumed quasi-independent nature of valence quark scattering.
The b-dependence of function f q is related to the quark formfactor behavior ∝ ( q 2 + m 2 q /ξ 2 ) −2 and has a simple form [3] f q ∝ exp(−m q b/ξ). Following the lines of the above considerations, the generalized reaction matrix in the pure imaginary case can be represented in the form
U(s, b) = iG(N − 1) N 1 + α √ s m q N exp(−Mb/ξ),(9)
where M = N q=1 m q . This expression allows one to get the scattering amplitude as a solution of Eq. 7 which reproduces the main regularities observed in elastic scattering at small and large angles and consider spin phenomena. For that purposes system of equations for helicity amplitudes has been solved and dynamical mechanism of quark scattering with and without helicity flip has been considered.
In particular spin of constituent quark in this model comes from the orbital moment of cloud of quark-antiquark pairs while the polarization of valence current quark and the polarization of the cloud ofqq pairs compensate each other, e.g. for z-component of spin it means S q = S q V + S {qq} + L {qq} = 1/2 − 1/2 + 1/2 = 1/2.
The above compensation occurs due to account of axial U(1) A anomaly in the framework of effective QCD. While considering the constituent quark as an extended object we can represent its spin as follows:
S q = L {qq} = ωI q ,
where ω is the angular velosity of quark matter inside the constituent quark and I q its moment of inertia. These notions on spin of constituent quark follows from consideration of spin in the framework of effective lagrangian approach to QCD [8], [9].
It should be noted that since spin of constituent quark is due to its orbital angular momentum the corresponding wave function should be equal to zero at r = 0 due to centrifugal barrier. Such picture was advocated for the proton as a whole by Ralston and Pire [10].
Quark helicity flip in the model is provided by the mechanism of quark exchange where valence quark is exchanged with the quark produced under interaction of condensates. These quarks have different helicities and therefore such mechanism can lead to helicity flip quark scattering. Helicity non-flip quark scattering has another origin resulting from optical type of interaction. The above difference of these mechanisms leads to the different energy dependence and different phases of helicity flip and non-flip quark scattering amplitudes. Helicity amplitudes at hadron level in this approach as it was already mentioned are obtained as solutions of coupled system of equations which accounts unitarity in direct channel. Analyzing power in the framework of this model does not decrease with energy and has a non-zero value at s → ∞. The value of analyzing power depends on the fraction of energy carried by valence quarks k and the phase difference ∆(s) ∝ (1 − k) √ s/m q and has the following form
A(s, θ) = 4 sin ∆(s) (1 − k)N f (θ) 1 + O m 2 q s ,(10)
where N is the total number of valence quarks in colliding hadrons and f (θ) is the known function of scattering angle. Asymmetry here results from interference of helicity amplitudes which occurs due to resonance type of quark helicity flip scattering and continuum type of quark helicity non-flip scattering. Analyzing power at √ s = 2 TeV and −t = 10 (GeV /c) 2 in p ↑p -elastic scattering is predicted to be 12% while at −t = 5 (GeV /c) 2 -A = 7%. Other non-perturbative models [11], [12] also predict non-zero values for analyzing power in TeV energy range. To summarize, the measurement of analyzing power in elastic scattering at high energies will allow • test perturbative QCD, mechanism, get knowledge on the region of applicability of perturbative QCD, study the transition from nonperturbative to perturbative phase of QCD;
• study of hadron structure and non-perturbative effects: spontaneous breaking of chiral symmetry and confinement.
AcknowledgementsWe are grateful to N. Akchurin, M. Anselmino, A. D. Krisch and J. P. Ralston for interesting discussions, one of us (S.T.) express also his gratitude to A. D. Krisch for support of the visit to this Simposium.
A D Krisch, Plenary lecture given at the 9th Intern. Symp. on High Energy Spin Physics. Bonn, GermanyA. D. Krisch, Plenary lecture given at the 9th Intern. Symp. on High Energy Spin Physics, Bonn, Germany 1990.
K Heller, Proc. of the 10th Intern. Symp. on High Energy Spin Physics. of the 10th Intern. Symp. on High Energy Spin PhysicsNagoya177K. Heller, Proc. of the 10th Intern. Symp. on High Energy Spin Physics, Nagoya, 1992, p. 177.
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| {'fraction_non_alphanumeric': 0.05294068294319993, 'fraction_numerical': 0.027183488547696956, 'mean_word_length': 4.085324232081911, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We consider theoretical background for experimental measurements of single-spin asymmetries. We stress the non-perturbative QCD aspects of observed asymmetries in hadronic reactions.', 'arxivid': 'hep-ph/9410202', 'author': ['S M Troshin \nInstitute for High Energy Physics\n142284Protvino, Moscow RegionRussia\n', 'N E Tyurin \nInstitute for High Energy Physics\n142284Protvino, Moscow RegionRussia\n'], 'authoraffiliation': ['Institute for High Energy Physics\n142284Protvino, Moscow RegionRussia', 'Institute for High Energy Physics\n142284Protvino, Moscow RegionRussia'], 'corpusid': 119424529, 'doi': '10.1063/1.48912', 'github_urls': [], 'n_tokens_mistral': 3597, 'n_tokens_neox': 3047, 'n_words': 1981, 'pdfsha': '84958f258d931d3cef09cc1feb58dd5660848b5c', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9410202v1.pdf'], 'title': ['Theoretical Aspects of Single-Spin Asymmetries Studies', 'Theoretical Aspects of Single-Spin Asymmetries Studies'], 'venue': []} |
arxiv |
Phonons as a Probe of the Low Temperature Metal Insulator Transition in Na 0.75 Co 0.95 Ni 0.05 O 2
M Premila
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
A Bharathi
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
P Yasodha
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
N Gayathri
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
Y Hariharan
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
C S Sundar
Materials Science Division
Indira Gandhi Centre for Atomic Research
Kalpakkam -603 102T.N
Phonons as a Probe of the Low Temperature Metal Insulator Transition in Na 0.75 Co 0.95 Ni 0.05 O 2
Nickel substitution at the cobalt site in Na x CoO 2 induces a Metal Insulator Transition (MIT) and the temperature at which this occurs (T MIT ) increases with Ni content. Low temperature far infrared measurements on polycrystalline samples of Na 0.75 CoO 2 and Na 0.75 Co 0.95 Ni 0.05 O 2 (T MIT = ~ 150K ) were carried out to look for signatures of this transition. While both the samples show an evident splitting of the high frequency Co-O mode on lowering the temperature, dramatic changes are observed in the low frequency sodium mode in Na 0.75 Co 0.95 Ni 0.05 O 2 clearly pointing out to an ordering of the sodium ions in the low temperature insulating state.
INTRODUCTION
Layered oxides of the form Na x CoO 2 have attracted the attention of researchers after the recent discovery of superconductivity at T c ~5K in the hydrated cobaltate Na .3 CoO 2 .1.3H 2 O [1], while the related anhydrous metallic compound Na .75 CoO 2 exhibits enhanced thermopower [2]. Crystal structures of these compounds involve CoO 2 layers formed from edge sharing CoO 6 octahedra, alternating with Na ions in two partially occupied sites -Na(1) and Na (2) (Fig.1) [3]. The physical properties in these materials strongly depend on the sodium concentration and for x=0. 5 the system undergoes a transition to the insulating state at ~53K. The observed insulating transition is said to be associated to a charge ordering in the CoO 2 planes which further induces an ordering of the sodium ions [4]. Substituting the Co sub lattice in Na .75 CoO 2 with nickel induces drastic changes in its electronic properties and the system undergoes a transition to the insulating state ( Fig.1) [5]. Since infrared spectra are sensitive to these local changes in the system associated with the charge ordered metal -insulator transition, it is of immense interest to follow the phonon modes across the metal -insulator transition. Here we report the low temperature (300K-77K) far infrared absorption measurements on polycrystalline samples of Na 0.75 CoO 2 and Na 0.75 Co 0.95 Ni 0 . 05 O 2 (T MIT ~150K).
EXPERIMENTAL DETAILS
Polycrystalline samples of Na 0.75 CoO 2 and Na 0.75 Co 0.95 Ni 0 . 05 O 2 were prepared by the conventional solid state route [5]. FTIR absorption measurements were carried out on well characterized powder samples using a Bomem DA8 spectrometer operating with a resolution of 4 cm -1 . Measurements in the far infrared range were done using a combination of the extended 6µm mylar beamsplitter and the DTGS detector in the range 200 -750 cm -1 on samples pelletised with CsI. Low temperature measurements were carried out on samples mounted inside a JANIS continuous flow cryostat in the temperature range 300 -77K.
RESULTS AND DISCUSSION
Room temperature infrared absorption spectra of the undoped Na 0.75 CoO 2 revealed the four expected infrared active modes -the two low frequency modes at 241 and 274 cm -1 corresponding to the in plane and out of plane vibration of sodium and a broad high frequency mode at 574 cm -1 comprising of both the out of plane and in plane vibrations of cobalt ( Fig.2) [6]. It is to be noted that these low frequency sodium modes have been discerned for the first time and this has been possible because the present measurements have been carried out in the absorption geometry. We try to assign these Na modes based on first principle calculations [6]. The in plane mode is found to occur at lower frequency and is insensitive to Na position, thus the unsplit 241cm -1 mode can be assigned to this vibration. The out of plane mode on the other hand is very sensitive to the sodium position [6], with the Na(2) position giving rise to the higher and Na(1) position giving rise to the lower frequency modes. We assign the modes observed at 277 cm -1 and 297 cm -1 to these vibrations. Neutron scattering data on the sample with Na 0.7 CoO 2 indicates a Na(1):Na(2) occupation ratio of 0.5:0.2. The observed intensity ratios of the Na(2) and Na(1) out of plane vibrations (see Fig.2) are also consistent with these site occupancy ratios. Although no new modes are seen to appear with nickel substitution, significant changes in the phonon modes are seen on lowering the temperature (Fig.2). In particular, the intensity of the out of plane sodium mode at 277 cm -1 in the Ni doped sample (Na(1)) reduces, whereas that of the 297 cm -1 mode corresponding to Na(2) out of plane vibrations builds up (see inset a of Fig.2 and Fig.3a.). These observations clearly indicate that at low temperature Na(2) becomes the preferred site for occupation and this changeover in occupation occurs at 150 K which correlates with T MIT (Fig.3a.) In addition the constituent phonon modes exhibit significant changes in their line shapes associated with this transition (Fig.3). Whereas it is evident from inset (b) in Fig.2. that the corresponding sodium mode at 274 cm -1 in the undoped sample follows a regular anharmonic temperature dependence. The evolution of the Na modes observed in Na 0.75 Co 0.95 Ni 0 . 05 O 2 can be rationalized as discussed below. It has been conjectured [4,5] that in these systems charge ordering of the Co ions drives the system to the insulating state. While there is no direct evidence for the ordering of the Co 4+ ions, the induced Na ordering is believed to be a consequence of the former [4]. Hence in the present work we that the abrupt change in the intensity of the out of plane Na(1) and Na(2) modes along with the associated step like anomalies in both the line position and width provide definitive experimental evidence for the MIT being driven by Co 3+ /Co 4+ charge ordering. It is also seen from Fig.2, that in addition to the changes in the sodium mode, the high frequency Co-O mode at ~570 cm -1 exhibits a significant splitting on lowering the temperature for both Na 0.75 CoO 2 and Na 0.75 Co 0.95 Ni 0 . 05 O 2 . Although such a splitting has earlier been attributed to the charge ordering in the CoO 2 planes [7], we are unable to rationalize this behavior on the same grounds due to the fact that a similar effect has also been observed for Na 0.75 CoO 2 -a system that remains metallic till the lowest of temperatures. Hence the origin of the splitting of the Co-O mode still remains unclear although one can always attribute such a splitting as arising due to a regular temperature effect that causes a better resolution of the phonons at low temperatures.
Fig. 1 .
1(a) Variation of resistivity with temperature. (b) The crystal structure of Na x CoO 2 showing the two sodium sites[3].
Fig. 2
2Room temperature and 77K FTIR absorption spectra of Na .75 CoO 2 and Na 0.75 Co 0.95 Ni 0 . 05 O 2 . Inset (a) shows the dramatic flipping of sodium mode at T MIT in the nickel doped sample while inset (b) shows a regular anharmonic hardening of the sodium mode due to temperature effects for the undoped sample.
Fig. 3 :
3(a) Dramatic flipping of the out of plane sodium mode intensities correlating with T MIT . Sharp changes in line shape parameters: (b) phonon frequency corresponding to Na(1) and Na(2) sites, (c) & (d) phonon linewidths, that are seen to correlate with T MIT ~150K
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| {'fraction_non_alphanumeric': 0.041244320167773504, 'fraction_numerical': 0.04974950483513923, 'mean_word_length': 3.9877977919814063, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Nickel substitution at the cobalt site in Na x CoO 2 induces a Metal Insulator Transition (MIT) and the temperature at which this occurs (T MIT ) increases with Ni content. Low temperature far infrared measurements on polycrystalline samples of Na 0.75 CoO 2 and Na 0.75 Co 0.95 Ni 0.05 O 2 (T MIT = ~ 150K ) were carried out to look for signatures of this transition. While both the samples show an evident splitting of the high frequency Co-O mode on lowering the temperature, dramatic changes are observed in the low frequency sodium mode in Na 0.75 Co 0.95 Ni 0.05 O 2 clearly pointing out to an ordering of the sodium ions in the low temperature insulating state.', 'arxivid': 'cond-mat/0510274', 'author': ['M Premila \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n', 'A Bharathi \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n', 'P Yasodha \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n', 'N Gayathri \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n', 'Y Hariharan \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n', 'C S Sundar \nMaterials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N\n'], 'authoraffiliation': ['Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N', 'Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N', 'Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N', 'Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N', 'Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N', 'Materials Science Division\nIndira Gandhi Centre for Atomic Research\nKalpakkam -603 102T.N'], 'corpusid': 118897393, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2634, 'n_tokens_neox': 2152, 'n_words': 1468, 'pdfsha': '956d4d5eb944e218d8be8748ee677153555cb8c2', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0510274v1.pdf'], 'title': ['Phonons as a Probe of the Low Temperature Metal Insulator Transition in Na 0.75 Co 0.95 Ni 0.05 O 2', 'Phonons as a Probe of the Low Temperature Metal Insulator Transition in Na 0.75 Co 0.95 Ni 0.05 O 2'], 'venue': []} |
arxiv |
On the Vertex Operators of the Elliptic Quantum Algebra U q,p ( sl 2 ) k
5 Dec 2008
Wen-Jing Chang
Institute of Applied Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
P.O.Box 2734100190BeijingP.R. China
Graduate School of Chinese Academy of Sciences
China
Xiang-Mao Ding
Institute of Applied Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
P.O.Box 2734100190BeijingP.R. China
On the Vertex Operators of the Elliptic Quantum Algebra U q,p ( sl 2 ) k
5 Dec 2008
A realization of the elliptic quantum algebra U q,p ( sl 2 ) for any given level k is constructed in terms of three free boson fields and their accompanying twisted partners. It can be viewed as the elliptic deformation of Wakimoto realization. Two screening currents are constructed; they commute or anti-commute with U q,p ( sl 2 ) modulo total q-differences. The free fields realization for two types vertex operators nominated as the type I and the type II vertex operators are presented. The twisted version of the two types vertex operators are also obtained. They all play crucial roles in calculating correlation functions.
Introduction
Infinite-dimensional symmetries, such as Virasoro algebra (W-algebra, for more general) and affine Lie algebra play a central role in the two-dimensional Conformal Field Theories (2D CFT) [1]. While for the non-conformal (off-critical) integrable theory, their roles are taken over by the so called quantum algebras. From the algebraic point of view, there are three kinds of quantum algebras, according to their different exchange properties, which are nominated as rational, trigonometric and elliptic quantum algebras respectively. The quantum algebras of the former two kinds could be regarded as certain degenerate cases of the last one. For example, the quantum affine algebra (trigonometric), which is also known as the quantum group, and the Yangian double with central (rational) could be obtained as a certain limited case for the elliptic quantum algebras. Various versions of elliptic quantum algebras, also known as elliptic quantum groups [2,3,4] have been introduced, through an attempt to understand elliptic face models of statistical mechanics, and in its semiclassical limit, a CFT of Wess-Zumino-Witten (WZW) models on tori. Their roles are similar to the Kac-Moody algebras in WZW models. From Hopf algebra point of view, the elliptic quantum groups are nothing but quantum affine algebras equipped with a co-product different from the original one by a certain kind of twisting, so they can be viewed as quasi-Hopf algebras in the sense of Drinfeld [5]. There are two types of elliptic quantum groups which correspond to different types of integrable models: the vertex type A q,p ( sl N ) and the face type B q,λ (G), where G is a Kac-Moody algebra associated with a symmetrizable generalized Cartan matrix [6]. The former is closely related to vertex models, for example, the XYZ model, or equivalently, the eight vertex model in the principal regime [7]; while some face models, such as the Andrew-Baxter-Forrester (ABF) models [8] which are 'solid-on-solid' (SOS) face models, possess symmetries corresponding to the face type elliptic algebras.
From mathematical, it is natural to study these algebraic objects' structures and their representations. In physical applications, their representations are also required. The standard scheme to study integrable models in field theories or statistical mechanics is solving the following basic problems: to diagonalize the given Hamiltonian, and then to compute the correlation functions. Usually, it is quite difficult to solve such problems directly. While it has been indicated that the algebraic analysis method is an extremely powerful tool in studying solvable lattice models, especially in deriving the correlation functions. This method is based on the infinite dimensional quantum group symmetry possessed by a model and the representation theory of such symmetry. As a result, if one expect to perform algebraic analysis of the above both types of elliptic lattice models, he should firstly study the corresponding elliptic quantum groups and their representations.
In practice, the Wakimoto realization, or the so called free field method, which is an infinite dimensional extension of the Heisenberg algebra, is a quite effective and useful approach to study complicated algebraic structures and their representations. The well known example is the realization of the affine Lie algebras [9,10,11]. It is also a common method to obtain representations of quantum affine algebras [12] and Yangian double [13]. In [14,15], the XXZ model in the anti-ferromagnetic regime was solved by applying the level one representation theory of the quantum affine algebra U q ( sl 2 ). In studying higher spin extension of the XXZ model, free field realization at level k > 1 is required, which was constructed by several authors, such as [16,17,18]. Furthermore, in [19], Wakimoto representation of U q ( sl N ) with arbitrary level k ≥ 1 was given, and it plays a central role in understanding higher rank extension of the XXZ model. Free field method is also a powerful way to study the integrable massive field theory [20]. Please see [21] for a nice review on quantum affine algebra, free field realization and their applications. The level k free field representation of Yangian double DY (sl 2 ) and application to physical problems were discussed in [22,23]. The level one free field realization of the Yangian double with central DY (sl N ) was constructed in [24]; while the level k representation of DY (sl N ) and DY (gl N ) were given in [25]. It should also be remarked that the Yangian double with central DY ( sl 2 ) is the symmetry possessed by the Sine-Gordon model, which is the field theory limit of the restricted SOS (RSOS) model [26,27]. The bosonization of the RSOS model was considered in [28].
So following the algebraic approach, it is also important to obtain the free fields realization of elliptic quantum algebras. For example, in studying the RSOS model and its higher spin extension (i.e. the k-fusion RSOS model), the free field representation of U q,p ( sl 2 ) with any given level k is needed, which has been presented in [29] and this construction corresponds to a deformation of special coset WZW model. In fact, the elliptic algebra U q,p ( sl 2 ) is the Drinfeld realization of B q,λ ( sl 2 ) showed in [30].
In fact, at classical level, there are various models of representation for the current algebra, each one has its significance in certain application. Here we just mention two of them: free boson representation (Wakimoto construction) [9,10], and parafermion realization [32,33,34]. The similar realizations have been extended to quantum affine algebras and Yangian double with central. But there is no similar extension for the elliptic quantum algebras, except the parafermion realization for U q,p ( sl 2 ) of higher level k [29], and the parafermion representation of U q,p ( sl N ) with level one [31]. In fact, the realization presented in [29] is obtained by twisting the parafermionic realization of the quantum affine algebra U q ( sl 2 ), which can be considered as the elliptic deformation of the parafermion realization. The su(2) parafermion currents can be identified with the coset WZW model of sl(2) k ⊗ sl(2) 1 / sl(2) k+1 . Although parafermion theory is important in physics [32,33,34] and in mathematics [35], it seems that it cannot be used directly to study the elliptic quantum algebra. In fact, the bosonization of non-local currents for higher rank algebras is a huge project even in the classical level. So if one want to deal with the elliptic quantum algebra of higher rank, through bosonization of the non-local currents is not a practical way.
It has special interest for the algebra of the intertwining operators in the WZW model. It is derived by Knizhnik and Zamolodchikov that the matrix coefficients of the intertwining operators for the WZW model satisfy certain holonomic differential equations, i.e., the Knizhnik-Zamolodchikov (KZ) equation [36]. There is an analogue holonomic q-difference equation for the quantum affine vertex operators. They satisfy the quantum (q-deformed) Knizhnik-Zamolodchikov (q-KZ) equation [37]. So it is also expected that the representations of the elliptic quantum algebras are likely to be helpful to construct the elliptic type solutions of quantum Knizhnik-Zamolodchikov-Bernard (q-KZB) equation, which is a higher genus extension of the q-KZ equation [38]. Furthermore, to consider a higher rank extension of the RSOS model, we should construct free boson realization of U q,p ( sl N ). However only in the level-one case, parafermion realization of it was given in [31], free boson realization of it with higher level is unknown at present.
In this paper, we present a new free boson representation of U q,p ( sl 2 ) with arbitrary level k. It is different from the known one in [29] which was constructed in terms of non-local currents. Our construction could be viewed as a twisted version of the quantum semi-infinite flag manifolds [10], which is called the elliptic version of Wakimoto realization. The realization of the quantum intertwining operators, such as the screening currents and the vertex operators are also given. They are necessary ingredients for calculating correlation functions and investigating the irreducible representations. The screening currents commute or anti-commute with U q,p ( sl 2 ), and the integrations of such currents give the screening charges. For U q,p ( sl 2 ), there are two types of intertwining operators, which are called the type I vertex operators (VO) and the type II vertex operators respectively with their different physical significance. The former is a local operator which describes the operation of adding one lattice site, and the formula of the correlation functions can be expressed as traces of the product of these operators over irreducible representation space; while the latter play the role of particle creation or annihilation operators. In fact, in this paper we construct two screening currents, and the two types of VO's as well as their twisted ones. Moreover we hope this construction can be generalized to other cases, which will be considered in future study [39].
The paper is organized as follows. In section 2 we fix notations and recall the definition of U q,p ( sl 2 ). We give our construction of the free boson realization in section 3. In section 4, two screening currents of it are constructed in terms of free bosons. Finally in section 5 we give the free boson realization of the type I VO's, the type II VO's and their twisted ones.
The definition of elliptic algebra U q,p ( sl )
Elliptic quantum algebras are introduced to study integrable models with elliptic Boltzmann weights. There are two types of them: the vertex type and the face type. Here we restrict to the face type B q,λ (G) with G = sl 2 . For the face type elliptic quantum algebras U q,p ( sl 2 ), it can be considered as the Drinfeld realization of B q,λ ( sl 2 ). In this section, we give a short review on its definition.
Let us introduce a pair of parameters p and p * :
p = q 2r , p * = q 2r * = pq −2c (r * = r − c; r, r * ∈ R >0 )
here c is the central element of the elliptic algebra U q,p ( sl 2 ) defined below. Throughout this paper, the complex number q 0, |q| < 1 is fixed.
Definition 1 . The associative algebra U q,p ( sl 2 ) is generated by the central element c and the operator-valued currents H ± (z), E(z) and F(z) of the complex variable z satisfying the following commutation relations:
H ± (z)H ± (w) = ( z w ) 2( 1 r * − 1 r ) Θ p (q −2 z w ) Θ p (q 2 z w ) Θ p * (q 2 z w ) Θ p * (q −2 z w ) H ± (w)H ± (z), (2.1) H + (z)H − (w) = q 2c( 1 r * + 1 r ) ( z w ) 2( 1 r * − 1 r ) Θ p (pq −2−c z w ) Θ p (pq 2−c z w ) Θ p * (p * q 2+c z w ) Θ p * (p * q −2+c z w ) H − (w)H + (z), (2.2) H ± (z)E(w) = q ± c r * −2 ( z w ) 2 r * Θ p * (q 2± c 2 z w ) Θ p * (q −2± c 2 z w ) E(w)H ± (z), (2.3) H ± (z)F(w) = q ± c r +2 ( z w ) − 2 r Θ p (q −2∓ c 2 z w ) Θ p (q 2∓ c 2 z w ) F(w)H ± (z), (2.4) [E(z), F(w)] = 1 (q − q −1 )zw [δ(q −c z w )H + (q − c 2 z) − δ(q c z w )H − (q − c 2 w)], (2.5) E(z)E(w) = q −2 ( z w ) 2 r * Θ p * (q 2 z w ) Θ p * (q −2 z w ) E(w)E(z), (2.6) F(z)F(w) = q 2 ( z w ) − 2 r Θ p (q −2 z w ) Θ p (q 2 z w ) F(w)F(z), (2.7) where δ(x) = n∈Z x n , Θ p (z) = (z; p) ∞ (pz −1 ; p) ∞ (p; p) ∞ , (z; t 1 , · · · , t k ) ∞ = n 1 ,···,n k ≥0 (1 − zt n 1 1 · · · t n k k ),
by the definition, Θ p (z) is the standard elliptic theta-function, up to a constant. In general, we can define Θ t (z) for any parameter t = q 2ν (ν ∈ C) as
Θ t (z) = (z; t) ∞ (tz −1 ; t) ∞ (t; t) ∞ .
It should also be remarked that this elliptic algebra U q,p ( sl 2 ) degenerates to the quantum affine algebra U q ( sl 2 ) in the p → 0 (or r → ∞) limit.
In order to rewrite the relations (2.1)-(2.7) in more elegant form, the following parameterization will be used:
q = e −πi/rτ , p = e −2πi/τ , p * = e −2πi/τ * z = q 2u = e −2πiu/rτ .
In fact, we can use the notation of Jacobi theta function
θ ν (u) = q u 2 ν −u Θ q 2ν (q 2u ) (q 2ν ; q 2ν ) 3 ∞ ;
however for simplicity, we denote θ r (u) as θ(u) and θ r * (u) as θ * (u), which satisfy the quasiperiodicity properties
θ(u + r) = −θ(u), θ(u + rτ) = −e −πτi−2πiu/r θ(u),
and similar relations hold for θ * (u) with r replaced by r * . Then it is obvious to see that (2.1)-(2.7) can be rewritten as follows:
H ± (u)H ± (v) = θ(u − v − 1) θ(u − v + 1) θ * (u − v + 1) θ * (u − v − 1) H ± (v)H ± (u), (2.8) H + (u)H − (v) = θ(u − v − c/2 − 1) θ(u − v − c/2 + 1) θ * (u − v + c/2 + 1) θ * (u − v + c/2 − 1) H − (v)H + (u), (2.9) H ± (u)E(v) = θ * (u − v ± c/4 + 1) θ * (u − v ± c/4 − 1) E(v)H ± (u),(2.
10)
H ± (u)F(v) = θ(u − v ∓ c/4 − 1) θ(u − v ∓ c/4 + 1) F(v)H ± (u), (2.11) [E(u), F(v)] = 1 (q − q −1 )zw [δ(u − v − c/2)H + (u − c/4) −δ(u − v + c/2)H − (v − c/4)], (2.12) E(u)E(v) = θ * (u − v + 1) θ * (u − v − 1) E(v)E(u), (2.13) F(u)F(v) = θ(u − v − 1) θ(u − v + 1) F(v)F(u), (2.14)
where the parameterizations z = q 2u and w = q 2v are implicit in the above expressions. In the following, we will use this parameterization without mentioning them if they do not make confusion. Note that the above exchange relations have nice periodicity property with the notations of the Jacobi theta functions.
Fock Realization of U q,p ( sl 2 ) currents
In this section, we construct an elliptic deformation of Wakimoto realization for U q,p ( sl 2 ) using the free fields representation of U q ( sl 2 ) for generic level k. We first fix some conventions and review the Wakimoto realization of U q ( sl 2 ); then give our construction of the realization for U q,p ( sl 2 ) currents.
Fock space and the quantum affine algebra U q ( sl 2 )
Three kinds free bosons a, b and c are needed to construct the realization of U q ( sl 2 ). Their commutation relations of modes are
[a n , a m ] = [(k + 2)n][2n] n δ n+m,0 , [p a , q a ] = 2(k + 2), [b n , b m ] = − [n] 2 n δ n+m,0 , [p b , q b ] = −1, [c n , c m ] = [n] 2 n δ n+m,0 , [p c , q c ] = 1,
and the others vanish, where k is generic with k −2. Throughout this paper, the following standard symbol [n] will be used: [n] = (q n − q −n )/(q − q −1 ). The vacuum state of the Fock space |0 ≡ |0, 0, 0 is set as a n |0 = b n |0 = c n |0 = 0 (n ≥ 0), and a state |l, m 1 , m 2 is produced through
|l, m 1 , m 2 ≡ exp{l q a /2(k + 2) + m 1 q b + m 2 q c }|0 .
Obviously |l, m 1 , m 2 is the highest weight state of the bosonic Fock space, which is uniquely characterized by: a n |l, m 1 , m 2 = b n |l, m 1 , m 2 = c n |l, m 1 , m 2 = 0 (n > 0),
p a |l, m 1 , m 2 = l|l, m 1 , m 2 , p b |l, m 1 , m 2 = −m 1 |l, m 1 , m 2 , p c |l, m 1 , m 2 = m 2 |l, m 1 , m 2 ,
then the Fock space F l,m 1 ,m 2 is generated by negative modes a n , b n and c n (n < 0) acting on the highest weight state |l, m 1 , m 2 . The dual Fock space could be constructed with the same matter. For convenience, we denote free boson fields a(z; α) with α ∈ C and a ± (z) as follows:
a(z; α) = − n 0 a n [n] q −α|n| z −n + q a + p a ln z, a ± (z) = ±((q − q −1 ) n>0 a ±n z ∓n + p a ln q), and a(z; 0) ≡ a(z) for simplicity. Similarly, the free boson fields b(z; α), b ± (z) and c(z; α), c ± (z) can also be given. Normal order prescription : : is set by moving a n (n > 0) and p a to right, while moving a n (n < 0) and q a to left. For example,
: exp(a(z)) := exp(− n<0 a n [n] z −n )e q a z p a exp(− n>0 a n [n] z −n ).
With the help of the above free bosonic fields, four fields ψ ± (z) and e ± (z) are introduced through their actions on the Fock space F l,m 1 ,m 2 . Let us fix the actions of these currents on the Fock space as: ψ ± (z): F l,m 1 ,m 2 → F l,m 1 ,m 2 , e + (z) : F l,m 1 ,m 2 → F l,m 1 −1,m 2 −1 and e − (z) : F l,m 1 ,m 2 → F l,m 1 +1,m 2 +1 , respectively. Then these currents can be expressed as follows:
ψ + (z) =: exp[b + (q k 2 z) + a + (qz) + b + (q k 2 +2 z)] :, ψ − (z) =: exp[b − (q − k 2 z) + a − (q −1 z) + b − (q −( k 2 +2) z)] :, e + (z) = −1 (q − q −1 )z : {exp[b + (z) − (b + c)(qz)] − exp[b − (z) − (b + c)(q −1 z)]} :, e − (z) = −1 (q − q −1 )z : {exp[(b + c)(q −(k+1) z)] exp[a − (q −( k+2 2 ) z) + b − (q −(k+2) z)] − exp[(b + c)(q k+1 z)] exp[a + (q k+2 2 z) + b + (q k+2 z)]} :;
and the following proposition is straightforward: Proposition 1 . The fields given above satisfy the following commutation relations [19]:
[ψ ± (z), ψ ± (w)] = 0, (3.1) (z − q 2−k w)(z − q −2+k w)ψ + (z)ψ − (w) = (z − q 2+k w)(z − q −2−k w)ψ − (w)ψ + (z), (3.2) (z − q ±(2− k 2 ) w)ψ + (z)e ± (w) = (q ±2 z − q ∓ k 2 w)e ± (w)ψ + (z), (3.3) (z − q ±(2− k 2 ) w)e ± (z)ψ − (w) = (q ±2 z − q ∓ k 2 w)ψ − (w)e ± (z), (3.4) [e + (z), e − (w)] = 1 (q − q −1 )zw [δ(q −k z w )ψ + (q − k 2 z) − δ(q k z w )ψ − (q − k 2 w)], (3.5) (z − q ±2 w)e ± (z)e ± (w) = (q ±2 z − w)e ± (w)e ± (z). (3.6)
As a result, the above bosonic expression of the fields ψ ± (z) and e ± (z) give the Wakimoto realization of the quantum affine algebra U q ( sl 2 ) for generic level k.
Free boson realization of U q,p ( sl 2 )
In this subsection, we will present a new free fields realization of U q,p ( sl 2 ) with given level k.
The elliptic algebra U q,p ( sl 2 ) can be realized as the tensor product of the elliptic currents Ψ ± (z), e(z), f (z) of U q ( sl 2 ) and a Heisenberg algebra [30]. The elliptic currents Ψ ± (z), e(z) and f (z) of U q ( sl 2 ) are the fields satisfying the following elliptic commutation relations:
Ψ ± (z)Ψ ± (w) = Θ p (q −2 z w ) Θ p (q 2 z w ) Θ p * (q 2 z w ) Θ p * (q −2 z w ) Ψ ± (w)Ψ ± (z), (3.7) Ψ + (z)Ψ − (w) = Θ p (pq −2−c z w ) Θ p (pq 2−c z w ) Θ p * (p * q 2+c z w ) Θ p * (p * q −2+c z w ) Ψ − (w)Ψ + (z), (3.8) Ψ ± (z)e(w) = q −2 Θ p * (q 2± c 2 z w ) Θ p * (q −2± c 2 z w ) e(w)Ψ ± (z), (3.9) Ψ ± (z) f (w) = q 2 Θ p (q −2∓ c 2 z w ) Θ p (q 2∓ c 2 z w ) f (w)Ψ ± (z), (3.10) [e(z), f (w)] = 1 (q − q −1 )zw [δ(q −c z w )Ψ + (q − c 2 z) − δ(q c z w )Ψ − (q − c 2 w)], (3.11) e(z)e(w) = q −2 Θ p * (q 2 z w ) Θ p * (q −2 z w ) e(w)e(z), (3.12) f (z) f (w) = q 2 Θ p (q −2 z w ) Θ p (q 2 z w ) f (w) f (z). (3.13)
In [30] the elliptic currents Ψ ± (z), e(z) and f (z) of U q ( sl 2 ) are realized by twisted the parafermion realization of quantum affine algebra U q ( sl 2 ). In the following, we will give another bosonic realization, which can be viewed as the twisted quantum version of the realization on flag manifold given by B. Feigin and E. Frenkel [10]. To get the bosonic representation of these elliptic currents, besides the bosonic fields introduced in the last subsection, we need some new ones, such as a * ± (z),
a * + (z) = − n>0 a n [rn] z −n , a * − (z) = n>0 a −n [r * n] z n .
Here we name them as the twisted partners of the fields a ± (z) respectively; and b * ± (z) are introduced with the same matter by replacing a ±n with b ±n in the above expressions. In terms of them we introduce two twisting currents U ± (z; r, r * ) depending on parameters r and r * as:
U + (z; r, r * ) = exp[a * − (q r * + k 2 −1 z) + b * − (q r * −1 (q + q −1 )z)], U − (z; r, r * ) = exp[a * + (q −(r− k 2 −1) z) + b * + (q −(r−k−1) (q + q −1 )z)]
. By twisting the fields ψ ± (z) and e ± (z) in subsection 3.1 with U ± (z; r, r * ), we obtain the fields Ψ ± (z), e(z) and f (z) as: Obviously, the actions of the fields Ψ ± (z), e(z) and f (z) on the Fock space F l,m 1 ,m 2 are the same as ψ ± (z), e + (z) and e − (z) respectively, and we have the following proposition:
Ψ + (z) = U + (q k 2 z; r, r * )ψ + (z)U − (q − k 2 z; r, r * ), (3.14) Ψ − (z) = U + (q − k 2 z; r, r * )ψ − (z)U − (qProposition 2 .
The fields Ψ ± (z), e(z) and f (z) obtained above with k = c satisfy the given elliptic commutation relations (3.7)- (3.13
exp(− n>0 x n n ) = 1 − x; (1 − x) −1 = n≥0 x n .
By this proposition, we state that the currents in (3.14)-(3.17) with k = c give a bosonization of the elliptic currents of U q ( sl 2 ). From their actions on the Fock space, we know that all the currents keep the "spin-l/2" representation. Furthermore, it should also be remarked that, in the
p → 0 limit, Ψ + (z) → (ψ − (q k z)) −1 , Ψ − (z) → (ψ + (q k z)) −1 , e(z) → q −(2p b +p a ) (ψ − (q k/2 z)) −1 e + (z)
and f (z) → e − (z)q 2p b +p a (ψ + (q k/2 z)) −1 give a new free fields realization of U q ( sl 2 ). It is different from the one given in subsection 3.1.
However, the exchange relations of the currents given by the above boson fields do not have good periodicity property. In order to touch that goal, i.e., to construct a free boson realization of the elliptic quantum algebra U q,p ( sl 2 ), we need introduce a Heisenberg algebra generated bŷ p andq such that [q,p] = 1, and they commute with a, b and c. With the help of them we have new fields
H ± (u) = Ψ ± (z)e 2q (q ∓ k 2 z) 2p b +pa r (q ±(r− k 2 ) z)ˆp −1 r −p −1 r * , (3.18) E(u) = e(z)e 2q z −p −1 r * , (3.19) F(u) = f (z)z 2p b +pa r zˆp −1 r . (3.20)
To see them more clearly, we turn to the Fock space structure. From the above expressions, the Fock space structure of H ± (u), E(u) and F(u) could be given as tensor product of Fock spaces of Ψ ± (z), e(z) and f (z) respectively, with certain ones generated byq. The results of their actions are H ± (u) :
F l,m 1 ,m 2 ⊗ F n → F l,m 1 ,m 2 ⊗ F n+2 , E(u) : F l,m 1 ,m 2 ⊗ F n → F l,m 1 −1,m 2 −1 ⊗ F n+2 and F(u) : F l,m 1 ,m 2 ⊗ F n → F l,m 1 +1,m 2 +1 ⊗ F n .
Here F n is a trivial Fock space generated by |n ≡ e nq |0 . Then by applying proposition 2 and direct calculation, we can verify the following theorem:
Theorem 1 .
The fields in Eqns. (3.18)- (3.20) with k = c satisfy the commutation relations (2.8)- (2.14).
Corollary 1 . H ± (u)
, E(u) and F(u) given above realize the elliptic algebra U q,p ( sl 2 ) with given level k = c.
Construction of the screening currents
In 2D CFT, screening current is a primary field of the energy-momentum tensor with conformal weight 1, and its integration gives the screening charge. It has the property that it commutes with the currents modulo a total differential of certain field. This property ensures that the screening charge may be inserted in the correlators by changing their conformal charges without affecting their conformal properties. In this section, using the bosons a, b and c, we construct two screening currents S I (z) and S II (z), which are integral parts in the free fields approach. The two currents in this section commute or anti-commute with the currents modulo a total qdifference of some fields, so they could be regarded as a quantum deformation of the screening currents in 2D CFT. Denote a sort of q-difference operator with a parameter n ∈ Z >0 by
n ∂ z X(z) ≡ X(q n z) − X(q −n z) (q − q −1 )z ,
which is called a total q-difference of a function X(z). Moreover to eliminate the total qdifference, one can define the Jackson integral as
s∞ 0 X(z)d p z ≡ s(1 − p) n∈Z X(sp n )p n ,
for a scalar s ∈ C\{0} and a complex number p such that | p |< 1. So that,
s∞ 0 ( n ∂ z X(z))d p z = 0,
if it is convergent and take p = q 2n . For simplicity, we denote boson fields with parameters L i and M j (i, j ∈ N) as follows:
A + (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z; α) = n>0 [L 1 n] · · · [L s n] [M 1 n] · · · [M s+1 n] a n (q α z) −n ,
A − (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z; α) = n>0 [L 1 n] · · · [L s n] [M 1 n] · · · [M s+1 n] a −n (q α z) n ,
and further abbreviate the notations as:
A ± (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z) = A ± (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z; 0),
A ± (M|z; α) = A ± (L 1 , · · · , L s ; L 1 , · · · , L s , M|z; α);
similarly the fields B ± (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z; α) and C ± (L 1 , · · · , L s ; M 1 , · · · , M s+1 |z; α) can also be given. Using these fields we obtain the screening currents as:
S I (z) =: exp{c(z) + q a 2 + q b − r * q } :, S II (z) = −1 (q − q −1 )z : exp{A + (k + 2|z; k + 2 2 ) − 1 k + 2 (q a + p a ln z) +A − (−(k + 2)|z; − k + 2 2 )} ×{exp[−b − (z) − (b + c)(qz)] − exp[−b + (z) − (b + c)(q −1 z)]} :,
since they have the following properties:
Theorem 2 : S I (z), S II (z) satisfy the following relations with the currents H ± (z), E(z) and F(z)
given by (3.18)- (3.20):
H ± (z)S I (w) = S I (w)H ± (z) = O(1), E(z)S I (w) = −S I (w)E(z) = 1 ∂ w [ 1 z − ws 1 (z)] + O(E(z)S II (w) = S II (w)E(z) = O(1), F(z)S II (w) = S II (w)F(z) = (k+2) ∂ w [ 1 z − ws 2 (z)] + O(1), S II (z)S II (w) = θ k+2 (u − v + 1) θ k+2 (u − v − 1) S II (w)S II (z); S II (z)S I (w) = −S I (w)S II (z) = 1 ∂ w [ 1 z − ws 3 (z)] + O(1),
where the symbol O(1) means regularity ands i (z) (i = 1, 2, 3) are given by:
s 1 (z) =: exp{A − (r * |z; r − k 2 − 1) + q a 2 + B − (−(r − k − 2); r * , 1|z; −1) −p b ln z + B + (1|z; −1) + (2 − r * )q −p − 1 r * ln z} :, s 2 (z) =: exp{A − (−(k + 2)|z; k + 2 2 ) + A + (r − k − 2; k + 2, r|z; k + 2 2 ) − 1 k + 2 (q a + p a ln z) − B + (2; 1, r|z; −(r − k − 1)) + 2p b + p a +p − 1 r ln z} :,s 3 (z) =: exp{A − (−(k + 2)|z; − k + 2 2 ) + A + (k + 2|z; k + 2 2 ) + 1 2(k + 2) (kq a − 2p a ln z) − b(z; 1) + q b − r * q } : .
Proof: Straightforward calculation. Here we only take the last relation as an example, we denote
S II (z) ≡ −1 (q − q −1 )z [A(z) − B(z)]
where
A(z) =: exp{A − (−(k + 2)|z; − k + 2 2 ) + A + (k + 2|z; k + 2 2 ) − 1 k + 2 (q a + p a ln z) − b − (z) − (b + c)(qz)} :, B(z) =: exp{A − (−(k + 2)|z; − k + 2 2 ) + A + (k + 2|z; k + 2 2 ) − 1 k + 2 (q a + p a ln z) − b + (z) − (b + c)(q −1 z)} :, then S II (z)S I (w) = −1 (q − q −1 )z [A(z)S I (w) − B(z)S I (w)],
and since the following relations hold:
A(z)S I (w) = 1 qz − w : A(z)S I (w) :, | z |>| w |; S I (w)A(z) = 1 w − qz : A(z)S I (w) :, | w |>| z |; B(z)S I (w) = 1 q −1 z − w : B(z)S I (w) :, | z |>| w |; S I (w)B(z) = 1 w − q −1 z : B(z)S I (w) :, | w |>| z |,
we obtain the following relation on the analytic continuations: then by the definition of the total q-difference given above, we get
S II (z)S I (w) = −S I (w)S II (z) = −1 (q − q −1 )z [ 1 qz − w : A(z)S I (w) : − 1 q −1 z − w : B(zS II (z)S I (w) = −S I (w)S II (z) = 1 ∂ w [ 1 z − ws 3 (z)] + O(1).
It is easy to see that the actions of screening currents on the Fock space are S I (z) : F l,m 1 ,m 2 ⊗F n → F l+(k+2),m 1 +1,m 2 +1 ⊗ F n−r * and S II (z) : F l,m 1 ,m 2 ⊗ F n → F l−2,m 1 −1,m 2 −1 ⊗ F n , respectively. Please note that the current S II (z) acts trivially on F n , so it is also the screening current of the elliptic currents Ψ ± (z), e(z) and f (z). On the other side, the current S I (z) is not screening operator of them, even if we remove the term involvingq in S I (z) by hand. In fact, using the expressions of S I (z) and S II (z) given above, we can calculate the cohomology of the algebra and study the irreducibility of modules of it, which will be discussed separately in the future.
Realization of the Vertex Operators
In fact, in 2D CFT, besides the screening currents, the other important object that one should discuss is the primary field. In WZW model, the primary fields could be realized as the highest weight representation of Kac-Moody algebra, which are commonly known as vertex operators (VOs)or intertwiner operators. They play crucial role in calculating correlation functions. For quantum affine algebra, there are two types of vertex operators [21] or intertwiner operators, in which the type I is a local operator and could be regarded as the quantum counterpart of the primary field in 2D CFT. In this section, we'll present a new realization of the two types vertex operators and their twisted ones, which are different from the ones given in [29], as they base on distinct free fields realization of U q,p ( sl 2 ). For the definitions of the VO's and the properties of them, please see [21,30] for more details.
The type I and the type II Vertex Operators
For U q,p ( sl 2 ), the type I vertex operators and the type II vertex operators are defined to be the operators:
Φ l (u) : F → F ⊗ V l,v (5.1) Ψ * l (u) : V l,v ⊗ F → F (5.2)
acting on the total Fock space F , where V l,v is the spin l 2 representation generated by vectors v l m (m = 0, · · · , l). For convenience, we set the components Φ l,m (u) and Ψ * l,m (u) (m = 0, · · · , l) of the VO's as
Φ l (u − 1 2 ) = l m=0 Φ l,m (u) ⊗ v l m , Ψ * l (u − k + 1 2 )(v l m ⊗ ·) = Ψ * l,m (u).
The fundamental property of the vertex operators is that they satisfy the intertwining relations. In fact, intertwining operators of the algebra could be used to define the Vertex operators in some sense. Here we just pay our attention to the intertwining relations for the highest components Φ l,l (v) and Ψ * l,l (v):
H ± (u)Φ l,l (v) = θ(u − v + l 2 ∓ k 4 ) θ(u − v − l 2 ∓ k 4 ) Φ l,l (v)H ± (u),(5.
3)
E(u)Φ l,l (v) = Φ l,l (v)E(u), (5.4) F(u)Φ l,l (v) = θ(u − v + l 2 ) θ(u − v − l 2 ) Φ l,l (v)F(u); (5.5) H ± (u)Ψ * l,l (v) = θ * (u − v − l 2 ± k 4 ) θ * (u − v + l 2 ± k 4 ) Ψ * l,l (v)H ± (u), (5.6) F(u)Ψ * l,l (v) = Ψ * l,l (v)F(u), (5.7) E(u)Ψ * l,l (v) = θ * (u − v − l 2 ) θ * (u − v + l 2 ) Ψ * l,l (v)E(u). (5.8)
It should be noted that all the expressions of the fields in this section are considered to be normal-ordered. Let us write currents V ± (w; r, r * ) as:
V + (w; r, r * ) = exp{−A + (l, r * ; 2, k, r|w; k + 2 2 ) − B + (l, r * ; 1, k, r|w; k + 1)}, V − (w; r, r * ) = exp{A − (−l, r; 2, k, r * |w; k − 2 2 ) + B − (−l, r; 1, k, r * |w; −1)}.
Using them and the parameterization given in the second section, we give a new realization of the type I and the type II VO's in the following theorem:
Theorem 3 . If we express Φ l,l (v) and Ψ * l,l (v) through:
Φ l,l (v) = V + (w; r, r * ) exp{A − (l; 2, k + 2|w; k + 2 2 ) + A + (l; k, k + 2|w; k + 2 2 ) + B + (l; 1, k|w; 1)} × exp{ l q a 2(k + 2) − l 2r (2p b +p) ln w}, Ψ * l,l (v) = V − (w; r, r * ) exp{−A − (−l, k + 1; 1, k, k + 2|w; − k + 2 2 ) + A + (−l; 2, k + 2|w; k + 2 2 )
Actually the lower components Φ l,m (v) and Ψ * l,m (v) (m = 0, · · · , l) can be completely determined by the highest ones Φ l,l (v) and Ψ * l,l (v), since they obey the following recursive relations:
Φ l,m−1 (v) = F + (v − l 2 ) θ(p + h + l − m) θ(p + h) Φ l,m (v) (m = 0, 1, · · · , l), (5.9) Ψ * l,m−1 (v) = Ψ * l,m (v)E + (v − l + k 2 − r * ) θ * (m)θ * (p − l + m − 2) θ * (l − m + 1)θ * (p − 2)
(m = 0, 1, · · · , l), (5.10) where E + (v), F + (v) are half currents defined by
E + (v) = ̺ * C * E(v ′ ) θ * (v − v ′ + k/2 −p + 1)θ * (1) θ * (v − v ′ + k/2)θ * (p − 1) dw ′ 2πiw ′ , F + (v) = ̺ C F(v ′ ) θ(v − v ′ +p + h − 1)θ(1) θ(v − v ′ )θ(p + h − 1) dw ′ 2πiw ′ ,
and h is one of the Drinfeld generators of U q ( sl 2 ).
Here the contours are C * : |p * q k w| < |w ′ | < |q k w|, C : |pw| < |w ′ | < |w|, and the constants ̺, ̺ * are chosen to satisfy ̺̺ * θ * (1) ξ(q −2 ; p * , q) ξ(q −2 ; p, q)
= q − q −1 ,
where the function ξ(z; p, q) is ξ(z; p, q) = (q 2 z; p, q 4 ) ∞ (pq 2 z; p, q 4 ) ∞ (q 4 z; p, q 4 ) ∞ (pz; p, q 4 ) ∞ .
The twisted Vertex Operators
In this subsection, we discuss another two vertex operators Φ t l (u) and Ψ * t l (u) for U q,p ( sl 2 ), which are called the twisted type I VO's and the twisted type II VO's (or twisted intertwiners) respectively. Their definitions are analogously to the non-twisted ones. It means that Φ t l (u) and Ψ * t l (u) are also the operators of the same type as (5.1)-(5.2) and they also have the similar decompositions, with their components denoted as Φ t l,m (u) and Ψ * t l,m (u) (m = 0, · · · , l). However, the crucial difference between them lies in that Φ t l (u) and Ψ * t l (u) satisfy the twisted intertwining relations. Here we also only consider them for the highest components Φ t l,l (v) and Ψ * t l,l (v):
H ± (u)Φ t l,l (v) = θ(u − v + l 2 ∓ k 4 ) θ(u − v − l 2 ∓ k 4 ) Φ t l,l (v)H ± (u), (5.11) E(u)Φ t l,l (v) + Φ t l,l (v)E(u) = 0, (5.12) F(u)Φ t l,l (v) = − θ(u − v + l 2 ) θ(u − v − l 2 )
Φ t l,l (v)F(u); (5.13)
H ± (u)Ψ * t l,l (v) = θ * (u − v − l 2 ± k 4 ) θ * (u − v + l 2 ± k 4 )
Ψ * t l,l (v)H ± (u), (5.14)
F(u)Ψ * t l,l (v) + Ψ * t l,l (v)F(u) = 0, (5.15) E(u)Ψ * t l,l (v) = − θ * (u − v − l 2 ) θ * (u − v + l 2 )
Ψ * t l,l (v)E(u). (5.16) In fact, there are a few degrees of freedom on choosing the zero-modes of the above nontwisted and twisted vertex operators. Here we only use the simplest ones.
Discussion
In this paper, we construct a new free boson realization of U q,p ( sl 2 ) k by twisting the flag manifold realization, which can be viewed as the elliptic deformation of Wakimoto realization. With this approach, the two important objects (screening currents and Intertwiner Operators or vertex operators) are also discussed in details. They all play important roles in calculating correlation functions. Of course the derivation of the multi-point correlation functions is a quiet interesting problem, but in view of its complexity and the length of the manuscript, we will discuss it in the future. Furthermore, it is an interesting problem to extend our results to the other types of Lie algebras.
z) = U + (z; r, r * )e + (z),(3.16) f (z) = e − (z)U − (z; r, r * ).(3.17)
F
(z)S I (w) = −S I (w)F(z) = O(1), S I (z)S I (w) = −S I (w)S I (z) = O(1); H ± (z)S II (w) = S II (w)H ± (z) = O(1),
)S I (w) :]; moreover, : A(z)S I (qz) :=: B(z)S I (q −1 z) :≡s 3 (z);
e A e B = e [A,B] e B e A , i f [A, B] commute with A and B;).
Proof: A straightforward but length OPE calculation verifies this proposition. Here we just
list some useful formulas:
AcknowledgmentsOne of the authors (Ding)thanks the Kavli Institute for Theoretical Physics China (KITPC) at the Chinese Academy of Sciences and the program of String Theory and Cosmology for hospitality as the write-up was completed. He is financially supported partly by the Natural Science Foundations of China through the grands No.10671196 and No.10231050. He is also supported partly by a key Fund of Chinese Academy of Sciences.The authors also thank the referee of our paper, whose suggestions make the expression in this paper more accurate and clearer.−C − (−l; 1, 1|w) + C + (−l; 1, 1|w)} × exp{ l q a 2(k + 2) + l(q b + q c ) − lq + l 2r * p ln w}, then they satisfy the intertwining relations(5.3)-(5.8).Proof: We take the relation (5.3) as an example. In fact, from the bosonic expression of Φ l,l (v) and(3.18), the following OPE's can be derived:The others can be derived similarly.It is easy to see that for the Cartan parts H ± (u), there is no difference for the twisted intertwining relations (5.3), (5.11) and the non-twisted ones (5.6), (5.14); while for E(u) and F(u), the difference between them is just a minus sign. With the notations introduced before, we haveSimilarly to the non-twisted case, we can prove that Φ t l,l (v) and Ψ * t l,l (v) obey the twisted intertwining relations (5.11)-(5.16).
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Feigin-Fuchs representation of su(2) k Kac-Moody algebras. D Nemeschansky, Phys. Lett. 224121D. Nemeschansky, Feigin-Fuchs representation of su(2) k Kac-Moody algebras, Phys. Lett. B224 (1989) 121.
Modular invariant partition functions for parafermionic field theories. D Gepner, Z Qiu, Nucl. Phys. 285423D. Gepner, Z. Qiu, Modular invariant partition functions for parafermionic field theories, Nucl. Phys. B285 (1987) 423.
New conformal field theories associated with Lie algebras and their partition functions. D Gepner, Nucl. Phys. 29010D. Gepner, New conformal field theories associated with Lie algebras and their partition functions, Nucl. Phys. B290 (1987) 10.
Generalized Vertex Algebras and Relative Vertex Operators. C Y Dong, J Lepowsky, Prog. Math. 112C.Y. Dong, J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Prog. Math. 112 Birhäuser, 1993.
Current algebra and Wess-Zumino models in two dimensions. V G Knizhnik, A B Zamolodchikov, Nucl. Phys. 24783V.G. Knizhnik, A.B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B247 (1984) 83.
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. W J Chang, X M Ding, in preparingW.J. Chang, X.M. Ding, in preparing.
| {'fraction_non_alphanumeric': 0.11852897673793196, 'fraction_numerical': 0.03516202769934113, 'mean_word_length': 3.2244627473255703, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 78, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A realization of the elliptic quantum algebra U q,p ( sl 2 ) for any given level k is constructed in terms of three free boson fields and their accompanying twisted partners. It can be viewed as the elliptic deformation of Wakimoto realization. Two screening currents are constructed; they commute or anti-commute with U q,p ( sl 2 ) modulo total q-differences. The free fields realization for two types vertex operators nominated as the type I and the type II vertex operators are presented. The twisted version of the two types vertex operators are also obtained. They all play crucial roles in calculating correlation functions.', 'arxivid': '0812.1140', 'author': ['Wen-Jing Chang \nInstitute of Applied Mathematics\nAcademy of Mathematics and Systems Science\nChinese Academy of Sciences\nP.O.Box 2734100190BeijingP.R. China\n\nGraduate School of Chinese Academy of Sciences\nChina\n', 'Xiang-Mao Ding \nInstitute of Applied Mathematics\nAcademy of Mathematics and Systems Science\nChinese Academy of Sciences\nP.O.Box 2734100190BeijingP.R. China\n'], 'authoraffiliation': ['Institute of Applied Mathematics\nAcademy of Mathematics and Systems Science\nChinese Academy of Sciences\nP.O.Box 2734100190BeijingP.R. China', 'Graduate School of Chinese Academy of Sciences\nChina', 'Institute of Applied Mathematics\nAcademy of Mathematics and Systems Science\nChinese Academy of Sciences\nP.O.Box 2734100190BeijingP.R. China'], 'corpusid': 14744612, 'doi': '10.1063/1.2905151', 'github_urls': [], 'n_tokens_mistral': 17275, 'n_tokens_neox': 15456, 'n_words': 8684, 'pdfsha': '88e1ef3d3816104cf7ace2c98f5e0d1950249126', 'pdfurls': ['https://arxiv.org/pdf/0812.1140v1.pdf'], 'title': ['On the Vertex Operators of the Elliptic Quantum Algebra U q,p ( sl 2 ) k', 'On the Vertex Operators of the Elliptic Quantum Algebra U q,p ( sl 2 ) k'], 'venue': []} |
arxiv |
Nonrelativistic Quantum Particle in a Curved Space as a Constrained System *
arXiv:hep-th/9411052v1 7 Nov 1994 September 1994
A Foerster
Instituto de Física
Universidade Federal do Rio Grande do Sul Caixa Postal
1505191501-970Porto AlegreRSBrazil
H O Girotti
Instituto de Física
Universidade Federal do Rio Grande do Sul Caixa Postal
1505191501-970Porto AlegreRSBrazil
P S Kuhn
Instituto de Física
Universidade Federal do Rio Grande do Sul Caixa Postal
1505191501-970Porto AlegreRSBrazil
Nonrelativistic Quantum Particle in a Curved Space as a Constrained System *
arXiv:hep-th/9411052v1 7 Nov 1994 September 1994* Supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparoà Pesquisa do Estado do Rio Grande do Sul (FAPERGS), Brazil.
The operator and the functional formulations of the dynamics of constrained systems are explored for determining unambiguously the quantum Hamiltonian of a nonrelativistic particle in a curved space. | {'fraction_non_alphanumeric': 0.019095477386934675, 'fraction_numerical': 0.07336683417085427, 'mean_word_length': 5.343949044585988, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The operator and the functional formulations of the dynamics of constrained systems are explored for determining unambiguously the quantum Hamiltonian of a nonrelativistic particle in a curved space.', 'arxivid': 'hep-th/9411052', 'author': ['A Foerster \nInstituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil\n', 'H O Girotti \nInstituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil\n', 'P S Kuhn \nInstituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil\n'], 'authoraffiliation': ['Instituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil', 'Instituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil', 'Instituto de Física\nUniversidade Federal do Rio Grande do Sul Caixa Postal\n1505191501-970Porto AlegreRSBrazil'], 'corpusid': 119424199, 'doi': '10.1016/0375-9601(94)90033-7', 'github_urls': [], 'n_tokens_mistral': 365, 'n_tokens_neox': 297, 'n_words': 138, 'pdfsha': '20e53d6690ccaadef245a77ef5a6d20d3ed8b72f', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9411052v1.pdf'], 'title': ['Nonrelativistic Quantum Particle in a Curved Space as a Constrained System *', 'Nonrelativistic Quantum Particle in a Curved Space as a Constrained System *'], 'venue': []} |
arxiv |
NEW CRITICALITY OF 1D FERMIONS
November 1993
Michael Lässig [email protected]
Institut für Festkörperforschung Forschungszentrum
52425JülichGermany
NEW CRITICALITY OF 1D FERMIONS
November 1993arXiv:cond-mat/9407045v1 8 Jul 1994 cond-mat/nnnmmyyy
One-dimensional massive quantum particles (or 1 + 1-dimensional random walks) with short-ranged multi-particle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite m-body forces (m = 3, 4, . . .), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g. in the Hubbard model, are also noted. PACS numbers: 5.70Jk, 5.40+j, 64.60Ak to appear in Phys. Rev. Lett.
Interacting quantum particles moving in one spatial dimension and imaginary time offer a unifying description of most 2D fluctuating systems. The trajectories of these particles represent (streched) polymers, domain walls or interfaces, steps on surfaces, magnetic flux lines, etc. Two ensembles have to be distinguished: (a) Vicinal surfaces [1] or systems at a 2D bulk critical point (e.g. Curie point, commensurate-to-incommensurate transition [2], surface reconstruction transition [3]) contain a finite density of such lines and are described by a massless quantum field theory which is generically isotropic and conformally invariant. (b) Systems with only a finite number of directed lines are ensembles of massive particles.
Such systems may exhibit critical behavior at delocalization transitions between a low-temperature dense phase and a high-temperature dilute phase [4].
In the dense phase, the lines are bound to a bundle of transversal extension ξ ⊥ . Their relative fluctuations are thus constrained; correlations in longitudinal direction decay on a scale ξ . This phase is a bound state of the quantum particles.
In the dilute phase, the lines fluctuate independently; the quantum particles are in a delocalized state. As the transition temperature is approached from below, the length scales ξ and ξ ⊥ = ξ ζ diverge. These transitions are generically anisotropic; the roughness exponent ζ equals 1/2 for temperature-driven transitions. Examples are wetting phenomena, polymer desorption, the helix-coil transition in DNA, and unbinding transitions of biomembrane bundles [5], which have gained considerable experimental interest recently [6]. Ensembles of interacting directed lines are also important as the replica formulation of polmers in random media [7]; those in turn are intimately related to theories of nonequilibrium directed growth.
This letter aims at a systematic understanding of delocalization phenomena as renormalized continuum field theories. An exact renormalization group (RG) based on the short-distance algebra of the interaction vertices [8,9] reveals the existence of a discrete series of universality classes that represent delocalization transitions of a finite number of interacting random walks. The possible existence of analogous massless (conformally invariant) field theories is discussed at the end.
One-dimensional quantum particles that interact only via two-body contact forces define the nonlinear Schrödinger model, which is exactly solvable by Bethe ansatz methods (see [12] for a review). This has been applied to unbinding transitions in refs. [10,11]. In real systems, the interactions are certainly more complicated than the simple pair force of the Schrödinger model. Typically, the force between two lines is screened or enhanced by the presence of further lines. Casimirtype many-body forces (which may be screened at some scale) arise from the coupling of the lines to the surrounding medium, e.g. a correlated fluid [13]. There is also experimental evidence for attractive forces between steps on vicinal surfaces.
When such interactions are taken into account, the exact solvability is lost, and we are led to study their effect on the delocalization transition by the RG.
Short-ranged multi-particle interactions are easily shown to be irrelevant in the sense of the RG (except 3-body forces for "bosons", see below). Hence weak forces do not alter the asymptotic behavior at large distances, but contribute only cor- In many of the applications above, the lines are effectively impenetrable objects and hence do not intersect. In one dimension, this constraint on their fluctuations is equivalent to the Pauli principle; the particles are fermions. Particles whose trajectories are free to intersect are bosons. Repulsive contact forces suppress intersections and hence generate a crossover from Bose statistics to a low-energy effective Fermi statistics [12]. The RG of this letter offers a unifying view on the interplay between dynamics and statistics: delocalization transitions of bosons and fermions fall into the same universality classes, the statistics merely corresponds to parametrizations of the space of interactions about two different fixed points. For the particular case of two-and three-particle interactions, the results are summarized in the RG flow diagram of fig. 1 and the resulting phase diagram of fig. 2. Depending on these interactions, the phase transition can be governed by two distinct fixed points, the free Bose and the "necklace" fixed point, which are discussed in detail below.
We stress that all these fixed points describe ensembles of an arbitrary number of lines; hence the critical exponents do not depend on their number. This result agrees with ref. [10] for the Bose fixed point and is presumably also consistent with the extensive numerical work of ref. [14] if the data are correctly interpreted [15].
Consider a d-dimensional system of p massive bosons coupled via forces that decay on some microscopic scale a. In the continuum limit a → 0, the Hamiltonian reads H (p)
B = 1 2 p α=1 ∂ 2 ∂r 2 α + p m=2 g m Φ (p) m ,(1)
where Φ
(p) 2 = p α<β δ d (r α − r β ), Φ (p) 3 = p α<β<γ δ d (r α − r β )δ d (r β − r γ )
, etc. are mparticle contact potentials. It describes the universal behavior in the scaling region ξ ⊥ ≫ a. In a system with long-ranged forces, (1) is still the correct continuum limit if these forces are irrelevant in the RG, i.e. decay with a power of the distance larger than 2. It is convenient to use a description in second quantization,
H B = (∂ r φ † (r, t))(∂ r φ(r, t))d d r + m≥2 g m Φ m (t) ,(2)
which is valid for an arbitrary number of lines. The operators φ and φ † obey canonical commutation relations and
Φ m (t) = 1 m! (φ † (r, t)) m (φ(r, t)) m d d r(3)
are normal-ordered m-particle vertices. With time as the basic scale, these vertices have canonical dimensions x m = (m − 1)d/2. Hence the conjugate coupling constants g m have dimensions
ε m = 1 − x m .(4)
The vertices form the short-distance algebra [16] Φ Hence consider first the series F (
k (t)Φ l (0) = k+l−1 m=max(k,l) C m kl |t| −(k+l−m−1)d/2 Φ m (0) + . . . ,(5)C m kl = m! (m − k)!(m − l)!(k + l − m)! k + l − m 2 −d/2 .(6)u 2 ) = F (0) + ∞ N =1 F N u N 2 , where F N = L 1−N ε 2 (−1) N N! Φ 2 (t 1 )Φ 2 (t 2 ) . . . Φ 2 (t N ) L dt 2 . . . dt N(7)
and . . . L denotes connected expectation values in the unperturbed ground state of an arbitrary particle number sector [17]. (The subsequent manipulations do not depend on the in-and out-states but only on the short-distance structure of the correlation functions.) In the series (7), a single primitive divergence
F 2 = L x 2 Φ 2 L C 2 22 L −ε 2 L 0 t −1+ε 2 dt + O(ε 0 2 ) = L x 2 Φ 2 L 1 ε 2 + O(ε 0 2 )(8)
occurs at ε 2 = 0 (i.e. d = 2). Hence the beta function in minimal subtraction is [18]
β 2 (U 2 ) = ε 2 U 2 − U 2 2 .(9)
This exact renormalizability is intimately related to the summability of the perturbation expansion in the nonlinear Schrödinger model [12]. Generically, (9) would make sense for ε 2 > 0, where U 2 is relevant at the Gaussian fixed point and generates a crossover to the infrared-stable fixed point U ⋆ 2 = ε 2 . As an exact one-loop equation, however, it continues to be valid for 0 > ε 2 > −1, where the ultraviolet divergences in F (u 2 ) can be absorbed in a single counterterm. U ⋆ 2 is then ultraviolet-stable. In the perturbation series for the correlation functions, singularities analogous to (8) at first order in u 2 lead to the beta functions
β m (U 2 , . . . , U m ) = ε m (U 2 )U m + O(U k U m )(10)
with (3 ≤ k ≤ m) and
ε m (U 2 ) = ε m − 2C m m2 U 2 + O(U 2 2 ) .(11)
For m ≥ 3, (11) does not terminate at first order. In d = 1, however, the combined contribution from higher orders turns out to vanish at the fixed point U ⋆ 2 = 1/2, so that the infrared dimensions resulting from (11) and (6),
x m = 1 − ε m (U ⋆ 2 ) = m 2 − 1 2 ,(12)
are the exact scaling dimensions of the fermionic operators (16) below. The full beta function for U 3 now follows in a similar way from the singularities in the series F (u 2 , u 3 ) at ε 3 = 0. Again, the only primitive singularity occurs at order u 2 3 , and hence (with C 3 ≡ C 3 33 )
β 3 (U 2 , U 3 ) = (ε 3 − 3U 2 )U 3 − C 3 U 2 3 .(13)
The RG flow of fig. 1 is given by (9) and (13) for d = 1. In the sequel, we discuss its three fixed points and the implications for the phase diagram.
Free Bosons (U 2 = U 3 = 0). The scale-invariant theory is characterized by algebraic finite-size effects Φ m L ∼ L −(m−1)/2 . For g 2 < 0, there is the well-known bound state with longitudinal correlation length ξ (g 2 ) ∼ |g 2 | −2 (14) and Φ m ∞ ∼ |g 2 | m−1 , while g 2 > 0 generates the crossover to free fermions with (14) now describing the scaling of the crossover length. A repulsive three-particle coupling g 3 > 0 is marginally irrelevant. For g 2 = 0, it leaves the particles infraredfree, but modifies the theory (e.g. the amplitudes Φ m L ) on scales smaller than
ξ (g 3 ) ∼ exp(−1/g 3 ) ;(15)
for g 2 ր0, it contributes logarithmic corrections to scaling [15], e.g. Φ m ∞ ∼ (g 2 /g 3 log |g 2 |) 2 . The marginally relevant g 3 < 0 leads to a bound state with (15) and Φ m ∞ ∼ exp((m − 1)/2g 3 ); the unbinding now takes place on the critical line g 2 = g c 2 (g 3 ) and is governed by the "necklace" fixed point described below. Free Fermions (U 2 = ε 2 , U 3 = 0). This fixed point describes the limit g 2 → ∞, Short-ranged interactions are instead described by the fermionic operators
Φ m (t) ≡ 1 m! m i=1 ψ † (r + a i )ψ(r + a i )dr ,(16)
where a i are fixed microscopic distances characterizing their range. These operators have scaling dimensionsx m = (m 2 −1)/2 [19] as given by (12) and form an operator algebra of the form (5). Hence for d = 1, the bosonic Hamiltonian (2) can be written in the equivalent fermionic form
H F = (∂ r ψ † (r, t))(∂ r ψ(r, t))dr + m≥2 g mΦm (t) .(17)
The fermionic RG equations are precisely of the form (9), (13) with coefficients ε m = 1 −x m andC 3 . Since all interactions (16) are irrelevant, both the bosonic fixed point (Ū 2 =ε 2 ,Ū 3 = 0 ) and the necklace fixed point (Ū 2 = 0,Ū 3 =ε 3 /C 3 ) are ultraviolet fixed points.
Necklace Theory (U 2 = ε 2 , U 3 =ε 3 /C 3 ). This theory describes the critical transition between the high-temperature phase of free fermions and the "necklace"
bound state [20] that forms forḡ 3 <ḡ c 3 < 0 and is named after the typical configurations of trajectories shown in fig 2. The transition temperature depends on the parameters a i and is nonuniversal. At this fixed point, the three-particle coupling is relevant. The one-loop RG predicts the exponent ε △ 3 = −ε 3 as long as ε 3 > −1. Since ε △ 3 cannot become > 1 (this would mean an unphysical divergence of Φ 3 ∞ (ḡ 3 ) ∼ ξ 1−ε 3 △ at the transition) we conclude ε △ 3 = 1 forε 3 < −1. This is confirmed by a mapping of the necklace theory onto a particular point of the critical line of wetting transitions [21,14]. Hence
ξ (ḡ 3 ) ∼ |ḡ 3 −ḡ c 3 | −1 ,(18)
and Φ 3 ∞ (ḡ 3 ) approaches a nonuniversal finite value asḡ 3 րḡ c 3 . This implies an unusual energy balance for the necklace bound state: its kinetic energy E kin and potential energy E pot remain separately finite as the total bound state energy E kin + E pot = −1/ξ approaches 0, while at the bosonic transition E kin ≃ −E pot /2 ≃ 1/ξ . A further interesting question is the existence of analogous fixed points for theories of massless relativistic fermions, where isotropy is restored through particleantiparticle processes. The simplest case is the critical point of the 2D Ising model, a theory of free Majorana fermions with action S = (ψ + ∂ − ψ + + ψ − ∂ + ψ − )d 2 r in terms of the chiral components ψ + and ψ − . The lowest-dimensional scalar interaction that is local in the Fermi fields is the irrelevant normal-ordered 4-particle vertex :ψ + ψ + ψ + ψ + ψ − ψ − ψ − ψ − : = T + T − (where T + and T − denote the components of the stress tensor). This interaction is known to be integrable and to generate a crossover whose ultraviolet "necklace" fixed point is the tricritical Ising model [22]. Thus it is tempting to associate the hierarchy of multi-particle interactions with the famous series of minimal conformal field theories [23]. Many other applications involve Dirac fermions with (marginal) local pair interactions. Examples are the ubiquitous Gaussian model with the fermionic action S = (ψ + ∂ − ψ + +ψ − ∂ + ψ − +g 2ψ+ ψ +ψ− ψ − )d 2 r, or the Hubbard model, a theory of two Dirac fermions coupled by similar pair forces (that is relevant to roughening of reconstructed surfaces [24]). In these cases, the effects of the higher interactions (e.g. T + T − ) are unknown, but since they have a self-coupling in the operator algebra, they are likely to generate similar transitions to massive strong-coupling phases. These multicritical Dirac theories would correspond to conformal field theories with central charge c > 1.
I am grateful to T.W. Burkhardt, H. Kinzelbach, R. Lipowsky, and R. Netz for useful discussions and comments.
rections to scaling. The new universality classes describe delocalization transitions at finite interaction strength. In a generic field theory, irrelevant vertices are unrenormalizable, i.e. new counterterms are necessary at every order in perturbation theory. Remarkably enough, this proliferation of counterterms does not take place here: the perturbation series remains renormalizable in an ε-expansion although the interaction is irrelevant. This expansion involves analytic continuation in the number d of transversal dimensions, see eq. (4) below.
However, in order to define correlation functions of these vertices, an infrared regularization is necessary. Here the range of each coordinate r i is compactified to a circle of radius L ζ ; this regularization preserves translational invariance in space and time. The scale L also serves to define the dimensionless bare couplings u m = g m L εm and the dimensionless free energy F (u 2 , u 3 , . . .) = LE 0 (g 2 , g 3 , . . . ; L) in terms of the ground state energy E 0 . The renormalization consists in absorbing the singularities in the perturbation expansion for F (u 2 , u 3 , . . .) into renormalized couplings U m . These singularities are encoded in the operator algebra[8]. By virtue of (5), the beta function β m (U 2 , U 3 , . . .) ≡ L∂ L U m depends only on the U k with k ≤ m.
g 3 = 0,where the particles obey the Pauli exclusion principle. Hence the operators Φ m vanish identically, as follows from the asymptotic crossover scaling of their correlation functions given by(9) and (10), e.g. Φ m L (g 2 ) ∼ g
From the foregoing RG analysis it transpires that the fixed pointŪ ⋆ 3 is just the first member of a whole family of fermionic necklace theories represented by fixed pointsŪ ⋆ m of the higher multi-particle interactionsΦ m . Thus the interplay between attractive and repulsive forces generates a rich scenario of universality classes of interacting directed walks. A detailed understanding of their correlation functions and the various crossover phenomena is within the reach of these RG methods butbeyond the scope of this letter. The Bethe ansatz yields the correct asymptotic scaling if and only if the Hamiltonian (2) is in the universality class of the free Bose or Fermi fixed point. Whether analogous methods of exact solution exist for the higher fixed pointsΦ m is an open question.
. See E G J Frohn, Phys. Rev. Lett. 673543See e.g. J. Frohn et al., Phys. Rev. Lett. 67 (1991), 3543.
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The authors of ref. [14] report a violation of (14), but their numerical data can probably be accounted for by corrections due to a repulsive 3-particle coupling implicitly present in their system. The authors of ref. [14] report a violation of (14), but their numerical data can probably be accounted for by corrections due to a repulsive 3-particle coupling implicitly present in their system.
The Φ m are normalized such that C 2 22 = 1. The terms omited on the r.h.s. contain gradients. The Φ m are normalized such that C 2 22 = 1. The terms omited on the r.h.s. contain gradients
2205, have shown that this interaction is renormalizable to all orders in general longitudinal dimensionality. F David, B Duplantier, E Guitter, Phys. Rev. Lett. 70F. David, B. Duplantier, and E. Guitter, Phys. Rev. Lett. 70 (1993), 2205, have shown that this interaction is renormalizable to all orders in general longitudinal dimensionality.
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The easiest derivation is to consider the expectation value in a m-particle state χ|Φ m |χ L = |χ(r + a 1 , . . . , r + a m )| 2 dr, which scales as L −(m−1)/2 | i<j ǫ ij | 2 ∼ L −xm with ǫ ij. This was first shown in ref. [20] without the explicit use of free fermions. −1/2 , due to the antisymmetry of the fermionic wave functionThis was first shown in ref. [20] without the explicit use of free fermions. The easiest deriva- tion is to consider the expectation value in a m-particle state χ|Φ m |χ L = |χ(r + a 1 , . . . , r + a m )| 2 dr, which scales as L −(m−1)/2 | i<j ǫ ij | 2 ∼ L −xm with ǫ ij ≡ (a i − a j )L −1/2 , due to the antisymmetry of the fermionic wave function.
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16031. Figure Captions 1. RG flow diagram. U 2 , U 3 andŪ 2 ,Ū 3 denote renormalized two-and threeparticle couplings about the fixed points of free bosons (•) and free fermions (•), respectively. The transition is governed by the Bose fixed point for U 3 ≥ 0 and by the necklace fixed point. L Balents, M Kardar, Phys. Rev. B. 46△) for U 3 < 0L. Balents, M. Kardar, Phys. Rev. B 46 (1992), 16031. Figure Captions 1. RG flow diagram. U 2 , U 3 andŪ 2 ,Ū 3 denote renormalized two-and three- particle couplings about the fixed points of free bosons (•) and free fermions (•), respectively. The transition is governed by the Bose fixed point for U 3 ≥ 0 and by the necklace fixed point (△) for U 3 < 0.
| {'fraction_non_alphanumeric': 0.07089361306228777, 'fraction_numerical': 0.03600502395683119, 'mean_word_length': 3.7467432104217266, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'One-dimensional massive quantum particles (or 1 + 1-dimensional random walks) with short-ranged multi-particle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite m-body forces (m = 3, 4, . . .), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g. in the Hubbard model, are also noted. PACS numbers: 5.70Jk, 5.40+j, 64.60Ak to appear in Phys. Rev. Lett.', 'arxivid': 'cond-mat/9407045', 'author': ['Michael Lässig [email protected] \nInstitut für Festkörperforschung Forschungszentrum\n52425JülichGermany\n'], 'authoraffiliation': ['Institut für Festkörperforschung Forschungszentrum\n52425JülichGermany'], 'corpusid': 119439397, 'doi': '10.1103/physrevlett.73.561', 'github_urls': [], 'n_tokens_mistral': 7048, 'n_tokens_neox': 5986, 'n_words': 3710, 'pdfsha': '92edabff15ad55f2aae86739c2d76171de8822a6', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/9407045v1.pdf'], 'title': ['NEW CRITICALITY OF 1D FERMIONS', 'NEW CRITICALITY OF 1D FERMIONS'], 'venue': []} |
arxiv |
27 Dec 2022
TWO MODES OF NONMONOTONIC CONSEQUENCE † ALEXEI MURAVITSKY
27 Dec 2022
We discuss two ways to implement a semantic approach to nonmonotonic consequence relations in an arbitrary propositional language. For one particular language, we also discuss the proof-theoretic framework that we connect with this semantic approach. This article is an addition to[Muravitsky, 2021]. † This is the text of my speech at the Logica Universalis webinar, which took place on May 11, 2022.1 First-order language is used only as an auxiliary tool. 2 Here is a full quote:1 Proposition 3.9. For any matrix M = A, D and a set X ∪ {α} ⊆ Fm L ,Thus, we obtain that X cM |= r M α. It remains to apply(22). Example 3.2 shows that the converse can fail.Proposition 3.10. Relation |= r * M is logical relative to |= M . Proof. Indeed, property (8) follows from Proposition 3.2 and Proposition 3.9.To prove property(9), we assume that X |= r * M α and α |= M β. In virtue of (22), X cM |= r M α. According to Proposition 3.2, X cM |= r M β. Then we apply(22). We should not expect that relation |= r * M will enjoy the weak monotonicity property (Definition 3.2). However, we obtain the following.Proposition 3.11. Given a matrix M, for any setM α and all other premises be fulfilled. Then, in virtue of (22),and X cM ⊆ Y cM , we can apply the property that |= r M is weakly monotonic (Proposition 3.3) in order to conclude that Y cM |= r M α, that is, Y |= r * M α. As to finitariness, we obtain the following proposition.Proposition 3.12. Given a matrix M, for any nonempty(22), X cM |= r M α. In virtue of Proposition 3.4, there is a nonempty X 0 ⋐ X cM such that V(X 0 )∩V(α) = V(X cM )∩ V(α) and X 0 |= r M α. It must be clear that for some nonempty Y ⋐ X, Y cM = X 0 . This implies that. The last part of the statement follows by Proposition 3.11. Now we will turn to M-r * -consequence. First of all, we observe the following.Proposition 3.13. Any M-r * -consequence is logical relative to the corresponding M-consequence.
What do we know and how do we know about nonmonotonic consequences?
We discuss non-monotony expressed in formalized languages. All main objective languages that will be involved are propositional. 1 Aristotle is credited with distinguishing between what we know and how we know. G. Kreisel who made this remark in [Kreisel, 2019], p. 42., adds,
The point I wish to stress, again as a matter of accumulated experience, is how much easier the former question is to answer (to our satisfaction) than the latter. Of the what, whose aspects strike the (mind's) eye, we have easily accessible practical knowledge (which constitutes the threshold for additions by further theory). We also would like to know about the how, but the price is high. (Ibid) We note that it is not only the possible easiness of answering the what question that prompts us to start with it but also the view that "knowing how we know is one small department of knowing what we know" (Russell). 2 Speaking of "easily accessible practical knowledge", Kreisel probably had in mind, first of all, modern mathematics and formal logic. As is known, both are based on monotonic reasoning. On the contrary, with respect to nonmonotonic reasoning, it seems that answering both what and how questions is not an easy task. We do not think that the following statement by D. Makinson can be taken seriously.
Of course, human have been reasoning nonmonotonically for as long as they have been reasoning at all. ([Makinson, 1994], section 1.1) I reverse the process which has been common in philosophy since Kant. It has been common among philosophers to begin with how we know and proceed afterwards to what we know. I think this is a mistake, because knowing how we know is one small department of knowing what we know. ([Russell, 1959], chapter II) Despite Makinson's remark, it is precisely because of the lack of rich contextual practices involving nonmonotonic reasoning that it is so difficult to answer what we know and how we know about it.
However, non-monotonicity does occur in colloquial speech. R. Stalnaker [Stalnaker, 1994] gives an example of nonmonotonic argument based on Grice's concept of (conversational) 'implicature'. 3 This example is remarkable only in one respect -it indicates the need to distinguish between the concepts of argumentation and logic. Logic deals exclusively with forms, while content can be involved in argumentation. An example of the latter is that of Stalnaker -the nonmonotonic effect of his example is based on the interaction of contents, more precisely, between implicit and explicit contents. Understanding logic as a science dealing with forms of judgment, we will pay attention to sentential variables and, as a consequence, to the role of substitution, understood both as an operation and as a rule of inference. But first, we outline the monotonicity-non-monotonicity dichotomy as such.
Before proceeding to a formal presentation, we note that we are using the concept of logic in its dialectical, and not demonstrative, manner. That is, we will consider the relationship between a set of premises and the conclusions that can be drawn from these premises according to some standards.
For this, we fix a propositional language, L, (later we will be more specific), whose formulas are denoted by letters α, β, etc.; sets of formulas by X, Y , etc.; and the set of all formulas by Fm L . Given sets X and Y of formulas, we write
X ⋐ Y if X is a finite subset of Y .
The operation of (uniform or simultaneous) substitution is understood in the usual way. We denote substitutions in L by σ, δ, . . . (with or without subscripts or other marks). The result of substitution σ in a formula α is denoted by σ(α); accordingly, σ(X) := {σ(α) : α ∈ X}. Let C be a mapping C : P(Fm L ) −→ P(Fm L ). Given a set X ⊆ Fm L , C(X) is understood as the set of all conclusions obtained from X.
The definition
X ⊢ C α df ⇐⇒ α ∈ C(X)
(1) will be very useful subsequently.
The last definition implies conversion; namely, given a relation ⊢⊆ P(Fm L ) × Fm L , we define:
α ∈ C ⊢ (X) df ⇐⇒ X ⊢ α.
(2) The definitions (1) and (2) allow to use relations of the above type and the corresponding operators interchangeably.
The following properties of the C operator have long been in the spotlight. 4 (con-1) X ⊆ C(X); (reflexivity)
(con-2) X ⊆ Y implies C(X) ⊆ C(Y ); (monotonicity) (con-3) C(C(X)) ⊆ C(X); (closedness) (con-4) C(X) ⊆ {C(Y ) : Y ⋐ X}; (finitariness or compactness) (con-5) X ⊆ C(Y ) implies C(X) ⊆ C(Y ); (cumulative transitivity) (con-6) X ⊆ Y ⊆ C(X) implies C(X) ⊆ C(Y ); (weak cumulative transitivity 5 ) (con-7) X ⊆ C(Y ) implies C(X ∪ Y ) ⊆ C(Y ); (strong cumulative transitivity) (con-8) {C(Y ) : Y ⋐ X} ⊆ C(X); (finitary inclusion) (con-9) if X = ∅, then C(X) ⊆ {C(Y ) : Y = ∅ and Y ⋐ X}. 6
(strong finitariness) A. Tarski, who introduced the operator C into the discourse in his address to the Polish Mathematical Society in 1928, 7 began with the following remark.
Our object in this communication is to define the meaning, and to establish the elementary properties, of some important concepts belonging to the methodology of deductive sciences, which, following Hilbert, it is customary to call metamathematics. ([Tarski, 1930]) In the published version of this address, Tarski formulated (con-1)-(con-3). Given a set X ⊆ Fm L , he called C(X) "the consequences of the set X," which suggests to call any operator C satisfying these properties a consequence operator. It was the rich variety of contextual practices in mathematics and the deductive sciences that prompted him to introduce this notion. In his own worlds,
The concept of logical consequence is one of those whose introduction into the field of strict formal investigation was not a matter of arbitrary decision on the part of this or that investigator; . . . ([Tarski, 1936]) Tarski also considered (con-4) but not other properties. One interesting property was introduced over a quarter of a century later in [ Loś and Suszko, 1958]. Namely, (con-10) α ∈ C(X) implies σ(α) ∈ C(σ(X)). (structurality)
As for the what / how questions about monotonic reasoning, at least with regard to its forms as presented in mathematics and the deductive sciences, it is generally agreed that the properties (con-1)-(con-3) seem to answer the first question, or at least can be regarded as "the threshold for additions by further theory" (Kreisel).
Frege and Tarski can be credited for pioneering to answer the second question. In particular, Tarski pointed out the following two ways of implementing monotonic reasoning in metamathematics. According to him, the first way is as follows.
From the sentences of any set X certain other sentences can be obtained by means of certain operations called rules of inference. ([Tarski, 1930]) The second way is through semantics. 8 In the most general terms, one can imagine a non-linguistic machinery M such that it is natural to say that the sentence α follows from the set of premises X, when M accepts α whenever M accepts all premises of X. For propositional languages, this leads to matrix consequences. 9 And although from this level of abstraction the implementation of this idea may vary, we will not discuss here all the alternatives. 10 Unfortunately, at this point, little can be said about answers to the what question in relation to nonmonotonic reasoning. To reiterate, we argue that it is the lack of contextual practices of nonmonotonic reasoning that has led to speculating about 6 We eliminate curly brackets for singleton premises; thus C(α) stands for C({α}). 7 He used the notation 'Cn' instead of 'C'. 8 Cf. [Tarski, 1936]. 9 See e.g. [Wójcicki, 1988], chapter 3, or [Citkin and Muravitsky, 2022], chapter 4. 10 See [Citkin and Muravitsky, 2022], section 4.3.3.
"the minimum conditions that a consequence relation should satisfy to represent a bona fide nonmonotonic logic . . ." 11 In contrast to this optimistic hope, one can see a growing confusion in the endeavor of trying to learn something about the objective world of reasoning through pure thinking. Another confusion, the whathow confusion, occurs when someone, in an attempt to give a general definition of nonmonotonic reasoning, gives examples of answers to the how question.
If contextual practices of nonmonotonic reasoning are lacking, what can the theory of nonmonotonic reasoning be based on? One possible answer is that it can be based on conceptualized practices of monotonic reasoning. Recall that algebraic logic began with George Boole's algebraization of Aristotle's syllogistic. And to give credit to Kreisel's insight, to answer the what question, we should simply propose a C operator that would not satisfy (con-2). However, there are many candidates for answering the how question, not all of them are indisputable. What seems certain is that any mode of nonmonotonic reasoning worth exploring should be considered together with its monotonic counterpart. 12 Following this observation, we introduce the notation 'Cn' to indicate that the last operator satisfies the properties (con-1)-(con-3). The seeming contradiction with the last paragraph is resolved by establishing relations between the operators Cn and C.
Logicality of (logical) consequences
First of all, we must explain the title of this section. Even in the case of monotonic reasoning, if the answer to the what question can be explicated in the form of conditions (con-1)-(con-3), any possible answer to the how question is fraught with danger. Here are two views on the issue. The first comes from the corner of constructive philosophy.
A logical inference is a movement from certain propositions (the premises) to a further proposition (the conclusion). We still need to know, however, which of these movements are to be called logical. ([Lorenzen, 1987], section 4)
The second point of view sounds from the corner of proof theory.
If we simply axiomatize or define the notion of logical consequence with the understanding that a logical consequence holds when this follows logically (!) from the axioms or the definitions, then one may rightly say whether anything really is achieved. ([Prawitz, 1974], section 2) Such sentiments sometimes, though rarely, appeared in the camp of researchers of nonmonotonic reasoning; in particular, in the second sentence of the following quote (if you forgive the what-how confusion in the first).
[. . .] what is wanted is not a specific technical definition for a specific non-monotonic logic, but a general account of what consequence is supposed to mean-of what concept some specific technical definition is trying to capture. And one should expect such an account to explain why it is reasonable to call the concept by the name "consequence". ( [Stalnaker, 1993]) Apparently, D. Makinson [Makinson, 1994], section 2.2, was the first to propose a clear criterion for the logicality of a nonmonotonic consequence; namely, he formulated that for two operators Cn (monotonic) and C (presumably nonmonotonic), the operator C is logical (with respect to Cn) if
CnC = C = CCn.(3)
It should be clarified that any standard on the logicality of nonmonotonic reasoning is an answer to the what question, not to the how question. In formulating such a standard, we want to prescribe the conditions of what we expect it to be.
To facilitate the analysis of (3), we introduce two relations similar to (1). So we define:
X ⊢ α df ⇐⇒ α ∈ Cn(X),(4)
and
X |∼ α df ⇐⇒ α ∈ C(X).(5)
We find that (3) is too strong. Let us consider first the inequality
CnC ≤ C,(6)
and express it as follows:
{β : X |∼ β} ⊢ α =⇒ X |∼ α,
or in a shorter form C(X) ⊢ α =⇒ X |∼ α. The last conditional does not look convincing; it is rather unjustifiably specific. 13 Turning to the inequality C ≤ CCn, the following generalization seems to be more flexible:
(X |∼ α and Y ⊆ Cn(X)) =⇒ X ∪ Y |∼ α.(7)
Besides, (7) can be seen as an attempt to make the relation | ∼ somewhat monotonic. As we will show in Section ??, in some interesting models, (7) does not hold. Nonetheless, as will be seen in the sequel, limited monotonicity can be achieved, although on a different path. We call a nonmonotonic relation | ∼ logical relative to a given monotonic consequence relation ⊢ if it satisfies the conditions:
X ⊢ α =⇒ X |∼ α, 14(8)
and (X |∼ α and α ⊢ β) =⇒ X |∼ β.
13 Although (6), and even (3), is fulfilled in any epsilon inference probabilistic operator (cf. [Makinson, 1994], observation 3.5.2), Makinson, however, makes the following remark.
It must be said, however, that there is a rather unsatisfying gap between the initial intuitions behind the epsilon approach and their technical formulation
[. . . ].
The conditions (8) and (9) can be formulated in terms of operators Cn and C that are defined by (4) and (5), respectively, as follows.
We say that an operator C is logical relative to Cn if, and only if, the following conditions are satisfied:
(log-1) Cn ≤ C; (log-2) α ∈ C(X) and β ∈ Cn(α) imply β ∈ C(X).
Continuing the discussion of the what question in relation to nonmonotonic reasoning, let us turn to the map of properties (con-1)-(con-9) presented in the next proposition. However, first, we will divide these properties into five (colored) groups: the reflexivity property (con-1), the closedness property (con-3), the monotonicity property (con-2), the cumulativity properties (con-5)-(con-7), and the finitariness properties (con-4), (con-8) and (con-9).
Proposition 2.1 (cf. [Citkin and Muravitsky, 2022], proposition 4.2.5). The following implications hold:
i) (con-1) and (con-6) imply (con-3); ii) (con-2) and (con-3) imply (con-5); iii) (con-5) implies (con-6); iv) (con-7) implies (con-6); v) (con-1) and (con-6) imply (con-7); vi) (con-1) and (con-7) imply (con-3); vii) (con-1) and (con-2) and (con-3) imply (con-7); viii) (con-9) implies (con-4); ix) (con-2) and (con-4) imply (con-9); x) (con-2) implies (con-8); xi) (con-4) and (con-8) imply (con-2). 15
As is clear from the last proposition, if we intend to challenge the monotonicity property (con-2), and we do so, we must be aware that other properties will be affected.
Note that (con-1) is always satisfied if the consequence operator in question is logical. We also observe that the cumulative properties and the finitariness properties do not interact. Further, in the presence of (con-1), if (con-7) does not hold, it is problematic that (con-3) holds.
In the mode of nonmonotonic reasoning that we are about to propose (thereby answering the how question), the closedness and cumulative properties that have been the focus of nonmonotonic research 16 will fail. Therefore, we will be interested in the finitariness properties. But even in this case, by Proposition 2.1-xi, we cannot have both (con-4) and (con-8). As is expected, (con-4) will take precedence. 15 The implication (xi) was noted by Makinson who wrote:
The compactness biconditional must [. . . ] fail in any nonmonotonic logic, and although the right-to-left half of it [. . .] if A 0 |∼ x for some A 0 included in A, then A |∼ x] seems the more clearly inappropriate, even its converse is rather dubious. . . ([Makinson, 1994], p. 41)
Matrix (monotonic) consequence and two restricted matrix consequences
In this section, we show some ways of passing from monotonic consequences to nonmonotonic consequences. We note that, although our starting point will be language-independent, in rejection of (con-2) we are forced to deal with specifics of the language involved. However, further, in showing which properties of the list (con-1)-(con-9) are preserved, we again can deal with a language-independent framework.
Fixing a language L, we also fix the following notations:
• V L is an infinite set of propositional variables;
• F L is a set of logical constants, also known as logical connectives, including 0-ary logical constants, also known as constants; • A L is the set of all atomic formulas, that is, the set of all variables and constants; • the universal algebra F L := Fm L , F L is called the L-formula algebra or simply formula algebra when L is unambiguous; • given a set X ⊆ Fm L , V(X) is the set of propositional variables occurring in all formulas from X. Interpreting elements of F L as operations on a nonempty set A, we obtain an algebra A := A, F L . Subsequently, we also use the notation |A| := A. We note that any mapping v : V L −→ A can be uniquely extended to a homomorphismv : F L −→ A. Because of this, we often regard v as a valuation in A 3.1. Matrix consequence. In defining a monotonic consequence, in order to keep a language-independent framework, we use one of Tarski's modes of semantic consequence (Section 1).
A system M := A, D , where D ⊆ |A| is called a (logical ) matrix. Any homomorphism v : F L −→ A is called a valuation in A; if A is part of a matrix M, v is
Given a matrix M = A, D , a relation |= M ⊆ P(Fm L ) × Fm L is defined as follows:
X |= M α df ⇐⇒ for any valuation v, v[X] ⊆ D implies v[α] ∈ D.(10)
The last relation can be generalized. Let M be a nonempty set of matrices. Then we define:
X |= M α df ⇐⇒ X |= M α for every M ∈ M.(11)
We will use the notations Cn M and Cn M , thereby indicating that the former is the operator corresponding to |= M and the latter is the operator corresponding to |= M , both in the light of definition (2). 17 These notations are justified by the following proposition.
Proposition 3.1. Any operator Cn M satisfies the conditions (con-1)-(con-3) and (con-10). 18 We note that the next equality is an immediate consequence of (11).
Cn M (X) = M∈M Cn M (X).(12)
3.2. M-r-consequence and M-r-consequence. Let V ⊆ V L and v be a valuation in an algebra A, regarded as a mapping on
V L . We call v ↾ V a restriction (of v) to V. Given V ⊆ V L , we denote by v V a restricted valuation with dom(v) = V, where dom(v) is the domain of v.
Given two restrictions v and w in the same algebra, we write v ≤ w (or w ≥ v) and say that w is an extension of v if dom(v) ⊆ dom(w).
In light of these notions, the relation |= M (see (10)) can be reformulated as follows:
X |= M α ⇐⇒ for any restriction v with dom(v) = V(X), if v[X] ⊆ D, then for any extension w ≥ v with V(X ∪ {α}) ⊆ dom(w), w[α] ∈ D.
Changing the right-hand side of the last equivalence, we obtain the following definition. 19
X |= r M α df ⇐⇒ for any restriction v with dom(v) = V(X), if v[X] ⊆ D, then there is an extension w ≥ v with V(X ∪ {α}) ⊆ dom(w), such that w[α] ∈ D.(13)
Generalizing the last definition to any nonempty set M of matrices, we obtain the following.
X |= r M α ⇐⇒ X |= r M α for every M ∈ M.(14)
The last definitions define M-r-consequence and M-r-consequence, respectively; we call them collectively restricted matrix consequences.
We denote by C r M and C r M the operators corresponding to |= r M and |= r M , respectively, and then we observe:
C r M (X) = M∈M C r M (X),(15)
which immediately follows from (13) To prove the next proposition, we borrow a definition from [Muravitsky, 2021] and then state a lemma.
X |= r M α ⇐⇒ (for any v V0 in M, v V0 M X =⇒ v V0 M α).(16)
Proposition 3.2. Any operator C r M is logical relative to Cn M . Proof. To prove the statement, we have to verify the conditions (log-1) and (log-2) for Cn M and C r M . It must be clear that for this, it suffices to verify (log-1) and (log-2) for Cn M and C r M , where M ∈ M. Let us fix such M = A, D . The condition (log-1), which is Cn M ≤ C r M , obviously is true. To prove (log-2), we use its form (9), that is, X |= r M α and α |= M β imply X |= r M β. Intending to apply (16), we denote V 0 = V(X) ∩ V(β) and assume that v V0 M X.
This implies that there is a restricted valuation v ≥ v V0 such that dom(v) = V(X) and v[X] ⊆ D. In virtue of the first assumption, there is a valuation v ′ ≥ v such that dom(v ′ ) = V(X) ∪ V(α) and v ′ [X ∪ {α}] ⊆ D. Now, given an arbitrary a ∈ |A|, we define a valuation w as follows:
w[q] := v ′ [p] if p ∈ V(X) ∪ V(α) a otherwise.
It is obvious that w validates α and hence, in virtue of the second assumption, w also validates β. Since v V0 ≤ w, we conclude that v V0 M β. It remains to apply Lemma 3.1.
The next two definitions were introduced in [Muravitsky, 2021].
Definition 3.2 (cf. [Muravitsky, 2021], definition 3.6). A relation ⊢⊆ P(Fm L ) × Fm L , or the operator C corresponding to this relation, is weakly monotonic if for any set
X ∪ Y ∪ {α} ⊆ Fm L with X ⊆ Y and V(X) ∩ V(α) = V(Y ) ∩ V(α), X ⊢ α implies Y ⊢ α.X, if X ⊢ α, then there is a nonempty Y ⋐ X with V(Y ) ∩ V(α) = V(X) ∩ V(α) such that Y ⊢ α.
The following properties of M-r-consequence were established earlier. In general, an operator C r M can fail (con-2), and the corresponding relation |= r M fails the transitivity property:
X |= r M β, for every β ∈ Y , and Y ∪ Z ⊢ C α imply X ∪ Z |= r M α, (transitivity) which is equivalent to the property:
Y ⊆ C r M (X) implies C r M (Y ∪ Z) ⊆ C r M (X ∪ Z).(17)
Example 3.1. Let A = A, ∧, ∨, ¬, 1 be a nontrivial Boolean algebra with a unit 1, and let M = A, {1} . Then, although p |= r M ¬q, we observe that p, q |= r M ¬q; and although p |= r M q and q |= r M ¬p, we observe that p |= r M ¬p. We note that (17) is a generalized form of (con-5). The property (con-6) also fails. Indeed, consider inclusion {p} ⊆ {p, q}. Although {p, q} ⊆ C r M (p), ¬q ∈ C r M (p) \ C r M ({p, q}). In virtue of Proposition 2.1-iv, (con-7) also fails.
Thus all cumulative properties can fail for some operators C r M , even when the matrix M is finite, while for all M, C r M is weakly monotonic, and for all finite M, C r M is very strongly finitary. 3.3. M-r * -consequence and M-r * -consequence. Although restricted matrix consequence exhibits interesting properties, there is a problem with it.
Example 3.2. Let A = A, →, 1 be a nontrivial implicative algebra; cf. [Rasiowa, 1974], chapter II. And let M = A, {1} . Then p |= r M q but p, q → q |= r M q, although |= M q → q and p, α |= r M q for any formula α which does not contain q. This observation leads to the following consideration.
Definition 3.4. Let M be a matrix and α be a formula. Given
p ∈ V(α), p is M- essential in (or of) α if there are valuations v and w in M such that v[α] = w[α], although for any q ∈ V(α) \ {p}, v[q] = w[q]; otherwise p is M-inessential. Given a family M of matrices, p is M-essential (in α) if it is M-essential for some M ∈ M; otherwise p is M-inessential.
Given a matrix M and formula α, we denote by V * M (α) the set of all M-essential variables of α; and denote:
V * M (X) := α∈X V * M (α).
We state the following two proposition that can be easily verified.
Proposition 3.5. Let M be a matrix. Given p ∈ V(α), if p is M-inessential, then v[α] = v[σ(α)]
for any valuation v in M and any substitution σ satisfying the condition: σ(q) = q whenever q = p.
Proof is obvious.
Proposition 3.6. Let α be a formula and v and v ′ be valuations in a matrix
M such that v ↾ V * M (α) = v ′ ↾ V * M (α). Then v[α] = v ′ [α]
. Proof can be carried out without much efforts by induction on the construction of α.
These two last propositions will be used without reference.
The concept of an inessential variable induces an additional concept.
Definition 3.5. Let M be a matrix and a ∈ |M|. We add a new constant c a to the language L along with the agreement that each valuation v in M, being extended to include c a in its domain, satisfies the condition: v[c a ] = a. Now, Given an L-formula α, any formula that is obtained from the former by replacing all Minessential variables of α by c a is called a c a -instance of α. If α does not contain M-inessential variables, a c a -instance of α is coincident with α.
[α] = v[α * ] = v[α * * ].
Proof can be carried out by induction on the complexity of α.
The last proposition induces the following definition.
Definition 3.6 (c-instance). Let M be a matrix. An arbitrary fixed c a -instance of a formula α is called simply a c M -instance of α is denoted by α cM . If X ⊆ Fm L , we denote:
X cM := {α cM : α ∈ X}.
The next equality and the following conditional will be used throughout without reference.
V
(X cM ) = V * M (X) for every matrix M. Y ⊆ X =⇒ Y cM ⊆ X cM for every matrix M.X |= r * M α df ⇐⇒ for any restricted v in M with dom(v) = V * M (X), there is an extension w ≥ v with V(X ∪ {α}) ⊆ dom(w) such that if v[X cM ] ⊆ D, then w[α] ∈ D.(18)
We note:
∅ |= r * M α ⇐⇒ ∅ |= r M α.(19)
Similarly to (14), we define the relation of M-r * -consequence.
X |= r * M α df ⇐⇒ X |= r * M α for every M ∈ M.(20)
The corresponding operators are denoted by C r * M and C r * M . Similarly to (15), we have:
C r * M (X) = M∈M C r * M (X).(21)
First, we focus on M-r * -consequence. From definition (18), there follows the following equivalence.
X |= r * M α ⇐⇒ X cM |= r M α.
(22) Proof is obvious.
We note that (22) could be taken as a definition of relation |= r * M .
It is also true that (23) can be extended to the following:
X |= r M σ(α) =⇒ X |= r M α and X |= r * M σ(α) =⇒ X |= r * M α, providing that V(X) ∩ V(α) = ∅.
The last implication can obviously be reduced to the case when M consists of a single matrix:
X |= r * M σ(α) =⇒ X |= r * M α, where V(X) ∩ V(α) = ∅.
Indeed, according to (22), the last implication is equivalent to the following:
X cM |= r M σ(α) ⇐⇒ X cM |= r M α. It remains to notice that V(X cM ) ∩ V(α) = ∅ whenever V(X) ∩ V(α) = ∅.
Logical friendliness relation and beyond
So far, the specifics of objective language have not been used in our discussion to achieve positive results, except that the language was sentential. But to get negative results, we had to turn to specific languages. However, specifying an objective language can also open up more possibilities for defining nonmonotonic consequence relations.
For the rest of this paper, we will use the language of Example 3.1, that is a language with the logical connectives ∧, ∨, →, ¬ and the constant ⊤. We denote this formal language by L • . Unspecified formulas of L • we denote by letters A, B, etc., and sets of such formulas by letters Γ, ∆, etc. We continue using the same notation for substitutions and valuations. Since all our matrices will be Heyting algebras with a greatest element 1, for each valuation v, we require the condition: v[⊤] = 1.
Logical friendliness.
Our interest to nonmonotonic relation was inspired by [Makinson, 2005b] and [Makinson, 2007], where the relation of logical friendliness was introduced. 20 To define logical friendliness, we use the matrix B 2 which is a two-element Boolean algebra with the filter {1}; namely
Γ |∼ F A df ⇐⇒ Γ |= r B2 A.(24)
Further, we define:
Γ |∼ F * A df ⇐⇒ Γ |= r * B2 A.(25)
We denote the operators corresponding to |∼ F and |∼ F * by C F and C F * , respectively.
In view of Example 3.1 and Example 3.2, we observe:
Cn B2 < C F < C F * .(26)
The following proposition later will play the role of a navigator, but it is convenient to discuss this property here. It reads that for defining logical friendliness, we can use any nontrivial Boolean algebra. 20 Makinson has proved that logical friendliness is strongly finitary with some additional property for monotonicity, which is however weaker than the one formulated in Proposition 5.1 of [Muravitsky, 2021].
Proposition 5.1. Let B be a nontrivial Boolean algebra that is in combination with a logical filter {1} forms a logical matrix B. Then
Γ |= r B A ⇐⇒ Γ |= r B2 A.(27)
Proof. We recall that B (understood as an algebra, not matrix) is a subdirect product B I 2 (the Cartesian product of copies of B 2 ), where B 2 is also regarded here as an algebra. This means that B is embedded to B I 2 so that each projection, g i , is an epimorphism. We also recall that each valuation in an algebra is a homomorphism from the corresponding formula algebra to the former. Now assume that Γ |= r B A and consider a valuation v in
B 2 such that v[Γ] ⊆ {1} and dom(v) = V(Γ).
We define a valuationv in B as follows.
v
[p] := 0 if v[p] = 0, 1 if v[p] = 1, for every p ∈ V(Γ). It is clear thatv[Γ] ⊆ {(1) I }. Therefore, by premise, there is an extension w ≥v such that w[Γ ∪ {α}] ⊆ {(1) I }. Now, v * := g i • w is a valuation in B 2 such that v * ≥ v and v * [Γ ∪ {A}] ⊆ {1}.
Next assume that Γ |= B2 A and that a valuation w in B validates Γ, that is,
w[Γ] ⊆ {(1) I }. Then, obviously, valuation v := g i • w validates Γ in B 2 , that is, v[Γ] ⊆ {1}. Then there is an extension v ′ ≥ v such that v ′ [Γ ∪ {A}] ⊆ {1}. Now we define a valuation w ′ in B (regarded as a subalgebra of B I 2 ) according to the rule: (w ′ [p]) i := v ′ [p]. It is clear that w ′ ≥ w and w ′ [Γ ∪ {A}] ⊆ {(1) I }.
5.2. Beyond logical friendliness. We aim to make logical friendliness the starting point for getting more logical nonmonotonic consequences. This time we choose the method of proof theory.
Let Int and Cl denote the classes of intuitionistic and classical propositional tautologies, respectively. An intermediate logic is a set L of formulas, satisfying the following conditions: Int ⊆ L ⊆ Cl, L is closed under uniform substitution and under modus ponens. The class of all intermediate logics is denoted by ExtInt.
Let L ∈ ExtInt. A finite sequence A 1 , A 2 , . . . , A n is called an L-derivation from a set Γ if for each A i , either A i ∈ L or A i ∈ Γ or A i can be obtained by modus ponens from two preceding formulas of the sequence. In addition, we say that the sequence is an L-derivation of the last formula of the sequence, that is, of A n . Next, we define:
Γ ⊢ L A df ⇐⇒ there is an L-derivation of A from Γ.
Each relation ⊢ L obviously is a monotonic consequence relations. We denote the corresponding consequence operator by Cn L .
We call a set Γ a Cn L -theory if Cn L (Γ) = Γ. (This concept will be employed later.)
We call any expression of the form Γ ⇒ A a sequent. Next, we provide the list of sequential L-axioms (or sL-axioms for short) and that of sequential L-rules (or sL-rules for short). sL-Axioms: axiom 1: ⇒ p, for any p ∈ V L • ; axiom 2: Γ ⇒ ⊤;
axiom 3: Γ ⇒ ∆ if ∆ ⋐ Γ ( ∆ := ⊤ if ∆ = ∅); axiom 4: Γ ⇒ A whenever Γ ⊢ L A.
sL-Rules: A finite nonempty list Γ 1 ⇒ A 1 , . . . , Γ n ⇒ A n is called an sL-derivation (of the last sequent on the list, i. Γ n ⇒ A n ) if each Γ k ⇒ A k is either a sL-axiom or can be obtained from preceding sequents by one of the rules 1-6. We say that Γ ⇒ A holds in sL-calculus if there is an sL-derivation ending with Γ ⇒ A. This leads to the following definition: for each l ∈ ExtInt,
rule 1: ⇒ A Γ ⇒ A , providing that V(Γ) ∩ V(A) = ∅; rule 2: Γ, A ⇒ C and ∆, B ⇒ C Γ ∪ ∆, A ∨ B ⇒ C , providing that V(Γ ∪ {A}) = V(∆ ∪ {B}); rule 3: Γ ⇒ σ(A) Γ ⇒ A ,Γ L A df ⇐⇒ Γ ⇒ L A holds in sl-calculus.(28)
The motivation for formulating the sL-calculi is based on the following observation.
Γ |∼ F A ⇐⇒ Γ Cl A (cf. [Muravitsky, 2007], theorems 4.5 and 4.8).
Reformulating the last equivalence, we obtain completeness theorem for the nonmonotonic consequence Cl :
Γ Cl A ⇐⇒ Γ |= r B2 A.
This suggests that there may be similar completeness theorems for other L relations. We will come back to this later, but now we will discuss what is interesting about the L relations.
According to [Muravitsky, 2021], proposition 7.3, each relation L is reflexive and nonmonotonic. Further, for any nonempty set Γ and any formula A with V(Γ) ∩ V(A) = ∅, if Γ L A, then there is a nonempty set ∆ ⋐ Γ such that ∆ L A. To this, we add the following.
Proposition 5.2. Each L is logical relative to ⊢ L .
Proof. Indeed, suppose Γ ⊢ L A. Then, by axiom 4, Γ ⇒ L A holds, that is, Γ L A.
Next, assume that Γ L A and A ⊢ L B. The first assumption implies that there is an sL-derivation of the sL-sequent Γ ⇒ L A. Then we apply rule 6. Thus there is an sL-derivation of the sL-sequent Γ ⇒ L B, that is, Γ L B. Now let us return to the question of the possibility of the completeness theorem for each relation L .
We recall that, in the terminology of [Citkin and Muravitsky, 2022], each ⊢ L is a unital assertional abstract logic. 21 For a fixed L ∈ ExtInt, we denote by Σ L the set of all L-theories. Then, given D ∈ Σ L , we denote by LT L [D] the Lindenbaum-Tarski algebra relative to D. Further, each LT L [D] is a homomorphic image of LT L [Cn L (∅)] and, hence, is a Heyting algebra. Moreover, each such algebra can be regarded as a unital matrix when the logical filter consists of one element. In each LT L [D], the designated element is the unit of the algebra, that is, its greatest element.
In the case L = Cl, each LT Cl [D] is a Boolean algebra. This circumstance will play an important role soon.
By proposition 6.3.5 of [Citkin and Muravitsky, 2022],
Γ ⊢ L A ⇐⇒ Γ |= LT L [D] A for every D ∈ Σ L .(29)
This allows us to say that each ⊢ L is the M L -consequence, where
M L := {LT L [D] : D ∈ Σ L }.
In the case L = Cl, we have:
Γ ⊢ Cl A ⇐⇒ Γ |= LT Cl [D]
A for every D ∈ Σ Cl . Now, when we move from M Cl consequence to M Cl -r-consequence, in virtue of Proposition 5.1, we obtain the conclusion: relation |∼ F is the M Cl -r-consequence. This suggests the following.
Conjecture 1. Each L is the M L -r-consequence.
Of course, from M L -r-consequence, we can move to M L -r * -consequence. Then, we define: 6. What has been achieved?
Γ L * A df ⇐⇒ Γ c L A,
We started our discussion about the concept of nonmonotonic reasoning from a philosophical point of view. Popular belief says that time can render the final verdict. But what is meant by saying this? We can suggest two possible answers.
The first is contained in the following quote from R. von Mises.
I aim at the construction of a rational theory, based on the simplest possible concepts, one which, although admittedly inadequate to represent the complexity of the real processes, is able to reproduce satisfactorily some of their essential properties. ( [von Mises, 1981], The inadequacy of theories)
The second opinion is more difficult to express. The success of the proposed approach is often provided by unexplored space for its further development. The larger this space, the more questions left unanswered, the more tools available to answer these questions, the more likely it is that this approach will become the main focus of research. Such an approach could be termed open, that is, open for further development.
This view of mine echoes another statement by von Mises.
[. . .] the value of a concept is not gauged by its correspondence with some usual group of notions, but only by its usefulness for further scientific development [. . .] ( [von Mises, 1981], Synthetic definitions)
To be open in the above sense means not to be sterile in the understanding of G. Kreisel, who wrote, Aristotle gives much attention to the matter of choice of abstractions. This indicates a shift of emphasis away from matters of principle (e.g. mere validity of analogies) to a focus (on, of course, valid cases) which provides safeguards against sterility (not merely against straight error). ([Kreisel, 2019], Choice of abstractions) Whether a (consistent) approach is open enough to avoid sterility can only be judged by time.
In a broader context, concerning the what and how questions, the difficulty of the task should not overshadow the more important level associated with the act of knowing. This thought is not mine, but a paraphrase of the thought of Thomas Aquinas, who wrote, When something is more difficult, it is not for that reason necessarily more worthwhile, but it must be more difficult in such a way, as also to be at a higher level of goodness. [Read: "the act of knowing" instead of "goodness"] (Quoted from [Pieper, 2009], chapter II.)
also a valuation in M. The result of the application of v to α is denoted by v[α]; and for X ⊆ Fm L , v[X] := {v[α] : α ∈ X}.
Definition 3. 1
1(comp. [Muravitsky, 2021], definition 3.5). Let M be a matrix. Given a nonempty set X of formulas and a valuation v in M, we say that v adopts X in M, in symbols v M X, if there is an extension w of v with dom(w) = V(X) ∪ dom(v) which validates all formulas of X. If X = {α}, we write v M α.
Given X ∪ {α} ⊆ Fm L , let us denote V 0 := V(X) ∩ V(α). Then for any matrix M,
Definition 3.3 (cf.[Muravitsky, 2021], definition 5.1). A relation ⊢⊆ P(Fm L ) × Fm L , or the operator C corresponding to this relation, is called very strongly finitary if for any nonempty set
Proposition 3.3 ([Muravitsky, 2021], corollary 3.1). Any M-r-consequence is weakly monotonic.
Proposition 3.4 ([Muravitsky, 2021], proposition 5.1). If M is a finite set of finite matrices, then M-r-consequence is very strongly finitary.
Proposition 3 . 7 .
37Let M be a matrix, a, b ∈ |M| and v be an arbitrary valuation in M extended by the conditions v[c a ] = a and v[c b ] = b. If α * and α * * are c a -instance and c b -instance of α, respectively, then v
Definition 3 . 7 .
37Let M = A, D be a matrix. A formula α is called an Mconstant if there is an element a ∈ |A| such that for any valuation v in M, v[α] = a. Given a family M of matrices, α is an M-constant if it is an Mconstant for each M ∈ M. Proposition 3.8. Let M be a family of matrices and α be a formula. If all variables of α are M-inessential, then α is an M-constant. Proof. Suppose M is an arbitrary matrix from M and v is a valuation in M. We denote: a := v[α]. Since each variable of α is M-inessential, it is not difficult to conclude that for any valuation v ′ in M, v ′ [α] = a. Therefore, α is M-constant. Refining definition (13), we arrive at the main definition of this subsection. Let M = A, D and X ∪ {α} ⊆ Fm L .
providing that V(Γ) ∩ V(A) = ∅; (Reverse substitution) rule 4: Γ ⇒ A, and A ⇒ B Γ ⇒ B , providing that either V(Γ) ⊆ V(A) or V(Γ) ∩ V(B) ⊆ V(A) ⊆ V(Γ);(Cut) rule 5: Γ, A ⇒ B and C ⇒ A Γ, C ⇒ B , providing that V(C) ⊆ V(Γ ∪ {A}); (Deductive replacement in antecedent) rule 6: Γ ⇒ A and A ⇒ B Γ ⇒ B . (Deductive replacement in consequent)
where Γ c := {B c LT L [X] : B ∈ Γ} and each B c LT L [X] is obtained from B by replacement of each occurrence of a variable inessential in LT L [Γ] with ⊤. (We recall that each LT L [Γ] is a Heyting algebra.) The following conjecture suggests itself. Conjecture 2. Γ L * A if, and only if, Γ |= r * M A for every M ∈ M L .
See [Grice, 1989], part I. 4 See e.g.[Makinson, 2005a]. 5 This property is also known as cautious monotony.
Cf.[Kraus et al., 1990], section 1.2; comp.[Gabbay, 1985]. 12 This idea was systematically implemented in[Makinson, 2005a], although only as an explanatory tool, and not as a methodological guide.
See comments about (8) in[Makinson, 2005a], section 1.3.
See e.g.[Makinson, 1989].
It must be clear that if M = {M}, |= M and |= M can be used interchangeably.
This statement goes back to[ Lukasiewicz and Tarski, 1930], theorem 3; For modern expositions, the reader is offered to consult[Wójcicki, 1988], theorem 3.1.3, or[Dunn and Hardegree, 2001], corollary 6.13.3, or[Citkin and Muravitsky, 2022], proposition 4.3.11. 19 The idea of this definition is due to D. Makinson, who used it with a two-element Boolean matrix; cf.[Makinson, 2005b, Makinson, 2007.
For definitions, the reader is offered to consult[Citkin and Muravitsky, 2022], chapter 6.
Proof. To verify (log-1), we notice that, in virtue of Proposition 3.10, for every M ∈ M, Cn M ≤ C r * M . Then, in virtue of (15) and(21), Cn M ≤ C r * M . To verify (log-2), assume that α ∈ C r * M (X) and β ∈ Cn M (α), that is, X |= r * M α and α |= M β. In virtue of Proposition 3.10, for every M ∈ M, X |= r * M β, that is, β ∈ C r * M (X).Proposition 3.14. If X |= r * M α, then for any set Y with X ⊆ Y andProof immediately follows from Proposition 3.11.Proposition 3.15. Let M be a finite family of finite matrices. For any nonemptyNext we denote:. . , n}. Applying Proposition 3.11, we conclude that, then, according to Proposition 3.14, Z |= r * Mi α for each M i ∈ M, that is, Z |= r * M α. Corollary 3.1. Let M be a finite family of finite matrices. Then M-r * -consequence is strongly finitary.SubstitutionAccording to Proposition 3.1, any Cn M operator is structural. In terms of consequence relations, this means that, given a matrix M, for any substitution σ,The structurality property can hardly be expected for |= r M or |= r * M relations. However, the following rule can be considered for these relations, although not without some restriction.providing that V(X) ∩ V(α) = ∅. (In Section 5.2 this property is formulated as the rule of reverse substitution.) It is obvious that (23) is true. Indeed, the restriction 'V(X) ∩ V(α) = ∅' allows us to reduce (23) to the implication: for any valuation v in M, if v validates σ(α), then v • σ validates α. On the other hand, if we drop the restriction, we face with the problem that when a valuation v validates X and its extension w validates σ(α), the valuation w • σ, which validates α, is not necessarily an extension of v.
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Comptes Rendus Séantes de la Société des Sciences et des Letters de Varsovie, Classe III. A Tarski ; Tarski, Sciences Mathéantiques et Physiques. 23English translation: On some fundamental concepts of metamathematics. Tarski, 1956. also in: [Tarski, 1983Tarski, 1930] Tarski, A. (1930).Üder einige fundamentale Begriffe der Metamathematik (Ger- man). Comptes Rendus Séantes de la Société des Sciences et des Letters de Varsovie, Classe III, Sciences Mathéantiques et Physiques, 23:22-29. English translation: On some funda- mental concepts of metamathematics, in: [Tarski, 1956], pp. 30-37; also in: [Tarski, 1983], pp. 30-37.
Über den Begriff der logischen Folgerung (German). A Tarski ; Tarski, Actes du Congrès International de Philosophie Scientifique. s du Congrès International de Philosophie Scientifique7English translation: On the concept of logical consequence. Tarski, 1956. Tarski, 1983Tarski, 1936] Tarski, A. (1936).Über den Begriff der logischen Folgerung (German). Actes du Congrès International de Philosophie Scientifique, 7:1-11. English translation: On the con- cept of logical consequence, in: [Tarski, 1956], pp. 409-420; [Tarski, 1983], pp. 409-420.
Logic, Semantics, Metamathematics. Papers from 1923 to 1938. A Tarski ; Tarski, J. H. WoodgerOxford at the Clarendon PressTarski, 1956] Tarski, A. (1956). Logic, Semantics, Metamathematics. Papers from 1923 to 1938. Oxford at the Clarendon Press. Translated by J. H. Woodger.
Logic, Semantics, Metamathematics. A Tarski ; Tarski, John CorcoranHackett Publishing CoIndianapolissecond editionTarski, 1983] Tarski, A. (1983). Logic, Semantics, Metamathematics. Hackett Publishing Co., Indianapolis, IN, second edition. Papers from 1923 to 1938, Translated by J. H. Woodger, Edited and with an introduction by John Corcoran.
Probability, Statistics and Truth. R Mises ; Von Mises, Dover Publicationssecond revsed English editionMises, 1981] von Mises, R. (1981). Probability, Statistics and Truth. Dover Publications, second revsed English edition.
Theory of logical calculi, volume 199 of Synthese Library. R Wójcicki, Kluwer Academic Publishers GroupDordrechtBasic theory of consequence operations, 1988] Wójcicki, R. (1988). Theory of logical calculi, volume 199 of Synthese Library. Kluwer Academic Publishers Group, Dordrecht. Basic theory of consequence operations.
| {'fraction_non_alphanumeric': 0.08003213527420187, 'fraction_numerical': 0.02456053099714991, 'mean_word_length': 3.8501716300213378, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We discuss two ways to implement a semantic approach to nonmonotonic consequence relations in an arbitrary propositional language. For one particular language, we also discuss the proof-theoretic framework that we connect with this semantic approach. This article is an addition to[Muravitsky, 2021]. † This is the text of my speech at the Logica Universalis webinar, which took place on May 11, 2022.1 First-order language is used only as an auxiliary tool. 2 Here is a full quote:1 Proposition 3.9. For any matrix M = A, D and a set X ∪ {α} ⊆ Fm L ,Thus, we obtain that X cM |= r M α. It remains to apply(22). Example 3.2 shows that the converse can fail.Proposition 3.10. Relation |= r * M is logical relative to |= M . Proof. Indeed, property (8) follows from Proposition 3.2 and Proposition 3.9.To prove property(9), we assume that X |= r * M α and α |= M β. In virtue of (22), X cM |= r M α. According to Proposition 3.2, X cM |= r M β. Then we apply(22). We should not expect that relation |= r * M will enjoy the weak monotonicity property (Definition 3.2). However, we obtain the following.Proposition 3.11. Given a matrix M, for any setM α and all other premises be fulfilled. Then, in virtue of (22),and X cM ⊆ Y cM , we can apply the property that |= r M is weakly monotonic (Proposition 3.3) in order to conclude that Y cM |= r M α, that is, Y |= r * M α. As to finitariness, we obtain the following proposition.Proposition 3.12. Given a matrix M, for any nonempty(22), X cM |= r M α. In virtue of Proposition 3.4, there is a nonempty X 0 ⋐ X cM such that V(X 0 )∩V(α) = V(X cM )∩ V(α) and X 0 |= r M α. It must be clear that for some nonempty Y ⋐ X, Y cM = X 0 . This implies that. The last part of the statement follows by Proposition 3.11. Now we will turn to M-r * -consequence. First of all, we observe the following.Proposition 3.13. Any M-r * -consequence is logical relative to the corresponding M-consequence.', 'arxivid': '2212.13355', 'author': ['\nTWO MODES OF NONMONOTONIC CONSEQUENCE † ALEXEI MURAVITSKY\n\n', '\nTWO MODES OF NONMONOTONIC CONSEQUENCE † ALEXEI MURAVITSKY\n\n'], 'authoraffiliation': ['TWO MODES OF NONMONOTONIC CONSEQUENCE † ALEXEI MURAVITSKY\n', 'TWO MODES OF NONMONOTONIC CONSEQUENCE † ALEXEI MURAVITSKY\n'], 'corpusid': 255186179, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 17337, 'n_tokens_neox': 15487, 'n_words': 9101, 'pdfsha': '1062dc4cad539756883a5c616cd2a58cac8c0498', 'pdfurls': ['https://export.arxiv.org/pdf/2212.13355v1.pdf'], 'title': [], 'venue': []} |
arxiv |
No-Cloning and No-Deleting theorems through the existence of Incomparable states under LOCC
24 Jun 2006
Amit Bhar
Department of Applied Mathematics
University of Calcutta
92, A.P.C. RoadKolkata-700009India
Indrani Chattopadhyay
Department of Applied Mathematics
University of Calcutta
92, A.P.C. RoadKolkata-700009India
Debasis Sarkar
Department of Applied Mathematics
University of Calcutta
92, A.P.C. RoadKolkata-700009India
No-Cloning and No-Deleting theorems through the existence of Incomparable states under LOCC
24 Jun 2006number(s): 0367Mn0367Hk Keywords: CloningDeletingIncomparability
No-Cloning and No-Deleting theorems are verified with the constraint on local state transformations via the existence of incomparable states. Assuming the existence of exact cloning or deleting operation defined on a minimum number of two arbitrary states, an incomparable pair of states of the joint system between two parties can be made to compare under deterministic LOCC. We have restricted our proof with the assumption that the machine states of the cloning or deleting operations do not keep any information about the input states. We use the same setting to establish the no-cloning and no-deleting theorems via incomparability that supports the reciprocity of the two operations in their operational senses. The work associates the impossibility of operations with the evolution of an entangled system by LOCC.
One of the most important task in quantum information processing is to detect the allowable set of operations performed on quantum systems. If someone wants to copy an arbitrary quantum information encoded in a quantum state then no-cloning theorem [1] restricts one to copy arbitrary quantum information exactly. Quite reverse to it, if we want to delete arbitrary quantum information then we have a similar kind of restriction [2,3]. According to the no-deletion theorem [2,3], it is not possible to delete arbitrary quantum information encoded in a quantum state to a standard one. On the other hand, manipulation of pure state entanglement provides us some other kind of restrictions on the evolution of quantum systems. Sometimes a specific state may be required to perform a specific information theoretic task. Then Nielsen's criterion [4] determines the possibility of inter-conversion of one pure entangled state shared between two spatially separated parties to another by deterministic LOCC. This result provides us a necessary and sufficient condition for converting a bipartite pure entangled state to another by LOCC with certainty. Now one may ask whether the no-go theorems and other impossibilities only restrict the specific tasks or may be useful for other kind of tasks that seems to be impossible otherwise? To search for a common origin of these impossibilities one have to find the possibility of interconnection between themselves within or outside the quantum formalism. Here we provide a connection between nocloning and no-deleting theorems with the incomparability of pure entangled states. The work shows, existence of either of the exact cloning or deletion machine that act perfectly on any set of non-orthogonal states, will imply local inter-conversion of a pair of incomparable states with certainty.
We begin with some necessary background to our work. In quantum information theory the no-go theorems are used to define intrinsic properties of quantum systems beyond their usual status of imposing restrictions over the systems. They allow quantum systems to perform some computational tasks which are rather impossible by using classical algorithms. In quantum cryptography [5], the possibility of detecting an eavesdropper having an access on the communication channel emerges out of the well known no-cloning [1] theorem. In terms of information processing, cloning can be viewed as the copying of information encoded in some systems to other systems [6]. If |ψ be the input state then we describe exact cloning operation as |ψ ⊗ |b ⇒ |ψ ⊗ |ψ , where |b is some suitably chosen blank state. Now quantum systems will not provide complete accuracy of performing those operations on arbitrary input states. Linearity of quantum operations establishes precisely the impossibility of existence of an Universal Exact Cloning Machine [1,7]. Unitarity of any quantum evolution also shows that Universal Exact Cloning operation is not physical in nature [8]. Linearity of allowable quantum operations further provides us another constraint which we termed as No-Deletion theorem [2,3]. Deletion is quite a reverse process than that of cloning. It is performed on two copies of an arbitrary input state and is not possible to delete exactly the information of one copy, keeping intact the information of the other copy. In other words, the operation |ψ ⊗ |ψ ⇒ |ψ ⊗ |b is not possible exactly for an arbitrary input state |ψ with certainty.
There are some other no-go theorems defined on single qubit systems, such as the no-flipping theorem [9]. From linearity of quantum operations we find further the restriction of no-partial-cloning, and other no-go theorems obtained from the concepts of various quantum gates [10]. Efforts are made to search the inter-relations between different no-go theorems and relate them with other theories. For example, no-signaling principle restricts any physical operation to evolve in such a way that can not be used to send a signal faster than the speed of light. Nosignaling condition preserves all the impossibilities cited above [11,12,13,14]. Again, the constraint of nonincrease of entanglement under LOCC, described in a quite similar way as that of the second law of thermodynamics. Applying any local operations on the subsystems of a quantum system together with classical communica-tions between distant parties, it is impossible to increase the entanglement of the joint system. The no-cloning [15], no-deleting [16], no-flipping [14] and many other impossibilities [16] are connected with this constraint of information theory. Also the interrelation between the cloning and flipping operations is revealed by the conservation laws of simple classical theory [17]. Now a very new kind of information theoretic restriction on allowable quantum operations observed through the existence of incomparable states [18]. This restriction retrieve the no-flipping theorem [19] and also detects impossibility of some general classes of local quantum operations [20].
Here we want to reveal a relation between this constraint with the very famous no-cloning and no-deleting theorems. The work proceeds to verify the reciprocity of the two no-principles by dealing them in a single setting without verifying them separately. The connection between all those no-go theorems of quantum systems with the impossibility of inter-conversion of incomparable states would support the existence of incomparable states beyond their mathematical status from Nielsen's criterion. It provides also the nature of allowable physical operations.
To present our work we need to define first the condition for a pair of states to be incomparable with each other. The notion of incomparability of a pair of bipartite pure entangled states is a consequence of Nielsen's [4] majorization criterion. Suppose we want to convert the pure bipartite state |Ψ to |Φ shared between two parties, say, Alice and Bob by deterministic LOCC. Consider the pair (|Ψ , |Φ ) in their Schmidt bases {|i A , |i B } with decreasing order of Schmidt coefficients:
|Ψ = d i=1 √ α i |i A i B , |Φ = d i=1 √ β i |i A i B , where α i ≥ α i+1 ≥ 0 and β i ≥ β i+1 ≥ 0, for i = 1, 2, · · · , d − 1, and d i=1 α i = 1 = d i=1 β i . The
Schmidt vectors corresponding to the states |Ψ and |Φ are λ Ψ ≡ (α 1 , α 2 , · · · , α d ) and λ Φ ≡ (β 1 , β 2 , · · · , β d ). Then Nielsen's criterion says |Ψ → |Φ is possible with certainty under LOCC if and only if λ Ψ is majorized by λ Φ , denoted by λ Ψ ≺ λ Φ and described as,
k i=1 α i ≤ k i=1 β i ∀ k = 1, 2, · · · , d(1)
It is interesting to note that as a consequence of nonincrease of entanglement by LOCC, if |Ψ → |Φ is possible under LOCC with certainty, then E(|Ψ ) ≥ E(|Φ ) [where E(·) denote the von-Neumann entropy of the reduced density operator of any subsystem and known as the entropy of entanglement]. Now in case of failure of the above criterion (1), it is usually denoted by |Ψ → |Φ . But it may happen that |Φ → |Ψ under LOCC. And if it happens that both |Ψ → |Φ and |Φ → |Ψ then we denote it by |Ψ ↔ |Φ and describe (|Ψ , |Φ ), as a pair of incomparable states. One of the peculiar feature of the existence of such incomparable pairs is that we are really unable to say which state has a greater amount of entanglement content than that of the other. For 2 × 2 systems there are no pair of incomparable pure entangled states as described above. For our purpose, we want to mention explicitly the criterion of incomparability for a pair of pure entangled states |Ψ , |Φ of m × n system where min{m, n} = 3. Suppose the Schmidt vectors corresponding to the two states are (a 1 , a 2 , a 3 ) and (b 1 , b 2 , b 3 ) respectively, where a 1 > a 2 > a 3 > 0 , b 1 > b 2 > b 3 > 0 , a 1 +a 2 +a 3 = 1 = b 1 +b 2 +b 3 . Then it follows from Nielsen's criterion that |Ψ , |Φ are incomparable [18] if and only if, either of the pair of relations
a 1 > b 1 & a 3 > b 3 b 1 > a 1 & b 3 > a 3(2)
will hold. Our paper concerns with one-to-two copy exact cloning operation on a minimum number of two arbitrary states |0 , |ψ in the following form
|0 |b −→ |0 |0 |ψ |b −→ |ψ |ψ (3)
where |b is a suitably chosen blank state. We concentrate entirely within the quantum formalism and for that reason we assume the machine states do not keep any information about the input states. So we drop the machine states in the definition of the cloning operation. No-cloning theorem then turns out to be the impossibility of this operation for arbitrary state |ψ . Now we consider that Alice and Bob, two spatially separated parties have a particular setting of a pure bipartite state in the form given below
|Ω i AB = 1 √ N i {|1 A |0ψ0ψ + ψ0ψ0 B + |2 A |0ψψ0 −ψ00ψ B + |3 A |00ψψ − ψψ00 B } ⊗ |b B
(4) This is a six particle state where Alice has one qutrit and Bob has four qubits entangled with Alice's system together with a separate qubit in the form of blank state |b . So the joint system is of 3 × 32 dimension, where N i = 2(3 − α 4 ) be the normalizing constant and |ψ = α|0 + β|1 be an arbitrary qubit with |α| 2 + |β| 2 = 1. As the arbitrary input state |ψ can be written in the form |ψ = cos θ 2 |0 + e −iφ sin θ 2 |1 , where θ, φ satisfy the following equations 0 ≤ φ ≤ 2π, − π 2 ≤ θ ≤ π 2 , hence without loss of generality, the parameter α is treated here as a real constant.
Tracing out Bob's local system we compute the initial reduced density matrix ρ i A on Alice's side in the following form
ρ i A = tr B [ |Ω i AB Ω i | ] = 1 N i {2(1 + |α| 4 )P [|1 ] + 2(1 − |α| 4 ) (P [|2 ] + P [|3 ])}(5)
where P [|j ] = |j j|, for any j. The Schmidt vector of the initial state can be written as λ i = (λ i 1 , λ i 2 , λ i 2 ) where λ i 1 = 1+α 4 3−α 4 and λ i 2 = 1−α 4 3−α 4 . Hence λ i max = max{λ i 1 , λ i 2 } = λ i 1 and λ i min = min{λ i 1 , λ i 2 } = λ i 2 . If the cloning operation defined in equation (3) exists and is applied on Bob's local system (say on his fourth qubit together with the blank state), the joint pure state shared between Alice and Bob could be exactly transformed to the pure state,
|Ω f AB = 1 √ N f {|1 A |0ψ0ψψ + ψ0ψ00 B + |2 A |0ψψ00 −ψ00ψψ B + |3 A |00ψψψ − ψψ000 B } (6) where N f = 2(3 − α 5 )
be the normalizing constant. The final reduced density matrix on Alice's side would be
ρ f A = tr B [ |Ω f AB Ω f | ] = 1 N f {2(1 + α 5 )P [|1 ] + 2(1 − α 5 )(P [|2 ] + P [|3 ]) − 2α 2 (1 − α)(|2 3| + |3 2|)} (7) Hence the Schmidt coefficients of the final state ρ f A are { 1+α 5 3−α 5 , (1+α 2 )(1−α 3 ) 3−α 5 , (1−α 2 )(1+α 3 ) 3−α 5 }. If we denote λ f 1 = 1+α 5 3−α 5 , λ f 2 = (1+α 2 )(1−α 3 ) 3−α 5 and λ f 3 = (1−α 2 )(1+α 3 ) 3−α 5
,
then λ f min = min{λ f 1 , λ f 2 , λ f 3 } = λ f 3 and thus λ f max = max{λ f 1 , λ f 2 }. Now using simple algebra we find, λ f 1 < λ i 1
and also λ f
2 < λ i 1 (if, λ f 1 < λ f 2 ), which implies that, λ f max < λ i max . Finally we get, λ f min = λ f 3 < λ i 2 = λ i min .
These inequalities clearly indicate the nature of incomparability of the pair of pure bipartite states |Ω i and |Ω f . The incomparability of the states imply that the final state |Ω f can not be achieved from the initial state |Ω i through LOCC with certainty. Thus we are compelled to conclude that the cloning operation performed on Bob's local system to implement the transformation |Ω i → |Ω f locally, is not a physical operation. In other words the exact cloning operation is not possible, for any pair of arbitrary non-orthogonal input states. This ensures the successful establishment of no-cloning theorem. Now if we further treat |Ω f as the initial pure bipartite state, shared between Alice and Bob and assume the existence of an exact deleting machine again defined on only two arbitrary input qubit |0 , |ψ as |0 |0 −→ |0 |b |ψ |ψ −→ |ψ |b (8) and apply this machine on Bob's local system the joint state |Ω f between them can be converted into the state |Ω i . Under the previous arguments it could be easily proved that |Ω i ↔ |Ω f , i.e., the transformation |Ω f → |Ω i is impossible by LOCC with certainty. This impossibility directly indicates that the deleting operation defined in equation (8) is not a valid physical operation for arbitrary input states. So this leads us to the formal no-deleting theorem.
In conclusion this work connects the two famous no-go theorems from a new viewpoint that restricts the possible evolution of any quantum system through local operations. It shows the physical reason behind the existence of "Incomparable Pair of Pure Bipartite States". This connection makes a bridge between two different aspects of information processing theory. Moreover the most interesting feature is that the no-cloning and no-deleting theorems are treated in the same platform and thus we see the reciprocity of the two theorems from an operational point of view. Although for simplicity we assume that the machine states for both the cloning and deleting operations do not contain any information about the input qubit state, one may not assume this restriction. The result also holds if we consider the general scenario. Another interesting part in our proof is that the state we have considered, has a peculiar kind of symmetry and we require 3 × 32 dimensional system to prove our result. However one may search for a proof in lower dimensional systems.
[email protected] 2 [email protected]
Acknowledgement. The authors thank the referee for his/her valuable comments and suggestions. I.C. also acknowledges CSIR, India for providing fellowship during this work.
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| {'fraction_non_alphanumeric': 0.06335869194958556, 'fraction_numerical': 0.03559668445554672, 'mean_word_length': 3.8220640569395017, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'No-Cloning and No-Deleting theorems are verified with the constraint on local state transformations via the existence of incomparable states. Assuming the existence of exact cloning or deleting operation defined on a minimum number of two arbitrary states, an incomparable pair of states of the joint system between two parties can be made to compare under deterministic LOCC. We have restricted our proof with the assumption that the machine states of the cloning or deleting operations do not keep any information about the input states. We use the same setting to establish the no-cloning and no-deleting theorems via incomparability that supports the reciprocity of the two operations in their operational senses. The work associates the impossibility of operations with the evolution of an entangled system by LOCC.', 'arxivid': 'quant-ph/0606206', 'author': ['Amit Bhar \nDepartment of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India\n', 'Indrani Chattopadhyay \nDepartment of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India\n', 'Debasis Sarkar \nDepartment of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India\n'], 'authoraffiliation': ['Department of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India', 'Department of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India', 'Department of Applied Mathematics\nUniversity of Calcutta\n92, A.P.C. RoadKolkata-700009India'], 'corpusid': 207228333, 'doi': '10.1007/s11128-006-0041-2', 'github_urls': [], 'n_tokens_mistral': 5711, 'n_tokens_neox': 4902, 'n_words': 3077, 'pdfsha': 'dab5db343ba8bcc68310dd68d10ef9aa464ef517', 'pdfurls': ['https://export.arxiv.org/pdf/quant-ph/0606206v1.pdf'], 'title': ['No-Cloning and No-Deleting theorems through the existence of Incomparable states under LOCC', 'No-Cloning and No-Deleting theorems through the existence of Incomparable states under LOCC'], 'venue': []} |
arxiv |
ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS
10 Mar 2023
Andrea Giorgini [email protected]
Politecnico di Milano Dipartimento di Matematica Via E. Bonardi 9
20133MilanoItaly
ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS
10 Mar 2023arXiv:2303.06013v1 [math.AP]
In this note, we consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the L ∞ metric) in the interval [−1 + δ, 1 − δ] on the time interval [τ, ∞) for any τ > 0, where δ only depends on the norms of the initial datum, τ and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.
INTRODUCTION AND MAIN RESULTS
We study the nonlocal Cahn-Hilliard equation (see [13,17,18])
∂ t φ = ∆ (F ′ (φ) − J * φ) in Ω × (0, ∞),(1.1)
where Ω is a smooth and bounded domain in R 3 . The state variable φ represents the difference of the concentrations of two fluids. This equation is commonly rewritten as
∂ t φ = ∆µ, µ = F ′ (φ) − J * φ in Ω × (0, ∞),(1.2)
which is equipped with the following boundary and initial conditions ∂ n µ = 0 on ∂Ω × (0, T ), φ(·, 0) = φ 0 in Ω, (1.3) where n is the outward normal vector on ∂Ω. The physically relevant form of the nonlinear function F is given by the convex part of the Flory-Huggins (also Boltzmann-Gibbs entropy) potential The function J : R 3 → R is a (sufficiently smooth) interaction kernel such that J(x) = J(−x). The notation (J * φ)(x) stands for Ω J(x − y)φ(y) dy. The system (1.2)-(1.3) is a gradient flow with respect to the metric of H 1 (0) (Ω) ′ , namely the dual of H 1 (Ω) with zero mean value, associated to the free energy
E N L (φ) = − 1 2 Ω×Ω J(x − y)φ(y)φ(x) dx dy + Ω F (φ(x)) dx = 1 4 Ω×Ω J(x − y)|φ(y) − φ(x)| 2 dx dy + Ω F (φ(x)) − a(x) 2 φ 2 (x) dx,(1.5)
where a(x) = (J * 1)(x) = Ω J(x − y) dy for x ∈ Ω. The function µ appearing in (1.2) is the so-called chemical potential, which corresponds to δE NL (φ)
δφ . The analysis of the nonlocal Cahn-Hilliard equation with logarithmic potential (1.20) (actually a more general class of singular potentials) has been firstly studied in [13] (see also [11] for another proof of existence and [15] for the viscous case). In particular, the authors in [13] proved the existence and uniqueness of global weak solutions and their propagation of regularity for positive times (see proof of Theorem 1.2 below for more details). Such solutions satisfy φ ∈ L ∞ (Ω × (0, ∞)) with |φ(x, t)| < 1 for a.e. x ∈ Ω, ∀ t > 0.
(1.6) Such property has an important physical meaning since the solution φ takes value in the significant interval [−1, 1] (cf. definition of φ). Concerning the regularity of the global solutions, a main task consists in establishing L p estimates of F ′′ (φ) and F ′′′ (φ), which are needed to prove the existence of classical solutions. This is a difficult question due to the growth conditions
F ′′ (s) ≤ Ce C|F ′ (s)| , |F ′′′ (s)| ≤ CF ′′ (s) 2 ,(1.7)
which prevent the possibility to control F ′′ (φ) or F ′′′ (φ) in L p spaces in terms of some L p norms of F ′ (φ) (as possible in the case of potential with polynomial growth). However, although L p estimates of F ′′ (φ) and F ′′′ (φ) can be useful, this is not sufficient (in many cases) to prove higher order regularity, and it is necesssary to show the instanteneous (also called strict) separation property: for any τ > 0, there exists δ = δ(τ ) ∈ (0, 1) such that
|φ(x, t)| ≤ 1 − δ, for all (x, t) ∈ Ω × (τ, ∞). (1.8)
We point out that the separation property is expected due to the gradient flow structure of the Cahn-Hilliard model, which drives the dynamics towards stationary states of the free energy consisting of separated functional minima. In [13], a first proof of (1.8) has been established in [13,Theorem 5.2] in the two dimensional case. The argument hinges upon an iterative Alikakos-Moser argument for the powers of F ′ (φ) combined with Gagliardo-Nirenberg interpolation inequalities and the Trudinger-Moser inequality. A new proof of such result admitting a more general class of singular potentials has been proposed in [14,Section 4]. The latter relies on a De Giorgi's iterative argument. This method is usually employed to obtain an L ∞ estimate of the solution to a second order PDE, thereby the main achievement in [14] was to recast the method in order to get a specific bound (cf. (1.8) with (1.6)). More recently, the separation property has been proven in three dimensions in [28], which allowed to show the convergence to stationary states. The author in [28] improved the method in [14] in two ways: the truncated functions φ n (see proof of Theorem 1.2 below) are shown to be bounded by 2δ (instead of 1 as in [14]) and a Poincaré type inequality for timedependent functions is employed to avoid the integrals of φ n (see term Z 2 in [14, Section 4]). However, a main drawback of the argument, which is due to the latter ingredient, is that the value of δ in (1.8) depends on the particular solution. More precisely, δ cannot be estimated only in terms of norm of the initial data and the parameters of the system. The purpose of this work is to demonstrate that the De Giorgi iterative scheme in [14] and the observation φ n L ∞ ≤ 2δ are sufficient to achieve (1.8) with a value δ which depends on τ , the initial energy E N L (φ 0 )) and the parameters of the system (e.g. F, Ω, J). Beyond its intrinsic interest, this allows us to improve the regularity of the global attractor for the dynamical system associated to the system (1.2)-(1.3).
In order to present the main results of this work, let us formulate the assumptions for the admissible class of potentials: The main result reads as follows
(A1) F ∈ C ([−1, 1]) ∩ C 2 (−1, 1) such that lim |s|→1 F ′ (s) = ±∞ and F ′′ (s) ≥ θ > 0 for all s ∈ (−1, 1). (A2) There exists ε 0 > 0 such that F ′′ is monotone non-decreasing on [1 − ε 0 , 1) and non-increasing in (−1, 1 + ε 0 ]. (A3) There exist ε 1 ∈ (0, 1 2 ) and C F ≥ 1 such that 1 F ′ (1 − 2δ) ≤ C F | ln(δ)| , 1 F ′ (−1 + 2δ) ≤ C F | ln(δ)| , ∀ 0 < δ ≤ ε 1 (1.9) and 1 F ′′ (1 − 2δ) ≤ C F δ, 1 F ′′ (−1 + 2δ) ≤ C F δ, ∀ 0 < δ ≤ ε 1 .Theorem 1.2. Assume that (A1)-(A3) hold. Let J be W 1,1 loc (R 3 ) such that J(x) = J(−x) for all x ∈ R 3 . Assume that φ 0 ∈ L ∞ (Ω) such that φ 0 L ∞ (Ω) ≤ 1 and |φ 0 | = |Ω| −1 Ω φ 0 (x) dx < 1.
Then, for any τ > 0, there exists δ ∈ (0, 1) such that the unique global solution to (1.2)-(1.3) satisfies
|φ(x, t)| ≤ 1 − δ, for a.e. (x, t) ∈ Ω × [τ, ∞).
(1.11)
In addition, there exists three positive constants C 1 , C 2 , C 3 and α ∈ (0, 1) such that
sup t≥τ µ(t) L ∞ (Ω) ≤ C 1 , sup t≥τ ∂ t µ L 2 (t,t+1;L 2 (Ω) ≤ C 2 ,(1.12)
and
|φ(x 1 , t 1 ) − φ(x 2 , t 2 )| ≤ C 3 |x 1 − x 2 | α + |t 1 − t 2 | α 2 , (1.13) for any (x 1 , t 1 ), (x 2 , t 2 ) ∈ Ω t = Ω × [t, t + 1], for any t ≥ τ .
The values of δ, C 1 , C 2 , C 3 and α only depend on τ , δ, the initial energy E N L (φ 0 ), the initial mean φ 0 and the parameters of the system (i.e. F , J, Ω).
Remark 1.3.
A combination of the separation property (1.11) and the Hölder regularity (1.13) gives the following stronger result
|φ(x, t)| ≤ 1 − δ, ∀ (x, t) ∈ Ω × [τ, ∞).
(1.14)
As a direct consequence of Theorem 1.2, we infer additional features of the longtime behavior of the solutions of system (1.2)-(1.3). Let us introduce the dynamical system associated with problem (1.2)-(1.3). For any given m ∈ (0, 1), we define the phase space
H m = φ ∈ L ∞ (Ω) : φ L ∞ (Ω) ≤ 1 and − 1 + m ≤ φ ≤ 1 − m (1.15) endowed with the metric d(φ 1 , φ 2 ) = φ 1 − φ 2 L 2 (Ω) . (1.16) The pair (H m , d) is a complete metric space. Then, we define the map S(t) : H m → H m , S(t)φ 0 = φ(t), ∀ t ≥ 0,
where φ is the global (weak) solution (see [13,Theorem 3.4]) originating from the initial condition φ 0 . It was shown in [13,Section 4] that (H m , S(t)) is a dissipative dynamical system and S(t) is a closed semigroup on the phase space H m (see [27]). Furthermore, the existence of the global attractor A m was proven in [13,Theorem 4.4]. In particular, it is shown that A m is a bounded set in H m ∩ H 1 (Ω). Our next result is concerned with the regularity of the global attractor A m .
Theorem 1.4. Let (A1)-(A3) hold. Assume that J ∈ W 1,1 loc (R 3 ) such that J(x) = J(−x) for all x ∈ R 3 . Consider the connected global attractor A m associated with the dynamical system (H κ , S(t)). Then, A m ⊂ B L ∞ (Ω) (0, 1 − δ) and is bounded in C α (Ω).
Before proceeding with the proofs of the main results, it worth presenting a wider picture about the validity of the separation property for other Cahn-Hilliard equations. First, we recall the nonlocal Cahn-Hilliard equation with non-constant degenerate mobility
∂ t φ = div (1 − φ 2 )∇µ , µ = F ′ (φ) − J * φ in Ω × (0, ∞), (1.17)
which is completed with (1.3). In this case, the separation property has been previously proven by [23] in both two and three dimensions (see also [10]). Next, we consider the (local) Cahn-Hilliard equation [2][3][4] (see also [9,26]) with constant mobility
∂ t φ = ∆ (−∆φ + Ψ ′ (φ)) in Ω × (0, T ),(1.18)
subject to the classical boundary and initial conditions
∂ n φ = ∂ n ∆φ = 0 on ∂Ω × (0, T ), φ(·, 0) = φ 0 in Ω, (1.19)
where Ψ is the Flory-Huggins potential defined by
Ψ(s) = F (s) − θ 0 2 s 2 = θ 2 (1 + s) ln(1 + s) + (1 − s) ln(1 − s) − θ 0 2 s 2 , s ∈ [−1, 1],(1.20)
with constant parameters θ and θ 0 fulfilling the conditions 0 < θ < θ 0 . The Cahn-Hilliard system (1.18) is the gradient flow with respect to the H 1 (0) (Ω) ′ metric of the total free energy
E L (φ) = Ω 1 2 |∇φ| 2 + Ψ(φ(x)) dx. (1.21)
The separation property (1.8) for (1.18)-(1.19) was first established in [7] and [25] in one and two dimensions, respectively. The argument has been subsequently simplified in [19] and [20]. More recently, it was extended to a more general class of potential in [14]. In three dimensions, the separation property has been shown only in [1] on the time interval [T s , ∞), where T s cannot be computed explicitly (see also [23] for a class of singular potentials different from (1.20)). However, it still remains a major challenge to demonstrate the separation property for (1.18)-(1.19) for all positive times in three dimensions. Finally, we mention some recent results regarding the nonlocal-to-local asymptotics obtained in [5,6,16], That is, the weak solution to the nonlocal Cahn-Hillliard equation converges to the weak solution of the local Cahn-Hilliard equation, under suitable conditions on the data of the problem and a rescaling of the interaction kernel J.
SEPARATION PROPERTY AND HÖLDER REGULARITY
In this section we provide an improved proof of the separation property for the nonlocal Cahn-Hilliard equation in three dimensional domains. Then, we derive some consequences on the regularity of the solution.
Let us first recall the following well known result.
Lemma 2.1. Let {y n } n∈N 0 ⊂ R + satisfy the relation y n+1 ≤ Cb n y 1+ǫ n , for some C > 0, b > 1 and ǫ > 0. Assume that y 0 ≤ C − 1 ǫ b − 1 ǫ 2 . Then, we have y n ≤ y 0 b − n ǫ , ∀ n ≥ 1.
In particular, y n → 0 as n → ∞. such that
∂ t φ, v + (∇µ, ∇v) = 0 ∀ v ∈ H 1 (Ω), a.e. in (0, ∞), (2.2) µ = F ′ (φ) − J * φ a.e. in Ω × (0, ∞),(2.3)
and φ(·, 0) = φ 0 (·) in Ω. Furthermore, for any τ ∈ (0, 1)
sup t≥τ ∂ t φ(t) (H 1 (Ω)) ′ + sup t≥τ ∂ t φ L 2 (t,t+1;L 2 (Ω)) ≤ C 0 √ τ , (2.4) sup t≥τ µ(t) H 1 (Ω) + sup t≥τ φ(t) H 1 (Ω) + sup t≥τ F ′ (φ) H 1 (Ω) + sup t≥τ µ L 2 (t,t+1;H 2 (Ω)) ≤ C 0 √ τ , (2.5) sup t≥τ ∇µ L q (t,t+1;L p (Ω)) + ∇φ L q (t,t+1;L p (Ω)) ≤ C 1 (τ ), where 3p − 6 2p = 2 q , ∀ p ∈ [2, 6],(2.6)
where the positive constant C 0 only depends on E N L (φ 0 ), φ 0 , Ω and the parameters of the system. The positive constant C 1 (τ ) also depends on the same quantities as C 0 , in addition to τ . Furthermore, the constants C 0 and C 1 are uniformly bounded in φ 0 if φ 0 lies in a compact set of (−1, 1).
In the first part of the proof, we show the separation property (1.11). To this end, we now introduce the iteration schemeà la De Giorgi devised in [14,Section 4]. Let τ > 0 be fixed. We consider three positive parameters T , τ and δ such that T − 3 τ ≥ τ 2 and δ ∈ 0, min{ ε 0 2 , ε 1 } (cf. assumption (A2)-(A3)). The precise value of τ and δ will be chosen afterwards. We define two sequences
t −1 = T − 3 τ t n = t n−1 + τ 2 n ∀ n ≥ 0, and k n = 1 − δ − δ 2 n , ∀ n ≥ 0. (2.7)
Notice that
t −1 < t n < t n+1 < T − τ , ∀ n ≥ 0, such that t n → t −1 + 2 τ = T − τ as n → ∞,(2.8)
and
1 − 2δ ≤ k n < k n+1 < 1 − δ, ∀ n ≥ 0, such that k n → 1 − δ as n → ∞. (2.9) For n ≥ 0, we introduce η n ∈ C 1 (R) such that η n (t) = 1, t ≥ t n 0, t ≤ t n−1 and |η ′ n (t)| ≤ 2 2 n τ .
(2.10)
Next, for n ≥ 0, we consider the function
φ n (x, t) = max{φ(x, t) − k n , 0} = (φ − k n ) + .
Consequently, we introduce the sets
I n = [t n−1 , T ] and A n (t) = {x ∈ Ω : φ(x, t) − k n ≥ 0}, ∀ t ∈ I n .
If t ∈ [0, t n−1 ), we set A n (t) = ∅. We observe that we have
I n+1 ⊆ I n , ∀ n ≥ 0, I n → [T − τ , T ] as n → ∞,(2.11)
and A n+1 (t) ⊆ A n (t), ∀ n ≥ 0, t ∈ I n+1 .
(2.12) The last ingredient is y n = In An(s) 1 dx ds, ∀ n ≥ 0.
For any n ≥ 0, we choose as test function v = φ n η 2 n in (2.2). Integrating over [t n−1 , t], where t n ≤ t ≤ T , we obtain the relation
t t n−1 ∂ t φ, φ n η 2 n ds + t t n−1 An(s) ∇F ′ (φ) · ∇φ n η 2 n dx ds = t t n−1 An(s) (∇J * φ) · ∇φ n η 2 n dx ds. (2.13)
Since F ′ (φ) ∈ L ∞ (τ, ∞; H 1 (Ω)) and |{x ∈ Ω : |φ(x, t)| = 1}| = 0 for all t ≥ τ , we deduce from [24] that
h k (F ′ (φ)) ∈ L ∞ (τ, ∞; H 1 (Ω) ∩ L ∞ (Ω)), where h k : R → R, h k (s) = k, s ≥ k, s, s ∈ (−k, k), k, s ≤ −k, ∀ k ∈ N.
Then, it follows that h k (F ′ (φ)) → F ′ (φ) almost everywhere in Ω and for all t ≥ τ , and ∇(h k (F ′ (φ))) = F ′′ (φ)∇φ 1 {|F ′ (φ)|<k} (·) → F ′′ (φ)∇φ almost everywhere in Ω and for all t ≥ τ . Thus, by the monotone convergence theorem, Ω |F ′′ (φ(t))∇φ(t)| 2 dx ≤ lim k→∞ h k (F ′ (φ(t))) 2
H 1 (Ω) = F ′ (φ(t)) 2 H 1 (Ω) < ∞,
for all t ≥ τ . As consequence, it is easily seen that ∇F ′ (φ) = F ′′ (φ)∇φ in distributional sense. Thanks to this, we rewrite (2.13) as
t t n−1 ∂ t φ, φ n η 2 n ds + t t n−1 An(s) F ′′ (φ)∇φ · ∇φ n η 2 n dx ds = t t n−1 An(s) (∇J * φ) · ∇φ n η 2 n dx ds. Notice that t t n−1 ∂ t φ, φ n η 2 n ds = 1 2 φ n (t) 2 L 2 (Ω) − t t n−1 φ n (s) 2 L 2 (Ω) η n ∂ t η n ds.
Also, by the choice of δ, the assumption (A2) and the fact A n (t) ⊆ A 0 (t) for t ≥ t n−1 , we have t t n−1 An(s)
F ′′ (φ)∇φ · ∇φ n η 2 n dx ds ≥ F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds.
Thus, we end up with
1 2 φ n (t) 2 L 2 (Ω) + F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds ≤ t t n−1 An(s) (∇J * φ) · ∇φ n η 2 n dx ds I 1 + t t n−1 φ n (s) 2 L 2 (Ω) η n ∂ t η n ds I 2 , ∀ t ∈ [t n , T ].
We now observe that
sup x∈Ω |(∇J * φ)(x)| = sup x∈Ω Ω ∇J(x − y)φ(y) dy ≤ sup x∈Ω Ω |∇J(x − y)| dy = sup x∈Ω x−Ω |∇J(z)| dz.
Since Ω is bounded, there exists M > 0 such that Ω ⊆ B M (0). Also, diam(Ω) < ∞. Then, there exists M 1 such that the set x − Ω ⊂ B M 1 (0) for any x ∈ Ω. It follows that
∇J * φ L ∞ (Ω) ≤ B M 1 (0) |∇J(z)| dz = ∇J L 1 (B M 1 (0)) .
(2.14)
For simplicity of notation, we will use B M 1 to denote B M 1 (0). A similar argument applies for J * φ L ∞ (Ω) . Concerning the first term I 1 , we obtain as in [14,Section 4] that
I 1 = t t n−1 An(s) (∇J * φ) η n · ∇φ η n dx ds ≤ 1 2 F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds + 1 2 1 F ′′ (1 − 2δ) t t n−1 An(s) |∇J * φ| 2 η 2 n dx ds ≤ 1 2 F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds + 1 2 1 F ′′ (1 − 2δ) t t n−1 ∇J * φ 2 L ∞ (Ω) An(s) 1 dx ds ≤ 1 2 F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds + 1 2 ∇J 2 L 1 (B M 1 ) F ′′ (1 − 2δ) In An(s) 1 dx ds ≤ 1 2 F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds + 1 2 ∇J 2 L 1 (B M 1 ) F ′′ (1 − 2δ) y n .
This is actually a correction of the argument in [13] and [28] where ∇J L 1 (Ω) appears in the estimate analogous to the one above, instead of ∇J L 1 (B M 1 ) . In order to handle the term I 2 , we recall the main observation in [28]:
0 ≤ φ n ≤ 2δ a.e. in Ω, ∀ t ∈ [T − 2 τ , T ].
(2.15) By exploiting (2.10) and (2.15), we simply have
I 2 ≤ 2 n+1 τ t t n−1 An(s) φ 2 n dx ds ≤ 2 n+1 τ
In An(s) (2δ) 2 dx ds = 2 n+3 τ δ 2 y n .
Collecting the above estimates together, we infer that
φ n (t) 2 L 2 (Ω) +F ′′ (1 − 2δ) t t n−1 ∇φ n 2 L 2 (Ω) η 2 n ds ≤ ∇J 2 L 1 (B M 1 ) F ′′ (1 − 2δ) y n +2 4 2 n τ δ 2 y n , ∀ t ∈ [t n , T ]. (2.16)
As a consequence,
max t∈I n+1 φ n (t) 2 L 2 (Ω) ≤ X n , F ′′ (1 − 2δ) I n+1 ∇φ n 2 L 2 (Ω) ds ≤ X n ,(2.17)
where X n := 2 n max ∇J 2
L 1 (B M 1 ) F ′′ (1 − 2δ) , 2 4 δ 2 τ y n . (2.18)
Now, in light of (A3), we observe that
∇J 2 L 1 (B M 1 ) F ′′ (1−2δ) ≤ C F δ ∇J 2 L 1 (B M 1 ) , thereby X n = 2 n ∇J 2 L 1 (B M 1 ) F ′′ (1 − 2δ) y n , provided that τ ≥ 2 4 δ C F ∇J 2 L 1 (B M 1 )
.
(2.19)
The latter constraint will be verified later on. Next, for t ∈ I n+1 and for almost every x ∈ A n+1 (t), following [14, Section 4], we observe that In order to proceed with the next step, we recall the following Gagliardo-Nirenberg inequality in three dimensions u Exploiting the definition of y n , (2.11) and (2.21), we have
φ n (x, t) = φ(x, t) − 1 − δ − δ 2 n = φ(x, t) − 1 − δ − δ 2 n+1 =φ n+1 (x,t)≥0 +δ 1 2 n − 1 2 n+1 ≥ δ 2 n+1 , which implies that I n+1 Ω |φ n | 10 3 dx ds ≥ I n+1 A n+1 (s) |φ n | 10 3 dx ds ≥ δ 2 n+1 10 3 I n+1 A n+1 (s) 1 dx ds = δ 2 n+1y n+1 δ 2 n+1 10 3 ≤ I n+1 An(s) |φ n | 10 3 dx ds ≤ C Ω I n+1 φ n 4 3 L 2 (Ω) ∇φ n 2 L 2 (Ω) + φ n 2 L 2 (Ω) ds ≤ C Ω I n+1 φ n 4 3 L 2 (Ω) ∇φ n 2 L 2 (Ω) ds A +C Ω I n+1 φ n 4 3 L 2 (Ω) φ n 2 L 2 (Ω) ds B .
As in [14], we infer from (2.17) that
A ≤ 1 F ′′ (1 − 2δ) max t∈I n+1 φ n (t) 4 3 L 2 (Ω) F ′′ (1 − 2δ) I n+1 ∇φ n 2 L 2 (Ω) ds ≤ 1 F ′′ (1 − 2δ) X 5 3 n .
On the other hand, by using (2.11) and (2.15), we notice that
B ≤ max t∈I n+1 φ n (t) 4 3 L 2 (Ω) In φ n 2 L 2 (Ω) ds ≤ (2δ) 2 X 2 3
n In An(s)
1 dx ds = (2δ) 2 X 2 3 n y n .
Thus, thanks to (2.19), and making use of (A3), we find which is equivalent to y n+1 ≤ 2 16 3 C Ω C 8 3 F C J δ 2 3 2 5n y 5 3 n .
y n+1 δ 2 n+1 10 3 ≤ C Ω ∇J 10 3 L 1 (B M 1 ) (F ′′ (1 − 2δ)) 8 3 2 5 3 n + 4C Ω δ 2 ∇J 4 3 L 1 (B M 1 ) (F ′′ (1 − 2δ)) 2 3 2 2 3 n y 5 3 n ≤ 4C Ω C 8 3 F max ∇J
An application of Lemma 2.1 with
C = 2 16 3 C Ω C 8 3 F C J δ 2 3 , b = 2 5 , ǫ = 2 3
entails that y n → 0 provided that
y 0 ≤ δ 2 8 C 3 2 Ω C 4 F C 3 2 J 1 2 45 4 = δ 2 77 4 C 3 2 Ω C 4 F C 3 2 J . (2.22)
We conclude from y n → 0 and y n → (x, t) ∈ Ω × [T − τ , T ] : φ(x, t) ≥ 1 − δ , as n → ∞, that
(φ − (1 − δ)) + L ∞ (Ω×(T − τ ,T )) = 0. (2.23)
We are left to show that (2.22) is satisfied. Recalling (A3), (2.5) and y 0 = T T −3 τ A 0 (s) 1 dx ds, we notice that (cf. [14,28]
) T T −3 τ A 0 (s) 1 dx ds ≤ T T −3 τ F ′ (φ(s)) L 1 (Ω) ds |F ′ (1 − 2δ)| ≤ 3 τ F ′ (φ) L ∞ ( τ 2 ,∞;L 1 (Ω)) C F | ln(δ)| = 3C F C(E N L (φ 0 ), τ ) τ | ln(δ)| .
(2.24)
Thus, we impose that
3C F C(E N L (φ 0 ), τ ) τ | ln(δ)| ≤ δ 2 77 4 C 3 2 Ω C 4 F C 3 2 J .
(2.25)
In light of (2.19) and (2.25), we choose δ sufficiently small such that τ satisfies the relations
2 4 δ C F ∇J 2 L 1 (B M 1 ) ≤ τ ≤ δ| ln(δ)| 3 2 77 4 C 3 2 Ω C 5 F C 3 2 J C(E N L (φ 0 ), τ )
.
(2.26)
Now, set T = τ + τ 2 . Up to eventually reducing δ to get τ even smaller, we clearly have τ − 5 τ 2 ≥ τ 2 . Therefore, by (2.23), we deduce that (φ − (1 − δ)) + L ∞ (Ω×(τ − τ 2 ,τ + τ 2 )) = 0. We point out that the value of τ is independent of the choice of T . Thus, repeating the same argument on intervals of size τ , we conclude that (φ − (1 − δ)) + L ∞ (Ω×(τ − τ 2 ,∞) = 0. Finally, repeating the same argument for (φ + (−1 + δ)) − , we arrive at the desired conclusion (1.11). It is important to highlight that the value of δ only depends on F , J, Ω, E N L (φ 0 ) and τ .
The rest of the proof is devoted to the additional regularity results (1.12) and (1.13) that can be inferred once the separation property is established. Firstly, by definition of µ in (1.2), we observe that
sup t≥τ µ(t) L ∞ ≤ sup t≥τ F ′ (φ(t)) L ∞ (Ω) + J * φ(t) L ∞ (Ω) ≤ |F ′ (1 − δ)| + J L 1 (B M 1 ) =: C 1 .
Let us observe that
∂ h t µ(·) = ∂ h t φ(·) 1 0 F ′′ (sφ(· + h) + (1 − s)φ(·)) ds − J * ∂ h t φ(·), 0 < t ≤ T − h. (2.27)
By (1.11), sφ(· + h) + (1 − s)φ(·) L ∞ (Ω×(τ,∞)) ≤ 1 − δ for all s ∈ (0, 1). Then, exploiting that ∂ h t φ L 2 (0,T −h;L 2 (Ω)) ≤ ∂ t φ L 2 (0,T ;L 2 (Ω)) , we infer from (2.4) that sup t≥τ ∂ h t µ L 2 (t,t+1;L 2 (Ω)) ≤ C 2 , where C 2 > 0 depends on C 0 , τ , δ and J, but is independent of h, . This implies that ∂ t µ ∈ L 2 (0, T ; L 2 (Ω)) for any T > 0, and sup t≥τ ∂ t µ L 2 (t,t+1;L 2 (Ω) ≤ C 2 .
Secondly, we study the Hölder continuity in both time and space. We notice that (1.1) is a quasilinear equation with principal part in divergence form. Following the notation in the book [21], we define a l (x, t, u, p) = F ′′ (u)p l − (∂ l J * φ(·, t))(x), where F is the restriction of F in [−1 + δ, 1 − δ]. In light of the convexity of F and | F ′′ (s)| ≤ |F ′′ (1 − δ)|, for all s ∈ [−1 + δ, 1 − δ], we deduce that
a l (x, t, u, p)p l ≥ θ 2 |p| 2 − 1 2θ ∇J L 1 (B M 1 ) , |a l (x, t, u, p)| ≤ |F ′′ (1 − δ)||p| + ∇J L 1 (B M 1 ) .
We also note that the solution φ satisfying (2.1)-(2.6) is a bounded generalized solution in the sense of [21,Chapter V]. Thus, by [21, Theorem 1.1, Chapter V], we deduce that (1.13) holds in Ω ′ × [t, t + 1] , for any t ≥ τ and Ω ′ ⊂ Ω separated from ∂Ω. In order to achieve (1.13) up to the boundary, we make use of [8,Corollary 4.2], which provides the desired conclusion under the same assumptions. It is worth noticing that the constant C 3 and the parameter α from both [21] and [8] only depends on δ, θ, ∇J L 1 (B M 1 ) and Ω. This completes the proof.
ON THE REGULARITY OF THE GLOBAL ATTRACTOR
This section is devoted to some regularity properties of the global attractor A m for the dynamical system (H m , S(t)) stated in Theorem 1.4.
Proof of Theorem 1.4. Let us consider φ ⋆ ∈ A m . It is clear that φ ⋆ L ∞ (Ω) ≤ 1 such |φ ⋆ | ≤ 1 − m and φ ⋆ H 1 (Ω) ≤ N 1 , where N 1 is a universal constant (namely, it does not depend on φ ⋆ ). We observe that |E N L (φ ⋆ )| ≤ N 2 , where N 2 is a universal constant depending only on J L 1 (B M 1 ) (cf. (2.14)) and max s∈[−1,1] |F (s)|. Then, applying Theorem 1.2, we deduce that
S(t)φ ⋆ L ∞ (Ω) ≤ 1 − δ, ∀ t ≥ [1, ∞). (3.28)
Here, δ depends on the constants in (2.26). In particular, since |φ ⋆ | ≤ 1 − m, it is easily seen that C(E N L (φ ⋆ ), 1) ≤ N 3 , where N 3 is a universal constant. This implies that δ is a universal constant. Thanks to the arbitrary of φ ⋆ in the above argument, we deduce that
A m = S(1)A m ⊂ B L ∞ (Ω) (0, 1 − δ).
Next, by the second part of Theorem 1.2, we infer from (1.13) and (1.14) that
S(t)φ ⋆ C α (Ω) = S(t)φ ⋆ C(Ω) + sup x,y∈Ω,x =y |(S(t)φ ⋆ )(x) − (S(t)φ ⋆ )(y)| |x − y| α ≤ 1 − δ + C 3 =: N 4 .
Notice that N 4 is a universal constant which depends only on N 2 , δ, m and the parameters of the system (namely, F , J, Ω). Thus, since the constant are independent of φ ⋆ , we conclude that A m = S(1)A m ⊂ B C α (Ω) (0, N 4 ). The proof is complete.
F
(
. 1 .
1The assumptions (A1)-(A3) are satisfied by the convex part of the Flory-Huggins potential (1.4).
Proof of Theorem 1.2. Let us report the well-posedness results from [13, Theorems 3.4 and 4.1]: there exists a unique weak solution φ : Ω × [0, ∞) → R to the system (1.2)-(1.3) satisfying φ ∈ L ∞ (Ω × (0, ∞)) : |φ(x, t)| < 1 a.e. in Ω, ∀ t > 0, φ ∈ L 2 loc (0, ∞; H 1 (Ω)) ∩ H 1 loc (0, ∞; H 1 (Ω) ′ ), µ ∈ L 2 loc (0, ∞; H 1 (Ω)), F ′ (φ) ∈ L 2 loc (0, ∞; H 1 (Ω)),
H 1
1(Ω) , ∀ u ∈ H 1 (Ω).(2.21)
Acknowledgment. AG is a member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), Istituto Nazionale di Alta Matematica (INdAM).
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| {'fraction_non_alphanumeric': 0.12106192497134537, 'fraction_numerical': 0.05207397540348812, 'mean_word_length': 3.0596076458752517, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 14, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this note, we consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the L ∞ metric) in the interval [−1 + δ, 1 − δ] on the time interval [τ, ∞) for any τ > 0, where δ only depends on the norms of the initial datum, τ and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.', 'arxivid': '2303.06013', 'author': ['Andrea Giorgini [email protected] \nPolitecnico di Milano Dipartimento di Matematica Via E. Bonardi 9\n20133MilanoItaly\n', 'Andrea Giorgini [email protected] \nPolitecnico di Milano Dipartimento di Matematica Via E. Bonardi 9\n20133MilanoItaly\n'], 'authoraffiliation': ['Politecnico di Milano Dipartimento di Matematica Via E. Bonardi 9\n20133MilanoItaly', 'Politecnico di Milano Dipartimento di Matematica Via E. Bonardi 9\n20133MilanoItaly'], 'corpusid': 257482279, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14305, 'n_tokens_neox': 12014, 'n_words': 6733, 'pdfsha': 'fcbd4c340a666352fbfd9fc901c506448e5dcef4', 'pdfurls': ['https://export.arxiv.org/pdf/2303.06013v1.pdf'], 'title': ['ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS', 'ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS', 'ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS', 'ON THE SEPARATION PROPERTY AND THE GLOBAL ATTRACTOR FOR THE NONLOCAL CAHN-HILLIARD EQUATION IN THREE DIMENSIONS'], 'venue': []} |
arxiv |
Growth and electronic structure of graphene on semiconducting Ge(110)
Julia Tesch
Fachbereich Physik
Universität Konstanz
78457KonstanzGermany
Elena Voloshina
Institut für Chemie
Humboldt-Universität zu Berlin
10099BerlinGermany
Mikhail Fonin
Fachbereich Physik
Universität Konstanz
78457KonstanzGermany
Yuriy Dedkov
Fachbereich Physik
Universität Konstanz
78457KonstanzGermany
Growth and electronic structure of graphene on semiconducting Ge(110)
(Dated: May 23, 2017)
The direct growth of graphene on semiconducting or insulating substrates might help to overcome main drawbacks of metal-based synthesis, like metal-atom contaminations of graphene, transfer issues, etc. Here we present the growth of graphene on n-doped semiconducting Ge(110) by using an atomic carbon source and the study of the structural and electronic properties of the obtained interface. We found that graphene interacts weakly with the underlying Ge(110) substrate that keeps graphene's electronic structure almost intact promoting this interface for future graphenesemiconductor applications. The effect of dopants in Ge on the electronic properties of graphene is also discussed.
I. INTRODUCTION
Presently, the main methods of the synthesis of graphene (gr), a purely 2D material consisting of carbon atoms, which can be scaled down in order to be used in further applications, are its preparation on semiconducting SiC [1][2][3] or on metallic substrates [4][5][6][7]. However, these methods have natural drawbacks like, e. g., the price of the high-quality SiC wafers and difficulty to control the thickness homogeneity of graphene on SiC. In case of graphene synthesis on metal substrates with the subsequent transfer onto the desired support, it was found that the level of the metal-atom contamination in the obtained graphene is not acceptable for modern microelectronics [8,9]. These as well as other fundamental problems limit the commercialization of graphene [10,11] and stimulate researchers to search for the new ways of graphene synthesis.
One possibility to implement graphene in modern microelectronics processing is to perform its synthesis directly on an insulating substrate. Here one option is to use h-BN, which can be grown on the metallic substrates, like Cu, Fe, or Ni, or on semiconductors, like Ge, thus allowing a CVD synthesis of graphene, make a tunnel barrier for the carrier injection in graphene, and to avoid a metal contamination of graphene [12][13][14].
Another approach comprises graphene synthesis directly on the semiconducting substrate.
The direct growth of graphene on Si is problematic due to its carbidic phase formation at high temperatures [15][16][17][18][19]. However, the recent progress in graphene synthesis reveals the possibility to grow single-and multilayer graphene on Ge and Ge/Si substrates [20][21][22][23][24]. While the Ge(001) surface is the most technologically relevant one, the faceting of the underlying Ge with with the Ge(107) facets upon graphene growth was found by means of scanning electron and tunneling microscopy (SEM and STM) [23,25,26], that limits further technological processing of this interface. Contrary to the previous case, graphene as well as the underlying Ge surface remain flat for the Ge(110) surface, which was confirmed by low-energy electron diffraction (LEED) and STM [20,22,23]. Despite the availability of a number of the intensive studies on the growth of graphene on Ge, the little is known about the electronic structure of this interface [27]. In this work the ex situ CVD grown graphene flakes on undoped Ge/Si(001) were investigated by means of micro-and nano-ARPES (angle-resolved photoelectron spectroscopy), which indicates the free-standing character of graphene maintaining the linear dispersion of the π states in the vicinity of the Fermi level (E F ) and its p-doping with the position of the Dirac point of E D = 0.185 eV above E F .
Here we present a complete in situ UHV preparation as well as structural and electronic structure study of a nearly full graphene layer epitaxially grown from an atomic carbon source on Ge(110). The presented LEED and STM results confirm the high quality of the prepared system indicating the existence of the reconstructed Ge(110) surface below graphene. Our x-ray photoelectron spectroscopy (XPS), normal-emission ARPES (NE PES), and energy-loss near-edge spectroscopy performed at the carbon K-edge (C K-edge ELNES) reveal the nearly free-standing behaviour of graphene on Ge(110). We also address the plasmon excitations in this system performing electron-energy loss spectroscopy (EELS). Our results were compared and analyzed with the available theoretical spectroscopic data for freestanding graphene and "strongly-interacting" gr/Ni(111) demonstrating good agreement with the former case.
III. RESULTS AND DISCUSSIONS
The growth of graphene on Ge(110) was characterized by means of STM, LEED, and XPS and these results are compiled in Figs. 1 and 2. The Ge(110) surface shows a large scale ordering as can be deduced from the STM [ Fig. 1(a,b)] and LEED images [ Fig. 1(f)].
According to previous studies this surface can be described as a faceted surface with {17 15 1} facets and c(8 × 10) reconstruction on the steps [28][29][30][31]. Deposition of carbon on Ge (110) Formation of the uniform graphene sp 2 structure is also confirmed by XPS data (Fig. 2).
High-temperature deposition of graphene on Ge(110) leads only to the damping of the Ge 2p XPS signal [ Fig. 2(a,b)] without indication of the formation of the Ge-C bonds as can be concluded from the analysis of the Ge-related XPS peaks. Our data reveal a single C 1s peak for gr/Ge(110) with a small shoulder at the low binding energies (due to the possible 4 bonds between carbon atoms and dopant atoms segregated at the interface) that confirms the homogeneity of the prepared gr/Ge(110) system.
The electronic structure of the grown graphene layer on Ge(110) was investigated by NE PES for the occupied valence band states below E F and by C K-edge ELNES for the unoccupied states above E F and these results are presented in Fig. 3(a,b), respectively.
From the comparison of the PES spectrum of gr/Ge(110) and the one for the graphite single crystal we can conclude that in the former case the graphene-derived π and σ states are of graphene [32], the recent theoretical works on the Sb intercalation in gr/SiC reveal the n-doping of graphene [33]. A similar effect of the n-doping of the free-standing graphene upon Sb adsorption was also observed in experiment [34].
The unoccupied electronic states of graphene on Ge(110) were probed by the C K-edge ELNES spectroscopy, which can be considered as a simplified version of the near-edge x-ray absorption spectroscopy (NEXAFS). Here we used an electron beam of energy E p = 700 eV and detected the signal originating from the energy losses due to the excitation of electrons from the C 1s core level of carbon atoms in graphene onto unoccupied states above E F .
Similarly to NEXAFS, this method is element-specific, i. e. the intensity of the loss-signal is proportional to the atom-projected partial density of unoccupied states of the element in the system, the core-level of which is involved in the process. In our case we will observe two structures, which can be assigned to the 1s → π * and 1s → σ * transitions and the respective density of states above E F [35][36][37][38].
The C K-edge ELNES spectrum of gr/Ge(110), collected in the specular-reflected electron-beam geometry, is shown in the lower part of [ Fig. 3(b)] and compared with the theoretical ELNES (middle part) [39] and NEXAFS (upper part) [40] spectra of graphene and the gr/Ni(111) system. [All theoretical spectra were shifted by the same energy value in order to have the first peak, corresponding to the 1s → π * transition in the theoretical ELNES spectra, energetically coincide with the same peak in the experimental spectrum.
The double-peak structure of the 1s → σ * transition in the NEXAFS spectrum is due to excitonic effects.] One can see that there is a very good agreement between experimental ELNES spectrum of gr/Ge(110) and theoretical ELNES spectrum for free-standing graphene (lower and middle parts): (i) both 1s → π * and 1s → σ * transitions exhibit a single peak at the respective threshold, that can be taken as a signature of the weak interaction between graphene and the Ge(110) surface, (ii) the energy splitting between two transitions in the experimental spectrum is almost identical to the one deduced from the theoretically calculated ELNES spectrum. As was shown in Refs. [39][40][41][42] the value of this splitting as well as the modification of the shape of the 1s → π * transition can be taken as an indication for the sp 2 − sp 3 rehybridization of carbon atoms, which can appear due to the graphene contact with substrate or due to the adsorption of different species on top of graphene [7].
Such example of the spectral shape modifications of the ELNES and NEXAFS spectra for the strongly interacting gr/Ni(111) interface is shown in Fig. 3(b). As was shown, besides the strong n-doping of graphene on Ni, there is a strong intermixing of the valence band states of graphene and Ni, leading to the strong modification of the energy distribution of the partial density of states of both elements. All discussed effects are clearly visible in ELNES as well as in the NEXAFS spectra, due to the similarity of the electron excitation processes.
In our EELS experiments on gr/Ge(110) we also address the plasmon excitations in the system. Figure 4 shows the energy-loss spectra for this system measured as a function of the primary electron beam energy (marked for every spectrum) and presented in the energy range around the elastic peak (zero energy-loss energy). These spectra reveal a series of peaks (≈ 17 eV, ≈ 33 eV), which can be clearly assigned to the bulk Ge plasmons, whereas the peak at ≈ 9.5 eV and low energy shoulders can be assigned to the surface-related transitions of Ge(110) [43][44][45][46].
Variation of the primary beam energy allows to change the surface sensitivity of EELS as can be seen from Fig. 4. This leads to the increase of the graphene-related signal in the EELS spectra as the energy of the electron beam is decreased, which manifests itself as an increase of the intensity in the energy range of 3.5 − 6.5 eV as well as in the increase of the overall background for the energies above 15 eV. The first feature is assigned to the so-called π plasmon [47][48][49], the energy of which is determined as 6.33 ± 0.25 eV by a curve fitting procedure. The second observation is connected to the increase of the intensity of the π + σ plasmon as well as the increase of the background of the low energy inelastically scattered electrons. The exact position of the π + σ plasmon cannot be extracted from these data.
IV. CONCLUSIONS
In conclusion, we demonstrate the growth of a high-quality graphene layer on Ge (110) by evaporation of atomic carbon on the hot Ge surface. Our STM and LEED data confirm the honeycomb sp 2 structure of the graphene layer. From the analysis of the electronic structure of the graphene layer by means of PES and ELNES we conclude the nearly freestanding character of graphene which was found to be n-doped due to the segregation of Sb dopant atoms at the gr/Ge interface during sample preparation routines. Such effect of the substrate-dopant segregation at the graphene-semiconductor interface can be used for a controllable doping of graphene that might influence its electron-and spin-transport properties.
II. EXPERIMENTAL DETAILSGrowth of graphene and all studies were performed in the surface science cluster tool (Omicron NanoTechnology; base pressure 1 × 10 −10 mbar). Prior to every experiment a Ge(110) substrate (G-materials (Germany), Sb doped, resistivity 0.35 Ω · cm) was cleaned via cycles of Ar + -sputtering (1.5 keV, p(Ar) = 1 × 10 −5 mbar) and annealing (T = 870 • C).Graphene was grown on the hot Ge(110) substrate (T = 860 − 870 • C) from the atomic carbon source (Dr. Eberl MBE-Komponenten GmbH) with the filament current of I = 70 A and maximal pressure of 2 × 10 −9 mbar during C-deposition. Cleanliness and quality of samples was controlled by LEED, STM (Omicron VT-SPM), NE PES (non-monochromatized He II line), and XPS (non-monochromatized Al K line) (energy analyzer Omicron EA 125 was set either in angle-resolved or in angle-integrated mode, respectively) after every preparation step. ELNES and EELS experiments were performed in the specularly-reflected electron beam mode with angular and energy resolution of 1 • and ≈ 1 eV, respectively. The primary electron energy is marked for every spectrum. Low-temperature (LT) STM experiments were performed in an Omicron Cryogenic STM on the gr/Ge(110) sample quickly transferred from the growth/characterization facility under N 2 -atmosphere. Following the transfer, gr/Ge(110) was annealed in UHV at 700 • C.
at T = 870 • C and subsequent cooling of the sample to room temperature lifts the previously observed reconstruction, however, producing an ordered underlying Ge surface as can be seen from the respective STM and LEED images [Fig. 1(c-e,g)]. The prepared graphene layer forms two types of domains rotated by 30 • with respect to each other as seen from LEED and demonstrates clear honeycomb sp 2 structure on the Ge(110) surface [Fig. 1(c-e,h)]. Our results on the observation of two graphene domains are consistent with the previously reported data for the CVD grown graphene on Ge(110) [23]. The observed alignment of the graphene lattices of two domains is different by ≈ 15 • compared to the one observed for the single-domain graphene growth in Ref. [22]. Similar to the results presented in this work, our growth methods rule out the influence of hydrogen on the alignment of graphene on Ge(110); however, further structural studies are required. Our atomically resolved STM images demonstrate clear signatures of quasiparticle scattering in the graphene layer due to imperfections in graphene as well as due to the presence of the scattering centres at the interface (segregated dopants, see discussion below). The interference of the scattering waves of the carriers in graphene leads to the formation of the corresponding ( √ 3× √ 3)R30 • structure with respect to the graphene atomic-related structure in the 2D Fast-Fourier-Transformation (FFT) map. The spots of these structures are marked in the inset of Fig. 1(e) by white rectangle and circle, respectively. This ( √ 3 × √ 3)R30 • structure in the FFT map is assigned to the so-called intervalley scattering between adjacent cones at K and K points of the graphene-derived Brillouin zone.
shifted to the higher binding energies by ≈ 1 eV and ≈ 0.5 eV, respectively. This shift indicates that in the present study the graphene layer is n-doped, which is opposite to the result presented in Ref.[27], where small p-doping of graphene was observed with the position of the Dirac point of E D = 0.185 eV above E F . This difference can be assigned to the different types of substrates used in the experiments: n-doped (Sb) Ge(110) in the present study and an undoped Ge-epilayer on Si(001) in Ref.[27]. Here we can conclude that the cleaning procedure of Ge(110) (cycles of the Ar + -sputtering and annealing) as well as the high temperature used during graphene growth can lead to the segregation of Sb atoms at the gr/Ge(110) interface, thus influencing the doping of the formed graphene layer. This is confirmed by our LT STM (T = 24 K) data of gr/Ge(110) which are presented as upper insets ofFig. 3(a), where one can clearly see the characteristic STM-signatures of such interface-trapped dopant atoms (one of them is circled). Although the first ARPES data on the Sb-atoms adsorption on graphene/SiC pointed towards the possible p-doping
AcknowledgementWe thank the German Research Foundation (DFG) for financial support within the Priority Programme 1459 "Graphene".
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gr/Ge(110graphene's atomic lattice and from the intervalley scattering in graphene, respectively. STM data were acquired at room temperature. Imaging parameters: (a) 500 × 500 nm 2 , U T = +2.5 V, I T = 1 nA, (b) 80 × 80 nm 2 , U T = +2.5 V, I T = 0.3 nA, (c) 400 × 400 nm 2 , U T = +0.5 V, I T = 5 nA. Leed Stm, Characterization Of Ge, d) 150 × 150 nm 2 , U T = +0.5 V, I T = 6 nA, (e) 30 × 30 nm 2 , U T = +1.5 V, I T = 0.8 nA. insetFIG. 1: STM and LEED characterization of Ge(110) (a,b,f) and gr/Ge(110graphene's atomic lattice and from the intervalley scattering in graphene, respectively. STM data were acquired at room temperature. Imaging parameters: (a) 500 × 500 nm 2 , U T = +2.5 V, I T = 1 nA, (b) 80 × 80 nm 2 , U T = +2.5 V, I T = 0.3 nA, (c) 400 × 400 nm 2 , U T = +0.5 V, I T = 5 nA, (d) 150 × 150 nm 2 , U T = +0.5 V, I T = 6 nA, (e) 30 × 30 nm 2 , U T = +1.5 V, I T = 0.8 nA (inset:
Electron beam energy is 38 eV for (f,g) and 73 eV for (h), respectively. FIG. 3: (a) NE PES spectra of Ge. × 7 nm 2 , U T = +0.02 V, I T = 8 nAintensity is scaled down by factor 5) and gr/Ge(110× 7 nm 2 , U T = +0.02 V, I T = 8 nA). Electron beam energy is 38 eV for (f,g) and 73 eV for (h), respectively. FIG. 3: (a) NE PES spectra of Ge(110) (intensity is scaled down by factor 5) and gr/Ge(110).
Imaging parameters: (left) 20 × 20 nm 2 , U T = +1.0 V, I T = 0.2 nA, (right) 10 × 10 nm 2 , U T = +0.5 V, I T = 0.9 nA. (b) Experimental and theoretical C K-edge ELNES and NEXAFS spectra of gr/Ge. Spectrum of graphite crystal is shown as a shaded area for comparison. Inset shows LT STM images of gr/Ge(110). where scattering features due to dopant atoms at the interface are clearly resolvedSpectrum of graphite crystal is shown as a shaded area for comparison. Inset shows LT STM images of gr/Ge(110), where scattering features due to dopant atoms at the interface are clearly resolved. Imaging parameters: (left) 20 × 20 nm 2 , U T = +1.0 V, I T = 0.2 nA, (right) 10 × 10 nm 2 , U T = +0.5 V, I T = 0.9 nA. (b) Experimental and theoretical C K-edge ELNES and NEXAFS spectra of gr/Ge(110), graphene, and gr/Ni(111).
EELS spectra of gr/Ge(110) obtained with different primary beams. The energy of the electron beam is marked for every spectra. 4Lower inset presents the geometry used in the EELS/ELNES experimentsFIG. 4: EELS spectra of gr/Ge(110) obtained with different primary beams. The energy of the elec- tron beam is marked for every spectra. Lower inset presents the geometry used in the EELS/ELNES experiments.
| {'fraction_non_alphanumeric': 0.07702930372579887, 'fraction_numerical': 0.04167702377993206, 'mean_word_length': 3.957962481172121, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "The direct growth of graphene on semiconducting or insulating substrates might help to overcome main drawbacks of metal-based synthesis, like metal-atom contaminations of graphene, transfer issues, etc. Here we present the growth of graphene on n-doped semiconducting Ge(110) by using an atomic carbon source and the study of the structural and electronic properties of the obtained interface. We found that graphene interacts weakly with the underlying Ge(110) substrate that keeps graphene's electronic structure almost intact promoting this interface for future graphenesemiconductor applications. The effect of dopants in Ge on the electronic properties of graphene is also discussed.", 'arxivid': '1705.07699', 'author': ['Julia Tesch \nFachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany\n', 'Elena Voloshina \nInstitut für Chemie\nHumboldt-Universität zu Berlin\n10099BerlinGermany\n', 'Mikhail Fonin \nFachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany\n', 'Yuriy Dedkov \nFachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany\n'], 'authoraffiliation': ['Fachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany', 'Institut für Chemie\nHumboldt-Universität zu Berlin\n10099BerlinGermany', 'Fachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany', 'Fachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany'], 'corpusid': 98924543, 'doi': '10.1016/j.carbon.2017.06.079', 'github_urls': [], 'n_tokens_mistral': 12999, 'n_tokens_neox': 10781, 'n_words': 5914, 'pdfsha': 'beb1c43e4f6fd29a181e0048365189b0a96b35c9', 'pdfurls': ['https://arxiv.org/pdf/1705.07699v1.pdf'], 'title': ['Growth and electronic structure of graphene on semiconducting Ge(110)', 'Growth and electronic structure of graphene on semiconducting Ge(110)'], 'venue': []} |
arxiv |
Mixed Noisy Network Coding and Cooperative Unicasting in Wireless Networks
May 2012
Arash Behboodi [email protected].
Department of Telecommunications, SUPELEC
Technische Universität Berlin
Einsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France
Pablo Piantanida [email protected].
Department of Telecommunications, SUPELEC
Technische Universität Berlin
Einsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France
Pablo Piantanida
Department of Telecommunications, SUPELEC
Technische Universität Berlin
Einsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France
Mixed Noisy Network Coding and Cooperative Unicasting in Wireless Networks
the 5th International Symposium on Communications, Control, and Signal Processing
May 20121 DRAFT arXiv:1307.0991v2 [cs.IT]Index Terms Cooperative unicastingwireless networkingdecode-and-forwardcompute-and-forwardquantize- map-and-forwardnoisy network codingconstant gapcomposite channeloutage capacity
We consider the problem of communicating a single message to a single destination in presence of multiple relay nodes that we refer to as cooperative unicast network. Basically, this paper consists of two parts. In the first part, we introduce "Mixed Noisy Network Coding" (MNNC) scheme, generalizing "Noisy Network Coding" (NNC) scheme, where relays are allowed to decode-and-forward (DF) the message while all relays (without exception) transmit noisy descriptions of their observations. These descriptions are exploited at the destination and the DF relays to decode the transmitted messages while creating full cooperation among the nodes. Moreover, the destination and the DF relays can independently select the set of descriptions that either will be decoded or simply treated as interference.We further extend the concept of MNNC to multi-hopping scenarios that we refer to as "Layered MNNC" (LMNNC) where DF relays are organized into disjoint groups, each of them representing one hop in the network. For cooperative unicast additive white Gaussian noise (AWGN) networks we show that, provided DF relays are properly chosen, MNNC improves over all previously established constant gaps to the cut-set bound. In the second part, we consider the composite cooperative unicast network, where the channel parameters are randomly drawn from a probability distribution before the communication and remain fixed during the transmission. Each draw is assumed to be unknown at the source and fully known at the destination, but only partly known at the relays. We introduce through MNNC scheme the concept of "Selective Coding Strategy" (SCS) that enables relays to decide dynamically whether, in addition to communicating noisy descriptions, it would be possible to decode and forward messages to The work of P. Piantanida is partially supported by the ANR grant (FIREFLIES) INTB 0302 01. The material in this paper was presented 2 the destination. It is demonstrated through the asymptotic average error probability of the slow-fading AWGN relay channel that SCS clearly outperforms conventional DF, compress-and-forward, amplifyand-forward, and hash-forward coding schemes.Index TermsCooperative unicasting, wireless networking, decode-and-forward, compute-and-forward, quantizemap-and-forward, noisy network coding, constant gap, composite channel, outage capacity.Gamal-Mohseni-Zahedi [3] developed an alternative version of CF scheme (not based on Wyner-Ziv coding and sequential decoding at the destination) which achieves the same rate that the original CF scheme[2]. In fact, both CF schemes can perform within a constant gap to the information-theoretic capacity of the AWGN relay channel, regardless of channel parameters [4].More recently, there has been a growing interest in cooperative networks with multiple relays and several attempts were made to develop cooperation strategies, e.g., for multiple access and broadcast relay channels (see [5]-[7] and references therein). The capacity of degraded unicast cooperative networks is derived in [5] by using a sequential DF scheme while the capacity of a class of modulo-sum relay channels is found in [8] by using a CF based scheme. Graphical multicast networks were studied in[9]where the "max-flow min-cut theorem" for network information flow was presented for the point-to-point communication network. Deterministic networks with no interference at the receivers were addressed in [10] whereas the capacity of wireless erasure multicast networks was determined in[11], and the scaling behavior of cooperative multicasting in wireless networks was studied in[12].A. Related WorkAn approximation approach to general networks via deterministic channels was introduced by Avestimehr-Diggavi-Tse [4]. This approach yields a novel improvement over CF scheme, referred to as "Quantize-Map-and-Forward" (QMF), which achieves performance within constant gap of capacity for unicast AWGN networks with arbitrary number of relays. This important feature guarantees the uniformity in the channel coefficients and hence the fading statistics. Relay nodes quantize their received signals at noise level, map them randomly to Gaussian codewords and forward them to the others nodes. The fundamental difference between CF and QMF schemes relies on the delay and CSI aspects. The standard CF scheme [2] requires successive decoding at the destination and forward channel knowledge at the relays while QMF uses joint decoding of descriptions and messages with only CSI at the destination. As a matter of fact, this approach has played a key role in the development of several further results on cooperative wireless networks.In[13], Nazer-Gastpar propose an ingenious coding scheme, referred to as compute-andforward, which aims at allowing the relays to decode and send noiseless functions -linear combinations-of the transmitted messages. By combining all these descriptions, the destination determines the original messages being sent. Indeed, due to the additive nature of the channel, June 3, 2014 DRAFT 4 each relay receives a linear combination of the lattice codewords [13] in addition to some additive noise, which have the property that any integer linear combination is still a codeword [14], [15]. The relays then decode the linear combination of the codewords and thus a noiseless function of the messages. Nevertheless, the lattice property requires a integer linear combination of codewords to guarantee that it is still a codeword, however the linear combination induced by wireless channels have arbitrary real (or complex) channel gains. In order to overcome this difficulty, [13] proposes to scale the received channel output so that the received signal is close to an integer linear combination. The tightness of this approximation relies on the scaling factor which introduces a tradeoff between closeness of approximation and noise amplification. Recent work [16] by Lim et al. generalizes QMF approach to arbitrary memoryless multicast networks by introducing the notion of "Noisy Network Coding" (NNC) scheme, which implies the previous inner bounds in [4], [11]. As a matter of fact, Yassaee et al. in [17]-[19] independently introduced for the first time the idea of NNC and derived the same achievable rate regions. In [16], relay nodes based on NNC scheme send the same -long-message over many blocks of equal length -repetitive encoding-and the descriptions at the relays do not require binning while their indices are non-uniquely decoded at the destination. While the same result was obtained by using short messages in [17]-[19]. The achievable region from NNC scheme is shown to be tight for specific cases, e.g., deterministic and erasure networks, and in particular, it achieves within constant gap of capacity for multicast AWGN relay networks. Further progress was made in [20] where authors showed that the gain in NNC comes from backward decoding and delaying the decoding procedure. The use of different message coding opens up the possibility of combining DF and NNC scheme. This approach was taken in [21], [22] and [23], [24], which is referred to as "Short-Message Noisy Network Coding" (SNNC). Transmission is done over (B + L) blocks, where the value B is the number of blocks in which a new message is being transmitted and the value L is the number of blocks in which the previous messages are repeated according to a specific pattern. Both (B, L) are required to be large enough in [20] while only B needs to be large enough in [21], and the destination uses backward decoding. In this case, relays are divided into two sets, the relays in the first set use NNC scheme while those in the second set use DF scheme. The previously mentioned works have neglected two aspects of cooperative unicast networks. First, all relay nodes are capable of collaborating with each other to increase their chances June 3, 2014 DRAFT 5 of decoding the source message, similarly as done in compute-and-forward [13], and second, the destination can benefit from noisy descriptions of all nodes which also includes DF relays. Actually, NNC and SNNC schemes have since then been exploited in various ways, e.g., multilevel DF schemes for DF relays are investigated in [25]-[27]where an aware source exploits the existence of a hierarchy of the relays based on their channel quality.B. Contribution and OutlineIn this paper, we investigate coding strategies for cooperative unicasting in wireless networks. This problem consists of a source that wishes to communicate a single message to a single destination in presence of multiple relay nodes. The focus is on wireless configurations where without CSI, the source cannot any longer agree with the relays to jointly select an adequate cooperative strategy for each specific draw of network parameters. Traditional approaches to deal with this scenario falls into composite models for networks [28] which, unlike compound models [29], channel uncertainty is addressed by introducing a probability distribution (PD) from which the current channel index (or vector of channels parameters) is drawn, but remains fixed during the communication. Composite cooperative AWGN networks have been studied beforehand via the notion of capacity versus outage (see [12], [30], [31] among other references). This setting prevents, in general, the source use of any hierarchical multi-level scheme [25], [26] to enhance cooperation among the nodes since without CSI at the source such approach would clearly result in performance degradation. We shall follow an approach similar to that of compute-and-forward [13] to the study of simultaneous coding strategies [32] that are capable of enabling all nodes to decide -depending on their instantaneous channel measurements-whether would be possible to decode-and-forward messages (e.g. the amount of available noisy descriptions provides enough information) and which nodes should cooperate with each other by decoding noisy descriptions of observations (or noisy functions of the transmitted messages). In the first part, we introduce "Mixed Noisy Network Coding" (MNNC) for memoryless unicast networks with perfect CSI at all nodes while in the second part, we focus on composite cooperative unicast networks where the channel parameters are assumed to be unknown at the source and fully known at the destination, but only partly known at the intermediate nodes. We introduce through MNNC scheme the concept of "Selective Coding Strategy" (SCS) that enables all relays to dynamically select, based on on
I. INTRODUCTION
Cooperation in multi-terminal networks is becoming the essential part of modern communication systems, e.g., wireless mobile systems, device-to-device (D2D) communications and network coding, sensor and ad-hoc networks. The increasing development of these networks during recent years has revitalized the interest in understanding the most basic informationtheoretic problems such as broadcast, interference and relay networks. A convenient wireless model for such scenarios, as has been widely adopted in the literature, is the slow-fading one in which the channel state information (CSI) is fully available to the receivers but not to transmitters and only partial available to intermediate nodes. In these cases, classical Shannon capacities are typically zero due to the non-zero probability of channels experiencing an arbitrarily deep fade, so performance is instead quantified in terms of maximum achievable rates subject to a constraint on the level of the error probability that can be tolerated [1].
The term "cooperation strategy" stands for the procedure used to forward information from source to destination in relay networks. In selecting a cooperative scheme for wireless scenarios, several factors must be considered in order to preserve the capability of relay nodes to deal with the physical and statistical nature of their channel disturbances. The main cooperation strategies have been first introduced by Cover-El Gamal [2] for the relay channel. Although these coding schemes were not shown to achieve capacity of the additive white Gaussian noise (AWGN) channel, Decode-and-Forward (DF) scheme has been shown to be well suitable for situations where the source-to-relay channel is stronger than the others channels while Compressand-Forward (CF) scheme is preferable for situations where the relay-to-destination channel is the strongest link. Essentially, a relay using DF scheme forwards information based on a hard estimate of the encoder's message whereas CF scheme is based on a soft estimate. Lately El the available CSI, both the cooperative strategy and the nodes to benefit from cooperation.
Our main contributions are summarized below: 1) We first introduce MNNC scheme where part of the nodes are allowed to decode-andforward (DF) messages while all nodes (without exception) transmit noisy descriptions of their observations (cf. Theorem 1 and Corollary 1, section II-B). Moreover, these descriptions can be exploited at the destination and the DF relays ends to decode the intended message -based on offset coding-while creating full cooperation among the nodes. A rather general achievable rate expression is derived from this cooperative mixed coding strategy which can be viewed as a generalization of several existing results on the literature of NNC (cf. section II-C). Indeed, this coding strategy differs from NNC [16] in at least two important aspects: (i) the use of short-messages coding [18], [20], [21], [23], [33] where transmission is done in (B + L) blocks and the relays retransmit the compression index of block (B + 2) in the last (L − 2) blocks while backward decoding is used at the destination;
(ii) the relays use simultaneous cooperative strategies [31] by decoding and forwarding messages in addition to communicate noisy descriptions of their observations.
2) We further extend the concept of MNNC to multi-hopping scenarios that we refer to as "Layered MNNC" (LMNNC) where DF relays are organized into disjoint groups, each of them representing one hop in the network (cf. section II-D). Furthermore, LMNNC performs at least as good as MNNC and improves upon the existent results in [27,Theorem 2].
3) For cooperative unicast AWGN networks we show that, provided DF relays are properly chosen, MNNC improves over all previously established constant gaps to the cut-set bound [4], [16] (cf. Proposition 2, section III-B). As a matter of fact, the presence of mixed cooperative strategies [22], [24] introduces considerable difficulty when attempting to compare such scheme to the original NNC scheme [16]. This issue is solved by the proposed MNNC scheme where all nodes are enable to simultaneously DF messages and transmit noisy descriptions of their observations. 4) We finally study composite cooperative unicast networks where the channel parameters are randomly drawn from a probability distribution (cf. section IV). Each random draw is assumed to be unknown at the source and fully known at the destination, but only partly known at the relay nodes. We introduce through MNNC the concept of SCS that enables relays to decide dynamically (e.g. based on their channel measurements) whether would be June 3, 2014 DRAFT possible to decode and forward messages in addition to communication of noisy descriptions to the destination and other nodes. It is demonstrated through the asymptotic average error probability of the slow-fading AWGN relay channel that SCS clearly outperforms conventional cooperative schemes (cf. section V).
Within the framework of wireless networks, the results of this paper are therefore useful to analyze the relationship between simultaneous cooperative strategies and the use of available CSI at nodes to dynamically select the coding strategy, transmission rate and the asymptotic error probability. A connection is established between the asymptotic error probability and the outage probability. As a matter of fact, assuming codes of sufficiently long block lengths, outage probability dominates from above and below the asymptotic error probability. Although our results are specific to cooperative unicasting, we believe that the framework is enough general to be useful more broadly in the analysis of user cooperation for multi-source/multicast networks.
Notation
The vector notation x stands for the collection of n samples (x 1 , . . . , x n ) while upper-case letters X n are used to denote a vector of random variables (RVs) (X 1 , . . . , X n ). The random channels parameters are denoted by θ and any specific draw is denoted by θ. Let N {1, . . . , N } denote a set of indices, then for any subset S ⊆ N the vector of random variables X S stands for the collection (X i ) i∈S ; and similarly X S c = (X i ) i∈N −S where "−" is understood as setminus.
The indicator function for the event A is denoted by 1 [A]. Differential entropy is denoted by h(·), and mutual information by I(· ; ·). Then, the entropy of X S is defined by H(X S ) = H ((X i ) i∈S )
and similarly with mutual information. Let X, Y and Z be three RVs on some alphabets with probability distribution p. If p(x|yz) = p(x|y) for each x, y, z, then they form a Markov chain, which is denoted by X − − Y − − Z. Finally we denote strong -typical and conditional strong -typical sets by A n (X) and A n (Y |X), respectively (see [34] for details). Logarithms are taken in base 2 and denoted by log(·).
II. MIXED NOISY NETWORK CODING
In this section, we introduce the problem of communicating a single message to a single destination in presence of multiple relay nodes. Through this section, we shall assume that all channel parameters involved in the network are known to all terminals. We introduce a novel June 3, 2014 DRAFT Y X P Y Y 1 ...Y N |XX 1 ...X N · · · WŴ · · · · · · · · · · · · (X j , Y j ) (X k , Y k ) Fig. 1: Cooperative unicast network with relay nodes (X j , Y j ), for all j ∈ N , source input X and destination Y. cooperation scheme referred to as "Mixed Noisy Network Coding" (MNNC) that yields a rather general achievable rate expression improving over several existing results in the literature which can be viewed as particular cases. Then we further extend this concept to multi-hopping scenarios that we refer to as "Layered MNNC" (LMNNC) where DF relays are organized into disjoint groups, each of them representing one hop in the network.
A. Definition of the Cooperative Unicast Network
The cooperative unicast network as depicted in Fig. 1 is defined by a conditional probability distribution (PD) characterizing the destination output Y and the observations at each of the relay nodes Y N (Y 1 , Y 2 , . . . , Y N ) given the source input X and all relay inputs X N (X 1 , X 2 , . . . , X N ),
P Y Y N |XX N : X × X 1 × · · · × X N −→ Y × Y 1 × · · · × Y N ,(1)
where N = {1, . . . , N }. This network is assumed to be memoryless and without channel feedback, and all alphabets are assumed to be finite. We let P Y n Y n N |X n X n N denote the PD of the n-memoryless extension for this network.
Definition 1 (code and achievability): A code-C(n, M n , n ) for the cooperative unicast network consists of the following mappings: Pr φ(Y n ) = w .
X Y (X i , Y i ) (X j , Y j ) (X k , Y k ) DF relays CF relays w b l i,b−1 l k,b−1 l j,b−1 (l k,b−1 , w b−1 ) (l j,b−1 , w b−1 )
(
A positive rate R is said to be achievable for the cooperative unicast network if there exists a code-C(n, M n , n ) defined as above such that lim inf
n→∞ 1 n log M n ≥ R(3)
and lim sup n→∞ n = 0 .
The supremum of all achievable rates is the capacity of the unicast multi-relay network.
B. Mixed Noisy Network Coding
The key ingredients behind MNNC scheme rely on the following ideas.
3) Selection of descriptions to be decoded at both destination and relays:
The destination selects the help of only the best subset T of nodes among all possible relays N . Similarly, the k-th DF relay in the set V c is allowed to exploit only the help of a selected subset of relays, which shall be denoted by T k .
The next theorem provides the corresponding achievable rate for this scheme.
Theorem 1 (Mixed Noisy Network Coding (MNNC)): All rates R satisfying the following inequality are achievable for the cooperative unicast network:
R ≤ max P ∈P max V⊆N min max T ∈Υ (N ) min V c ⊆S⊆T R T (S) , min k∈V c max T k ∈Υ k (N ) min S⊆T k R (k) T k (S) ,(5)
where
R T (S) I(XX S ;Ŷ S c Y |X S c Q) − I(Ŷ S ; Y S |XX TŶS c Y Q) ,(6)R (k) T k (S) I(X;Ŷ T k Y k |V X k X T k Q) + I(X S ; Y k |V X k X S c Q) − I(Ŷ S ; Y S |V X k X T kŶ S c Y k Q) ,(7)
with S c T − S in (6) and S c T k − S in (7). Moreover, T ⊆ N , T k ⊆ N − {k} and V c = N − V, and Υ (N ) and Υ k (N ) are defined as follows: T (S) are used to denote:
Υ (N ) T ⊆ N : Q T (S) ≥ 0 ∀ S ⊆ T ,(8)Υ k (N ) T ⊆ N − {k} : Q (k) T (S) ≥ 0 ∀ S ⊆ T ,(9)Q T (S) I(X S ;Ŷ S c Y |V XX S c Q) − I(Ŷ S ; Y S |V XX TŶS c Y Q) ,(10)Q (k) T (S) I(X S ; Y k |V X k X S c Q) − I(Ŷ S ; Y S |V XX k X TŶS c Y k Q) .(11)
The set of all admissible input distributions P is defined by
P P QV XX N Y NŶN Y = P QV X P Y Y N |XX N j∈V c P X j |V Q PŶ j |V X j Y j Q j∈V P X j |Q PŶ j |X j Y j Q .(12)
Proof: The proof of this theorem is provided in Appendix A.
Remark 1: By following similar arguments to those used in [23], it is not difficult to check that the optimization of the term R T (S) in expression (5) can be taken over T ⊆ N instead of a subset T ∈ Υ (N ). To this end, it is enough to show that for every T ⊆ N , if there is a subset A ⊆ T such that Q T (A) < 0, then there must be another subset T ⊂ T ⊆ N such that the region with respect to T is increased. Therefore, for each S ⊆ T there is a unique
S ⊆ T such that R T (S) ≤ R T (S ) whichR T (S) = R T ∩A c (S ∩ A c ) + Q T (A) ,(13)
for each S ⊆ T ∩A c . Hence, for every set T ⊆ N , if there is a set A ⊆ T such that Q T (A) < 0, then it can be seen from (13) that
R T (S ∪ A) < R T ∩A c (S) ,(14)
which implies the final rate is increased by replacing T with T ∩ A c . For instance, for each T ⊆ N and T ∈ Υ (N ) c , we can find a subset T ⊂ T ⊆ N that is not necessarily in Υ c (N )
such that the region with respect to T is enlarged and this proves the claim.
A direct consequence of the above observation is that, for every T and A ⊆ T such that Q T (A) < 0, it is enough to ignore -not looking at their description indices-the relay nodes in A. Thus, the next achievable rate simply follows from this replacement:
R ≤ max P ∈P max V⊆N min max T ⊆N min V c ⊆S⊆T R T (S) , min k∈V c max T k ∈Υ k (N ) min S⊆T k R (k) T k (S) .(15)
June 3, 2014 DRAFT It is worth mentioning here that by setting V = N , the rate expression in Theorem 1 reduces to that of SNNC derived in [20], [23], which was shown to be equivalent to NNC first introduced in [16]. Therefore, Theorem 1 can be seen as a generalization that includes the previous results based on NNC schemes while it also provides a potentially larger rate (e.g. it achieves the capacity of Degraded Relay Channels which is not the case of NNC region).
We also note that since DF relays require the use of "forward decoding", the rate R (k)
T k (S) is clearly expected to be smaller compared to the situation where all relays are allowed to use "backward decoding". The reason for this, as it was also pointed out in [20], is that the gain of NNC is achieved by delaying decoding until the last block. However, postponing decoding to the last block would not be possible for those relays cooperating via DF scheme which brings the rate loss we mentioned. In order to better explore this rate loss, let us assume the unicast relay network where all relays are forced to use CF scheme, but the destination decodes based on "forward decoding" -instead of "backward decoding"-, i.e., the same decoding method as DF relays in Theorem 1. As a consequence of Theorem 1, we can obtain the following corollary that provides an achievable rate based on "forward decoding".
Corollary 1 (Forward decoding NNC): Assuming that all nodes are forced to use "forward decoding", then all rates R satisfying the following inequality are achievable:
R ≤ max P ∈P max T ⊆Υ (N ) min S⊆T R FD T (S) ,(16)
where
R FD T (S) I(X;Ŷ T Y |X T Q) + I(X S ; Y |X S c Q) − I(Ŷ S ; Y S |X TŶS c Y Q)(17)
with S c T − S and Υ (N ) defined by
Υ (N ) T ⊆ N : I(X S ; Y |X S c Q) − I(Ŷ S ; Y S |XX TŶS c Y Q) ≥ 0 ∀ S ⊆ T .(18)
The first observation from the above rate is that "forward decoding" at the destination does not perform in general as good as NNC. Nevertheless, it potentially improves on the use of "binning" and other "forward decoding" techniques [20]. Particularly, the condition which determines the optimization region in [20], i.e.,
I(X S ; Y |X S c Q) − I(Ŷ S ; Y S |X TŶS c Y Q) ≥ 0 ,(19)
creates a smaller optimization region than Υ (N ) because
I(Ŷ S ; Y S |X TŶS c Y Q) ≥ I(Ŷ S ; Y S |XX TŶS c Y Q) .(20)
Actually, the use of "joint-forward decoding" without "binning" performs potentially better respect to "joint-forward decoding" with "binning".
Remark 2:
The reason for the sub-optimality of the current forward-decoding scheme is two fold and can be easily understood from the proof. The DF relays decode the compression indices of others relays and also the source message by using the typical sets (172). The destination, however, does the same but using the typical set given by (196). By comparing these decoding rules, it is not difficulty to see that the destination decodes jointly the compression index and the source message using a typical set involving: the descriptionŶ T , the compression index of the other relays and the source codeword X. Whereas these terms are absent from the second decoding rule of DF relays in (172). Indeed, DF relays decode jointly the source message and compression index of other relays of block b based on the consecutive blocks b and b + 1. In the decoding block b + 1, DF relays cannot dispose of any information about the fresh compression index and source's message of the same block. Intuitively, these relays cannot benefit fully from the presence of the descriptionsŶ T and X. In contrast with this, when backward decoding is used decoding is performed in a single block and full cooperation can be obtained.
This problem may be partially fixed by using the layering coding introduced in [19]. In this case, the set of DF relays is partitioned into a set of layers {L 1 , . . . , L k }, representing the order in which the compression indices are decoded. Therefore, the decoding process is delayed at a
given layer for few blocks, and then by proper choice of layers one can achieve each corner point of the original region. This is possible because DF relays at the layer b are able to useŶ L i for all blocks i ≤ b − 1 while decoding the compression index. The drawback, however, is that this would require new layering and thus a new coding to achieve each of the corner points.
Specifically, in communication scenarios where the current channel is unknown at the source is beneficial to have an oblivious coding schemes for which the source code does not require to be aware of the cooperation strategy. As a matter of fact, the proper choice of layering at the source to achieve a certain corner point is mostly dependent on the channels parameters. As we will discuss later, the same problem occurs in multi-hopping settings where the optimal choice and number of hops strongly depends on the channels parameters.
Although this layering coding can improve on the results of Corollary 1, still there are some problems that cannot be completely solved. In our setting, the single source does not observe any output and thus, it can only cooperate with DF relays by helping them in decoding the compression indices. As a matter of fact, by looking at the proof together with expression (171), it is easy to check that the message index of source at block b + 1 must be unknown to the other relays when decoding the messages of block b. As a consequence of this, the source codewords cannot be fully exploited in the decoding rule. On one hand, since the source does not have any channel observation and hence cannot cooperate by transmitting its compression index while on the other, the source is the only node having new messages and therefore, its fresh information must be superimposed on the last layer of the relays. Otherwise, the relays would need to know the message beforehand which is not possible.
We consider now another scenario. Let us assume that each node in the network, including the destination and all relays, decides to use the help of all nodes. In other words, we set T N and T k N − {k}, then the following corollary easily follows from Theorem 1.
Corollary 2 (Fully cooperative MNNC):
Assuming that all nodes cooperate each other in the cooperative unicast network, then all rates R satisfying the following inequality are achievable:
R ≤ max P ∈P max V⊆N min min V c ⊆S⊆N R N (S) , min k∈V c min S⊆N −{k} R (k) N (S) ,(21)
where
R N (S) I(XX S ;Ŷ S c Y |X S c Q) − I(Ŷ S ; Y S |XX NŶS c Y Q) ,(22)R (k) N (S) I(X;Ŷ N −{k} Y k |V X N Q) + I(X S ; Y k |V X k X S c Q) − I(Ŷ S ; Y S |V X NŶS c Y k Q)(23)
with S c N − S in expression (22), S c N − (S ∪ {k}) in expression (23) and V c N − V satisfying the constraints:
I(X S ;Ŷ S c Y |V XX S c Q) − I(Ŷ S ; Y S |V XX NŶS c Y Q) ≥ 0 , ∀S ⊆ N ,(24)I(X S ; Y k |V X k X S c Q) − I(Ŷ S ; Y S |V XX NŶS c Y k Q) ≥ 0 , ∀S ⊆ N − {k} , ∀ k ∈ V c . (25)
The set of all admissible input distributions P is again defined by (12).
X Y (X i , Y i ) (X j , Y j ) (X k , Y k ) DF relays CF relays w b l i,b−1 w b−1 w b−1
C. Variations of Mixed Noisy Network Coding
For the rest of this section, we shall focus on some variations of MNNC scheme. The cooperative strategy that yields Theorem 1 allows all DF relays to cooperate with each other and with CF relays via the exchanges of their compression indices. It appears clear that, decoding of the source message at any node becomes dependent on all other relays which in many cases will lead to rather complex optimization problems. In order to further simplify the coding scheme and thus to avoid such complex dependences, DF relays can be constrained to decode messages as stand alone terminals, i.e., without the use of additional description indices, as it is shown in Fig. 3. Moreover, this simplification can also avoid decoding delay at all DF relays, which was necessary for decoding based on the compression indices of other relays. The next theorem provides the achievable rate of this simplified coding scheme that is referred to as "non-cooperative MNNC", which was independently derived in [27].
Theorem 2 (Non-cooperative MNNC): Assuming that no cooperation is allowed among the relay nodes, then all rates R satisfying the following inequality are achievable:
R ≤ max P ∈P max V⊆N , T ∈Υ (V) min min S⊆T R T (S) , min i∈V c I(X; Y i |X V c Q) ,(26)
where
R T (S) I(XX V c X S ;Ŷ S c Y |X S c Q) − I(Y S ;Ŷ S |XX T ∪V cŶ S c Y Q)(27)
with T ⊆ V ⊆ N , and V c N − V, and S c T − S. For notation convenience, we use
min ∅ (·) +∞. Moreover, Υ (V) is defined by Υ (V) T ⊆ V I(X S ;Ŷ S c Y |XX S c ∪V c Q) − I(Y S ;Ŷ S |XX T ∪V cŶ S c Y Q) ≥ 0 ∀ S ⊆ T .(28)
The set of all admissible input distributions P is defined as follows:
P P QXX N Y NŶN Y = P Q P XX V c |Q P Y Y N |XX N j∈V P X j |Q PŶ j |X j Y j Q .(29)
Proof: The proof of this theorem is provided in Appendix B.
As we have previously stated in Theorem 1, all DF relay nodes forward simultaneously the source message and their description indices. However, we can reduce the complexity of relaying functions by forcing partial cooperation, i.e., all DF relays in V c to use only the help of CF relays in V. This simplified scheme yields the following corollary, which is a special case of Theorem 1.
Corollary 3 (Partially cooperative MNNC):
Assuming that only partial cooperation is allowed among the relays, then all rates R satisfying the following inequality are achievable:
R ≤ max P ∈P max V⊆N min max T ∈Υ (V) min S⊆T R T (S) , min k∈V c max T k ∈Υ k (V) min S⊆T k R (k) T k (S) ,(30)
where
R T (S) I(XX V c X S ;Ŷ S c Y |X S c Q) − I(Y S ;Ŷ S |XX T ∪V cŶ S c Y Q) ,(31)R (k) T k (S) I(X;Ŷ T k Y k |X V c X T k Q) + I(X S ; Y k |X V c ∪S c Q) − I(Ŷ S ; Y S |X V c ∪T kŶ S c Y k Q)(32)
with sets T , and T k ⊆ V ⊆ N , and S c T − S, and S c T k − S, and V c N − V. Moreover,
Υ (V) and Υ k (V) are defined by Υ (V) T ⊆ V I(X S ;Ŷ S c Y |XX S c ∪V c Q) − I(Y S ;Ŷ S |XX T ∪V cŶ S c Y Q) ≥ 0 ∀ S ⊆ T ,(33)Υ k (V) T ⊆ V I(X S ; Y k |X V c ∪S c Q) − I(Ŷ S ; Y S |XX V c ∪TŶS c Y k Q) ≥ 0 ∀ S ⊆ T .(34)
The set of all admissible input distributions P is defined by (12).
D. Layered Mixed Noisy Network Coding
In this section, we extend the previous results to multi-hopping scenarios. As before, the relays are divided into two disjoint subsets V and V c . Nodes in V c are purely dedicated to use DF scheme. Whereas the main difference relies on the fact that DF relays are organized into disjoint groups where each of them represents one hop in the network. Each of these groups are referred to as "layer" and are denoted by an ordered pair L j : j = [1 : T ] , where T represents the number of hops present in the scheme. We denote the set of all such ordered partitions of an arbitrary set X by Π o (X ), i.e., the set Π o (X ) contains all possible hops and layers for all relays. As it is the case for MNNC, decoding at DF relays is delayed by one block in order to benefit from the compression indices of other CF relays. Moreover, decoding at DF relays is sequentially performed, such that DF relays at a higher layer start to decode sooner than the other lower layers. In this case, DF relays at each layer can enjoy the help of higher relays which have already decoded the message. We next introduce notation needed for the rest of this section:
L ≤d t≤d L t and L >d t>d L t ,
where L ≤T = V c . For any sequence of RVs {X i } with i = {1, . . . , n}, we define
V ≤m (V 1 , V 2 , . . . , V m ) ,
and similarly define x L (x k ) k∈L .
We next present an achievable rate for Layered Mixed Noisy Network Coding (LMNNC).
Theorem 3 (Layered Mixed Noisy Network Coding (LMNNC)): All rates R satisfying the following inequality are achievable:
R ≤ max P ∈P max V⊆N max L j ,j∈[1:T ] ⊆Πo(V c ) min max T ∈Υ (V) min S⊆T R T (S) , min i∈[1:T ] min k∈L i max T k ∈Υ k (V) min S⊆T k R (k) T k (L i , S)(35)
where
R T (S) I(XX V c ∪S ;Ŷ S c Y |X S c Q) − I(Ŷ S ; Y S |XX V c ∪TŶS c Y Q) ,(36)R (k) T k (L t , S) I(XV >t X L>t ; Y k |V ≤t X L ≤t Q) + I(X S ; Y k |V ≤T X V c ∪S c Q) −I(Ŷ T k ; Y S |V ≤T X V c ∪T k Y k Q)(37)
with S c T − S in (36) and S c T k − S in (37). Moreover, T , T k ⊆ N , and V c = N − V, and Υ (V) and Υ k (V) are defined as follows:
Υ (V) T ⊆ V : Q T (S) ≥ 0 ∀ S ⊆ T ,(38)Υ k (V) T ⊆ V : Q (k) T (S) ≥ 0 ∀ S ⊆ T ,(39)
where Q T (S) and Q (k)
T (S) are used to denote:
Q T (S) I(X S ;Ŷ S c Y |V ≤T XX V c X S c Q) − I(Ŷ S ; Y S |V ≤T XX V c X TŶS c Y Q) ,(40)Q (k) T (S) I(X S ; Y k |V ≤T X V c X S c Q) − I(Ŷ S ; Y S |V ≤T XX V c X T kŶ S c Y k Q) .(41)
The set of all admissible input distributions P is defined by
P P QV 1 ...V T XX N Y NŶV Y = P Q T t=1 P Vt|V t−1 Q P Xt|VtQ P Y Y N |XX N Q j∈V P X j |Q PŶ j |X j Y j . (42)
Proof: The proof of this theorem is provided in Appendix C.
We first remark that, by comparing the decoding condition (37) with (32), the contribution of having different layers brings the term (V >t , X L>t ) in the mutual information, which corresponds to the help of higher layers from DF relays, i.e., the t-th layers shared at the source. In other words, the source superimposes the fresh information over the layers (V 1 , . . . , V T ). Moreover, the rate region presented in Theorem 3 performs at least as good as partially cooperative MNNC while by exploiting the help of CF relays it improves upon the existent results in [27,Theorem 2]. Furthermore, this multi-hopping scheme achieves the capacity of some networks, e.g., line networks where relays can be ordered in a way that the observation of lower layer nodes is a physically degraded version of that of higher nodes.
In networks with random parameters, e.g., wireless networks, it is hard to assume a fixed hierarchy between relays for all channel draws and thus the degradedness assumption does not usually hold. Even the optimal number of hops T depends on the specific channels realizations.
Hence the source cannot adaptively change the number of hops T and set a priori coding based on a hierarchy. Nevertheless, this problem can be partially addressed through the adaptive use of (V 1 , . . . , V T ) where after the source transmission, DF relays can choose a set of layers by looking at their channels and superimpose the information over the corresponding layers, generating a conditional codebook. But the number of hops must be selected in advance.
III. CAPACITY OF COOPERATIVE UNICAST AWGN NETWORKS WITHIN A CONSTANT GAP
In this section, we study the characterization of the capacity of cooperative unicast additive Gaussian noise (AWGN) networks within a constant gap with respect to the cut-set bound. In particular, we show that MNNC under certain conditions -provided that DF relays are chosen
properly-can achieve a tighter "constant gap" than the standard NNC.
A. Single-Relay AWGN Channel
We first review the constant gap of DF rate for the single AWGN relay channel while CF constant gap follows along the same lines as shown in [16]. Consider the AWGN relay channel defined by the channel outputs:
Y = g 3 X + g 2 X 1 + V 1 , Y 1 = g 1 X + V 2 ,(43)
where the inputs are constrained to satisfy the average power E[X 2 ] ≤ P , E[X 2 1 ] ≤ P , and the Gaussian noises V 1 and V 2 are zero-mean of equal variance N ; the channel gains (g 1 , g 2 , g 3 ) are assumed to take arbitrary real values. Lower bounds on the capacity of this channel are well-known from literature [2]. The lower bound given by DF rate can be written as follows
R DF max β∈[0,1] min C g 2 1 βP N , C g 2 3 P + g 2 2 P + 2 βg 2 2 g 2 3 P N (44)
and the cut-set bound (CB) reads as
C CB max β∈[0,1] min C g 2 1 βP + g 2 3 βP N , C g 2 3 P + g 2 2 P + 2 βg 2 2 g 2 3 P N ,(45)
where C(x) 1 2 log (1 + x) and β is the correlation coefficient. Observe that the second term in (44) appears unchnaged in (45), and let us assume that β is the value maximizing the CB in (45) that is also chosen to evaluate the achievable rate in (44). Hence, only the difference between the first two terms affects the gap that is bounded as follows:
∆(C CB , R DF ) C CB − R DF ≤ C g 2 1 β P + g 2 3 β P N − C g 2 1 β P N (46) = 1 2 log N + β g 2 1 P + β g 2 3 P N + β g 2 1 P (47) = C g 2 3 g 2 1 βP N g 2 1 + βP (48) ≤ C g 2 3 g 2 1 .(49)
From our previous analysis we remark that -unlike the conventional NNC-the gap for DF rate cannot be made independent of the channel gains. For instance, if the channel gain sourceto-relay is enough strong with respect to that of source-to-destination, the capacity gap can be made arbitrarily small. Furthermore, we may expect that in general, as will be the case later, the performances of DF based schemes are heavily related to channel conditions and therefore cannot be evaluated independently. Also. it can be seen that when the quality of source-destination channel is better than the quality of source-relay channel, namely g 3 > g 1 , then the direct transmission and therefore CF scheme perform better than DF scheme.
B. Cooperative Unicast AWGN Networks
Consider a cooperative unicast AWGN network composed of N relay nodes, a single source and single destination node, which yields in total to N + 2 nodes. The relays are indexed as before with index belonging to the set N {1, . . . , N } but for simplicity, we will also associate the source with index 0 and the destination with index N +1, i.e., The channel gain from node i to node j is denoted by {g ij }, and V j denotes the noise at node j, which is assumed to follow a Gaussian distribution of zero-mean and unit variance. The channel outputs at the different nodes are given by
Y D = G(D, T )X M + V D , Y N = Y N +V N ,(50)
where
Y D Y 1 . . . Y N Y , Y N Y 1 . . . Y N , X M X 0 . . . X N , V D V 1 . . . V N +1 ,(51)
and G(D, T ) denotes the channel matrix with the corresponding channel gains, where we use the definition g ii 0 for all i ∈ D; andV N denotes the compression noise vector that is chosen to follow the same statistic as the channel noise, i.e., Gaussian distribution of zero-mean and unit variance. All through this section, the notation G(S 1 , S 2 ) is used to indicate the set of channel gains g ij | i ∈ S 1 , j ∈ S 2 . We simply use G when the respective sub-matrices can be understood implicitly. All channel inputs are constrained to satisfy average power E[X 2 i ] ≤ P , for all i ∈ M. The covariance matrix of any subset X S of channel inputs is denoted by Σ(S) = [P ρ ij ] for all i, j ∈ S with corresponding correlation coefficients ρ ij between the input components (X i , X j ).
Similarly, we have Σ(S 1 , S 2 ) = [P ρ ij ] for all i ∈ S 1 and j ∈ S 2 . Also I denotes the identity matrix.
We first recall the capacity within a constant gap which has been derived in [16] based on NNC scheme. Proposition 1 (Capacity within a constant gap from NNC [16]): A constant gap between NNC rate and the cut-set bound, for the Gaussian network with N relays, one transmitter and one receiver is given by
∆ * (C CB , R NNC ) 0.63(N + 2) .(52)
We proceed to evaluate the achievable rate given in Theorem 1 from which we shall derive a novel constant gap to the capacity. Let us assume that T = N , which implies that the destination decodes the compression indices of all relays. Channel inputs are chosen to be Gaussian random variables of zero-mean and unit variance satisfying the corresponding average power constraints.
The set V c denotes the index set of all relays using DF scheme and V those using CF scheme.
Based on these settings, we need to evaluate the next expression:
R MNNC sup P ∈P min min V c ⊆S⊆N R N (S) , min k∈V c max T k ∈Υ k (N ) min S⊆T k R (k) T k (S)(53)June 3, 2014 DRAFT where R N (S) I(XX S ;Ŷ S c Y |X S c ) − I(Ŷ S ; Y S |XX NŶS c Y ) ,(54)R (k) T k (S) I(X;Ŷ T k Y k |V X k X T k Q) + I(X S ; Y k |V X k X S c ) − I(Ŷ S ; Y S |V X k X T kŶ S c Y k ) . (55)
In order to evaluate expression (53) and thus compute the gap from capacity based on MNNC,
we first need to evaluate the cut-set bound in an more convenient manner.
Lemma 1 (Cut-set bound):
The capacity of the unicast cooperative AWGN network is upper bounded by
C CB max Σ(·) min V c ⊆S⊆N 1 2 log det I(S c ∪ {N + 1}) + 1 2 G Σ({0} ∪ V c ) 0 0 P I(S ∩ V) G T + 1 + min{|S c |, |S|} 2 log (4 max(1, |S ∩ V|)) ,(56)
for an arbitrary set of nodes V ⊆ N , where the maximum is taken over all covariance matrices Σ(·) satisfying the corresponding inputs constraints. Indeed, the set V c N − V can be seen as the set of relays already having or decoding the source message.
Proof: Let A and B be two positive semidefinite matrices such that
A = A 11 A 12 A 21 A 22 0 and B = A 11 0 0 A 22 0 , then 2B A .(57)
It is enough to check that the matrix
A (−) A 11 −A 12 −A 21 A 22 is positive semidefinite and hence A (−) + A = 2B.
On the other hand, the cut-set bound reads as
C CB max P ∈P min S⊆N I(XX S ; Y S c Y |X S c ) .(58)
By convenience, we define the sets
S c * S c ∪ {N + 1}, V c * V c ∪ {0} and S CF S ∩ V.
The following inequalities hold true:
I(XX S ; Y S c Y |X S c ) ≤ h(Y S c Y ) − h(Y S c Y |XX N ) (59) = 1 2 log det I(S c * ) + G Σ(V c * ) Σ(V c * , S CF ) Σ(S CF , V c * ) Σ(S CF ) G T (60) ≤ 1 2 log det 2I(S c * ) + 2G Σ(V c * ) 0 0 Σ(S CF ) G T (61) ≤ 1 2 log det 2I(S c * ) + 2G Σ(V c * ) 0 0 Tr(S CF )I(S CF ) G T (62) ≤ 1 2 log det 2I(S c * ) + 2 max(1, |S CF |)G Σ(V c * ) 0 0 P I(S CF ) G T (63) ≤ 1 2 log det I(S c * ) + 1 2 G Σ(V c * ) 0 0 P I(S CF ) G T + |S c | + 1 2 log 4 max(1, |S CF |)(64)
where (61)
∆(C CB , R MNNC ) max V c ⊆S⊆N |S| 2 + 1 + min{|S|, |S c |} 2 log (4 max(1, |V| − |S c |)) .(65)
Furthermore, if all source-to-relay channels are good enough to select DF then the constant gap verifies:
∆(C CB , R MNNC ) ≤ 0.5N + 0.7 (66) < ∆(C CB , R NNC ) ,(67)
which leads to a strictly tighter gap than that of NNC scheme [16].
It should be emphasized that the gains in terms of "constant gap" provided by MNNC respect to NNC scheme are obtained by taking empty sets T k . However, the original rate R (k)
T k in Theorem 1 can be maximized over all general (non necessarily empty) sets T k which may improve the final rate. Although we have assumed that the gap incurred by restricting this maximization is not significant (at least in terms of the notion of constant gap), the main outcome of our analysis is that the constant gap can be improved, provided the relays using DF scheme are adequately chosen.
Proof: Consider the first term of MNNC rate given by expression (53). This can be lower bounded, for any set S ⊆ N , as follows:
R N (S) = I(XX S ;Ŷ S c Y |X S c ) − I(Ŷ S ; Y S |XX TŶS c Y ) (68) ≥ I(XX S ;Ŷ S c Y |X S c ) − I(Ŷ S ; Y S |XX N ) (69) = I(XX S ;Ŷ S c Y |X S c ) − |S| 2 . (70) By convenience, we select the sets S c * S c ∪ {N + 1}, V c * V c ∪ {0} and S CF S ∩ V.
Since all channel inputs among the nodes in V c are not necessarily independent, we have
I(XX S ;Ŷ S c Y |X S c ) ≥ I(XX S ;Ŷ S cŶ |X S c ) (71) = h(Ŷ S cŶ |X S c ) − h(Ŷ S cŶ |XX N ) (72) = 1 2 log det I(S c * ) + 1 2 G Σ(V c * ) 0 0 P I(S CF ) G T ,(73)
where the covariance matrix Σ(V c * ) is the one that maximizes the cut-set bound in (56). Hence, the maximum rate R N (S) is lower bounded by
R N (S) ≥ 1 2 log det I(S c * ) + 1 2 G Σ(V c * ) 0 0 P I(S CF ) G T − |S| 2 .(74)
Finally, from (74) the gap between MNNC rate (53) and the cut-set bound (56) is bounded by
∆ 1 max V c ⊆S⊆N |S| 2 + 1 + min{|S c |, |S|} 2 log (4 max(1, |S ∩ V|)) .(75)
The remanning part of the MNNC rate, which is related to all relays using DF scheme, can be bounded as follows. We remark that the channel output Y is absent in the rate expression while it is present in the cut-set bound. Hence, any bound on the gap between the achievable rate and the cut-set bound -no matter how tight it is-will depend on the channel gains between the output Y and all inputs. For sake of simplicity, we shall assume that each DF relay is decoding the source message without looking at the compression indices of others relays, which yields
T k = ∅. Then, the rate R (k) T k is simply reduced to R (k) DF = I(X; Y k |V X k )
and it can be bounded using the same steps as before. The outputs Y k can be re-written as
Y k = g 0k X + G({k}, N )X N + V k (76) = g 0k X + i∈V g ik X i + i∈V c g ik X i + V k ,(77)
where the relays in the set V use CF scheme. In order to evaluate the conditional entropy h(Y k |V X k ), we can use the standard linear decomposition X i =X i +α i V , based on independent descriptionsX i and V that satisfy the power constraints. It is easy to check that
h(Y k |V X k ) = 1 2 log(2πe) g 2 0k (1 − ρ 2 0k )P + i∈V g 2 ik P + i∈V c g 2 ik (1 − ρ 2 ik )P + 1 ,(78)
and similarly
h(Y k |V XX k ) = 1 2 log(2πe) i∈V g 2 ik P + i∈V c g 2 ik (1 − ρ 2 ik )P + 1 .(79)
From which the mutual information decomposes as
I(X; Y k |V X k ) = 1 2 log 1 + g 2 0k (1 − ρ 2 0k )P i∈V g 2 ik P + i∈V c g 2 ik (1 − ρ 2 ik )P + 1 .(80)
By evaluating the the cut-set bound
I(XX N −{k} ; Y k Y |X k ) = 1 2 log det I({2}) + GΣ(M − {k})G T (81) where G = G({k, N + 1}, M − {k}) = g 0k g 1k . . . g N k g 0(N +1) g 1(N +1) . . . g N (N +1) ,(82)
the gap between the achievable rate in this case and the cut-set bound can be bounded by
∆ 2 max k∈V c 1 2 log det I({2}) + GΣ(M − {k})G T i∈V g 2 ik P + i∈V c g 2 ik (1 − ρ 2 ik )P + 1 g 2 0k (1 − ρ 2 0k )P + i∈V g 2 ik P + i∈V c g 2 ik (1 − ρ 2 ik )P + 1 . (83)
As it was expected, this gap does depend on channel gains. From expressions (75) and (83), the final gap reads:
∆(C CB , R MNNC ) max {∆ 1 , ∆ 2 } .(84)
Nevertheless, assuming that all relays using DF scheme are chosen appropriately, the first term ∆ 1 of the gap in (84), which is independent of the channel gains is expected to be dominant.
For instance, the question that arises here is what is the most appropriate condition to select the set of DF relays. A simple way to deal with this issue is to pick all relay nodes whose channel gains {g 0k } are enough large compared to those of the other nodes. This selection rule only requires the optimization of the first term in (84) which yields the desired gap in (65).
Therefore, it is of interest to understand how the gap of MNNC is compared with that of NNC [16]. As it was previously discussed, there is no definitive answer for general cases.
IV. COOPERATIVE UNICASTING IN WIRELESS NETWORKS
In this section, we consider a direct application of MNNC to the problem of cooperative unicasting in wireless networks where a single source wishes to communicate with a destination in presence of multiple relay nodes. A composite unicast network is assumed where the channel parameters are randomly drawn and this draw is assumed to be unknown at the source, fully known at the destination and only partly known at the relay nodes. We exploit MNNC scheme to introduce a novel transmission scheme that enables the relays to select -based on their channel measurements-the best cooperative strategy. Bounds on the asymptotic average error probability of this class of networks are derived.
A. Composite Relay Channels
Consider the composite relay channel described in Fig. 4, where the probability distribution characterizing the channel is indexed with parameters θ. This channel can be defined as a set of memoryless probability distributions:
W Θ P Y n Y n 1 |X n X n 1 ;θ (y, y 1 |x, x 1 ; θ) x ∈ X n , x 1 ∈ X n 1 , y 1 ∈ Y n 1 , y ∈ Y n , θ ∈ Θ ∞ n=1 .(85)
This family of channels corresponds to the definition of the compound relay channel for which the channel is chosen in an arbitrary manner but remains fixed during the communication. Whereas, the composite relay channel introduces a probability measure P θ on Θ to handle the channel selection and thus an index θ is present with probability P θ (θ) but also remains fix during the
communication. The index θ represents vectors of parameters θ = (θ d , θ r ) with (θ d , θ r ) ∈ Θ,
where θ r ∈ Θ r denotes all parameters affecting the relay output and θ d ∈ Θ d are the remaining parameters involved in the communication. More precisely, the marginal distributions read as:
P Y n 1 |X n X n 1 ;θ = P Y n 1 |X n X n 1 ;θr ,(86)P Y n |X n X n 1 ;θ = P Y n |X n X n 1 ;θ d .(87)
The specific draw θ = (θ d , θ r ) is assumed to be unknown at the source and fully known at the destination while the relay only knows θ r . The notion of capacity-versus-outage shall be used to characterize the performance of this channel 1 . Definition 2 (code and achievability): A code-C(n, M n , n,θ ) for the composite relay channel with (W Θ , P θ ) defined as before consists of:
X Y Y 1θ r X 1θ r θ r θ d θ ∼ P θ• An encoder mapping {ϕ : M n −→ X n }, • A decoder mapping {φ : Y n × Θ −→ M n }, • A set of relay functions f i : Y i−1 1 × Θ r −→ X 1 n i=1
, for some set of uniformly distributed messages W ∈ M n = 1, . . . , M n . For each θ, the average error probability is defined as:
n,θ = Pr φ(Y n , θ) = W θ .
An error probability 0 ≤ < 1 is said to be r-achievable, or the rate r is said to be -achievable, if there exists a code-C(n, M n , r) with rate satisfying
lim inf n→∞ 1 n log M n ≥ r(88)
and average error probability
lim sup n→∞ E θ Pr φ(Y n , θ) = W θ ≤ .(89)
The infimum of all r-achievable error probabilities¯ (r) is defined as
(r) inf {0 ≤ < 1 : is r-achievable} .(90)
Remark 3: It is important to remark that the reliability function of the composite relay channel may be defined in different ways. If the expectation of the error probability is chosen as the reliability function (89), then the definition remains the same as that of averaged channels in [35], [36]. The notion of -achievability stays the same as the previous definition where the supremum of all -achievable rates is refereed to as -capacity of the averaged channel
C sup {r ≥ 0 : r is -achievable} .(91)
Indeed, composite channels provide more general models since the reliability function unlike averaged channels is not uniquely determined.
We aim at characterizing the smallest possible average error probability (89) as a function of the coding rate r. In wireless scenarios, the notion of outage probability is extensively used to characterize the average error probability. To properly define this notion, let us assume that the decoder is equipped with an outage identification function [28]:
I : Θ −→ {0, 1}(92)
such that I(θ) equal to one indicates that the decoder is able to recover the message, i.e.,
lim n→∞ Pr φ(Y n , θ) = W θ = 0 ,(93)
otherwise if I(θ) is zero, the decoder does not decode the message and declares an outage event.
The outage probability is then defined by
P out Pr{I(θ) = 0} .(94)
Hence, for any code with outage probability P out we know that if I(θ) is equal to one the error probability tends to zero as n goes to infinity and thus (89) can be upper bounded by the outage probability, i.e.,
lim sup n→∞ E θ Pr φ(Y n , θ) = W θ ≤ P out .(95)
On the other hand, for I(θ) = 0 the error probability can be only bounded away from zero. Let us now assume that the decoder is provided with the full error genie aided function:
J : Θ −→ {0, 1} ,(96)
where J(θ) equal to zero indicates that the error probability tends to one, i.e.,
lim sup n→∞ Pr φ(Y n , θ) = W θ = 1 ,(97)
and the value one is assigned to indicate that the error probability tends not to one, but neither necessarily to zero. Thus, we have that the average error probability is bounded away from the probability that J(θ) equals zero, yielding the lower bound:
lim inf n→∞ E θ Pr φ(Y n , θ) = W θ ≥ Pr{J(θ) = 0} .(98)
In a recent work [31], it has been shown that any rate bigger than the cut-set bound will produce an error probability tending to one. This implies that all relay channels for which the cut-set bound is tight satisfy the strong converse property. Furthermore, it also implies that the next genie aided function is a full error identification function:
J(θ) 1[r > C CB (θ)] ,(99)
where C CB (θ) is the cut-set bound indexed by θ,
C CB (θ) max p(x,x 1 ) min {I θ (X; Y Y 1 |X 1 ), I θ (XX 1 ; Y )} .
By using the previous inequalities, the average error probability¯ (r) can be bounded as follows
P θ {r > C CB (θ)} ≤¯ (r) ≤ P out (r)(100)
and P out (r) is the outage probability of a given coding strategy (e.g. DF and CF schemes). The use of DF scheme yields an outage probability given by
P DF out (r) min p(x,x 1 ) P θ r > min{I θr (X; Y 1 |X 1 ), I θ (XX 1 ; Y )} ,(101)
where I θ denotes the mutual information for a given θ. Notice that since the source is unaware of θ = (θ r , θ d ), and p(x, x 1 ) must be known at both source and relay end, then p(x 1 ) cannot be independently optimized on θ r to minimize the outage probability.
Consider now the case of CF, for which the source does not need to know p(x 1 ), so the relay can choose p(x 1 ) to minimize the outage probability conditioned on each value θ r . This requires two steps of optimization, the outage probability of CF scheme [16] reads as:
P CF out (r) min p(x,q) E θr min p(x 1 |q)p(ŷ 1 |x 1 ,y 1 ,q) P θ|θr r > min{I θ (X;Ŷ 1 Y |X 1 Q), I θ (XX 1 ; Y |Q) − I θ (Y 1 ;Ŷ 1 |XX 1 Y Q)} θ r .(102)
Moreover, from (101) and (102) the selection of the best strategy minimizing the outage probability provides the tightest bound on the error probability, The central question that arises here is whether the achievable error probability (103) can be improved by some kind of smart coding strategy in which the relay selects instantaneously the best strategy between DF or CF, according to its channel measurement θ r . To this purpose, the source code should be capable of being used simultaneously with DF and CF schemes, as shown in Fig. 5(a). Nevertheless, the source is not aware of the channel measurement at the relay and hence it is not able to know which strategy is going to be selected at the relay. This is an example of simultaneous relay channel with two possible situation, DF relay and CF relay.
P out (r) min P DF out (r), P CF out (r) . (103) Y ✓ Y 1✓ r X 1✓ r ✓ r ✓ d ✓ ⇠ P ✓ X Y 1 X 1 DF X Y 2 X 2 CF (a) Selective Coding Strategy (SCS). X(source) Y (destination) Y 1 Y 2 X 2 X 1 DF CF (b) Two-relay network.
As it is discussed previously in [37], this problem can be studied using an equivalent model. The source must consider the two-relay network where one relay node employs DF scheme while the other one uses CF scheme, as shown in Fig. 5(b). Now the source code should be designed to account for both relays. Also in the equivalent model, the relays cannot collaborate since only one of them is present at once. This model sheds light on the proper code design for composite setting due to the simultaneous presence of relays with heterogeneous cooperative strategy.
The next corollary is a special case of Theorem 2 for a two-relay network where the DF relay decodes directly the source message without the help of the other relay.
Corollary 4 (Two-relay network): A lower bound on the capacity of the two-relay network is given by all rates satisfying
R ≤ max P ∈P min I(X; Y 1 |X 1 Q) , max I(XX 1 ; Y |Q), min I(XX 1 ;Ŷ 2 Y |X 2 Q), I(XX 1 X 2 ; Y |Q) − I(Y 2 ;Ŷ 2 |Y XX 1 X 2 Q)(104)
June 3, 2014 DRAFT and the set of all admissible input distributions P is defined as
P P QX 2 X 1 XY Y 1 Y 2Ŷ2 = P Q P X 2 |Q P XX 1 |Q P Y Y 1 Y 2 |XX 1 X 2 PŶ 2 |X 2 Y 2 Q .(105)
The maximum in (104) determines whether the second relay that uses CF scheme is increasing the rate or would be better to treat its transmission as interference. It is not difficult to check that the second relay increases the rate provided the following condition is satisfied:
I(X 2 ; Y |XX 1 Q) ≥ I(Y 2 ;Ŷ 2 |Y XX 1 X 2 Q) .(106)
The last two terms in (104) represent the condition of successful decoding at the destination while the first term is the condition of successful decoding of X 1 at the first relay. By comparing the last two terms with the standard expression of CF rate, it is easy to see that these present similar behavior with the minor difference that the relay codeword has been replaced with (X, X 1 ). It is also worth mentioning that by treating the CF relay as noisy, e.g., whenever its link is too noisy, or by using NNC which improves the constraint (106), Corollary 4 improves over the results in [5].
Based on Corollary 4 we can state an achievable result for the composite relay channel that is a direct consequence of Corollary 4 and some additional subtleties which are addressed in Appendix D. First, we emphasize on the fact that the coding strategy used in Corollary 4
is also well adapted to the composite relay channel. Basically, the relay may dispose of two set of codebooks, namely X 1 and X 2 , and it sends either X 1θr = X 1 (corresponding to DF scheme) when condition θ r ∈ D DF holds or X 1θr = X 2 (corresponding to CF scheme) elsewhere.
Therefore, since the error probability is made arbitrary small simultaneously for both relays, the source does not need to know the specific relay function implemented. With this coding, the relay can select the coding strategy according to its instantaneous channel measurement θ r . Secondly, we remark that for the CF relay there may be the additional condition (106) for decoding. The destination is assumed to know θ and consequently is aware if condition (106) does hold or not.
In the case where it fails, destination will treat the relay input as interference -without perform its decoding-and then the condition for unsuccessful decoding simple becomes 1 {r > I θ (X; Y )}.
We refer to this coding scheme as to "selective coding strategy" (SCS). In the next section, we
show that it can further improve the asymptotic error probability.
June 3, 2014 DRAFT
Proposition 3 (SCS with partial CSI at relay): The average error probability of the composite relay channel with partial CSI θ r at the relay can be upper bounded bȳ
(r) ≤ min p(x,x 1 ,q) inf D DF ⊆Θr E θr P θ|θr r > I DF (θ) , θ r ∈ D DF θ r + min p(x 2 |q)p(ŷ 2 |x 2 ,y 1 ,q) P θ|θr r > I CF (θ) , θ r / ∈ D DF θ r ,(107)
where (X 1 , X 2 ) denote the relay inputs corresponding to each strategy selected as follows
X 1θr = X 1 if θ r ∈ D DF X 2 if θ r / ∈ D DF(108)
and the quantities I DF , I CF are defined by
I DF (θ) min I θr (X; Y 1 |X 1 Q) , I θ (XX 1 ; Y |Q) ,(109)I CF (θ) max min I θ (X;Ŷ 2 Y |X 2 Q), I θ (XX 2 ; Y ) −I θr (Y 1 ;Ŷ 2 |Y XX 2 Q) , I θ (X; Y ) .(110)
Consider an index draw θ r such that 1 {r > I θr (X; Y 1 |X 1 )} = 1, i.e., the relay is not able to decode the message. Then, DF scheme would lead to an outage event while CF scheme does not necessarily yield to such event and so the best guess of the relay would be to use CF scheme.
The question that arises here is what the proper guess would be if the relay can decode the message. As a matter of fact, if the relay decodes and uses DF scheme, an outage event may still occur if 1 {r > I θ (XX 1 ; Y )} = 1. However, since X 2 is independent of X while X is in general dependent on X 1 , for Gaussian inputs, we have that I θ (XX 1 ; Y ) ≥ I θ (XX 2 ; Y ). This implies that if an outage event occurs with DF scheme while the relay has the message then the event will happen anyway with CF scheme. Note that the preceding inequality is not true in general, e.g., consider the case of binary RVs where correlation does not necessarily increase mutual information.
Remark 4 (Optimizing the decision region): The optimal decision region when the inputs (107) are jointly Gaussian is given by the set
D DF θ r ∈ Θ r I θr (X; Y 1 |X 1 Q) > r .(111)
Although the knowledge of θ r at the relay is enough to select the adequate coding strategy, full CSI (θ r , θ d ) further improves the description that the relay sends to the destination and yields the following extension of Proposition 3.
X Y (X j , Y j ) (X k , Y k ) DF relays CF relays ✓ r ✓ d ✓ r ✓ ⇠ P ✓(r) ≤ min p(x,x 1 ,q) inf D DF ⊆Θr P θ r > I DF (θ) , θ r ∈ D DF + P θ r > I CF (θ) , θ r / ∈ D DF ,(112)
where (X 1 , X 2 ) denote the relay inputs corresponding to each strategy selected as follows
X 1θr = X 1 if θ r ∈ D DF X 2 if θ r / ∈ D DF(113)
and the quantities I DF , I CF are defined by
I DF (θ) min I θr (X; Y 1 |X 1 Q) , I θ (XX 1 ; Y |Q) ,(114)I CF (θ) max p(x 2 |q)p(ŷ 2 |x 2 ,y 1 ,q) min I θ (X;Ŷ 2 Y |X 2 Q) , I θ (XX 2 ; Y |Q) −I θr (Y 1 ;Ŷ 2 |Y XX 2 Q) .(115)
The proof of this proposition follows along the same lines than that of Proposition 3. It is worth mentionning here that since full CSI is available at the relay, the relay input can be optimized over θ = (θ r , θ d ) and then I CF can never be less than the capacity of the source-to-destination channel. Similarly, the optimal decision region assuming Gaussian inputs reads as (111).
B. Composite Cooperative Unicast Networks
Consider the composite cooperative unicast network as described in Fig. 6, where the probability distribution characterizing the network is indexed with parameters θ ∈ Θ. This network June 3, 2014 DRAFT can be defined as a set of memoryless probability distributions:
W Θ P Y n Y n 1 ...Y n N |X n X n 1 .
..X n N ;θ (y, y 1 , . . . , y N |x, x 1 , . . . , x N ; θ) x ∈ X n , x 1 ∈ X n 1 , . . . , x N ∈ X n N ,
y ∈ Y n , y 1 ∈ Y n 1 , . . . , y N ∈ Y n N , θ ∈ Θ ∞ n=1 .(116)
This family of networks corresponds to the definition of the compound cooperative unicast network for which the channels are chosen in an arbitrary manner but remain fix during the communication. Similar to the case of the composite relay channel, we introduce a probability measure P θ over Θ, then each index θ is present with probability P θ . The vectors of parameters is θ = (θ d , θ r ) ∈ Θ with θ r denoting all parameters that affect the relays' outputs, and θ d are the remaining parameters involved in the communication, as shown in Fig. 6. More precisely, the marginal distributions read as:
P Y n 1 ...Y n N |X n X n 1 ...X n N ;θ = P Y n 1 ...Y n N |X n X n 1 ...X n N ;θr ,(117)P Y n |X n X n 1 ...X n N ;θ = P Y n |X n X n 1 ...X n N ;θ d .(118)
The specific draw of θ = (θ d , θ r ) is assumed to be unknown at the source and fully known at the destination while the relays only know θ r . Again the notion of capacity-versus-outage shall be used to characterize the performance of this network.
Definition 3 (code and achievability):
A code-C(n, M n , r) for the composite cooperative unicast network with (W Θ , P θ ) consists of:
• An encoder mapping {ϕ : M n −→ X n },
• A decoder mapping {φ : Y n × Θ −→ M n }, • A set of relay functions f (k) i : Y i−1 k × Θ r −→ X k n i=1
for k ∈ N .
An error probability 0 ≤ < 1 is said to be r-achievable, or the rate r is said to be -achievable,
if there exists a code-C(n, M n , r) with rate satisfying
lim inf n→∞ 1 n log M n ≥ r(119)
and average error probability
lim sup n→∞ E θ Pr φ(Y n θ , θ) = W θ ≤ .(120)
The infimum of all r-achievable error probabilities¯ (r) is defined as
(r) inf {0 ≤ < 1 : is r-achievable} .(121)
It should be worth mentioning here that the definition -achievability based on (120) becomes equivalent to that of averaged channels [35], [36], where
C sup {0 ≤ r : r is -achievable} .(122)
Assume that the destination is equipped with outage and full error identification functions I and J, given by expressions (92) and (96), respectively. The average error of a given code can be bounded by following the same arguments as before and thus,
Pr{J(θ) = 0} ≤ lim sup n→∞ E θ Pr φ(Y n , θ) = W θ (123) ≤ Pr{I(θ) = 0} .(124)
Because the preceding lower bound is valid for arbitrary codes, it provides a converse bound on (r).
Let us assume any code with rate r and full error identification function J such that Pr{J(θ) = 0} > , then we know that the code is not -achievable. This condition does not imply any specific connection between the rate r and the -capacity because there may exist another code with rate r which is -achievable. To remove this possibility, we can consider only those rates with Pr{J(θ) = 0} > such that that there is no code with this rate which is -achievable. The set of these rates is non-empty in general because there is certain rate limit over which every code with that rate will have the error probability asymptotically tending to one. For instance, every rate beyond the capacity of discrete memoryless satisfies such condition and hence it is not -achievable for all < 1. Formally, if the rate r is such that for all code-C(n, M n , r) with lim inf n→∞ 1 n log M n ≥ r the error probability satisfies Pr{J(θ) = 0} > , then r provides an upper bound on the -capacity of the composite cooperative unicast network, i.e.,
C ≤ inf r ≥ 0 : ∀ code-C(n, M n , r) if lim inf n→∞ 1 n log M n ≥ r then Pr{J(θ) = 0} > .(126)
For all codes with rate at least r, an identification function is given by [38] J
(θ) 1[r > C CB (θ)] ,(127)
where C CB (θ) denotes the cut-set bound of the cooperative unicast network with index θ. The -capacity of the averaged composite cooperative unicast network is bounded by
C ≤ inf {r ≥ 0 : Pr{r > C CB (θ)} > } .(128)
In the rest of this section, we upper bound the average error probability based on the outage probability of the "Selective Coding Strategy" (SCS). Indeed, we first derive an upper bound using Theorem 1. Let us select a set of nodes V ⊆ N and a probability distribution P QV X that is independent of the specific draw θ, which is not available at the source. Besides relays can adapt {P X k |QV } to the parameters involved in θ r for which the identification function reads as:
I(θ) 1[r ≤ I MNNC (V, θ)] ,(129)
where I MNNC (V, θ) is defined by
I MNNC (V, θ) min max T ∈Υ (N ) min V c ⊆S⊆T R T (S, θ) , min k∈V c max T k ∈Υ k (N ) min S⊆T k R (k) T k (S, θ r ) ,(130)
with V c T − V and
R T (S, θ) I θ (XX S ;Ŷ S c Y |X S c Q) − I θ (Ŷ S ; Y S |XX TŶS c Y Q) ,(131)R (k) T k (S, θ r ) I θr (X;Ŷ T k Y k |V X k X T k Q) + I θr (X S ; Y k |V X k X S c Q) −I θr (Ŷ S ; Y S |V X k X T kŶ S c Y k Q) .(132)
The sets Υ (N ) and Υ k (N ) are given by T (S, θ r ) are defined as:
Υ (N ) {T ⊆ N : ∀ S ⊆ T , Q T (S, θ) ≥ 0} ,(133)Υ k (N ) T ⊆ N − {k} : ∀ S ⊆ T , Q (k) T (S, θ r ) ≥ 0 ,(134)Q T (S, θ) I θ (X S ;Ŷ S c Y |V XX S c Q) − I θ (Ŷ S ; Y S |V XX TŶS c Y Q) ,(135)Q (k) T (S, θ r ) I θr (X S ; Y k |V X k X S c Q) − I θr (Ŷ S ; Y S |V XX k X TŶS c Y k Q) .(136)
The following upper bound on the expected error probability of the composite cooperative unicast network with partial CSI θ r at the relays holds:
(r) ≤ min p(x,v,q) inf V⊆N E θr min p(·)∈Q P θ|θr I MNNC (V, θ) θ r ,(137)
where the set of all admissible distributions Q is given by
p(·) = j∈V p(x j |q)p(ŷ j |x j y j q) j∈V c p(x j |vq)p(ŷ j |vx j y j q) .(138)
We can also further exploit the SCS presented in the previous subsection. Again the general idea is that, based on the channel parameters θ r , each relay is allowed to use either CF or DF scheme. In this setting, the relays must have two set of codebooks: one set is intended to the case in which the relay uses CF scheme and the other set is for the case of DF scheme. Without loss of generality, we assume that θ r is known to all relay terminals. Each relay has a decision region, say D
D V k∈V c D (k) DF ∩ k∈V D (k) DF c .(139)
By consequence, if θ r ∈ D V then θ r / ∈ D (k) DF for all k ∈ V, and θ r ∈ D (k) DF for all k ∈ V c . Relay k, for k ∈ V, uses CF scheme while relay k , for k ∈ V c , employes DF scheme. The ensemble of decision regions of the relays given by the regions D V , which are mutually disjoint, form a partition of the space of parameters Θ r . Notice that for some V, the set D V may be empty.
Indeed, a decision region {D V : V ⊆ N } is a set of individual decision region D V satisfying:
V⊆N D V = Θ r and for V = V with D V ∩ D V = ∅ .(140)
The indexed partitions over Θ r which are at most 2 N subsets 2 are refereed to as the decision
June 3, 2014 DRAFT Y X DF relays CF relays X (1) k i X (1) k 2 X (2) k j (X (1) k 1 , X (2) k 1 ) (✓ r / 2D k 1 DF ) X (2) k 1 (X (1) k i , X (2) k i ) (✓ r 2D k i DF ) P ✓ (X (1) k 2 , X (2) k 2 ) (✓ r 2D k 2 DF ) (X (1) k j , X (2) k j ) (✓ r / 2D k j DF )
(k) (CF scheme) is transmitted. Notice that unlike DF, x
(k) is independent of the source codewords and so its probability distribution can be chosen adaptively based on θ r . The optimization over D V would potentially improve the outage probability compared to the case in which every relay uses a fix coding strategy for all θ r . A careful evaluation of Theorem 1 yields the following proposition. where the set of all admissible PDs Q is given by all distributions
p(·) = j∈V p(x j |q)p(ŷ j |x j y j q) j∈V c p(x j |vq)p(ŷ j |vx j y j q)(142)
and Π (Θ r , N ) is the set of all indexed partitions over Θ r into at most 2 N disjoint sets and I MNNC (V, θ) is defined by expression (130).
Proof: This proposition is shown in Appendix E.
It is worth noting that (141) reaches at least the same performance as (137). In (137), the set V is chosen beforehand independent of θ r . Now if one choose D V = Θ r for this V and (141), then (137) is obtained as special case. The advantage of (141) is in choosing the set of CF relays, V, adaptively based on channel state information of relays.
D V = ∅ for all V = V in
Finally, if full CSI is available at all relays then this will simply give the following proposition.
Proposition 6 (SCS with full CSI): The average error probability of the composite cooperative network with full CSI θ r at the relays can be upper bounded bȳ
(r) ≤ min p(x,v,q) inf {D V : V⊆N }∈Π(Θr,N ) V⊆N P θ r > I MNNC (V, θ), θ r ∈ D V ,(143)
where Π (Θ r , N ) is the set of all indexed partitions over Θ r into at most 2 N disjoint sets and
I MNNC (V, θ) is defined by I MNNC (V, θ) max p(·)∈Q min max T ∈Υ (N ) min T −V⊆S⊆T R T (S, θ) , min k∈V c max T k ∈Υ k (N ) min S⊆T k R (k) T k (S, θ r ) (144)
where the set of all admissible distributions Q is given by
p(·) = j∈V p(x j |q)p(ŷ j |x j y j q) j∈V c p(x j |vq)p(ŷ j |vx j y j q) .(145)
Finally, we present a similar result to that of Theorem 2 where there is no cooperation among the relay nodes.
Proposition 7 (SCS with partial CSI):
The average error probability of the composite noncooperative unicast network with partial CSI θ r at the relays can be upper bounded bȳ
(r) ≤ min p(x,x (1) N ,q) inf {D V ,V⊆N }∈Π(Θr,N ) V⊆N E θr min p(·) P θ|θr r > I MNNC (V, θ), θ r ∈ D V θ r ,(146)
where the set of all admissible PDs Q is given by all PDs decomposing as:
p(·) = j∈V p(x(2)
j |q)p(ŷ j |x (2) j y j q)
I MNNC (V, θ) max T ⊆V min min S⊆T R T (S, θ) , min i∈V c I θr (X; Y i |X (1) N Q) (148) with R T (S, θ) I θ (XX (1) V c X (2) S ;Ŷ S c Y |X (2) S c Q) − I θ (Y Sθr ;Ŷ S |XX (2) T X (1) V cŶS c Y Q) ,(149)
where (X (1) k , X
k ) denote the corresponding relay inputs selected as follows
X kθr = X (1) k if θ r ∈ D k DF X (2) k if θ r / ∈ D k DF with D k DF = V⊂N , k / ∈V D V .(150)
For θ r ∈ D V , the following Markov chain holds:
(X (1) V , X (2) V c ) (X, X (1) V c , X (2) V ) (Y, Y N ) .(151)
Proof: The proof is presented in Appendix F.
It can be checked that the use of superposition coding does not change the rate R T (S, θ), but unlike the case of the composite relay channel, the condition of correct decoding at DF relays is changed from I θr (X;
Y i |X (1) V c Q) to I θ (X; Y i |X(1)
N Q).
V. NUMERICAL EXAMPLE: GAUSSIAN FADING RELAY CHANNEL
We now consider an application example of SCS to the fading Gaussian relay channel defined by the following destination and relay output, respectively,
Y = g 1 X + g 3 X 1 + V 1 ,(152)Y 1 = g 2 X + V 2 ,(153)
where V 1 and V 2 are independent complex Gaussian noises with zero-mean and unit variance; the channel gains (g 1 , g 2 , g 3 ) are independent complex Gaussian with zero-mean and unit variance; and the inputs must not exceed the average powers P and P 1 , respectively. It is assumed that the source is not aware of the channel measurements θ (g 1 , g 2 , g 3 ), the relay only knows θ r g 2 and the destination is fully aware of all fading coefficients θ, and thus θ d (g 1 , g 3 ). This model is special case of the composite relay channel described in Section IV-A. We aim to evaluate the asymptotic error probability based on the bounds derived in Propositions 3 and 4, and compare them to the upper bounds corresponding to DF and CF schemes (103), and the cut-set based lower bound. In this case, the expression for the DF rate reads as:
I DF (θ) min C β|g 2 | 2 P , C |g 1 | 2 P + |g 3 | 2 P 1 + 2 βP P 1 Re{g 1 g 3 }(154)
where 0 ≤ β ≤ 1 and C(x) log 2 (1 + x). The CF rate is given by
I CF (θ) max I CF (θ), C |g 1 | 2 P |g 3 | 2 P 1 + 1 ,(155)I CF (θ) min C |g 1 | 2 P + |g 2 | 2 P N 2 + 1 , C |g 1 | 2 P + |g 3 | 2 P 1 − C 1 N 2 ,(156)
where the descriptionŶ 1 is generated by adding independent complex Gaussian noise with zeromean and variance N 2 . Finally, the asymptotically error probability based on DF and CF schemes can be easily derived by using expressions (101) and (102), respectively. If full CSI is available at the relay, then N 2 can be optimally chosen as:
N opt 2 P (|g 1 | 2 + |g 2 | 2 ) + 1 |g 3 | 2 P 1 .(157)
Whereas, if only g 2 is available at the relay, a constant N 2 must be selected to minimize the outage probability. We have shown that based on |g 2 |, SCS allows the relay to select the proper coding strategy. In this case, if the source-to-relay channel is not good enough for decoding the message, the relay uses CF scheme while otherwise DF scheme would be the best choice. It turns out that the optimum decision region D DF is given by the set D DF {g 2 : r ≤ C (β|g 2 | 2 P )}. Fig. 8 presents numerical plots of the asymptotic error probability with P 1 = P = 1. For the case of partially CSI at the relay, we observe that SCS clearly outperforms the naive DF and CF schemes. Moreover, notice that full CSI (g 1 , g 2 , g 3 ) at the relay improves the error probability only through the choice of the best possible compression noise N 2 . Besides this guarantee that CF scheme can never perform less good than direct transmission. Finally, it can be seen that the upper bound (achievable error probability) resulting from SCS is close to the cut-set lower bound, and so to the best average error probability. Fig. 9 presents bounds on -capacity of the corresponding averaged channel for = 0.01 based on the signal-to-noise ratio (SNR). We set P 1 = 1 while P is varying with SNR. Indeed, thecapacity represents the maximum achievable rate subject to satisfy an expected error probability less or equal than . The bounds illustrated in (125) and (126) have been used here. Observe that again the -capacity is clearly enlarged by using our SCS and is not far from the upper bound.
VI. SUMMARY AND CONCLUDING REMARKS
In this paper, we investigated the problem of communicating a single message to a single destination in presence of multiple relay nodes. We considered a general framework of composite cooperative networks where the channel parameters are randomly drawn from a probability distribution and each draw is assumed to be unknown at the source and fully known at the destination, but only partly known at the relay nodes. Within this framework, we introduced "Mixed Noisy Network Coding" (MNNC) where nodes are allowed to decode and forward messages while all nodes transmit noisy descriptions of their observations. We further extended MNNC to multi-hopping networks that we referred to as "Layered MNNC" (LMNNC) where DF relays are organized into disjoint groups, each of them representing one hop in the network.
Perhaps the main feature of MNNC scheme relies on the meaningful concept of "Selective Coding Strategy" (SCS) [21] that enables relays to decide dynamically, e.g., based on the channel measurements, whether in addition to communicate descriptions that are potentially exploited at the destination and the DF relays, would be possible to decode and forward messages. This guarantees full cooperation among all nodes, and in particular, without the requirement of any hierarchy between them. It is demonstrated for additive white Gaussian noise (AWGN) networks that MNNC improves over all previously established constant gaps to the cut-set bound, and for the slow-fading AWGN relay channel that SCS clearly outperforms the asymptotic average error probability of previous coding schemes.
An important direction of future work is to investigate more sophisticated wireless models within our composite framework, e.g., the effects of user mobility, network geometry and shadowing can be incorporated into the model [39]. It remains to investigate network scaling exponents [12] in the limit of large network size (number of nodes) where MNNC may substantially improve the scaling of the capacity with the number of nodes. In this setting, the multi-hopping scenario is in general challenging and needs further study since it is not clear how DF relays should choose their layering according to partial channel observation. Another important direction is to investigate how MNNC could be exploited in half-duplex networks. With half-duplex constraints, the nodes are permitted -within each slot of time-to either transmits or receives information. Therefore, it would be of interest to study cooperation protocols based on MNNC with slots of variable time durations, where nodes can dynamically select not only whether decode and forward messages in addition to send noisy descriptions of their observations, but also the time allowed to receive and transmit data [40], [41]. We believe this problem may be quite rich because it yields several other scenarios of practical importance. For simplicity, we adopt the following notation:
T DF k T k ∩ V c ,(158)T CF k T k ∩ V ,(159)T DF T ∩ V c ,(160)T CF T ∩ V .(161)
Moreover, for any arbitrary message w i , the notation w S is used to denote the set of indices
{w i : i ∈ S}.
We now provide the code generation, encoding and decoding procedures. source.
x and y are source codeword and destination received message. As discussed before, the DF relay delays the decoding of source message at b = 1 to the end of block b = 2 so that it can jointly decode the compression index l 11 and w 1 using y 1 (1) and y 1 (2). The destination on the other hand decodes backwardly. It first finds l 1(B+2) and l 2(B+2) and then jointly decodes (w B , l 1(B+1) , l 2(B+1) ). It continues this process until it finds all the messages. Code Generation:
b = 1 b = 2 b = 3 . . . b = B + 2 b = B + 3 . . . b = B + L v(1) v(1) v(w1) . . . v(wB) v(1) . . . v(1) x 1 (1, 1) x 1 (1, l11) x 1 (w1, l12) . . . x 1 (wB, l 1(B+1) ) x 1 (1, l 1(B+2) ) . . . x 1 (1, l 1(B+2) ) x(1, w1) x(1, w2) x(w1, w3) . . . x(wB, 1) x(1, 1) . . . x(1, 1) x 2 (1) x 2 (l21) x 2 (l22) . . . x 2 (l 2(B+1) ) x 2 (l 2(B+2) ) . . . x 2 (l 2(B+2) ) y 1 (1, 1, l11)ŷ 1 (1, l11, l12)ŷ 1 (w1, l12, l13) . . .ŷ 1 (wB, l 1(B+1) , l 1(B+2) ) ŷ 2 (1, l21)ŷ 2 (l21, l22)ŷ 2 (l22, l23) . . .ŷ 2 (l 2(B+1) , l 2(B+2) ) y 1 (1) y 1 (2) y 1 (3) . . . y 1 (B + 2) y 1 (B + 3) . . . y 1 (B + L) y 2 (1) y 2 (2) y 2 (3) . . .
(i) Randomly and independently generate 2 nR sequences v drawn i.i.d. from
P n V (v) = n j=1 P V (v j ) .(162)
Index them as v(w 0 ) with index w 0 ∈ 1, 2 nR .
(ii) For each k ∈ V c and each v(w 0 ), randomly and independently generate 2 nR k sequences x k drawn i.i.d. from
P n X k |V (x k |v(w 0 )) = n j=1 P X k |V (x kj |v j (w 0 )) .(163)
Index them as x k (w 0 , l 0k ), where l 0k ∈ 1, 2 nR k forR k I(Y k ;Ŷ k |X k , V )+ with k ∈ V c . This is the codebook for DF relays.
June 3, 2014 DRAFT (iii) For each k ∈ V, randomly and independently generate 2 nR k sequences x k drawn i.i.d. from
P n X k (x k ) = n j=1 P X k (x kj ) .(164)
Index them as x k (l 0k ), where l 0k ∈ 1, 2 nR k forR k I(Y k ;Ŷ k |X k ) + with k ∈ V. This is the codebook for CF relays.
(iv) For each v(w 0 ), randomly and conditionally independently generate 2 nR sequences x drawn
i.i.d. from P n X|V (x|v(w 0 )) = n j=1 P X|V (x j |v j (w 0 )) .(165)
Index them as x(w 0 , w), where w ∈ 1, 2 nR . This is the source codebook.
(v) For each k ∈ V c and each v(w 0 ), x k (w 0 , l 0k ), randomly and conditionally independently generate 2 nR k sequencesŷ k each with probability
P n Y k |X k V (ŷ k |x k (w 0 , l 0k ), v(w 0 )) = n j=1 PŶ k |X k V (ŷ kj |x kj (w 0 , l 0k ), v j (w 0 )) .(166)
Index them asŷ k (w 0 , l 0k , l k ), where l k ∈ 1, 2 nR k with k ∈ V c . This is the compressed version of DF relays output.
(vi) For each k ∈ V and each x k (l 0k ), randomly and conditionally independently generate 2 nR k sequencesŷ k each with probability
P n Y k |X k (ŷ k |x k (l 0k )) = n j=1 PŶ k |X k (ŷ kj |x kj (l 0k )) .(167)
Index them asŷ k (l 0k , l k ), where l k ∈ 1, 2 nR k with k ∈ V. This is the compressed version of CF relays output.
(vii) Provide the corresponding codebooks to the relays, the encoder and the decoder ends.
Encoding:
v(w (i−2) ), x k (w (i−2) , l k(i−1) ), y k (i),ŷ k (w (i−2) , l k(i−1) , l ki ) ∈ A n [V X k Y kŶk ] .(168)
The probability of finding such l ki goes to one as n goes to infinity due to our adequate choice of the rateR k for compression.
(iii) For i = [1 : B + 2] and k ∈ V c , relay k knows from the previous block l k(i−1) and w (i −2) and it sends x k (w (i−2) , l k(i−1) ). Moreover, k-th relay repeats l k(B+2) for i = [B + 3 :
B + L],
i.e. for L − 2 blocks.
(iv) For each i = [1 : B + 2], each k ∈ V, the relay k after receiving y k (i), searches for at least one index l ki with l k0 = 1 such that
x k (l k(i−1) ), y k (i),ŷ k (l k(i−1) , l ki ) ∈ A n [X k Y kŶk ] .(169)
The probability of finding such l ki goes to one as n goes to infinity due to our adequate choice of the rateR k for compression.
(v) For i = [1 : B + 2] and k ∈ V, relay k knows from the previous block l k(i−1) and it sends indices up to block i − 1 have been correctly decoded. We emphasize that there are two kind of relays in T k , those who employ DF scheme and those who are using CF scheme.
Relay k knows the message (w (i−2) , w (i−1) ), and so v w (i−2) and v w (i−1) . Let us define the sequences:
E k ŵ b ,l T k b x(w (b−2) ,ŵ b ), v(w (b−2) ), x k (w (b−2) , l k(b−1) ), y k (b), x i (l i(b−1) ),ŷ i (l i(b−1) ,l ib ) i∈T CF k , x i (w (b−2) , l i(b−1) ),ŷ i (w (b−2) , l i(b−1) ,l ib ) i∈T DF k ,(170)E k l T k b v(w (b−1) ), x k (w (b−1) , l kb ), y k (b + 1), x i (w (b−1) ,l ib ) i∈T DF k , x i (l ib ) i∈T CF k .(171)
By looking at two consecutive blocks (b, b + 1), the k-th relay searches for the unique indices (ŵ b ,l T k b ) such that:
E k (ŵ b ,l T k b ) ∈ A n [V XX k X T kŶ T k Y k ] and E k (l T k b ) ∈ A n [V X T k X k Y k ] .(172)
Given the sets S ⊆ T k , and S c T k − S and assuming that the correct messages were (w b , l T k b ), we define the following events:
E 0 : E k (w b , l T k b ) / ∈ A n [V XX k X T kŶ T k Y k ] or E k (l T k b ) / ∈ A n [V X T k X k Y k ] , (173) E S : E k (w b ,l T k b ) ∈ A n [V XX k X T kŶ T k Y k ] and E k (l T k b ) ∈ A n [V X T k X k Y k ]
for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c , (174)
E w,S : E k (ŵ b ,l T k b ) ∈ A n [V XX k X T kŶ T k Y k ] and E k (l T k b ) ∈ A n [V X T k X k Y k ] for someŵ b = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c . (175)
Hence, the probability of error can be bounded as follows:
Pr (ŵ b ,l T k b ) = (w b , l T k b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] ,(176)
where Pr(E 0 ) goes to zero as n → ∞ , provided by the code generation and the encoding process. As the next step, we bound the probability:
Pr(E S ) ≤ l kb =l kb , k∈S Pr E k (w b ,l T k b ) ∈ A n [V XX k X T kŶ T k Y k ] and E k (l T k b ) ∈ A n [V X T k X k Y k ] (177) ≤ j∈S 2 nR j − 1 2 n(∆ 1 +∆ 2 ) ,(178)
where
∆ 1 H(V XY k X k X T kŶ T k ) − j∈S∩T CF k H(Ŷ j |X j ) − j∈S∩T DF k H(Ŷ j |X j V ) −H(V XX k Y k X T kŶ S c ) ,(179)∆ 2 H(V X k Y k X T k ) − j∈S∩T CF k H(X j ) − j∈S∩T DF k H(X j |V ) −H(V X k Y k X S c ) + 1 .(180)
To gaurantee that the probability Pr(E S ) is arbitrarily small, the following inequality needs to hold:
j∈S∩T CF k I(Ŷ j ; Y j |X j ) + j∈S∩T DF k I(Ŷ j ; Y j |X j V ) + 1 < j∈S∩T CF k H(Ŷ j |X j ) + j∈S∩T CF k H(X j ) + j∈S∩T DF k H(Ŷ j |X j V ) + j∈S∩T DF k H(X j |V ) + H(V XX k Y k X T kŶ S c ) + H(V X k Y k X S c ) − H(V XY k X k X T kŶ T k ) − H(V X k Y k X T k )(181)
from which we can have that
1 < j∈S∩T CF k H(Ŷ j |Y j X j ) + j∈S∩T DF k H(Ŷ j |Y j X j V ) + I(X S ; V X k X S c Y k ) −H(Ŷ S |V XX k X T kŶ S c Y k ) .(182)
Indeed, (182) can be further simplified by using the fact thatŶ j is independent of all the other random variables, given
(X j , Y j ) for j ∈ T CF k and (V, X j , Y j ) for j ∈ T DF k : j∈S∩T CF k H(Ŷ j |Y j X j ) + j∈S∩T DF k H(Ŷ j |Y j X j V ) + I(X S ; V X k X S c Y k ) −H(Ŷ S |V XX k X T kŶ S c Y k ) (183) = j∈S H(Ŷ j |Y j X j V ) + I(X S ; V X k X S c Y k ) − H(Ŷ S |V XX k X T kŶ S c Y k ) (184) = j∈S H(Ŷ j |Y S X S V ) + I(X S ; V X k X S c Y k ) − H(Ŷ S |V XX k X T kŶ S c Y k ) (185) = |S| j=1 H(Ŷ o(j) |Y S X S V ) + I(X S ; V X k X S c Y k ) − H(Ŷ S |V XX k X T kŶ S c Y k ) (186) = |S| j=1 H(Ŷ o(j) |Ŷ o(1) . . .Ŷ o(j−1) Y S X S V ) + I(X S ; V X k X S c Y k ) −H(Ŷ S |V XX k X T kŶ S c Y k ) (187) = H(Ŷ S |Y S X S V ) + I(X S ; V X k X S c Y k ) − H(Ŷ S |V XX k X T kŶ S c Y k ) ,(188)
where o : [1 : |S|] −→ S is an arbitrary ordering over S. This manipulation provides us June 3, 2014 DRAFT the next condition that must be satisfied:
1 < I(X S ; Y k |V X k X S c ) − I(Ŷ S ; Y S |V XX k X T kŶ S c Y k ) .(189)
However, given the fact that T k ∈ Υ k (N ), inequality (189) holds for every subset S ⊆ T k .
Finally, the probability Pr(E w,S ) can be bounded by following the very same steps as before and thus Pr(E w,S ) goes to zero as n → ∞ provided that
R + j∈S∩T CF k I(Ŷ j ; Y j |X j ) + j∈S∩T DF k I(Ŷ j ; Y j |X j V ) + 2 < j∈S∩T CF k H(Ŷ j |X j ) + j∈S∩T CF k H(X j ) + j∈S∩T DF k H(Ŷ j |X j V ) + j∈S∩T DF k H(X j |V ) + H(X|V ) + H(V X k Y k X T kŶ S c ) + H(V X k Y k X S c ) −H(V XY k X k X T kŶ T k ) − H(V X k Y k X T k ) . (190)
From which we obtain the last condition:
R + 2 < I(X;Ŷ S c Y k |V X k X T k ) + I(X S ; Y k |V X k X S c ) −I(Ŷ S ; Y S |V XX k X T kŶ S c Y k ) (191) = I(X;Ŷ T k Y k |V X k X T k ) + I(X S ; Y k |V X k X S c ) −I(Ŷ S ; Y S |V X k X T kŶ S c Y k ) .(192)
(ii) Decoding at the destination is done backwardly. After the last block, the decoder jointly searches for the unique indices l k(B+2) k∈T such that for all b = [B + 3 : B + L] the following condition holds:
x k (l k(B+2) ) k∈T CF , x k (1,l k(B+2) ) k∈T DF , x(1, 1), v(1), y(b) ∈ A n [V XX T Y ] . (193)
The probability of error goes to zero as n goes to infinity provided that
k∈S∩T CF I(Ŷ k ; Y k |X k ) + k∈S∩T DF I(Ŷ k ; Y k |X k V ) + 2 ≤ (L − 2)I(X S ; V XX S c Y ) ,(194)
for all subsets S ⊆ T .
(iii) After finding the correct index l k(B+2) , for each k ∈ T , and using the fact that w (B+1) = 1, the destination decodes jointly the message and all compression indices (w b , l T (b+1) ), for each block b = [1 : B], where we define l T b (l kb ) k∈T . Decoding is performed backwardly with the assumption that (w b+2 , l T (b+2) ) have been correctly decoded. Let us define the following sequence:
E(ŵ b ,l T (b+1) ) x(ŵ b , w (b+2) ), v(ŵ b ), y(b + 2), x k (l k(b+1) ),ŷ k (l k(b+1) , l k(b+2) ) k∈T CF , x k (ŵ b ,l k(b+1) ),ŷ k (ŵ b ,l k(b+1) , l k(b+2) ) k∈T DF .(195)
The destination finds the unique pair of indices (ŵ b ,l T (b+1) ) such that
E(ŵ b ,l T (b+1) ) ∈ A n [V XX TŶT Y ] .(196)
For every S ⊆ T and S c T − S, we consider the error events associated with this step which are given by
E 0 : E(w b , l T (b+1) ) / ∈ A n [V XX TŶT Y ],(197)E S : E(w b ,l T (b+1) ) ∈ A n [V XX TŶT Y ] for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,(198)E w,S : E(ŵ b ,l T (b+1) ) ∈ A n [V XX TŶT Y ] for someŵ b = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,(199)
where E S denotes the event that there exist joint typical sequences for relays in S with correct message index but with wrong compression indices while E w,S denotes the event that there exist joint typical sequences for relays in S with both wrong, message and compression indices. Hence, the error probability of this step is bounded by
Pr (ŵ b ,l T b ) = (w b , l T b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] .(200)
The first probability, on the right-hand side goes to zero as n → ∞. On the other hand,
Pr(E S ) goes to zero as n → ∞ provided that
j∈S∩T CF I(Ŷ j ; Y j |X j ) + j∈S∩T DF I(Ŷ j ; Y j |X j V ) + 3 < j∈S∩T CF H(Ŷ j X j ) + j∈S∩T DF H(Ŷ j X j |V ) + H(V XX S cŶ S c Y ) − H(V XX TŶT Y ) ,(201)
which also reads as:
3 < j∈S∩T CF H(Ŷ j |Y j X j ) + j∈S∩T DF H(Ŷ j |Y j X j V ) + H(XX S cŶ S c Y |V ) − H(XX S cŶ T Y |X S V ) .(202)
Indeed, inequality (202) can be further simplified by using the same method as before:
j∈S∩T CF H(Ŷ j |Y j X j ) + j∈S∩T DF H(Ŷ j |Y j X j V ) + H(XX S cŶ S c Y |V ) − H(XX S cŶ T Y |X S V ) = H(Ŷ S |Y S X S V ) + H(XX S cŶ S c Y |V ) − H(XX S cŶ T Y |X S V ) .(203)
This manipulation yields the following expression:
3 < H(Ŷ S |Y S X S V ) + H(XX S cŶ S c Y |V ) − H(XX S cŶ T Y |X S V ) = H(Ŷ S |Y S X S V ) − H(Ŷ S |XX S cŶ S c Y X S V ) + H(XX S cŶ S c Y |V ) − H(XX S cŶ S c Y |X S V ) = I(X S ; XX S cŶ S c Y |V ) − I(Ŷ S ; Y S |V XX TŶS c Y ) = I(X S ;Ŷ S c Y |XX S c V ) − I(Ŷ S ; Y S |V XX TŶS c Y ) .(204)
Given the fact that T ∈ Υ (N ), inequality (204) holds for each S ⊆ T . Finally, Pr(E w,S ) tends to zero as n goes to infinity provided that:
R + j∈S∩T CF I(Ŷ j ; Y j |X j )+ j∈S∩T DF I(Ŷ j ; Y j |X j V ) + 4 < j∈S∩T CF H(Ŷ j X j ) + j∈T DF H(Ŷ j X j |V ) + H(V X) + H(X T CF ∩S cŶ T CF ∩S c Y ) − H(V XX TŶT Y ) . (205)
It is worth mentioning here that the right-hand side of (205) is independent of S ∩ T DF and thus, if we take the set S such that S ∩ T DF = T DF , then (205) implies similar inequalities for all other S with S ∩ T DF ⊂ T DF . In fact, we continue the proof based on this choice that leads to
R + j∈S∩T CF I(Ŷ j ; Y j |X j ) + j∈T DF I(Ŷ j ; Y j |X j V ) + 4 < j∈S∩T CF H(Ŷ j X j ) + j∈T DF H(Ŷ j X j |V ) + H(V X) + H(X T CF ∩S cŶ T CF ∩S c Y ) − H(V XX TŶT Y ) ,(206)
and thus
R + 4 < j∈S∩T CF H(Ŷ j |Y j X j ) + j∈T DF H(Ŷ j |Y j X j V ) + H(X|V ) + H(X T CF ∩S cŶ T CF ∩S c Y ) −H(XX S c ∩T CFŶ T Y |X S∪T DF V ) (207) = H(Ŷ S∪T DF |Y S∪T DF X S∪T DF V ) + H(X|V ) + H(X T CF ∩S cŶ T CF ∩S c Y ) −H(XX S c ∩T CFŶ T Y |X S∪T DF V ) (208) = H(Ŷ S∪T DF |Y S∪T DF X S∪T DF V ) − H(Ŷ S∪T DF |X S∪T DF V XX S c ∩T CFŶ T CF ∩S c Y ) +H(X T CF ∩S cŶ T CF ∩S c Y ) − H(X T CF ∩S cŶ T CF ∩S c Y |XX S∪T DF V ) (209) = I(V XX S∪T DF ;Ŷ T CF ∩S c Y |X T CF ∩S c ) −I(Ŷ S∪T DF ; Y S∪T DF |X S∪T DF V XX S c ∩T CFŶ T CF ∩S c Y ) (210) = I(XX S∪T DF ;Ŷ T CF ∩S c Y |X T CF ∩S c ) − I(Ŷ S∪T DF ; Y S∪T DF |XX TŶT CF ∩S c Y ) (211) = I(XX S ;Ŷ S c Y |X S c ) − I(Ŷ S ; Y S |XX TŶS c Y ) ,(212)
where step (211) comes from the fact S has been selected satisfying T DF ⊆ S. Let p be an arbitrary probability distribution satisfying the conditions in expression (26), and let two sets V and T maximizing the right-hand side of (26). Let M n be a set of messages of size 2 nR with an index W to be transmitted. Transmission is done in B + L blocks, each of them of length n, and decoding at the destination is done backwardly. At the last L − 1 blocks, the last compression index is first decoded and then all compression indices and transmitted messages are jointly decoded. By the vector notation x S we denote the collection (x i ) i∈S .
June 3, 2014 DRAFT
Code generation:
(i) Randomly and independently generate 2 nR sequences x V c drawn i.i.d. from
P n X V c (x V c ) = n j=1 P X V c x V c j .(213)
Index them as x V c (w 0 ) with index w 0 ∈ 1, 2 nR . This step will provide |V c | different codebooks x k (w 0 ), w 0 ∈ [1, 2 nR ] for each k ∈ V c , every having 2 nR codewords. However, the codewords in each codebook corresponding to an index are jointly generated based on P n X V c and in general are not independent. (ii) For each x V c (w 0 ), randomly and conditionally independently generate 2 nR sequences x
drawn i.i.d. from P n X|X V c (x|x V c (w 0 )) = n j=1 P X|X V c x j |x V c j (w 0 ) .(214)
Index them as x(w 0 , w) with w ∈ 1, 2 nR .
(iii) For each k ∈ T , randomly and independently generate 2 nR k sequences x k drawn i.i.d. from
P n X k (x k ) = n j=1 P X k x kj .(215)
Index them as x k (l 0k ) with l 0k ∈ 1, 2 nR k forR k I(Y k ;Ŷ k |X k ) + .
(iv) For each k ∈ T and each x k (l 0k ), randomly and conditionally independently generate 2 nR k sequencesŷ k each with probability
P n Y k |X k (ŷ k |x k (l 0k )) = n j=1 PŶ k |X k ŷ kj |x kj (l 0k ) .(216)
Index them asŷ k (l 0k , l k ) with l k ∈ 1, 2 nR k .
(v) Provide the corresponding codebooks to the relays, the encoder and the decoder ends.
Encoding: (iii) For each i = [1 : B + 1], each k ∈ T , relay k after receiving y k (i), searches for at least one index l ki with l k0 = 1 such that
x k (l k(i−1) ), y k (i),ŷ k (l k(i−1) , l ki ) ∈ A n [X k Y kŶk ] .(217)
The probability of finding such l ki goes to one as n goes to infinity provided by our choice of the rateR k . Relay k searches for the unique indexŵ i ∈ M n such that:
x w (i−1) ,ŵ i , x V c w (i−1) , y k (i) ∈ A n [XX V c Y 1 ] .(218)
By following similar arguments to those in [2], the probability of error goes to zero as n goes to infinity provided that:
R < I(X; Y k |X V c ) .(219)
(ii) Decoding at destination is done backwardly. First, the destination decodes all last compression indices sent by the relays in T then it waits until the last block to jointly search for unique indices l k(B+1) k∈T such that for all b = [B + 2 : B + L] the following condition holds:
x k (l k(B+1) ) k∈T , x(1, 1), x V c (1), y(b) ∈ A n [XX T X V c Y ] .(220)
Let us define the following events that can cause an error in the previous decoding step:
E 0 : x k (l k(B+1) ) k∈T , x(1, 1), x V c (1), y(b) / ∈ A n [XX T X V c Y ] ,(221)E S : x k (l k(B+1) ) k∈S , x k (l k(B+1) ) k∈S c , x(1, 1), x V c (1), y(b) ∈ A n [XX T ∪V c Y ]
for somel k(B+1) = l k(B+1) , and all b = [B + 2 :
B + L] .(222)
The last event is the event that there exist joint typical sequences that have correct indices for the relays in S c = T − S and wrong indices for the relays in S. The probability of error is bounded as follows
Pr l B+1 = l B+1 ≤ Pr(E 0 ) + S⊆T Pr(E S ) .(223)
The first probability on the right-hand side goes to zero as n → ∞, and the second probability can be bounded as follows:
Pr(E S ) ≤ l k(B+1) =l k(B+1) , k∈S Pr b=[B+2:B+L]
x k (l k(B+1) ) k∈S , x k (l k(B+1) ) k∈S c ,
x(1, 1), x V c (1), y(b) ∈ A n [XX T X V c Y ] (224) ≤ k∈S 2 nR k − 1 2 −n(I(X S ;XX S c ∪V c Y )− 1 ) L−1 .(225)
This probability goes to zero as n goes to infinity provided that for all S ⊆ T :
k∈S I(Ŷ k ; Y k |X k ) + 2 ≤ (L − 1)I(X S ; XX S c ∪V c Y ) .(226)
(iii) After finding correct index l k(B+1) for all k ∈ T and since w (B+1) = 1, the destination decodes jointly the message and all the compression indices (w b , l T b ) for each b = [1 :
B] where l T b = (l kb ) k∈T .
Decoding is performed backwardly with the assumption that (w b+1 , l T (b+1) ) have been correctly decoded. Define the following event:
E(ŵ b ,l T b ) x(ŵ b , w (b+1) ), x V c (ŵ b ), y(b + 1), x k (l kb ),ŷ k (l kb , l k(b+1) ) k∈T .(227)
The destination finds the unique pair of indices (ŵ b ,l T b ) such that
E(ŵ b ,l T b ) ∈ A n [XX T ∪V cŶ T Y ] .(228)
Consider the following error events associated with this step (S ⊆ T , S c = T − S):
E 0 : E(w b , l T b ) / ∈ A n [XX T ∪V cŶ T Y ],(229)E S : E(w b ,l T b ) ∈ A n [XX T ∪V cŶ T Y ] for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,(230)E w,S : E(ŵ b ,l T b ) ∈ A n [XX T ∪V cŶ T Y ] for someŵ = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c .(231)
The event E S represents the event that there exist jointly typical sequences with correct message index but wrong compression indices for the relays in S. On the other hand E w,S is the event that there is jointly typical codes with wrong message index and wrong compression indices for the relays in S. The error probability of this step is bounded by
Pr (ŵ b ,l T b ) = (w b , l T b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] .(232)
The first term on the right-hand side goes to zero as n → ∞ and Pr(E S ) goes to zero as
n → ∞ provided that k∈S I(Ŷ k ; Y k |X k ) + 3 < k∈S H(Ŷ k |X k ) + H(XX V c ∪S cŶ S c Y ) − H(XX V c ∪S cŶ T Y |X S ) ,(233)
which can be written as:
3 < k∈S H(Ŷ k |Y k X k ) + H(XX V c ∪S cŶ S c Y ) − H(XX V c ∪S cŶ T Y |X S ) .(234)
The preceding inequality can be simplified by using the fact thatŶ k is independent of the other random variables given (X k , Y k ). The standard manipulation introduced before gives us the following:
3 < I(XX V c ∪S cŶ S c Y ; X S ) − I(Ŷ S ; Y S |XX V c ∪TŶS c Y ) = I(Ŷ S c Y ; X S |XX V c ∪S c ) − I(Ŷ S ; Y S |XX V c ∪TŶS c Y ) .(235)
Given the fact that T ∈ Υ (V), the last inequality holds for each S ⊆ T . As the next step, we bound the probability Pr(E w,S ) as follows:
Pr(E w,S ) ≤ l kb =l kb ,ŵ =w b Pr x(ŵ b , w (b+1) ), x V c (ŵ b ), y(b + 1), x k (l kb ),ŷ k (l kb , l k(b+1) ) k∈S , x k (l kb ),ŷ k (l kb , l k(b+1) ) k∈S c ∈ A n [XX T ∪V cŶ T Y ](236)≤ 2 nR − 1 k∈S 2 nR k − 1 2 n∆ 3 ,(237)
where
∆ 3 H(XX V c ∪S c YŶ T |X S ) − H(XX V c ) − H(YŶ S c X S c ) − k∈S H(Ŷ k |X k ) + 4 ) .(238)
From the following inequality, the last probability also tends to zero as n → ∞,
R + k∈S I(Ŷ k ; Y k |X k )) + 5 ≤ −H(XX V c ∪S c YŶ T |X S ) + H(XX V c ) + H(YŶ S c X S c ) + k∈S H(Ŷ k |X k ) .(239)
Inequality (239) is then simplified and reads as
R + 5 ≤ −H(XX V c ∪S c YŶ T |X S ) + H(XX V c ) +H(YŶ S c X S c ) + k∈S H(Ŷ k |Y k X k ) (240) ≤ −H(XX V c ∪S c YŶ T |X S ) + H(XX V c ) +H(YŶ S c X S c ) + H(Ŷ S |Y S X S ) (241) ≤ I(XX V c X S ; YŶ S c |X S c ) − H(Ŷ S |XX V c ∪T YŶ S c ) + H(Ŷ S |Y S X S ) (242) ≤ I(XX V c X S ; YŶ S c |X S c ) − I(Ŷ S ; Y S |XX V c ∪T YŶ S c ) .(243)b = 1 b = 2 b = 3 b = B + 2 b = B + 3 b = B + L + 1 v 1 (1) v 1 (1) v 1 (1) v 1 (wB−1) v 1 (wB) v 1 (1) x 1 (1) x 1 (1) x 1 (1) x 1 (wB−1) x 1 (wB) x 1 (1) v 2 (1, 1) v 2 (1, 1) v 2 (1, w1) v 2 (wB−1, wB) v 2 (wB, 1) v 2 (1, 1) x 2 (1, 1) x 2 (1, 1) x 2 (1, w1) x 2 (wB−1, wB) x 2 (wB, 1) x 2 (1, 1) x(1, 1, w1) x(1, 1, w2) x(1, w1, w2) x(wB−1, wB, 1) x(wB, 1, 1) x(1, 1, 1) x 3 (1) x 3 (l31) x 3 (l32) x 3 (l 3(B+1) ) x 3 (l 3(B+2) ) x 3 (l 3(B+3) ) y 3 (1, l31)ŷ 3 (l31, l32)ŷ 3 (l32, l33)ŷ 3 (l 3(B+1) , l 3(B+2) )ŷ 3 (l 3(B+2) , l 3(B+3) ) y 1 (1) y 1 (2) y 1 (3) y 1 (B + 2) y 1 (B + 3) y 1 (B + L + 1) y 2 (1) y 2 (2) y 2 (3) y 2 (B + 2) y 2 (B + 3) y 2 (B + L + 1) y 3 (1) y 3 (2) y 3 (3) y 3 (B + 2) y 3 (B + 3) y 3 (B + L + 1) y(1) y(2) y(3) y(B + 2) y(B + 3) y(B + L)
of hops. The relays in the second layer starts to decode at the end of block B + 1 and those in the first layer at the end of block B + 2.
At the of block B + 3, the first relay sends w 1 while the second relay transmits both messages (w 1 , w 2 ). When the number of hops is larger than two, Table III shows the order of message transmission across the different hops. Briefly, the DF relays with higher layers start to decode sooner and transmit more messages in each transmission round.
Layers Block b = T + 2 Block b L1 x 1 (w1) x 1 (w b−T −1 ) L2 x 2 (w1, w2) x 2 (w b−T −1 , w b−T ) . . . . . . . . . Lt x t (w1, . . . , wt) x t (w b−T −1 , . . . , w b−T −2+t ) . . . . . . . . . LT x T (w1, . . . , wT ) x T (w b−T −1 , . . . , w b−2 ) Source x(w1, . . . , wT , wT +2) x(w b−T −1 , . . . , w b−2 , w b ) June 3, 2014 DRAFT
Code generation:
(i) Randomly and independently generate 2 nR sequences v 1 drawn i.i.d. from
P n V 1 (v 1 ) = n j=1 P V 1 (v 1j ) .(244)
Index them as v(w 1 ) with index w 1 ∈ 1, 2 nR .
(ii) For all hops t = [2 : T ], randomly and independently generate 2 nR sequences v t based on
v 1 (w 1 ), . . . v t−1 (w 1 , . . . , w t−1 ) i.i.d. from P n Vt|V ≤t−1 v t |v 1 (w 1 ), . . . v t−1 (w 1 , . . . , w t−1 ) = n j=1 P Vt|V ≤t−1 v tj |v 1j (w 1 ), . . . v (t−1)j (w 1 , . . . , w t−1 ) .(245)
Index them as v t (w 1 , . . . , w t ) with index w t ∈ 1, 2 nR .
(iii) For each k ∈ L t and each tuple of codewords v 1 (w 1 ), . . . , v t (w 1 , . . . , w t ) , randomly and independently generate x k drawn i.i.d. from
P n X k |V ≤t x k |v 1 (w 1 ), . . . , v t (w 1 , . . . , w t ) = n j=1 P X k |V ≤t x kj |v 1j (w 1 ), . . . , v tj (w 1 , . . . , w t ) .(246)
This is the codebook for DF relays in L t .
(iv) For each k ∈ V, randomly and independently generate 2 nR k sequences x k drawn i.i.d. from
P n X k (x k ) = n j=1 P X k (x kj ) .(247)
Index them as x k (l 0k ), where l 0k ∈ 1, 2 nR k forR k I(Y k ;Ŷ k |X k ) + with k ∈ V. This is the codebook for CF relays.
(v) For each tuple of codewords v 1 (w 1 ), . . . , v T (w 1 , . . . , w T ) , randomly and conditionally independently generate 2 nR sequences x drawn i.i.d. from
P n X|V ≤T x|v 1 (w 1 ), . . . , v T (w 1 , . . . , w T ) = n j=1 P X|V ≤T x j |v 1j (w 1 ), . . . , v T j (w 1 , . . . , w T ) .(248)
Index them as x(w 1 , . . . , w T , w), where w ∈ 1, 2 nR . This is the source codebook.
(vi) For each k ∈ V and each x k (l 0k ), randomly and conditionally independently generate 2 nR k sequencesŷ k each with probability P n Y k |X k (ŷ k |x k (l 0k )) = n j=1 PŶ k |X k (ŷ kj |x kj (l 0k )) .
Index them asŷ k (l 0k , l k ), where l k ∈ 1, 2 nR k with k ∈ V. This is the cookbook for the descriptions sent by CF relays.
(vii) Provide the corresponding codebooks to all relays, the encoder and the decoder ends.
Encoding: (iii) For i = [1 : B + T + 1], and each k ∈ V, the k-th relay -after receiving y k (i)searches for at least one index l ki with l k0 = 1 such that
x k (l k(i−1) ), y k (i),ŷ k (l k(i−1) , l ki ) ∈ A n [X k Y kŶk ] .(250)
The probability of finding such l ki goes to one as n goes to infinity due to our adequate choice of the rateR k for compression. The k-th relay knows the tuple of message w (i−T −2) , . . . , w (i−1) , and so it knows all codewords indexed by them. We next define the sequences:
E k ŵ b ,l T k b , b x(w (b−T −1) , . . . , w (b−2) ,ŵ b ), x i (l i(b−1) ),ŷ i (l i(b−1) ,l ib ) i∈T k , y k (b) v t (w (b−T −1) , . . . , w (b−T −2+t) ), x Lt (w (b−T −1) , . . . , w (b−T −2+t) ) t∈[1:T ] , E k l T k b , b + 1 v t (w (b−T ) , . . . , w (b−T −1+t) ), x Lt (w (b−T ) , . . . , w (b−T −1+t) ) t∈[1:T ] , y k (b + 1), x i (l ib ) i∈T k .
Therefore, the k-th relay, by looking at two consecutive blocks (b, b + 1), searches for the unique indices (ŵ b ,l T k b ) such that:
E k (ŵ b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ] .(251)
We define the following error events:
E 0 : E k (w b , l T k b , b) / ∈ A n [V ≤T XX V c X T kŶ T k Y k ] or E k (l T k b , b + 1) / ∈ A n [V ≤T X V c X T k Y k ] , E S : E k (w b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ]
for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,
E w,S : E k (ŵ b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ] for someŵ b = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c .(252)
The probability of error for these relays can be bounded as follows:
Pr (ŵ b ,l T k b ) = (w b , l T k b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] .(253)
The probability Pr(E 0 ) goes to zero as n → ∞ given the code generation and the encoding process. We bound the other probabilities using the same technique as we already did in Appendix A. First of all, the probability Pr(E S ) is bounded as:
Pr(E S ) ≤ k∈S 2 nR k − 1 2 n(∆ 1 +∆ 2 ) ,(254)
where
∆ 1 H(V ≤T XX V c X T kŶ T k Y k ) − j∈S H(Ŷ j |X j ) − H(V ≤T XX V c X T kŶ S c Y k ) ,(255)∆ 2 H(V ≤T X V c X T k Y k ) − j∈S H(X j )−H(V ≤T X V c X S c Y k ) + 1 .(256)
The probability Pr(E S ) goes to zero as n goes to infinity if the exponent of the right hand side is also negative which yields:
j∈S I(Ŷ j ; Y j |X j ) + 1 < j∈S H(Ŷ j |X j ) + j∈S H(X j ) + H(V ≤T XX V c X T kŶ S c Y k ) + H(V ≤T X V c X S c Y k ) − H(V ≤T XX V c X T kŶ T k Y k ) − H(V ≤T X V c X T k Y k ) .(257)
After simplification of both sides, in order to guarantee that the probability Pr(E S ) is arbitrarily small, we need to add the next constraint:
1 < I(X S ; Y k |V ≤T X V c ∪S c ) − I(Ŷ S ; Y S |V ≤T XX V c X T kŶ S c Y k ) .(258)
However, given the fact that T k ∈ Υ k (N ), inequality (258) holds for every subset S ⊆ T k .
As the next step, the probability Pr(E w,S ) can be bounded by following the very same steps as before and thus Pr(E w,S ) goes to zero as n → ∞ provided that
R + j∈S I(Ŷ j ; Y j |X j ) + 2 < j∈S H(Ŷ j |X j ) + j∈S H(X j ) + H(X|V ≤T X V c ) + H(V ≤T X V c X T kŶ S c Y k ) + H(V ≤T X V c X S c Y k ) − H(V ≤T XX V c X T kŶ T k Y k ) − H(V ≤T X V c X T k Y k ) . (259)
From this, we obtain the following condition:
R + 2 < I(X;Ŷ S c Y k |V ≤T X V c X T k ) + I(X S ; Y k |V ≤T X V c ∪S c ) −I(Ŷ S ; Y S |V ≤T XX V c X T kŶ S c Y k ) (260) = I(X;Ŷ T k Y k |V ≤T X V c X T k ) + I(X S ; Y k |V ≤T X V c ∪S c ) −I(Ŷ S ; Y S |V ≤T X V c X T kŶ S c Y k ) .(261)
(ii) Decoding at DF relays in L t is done as follows. As we already mentioned, these relays start to decode the fresh source message after all relays have decoded the same messages corresponding to the higher layers, namely {L t+1 , . . . , L T }. Therefore, those relays in higher layers act as relay for the relays in L t . Whereas the lower layers cannot help the relays in L t , having decoded only the previous source messages. Indeed, they act as side information in the decoding process.
After the transmission of the block i + 1 + T − t ∈ {2 + T − t, . . . , B + 1 + T − t} and for each k ∈ L t , the k-th relay decodes the message w i and the compression index l T k i , i.e., the compression indices for block i of all relays in T k , with the assumption that all messages and compression indices up to block i − 1 have been correctly decoded. To this end, this relay uses the blocks [b : b + t]. Let us define the sequences:
E k ŵ b ,l T k b , b x(w (b−T −1) , . . . , w (b−2) ,ŵ b ), x i (l i(b−1) ),ŷ i (l i(b−1) ,l ib ) i∈T k , y k (b) v t (w (b−T −1) , . . . , w (b−T −2+t) ), x Lt (w (b−T −1) , . . . , w (b−T −2+t) ) t∈[1:T ](262)E k l T k b , b + 1 v t (w (b−T ) , . . . , w (b−T −1+t) ), x Lt (w (b−T ) , . . . , w (b−T −1+t) ) t∈[1:T ] , y k (b + 1), x i (l ib ) i∈T k (263) E k (ŵ b , b + 2) v t (w (b−T +1) , . . . , w (b−T +t) ), x Lt (w (b−T +1) , . . . , w (b−T +t) ) t∈[1:T −1] , v T (w (b−T +1) , . . . ,ŵ b ), x L T (w (b−T +1) , . . . ,ŵ b ) , y k (b + 2) (264) E k (ŵ b , b + j) v t (w (b−T +j−1) , . . . , w (b−T −2+j+t) ), x Lt (w (b−T +j−1) , . . . , w (b−T −2+j+t) ) t∈[1:T −j+1] , v T −j+2 (w (b−T +j−1) , . . . ,ŵ b ), x L T −j+2 (w (b−T +j−1) , . . . ,ŵ b ) , y k (b + j) (265) E k (ŵ b , b + 1 + T − t) v t (w (b−t) , . . . , w (b−1) ), x Lt (w (b−t) , . . . , w (b−1) ) t∈[1:t] , v t+1 (w (b−t) , . . . ,ŵ b ), x L t+1 (w (b−t) , . . . ,ŵ b ) , y k (b + 1 + T − t) .(266)
In this case, by looking at t + 1 consecutive blocks (b, . . . , b + t), the k-th relay finds the unique pair of indices (ŵ b ,l T k b ) such that:
E k (ŵ b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ] .(267)E k (ŵ b , b + j) ∈ A n [V ≤T −j+2 X L ≤T −j+2 Y k ] for j = [2 : T − t + 1] .(268)
We define the following error events:
E 0 : E k (w b , l T k b , b) / ∈ A n [V ≤T XX V c X T kŶ T k Y k ] or E k (l T k b , b + 1) / ∈ A n [V ≤T X V c X T k Y k ] or E k (w b , b + j) / ∈ A n [V ≤T −j+2 X L ≤T −j+2 Y k ] for j = [2 : T − t + 1] .(269)E S : E k (w b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ]
for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,
E w,S : E k (ŵ b ,l T k b , b) ∈ A n [V ≤T XX V c X T kŶ T k Y k ] and E k (l T k b , b + 1) ∈ A n [V ≤T X V c X T k Y k ] and E k (ŵ b , b + j) ∈ A n [V ≤T −j+2 X L ≤T −j+2 Y k ] for j = [2 : T − t + 1] . for someŵ b = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c .(270)
Hence, the probability of error can be bounded as follows:
Pr (ŵ b ,l T k b ) = (w b , l T k b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] .(272)
It is easy to check that the probability Pr(E 0 ) goes to zero as n → ∞. As the next step, the probability Pr(E S ) goes to zero with the exact same condition as (258), namely:
1 < I(X S ; Y k |V ≤T X V c ∪S c ) − I(Ŷ S ; Y S |V ≤T XX V c X T kŶ S c Y k ) .(273)
The probability Pr(E w,S ) can be bounded by following the very same steps as before and thus Pr(E w,S ) goes to zero as n → ∞ provided that
R + j∈S I(Ŷ j ; Y j |X j ) + 2 < j∈S H(Ŷ j |X j ) + j∈S H(X j ) + H(X|V ≤T X V c ) + H(V ≤T X V c X T kŶ S c Y k ) + H(V ≤T X V c X S c Y k ) − H(V ≤T XX V c X T kŶ T k Y k ) − H(V ≤T X V c X T k Y k ) + T −t+1 j=2 I(V T −j+2 X L T −j+2 ; Y k |V ≤T −j+1 X L ≤T −j+1 ) ,(274)
which is simplified to its equivalent condition given by
R + 2 < I(X;Ŷ S c Y k |V ≤T X V c X T k ) + I(X S ; Y k |V ≤T X V c X S c ) I(V >t X L>t ; Y k |V ≤t X L ≤t ) − I(Ŷ S ; Y S |V ≤T XX V c X T kŶ S c Y k ) = I(X;Ŷ T k Y k |V ≤T X V c X T k ) + I(X S ; Y k |V ≤T X V c X S c ) I(V >t X L>t ; Y k |V ≤t X L ≤t ) − I(Ŷ S ; Y S |V ≤T X V c X T kŶ S c Y k ) (275) = I(XV >t X L>t ; Y k |V ≤t X L ≤t ) + I(X S ; Y k |V ≤T X V c X S c ) −I(Ŷ T k ; Y S |V ≤T X V c X T k Y k ) .(276)
(iii) Decoding at the destination is done backwardly and it first starts to decode the last compression index, namely the decoder jointly searches for the indices l k(B+T +1) k∈T such that for all b = [B + T + 2 : B + T + L − 1] the following condition holds:
x k (l k(B+T +1) k∈T , v t (1, . . . , 1), x Lt (1, . . . , 1) t∈[1:T ] , x(1, . . . , 1), y(b) ∈ A n [V ≤T X V c X T Y ] . (277)
The probability of error is calculated similarly to previous theorems and it goes to zero as n goes to infinity provided by
k∈S∩T I(Ŷ k ; Y k |X k ) + 2 ≤ (L − 2)I(X S ; V ≤T XX V c ∪S c Y ) ,(278)
for all subsets S ⊆ T .
(iv) After finding the correct index l k(B+T +1) , for each k ∈ T , the destination starts decoding from the block B + T + 1 backward. It decodes jointly the message and all compression indices (w b , l T (b+T ) ), for each block b = [1 : B]. The message w b is decoded at the block b+T +1 and it is assumed that (w b+1 , . . . , w b+T +1 , l T (b+T +1) ) have been correctly decoded.
Let us define the following event:
E(ŵ b ,l T (b+1) ) x(ŵ b , . . . , w b+T −1 , w (b+T +1) ), x k (l k(b+1) ),ŷ k (l k(b+1) , l k(b+2) ) k∈T , v t (ŵ b , . . . , w b+t−1 ), x Lt (ŵ b , . . . , w b+t−1 ) t∈[1:T ] , y(b + T + 1) .(279)
The destination finds the unique pair of indices (ŵ b ,l T (b+1) ) such that
E(ŵ b ,l T (b+1) ) ∈ A n [V ≤T XX V c ∪TŶT Y ] .(280)
The error events associated with this step are characterized as follows (S ⊆ T ):
E 0 : E(w b , l T (b+1) ) / ∈ A n [V ≤T XX V c ∪TŶT Y ] ,(281)E S : E(w b ,l T (b+1) ) ∈ A n [V ≤T XX V c ∪TŶT Y ] for somel kb = l kb , k ∈ S andl kb = l kb , k ∈ S c ,(282)E w,S : E(ŵ b ,l T (b+1) ) ∈ A n [V ≤T XX V c ∪TŶT Y ] for someŵ b = w b ,l kb = l kb , k ∈ S andl kb = l kb , k ∈ S c .(283)
The error probability can be bounded as follows:
Pr (ŵ b ,l T b ) = (w b , l T b ) ≤ Pr(E 0 ) + S⊆T [Pr(E S ) + Pr(E w,S )] .(284)
These probabilities are bounded using the same technique in Appendix A and therefore we omit the details here. The probability Pr(E 0 ) goes to zero as n → ∞. On the other hand,
Pr(E S ) goes to zero as n → ∞ provided that:
k∈S I(Ŷ k ; Y k |X k ) + 3 < k∈S H(Ŷ k X k ) + H(V ≤T XX V c ∪S cŶ S c Y ) − H(V ≤T XX V c ∪TŶT Y ) ,(285)
which is equivalent to the following inequality after standard manipulation: This manipulation yields the following expression:
3 < I(X S ;Ŷ S c Y |V ≤T XX V c ∪S c ) − I(Ŷ S ; Y S |V ≤T XX V c ∪TŶS c Y ) .(286)
By the choice T ∈ Υ (V), inequality (286) holds for each S ⊆ T . Now, the probability
Pr(E w,S ) tends to zero as n goes to infinity provided that:
R + k∈S I(Ŷ k ; Y k |X k ) + 4 < k∈S H(Ŷ k X k ) + H(V ≤T XX V c ) + H(X S cŶ S c Y ) − H(V ≤T XX V c ∪TŶT Y ) .(287)
We apply once again the standard simplification used in Appendix A from which we get:
R + 4 < I(XX V c ∪S ;Ŷ S c Y |X S c ) − I(Ŷ S ; Y S |XX V c ∪TŶS c Y ) .(288)
Notice that we can see that DF relays X V c appear in the mutual information part of the rate (288), which means that they contribute to the increase the final rate by their Consider now the composite relay channel with given parameters θ = (θ d , θ r ) and target rate r. Transmission takes place over B + L blocks of length n.
Code generation:
(i) The relay disposes of two different codebooks. First, randomly and independently generate
2 nr sequences x 1 drawn i.i.d. from P n X 1 (x 1 ) = n j=1 P X 1 (x 1j ) .(289)
Index them as x 1 (w 0 ) with index w 0 ∈ [1, 2 nr ]. This codebook must be given to the source so it cannot depend on the specific draw θ r . Then, for each θ r ∈ Θ r randomly and independently generate 2 nR θr sequences x 2 drawn i.i.d. from P n X 2 |θr (x 2 |θ r ) = n j=1 P X 2 |θr (x 2j |θ r )
Index them as x 2 (l 0 ), where l 0 ∈ 1, 2 nR θr forR θr I θr (Y 2 ;Ŷ 2 |X 2 ) + .
(ii) For each x 2 (l 0 ), randomly and conditionally independently generate 2 nR θr sequencesŷ 2 each with probability P n Y 2 |X 2 ;θr (ŷ 2 |x 2 (l 0 ); θ r ) = n j=1 PŶ 2 |X 2 ;θr (ŷ 2j |x 2j (l 0 ); θ r ) .
Index them asŷ 2 (l 0 , l), where l ∈ 1, 2 nR θr .
(iii) For each x 1 (w 0 ), randomly and conditionally independently generate 2 nr sequences x drawn i.i.d. from
P n X|X 1 (x|x 1 (w 0 )) = n j=1 P X|X 1 (x j |x 1j (w 0 )) .(292)
Index them as x(w 0 , w), where w ∈ [1, 2 nr ]. This is the source codeword independent of the specific draw of θ. Again note that x 1 (r 1 ), which is used to generate the source code, does not depend on θ.
(iv) Provide the codebooks to every node available except for the collection of codebooks {x 2 (l 0 )} that cannot be known to the source.
Encoding: (ii) The parameter θ r is available to the relay. If θ r ∈ D DF the relay sends the codeword x 1 from the first codebook and uses it for the rest of the communication. In other words, the relay function for this choice is DF scheme. In block i, the relay uses its decoder output w (i−1) (w 0 = 1) and sends the codeword x 1 ŵ (i−1) . Otherwise, if θ r / ∈ D DF , then the relay picks the codebook of codewords x 2 corresponding to θ r . The relay function in this case is CF scheme. After receiving the corresponding output, namely y 1 (i), the relay searches for at least one index l i , where l 0 = 1, such that
x 2 (l (i−1) ), y 1 (i),ŷ 2 (l (i−1) , l i ) ∈ A n [X 2 Y 1Ŷ2 |θ r ] ,(293)
where A n [X 2 Y 1Ŷ2 |θ r ] is the joint typical set indexed with θ r . The probability of finding such l i goes to one as n goes to infinity. We remark that the typical set used for such coding is known to the relay because it knows θ r . For i = [1 : B + 1], the relay knows from the previous block l (i−1) and it sends x 2 (l (i−1) ). Moreover, the relay repeats l B+1 for i = [B + 2 : B + L].
Decoding:
We assume that destination is equipped with an outage identification function I that is given as follows:
1) For every block i = [1 : B + L], the relay decodes w i exactly similar to Theorem 2. For a fix r and given θ r , if the next condition is satisfied the probability of error event will asymptotically tend to zero, i.e., r ≤ I θr (X; Y 1 |X 1 ) .
On the contrary, an error occurs when r > I θr (X; Y 1 |X 1 ) .
In this case, the relay will transmit a random messageŵ and the destination that knows θ will set I(θ) = 0.
2) The decoder knows the channel index θ and hence θ r . If θ r ∈ D DF and inequality (294) is satisfied, the destination decodes the message using the DF codebook. Thus, decoding of this would not be successful if
r > I θ (X, X 1 ; Y ) ,(296)
for which the destination sets I(θ) = 0 and declares an outage event. Therefore, if θ r ∈ D DF , an outage event is declared when r > I DF (θ) with I DF (θ) min I θr (X; Y 1 |X 1 ), I θ (X, X 1 ; Y ) .
Consider the step for the case θ r / ∈ D DF where the relay input is chosen from the second set of codebooks with distribution X 2 . As a matter of fact, for all draws θ yielding this case we have
X 1 X (Y, Y 1 , X 2 ) .(298)
Moreover, the decoder knows whether the next inequality is satisfied subject to the Markov chain (298)
I θ (X 2 ; Y |XX 1 ) ≥ I θ (Y 1 ;Ŷ 2 |Y XX 1 X 2 ) .(299)
The decoder applies the exact same decoding procedure as in Theorem 2. It can be seen that the decoding conditions at destination do not change even if X 1 is not really transmitted. The only change is in the Markov chains, it can be seen that the previous inequality corresponds to (106) for the composite relay channel. The destination declares an outage event and I(θ) = 0 for θ r / ∈ D DF if r > I CF (θ), where I CF (θ) max min I θ (XX 1 ; YŶ 2 |X 2 ), I θ (XX 1 X 2 ; Y ) −I θ (Ŷ 2 ; Y 1 |Y XX 1 X 2 ) , I θ (X; Y ) .
Using (297) and (300), the outage event denoted by the indicator function 1 E is as follows 1 E 1[θ r ∈ D DF and r > I DF (θ)] + 1[θ r / ∈ D DF and r > I CF (θ)] .
Taking the expected value from both sides lead to the outage probability. Indeed, the expected value is taken in two steps. For each θ r , the expected value is calculated using P θ|θr . The relay, for each θ r chooses the joint distribution of (X 2 ,Ŷ 2 ) to minimize:
At the next step, the expectation is taken over θ r and is minimized over D DF and p(x, x 1 ), yielding (r) = min p(x,x 1 ) inf D DF ⊆Θr E θr P θ|θr r > I DF , θ r ∈ D DF |θ r + min p(x 2 )p(ŷ 2 |x 2 ,y 1θr ) P θ|θr r > I CF , θ r / ∈ D DF |θ r .
Finally, a time sharing random variable Q can be added to the region, however the optimization should be done outside the expectation.
APPENDIX E OUTLINE OF THE PROOF OF PROPOSITION 5
Consider the composite unicast network with parameters θ = (θ d , θ r ). Transmission is done over B + L blocks. Suppose that every relay knows D V , i.e., the decision region of the other relays for each θ r .
Code generation:
1) Randomly and independently generate 2 nR sequences v drawn i.i.d. from
P n V (v) = n j=1 P V (v j ) .(304)
Index them as v(w 0 ) with index w 0 ∈ 1, 2 nR . This codebook must be given to the source so it cannot depend on the specific draw θ r .
2) Since all relays know θ r , for each θ r generate two sets of codebooks: a) Each codebook for θ r in the first set is generated as follows. For each v(w 0 ), randomly and conditionally independently generate 2 nR kθr sequences x k drawn i.i.d. from P n X k |V ;θr (x k |v(w 0 ), θ r ) = n j=1 P X k |V ;θr (x kj |v j (w 0 ), θ r ) .
Index them as x k (w 0 , l 0k ) with index l 0k ∈ 1, 2 nR kθr forR kθr I θr (Y k ;Ŷ k |X k V ) + .
b) For each v(w 0 ), x k (w 0 , l 0k ) and θ r , randomly and conditionally independently generate 2 nR kθr sequencesŷ k each with probability P n Y k |X k V ;θr (ŷ k |x k (w 0 , l 0k ), v(w 0 ), θ r ) = n j=1 PŶ k |X k V ;θr (ŷ kj |x kj (w 0 , l 0k ), v j (w 0 ), θ r ) .
Index them asŷ k (w 0 , l 0k , l k ), where l k ∈ 1, 2 nR kθr forR kθr I θr (Y k ;Ŷ k |X k V ) + . c) As for the second set of codebooks, for each θ r randomly and independently generate 2 nR kθr sequences x k drawn i.i.d. from P n X k |θr (x k |θ r ) = n j=1 P X k |θr (x kj |θ r ) .
Index them as x k (l 0k ), where l 0k ∈ 1, 2 nR kθr forR kθr I θr (Y k ;Ŷ k |X k ) + . d) For each x k (l 0k ), randomly and conditionally independently generate 2 nR k sequencesŷ k each with probability P n Y k |X k ;θr (ŷ k |x k (l 0k ), θ r ) = n j=1 PŶ k |X k ;θr (ŷ kj |x kj (l 0k ), θ r ) .
Index them asŷ k (l 0k , l k ), where l k ∈ 1, 2 nR kθr forR kθr I θr (Y k ;Ŷ k |X k ) + .
Note that the rateR kθr depends on the relay strategy, i.e., CF or DF scheme, and so the relay is superimposing the compression index over the DF code. Therefore, the rateR kθr varies according to each scenario.
3) For each v(w 0 ), randomly and conditionally independently generate 2 nR sequences x drawn i.i.d. from P n X|V (x|v(w 0 )) = n j=1 P X|V (x j |v j (w 0 )) .
Index them as x(w 0 , w), where w ∈ 1, 2 nR . 4) Provide the codebooks to every node available except for the collection of codebooks {x k (l 0k ), x k (w 0 , l 0k )} that cannot be known to the source. (ii) Since all relays know θ r , if θ r ∈ D (k) DF for every block i = [1 : B + L], the relay k knows w (i−2) by assumption and w 0 = 1.
Moreover, it searches in the codebook for θ r in its first set of codebooks for at least one index l ki with l k0 = 1 such that v(w (i−2) ), x k (w (i−2) , l k(i−1) ), y k (i),ŷ k (w (i−2) , l k(i−1) , l ki ) ∈ A n [V X k Y kŶk |θ r ] . (310)
The probability of finding such l ki goes to one as n goes to infinity by a choice of the rateR kθr . Relay k sends x k (w (i−2) , l k(i−1) ) in block i and it repeats l k(B+2) for all blocks i = [B + 3 : B + L].
If θ r / ∈ D (k) DF , the relay k after receiving y k (i), searches in the codebook indexed with θ r in its second set of codebooks for at least an index l ki with l k0 = 1 such that x k (l k(i−1) ), y k (i),ŷ k (l k(i−1) , l ki ) ∈ A n [X k Y kŶk |θ r ] .
The probability of finding such l ki goes to one as n goes to infinity. The relay k knows from the previous block l k(i−1) and it sends x k (l k(i−1) ). Moreover, relay k repeats l k(B+1)
for all blocks i ∈ [B + 2 : B + L].
Decoding:
If θ r ∈ D V , then relays k ∈ V use CF scheme while the others relays use DF scheme. It means that the relays with k ∈ V use the codebook for θ r in the second set of codebooks and the others relays use the codebook for θ r in the first set of codebooks. Now the outage indicator function can be considered as follows:
1 E V⊆N 1[r ≤ I MNNC (V, θ) , θ r ∈ D V ] .(312)
Therefore, if r is less or equal than I MNNC (V, θ) it can be achieved and the probability of error tends to zero. Hence, the outage probability is calculated easily from (312) by considering the optimization over all probability distributions and taking the expected value from both sides.
APPENDIX F OUTLINE OF THE PROOF OF PROPOSITION 7
Consider the composite unicast network with parameters θ = (θ d , θ r ). Transmission is done over B + L blocks. It is assumed that every relay knows D V , i.e., the decisions regions of all others relays.
Code generation:
(i) The relay k knows θ r and for each θ r it generates two codebooks x (1) k , x
Index them as x (1) N (r) with index r ∈ 1, 2 nR . Since this codebook must be also given to the source it cannot depend on θ r . b) Randomly and independently generate 2 nR kθr sequences x (2) k drawn i.i.d. from P n X (2) k |θr x (2) k |θ r = n j=1 P X (2) k |θr x (2) kj |θ r .
Index them as x (2) k (r k ), where r k ∈ 1, 2 nR kθr forR kθr I θr (Y k ;Ŷ k |X (2) k ) + . c) For each x (2) k (r k ), randomly and conditionally independently generate 2 nR kθr sequenceŝ y k each with probability P n Y k |X (2) k ;θr ŷ k |x (2) k (r k ), θ r = n j=1 PŶ k |X (2) k ;θr ŷ kj |x (2) kj (r k ), θ r .
Index them asŷ k (r k ,ŝ k ), whereŝ k ∈ 1, 2 nR kθr .
(ii) For each x
Index them as x(r, w), where w ∈ 1, 2 nR . Indeed, this codebook must be also given to the source so it cannot depend on θ r .
After finding the correct index l k(B+1) for all k ∈ T and since w (B+1) = 1, the destination decodes jointly the message and all compression indices (w b , l T b ), for each b = [1 : B],
where l T b = (l kb ) k∈T . Indeed, decoding is performed backwardly with the assumption that (w b+1 , l T (b+1) ) have been correctly decoded. The destination finds the unique pair of indices (ŵ b ,l T b ) such that
x(ŵ b , w (b+1) ), x V c (ŵ b ), y(b + 1), x k (l kb ),ŷ k (l kb , l k(b+1) ) k∈T ∈ A n [XX T ∪V cŶ T Y |θ] .
It can be seen from Theorem 2 that an error occurs if:
r > min S⊆T R T (S, θ) ,(322)
where R T (S, θ) I θ (XX
(1) V c X (2) S ;Ŷ S c Y |X (2) S c ) − I θ (Y S ;Ŷ S |XX (2) T X (1) NŶ S c Y ) .(323)
Note that T is chosen in such a way that the right-hand side achieves its maximum value.
From our previous discussion on expression (15), we know that this set belongs to Υ (V) and so Q T (A) ≥ 0 for each A ⊆ T .
3) Using (319) and (322), the outage indicator function can be defined as
As before, the expected value is taken in two steps. For each θ r , the expected error is calculated with P θ|θr . The relays chose the distribution j∈V p(x (2) j )p(ŷ j |x (2) j y j ) to minimize the conditional expectation for each θ r and D V . This will lead to the following: At the end, a time sharing random variable Q can be added to the region, however the optimization should be done outside the expectation.
E θ|θr [1 E ] V⊆N min j∈V p(x (2) j )p(ŷ j |x
Fig. 2 :
2Mixed Noisy Network Coding (MNNC). • A decoder mapping: φ : Y n −→ M n , • A sequence of relay functions: f
1 )
1Simultaneous use of different cooperative strategies among the relay nodes: Relay nodes are divided into two disjoint groups of relays denoted by (V, V c ) satisfying V ∪ V c = N . As it is shown in Fig. 2, relays in the group V c with nodes indices (j, k) are simultaneously employing partly DF and CF scheme as cooperation strategy while relays in the group V, denoted by index i, simply employ CF scheme. in the network is transmitting the compressed version of its observation and thus all DF relays take advantage of the compression indices of the other relays in the network, which clearly improves the decoding of source messages. In this setting, transmission takes place via block-Markov coding in B + L blocks each of length n, where DF relays for each block b ≤ B + 2 forward the source message of the (b − 2)-th block. We remark that this is slightly different from the conventional DF scheme [2], where the relay forwards the message of the previous block. Indeed, the compression index of the b-th block is transmitted only in block b + 1 and thus DF relays have to wait until the end of block b + 1 to decode the compression index as well as the message of block b. Hence, they can forward the message of block b only after block b + 1.
Fig. 3 :
3Non-cooperative Mixed Noisy Network Coding (MNNC) scheme.
X 0 X
0and Y N +1 Y . Thus, there is a bijection from the set of nodes to the set {0, 1, . . . , N, N + 1}. The transmitters' set is denoted by M {0, 1, . . . , N } and the receivers' set is denoted by D {1, . . . , N, N + 1}.
follows from the identity (57), and (62) follows by noting that Tr(S CF )I(S CF ) Σ(S CF ). For |S CF | = 0, (63) and (64) follow from Tr(S CF ) = P |S CF | and basic matrix operations. For |S CF | = 0, one can bound (62) directly by |S c |+1 2 log(4). By rewriting expression (64), it is not difficult to check that (64) implies (56) which concludes the proof. Proposition 2 (Capacity within a constant gap from MNNC): Provided source-to-relays channels allow decoding at all relay nodes in V c N − V for some set V ⊆ N , capacity can be stated within a constant gap from MNNC rate satisfying
However, if a considerable amount of relays are well positioned to perform DF scheme, then the constant gap can be strictly improved. Notice the interest behind (65) is to emphasize that MNNC can improve the gap to the capacity. The improvement of this gap is two fold. Firstly, the gap decreases logarithmically as the number of DF relays increases. Moreover, the expression of constant gap is optimized over all sets S such that V c ⊆ S ⊆ N . This leads to a reduced optimization space which eventually can lead to a better gap. Secondly, all this is conditioned by whether the DF relays affect the gap or not. For instance, if all relays are good enough then the encoder chooses V c = N and thus the optimization set is reduced to a single element N , and the constant gap becomes 0.5N + 0.7 which is strictly better than (52). More precisely, the gap can be bounded by 0.5N + C(k), where the value of C(k) is independent of N and k is the number of CF relays provided that N − k relays perform DF scheme without degrading the rate. Once again, it should be emphasized that the constant gap results are dependent on the assumption of good source-to-relay channel gains.
Fig. 4 :
4Composite Relay Channel.
Fig. 5 :
5Selective Coding Strategy (SCS) and Two-relay network.
Fig. 6 :
6Composite Cooperative Unicast Network. Proposition 4 (SCS with full CSI at relay): The average error probability of the composite relay channel with full CSI θ = (θ r , θ d ) at the relay can be upper bounded bȳ
Remark 5 :
5The outage and full error identification functions I and J can be equally used to bound the -capacity of averaged networks with parameter θ. Consider a code with maximal achievable rate r and outage probability given by Pr{I(θ) = 0}. Since the outage probability provides an upper bound on the expected error of the code, the rate r is -achievable for all 0 ≤ < 1 exceeding the outage probability. Hence, all codes with outage probability less than are -achievable, i.e., C ≥ sup r ≥ 0 : ∃ a code-C(n, M n , r) with Pr{I(θ) = 0} ≤ and lim inf n→∞ 1 n log M n ≥ r .
DF . So that relay k can decide for θ r ∈ D (k) DF to use DF scheme and otherwise it would use CF scheme. Let V ⊆ N and define D V as follows:
Fig. 7 :
7Selective coding strategy (SCS) over cooperative unicast networks.region. In other words, the very same partitioning π(Θ r ) over Θ r can be indexed differently with subsets of N and so it can lead to different sets of decision regions. We denote by Π (Θ r , N ) the set of all possible indexed partitions on Θ r . Hence, if θ r ∈ D V we have a cooperative unicast network where the relays in V use CF scheme while the others relays use DF scheme, as shown inFig. 7. Each relay has two codebooks based on x ) (DF scheme) is transmitted when θ r ∈ D (k) DF , so relay k decodes the source message and transmits it to the destination with the index according to MNNC scheme. The source not knowing whether the relay k is decoding or not uses superposition coding over x
Proposition 5 (
5SCS with partial CSI): The average error probability of the composite unicast network with partial CSI θ r at the relays can be upper bounded bȳ(r) ≤ min p(x,v,q) inf {D V : V⊆N }∈Π(Θr,N ) V⊆N E θr min p(·)∈Q P θ|θr r > I MNNC (V, θ), θ r ∈ D V θ r ,(141)2 It is due to the fact that each partition subset is indexed by V.
(Θ r , N ) is the set of all indexed partitions over Θ r into at most 2 N disjoint sets andI MNNC (V, θ) is defined by
Fig. 8 :
8Bounds on the asymptotic error probability¯ (r) vs. the coding rate r.
Fig. 9 :
9Bounds on the -capacity vs. SNR.
we divide the relay nodes into two disjoint groups, namely, V and V c = N −V, as shown inFig. 10. The relays in V will use Compress-and-Forward (CF) scheme while the others relays will use Decode-and-Forward (DF) scheme, which are simply refereed to DF and CF relays.The DF relays transmit the compressed version of their observations, superimposed over the source message of the previous block. In this sense, the compressed version of the observation of each relay is transmitted to the other nodes. The k-th DF relay with k ∈ V c decodes the source message of block i by exploiting the descriptions sent by the others relays. Because the relays transmit the descriptions (or compression index) related to the block i in block i + 1, the k-th relay has to wait until the end of block i + 1 to decode it and therefore DF relays has to wait until the block i + 2 to forward the source message of the i-th block. Moreover, the k-th relay exploits only the compression index of relays in T k ⊆ N − {k}. Similarly, the destination decodes only the compression index of relays in T ⊆ N . It is shown that, by selecting a subset of relays, we contribute to increase the total rate provided that certain conditions are satisfied.
. . . y(B + 2) y(B + 3). . . y(B + L)
( i )
iIn every block i = [1 : B], the source sends w i using x w (i−2) , w i , where we have defined w 0 = w −1 = 1. Moreover, for blocks i = [B + 1 : B + L], the source sends the dummy message w i = 1 known to all users. (ii) For every block i = [1 : B + L], and each k ∈ V c , the relay k knows w (i−2) by assumption and w 0 = w −1 = 1, so it picks up v w (i−2) . For each i = [1 : B + 2], the relay k after receiving y k (i), searches for at least one index l ki with l k0 = 1 such that
x k (l k(i−1) ). Moreover, k-th relay repeats l k(B+2) for i = [B + 3 : B + L], i.e., for L − 2 blocks.Decoding:(i) After the transmission of the block i + 1 = [1 : B + 1] and for each k ∈ V c , the relay k decodes the message w i and the compression index l T k i , i.e., the compression indices for the block i of all relays in T k , with the assumption that all messages and compression
By choosing finite L but large enough, inequalities (189), (204), (212) and (192) prove Theorem 1, where the final rate is achieved by letting (B, n) tend to infinity. At the end, a time sharing random variable Q can be added over all expressions, concluding the proof.
( i )
iIn every block i = [1 : B], the source sends w i based on x w (i−1) , w i . Moreover, for blocks i = [B + 1 : B + L], the source sends the dummy message w i = 1 known to all nodes. June 3, 2014 DRAFT (ii) For every block i = [1 : B + L], and each k ∈ V c , relay k knows w (i−1) by assumption and w 0 = 1, so it sends x k w (i−1) .
(
iv) For i = [1 : B + 1] and k ∈ T , relay k knows the index l k(i−1) from the previous block and it sends x k (l k(i−1) ). Moreover, relay k repeats l k(B+1) for all i = [B + 2 : B + L].Decoding:(i) After transmission of the block i = [1 : B] is accomplished and for each k ∈ V c , relay k decodes the message of block i with the assumption that all messages up to block i−1 have been correctly decoded. Since the relay k knows the message w (i−1) and so x k w (i−1) , it also knows because of the code generation all others codewords x k w (i−1) for k ∈ V c .
By choosing finite L but large enough, inequalities (219) and (243) prove Theorem 2, where the rate is achieved by letting (B, n) tend to infinity. At the end, a time sharing random variable Q can be added. rest of this section we shall assume that relays are ordered by {L 1 , . . . , L T }. The transmission is done in B + T + L − 1 blocks where at the end of block B the source finished of transmitting messages. At the end of this block, the relays at the layer L t had only repeated messages until the block B − T − 2 + t, and therefore it continues transmission until the end of block B + T + 2 − t. Therefore, we need B + T + 1 blocks for all relays to transmit all messages. From B + T + 2 up to block B + T + L − 1, namely for L − 2 blocks, CF relays repeat their compression index of their last block B + T + 2. In the
( i )
iIn every block i ∈ [1 : B], the source sends w i using x w (i−T −1) , . . . , w (i−2) , w i , where we have defined w i = 1 for i ≤ 0. Moreover, for blocks i = [B + 1 : B + T + L − 1], the source sends the dummy message w i = 1 known to all users. (ii) For i ∈ [1 : B + T + 2 − t] and each k ∈ L t , the k-th relay knows w i−T −1 , . . . , w i−T −2+t in block i, and it sends x k (w i−T −1 , . . . , w (i−T −2+t) ). Notice that at the of block B + T + 1, all DF relays are transmitting x k (w B , 1, . . . , 1).
(
iv) For i = [1 : B + T + 1] and k ∈ V, relay k knows from the previous block l k(i−1) and it sends x k (l k(i−1) ). Moreover, k-th relay repeats l k(B+T +1) for i = [B +T +2 : B +T +L−1], i.e., for L − 2 blocks. Decoding: (i) The DF relays in L T are first to start decoding messages. After the transmission of the block i + 1 = [2 : B + 1] and for each k ∈ L T , the k-th relay decodes the message w i and the compression index l T k i of block i for all relays in T k , with the assumption that all messages and compression indices up to block i − 1 have been correctly decoded. We emphasize that in this case there are only CF relays in T k .
( i )
iIn every block i ∈ [1 : B], the source sends w i ∈ [1, 2 nr ] based on x w (i−1) , w i . Moreover, for blocks i ∈ [B + 1 : B + L], the source sends the dummy message w i = 1 known to all nodes.
P
θ|θr [r > I DF (θ)|θ r ] θ r ∈ D DF min p(x 2 )p(ŷ 2 |x 2 ,y 1 ) P θ|θr [r > I CF (θ)|θ r ] θ r / ∈ D DF .
every block i = [1 : B], the source sends w i based on x w (i−1) , w i . Moreover, for blocks i = [B + 1 : B + L], the source sends the dummy message w i = 1 known to all nodes.
N
(r), randomly and conditionally independently generate 2 nR sequences x drawn i.i.
1R
[θ r ∈ D V and r > I MNNC (V, θ)] , T (S, θ), min i∈V c I θr (X; Y i |X (1) N ) .
P
θ|θr r > I MNNC (V, θ) , θ r ∈ D V θ r . (326)At the next step, the expected value is taken over θ r and is minimized over all decision regions D V and p(x, x
{D V ,V⊆N }∈Π(Θr,N ) V⊆N E θr min j∈V p(x (2) j )p(ŷ j |x(2)j y j ) P θ|θr r > I MNNC (V, θ), θ r ∈ D V θ r . (327)
yields the desired inequality. Interestingly, it appears that this new set is obtained by removing from the set T the relays that are present in A. Thus, for all A ⊆ S ⊆ T , it holds that
Table I
Ipresents details of the transmission schedule for a case with one DF relay and one CF relay. x 1 and x 2 represent the code of DF and CF relays withŷ 1 andŷ 2 as the compressed version of their observation. v is used for coherent transmission between the relays and the
TABLE I :
ITransmission schedule for "Mixed Noisy Network Coding" (MNNC).
Table II ,
IIwe illustrate this procedure
by considering a two hop network with a single CF relay. As it can be seen, the codewords
denoted by x 2 of the second layer among DF relays are superimposed on those denoted by x 1
of the first layer among DF relays. Notice that the number of DF relays can exceed the number
TABLE II :
IITransmission schedule for "Layered Mixed Noisy Network Coding" (LMNNC).
TABLE III :
IIICoding for DF relays at each layer.
cooperation. The proof is finalized by choosing finite but large enough L, and then the inequalities (258), (286), (288), (276), and (261) prove Theorem 3, where the final rate is achieved by letting (B, n) tend to infinity. At the end, a time sharing random variable Q can be added over all expressions which concludes the proof.APPENDIX D
PROOF OF PROPOSITION 3
Notice that non-zero rate cannot always be guaranteed for the compound model which means that the weak capacity of many compound models would be zero.June 3, 2014 DRAFT
ACKNOWLEDGMENTThe authors are grateful to Prof. Abbas El Gamal for his valuable advice at the early stage of this work. They are also grateful to Prof. Gerhard Kramer for valuable comments and suggestions.Encoding:(i) In every block i = [1 : B], the source sends w i based on x w (i−1) , w i . Moreover, for all blocks i = [B + 1 : B + L], the source sends the dummy message w i = 1 known to all nodes.(ii) If θ r ∈ D (k) DF , for every block i = [1 : B + L], the relay k knows w (i−1) by assumption andDF , the relay k after receiving y k (i), searches for at least one index l ki with l k0 = 1 such thatThe probability of finding such l ki goes to one as n goes to infinity. The relay k knows from the previous block l k(i−1) and it sends x DF the k-th relay decodes the message of block i with the assumption that all messages up to block i − 1 have been correctly decoded. The relay k searches for the unique indexŵ i ∈ M n such that:The outage event is occurred whenWe emphasize that not all XN are sending description but only codewords X2) Decoding at the destination is done backwardly. It knows θ, D V and therefore V, i.e., it is aware of the strategy of each relay (e.g. using DF or CF scheme). Moreover, it chooses T to maximize (148). It first decodes the last compression indices sent by the relays in T . It jointly searches for the unique indices l k(B+1) k∈T such that for all b = [B + 2 : B + L] the following condition holds:x k (l k(B+1) ) k∈T , x(1, 1),
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| {'fraction_non_alphanumeric': 0.08680716083628705, 'fraction_numerical': 0.025723018441465042, 'mean_word_length': 3.1773825809393523, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 49, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 178, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We consider the problem of communicating a single message to a single destination in presence of multiple relay nodes that we refer to as cooperative unicast network. Basically, this paper consists of two parts. In the first part, we introduce "Mixed Noisy Network Coding" (MNNC) scheme, generalizing "Noisy Network Coding" (NNC) scheme, where relays are allowed to decode-and-forward (DF) the message while all relays (without exception) transmit noisy descriptions of their observations. These descriptions are exploited at the destination and the DF relays to decode the transmitted messages while creating full cooperation among the nodes. Moreover, the destination and the DF relays can independently select the set of descriptions that either will be decoded or simply treated as interference.We further extend the concept of MNNC to multi-hopping scenarios that we refer to as "Layered MNNC" (LMNNC) where DF relays are organized into disjoint groups, each of them representing one hop in the network. For cooperative unicast additive white Gaussian noise (AWGN) networks we show that, provided DF relays are properly chosen, MNNC improves over all previously established constant gaps to the cut-set bound. In the second part, we consider the composite cooperative unicast network, where the channel parameters are randomly drawn from a probability distribution before the communication and remain fixed during the transmission. Each draw is assumed to be unknown at the source and fully known at the destination, but only partly known at the relays. We introduce through MNNC scheme the concept of "Selective Coding Strategy" (SCS) that enables relays to decide dynamically whether, in addition to communicating noisy descriptions, it would be possible to decode and forward messages to The work of P. Piantanida is partially supported by the ANR grant (FIREFLIES) INTB 0302 01. The material in this paper was presented 2 the destination. It is demonstrated through the asymptotic average error probability of the slow-fading AWGN relay channel that SCS clearly outperforms conventional DF, compress-and-forward, amplifyand-forward, and hash-forward coding schemes.Index TermsCooperative unicasting, wireless networking, decode-and-forward, compute-and-forward, quantizemap-and-forward, noisy network coding, constant gap, composite channel, outage capacity.Gamal-Mohseni-Zahedi [3] developed an alternative version of CF scheme (not based on Wyner-Ziv coding and sequential decoding at the destination) which achieves the same rate that the original CF scheme[2]. In fact, both CF schemes can perform within a constant gap to the information-theoretic capacity of the AWGN relay channel, regardless of channel parameters [4].More recently, there has been a growing interest in cooperative networks with multiple relays and several attempts were made to develop cooperation strategies, e.g., for multiple access and broadcast relay channels (see [5]-[7] and references therein). The capacity of degraded unicast cooperative networks is derived in [5] by using a sequential DF scheme while the capacity of a class of modulo-sum relay channels is found in [8] by using a CF based scheme. Graphical multicast networks were studied in[9]where the "max-flow min-cut theorem" for network information flow was presented for the point-to-point communication network. Deterministic networks with no interference at the receivers were addressed in [10] whereas the capacity of wireless erasure multicast networks was determined in[11], and the scaling behavior of cooperative multicasting in wireless networks was studied in[12].A. Related WorkAn approximation approach to general networks via deterministic channels was introduced by Avestimehr-Diggavi-Tse [4]. This approach yields a novel improvement over CF scheme, referred to as "Quantize-Map-and-Forward" (QMF), which achieves performance within constant gap of capacity for unicast AWGN networks with arbitrary number of relays. This important feature guarantees the uniformity in the channel coefficients and hence the fading statistics. Relay nodes quantize their received signals at noise level, map them randomly to Gaussian codewords and forward them to the others nodes. The fundamental difference between CF and QMF schemes relies on the delay and CSI aspects. The standard CF scheme [2] requires successive decoding at the destination and forward channel knowledge at the relays while QMF uses joint decoding of descriptions and messages with only CSI at the destination. As a matter of fact, this approach has played a key role in the development of several further results on cooperative wireless networks.In[13], Nazer-Gastpar propose an ingenious coding scheme, referred to as compute-andforward, which aims at allowing the relays to decode and send noiseless functions -linear combinations-of the transmitted messages. By combining all these descriptions, the destination determines the original messages being sent. Indeed, due to the additive nature of the channel, June 3, 2014 DRAFT 4 each relay receives a linear combination of the lattice codewords [13] in addition to some additive noise, which have the property that any integer linear combination is still a codeword [14], [15]. The relays then decode the linear combination of the codewords and thus a noiseless function of the messages. Nevertheless, the lattice property requires a integer linear combination of codewords to guarantee that it is still a codeword, however the linear combination induced by wireless channels have arbitrary real (or complex) channel gains. In order to overcome this difficulty, [13] proposes to scale the received channel output so that the received signal is close to an integer linear combination. The tightness of this approximation relies on the scaling factor which introduces a tradeoff between closeness of approximation and noise amplification. Recent work [16] by Lim et al. generalizes QMF approach to arbitrary memoryless multicast networks by introducing the notion of "Noisy Network Coding" (NNC) scheme, which implies the previous inner bounds in [4], [11]. As a matter of fact, Yassaee et al. in [17]-[19] independently introduced for the first time the idea of NNC and derived the same achievable rate regions. In [16], relay nodes based on NNC scheme send the same -long-message over many blocks of equal length -repetitive encoding-and the descriptions at the relays do not require binning while their indices are non-uniquely decoded at the destination. While the same result was obtained by using short messages in [17]-[19]. The achievable region from NNC scheme is shown to be tight for specific cases, e.g., deterministic and erasure networks, and in particular, it achieves within constant gap of capacity for multicast AWGN relay networks. Further progress was made in [20] where authors showed that the gain in NNC comes from backward decoding and delaying the decoding procedure. The use of different message coding opens up the possibility of combining DF and NNC scheme. This approach was taken in [21], [22] and [23], [24], which is referred to as "Short-Message Noisy Network Coding" (SNNC). Transmission is done over (B + L) blocks, where the value B is the number of blocks in which a new message is being transmitted and the value L is the number of blocks in which the previous messages are repeated according to a specific pattern. Both (B, L) are required to be large enough in [20] while only B needs to be large enough in [21], and the destination uses backward decoding. In this case, relays are divided into two sets, the relays in the first set use NNC scheme while those in the second set use DF scheme. The previously mentioned works have neglected two aspects of cooperative unicast networks. First, all relay nodes are capable of collaborating with each other to increase their chances June 3, 2014 DRAFT 5 of decoding the source message, similarly as done in compute-and-forward [13], and second, the destination can benefit from noisy descriptions of all nodes which also includes DF relays. Actually, NNC and SNNC schemes have since then been exploited in various ways, e.g., multilevel DF schemes for DF relays are investigated in [25]-[27]where an aware source exploits the existence of a hierarchy of the relays based on their channel quality.B. Contribution and OutlineIn this paper, we investigate coding strategies for cooperative unicasting in wireless networks. This problem consists of a source that wishes to communicate a single message to a single destination in presence of multiple relay nodes. The focus is on wireless configurations where without CSI, the source cannot any longer agree with the relays to jointly select an adequate cooperative strategy for each specific draw of network parameters. Traditional approaches to deal with this scenario falls into composite models for networks [28] which, unlike compound models [29], channel uncertainty is addressed by introducing a probability distribution (PD) from which the current channel index (or vector of channels parameters) is drawn, but remains fixed during the communication. Composite cooperative AWGN networks have been studied beforehand via the notion of capacity versus outage (see [12], [30], [31] among other references). This setting prevents, in general, the source use of any hierarchical multi-level scheme [25], [26] to enhance cooperation among the nodes since without CSI at the source such approach would clearly result in performance degradation. We shall follow an approach similar to that of compute-and-forward [13] to the study of simultaneous coding strategies [32] that are capable of enabling all nodes to decide -depending on their instantaneous channel measurements-whether would be possible to decode-and-forward messages (e.g. the amount of available noisy descriptions provides enough information) and which nodes should cooperate with each other by decoding noisy descriptions of observations (or noisy functions of the transmitted messages). In the first part, we introduce "Mixed Noisy Network Coding" (MNNC) for memoryless unicast networks with perfect CSI at all nodes while in the second part, we focus on composite cooperative unicast networks where the channel parameters are assumed to be unknown at the source and fully known at the destination, but only partly known at the intermediate nodes. We introduce through MNNC scheme the concept of "Selective Coding Strategy" (SCS) that enables all relays to dynamically select, based on on', 'arxivid': '1307.0991', 'author': ['Arash Behboodi [email protected]. \nDepartment of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France\n', 'Pablo Piantanida [email protected]. \nDepartment of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France\n', 'Pablo Piantanida \nDepartment of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France\n'], 'authoraffiliation': ['Department of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France', 'Department of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France', 'Department of Telecommunications, SUPELEC\nTechnische Universität Berlin\nEinsteinufer 25FT 5 10587, 91192Berlin, Gif-sur-YvetteGermany, France'], 'corpusid': 379493, 'doi': '10.1109/tit.2014.2368564', 'github_urls': [], 'n_tokens_mistral': 56495, 'n_tokens_neox': 51394, 'n_words': 31773, 'pdfsha': '953ac9ddada587b2ef8b1902262989ef6022e19b', 'pdfurls': ['https://arxiv.org/pdf/1307.0991v3.pdf'], 'title': ['Mixed Noisy Network Coding and Cooperative Unicasting in Wireless Networks', 'Mixed Noisy Network Coding and Cooperative Unicasting in Wireless Networks'], 'venue': ['the 5th International Symposium on Communications, Control, and Signal Processing']} |
arxiv |
Observation of the Decays B
28 Jun 2004
K +
P Chang
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
K Abe
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
K Abe
T Abe
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H Aihara
Y Asano
V Aulchenko
Budker Institute of Nuclear Physics
NovosibirskRussia
T Aushev
Institute for Theoretical and Experimental Physics
MoscowRussia
T Aziz
S Bahinipati
University of Cincinnati
CincinnatiOHUSA
A M Bakich
Y Ban
A Bay
Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne
I Bedny
Budker Institute of Nuclear Physics
NovosibirskRussia
U Bitenc
J. Stefan Institute
LjubljanaSlovenia
I Bizjak
J. Stefan Institute
LjubljanaSlovenia
A Bondar
Budker Institute of Nuclear Physics
NovosibirskRussia
A Bozek
M Bračko
J. Stefan Institute
LjubljanaSlovenia
University of Maribor
MariborSlovenia
J Brodzicka
T E Browder
University of Hawaii
HonoluluHIUSA
M.-C Chang
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
Y Chao
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
B G Cheon
Chonnam National University
KwangjuSouth Korea
R Chistov
Institute for Theoretical and Experimental Physics
MoscowRussia
S.-K Choi
Gyeongsang National University
ChinjuSouth Korea
Y Choi
S Cole
M Danilov
Institute for Theoretical and Experimental Physics
MoscowRussia
M Dash
L Y Dong
Institute of High Energy Physics
Chinese Academy of Sciences
BeijingPR China
A Drutskoy
University of Cincinnati
CincinnatiOHUSA
S Eidelman
Budker Institute of Nuclear Physics
NovosibirskRussia
V Eiges
Institute for Theoretical and Experimental Physics
MoscowRussia
Y Enari
Nagoya University
NagoyaJapan
Women's University
NaraJapan
F Fang
University of Hawaii
HonoluluHIUSA
S Fratina
J. Stefan Institute
LjubljanaSlovenia
N Gabyshev
Budker Institute of Nuclear Physics
NovosibirskRussia
A Garmash
T Gershon
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
G Gokhroo
B Golob
J. Stefan Institute
LjubljanaSlovenia
University of Ljubljana
LjubljanaSlovenia
R Guo
National Kaohsiung Normal University
KaohsiungTaiwan
N C Hastings
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H Hayashii
M Hazumi
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
T Higuchi
L Hinz
Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne
T Hokuue
Nagoya University
NagoyaJapan
Women's University
NaraJapan
Y Hoshi
W.-S Hou
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
Y B Hsiung
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
H.-C Huang
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
K Inami
Nagoya University
NagoyaJapan
Women's University
NaraJapan
A Ishikawa
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
R Itoh
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H Iwasaki
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
M Iwasaki
Y Iwasaki
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
J H Kang Au
J S Kang
Korea University
SeoulSouth Korea
S U Kataoka
N Katayama
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H Kawai
Chiba University
ChibaJapan
T Kawasaki Aa
H R Khan
H Kichimi
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H J Kim
Kyungpook National University
TaeguSouth Korea
J H Kim
S K Kim
T H Kim Au
P Koppenburg
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
S Korpar
J. Stefan Institute
LjubljanaSlovenia
University of Maribor
MariborSlovenia
P Križan
J. Stefan Institute
LjubljanaSlovenia
University of Ljubljana
LjubljanaSlovenia
P Krokovny
Budker Institute of Nuclear Physics
NovosibirskRussia
A Kuzmin
Budker Institute of Nuclear Physics
NovosibirskRussia
Y.-J Kwon Au
S E Lee
T Lesiak
J Li
Ag
S.-W Lin
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
J Macnaughton
Institute of High Energy Physics
ViennaAustria
G Majumder
F Mandl
Institute of High Energy Physics
ViennaAustria
D Marlow
T Matsumoto
A Matyja
W Mitaroff
Institute of High Energy Physics
ViennaAustria
H Miyake
H Miyata Aa
R Mizuk
Institute for Theoretical and Experimental Physics
MoscowRussia
D Mohapatra
T Mori
T Nagamine
Y Nagasaka
Hiroshima Institute of Technology
HiroshimaJapan
E Nakano
M Nakao
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
H Nakazawa
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
Z Natkaniec
S Nishida
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
O Nitoh Ar
S Ogawa
T Okabe
Nagoya University
NagoyaJapan
Women's University
NaraJapan
S Okuno
Kanagawa University
YokohamaJapan
S L Olsen
University of Hawaii
HonoluluHIUSA
W Ostrowicz
H Ozaki
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
P Pakhlov
Institute for Theoretical and Experimental Physics
MoscowRussia
H Park
Kyungpook National University
TaeguSouth Korea
N Parslow
L S Peak
L E Piilonen
A Poluektov
Budker Institute of Nuclear Physics
NovosibirskRussia
N Root
Budker Institute of Nuclear Physics
NovosibirskRussia
M Rozanska
H Sagawa
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
Y Sakai
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
O Schneider
Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne
J Schümann
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
S Semenov
Institute for Theoretical and Experimental Physics
MoscowRussia
K Senyo
Nagoya University
NagoyaJapan
Women's University
NaraJapan
M E Sevior
University of Melbourne
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H Shibuya
V Sidorov
Budker Institute of Nuclear Physics
NovosibirskRussia
A Somov
University of Cincinnati
CincinnatiOHUSA
R Stamen
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
S Stanič
M Starič
J. Stefan Institute
LjubljanaSlovenia
K Sumisawa
T Sumiyoshi
S Suzuki
O Tajima
F Takasaki
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
K Tamai
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
N Tamura
M Tanaka
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
Y Teramoto
T Tsukamoto
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
S Uehara
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
K Ueno
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
T Uglov
Institute for Theoretical and Experimental Physics
MoscowRussia
Y Unno
Chiba University
ChibaJapan
S Uno
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
G Varner
University of Hawaii
HonoluluHIUSA
K E Varvell
S Villa
Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne
C C Wang
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
C H Wang
National United University
Miao LiTaiwan
M.-Z Wang
Department of Physics
Institute of Nuclear Physics
National Taiwan University
Taipei, KrakowTaiwan, Poland
M Watanabe Aa
atB D Yabsley
Y Yamada
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
A Yamaguchi
Y Yamashita
Nihon Dental College
NiigataJapan aa
Niigata University
NiigataJapan ab
Osaka City University
OsakaJapan
ac Osaka University
OsakaJapan
ad Peking University
BeijingPR
China ae Princeton University
PrincetonNJUSA
af Saga University
SagaJapan
ag University of Science and Technology of China
HefeiPR China
ah Seoul National University
SeoulSouth Korea
ai Sungkyunkwan University
SuwonSouth Korea
aj University of Sydney
SydneyNSWAustralia
Tata Institute of Fundamental Research
BombayIndia
aℓ Toho University
FunabashiJapan am
Tohoku Gakuin University
TagajoJapan
Department of Physics
an Tohoku University
SendaiJapan
University of Tokyo
TokyoJapan ap
Tokyo Institute of Technology
TokyoJapan
ar Tokyo University of Agriculture and Technology
aq Tokyo Metropolitan University
Tokyo, TokyoJapan, Japan
University of Tsukuba
TsukubaJapan at
Virginia Polytechnic Institute and State University
BlacksburgVAUSA
au Yonsei University
SeoulSouth Korea
M Yamauchi
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
J Ying
C C Zhang
Institute of High Energy Physics
Chinese Academy of Sciences
BeijingPR China
J Zhang
High Energy Accelerator Research Organization (KEK)
TsukubaJapan
Z P Zhang Ag
D Žontar
J. Stefan Institute
LjubljanaSlovenia
University of Ljubljana
LjubljanaSlovenia
Nara
H Niewodniczanski
Belle Collaboration
Observation of the Decays B
28 Jun 2004Preprint submitted to Elsevier Preprint 25 March 2022BELLE KEK Preprint 2004-25 Belle Preprint 2004-18PACS: 1325Hw, 1440Nd
We report the observation of B 0 decays to the K + π − π 0 final state using a data sample of 78 fb −1 collected by the Belle detector at the KEKB e + e − collider. With no assumptions about intermediate states in the decay, the branching fraction is measured to be (36.6 +4.2 −4.3 ± 3.0) × 10 −6 . We also search for B decays to intermediate two-body states with the same K + π − π 0 final state. Significant B signals are
observed in the ρ(770) − K + and K * (892) + π − channels, with branching fractions of (15.1 +3.4+1.4+2.0 −3.3−1.5−2.1 ) × 10 −6 and (14.8 +4.6+1.5+2.4 −4.4−1.0−0.9 ) × 10 −6 , respectively. The first error is statistical, the second is systematic and the third is due to the largest possible interference. Contributions from other possible two-body states will be discussed. No CP asymmetry is found in the inclusive K + π − π 0 or ρ − K + modes, and we set 90% confidence level bounds on the asymmetry of −0.12 < A CP < 0.26 and −0.18 < A CP < 0.64, respectively. PACS: 13.25.Hw, 14.40.Nd
Introduction
Recently, observations of large branching fractions for three-body charmless hadronic decays of B mesons have been reported by the B factory experiments [1,2,3,4,5,6]. In the mesonic decays B → Kπ + π − and B → KK + K − , a large fraction of the decays proceed through intermediate two-body decay processes, such as B + → K * (892) 0 π + , K * (892) 0 → K + π − and B + → φK + , φ → K + K − . However, higher mass K + π − , π + π − and K + K − states may contribute but are not clearly identified due to limited statistics. Moreover, the broad K + K − mass spectrum above 1.5 GeV/c 2 in B + → K + K + K − suggests a large nonresonant B + → K + K + K − contribution. In the baryonic decay B + → ppK + , a simple phase-space model fails to describe the pp mass spectrum, which may be explained by a baryonic form factor model [7] or by an additional, unknown resonance around 2 GeV/c 2 . These studies of three-body decays have provided new information on the mechanism of B meson decay, and provide opportunities to search for unknown B meson decays and to understand the interference between them. In this paper we report on a study of B meson decays to K + π − π 0 , independently of possible intermediate states. In addition, we also present results of a search for quasi-two-body intermediate states.
Inclusion of charge conjugate modes is always implied in this letter unless otherwise specified. The results are obtained from data collected by the Belle detector [8] at the KEKB asymmetric e + e − storage ring [9]. The data sample corresponds to an integrated luminosity of 78 fb −1 and contains 85.0 million BB pairs at the Υ(4S) resonance.
Apparatus and Event Selection
The Belle detector is a large-solid-angle general purpose spectrometer based on a 1.5 T superconducting solenoidal magnet. Charged tracks are reconstructed with a three layer double-sided silicon vertex detector (SVD) and a central drift chamber (CDC) that consists of 50 layers segmented into 6 axial and 5 stereo superlayers. The CDC covers the polar angle range between 17 • and 150 • in the laboratory frame and, together with the SVD, gives a transverse momentum resolution of (σ pt /p t ) 2 = (0.0019 p t ) 2 + (0.0030) 2 , where p t and σ pt are in GeV/c. Charged hadron identification is performed using a combination of three devices: an array of 1188 aerogelČerenkov counters (ACC) covering the momentum range 1-4 GeV/c, a time-of-flight scintillation counter system (TOF) for track momenta below 1.5 GeV/c, and dE/dx information from the CDC for particles with low or high momenta. Situated between these devices and the solenoid coil is an electromagnetic calorimeter (ECL) consisting of 8736 CsI(Tℓ) crystals with a typical front-surface cross-section of 5.5 × 5.5 cm 2 and a depth of 16.2 X 0 . The ECL provides a photon energy resolution of (σ E /E) 2 = 0.013 2 + (0.0007/E) 2 + (0.008/E 1/4 ) 2 , where E and σ E are in GeV. An instrumented iron flux return outside the solenoid coil is used for muon and K L detection. A detailed description of the Belle detector can be found in Ref. [8].
Charged tracks are required to come from the collision point and have transverse momenta, p t , above 100 MeV/c. The accepted tracks are then refitted with their vertex position constrained to the run-averaged profile of B meson decay vertices in the transverse plane. Charged K and π mesons are identified by combining information from the CDC (dE/dx), the TOF and the ACC to form a K(π) likelihood L K (L π ). Discrimination between kaons and pions is achieved through the likelihood ratio L K /(L π + L K ). The performance of the charged hadron identification is studied using a kinematically selected high momentum D * + data sample, where D * + → D 0 π + , D 0 → K − π + . We measure the pion and kaon identification efficiencies and their fake rates as functions of track momentum. The typical kaon and pion identification efficiencies for 1 GeV/c tracks are (87.9 ± 0.6)% and (89.4 ± 0.6)%, respectively. The rate for true pions to be misidentified as kaons is (9.0 ± 0.5)%, while the rate for true kaons to be misidentified as pions is (10.0 ± 0.6)%. Charged tracks which are positively identified as electrons and muons are rejected. Candidate neutral pions are selected by requiring the two-photon invariant mass to be in the mass window 0.118 GeV/c 2 < M(γγ) < 0.150 GeV/c 2 , corresponding to ±2.5σ mass resolution with momentum above 2 GeV/c. The momentum of each photon is then readjusted, constraining the mass of the photon pair to be the nominal π 0 mass. To reduce the background from soft photons, each photon is required to have energy above 50 MeV and the minimum π 0 momentum is 200 MeV/c. Candidate B mesons are identified using the beam constrained mass, M bc = E 2 beam − P 2 B , and the energy difference, ∆E = E B − E beam , where E beam is run-dependent and determined from B → D ( * ) π events, and P B and E B are the momentum and energy of the B candidate in the Υ(4S) rest frame. The parameterizations of the signal in M bc and ∆E are determined by a GEAN-based Monte Carlo (MC) [10] simulation of non-resonant B 0 → K + π − π 0 decays, and various quasi-two-body decays to the K + π − π 0 final state. The signal parameterization is verified using the data and MC samples of B + → D 0 π + , D 0 → K + π − π 0 candidates. Our MC overestimates the M bc resolution by 8% but underestimates the ∆E resolution by 9% to 15%, depending on the kinematics of the K + π − π 0 events. The MC based signal probability density functions (PDF) are readjusted accordingly.
The Gaussian width of the signal in M bc is about 3.0 MeV/c 2 , which is primarily due to the beam energy spread. The ∆E distribution is found to be asymmetric, with a tail on the lower side due to γ interactions with material in front of the calorimeter, and shower leakage out of the back side of the crystals. As a result, the ∆E resolution and the tail distribution strongly depend on the π 0 energy; the ∆E width ranges from 20 MeV to 33 MeV. In the inclusive K + π − π 0 study, since the π 0 energy distribution for the signal is not known a priori, the data is divided into three samples: P (π 0 ) < 0.5 GeV/c, 0.5 GeV/c < P (π 0 ) < 1.5 GeV/c and P (π 0 ) > 1.5 GeV/c. The ∆E distribution in each sample is modeled with a Crystal Ball lineshape [11] with parameters determined from MC. Events with M bc > 5.2 GeV/c 2 and |∆E| < 0.3 GeV are selected for the final analysis. The signal region is defined as M bc > 5.27 GeV/c 2 and −0.10 GeV < ∆E < 0.08 GeV. Events located in the region M bc < 5.265 GeV/c 2 are defined as sideband events and are used for background studies. When more than one B 0 candidate is found in an event, the candidate having the smallest sum of the χ 2 from the vertex fit and π 0 mass constrained fit is selected.
The dominant background for three-body B decay events comes from the e + e − → qq continuum, where q = u, d, s or c. In order to reduce this background, several shape variables are chosen to distinguish spherical BB events from jet-like continuum events. Five modified Fox-Wolfram moments [12] and a measure of the momentum transverse to the event thrust axis (S ⊥ ) [13] are combined into a Fisher discriminant. The PDFs for this discriminant and cos θ B , where θ B is the angle between the B flight direction and the beam direction in the Υ(4S) rest frame, are obtained using events in the signal and sideband regions from MC simulations for signal and qq background. These two variables are then combined to form a likelihood ratio R = L s /(L s + L qq ), where L s(qq) is the product of signal (qq) probability densities. Continuum background is suppressed by requiring R > 0.9, based on a study of the signal significance (N S / √ N S + N B ) using a MC sample, where N S and N B are signal and background yields, respectively. This R requirement retains 45% of the signal and removes 97% of the continuum events. The effect of the R cut is studied by comparing B + → D 0 π + in data and MC, for different values of R. A systematic error of 3% is obtained for the R cut. Figure 1 shows the Dalitz plot distribution of K + π − π 0 candidates in the ∆E − M bc signal region and M bc sideband region. K + π − π 0 candidates populate the three edges of the Dalitz plot, indicating the existence of quasi-two-body intermediate states. Moreover, there is an enhancement near M 2 (K + π − ) = 3.5 (GeV/c 2 ) 2 in Fig. 1(a), which is due to the decay B 0 → D 0 π 0 , D 0 → K + π − . To restrict the study to charmless B decays, events with a K + π − mass within 50 MeV/c 2 of the nominal D 0 mass are rejected.
(K + π − ) = 3.5 (GeV/c 2 ) 2 is from B 0 → D 0 π 0 events.3 Inclusive K + π − π 0 Yield
The final signal yields are obtained from fits to the ∆E and M bc distributions. In addition to continuum background, the final sample contains background from Υ(4S) → BB events. The ∆E and M bc shapes of this BB background are modeled with smooth histograms, generated from a large GEANT based BB MC sample, which includes b → c transitions and charmless B decays. The continuum ∆E background shape is modeled by either a first or second order polynomial, determined from the M bc sideband data. The continuum M bc background shape is modeled with an ARGUS function [14] with parameters determined from events outside the ∆E signal region. One-dimensional binned likelihood fits to ∆E and M bc are performed using signal, continuum background and BB background PDFs for events in the M bc and ∆E signal region, respectively. Since the M bc shapes from BB background are difficult Table 1 Fit results for inclusive K + π − π 0 events. Column 3 and 4 list the signal yield and the statistical significance. The BB fraction is fixed to the MC expectation in the M bc fit, while it is allowed to float in the ∆E fit. The last row shows the sum of the three ∆E fits. to distinguish from signal shapes, the signal yields are estimated using the ∆E fit and cross checked by the M bc fit, where the BB background fraction is fixed to the MC expectation. Table 1 summarizes the fit result of the inclusive K + π − π 0 sample with the statistical significance (Σ) defined as −2 ln(L 0 /L max ), where L 0 and L max denote the likelihood values at zero yield and the best fit numbers, respectively. The sum of the signal yield from the ∆E fits to the three subsamples, 386 ± 44, is consistent with the yield from the M bc fit, 369 ± 35, which has a smaller signal efficiency than the ∆E fit due to the tighter ∆E requirement. The corresponding projections of the fits are shown in Fig. 2. Furthermore, a consistent result is obtained when the BB fraction is fixed according to the MC expectation. 3. B yields from ∆E fits as a function of (a) K + π − , (b) K + π 0 and (c) π − π 0 . Each two-body mass is examined after requiring the other two-body masses to be large (> 1.6 GeV/c 2 for Kπ and > 1.1 GeV/c 2 for π − π 0 ). Dotted points are data; the superimposed curves in (b) and (c) are the projection curves based on K * (892) + and ρ(770) − , The enhancements between 1.1 to 1.5 GeV/c 2 in (a) and (b) are modeled with Breit-Wigner functions.
π 0 momentum (GeV/c) Signal
Two-body Intermediate States
We perform a search for quasi-two-body decays in the K + π − π 0 final state, including B decays to a pseudoscalar (K or π) and a vector meson (ρ(770) or K * (892)) and other possible intermediate states with higher mass resonances. Three pseudoscalar-vector (PV) modes are considered: K * (892) 0 π 0 , K * (892) + π − , and ρ(770) − K + . Figure 3 shows the B signal yields from the ∆E fit as functions of K + π − , K + π 0 and π − π 0 masses. To eliminate crosstalk between decay modes, each two-body mass is examined after requiring the other two-body masses to be large (> 1.6 GeV/c 2 for Kπ and > 1.1 GeV/c 2 for π − π 0 ). The ∆E signal PDFs are obtained from MC simulations of B 0 → K * 0 (1430) 0 π 0 , B 0 → K * 0 (1430) + π − and B 0 → ρ(770) − K + for the K + π − , K + π 0 and π − π 0 cases, respectively.
In the K + π − sample, a large enhancement is observed between 1.0 and 1.6 GeV/c 2 , peaking around 1.2 to 1.4 GeV/c 2 . More structure is observed in the K + π 0 spectrum. An enhancement is seen in the K * (892) + mass region, in the region from 1.2 to 1.4 GeV/c 2 and possibly between 1.8 and 2.1 GeV/c 2 . In the π − π 0 sample, a clear excess is seen in the ρ(770) − signal region. Although the enhancement between 1.1 and 1.6 GeV/c 2 is observed in both K + π − and K + π 0 spectra, these higher mass Kπ states cannot be identified without performing an angular analysis that requires much more data. Possible candidates are K * (1410), K * 0 (1430) and K * 2 (1430). Earlier studies of B + → K + π + π − [1] and B 0 → K 0 π + π − [5] decays also observed large quasi-two-body B decays with K * x (Kπ) mesons in the final state. If the enhancements observed in the K + π 0 mass spectrum are due to such K * x mesons, one would expect the same resonances to appear in the K 0 π + mode. The first two enhancements seen in the K + π 0 mode ( Fig. 3(b)) are indeed observed in the B 0 → K 0 π + π − analysis [2]. However, the third enhancement, between 1.8 and 2.1 GeV/c 2 , does not appear in the K 0 π + π − mode. This enhancement, which has a signal yield of 22 +9 −8 events and a significance of 3σ, may be either a statistical fluctuation, or originate from K 2 (1820), K * 4 (2045) or from a doubly Cabibbo suppressed D + decay. More data are needed to clarify the current situation.
To further understand the possible resonances, we study the distributions of cos θ H , where the helicity angle θ H is defined as the angle between the direction of the candidate B meson and the K + (π 0 ) direction in the K * (ρ) rest frame. Figure 4 shows the B yields as a function of cos θ H for events in the K * (892) + , ρ(770) − , K * x (Kπ) signal region. The cos θ H distributions for the first two modes are consistent with those of B 0 → pseudoscalar vector (PV) decays, as expected for B 0 → K * (892) + π − and B 0 → ρ(770) − K + . Note that the asymmetry in the cos θ H distributions is due to the inefficiency of low momentum π 0 reconstruction. Since π 0 s from B 0 → ρ − K + decays are more energetic than those from B 0 → K * (892) + π − , the asymmetric effect is less pronounced. The cos θ H distributions for the K * 0
x and K * + x modes favor a scalar behavior.
Yield for Various States
We measure B 0 → K + π − π 0 decay rates for the three PV modes, events in the two K * x π regions, and the central region of the Dalitz plot with twobody masses above 2.0 GeV/c 2 . Candidate K * (892) and ρ(770) − mesons are identified by requiring the K + π −(0) and π − π 0 masses to be in the range 820-980 MeV/c 2 and 570-970 MeV/c 2 , respectively. To further reduce background, a selection of | cos θ H | > 0.3 is applied to the vector meson candidates. In the two K * 0(+)
x π regions, we require 1.1 GeV/c 2 < M(K + π −(0) ) < 1.6 GeV/c 2 . B meson candidates are then selected from the inclusive K + π − π 0 events after applying all analysis cuts, including the appropriate two-body mass vetos to avoid cross talk. The signal PDFs are obtained from MC simulations for all six channels, where a scalar hypothesis is used to model K *
x π. Figure 5 shows the M bc and ∆E distributions, and their corresponding fit curves, for the three PV modes. No signal yield is seen in the K * 0 π 0 channel but significant signals are observed for the K * + π − and ρ − K + modes; the yields measured from the ∆E fits are 38 ±11 and 77 +18 −17 events, with statistical significances of 3.8σ and 4.9σ, respectively. As for events in the central region of the Dalitz plot and the two K * x regions, Fig. 6 shows their M bc and ∆E distributions with the fit curves superimposed. Based on the ∆E fit, there are 67 ± 17 and 52 ± 15 signal events in K * 0
x π 0 and K * + x π − , respectively. Since events in these K * x π regions cannot be positively identified, their reconstruc- tion efficiencies are determined without assumptions about the intermediate two-body states. Although a yield of around 20 events is obtained from both the ∆E and M bc fits for the central region of the Dalitz plot, the statistical significance is below 3σ and, hence, an upper limit is reported. The reconstruction efficiency is obtained from a phase-space decay model.
The number of feed-across events from high mass K *
x to the K * (892) region is estimated from the B yields in the 1.1 GeV/c 2 < M(K + π 0 ) < 1.6 GeV/c 2 region, and the Breit-Wigner distribution, modeled with a mass of 1.326 GeV/c 2 and a mean of 252 MeV/c 2 . We find a contribution of 1 event. Assuming no interference, the K * (892) + π − yield is estimated to be 37 ± 11. The possible effect of interference is studied using a Monte Carlo simulation which assumes the three P V decays and two K * x π states. We compare the yields with and without interference. After varying the relative phase of each channel, and tak- ing into account the non-uniform reconstruction efficiency over the Dalitz plot, the largest deviation is +16% −6% for K * (892) + π − and +13% −14% for ρ − K + . These two numbers are used to estimate the systematic error arising from interference.
Systematic Uncertainties
The systematic error for each signal yield is estimated by varying each parameter of the fit functions by ±1σ from the measured values. The shifts in signal yield are then added in quadrature. The typical fit systematic error for the inclusive decay is around 4%. Signal efficiencies are first obtained from MC simulations, and then corrected by comparing data and MC predictions for other processes. The efficiency for the inclusive K + π − π 0 signals is estimated Table 2 Summary of the B 0 → K + π − π 0 search. We present signal yields, efficiencies, and their statistical significances for the inclusive mode, the three intermediate channels of B → PV decays and the other three regions of the Dalitz plot. Branching fractions and/or upper limits are shown in the last two columns. In the branching fractions, the first error is statistical and the second systematic. The third error for K * (892) + π − and ρ − K + corresponds to the largest uncertainty from the interference between different states. The last channel, K + π − π 0 N R , indicates the non-resonant B 0 → K + π − π 0 decay. from the weighted sum of the efficiencies for the possible two-body intermediate states shown in Fig. 3, where the sub-decay branching fraction for each two-body state is not included. The uncertainty on this inclusive efficiency is 5%, determined by checking the reconstruction efficiencies on various twobody modes. The π 0 reconstruction efficiency is verified by comparing the π 0 decay angular distribution with the MC prediction, and by measuring the ratio of the branching fractions of two η decay channels: η → γγ and η → π 0 π 0 π 0 . The typical systematic error for π 0 detection is 3%. The systematic errors on the charged track reconstruction are estimated to be ∼ 2% using partially reconstructed D * events, and verified by comparing the ratio of η → π + π − π 0 to η → γγ in data with MC expectations. The final systematic errors on the reconstruction efficiencies, including charged particle and π 0 detection, particle identification and the R cut, range from 7 to 14% for the quasi-two-body decays and is 8.2% for the inclusive K + π − π 0 channel. , where the first error is statistical, the second is systematic and the third corresponds to the largest uncertainty from the interference between different states. Finally, the B 0 → K + π − π 0 decay branching fractions in the K * 0
Channel
Branching Fractions
x π 0 and K * +
x π − regions are measured to be (6.1 +1.6+0.5 −1.5−0.6 )×10 −6 and (5.1±1.5 +0.6 −0.7 )×10 −6 , respectively.
Using the large signals observed in the inclusive B 0 → K + π − π 0 and B 0 → ρ − K + modes, we search for direct CP violation by dividing the data into two subsets, according to the charge of the kaon. The asymmetry, defined as
A CP = NB −N B
NB +N B , is then computed using the B signal yields obtained from ∆E fits. Following the same fitting procedure, we observe 179 +31 −30 K + π − π 0 events and 207 +32 −31 K − π + π 0 events, while the ∆E fit yields 30 +12 −11 ρ − K + events and 47 +13 −12 ρ + K − events (see Fig.7). The possible reconstruction bias in A CP is studied by checking the inclusive D 0 → K + π − and D 0 → K − π + yields in data. The obtained systematic error is 0.5%. Adding this 0.5% error in quadrature with the fitting systematic error, obtained by varying each parameter in the PDFs by 1σ, gives the total systematic error. Finally, the CP asymmetry is calculated to be A CP = 0.07 ± 0.11 ± 0.01 for the inclusive mode, and A CP = 0.22 +0.22+0.06 −0.23−0.02 for B 0 (B 0 ) → ρ ∓ K ± . We also set 90% confidence intervals on the asymmetry of −0.12 < A CP < 0.26 for the inclusive mode, and −0.18 < A CP < 0.64 for B 0 → ρ ∓ K ± .
Conclusions
In summary, we have studied the charmless hadronic decays, B 0 → K + π − π 0 , which is observed for the first time. Our results show that the branching fraction of B 0 → K + π − π 0 is (64 ± 10)% and (78 ± 15)% of that of B + → K + π − π + [2,4] and B 0 → K 0 π − π + [2,5], respectively. The K + π − π 0 signal candidates populate the edge of the Dalitz plot, indicating the existence of quasi-two-body states. For the K + π − π 0 final state, we have observed signals in the K * + π − and ρ − K + samples but no significant K * 0 π 0 signal is seen. The ρ − K + branching fraction is close to the K * + π − branching fraction, where our measurement is consistent with the earlier CLEO result [5]. However, our ρ − K + result is twice that of BaBar's measurement [6]. We also report the B decay rates in other regions of the Dalitz plot without assumptions about the presence of two-body intermediate states. In the future, significantly more data will be collected at Belle, which will enable us to perform a full Dalitz analysis, allowing us to identify other quasi-two-body states and extract their relative phases. Finally, we performed a search for direct CP violation in the inclusive and B 0 → ρ − K + channels. No evidence of CP violating asymmetry is seen and 90% C.L. limits on A CP are set.
Fig. 1 .
1Dalitz plot of B 0 → K + π − π 0 candidates from (a) the B signal region and (b) the M bc sideband region. The enhancement around M 2
Fig. 2 .
2(a) M bc and (b) ∆E distributions for inclusive K + π − π 0 events. Solid lines in the figures represent the result of the fit and the signal contribution. Dashed lines show the total background contribution, while dotted lines indicate the continuum contribution. The ∆E lines in (b) show the sum of the fit results to the three subsamples. A sizeable B 0 → K + π − and B + → K + π 0 feed-down at ∆E > 0.2 GeV is found in the MC. This contribution is included in the fit.
Fig.
Fig. 3. B yields from ∆E fits as a function of (a) K + π − , (b) K + π 0 and (c) π − π 0 . Each two-body mass is examined after requiring the other two-body masses to be large (> 1.6 GeV/c 2 for Kπ and > 1.1 GeV/c 2 for π − π 0 ). Dotted points are data; the superimposed curves in (b) and (c) are the projection curves based on K * (892) + and ρ(770) − , The enhancements between 1.1 to 1.5 GeV/c 2 in (a) and (b) are modeled with Breit-Wigner functions.
Fig. 4 .
4B signal yields as a function of cos θ H for events in (a) K * (892) + , (b) ρ(770) − , (c) K * 0 x and (d) K * + x regions. Points are data and histograms are the expectations from (a) B 0 → K * (892) + π − , (b) B 0 → ρ(770) − K + and (c,d) B decays to a scalar and a pseudoscalar meson.
Fig. 5 .
5M bc and ∆E distributions for (a,b) K * 0 π 0 , (c,d) K * + π − and (e,f) ρ + K − events. The superimposed solid curves represent the corresponding fits. Dashed lines are the background projections and the dotted lines show the continuum qq contribution. Enhancements in the projection curves at ∆E > 0.2 GeV in (d) and (f) are due to B → Kπ decays.
Fig. 6 .
6M bc and ∆E distributions for events in different regions of the Dalitz plots: (a,b) the central region, (c,d) K * 0 x π 0 region and (e,f) K * + x π − region. The superimposed solid curves represent the corresponding fits. Dashed lines are the background projections and the dotted lines show the continuum qq contribution. Enhancements in the projection curves at ∆E > 0.2 GeV in (d) and (f) are due to B → Kπ decays
Fig. 7 .
72 summarizes the fit results for each reconstructed decay channel. The branching fractions and upper limits are calculated assuming that B + B − and B 0 B 0 are produced with equal probability. The systematic errors on the branching fractions combine the systematic errors for the reconstruction efficiencies and the ∆E fit with the uncertainty from the number of BB events. Since no signal is seen in the K * 0 π 0 mode, and the signal yield in the central region of the Dalitz plot is not significant, upper limits are computed at the 90% confidence level (C.L.) based on the observed number of events in the sig-∆E distributions for the (a) K + π − π 0 , (b) K − π + π 0 , (c) ρ − K + and (d) ρ + K − samples. The superimposed solid curves represent the corresponding fits and signal projections. Dashed lines are the background projections and dotted lines show the continuum qq contribution. Enhancements in the projection curves at ∆E > 0.2 GeV are due to B → Kπ decays nal region, and the background level found by the fit; both the statistical and systematic errors are taken into account[15]. With 85.0 million BB events, we measure the branching fraction to be (36.6 +4.2 −4.1 (stat.)±3.0(syst.))×10 −6 for the inclusive B 0 → K + π − π 0 decay, without assumptions about intermediate twobody states. The branching fractions of B 0 → K * (892) + π − and B 0 → ρ − K + decay are measured to be (14.8 +4.6+1.5+2.4 −4.4−1.0−0.9 ) × 10 −6 and (15.1 +3.4+1.4+2.0 −3.3−1.5−2.1 ) × 10 −6
Table
ucation ,
ucationCulture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Education, Science and Training; the National Science Foundation of China under contract No. 10175071; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation; the Polish State Committee for Scientific Research under contract No. 2P03B 01324; the Ministry of Science and Technology of the Russian Federation; the Ministry of Education, Science and Sport of the Republic of Slovenia; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy.
on leave from Fermi National Accelerator Laboratory, Batavia, IL, USA 2 on leave from Nova Gorica Polytechnic, Nova Gorica, Slovenia
AcknowledgementsWe wish to thank the KEKB accelerator group for the excellent operation of the KEKB accelerator. We acknowledge support from the Ministry of Ed-
. A Garmash, Belle CollaborationPhys. Rev. D. 6592005Belle Collaboration, A. Garmash et al., Phys. Rev. D 65 (2002) 092005.
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. E Eckhart, CLEO CollaborationPhys. Rev. Lett. 89251801CLEO Collaboration, E. Eckhart et al., Phys. Rev. Lett. 89 (2002) 251801 .
. B Aubert, BaBar CollaborationPhys. Rev. Lett. 91201802BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. 91 (2003) 201802.
. C.-K Chua, W.-S Hou, S.-Y Tsai, Phys. Rev. D. 6534003C.-K. Chua, W.-S. Hou, and S.-Y. Tsai, Phys. Rev. D 65 (2002) 034003;
. H.-Y Cheng, K.-C Yang, Phys. Rev. D. 6614020H.-Y. Cheng and K.-C. Yang, Phys. Rev. D 66 (2002) 014020.
. A Abashian, Belle CollaborationNucl. Inst. and Meth. A. 479117Belle Collaboration, A. Abashian et al., Nucl. Inst. and Meth. A 479 (2002) 117.
. S Kurokawa, E Kikutani, Nucl. Inst. Meth. A. 499S. Kurokawa and E. Kikutani, Nucl. Inst. Meth. A 499 ( 2003) 1, and other papers included in this Volume.
. R Brun, GEANT 3.21, CERN Report No. DD/EE/84-1R. Brun et al., GEANT 3.21, CERN Report No. DD/EE/84-1 (1987).
. J E Gaiser, Crystal Ball CollaborationPhys. Rev. D. 34711Crystal Ball Collaboration, J.E. Gaiser et al., Phys. Rev. D 34 (1986) 711.
41 (1978) 1581. The modified Fox-Wolfram moments were described in K. G Fox, S Wolfram, Abe, Phys. Rev. Lett. 511151Phys. Lett. BThe Fox-Wolfram moments were introduced in G. Fox and S. Wolfram, Phys. Rev. Lett. 41 (1978) 1581. The modified Fox-Wolfram moments were described in K. Abe et al., Phys. Lett. B 511 (2001) 151.
. R Ammar, CLEO CollaborationPhys. Rev. Lett. 71674CLEO Collaboration, R. Ammar et al., Phys. Rev. Lett. 71 (1993) 674.
. H Albrecht, ARGUS CollaborationPhys. Lett. B. 241278ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B 241 (1990) 278.
For the K * (892) 0 π 0 limit we follow the approach of. G J Feldman, R D Cousins, ; J Conrad, Phys. Rev. D. 57Phys. Rev. D. 012002 to include the systematic errors. For the K + π − π 0For the K * (892) 0 π 0 limit we follow the approach of G.J. Feldman and R.D. Cousins, Phys. Rev. D 57 (1998) 3873, using the implementation of J. Conrad et al., Phys. Rev. D 67 (2003) 012002 to include the systematic errors. For the K + π − π 0
N R mode this method returns a two-sided interval. 2.0, 9.4]×10 −6 . Because the significance is less than three, we discard the lower edge of this interval and (conservatively) quote an upper limit of 9.4 × 10 −6N R mode this method returns a two-sided interval [2.0, 9.4]×10 −6 . Because the significance is less than three, we discard the lower edge of this interval and (conservatively) quote an upper limit of 9.4 × 10 −6 .
| {'fraction_non_alphanumeric': 0.053994503998217515, 'fraction_numerical': 0.032901740400564455, 'mean_word_length': 3.7939710420128177, 'pattern_counts': {'":': 0, '<': 21, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We report the observation of B 0 decays to the K + π − π 0 final state using a data sample of 78 fb −1 collected by the Belle detector at the KEKB e + e − collider. With no assumptions about intermediate states in the decay, the branching fraction is measured to be (36.6 +4.2 −4.3 ± 3.0) × 10 −6 . We also search for B decays to intermediate two-body states with the same K + π − π 0 final state. Significant B signals are', 'arxivid': 'hep-ex/0406075', 'author': ['K + ', 'P Chang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'K Abe \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'K Abe ', 'T Abe \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Aihara ', 'Y Asano ', 'V Aulchenko \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'T Aushev \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'T Aziz ', 'S Bahinipati \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'A M Bakich ', 'Y Ban ', 'A Bay \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', 'I Bedny \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'U Bitenc \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'I Bizjak \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'A Bondar \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Bozek ', 'M Bračko \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Maribor\nMariborSlovenia\n', 'J Brodzicka ', 'T E Browder \nUniversity of Hawaii\nHonoluluHIUSA\n', 'M.-C Chang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'Y Chao \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'B G Cheon \nChonnam National University\nKwangjuSouth Korea\n', 'R Chistov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'S.-K Choi \nGyeongsang National University\nChinjuSouth Korea\n', 'Y Choi ', 'S Cole ', 'M Danilov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'M Dash ', 'L Y Dong \nInstitute of High Energy Physics\nChinese Academy of Sciences\nBeijingPR China\n', 'A Drutskoy \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'S Eidelman \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'V Eiges \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', "Y Enari \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'F Fang \nUniversity of Hawaii\nHonoluluHIUSA\n', 'S Fratina \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'N Gabyshev \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Garmash ', 'T Gershon \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'G Gokhroo ', 'B Golob \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Ljubljana\nLjubljanaSlovenia\n', 'R Guo \nNational Kaohsiung Normal University\nKaohsiungTaiwan\n', 'N C Hastings \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Hayashii ', 'M Hazumi \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'T Higuchi ', 'L Hinz \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', "T Hokuue \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'Y Hoshi ', 'W.-S Hou \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'Y B Hsiung \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'H.-C Huang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', "K Inami \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'A Ishikawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'R Itoh \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Iwasaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'M Iwasaki ', 'Y Iwasaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'J H Kang Au ', 'J S Kang \nKorea University\nSeoulSouth Korea\n', 'S U Kataoka ', 'N Katayama \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Kawai \nChiba University\nChibaJapan\n', 'T Kawasaki Aa ', 'H R Khan ', 'H Kichimi \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H J Kim \nKyungpook National University\nTaeguSouth Korea\n', 'J H Kim ', 'S K Kim ', 'T H Kim Au ', 'P Koppenburg \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'S Korpar \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Maribor\nMariborSlovenia\n', 'P Križan \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Ljubljana\nLjubljanaSlovenia\n', 'P Krokovny \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Kuzmin \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'Y.-J Kwon Au ', 'S E Lee ', 'T Lesiak ', 'J Li ', 'Ag ', 'S.-W Lin \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'J Macnaughton \nInstitute of High Energy Physics\nViennaAustria\n', 'G Majumder ', 'F Mandl \nInstitute of High Energy Physics\nViennaAustria\n', 'D Marlow ', 'T Matsumoto ', 'A Matyja ', 'W Mitaroff \nInstitute of High Energy Physics\nViennaAustria\n', 'H Miyake ', 'H Miyata Aa ', 'R Mizuk \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'D Mohapatra ', 'T Mori ', 'T Nagamine ', 'Y Nagasaka \nHiroshima Institute of Technology\nHiroshimaJapan\n', 'E Nakano ', 'M Nakao \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Nakazawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Z Natkaniec ', 'S Nishida \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'O Nitoh Ar ', 'S Ogawa ', "T Okabe \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'S Okuno \nKanagawa University\nYokohamaJapan\n', 'S L Olsen \nUniversity of Hawaii\nHonoluluHIUSA\n', 'W Ostrowicz ', 'H Ozaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'P Pakhlov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'H Park \nKyungpook National University\nTaeguSouth Korea\n', 'N Parslow ', 'L S Peak ', 'L E Piilonen ', 'A Poluektov \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'N Root \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'M Rozanska ', 'H Sagawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Y Sakai \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'O Schneider \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', 'J Schümann \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'S Semenov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', "K Senyo \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'M E Sevior \nUniversity of Melbourne\nVictoriaAustralia\n', 'H Shibuya ', 'V Sidorov \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Somov \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'R Stamen \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'S Stanič ', 'M Starič \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'K Sumisawa ', 'T Sumiyoshi ', 'S Suzuki ', 'O Tajima ', 'F Takasaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'K Tamai \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'N Tamura ', 'M Tanaka \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Y Teramoto ', 'T Tsukamoto \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'S Uehara \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'K Ueno \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'T Uglov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'Y Unno \nChiba University\nChibaJapan\n', 'S Uno \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'G Varner \nUniversity of Hawaii\nHonoluluHIUSA\n', 'K E Varvell ', 'S Villa \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', 'C C Wang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'C H Wang \nNational United University\nMiao LiTaiwan\n', 'M.-Z Wang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'M Watanabe Aa ', 'atB D Yabsley ', 'Y Yamada \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'A Yamaguchi ', 'Y Yamashita \nNihon Dental College\nNiigataJapan aa\n\nNiigata University\nNiigataJapan ab\n\nOsaka City University\nOsakaJapan\n\nac Osaka University\nOsakaJapan\n\nad Peking University\nBeijingPR\n\nChina ae Princeton University\nPrincetonNJUSA\n\naf Saga University\nSagaJapan\n\nag University of Science and Technology of China\nHefeiPR China\n\nah Seoul National University\nSeoulSouth Korea\n\nai Sungkyunkwan University\nSuwonSouth Korea\n\naj University of Sydney\nSydneyNSWAustralia\n\nTata Institute of Fundamental Research\nBombayIndia\n\naℓ Toho University\nFunabashiJapan am\n\nTohoku Gakuin University\nTagajoJapan\n\nDepartment of Physics\nan Tohoku University\nSendaiJapan\n\nUniversity of Tokyo\nTokyoJapan ap\n\nTokyo Institute of Technology\nTokyoJapan\n\nar Tokyo University of Agriculture and Technology\naq Tokyo Metropolitan University\nTokyo, TokyoJapan, Japan\n\nUniversity of Tsukuba\nTsukubaJapan at\n\nVirginia Polytechnic Institute and State University\nBlacksburgVAUSA\n\nau Yonsei University\nSeoulSouth Korea\n', 'M Yamauchi \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'J Ying ', 'C C Zhang \nInstitute of High Energy Physics\nChinese Academy of Sciences\nBeijingPR China\n', 'J Zhang \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Z P Zhang Ag ', 'D Žontar \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Ljubljana\nLjubljanaSlovenia\n', 'Nara ', 'H Niewodniczanski ', '\nBelle Collaboration\n\n', 'K + ', 'P Chang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'K Abe \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'K Abe ', 'T Abe \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Aihara ', 'Y Asano ', 'V Aulchenko \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'T Aushev \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'T Aziz ', 'S Bahinipati \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'A M Bakich ', 'Y Ban ', 'A Bay \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', 'I Bedny \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'U Bitenc \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'I Bizjak \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'A Bondar \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Bozek ', 'M Bračko \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Maribor\nMariborSlovenia\n', 'J Brodzicka ', 'T E Browder \nUniversity of Hawaii\nHonoluluHIUSA\n', 'M.-C Chang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'Y Chao \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'B G Cheon \nChonnam National University\nKwangjuSouth Korea\n', 'R Chistov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'S.-K Choi \nGyeongsang National University\nChinjuSouth Korea\n', 'Y Choi ', 'S Cole ', 'M Danilov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'M Dash ', 'L Y Dong \nInstitute of High Energy Physics\nChinese Academy of Sciences\nBeijingPR China\n', 'A Drutskoy \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'S Eidelman \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'V Eiges \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', "Y Enari \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'F Fang \nUniversity of Hawaii\nHonoluluHIUSA\n', 'S Fratina \nJ. Stefan Institute\nLjubljanaSlovenia\n', 'N Gabyshev \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Garmash ', 'T Gershon \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'G Gokhroo ', 'B Golob \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Ljubljana\nLjubljanaSlovenia\n', 'R Guo \nNational Kaohsiung Normal University\nKaohsiungTaiwan\n', 'N C Hastings \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Hayashii ', 'M Hazumi \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'T Higuchi ', 'L Hinz \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', "T Hokuue \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'Y Hoshi ', 'W.-S Hou \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'Y B Hsiung \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'H.-C Huang \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', "K Inami \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'A Ishikawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'R Itoh \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Iwasaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'M Iwasaki ', 'Y Iwasaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'J H Kang Au ', 'J S Kang \nKorea University\nSeoulSouth Korea\n', 'S U Kataoka ', 'N Katayama \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Kawai \nChiba University\nChibaJapan\n', 'T Kawasaki Aa ', 'H R Khan ', 'H Kichimi \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H J Kim \nKyungpook National University\nTaeguSouth Korea\n', 'J H Kim ', 'S K Kim ', 'T H Kim Au ', 'P Koppenburg \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'S Korpar \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Maribor\nMariborSlovenia\n', 'P Križan \nJ. Stefan Institute\nLjubljanaSlovenia\n\nUniversity of Ljubljana\nLjubljanaSlovenia\n', 'P Krokovny \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Kuzmin \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'Y.-J Kwon Au ', 'S E Lee ', 'T Lesiak ', 'J Li ', 'Ag ', 'S.-W Lin \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'J Macnaughton \nInstitute of High Energy Physics\nViennaAustria\n', 'G Majumder ', 'F Mandl \nInstitute of High Energy Physics\nViennaAustria\n', 'D Marlow ', 'T Matsumoto ', 'A Matyja ', 'W Mitaroff \nInstitute of High Energy Physics\nViennaAustria\n', 'H Miyake ', 'H Miyata Aa ', 'R Mizuk \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'D Mohapatra ', 'T Mori ', 'T Nagamine ', 'Y Nagasaka \nHiroshima Institute of Technology\nHiroshimaJapan\n', 'E Nakano ', 'M Nakao \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'H Nakazawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Z Natkaniec ', 'S Nishida \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'O Nitoh Ar ', 'S Ogawa ', "T Okabe \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'S Okuno \nKanagawa University\nYokohamaJapan\n', 'S L Olsen \nUniversity of Hawaii\nHonoluluHIUSA\n', 'W Ostrowicz ', 'H Ozaki \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'P Pakhlov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'H Park \nKyungpook National University\nTaeguSouth Korea\n', 'N Parslow ', 'L S Peak ', 'L E Piilonen ', 'A Poluektov \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'N Root \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'M Rozanska ', 'H Sagawa \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'Y Sakai \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'O Schneider \nSwiss Federal Institute of Technology of Lausanne, EPFL, Lausanne\n\n', 'J Schümann \nDepartment of Physics\nInstitute of Nuclear Physics\nNational Taiwan University\nTaipei, KrakowTaiwan, Poland\n', 'S Semenov \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', "K Senyo \nNagoya University\nNagoyaJapan\n\nWomen's University\nNaraJapan\n", 'M E Sevior \nUniversity of Melbourne\nVictoriaAustralia\n', 'H Shibuya ', 'V Sidorov \nBudker Institute of Nuclear Physics\nNovosibirskRussia\n', 'A Somov \nUniversity of Cincinnati\nCincinnatiOHUSA\n', 'R Stamen \nHigh Energy Accelerator Research Organization (KEK)\nTsukubaJapan\n', 'S Stanič ', 'M Starič \nJ. 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arxiv |
Tighter inapproximability for set cover
6 Dec 2016
David G Harris
Tighter inapproximability for set cover
6 Dec 2016
Set Cover is a classic NP-hard problem; as shown by Slavík (1997) the greedy algorithm gives an approximation ratio of ln n − ln ln n + Θ(1). A series of works byLund & Yannakakis (1994), Feige (1998,Moshkovitz (2015)have shown that, under the assumption P = N P , it is impossible to obtain a polynomial-time approximation ratio with approximation ratio (1 − α) ln n, for any constant α > 0.In this note, we show that under the Exponential Time Hypothesis (a stronger complexitytheoretic assumptions than P = N P ), there are no polynomial-time algorithms achieving approximation ratio ln n−C ln ln n, where C is some universal constant. Thus, the greedy algorithm achieves an essentially optimal approximation ratio (up to the coefficient of ln ln n).
Introduction
Set Cover is a classic NP-hard problem. In its simplest form, it can be stated as follows: given a collection S = {S 1 , . . . , S m } of subsets of [n], find a subset T ⊆ S such that S∈T S = [n], and such that |T | is minimized. There are many variants and generalizations of this basic framework.
It has long been known that the greedy algorithm (repeatedly add to T the element of S which covers the largest number of uncovered elements of the ground set) gives a ln n approximation to this problem [3]. Slavík [6] later showed that in fact the greedy algorithm achieves an approximation ratio of ln n − ln ln n + Θ(1). An alternative algorithm based on LP relaxation and randomized rounding yields a similar approximation ratio [7].
On the other hand, a series of works [4,1] have shown lower bounds on the approximation ratio available for polynomial-time algorithms for Set Cover; most recently, Moshkovitz [5] has shown that, under the assumption P = N P , it is impossible to obtain an approximation ratio of (1−α) ln n for any fixed constant α > 0.
There still remains a large gap between the lower bound of (1 − Ω(1)) ln n and the upper bound of ln n − ln ln n + Θ(1). In this short note, we show that under stronger complexity-theoretic assumptions, we can mostly close this gap. Namely, there is some universal constant C such that no polynomial-time algorithm can achieve an approximation ratio of ln n − C ln ln n.
Hardness of Set Cover from Exponential Time Hypothesis
The result of [5] was based on the hypothesis that P = N P , i.e. that algorithms deciding satisfiability have a time-complexity at least n ω (1) . In fact, it is believed that such SAT algorithms must be much slower -essentially they must perform a brute-force enumeration over their n variables. The Exponential Time Hypothesis, first introduced in [2], formalizes this conjecture. There are several formulations of the ETH; we state one version here: Conjecture 2.1 (Exponential Time Hypothesis). Let k > 2 be an integer and let A be an algorithm which solves k-SAT instances on n variables in time at most f (n). Then there is some parameter γ > 0 (which may depend on k, A) such that f (n) > 2 γn for infinitely many n.
We will show a lower bound on the approximation ratio for Set Cover, assuming Conjecture 2.1. We begin by quoting a result of [5]. We note that the size of any set cover instance is at least equal to the size of its ground set n. Thus, if we have any algorithm which guarantees an approximation ratio β(n) for problem instances with ground set n, it also guarantees an approximation ratio β(t) for problem instances of size t.
We now obtain our main result: Theorem 2.3. Let A be an polynomial-time approximation algorithm for Set Cover, which guarantees an approximation ratio β(n) on problem instances with ground set n. Then there is some universal constant C ≥ 0, such that for infinitely many n we have β(n) ≥ ln n − C ln ln n.
Proof. Suppose that there is an algorithm A which takes a Set Cover of size v and runs in time v k for some parameter k > 0. In order for this to be a polynomial-time algorithm, k must be independent of v, but k is not necessarily bounded by a universal constant. We can assume that k ≥ c 3 for any fixed constant c 3 .
Set c = 6c 1 , and suppose for contradiction that β(n) ≤ ln n − c ln ln n for all n > n 0 , where n 0 is a parameter which may depend on A.
Any 3-SAT instance on n variables has size at most n 3 . By Theorem 2.2, in order to solve such a SAT instance of size n, it suffices to approximate Set Cover to within (1 − α) ln N for a problem instance of size N ≤ n 3c 1 /α , for any α > 0. We will choose to set α = 3c 1 n −3c 1 /c ln n Observe that α ∈ (0, 1) for n sufficiently large.
So N = e n 3c 1 /c and (1 − α) ln N = n 3c 1 /c − 3c 1 ln n = ln N − c ln ln N . The Set Cover instance has size N , so A guarantees an approximation ratio of β(N ); as N grows as a function of n, we have N ≥ n 0 for n > n 1 where n 1 is some parameter depending on A.
Hence, for n > n 1 , the guaranteed approximation ratio β(N ) satisfies β(N ) ≤ ln N − c ln ln N .
Thus, by Theorem 2.2, A can be used as a building-block to solve the 3-SAT instance. Let us denote the resulting 3-SAT algorithm B. The run-time of A is at most N k ; other steps in the reduction from SAT to Set Cover run in time poly(n); thus, if we take k to be sufficiently large, the overall run-time of algorithm B at most 2N k .
By Conjecture 2.1, there must be some γ > 0 such that 2N k ≥ 2 γn for infinitely many values of n. This implies that for infinitely many values of n we have ln 2 + kn 3c 1 /c ≥ γn ln 2 However, 3c 1 /c = 1/2, and so the RHS dominates the LHS for all sufficiently large n, a contradiction.
We note that there a several versions of the ETH which are stronger than Conjecture 2.1. One form of the conjecture states for any fixed integer k > 2 there is some universal constant γ k such that any k-SAT algorithm requires time 2 γ k n for infinitely many n. (In particular, γ k is not allowed to depend on the k-SAT algorithm.) A yet stronger form of the conjecture, known as the Strong Exponential Time Hypothesis (SETH), states that γ k → 2 as k → ∞. However, the argument used in Theorem 2.3 does not lead to stronger inapproximability bounds from these stronger hypotheses.
Theorem 2.2 ([5]). For every 0 < α < 1, (exact) SAT on inputs of size n can be reduced in polynomial time to approximating Set Cover to within (1 − α) ln N on inputs of size N = n c 1 /α , where c 1 > 0 is a constant.
AcknowledgementsThanks to Aravind Srinivasan and Dana Moshkovitz, for helpful discussions and references to literature.
A threshold of ln n for approximating set cover. U Feige, Journal of the ACM 45-4. Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45-4, pp. 634-652 (1998).
On the complexity of k-SAT. R Impagliazzo, R Paturi, Journal of Compuyter and System Sciences. 62Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. Journal of Compuyter and System Sciences 62, pp. 367-375 (2001).
Approximation algorithms for combinatorial problems. D Johnson, Journal of Computer and System Sciences. 93Johnson, D.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9-3, pp. 256-278 (1974).
On the hardness of approximating minimization problems. C Lund, M Yannakakis, Journal of the ACM. 5Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. Journal of the ACM 41-5, pp. 960-981 (1994).
The projection games conjecture and the NP-hardness of ln n-approximating set-cover. D Moshkovitz, Theory of Computing. Moshkovitz, D.: The projection games conjecture and the NP-hardness of ln n-approximating set-cover. Theory of Computing 11-7, pp. 221-235 (2015).
A tight analysis of the greedy algorithm for set cover. P Slavík, Journal of Algorithms. 252Slavík, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25-2, pp. 237-254 (1997).
Improved approximations guarantees for packing and covering integer programs. A Srinivasan, SIAM Journal on Computing. 2Srinivasan, A.: Improved approximations guarantees for packing and covering integer programs. SIAM Journal on Computing 29-2, pp. 648-670 (1999).
| {'fraction_non_alphanumeric': 0.05353868720205354, 'fraction_numerical': 0.027258281383693926, 'mean_word_length': 3.7542126670540386, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Set Cover is a classic NP-hard problem; as shown by Slavík (1997) the greedy algorithm gives an approximation ratio of ln n − ln ln n + Θ(1). A series of works byLund & Yannakakis (1994), Feige (1998,Moshkovitz (2015)have shown that, under the assumption P = N P , it is impossible to obtain a polynomial-time approximation ratio with approximation ratio (1 − α) ln n, for any constant α > 0.In this note, we show that under the Exponential Time Hypothesis (a stronger complexitytheoretic assumptions than P = N P ), there are no polynomial-time algorithms achieving approximation ratio ln n−C ln ln n, where C is some universal constant. Thus, the greedy algorithm achieves an essentially optimal approximation ratio (up to the coefficient of ln ln n).', 'arxivid': '1612.01610', 'author': ['David G Harris '], 'authoraffiliation': [], 'corpusid': 17864641, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2498, 'n_tokens_neox': 2152, 'n_words': 1471, 'pdfsha': '3eb153eb049d0d7a89f8ce2b27e08ac4c5ef11af', 'pdfurls': ['https://arxiv.org/pdf/1612.01610v1.pdf'], 'title': ['Tighter inapproximability for set cover', 'Tighter inapproximability for set cover'], 'venue': []} |
arxiv |
Springer Nature 2021 L A T E X template Bayesian Testing Of Granger Causality In Functional Time Series
Rituparna Sen
Applied Statistics Unit
Indian Statistical Institute
Mysore Rd8thMile
RVCE Post
560059BangaloreKAIndia
Anandamayee Majumdar
Stock Assessment Program
Inter-American Tropical Tuna Commission
San DiegoCAUSA
Shubhangi Sikaria [email protected]
Department of Mathematics
Indian Institute of Technology Madras
ChennaiTNIndia
Springer Nature 2021 L A T E X template Bayesian Testing Of Granger Causality In Functional Time Series
1 arXiv:2112.15315v1 [stat.ME] 31 Dec 2021 Springer Nature 2021 L A T E X template 2 Bayesian Testing Of Granger Causality In Functional Time SeriesMultivariate functional time seriesdynamic linear modelGranger causalityBayesian Analysis
We develop a multivariate functional autoregressive model (MFAR), which captures the cross-correlation among multiple functional time series and thus improves forecast accuracy. We estimate the parameters under the Bayesian dynamic linear models (DLM) framework. In order to capture Granger causality from one FAR series to another we employ Bayes Factor. Motivated by the broad application of functional data in finance, we investigate the causality between the yield curves of two countries. Furthermore, we illustrate a climatology example, examining whether the weather conditions Granger cause pollutant daily levels in a city.
Introduction
A functional time series arises when at each point in time we observe a random function, generally at discrete points on a dense support. Examples of functions include satellite images(Guyet & Nicolas [1]), fMRI data(Stoehr, Aston & Kirch [2]) and yield curves ( Sen & Klüppelberg [3]). Another common practice that leads to functional time series is to slice high-frequency temporal sequence records into curves over logical consecutive time intervals. Examples of such curve time series include the daily price curve of financial transactions ( Horváth, Kokoszka & Rice [4]; Li, Robinson & Shang [5]), electricity price curves as a function of day-of-year ( Chen & Li [6]; Chen, Marron & Zhang [7]), daily pollution levels as a function of the time of the day (Aue, Norinho & Hörmann [8]), and daily patterns of environmental data (Besse, Cardot & Stephenson [9], Ramsay & Dalzell [10]). The functional autoregressive (FAR) model, developed by Bosq [11], plays a central role in modeling and predicting functional time series. The FAR model extends the vector autoregressive (VAR) model to the infinite-dimensional space and considers the serial correlation between the functional observations.
Most of the existing literature relies on functional principal component analysis (FPCA) to treat multivariate functional data. In a multivariate setting, along with serial correlation, one needs to study the correlation among the data series. Ramsay & Silverman [12] introduced a classical multivariate FPCA that concatenates the multiple functional data and performs univariate FPCA on a single long vector. However, this model is restricted to multivariate random functions on the same finite, one-dimensional interval. Chiou, Chen & Yang [13] discussed a normalized multivariate functional principal component analysis (mFPCA n ) which accounts for data measured in different units using normalized covariance operators. Although mFPCA n produces sound estimates on dense grids without calculation errors, it leads to inadmissible results with significant measurement errors in the case of sparse settings.
Apart from FPCA, there is abundant literature that models multivariate functional data using regression analysis. Zhu, Li & Kong [14] developed a multivariate varying coefficients model (MVCM) to estimate covariance function. They also establish the uniform convergence rate of this covariance function. Li et al. [15] extended the previous varying-coefficient single index model to perform the regression on functional response data. Zhu et al. [16] performed linear transformation and adopted the Bayesian approach to sample the model parameters in the transformed space. Zhu, Strawn, & Dunson [17] introduced a notion of graphical models in multivariate functional data by constructing Bayesian inference of undirected, decomposable graphs based on Markov distribution. The literature mentioned above mainly focuses on dense functional data. Li, Xiao & Luo [18] adopted the fast covariance-based FPCA method for multivariate sparse functional data, approximating covariance functions using Tensor-product B-splines. The underlying assumption while using FPCA is that the observable process with measurement errors follows the multivariate functional autoregressive (MFAR) process. We later prove that such an observable process follows a multivariate functional autoregressive moving average (FARMA) process. Thus, existing approaches based on FPCA produce inefficient estimates. This paper proposes a hierarchical model for multivariate functional time series, which is the extension of Kowal, Matteson & Ruppert [19] to the multivariate setting. This model will overcome the issue of inconsistent solutions arising in the FPCA approach and produce efficient results in a sparse setting. To estimate the MFAR model, we firstly form a two-level hierarchical model using the observed values. For computation, we later represent the hierarchical model as a dynamic linear model (DLM). We use observation equations to discretize the functional data and obtain measurement error. The evolution equations take into account the correlation among the functional time series and define the process model. The latent process in our case is the multivariate functional autoregressive model of order p, written as MFAR(p). A dynamic functional factor model further specifies the dynamic innovation process. We employ an efficient Gibbs sampling algorithm to perform inference and prediction for multivariate time series.
While studying the cross-correlation among the bivariate functional time series, a natural question is whether there exists a directionality of information. We are interested in analyzing the causal relations among a set of variables. As discussed by Granger [20], the first time series X is said to have a causal influence on the second time series Y if the prediction of second series Y could be improved by the inclusion of past measurements from the first time series X. Later, classical Granger causality was extended to different types of time series model. Boudjellaba, Dufour, & Roy [21] derived the necessary and sufficient condition for non-causality in stationary vector autoregressive (VAR) process. Comte & Lieberman [22] investigated second-order non-causality in the framework of the VARMA process with generalized autoregressive conditional heteroscedasticity (GARCH) type errors. This unified treatment of first and second-order causality is widely used with some modification in various fields (see, e.g., Hafner & Herwartz [23], Woźniak [24], Chen, Marron & Zhang [7]). For most of these models, restrictions on parameters are linear functions. Droumaguet & Woźniak [25] applied Markov-switching VAR models for which the restrictions for non-causality in variance are a non-linear function of parameters and used the Bayesian approach to test non-causality. Allen & Hooper [26] used Granger causality generalized measures of correlation (GcGMC) to analyze the causal relations between VIX and S&P 500.
Above mentioned literature has applied some modified versions of Granger causality to multivariate time series. Saumard [27] investigated the causality of functional time series by testing the equality of covariance operators for dependent processes. Recently, Shang, Ji & Beyaztas [28] extended the GcGMC to bivariate functional time series. This paper will address the Granger causality for functional time series utilizing the Bayes factor technique introduced by Kass & Raftery [29]. Bayes factor provides a way of evaluating evidence in favor of the null hypothesis; in our case, it tests the non-causality. Following Geweke [30], we use a sample of the posterior distribution to evaluate the model's marginal likelihood, estimate the integral over the posterior sample, and calculate the modified harmonic mean (MHM). Using the marginal density of data of the two models under consideration, we calculate and use the Bayes factor to choose the preferred model. The rest of the paper is organized as follows. In Section 2 we illustrate the MFAR model of order p and its representation in a dynamic linear model (DLM), which is further used for inference and prediction. We then generalize the concept of Granger causality to functional time series. Section 3 provides the methodology followed in this paper including a detailed explanation of the Bayes factor and the method of obtaining the same. Section 4 presents the empirical results of checking the causal relationship among the bond yields of different countries and effect of meteorological factors on air pollution level. Section 5 presents the concluding remarks.
The Model
Suppose we have K functional time series defined as {Y n t }, n = 1, · · · , K on some
compact index set T ⊂ R D , typically D = 1. Consider {Y Y Y t } as K−variate functional time series represented as: Y Y Y t = [Y 1 t , · · · ,Y K t ]
where Y n t ∈ L 2 (T ) for n = 1, 2, · · · , K.
Multivariate Functional Autoregressive Model
Multivariate functional autoregressive model of order p, written as MFAR(p):
Y Y Y t − µ µ µ = p ∑ l=1 Ψ Ψ Ψ l (Y Y Y t−l − µ µ µ) + ε ε ε t .(1)
where
µ µ µ = [µ 1 , µ 2 , · · · , µ K ] is mean of Y Y Y t under stationarity, ε ε ε t = [ε 1 t , ε 2 t , · · · , ε K t ]
is a vector of innovation functions with each ε n t being a sequence of independent mean zero random functions with E || ε n t || 2 < ∞ for n = 1, · · · , K and Ψ Ψ Ψ l = [Ψ i, j l ] K i, j=1 where each Ψ i, j l is a bounded linear operator on L 2 (T ) for i, j = 1, · · · , K and l = 1, 2, · · · , p. We assume that each function Y n t (τ n ), n = 1, · · · , K is observed at unequallyspaced points τ n ∈ T . We restrict to p = 1 and consider only integral operators for better interpretability and computational convenience. In practice, the functional observations {Y t } are observed via discrete samples of each curve with some measurement error. Suppose we observe y n i,t ∈ R sampled with measurement error v n i,t from {Y n t } ∈ L 2 (T ), n = 1, · · · , K:
y n i,t = Y n t (τ n i,t ) + v n i,t(2)
where τ n i,t for i = 1, · · · , m n t are observation points of Y n t and v n i,t is a mean zero measurement error with finite variance. Assume α α α t = Y Y Y t − µ µ µ, then two level hierarchical model is given as:
y n i,t = µ n (τ n i,t ) + α n t (τ n i,t ) + v n i,t , i = 1, · · · , m n t , α n t = K ∑ m=1 ψ n,m (τ n , u)α m t−1 (u)du + ε n t (τ n ), ∀τ ∈ T and n = 1, · · · , K,(3)
for t = 2, · · · , T, where ψ n,m are integral operators and we assume that {v v v i,t } and {ε ε ε t } are mutually independent.
The presence of measurement error in model (3) increases the estimation error of Ψ Ψ Ψ and produces inefficient forecasts. The following proposition illustrates that MFAR(p) model for the observables is inappropriate.
Proposition 1: Let Y Y Y t − µ µ µ = ∑ p l=1 Ψ Ψ Ψ(Y Y Y t−l + ε ε ε t )
and suppose we observe y y y t = Y Y Y t + v v v t are independent white noise processes. Then the observable process {y y y t } follows a multivariate functional autoregressive moving average (FARMA) process of order (p, p). For proof see Kowal et al. [19].
To overcome this model mis-specification, we separate the observed data into functional process and measurement errors in the dynamic linear model (DLM).
Dynamic Linear Models for MFAR(p)
For practical implementation of model (3), we select a finite set of evaluation points for each of functional time series, T n e = {τ n 1 , · · · , τ n M } ⊂ T for n = 1, · · · , K. We approximate the integral in (3) using quadrature methods with Q n,m for n, m = 1, · · · , K, as a known quadrature weight matrix. Let Z n t be m n t × M, n = 1, · · · , K, incidence matrix that identifies the observation points observed at time t for n-th functional times series. We can formulate the dynamic linear model (DLM) corresponding to the hierarchical model (3) as:
y y y t = Z Z Z t µ µ µ + Z Z Z t α α α t + v v v t , [v v v t | Σ v ] indep ∼ N(0 0 0, Σ v ) for t = 1, · · · , T, α α α t = ψ ψ ψQ Q Qα α α t−1 + ε ε ε t , [ε ε ε t | K K K ε ] indep ∼ N(0 0 0, K K K ε ) for t = 2, · · · , T, α α α 1 ∼ N(0 0 0, K K K ε )(4)
where y y y t = [y n i,t ], is a discrete sample of K-variate functional time series {Y Y Y t } for i = 1, · · · , m n t and n = 1, · · · , K. Here,
µ µ µ = [µ 1 (τ 1 ), · · · , µ K (τ K )] and α α α t = [α 1 t (τ 1 ), · · · , α K t (τ K )] where µ i (τ i ) = [µ i (τ i 1 ), · · · , µ i (τ i M )] and α i t (τ i ) = [α i t (τ i 1 ), · · · , α i t (τ i M )] for i = 1, · · · , K.
Other matrices are defined as:
Z Z Z t = Z 1 t 0 · · · 0 . . . . . . . . . 0 0 · · · Z K t , ψ ψ ψ = {ψ 1,1 (τ i , τ j )} M i, j=1 · · · {ψ 1,K (τ i , τ j )} M i, j=1 . . . . . . . . . {ψ 1,K (τ i , τ j )} M i, j=1 · · · {ψ K,1 (τ i , τ j )} M i, j=1 and Q Q Q = Q 1,1 · · · Q 1,K . . . . . . . . . Q 1,K · · · Q K,1 .
Model (4) can be extended to multiple lags to the MFAR(p) by replacing the evolution equation with α α α t = ∑ p l=1 ψ ψ ψ l Q Q Qα α α t−l + ε ε ε t .
Granger Causality in Functional Autoregressive Model
Here, we investigate the causal relationship between two functional time series data Y t and X t , t ∈ T observed on unequally spaced points τ ∈ T where T ⊂ R is a compact index set. Firstly, we model Y t as functional autoregressive model of order 1, written as FAR (1), and partition the linear projection of Y t on Y t−1 and X t−1 to account for the Granger causality as
Y t − µ Y X t − µ X = Ψ YY Ψ Y X 0 Ψ XX Y t−1 − µ Y X t−1 − µ X + ε Y t ε X t (M1)
In second model, we model Y t as linear projection of Y t−1 only as
Y t − µ Y X t − µ X = Ψ YY 0 0 Ψ XX Y t−1 − µ Y X t−1 − µ X + ε Y t ε X t (M2)
where Y t and X t ∈ L 2 (T ) and µ Y and µ X are mean of {Y t } and {X t } respectively.
Consider ε t = (ε Y t ε X t ) and ε t = (ε Y t ε X t )
as zero mean independent random functions with covariance matrices K ε and K ε respectively. In model (M1), Y t depends on both Y t−1 and X t−1 and is considered as unrestricted model while model (M2) is considered as restricted as Y t depends only it's past values. In this setting, X is said to Granger cause Y if model (M1) is significantly better than model (M2) at forecasting future values of Y . Model (M1) and (M2) are in MFAR form, but for the computation purpose, we need to convert the MFAR models into DLM form as described in equation (4). Let Θ 1 and Θ 2 be the vector of parameters corresponding to model (M1) and (M2), respectively. Now,
Θ 1 = (µ, Σ ν , Ψ, K ε ) and Θ 2 = (µ, Σ ν , Ψ , K ε ) where, µ = µ Y µ X , Σ ν = σ 2 ν Y 0 0 σ 2 ν X , Ψ = Ψ YY Ψ Y X 0 Ψ XX and Ψ = Ψ YY 0 0 Ψ XX .
Methodology
We hypothesize that the unrestricted model (M1) yields better prediction in contrast to the restricted model (M2). The null hypothesis indicates no additional contribution from the past of X to the future evolution of Y once the past of Y has been take into account. The alternative is that there is an additional contribution. In the univariate setting the usual way to test for Granger causality is to look at the forecast error sum of squares from each mode land compare them through an F test. For multivariate data, this can be extended by taking the trace or determinant of the covariance matrices under each model. As the dimension of the data becomes large, the F test becomes more unreliable as the estimation of the covariance matrix in higher dimensions is not efficient. For functional data, which is infinite dimensional, this procedure is unusable. Hence we look into Bayesian procedures which concentrate on the dimension of the parameters and not of the data. The Bayes factor is a central quantity of interest in Bayesian hypothesis testing. The ratio of the marginal data densities for both hypothesis-specific models is known as the Bayes factor. Let p(Y t | M) be the marginal data density for model M; then the Bayes factor would be
B 12 = p(Y t | M1) p(Y t | M2) .(5)
Bayesian hypothesis testing begins by specifying different prior distributions on the parameters involved in the models. We also have the likelihood for the data given the prior values of the parameter. Using these, the crucial step is the computation of marginal data densities, p(Y t | M) for both models. For computing this value, we apply the Modified Harmonic Mean (MHM) method. Let Θ be a K-variate vector of parameters corresponding to the model. Firstly, obtain the sample of draws {Θ (i) } S i=1 , where S is the number of samples, using the posterior distribution of the parameters p(Θ | Y t ) for each model. Now, the marginal density of data is computed as:
p(Y t | M) = S −1 S ∑ i=1 h(Θ (i) ) L (Y t ; Θ (i) )p(Θ (i) ) −1(6)
where, L (Y t ; Θ (i) ) is the likelihood function and p(Θ (i) ) is the prior density function of parameters obtained by substituting sample of draws {Θ (i) } S i=1 . Also, h(Θ (i) ) is a K-variate truncated Normal distribution with mean equal to posterior mean of Θ and variance set to the posterior covariance matrix of Θ.
In the rest of this section we elaborate the various steps involved in this procedure. In sections 3.1 and 3.2 we present the assumed prior distribution and the likelihood of the data under the two models. In section 3.3 we describe the method to draw samples from the posterior distribution of the parameter. Finally in section 3.4 we present the algorithm to compute the Bayes factor.
Prior Distribution
We specify the prior distributions for each of parameter in the vector Θ = (µ, Σ ν , Ψ, ε t ) below.
1. Prior distribution for mean function, µ:
µ(τ) = b µ (τ)θ µ , for τ ∈ T , T is the index set,
where, b µ is low rank thin plate spline basis and
θ µ ∼ N(0, Λ µ ) s.t. Λ µ = diag(10 8 , 10 8 , λ −1 µ , λ −1 µ , · · · , λ −1 µ ) and λ −1/2 µ ∼ Uniform(0, 10 4 ).
2. Prior distribution for observed covariance matrix, Σ ν :
Σ ν = σ 2 ν Y 0 0 σ 2 ν X
where, σ 2 ν Y ∼ Gamma(10 −3 , 10 −3 ) and σ 2 ν X ∼ Gamma(10 −3 , 10 −3 ).
3. Prior distribution for FAR kernel, Ψ:
Ψ(τ, u) = (b Ψ (u) ⊗ b Ψ (τ))θ Ψ , for τ, u ∈ T , where, b Ψ is cubic B-spline basis and [θ Ψ | λ Ψ ] ∼ N(0, λ −1 Ψ Ω −1 Ψ ) (θ Ψ induces a Gaussian prior on Ψ) s.t. λ Ψ =ζ −2 Ψλ Ψ ,
where,ζ Ψ ∼ N(0, 10 6 ) andλ Ψ ∼ Gamma(1/2, 1/2) and N(0, 4), Ω 0 = Ψ(τ, u)dτdu and
Ω Ψ = Ω 2 + κΩ 0 , where, log(κ) ∼Ω 2 = d 2 dτ 2 Ψ(τ, u) + d 2 dτdu Ψ(τ, u) + d 2 du 2 Ψ(τ, u) dτdu.
4. Prior distribution for evolution error, ε t : We use the functional dynamic linear model (FDLM) approach of Kowal et.al. (2016), in which they decomposes the evolution error ε t into factor loading curves φ j , and time-dependent factors e j,t ∈ R, for j = 1, · · · , J ε :
ε t (τ) = J ε ∑ j=1 e j,t φ j (τ) + η t (τ), ∀τ ∈ T ,(7)
where, J ε is the number of factors and {η t } is mean zero approximation error. Priors for each of the parameters in equation (7)
e t ∼ N(0, Σ e ) where, Σ e = diag({σ 2 j } J ε j=1 ). Joint distribution of [σ −2 1 , · · · , σ −2 J ε ] = [σ −2 J ε ] J ε −1 ∏ j=1 [σ −2 j | σ −2 j+1 , · · · , σ −2 J ε ] where, σ −2 J ε ∼ Gamma(10 −3 , 10 −3 ) and [σ −2 j | σ −2 j+1 , · · · , σ −2 J ε ] = [σ −2 j | σ −2 j+1 ] ∼ Uniform(0, σ −2 j+1 ) for j = 1, · · · , J ε − 1.
(c) Prior distribution for φ j for j = 1, · · · , J ε :
φ j (τ) = b φ (τ)θ φ , for τ ∈ T ,
where, b φ is low rank thin plate spline basis and
θ φ ∼ N(0, Λ φ )
s.t. Λ φ = diag(10 8 , 10 8 , λ −1 φ , λ −1 φ , · · · , λ −1 φ ) and λ −1/2 φ ∼ Uniform(0, 10 4 ).
Likelihood Function
We use the dynamic linear model described in equation (4) to form the likelihood function for Y t . At initial time i = 1, we have
Y Y Y 1 ∼ N(Z Z Z 1 µ µ µ (1) + Z Z Z 1 α α α 1 , Σ (1) v ) α α α 1 ∼ N(0 0 0, K K K (1) ε )
Now, for i = 2, · · · , n, one-step ahead distribution of Y t is given as:
Y Y Y i|i−1 ∼ N(Z Z Z i µ µ µ (i) + Z Z Z i α α α i|i−1 , Σ (i) v + Z Z Z i Ξ Ξ Ξ i|i−1 Z Z Z T i ) α α α i|i−1 = ψ ψ ψ (i) α α α i−1|i−1 , and Ξ Ξ Ξ i|i−1 = ψ ψ ψ (i) Ξ Ξ Ξ i−1|i−1 [ψ ψ ψ (i) ] T + K K K (i) ε )
Priors for each time is given as:
α α α i|i = α α α i|i−1 + K K K i v v v i , and Ξ Ξ Ξ i|i = (I M − K K K i Z Z Z i )Ξ Ξ Ξ i|i−1 , where, K i = Ξ Ξ Ξ i|i−1 Z Z Z T i (Z Z Z i Ξ Ξ Ξ i|i−1 Z Z Z T i + Σ v ) −1 , and v i = (Y Y Y i|i−1 − Z Z Z i µ µ µ (i) ) − Z Z Z i α α α i|i−1 .
Posterior Sampling Distribution
We use Gibbs sampling algorithm to obtain the posterior sampling distribution. Firstly, we initialize the DLM (4) parameters and later we present the sampling distributions.
Initialization
Step 1: We firstly initialize µ = [µ Y µ X ] as smooth mean of [{y n t } {x n t }] T t=1 and fit a spline to each [Y t − µ Y X t − µ X ] to estimate α α α t . Thus, we obtain the estimates of E[Y t X t ] , which is further used in estimating the measurement error variance-covariance matrix Σ v v v :
Σ v v v = σ −2 v Y 0 0 σ −2 v X , where σ −2 v Y = ∑ T t=1 (y t − E[Y t ]) and σ −2 v X = ∑ T t=1 (x t − E[X t ]).
Step 2: Now, we use α α α t to initialize the FAR kernels ψ ψ ψ. For ψ ψ ψ = B ψ ψ ψ θ ψ ψ ψ , where B ψ ψ ψ = (b ψ ψ ψ (τ 1 ), · · · , b ψ ψ ψ (τ M )) for M number of evaluation points and θ ψ ψ ψ is sampled from
[θ ψ ψ ψ | · · · ] ∼ N(A ψ ψ ψ a ψ ψ ψ , A ψ ψ ψ ), where A −1 ψ ψ ψ = λ ψ ψ ψ Ω ψ ψ ψ + [B ψ ψ ψ { T ∑ t=p+1 α α α t−1 α α α t−1 }B ψ ψ ψ ][B ψ ψ ψ B ψ ψ ψ ], a ψ ψ ψ = vec B ψ ψ ψ { T ∑ t=p+1 α α α t−1 α α α t−1 }B ψ ψ ψ
where, B ψ ψ ψ is a cubic B-spline basis functions with equally spaced knots, Ω ψ ψ ψ is a prior precision matrix and λ ψ ψ ψ = 1, is a smoothing parameter.
Step 3: Using estimates for FAR kernels in second equation of model (4), we compute the innovations ε ε ε t . We decompose the innovations into factor loading curves (FLCs) Φ = (φ 1 , · · · , φ J ε ε ε ), and time-dependent factors, e t = (e 1,t , · · · , e J ε ,t ) as described in equation (7). For simplicity, we assume
[e t | Σ e ] indep ∼ N(0, Σ e ), with Σ e = diag({σ 2 j } J ε ε ε j=1 ),
where J ε ε ε is the number of factors. We initialize the FDLM parameters using the singular value decomposition (SVD) of (ε ε ε 1 , · · · , ε ε ε T ) = U e D e V e . We set Φ equal to the first J ε ε ε columns of V e and the factors (e 1 , · · · , e T ) equal to the first J ε ε ε columns of (U e D e ). Further, we estimate {σ 2 j } and σ 2 η using the conditional maximum likelihood estimators and {η t } is the mean zero approximation error with η t iid ∼ GP(0, σ 2 η ).
Step 4: The implied covariance matrix for ε ε ε t is K ε ε ε = ΦΣ e Φ + σ 2 η I M , conditional on φ j , σ 2 j and σ 2 η . We obtain the inverse of K ε ε ε using the Woodbury identity:
K −1 ε ε ε = σ −2 η I M − σ −2 η ΦΣ e Φ , whereΣ e = diag({σ 2 j /(σ 2 η + σ 2 j )} J ε ε ε j=1 ).
Sampling Distribution
Step 1: We sample the FAR kernels ψ ψ ψ, ψ ψ ψ = B ψ ψ ψ θ ψ ψ ψ , , using parametrization of θ ψ ψ ψ =ξ ψ ψ ψθψ ψ ψ , where we sampleθ ψ ψ ψ from [θ ψ ψ ψ | · · · ] ∼ N(A ψ ψ ψ a ψ ψ ψ , A ψ ψ ψ ), where
A −1 ψ ψ ψ = λ ψ ψ ψ Ω ψ ψ ψ + s lξ 2 ψ ψ ψ l [(B ψ ψ ψ Q Q Q){ T ∑ t=p+1 α α α t−1 α α α t−1 }(B ψ ψ ψ Q Q Q) ] ⊗ [B ψ ψ ψ K −1 ε ε ε B ψ ψ ψ ], a ψ ψ ψ =ξ ψ ψ ψ vec B ψ ψ ψ { T ∑ t=p+1 α α α t−1 α α α t−1 }(B ψ ψ ψ Q Q Q) .
Sample [ξ ψ ψ ψ | · · · ] ∼ N(Aξ ψ ψ ψ aξ ψ ψ ψ , Aξ ψ ψ ψ ), where
A −1 ξ ψ ψ ψ = 10 −6 +θ ψ ψ ψ (B ψ ψ ψ Q Q Q) T ∑ t=p+1 α α α t−1 α α α t−1 (B ψ ψ ψ Q Q Q) ⊗ B ψ ψ ψ K −1 ε ε ε B ψ ψ ψ θ ψ ψ ψ a ψ ψ ψ =θ ψ ψ ψ vec B ψ ψ ψ { T ∑ t=p+1 α α α t t t − ∑ k =1 s k α α α t−k α α α t−1 }(B ψ ψ ψ Q Q Q)
and sample [λ ψ ψ ψ | · · · ] ∼ Gamma 1/2 + J 2 ψ ψ ψ /2, 1/2 + θ ψ ψ ψ Ω ψ ψ ψ θ ψ ψ ψ /2 . Step 2: We compute the innovations using the estimate of FAR kernels. Under FDLM, we decompose the innovations into FLCs and time-dependent factors.
(a) The factors {e t } T t=1 : sample [e t | · · · ] ∼ N(A e a e t , A e ), where
A −1 e = diag {σ 2 η + σ 2 j )} J ε ε ε j=1
,
a e t = σ −2 η Φ ε ε ε t . (b) The factor precision, σ −2 j : sample [σ −2 J ε ε ε | · · · ] ∼ Gamma 10 −3 + T 2 , 10 −3 + 1 2 T ∑ t=1 e 2 J ε ε ε ,t .
(c) The approximation error precision, σ −2 η : sample
[σ −2 η | · · · ] ∼ Gamma 10 −3 + T M 2 , 10 −3 + 1 2 T ∑ t=1 || ε ε ε t − Φe t || 2 ,
where || · || 2 denotes the Euclidean distance and M is the number of evaluation points . (d) The factor loading curves Φ = B φ Ξ, where B φ is the matrix of basis functions and Ξ = (ξ 1 , · · · , ξ J ε ε ε ).
For j = 1, · · · , J ε ε ε , sample ξ j ∼ N(A ξ j a ξ j , A ξ j ), where A −1 ξ j = σ −2 η T ∑ t=1 e 2 j,t B φ Bφ , a ξ j = σ −2 η B φ T ∑ t=1 e j,t ε ε ε t − B φ ∑ k = j ξ k e k,t .
Step 3: We obtain the covariance matrix K ε ε ε and its inverse K −1 ε ε ε using the similar approach as in Step 4 of initialization.
Step 4: We sample measurement error precision, σ −2 ν Y and σ −2 ν X as follows:
[σ −2 ν Y | · · · ] ∼ Gamma 10 −3 + 1 2 T ∑ t=1 m t , 10 −3 + 1 2 T ∑ t=1 (y t − µ Y − α t ) 2 , [σ −2 ν X | · · · ] ∼ Gamma 10 −3 + 1 2 T ∑ t=1 m t , 10 −3 + 1 2 T ∑ t=1 (x t − µ X − α t ) 2 .
Step 5: Form the DLM and sample [α α α T t=1 | · · · ], using KFAS package in R. We sample the
mean function µ = B φ θ µ , where [θ µ | · · · ] ∼ N(A µ a µ , A µ ) where A −1 µ = Λ −1 µ + σ −2 ν T ∑ t=1 B φ (Z t ) Z t B φ , a µ = σ −2 ν T ∑ t=1 B φ (Z t ) (w t − Z t α t ),
where, w t = [y t x t ] , Λ µ = diag(10 8 , 10 8 , λ −1 µ , · · · , λ −1 µ ) and [λ µ | · · · ] ∼ Gamma 1 2 (J µ − 3), 1 2 ∑ J µ j=3 θ 2 µ, j .
Algorithm and Interpretation of Bayes Factor
Algorithm to compute Bayes factor: (µ, Σ ν , Ψ , ε t ) to obtain the marginal data density corresponding to model M2, p(Y t | M2). 5. Compute Bayes factor B 12 using the marginal data densities of model M1 and M2 and equation (5).
Interpretation of Bayes factor:
Kass & Raftery [29] provided the interpretation of the Bayes factor values in terms of weighing evidence against the null hypothesis, presented in Table 1. A positive logarithm of the Bayes factor is interpreted as evidence in favor of the unrestricted model. Symmetrically, a negative logarithm of the Bayes factor indicates that the data prefer the restricted model.
Simulation study
In this section, we compared the predicted results of simulated functional time series using the proposed method for MFAR relative to FAR. We are particularly interested in one-step and five-step forecasting and finding the causal each kernel is rescaled to pre-specified squared norm, C Φ = Ψ 2 (τ, u)dτdu, with C Φ < 1. For FAR(1) we select C Ψ = 0.5 and for MFAR(1) we select C Ψ Y ,C Ψ X = (0.5, 0.1). Thus, functional time series {Y t } generated from MFAR(1) kernels is correlated to functional time series {X t } while {Y t } generated from FAR(1) kernel is independent of {X t }. We use smooth Gaussian process for innovation process ε t . In our experiment, we consider dense sampling design using m t = 30 equally-spaced observation points on [0, 1] for all t, however the model works well with sparse sampling design also. Data sets generated using FAR(1) kernel and MFAR(1) kernels are applied to proposed model (MFAR) as well as to univariate model proposed by Kowal et. al. (FAR) [19]. Table 2 shows the one-and five-step root mean squared forecasting errors (RMSFEs) for both the models. When {Y t } is dependent on {X t },we obtain lower root mean squared forecasting errors for the MFAR model than the FAR model suggesting MFAR performs better than FAR in forecasting the functional time series.
To acknowledge the causal relationship between {X t } and {Y t }, we calculate the Bayes factor as described in section 3.4. We consider three different simulated cases: (1) {Y t } is generated from MFAR and {X t } is generated from FAR, and we check whether the {X t } series causes {Y t } series, (2) {X t } is generated from MFAR and {Y t } is generated from FAR, and we check for causality in the opposite direction, and (3) {X t } and {Y t } both are generated from FAR. Table 3 presents the Bayes factor B 12 and logarithm of Bayes factor ln(B 12 ) values. Based on ln(B 12 ), we conclude that there is statistical evidence in favor of a causal relationship in first two cases and no such relationship exists when {X t } and {Y t } are independent.
We now repeat the analysis presented in Table 3 forty times for each of the dependent and independent cases and compute the Bayes factor. The boxplot of the logarithm of these Bayes factors is presented fin Figure 1. It can be seen that the distributions are well-separated. Therefore the method is useful in detecting whether the data arises from the dependent or the independent model.
Application to Real Data
We apply the proposed method to two real datasets. The first application is to daily data of yield curves over different maturities for two countries. For the second application, the response variable is still functional but the predictor variable is multivariate. The response is the daily curve of particulate matter in a city while the predictors are several meteorological variables.
Testing causality between yield curves of two countries
To assess the performance of the proposed model, we examine the causal relationship between the yield curves of two countries: USA and UK. The yield curve depicts the relationships of bond yields with different maturities. Yield curves have a significant impact on the money supply within the economy and thus give insights into the country's economic condition.
Testing whether meteorological factors cause air pollution level
Numerous pieces of literature have been forecasting air quality using meteorological features (see, e.g., Aneja et al. [31], Kumar & Goyal [32], Meng et al. [33]). Here, we investigate the causal effect of meteorological factors such as temperature, humidity, wind speed, and solar radiation on air pollution concentration. To compute particle pollution levels, we use particulate matter with diameters 10 micrometers and smaller, denoted by PM10. For our study, we used daily data from Jan 2015 to June 2020 in Delhi, India, consisting of 2008 days. However, after data cleansing we are left with 1670 days of data. We downloaded the data from official Central Pollution Control Board of India website. We take Y t as PM10 in Delhi for each hour of a day; that is, [0,23] is the support of each function. Figure 2 provides with the surface plot depicting the pollution level in Delhi over the years 2015 and 2020. For X, we take the daily average of three meteorological factors: temperature, relative humidity, and wind speed obtained from meteoblue.com. In this case, X is a vector instead of a function. Using X t and Y t , we obtain unrestricted and restricted models and apply the algorithm to compute the Bayes factor as mentioned in section 3.4. In this case, the estimated parameter space Θ is of length 361. The logarithm of the Bayes factor is obtained as ln(B 12 ) = 78.61206652. Based on the interpretation of ln(B 12 ) from table (5), we conclude that there is strong evidence that meteorological factors Granger cause air pollution.
Conclusion
The paper introduces the two-level hierarchical Gaussian process for modeling time series of multivariate functional data. We examine the Granger causality among the functional responses utilizing Bayesian techniques. We compare the marginal data densities of the restricted and unrestricted model to form a conclusion on the causal relationship among the functional time series. We implement the procedure on two real datasets. For the data on yield curves we find that the yield curve of USA is not Granger caused by that of UK. Nor is the yield curve of UK Granger caused by that of USA. In the pollution example we find that daily pollution level curves are indeed Granger caused by meteorological factors. The details of simulations and applications to actual data can be accessed through the GitHub repository Bayesian-Testing-Of-Granger-Causality-In-Functional-Time-Series.
are as follows: (a) Prior distribution for {η t }:η t iid ∼ G P(0, K η ) where, K η (τ, u) = σ 2 η 1 1 1(τ = u) s.t. σ −2η ∼ Gamma(10 −3 , 10 −3 ) and 1 1 1(.) is the indicator function.(b) Prior distribution for e t :
1 .
1Translate the models M1 and M2 into DLM form as described in equation (4). 2. Consider model M1. Initialize the parameter space Θ 1 = (µ, Σ ν , Ψ, ε t ) using the steps mentioned in Section 3.3 under Initialization. 3. For i = 1, · · · , S, where, S is the number of simulations: (a) Obtain the sample of {Θ the likelihood function L (Y t ; Θ (i) 1 ) as described in Section 3.2 and obtain a truncated Normal distribution with mean equal to posterior mean of Θ 1 and variance set to the posterior covariance matrix of Θ 1 , denoted by h(Θ to compute the marginal data density corresponding to model M1, p(Y t | M1). 4. Repeat step 2 and 3, considering model M2 and parameter space as Θ 2 =
exp −(τ − 0.7) 2 /(0.3) 2 − (u − 0.8) 2 /(0.4) 2 and MFAR(1) kernels used for model (M1) are also Bimodal-Gaussian kernal, Ψ(τ, u) = Ψ Y (τ, u) and Ψ X (τ, u) ∝ 0.75 π(0.35)(0.35) exp −(τ − 0.5) 2 /(0.35) 2 − (u − 0.3) 2 /(0.35) 2 + 0.45 π(0.35)(0.35) exp −(τ − 0.75) 2 /(0.35) 2 − (u − 0.6) 2 /(0.35) 2 . For stationarity,
Fig. 1
1Boxplot of the logarithm of the Bayes factor for the dependent and independent cases.
Fig. 2
2The surface plot displays the evolution of the Pollution level in Delhi between the years 2015 and 2020, observed at every hour of a day.
Table 1
1Interpretation of Bayes Factorln(B 12 ) B 12
Evidence against H 0
0 to 1
1 to 3
Not worth more than a bare mention
1 to 3
3 to 20
Substantial
3 to 5
20 to 150 Strong
>5
>150
Decisive
Table 2
2Simulation Results showing h-step ahead RMSFEsh=1
h=5
Data Generation
RMSFE
RMSFE
RMSFE
RMSFE
(MFAR)
(FAR)
(MFAR)
(FAR)
{Y t } is dependent on {X t }
0.009160474 0.009312481
0.01064169 0.0110255
{Y t } is independent of {X t } 0.009157562 0.00964527
0.01069097 0.008994967
Table 3
3Bayes Factor Values of Simulated Series{X t } Cause {Y t } {Y t } cause {X t } {Y t } and {X t } are independentB 12
8.70756E+33
1.35857E+52
0.007122615
ln(B 12 ) 78.14949918
120.0408611
-4.94448041
relationship between two functional time series. Throughout the experiments,
we consider two scenarios based on model (M1) and model (M2) with varying
sample size T . The FAR(1) kernel used for model (M2) is the Bimodal-
Gaussian kernel, Ψ(τ, u) ∝
Time series of yield curves have been studied by several authors. See Sen & Klüppelberg[3] for an introduction and exploratory plots. For our study, we considered 5-month data starting from Jan 2021 till May 2021, which includes 82 working days data. Values of the yield curve are available for 9 maturities: 3, 6, 12, 24, 36, 60, 84, 120, 360 months. We obtain the USA bond yield data for T maturities from the U.S. Department of the Treasury website and the UK bond yield data from investing.com. The objective is to investigate whether UK yield curve Granger cause USA yield curve and vice-versa. We denote the USA yield curves as
Table 4
4Bayes Factor Values of Yield Curves Example UK Cause USA USA cause UK Bayes Factor) ln(B 12 ) -44.89252354 -36.78012961 Y t and the UK yield curve as X t , substituting them in models (M1) and (M2). Once we have the unrestricted and restricted models, we apply the Bayesian analysis to calculate the Bayes factor B 12 as defined in the equation (5). We perform 1000 simulations to generate posterior samples of Θ of dimension 123. Looking at the logarithm of Bayes factor, ln(B 12 ) in table 4, we conclude that it is not statistically significant to say that the UK yield curve Granger cause USA yield curve. We also check whether the reverse relation exists or not, i.e., whether USA yield curve cause UK yield curve. In this case, we take UK yield curves as Y t and the USA yield curve as X t in model (M1) and (M2). Based on ln(B 12 ) (obtained from table 4), we conclude that USA yield curve does not Granger cause UK yield curve. Now we reverse the predictor and response variables and repeat the analysis. We observe that the Bayes Factor is low in this case also and hence conclude that the UK yield curve does not Granger cause the USA yield curve.Bayes Factor (B 12 )
3.18731E-20
1.06315E-16
Logarithm(
Acknowledgments. The research was supported by SERB-MATRICS grant MTR/2020/000324.
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arxiv |
Non-Associativity of Lorentz Transformation and Associative Reflection Symmetric Transformation
Mushfiq Ahmad [email protected]
Department of Physics
Department of Physics
Rajshahi University
RajshahiBangladesh
M Shah
Shah Jalal University of Science and Technology
SylhetBangladesh
Alam [email protected]
Shah Jalal University of Science and Technology
SylhetBangladesh
Non-Associativity of Lorentz Transformation and Associative Reflection Symmetric Transformation
Each of the two moving observers observes the relative velocity of the other. The two velocities should be equal and opposite. We have shown that this relativistic requirement is not fulfilled by Lorentz transformation. We have also shown that the reason is that Lorentz transformation is not associative. Reciprocal symmetric transformation is associative and fulfills relativistic requirements.
Introduction
W V U Two observers are moving with velocities U and V. Their relativity velocity according to one observer should be W. According to the other observer it should be -W. Arrows give the positive directions of the vectors. Negative vectors have the opposite directions. According to triangular law of addition of vectors, we must have and
U V W + − =) ( V U W + − = −) ( . Therefore, U} V { + − −) ( V U + − =) ( (1.1) If is non-commutative + U} V { + − −) ( ) ( U V − + ≠ (1.2)
Lorentz-Einstein Transformation
If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity, , according to Lorentz-Einstein transformation is given by.
E L− W [ ] 2 V.U 1 V V U.V V U V U V W c c c E L − − − − + − = + − = − 1 ) / ( 1 1 ) / ( 1 ) ( 2 2 2 r r (2.1)
We have used the symbol to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity,
+ E L− W , is given by [ ] 2 2 2 2 V.U 1 U U U.V ) (U/ 1 1 V ) (U/ 1 V (-U) W c c c E L − − − − + − = + = − 1 r (2.2)
In general they are not equal and opposite.
E L E L − − − ≠ W W (2.
3) An exception is when U and V are parallel. In that case they are equal.
(2.1) and (2.2) show that
E L E l − − ≠ − = − + W W U V) ( r (2.4)
Non-Associativity of Lorentz Transformation
Consider 3 vectors U, V and W using definition (2.1) of + r , we can calculate r r and r and show that in general
W V) (U + + W) (V U + + r r ≠ + + W V) (U r W) (V U + + r r
(3.1) We can by pass the cumbersome calculation and prove (3.1) by using the following theorem.
Theorem:
If operation is associative +
) ( ) ( ) ( U V V U − + − = + − (4.1a) And if operation is also non-commutative + ) ( ) ( ) ( V U V U − + − ≠ + −
(4.1b) (4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity.
Proof:
Consider ) ( ) ( } { U V V U − + − + +
(4.2) Using associativity and (4.2) becomes 0
) ( = − + V V = − + − + + ) ( ) ( } { U V V U 0 ) ( )} ( { = − + − + + U V V U (4.3) Therefore, 2 0 ) ( ) ( = − + − + + } U V { V} {U
(4.5) Comparing (4.5) with 0
= + − + + V) (U V) (U (4.6) We find that ) ( ) ( ) ( U V V U − + − = + − (4.7) If is also non-commutative + U} {V + −ˆ) ( ) ( U V − + − ≠ (4.8)
Non-Associativity of Lorentz Transformation
(2.4) contradicts (4.7) and (4.8). Therefore, Lorentz transformation is not associative.
Associativity of Reciprocal Symmetric Transformation
Relative velocity according to reciprocal transformation is defined as
2 V.U VxU U V U (-V) W c c i RS − − + − = + = 1 1 (6.1)
A direct calculation will show that for any U, V and W = + + W V) (UˆW) (V U + +ˆ (6.2) (6.2) can also be seen from the fact that (6.1) is built from Pauli quaternion 1 and Pauli quaternion is associative. Definition (6.1) fulfills requirement (4.1)
) ( ) ( ) ( V U U V − + − = + − (6.3)
Comparison between Lorentz and Reciprocal Symmetric Transformations
Lorentz transformation and reciprocal symmetric transformations have the same magnitude. | ) | | | V U U V + = + v (7.1) Reciprocal symmetric transformation has a complex x-product term. This x-product term ensures (6.3). In Lorentz transformation (2.1) the product term comes as a dot-product, which does not change sing when order is changed. We have already seen 2 the complex part explicitly shows the rotation hidden in Lorentz transformation. We have also seen 3 that the complex part explicitly shows the origin of spin.
Conclusion
We have shown that Lorentz Transformation is not associative and that Reciprocal Symmetric transformation is associative. Reciprocal Symmetric transformation fulfills requirement of Lorentz invariance 4 . It is complex. This complex part gives rise to spin 5 .
Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 2 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 3 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 4 Mushfiq AhmadReciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical Concepts. http://www.arxiv.org/abs/math-ph/0611024. 5 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043
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arxiv |
Quantum solvability of a nonlinear δ-type mass profile system: Coupling constant quantization
V Chithiika Ruby
V K Chandrasekar
Department of Physics
Centre for Nonlinear Science and Engineering
School of Electrical and Electronics Engineering
SASTRA Deemed University
Thanjavur-613 401TamilnaduIndia
M Lakshmanan
Department of Nonlinear Dynamics
School of Physics
Bharathidasan University
Tiruchirapalli -620 024TamilnaduIndia
Quantum solvability of a nonlinear δ-type mass profile system: Coupling constant quantization
In this paper, we discuss the quantum dynamics of a nonlinear system that admits temporally localized solutions at the classical level. We consider a general ordered position-dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. The mass function here is singular at the origin. We observe that the quantum system admits bounded solutions but importantly the coupling parameter of the system gets quantized which has also been confirmed by the semiclassical study as well.
Introduction
Several studies on physical systems with position-dependent effective mass have emerged in recent years due to their wide applications in the study of electronic properties of semiconductors [1], inhomogeneous crystals, quantum dots, quantum liquids [2][3][4] and so on. The time-independent Schrödinger equation gets generalized when the effective mass depends on the position and it is solved using both numerical and analytical techniques. Though difficult, it is of general interest to get exact solutions for such position-dependent mass Schrödinger equation (PDMSE) for specific potentials. Certain nonlinear systems, specifically quadratic Liénard type nonlinear oscillators, are found to possess position-dependent mass Hamiltonians. For example, Mathews-Lakshmanan oscillator and Higgs oscillator are considered to describe the dynamics of harmonic oscillators in curved space [5,6]. Different studies have been carried out on these systems in the literature since their introduction in the literature [7][8][9][10]. While quantizing these position-dependent mass (PDM) quantum systems, one should consider (i) the possible choices of ordering between momentum and mass operators in their kinetic energy term and (ii) appropriate modification on the boundary conditions. The ordering may lead to Hermitian or non-Hermitian Hamiltonians. The most general ordering form had been introduced by Trabelsi et al [11]. In a recent study, it has been shown that the Mathews-Lakshmanan oscillator is exactly solvable for the general ordered form [12]. Motivated by the problem of ordering ambiguity of position-dependent mass Hamiltonian, two of the present authors studied the quantum dynamics of the Higgs oscillator and a kdependent nonpolynomial oscillator by considering the general ordered form introduced by Trabelsi et al, in Ref. [13].
Classically both the systems, Mathews-Lakshmanan oscillator and Higgs oscillator admit non-isochronous solutions. It is recently reported that certain quadratic Liénard type nonlinear oscillators can possess isochronous solutions as well [14]. We solved these nonlinear oscillators quantum mechanically and discussed their exact and quasi-exact solvable nature [15]. It is also worth mentioning that one can also derive a conservative description for the nonlinear oscillators of position dependent linearly damped Liénard type systems classically. Such studies have been carried out on generalized modified Emden equation in Ref. [16,17]. The associated Hamiltonians obtained are nonstandard. The Hamiltonian description for such a nonlinear oscillator, governed by a modified Emden equation with certain constraints on its parameters, paves a way to solve the system quantum mechanically. It is also shown that the Hamiltonian is invariant under combined coordinate reflection and time reversal transformation and exhibits linear energy spectrum as that of the standard harmonic oscillator [18].
Based on all these studies, we are here interested to study the quantum dynamics of a quadratic Liénard type nonlinear oscillator which shows a special behavior at its classical level. In this work, we consider such a type of nonlinear system that exhibits temporally localized solutions [14]. It is observed that the associated Hamiltonian is of the form of position-dependent mass type. The mass profile has a resemblance to a δ-function form. A related model that has been used for describing electron systems in δ-doped semiconductors in the Thomas-Fermi field has been shown to be quantum mechanically exactly solvable [19]. In our work, we use a general ordering procedure to write down the appropriate quantum Hamiltonian in order to solve the underlying generalized Schrödinger equation. We also study the role of ordering parameters on obtaining well defined eigenfunctions as the mass function is not a continuous one here.
In this paper, we discuss the classical solvability of the system in section 2. In section 3, we implement a semiclassical quantization rule to analyze the quantum solvability of the system and find that the coupling parameter of the system gets quantized. The system is observed as a position-dependent mass one. We consider the generalized Schrödinger equation corresponding to a non-Hermitian ordered form to analyze the quantum solvability of the system which is discussed in section 4. Finally, we summarize our results.
A δ-type mass system and its classical dynamics
Consider a Hamiltonian of the form studied by Tiwari et al. [14],
H = x 4 p 2 4 + λx 2(1)
and the corresponding Lagrangian is
L =ẋ 2 x 4 − λx 2(2)
It is of the position-dependent mass form, H = p 2 2 m(x) + V (x), where the mass profile is of the form
m(x) = 2 x 4 and V (x) = λx 2 .(3)
Here the mass is singular at x = 0. The equation of motion for the Hamiltonian H in (1) reads as
x − 2 xẋ 2 + λx 5 = 0.(4)
It can be integrated once on using the integrating factor, say 2ẋ
x 4 , aṡ x 2 x 4 + λx 2 = C 1 ,(5)
where C 1 is an integration constant. Integrating this equation (5) once more, we find that equation (4) admits the general solution,
x(t) = 1 λ C 1 + (C 2 + √ C 1 t) 2 ,(6)
where C 2 is the second integration constant. For λ > 0, we have a temporally localized solution. And for λ < 0, we have a singular solution when t = 1
√ C 1 |λ| C 1 − C 2 in
which case we consider that C 1 and C 2 are positive. The plot of x(t) against t is depicted in figure 1 (i) for certain values of C 1 , C 2 and λ. The figure 1 (ii) depicts the contour plot of x(t) given in Eq. (6) for various values of λ with C 1 = 1, and C 2 = −5 .
Semiclassical quantization
To understand the possibility of quantization of the above type of position-dependent mass system, we first apply the semiclassical quantization procedure to the system. The standard leading order WKB quantization condition for the potential having two turning points is [20],
x 2 x 1 pdx = n + 1 2 π, n = 0, 1, 2, ...,(7)
where x 1 and x 2 are the classical turning points and the conjugate momentum,
p = 2m(x) (E − V (x)). Here, = h 2π , where h is Planck's constant. From the
Hamiltonian (1), with H = E, one can express the momentum as
p = 4E x 4 − 4λ x 2 .(8)
At the turning points, say (x 1 , x 2 ) = (−A, A), the momentum is zero, which is shown in the figure 2. Hence, from (1), the total energy, H = E = λA 2 and the integral (7) becomes,
2 √ λ A −A √ A 2 − x 2 x 2 dx = n + 1 2 π, n = 0, 1, 2, ....(9)
To evaluate (9), consider the integral
I = A −A √ A 2 − x 2 x 2 dx.(10)
One can also use the classical solution x(t), (vide (6)) and evaluate the closed integral around contour C (given in Fig.2) in the modified Bohr-Sommerfeld quantization rule [21],
pdx = n + 1 2 h.(11)
Here, the momentum, p(t), takes the form as
p(t) = 2ẋ(t) x(t) 4 = −2 C 1 (C 2 + C 1 t) λ C 1 + (C 2 + C 1 t) 2 .(12)
We integrate the integral (10) by considering u = √ A 2 − x 2 and dv = 1 x 2 dx and get
I = √ A 2 − x 2 x A −A − A −A dx √ A 2 − x 2 , = 0 − arcsin x A A −A , I = − π.(13)
On substituting the integral (13) in (9), one obtains the following relation on the coupling parameter, λ, as
λ = n + 1 2 2 2 4 , n = 0, 1, 2, 3, ....(14)
Hence, the coupling parameter λ gets related with the quantum number n, as in (14). While studying the quantum dynamics of the above type of position-dependent mass system (1) with a singular mass function, we meet with two difficulties: (i) how to define the configuration space and (ii) how to ensure the continuity of the eigenfunctions of the corresponding Schrödinger equation? We proceed to incorporate these two aspects in our further study as indicated below.
Quantization: general ordered form of Hamiltonian
We now consider the most general form of the associated Hamiltonian operator that provides a complete classification of Hermitian and non-Hermitian orderings [11],
H = 1 2 N i=1 w i m α ip m β ip m γ i + V (x),(15)
where N is an arbitrary positive integer andp is the one dimensional momentum operator. The ordering parameters should satisfy the constraints,
α i + β i + γ i = −1, i = 1, 2, 3, .
..N, and w i 's are real weights which are summed to be 1. The above form globally connects all the Hermitian orderings and also provides a complete classification of Hermitian and non-Hermitian orderings [11]. The operatorĤ in (15) possesses 2N free ordering parameters, after taking into account the above constraints. The corresponding Hamiltonian for the potential V can be written aŝ
H = 1 2p 1 mp + (γ −ᾱ) i 2 d dx 1 m p + 2 2 γ d 2 dx 2 1 m + αγ m 2 m 3 + V,(16)wherep = −i d dx .
In (16), the over bar over the parameters represent their total value,
X = N i w i X i .
The study on the effective-mass Hamiltonians for abrupt heterojunctions indicates that the single-term ordering forms of kinetic energy operator are viable candidates that ensure continuity of the associated matching conditions [22]. As the mass m(x) is singular at x = 0, we use the single term of the general ordered form of the Hamiltonian asĤ
= 1 2 m α 1p m β 1p m γ 1 + V (x), α 1 + β 1 + γ 1 = −1.(17)
Here, we are considering non-Hermitian ordered form of the Hamiltonian (16) as the non-Hermitian ordered form can be related with the Hermitian ordered form through similarity transformation [23] aŝ
H her = m ηĤ m −η , 2 η = γ 1 − α 1 , .(18)
Consequently, for (18) we havê
H her = 1 2 m γ 1 +α 1 2p m β 1p m γ 1 +α 1 2 + V (x).(19)
As the non-Hermitian ordered form (16) is being related with the Hermitian ordered form through similarity transformation (18), we use the non-Hermitian ordered form of the Hamiltonian in this present work and analyze the possibility of obtaining a complete set of solutions of the operator (16).
The time-independent Schrödinger equation for the non-Hermitian ordered Hamiltonian (17),Ĥψ = Eψ, can be written as
ψ + (γ 1 − α 1 − 1) m m ψ + γ 1 m m − (α 1 γ 1 + 2γ 1 ) m 2 m 2 ψ + 2m 2 (E − V (x)) ψ = 0,(20)
where = d dx . As the above Hamiltonian depicts the dynamics of the one dimensional potential (1), we use the generalized position-dependent mass Schrödinger equation resulting from the non-Hermitian ordering (17), to study the solvability of the system (1). It results that
ψ + 4 (1 + α 1 − γ 1 ) x ψ + 4 E 2 x 4 − 16α 1 γ 1 + 12γ 1 + 4λ 2 x 2 ψ = 0.(21)
By using the transformation, ψ(x) = x d φ(x), where d is a parameter to be determined, we can reduce the equation (21) to the form
φ + 2d + 4 (1 + α 1 − γ 1 ) x φ + d(d + 3 + 4(α 1 − γ 1 )) − 16α 1 γ 1 + 12γ 1 + 4λ 2 x 2 + 4 E 2 x 4 φ = 0.(22)
We further use the transformation, g(x) = −1 2 x , so that Eq. (22) can be rewritten as
g 2 φ gg + 2g [(2γ 1 − 2α 1 − 1 − d)] φ g + d(d + 3 + 4(α 1 − γ 1 )) − 16α 1 γ 1 + 12γ 1 + 4λ 2 + 16 E 2 g 2 φ = 0,(23)
where φ g = d φ dg .
In order to map Eq. (23) to the known form, we again use the transformation,
τ = 4 √ E g,(24)with d = 2γ 1 − 2α 1 − 3 2 ,(25)
to transform equation (23) as
τ 2 φ τ τ + τ φ τ + τ 2 − ν 2 φ = 0,(26)
where
ν 2 = 2α 1 + 2γ 1 + 3 2 2 + 4λ 2 .(27)
Eq. (23) is of the form of Bessel's differential equation. Hence, the corresponding general solution is
φ ν (τ ) = CJ ν (τ ) + DY ν (τ ),(28)
where J ν (τ ) and Y ν (τ ) are the first and second kind of Bessel polynomials [24] and C and D are arbitrary constants. Now we can obtain the general solution for the equation (21) for the region x ∈ (0, ∞) as
ψ ν (x) = ψ (+) ν (x) = x d CJ ν 2 √ E x + DY ν 2 √ E x , x ∈ (0, ∞).(29)
And we can write down the general solution for the region x < 0, as
ψ (−) ν (x) = (−|x|) d C J ν 2 √ E x +DY ν 2 √ E x , x ∈ (−∞, 0). (30)
whereC andD are arbitrary constants and d (vide Eq. (25)).
Here we are interested to derive bounded solutions for the system (1) and so analyze the boundary conditions for the Bessel polynomials.
By choosing d = 2γ 1 − 2α 1 − 1, equation (23) can now be reduced to the constant mass Schrödinger equation as
φ gg + 16 E 2 − 4λ 2 + (2α 1 + 2γ 1 + 2) (2α 1 + 2γ 1 + 1)) g 2 φ = 0.(31)
This equation can also be deduced by means of a point canonical transformation method, which relates the PDM Schrödinger equation with the canonical form of constant mass Schrödinger equation and it is a widely used method in solving position-dependent mass Schrödinger equations [25]. The potential of (31), U (g) ∝ 1 g 2 , is similar to the effective potential that arose while studying the Efimov effect in the quantum three body system that describes the dynamics of two heavy particles interacting through a light particle [26].
Boundary conditions
In Eq. (29), when x → ∞ the polynomials J ν become zero for positive values of ν and become complex infinity for ν < 0. And Y ν becomes ∞ provided ν = 0. Hence, we take D = 0 and ν > 0 to get the solutions which are bounded as x → ∞.
To proceed further, we now expand (29) around x = ∞,
ψ (+) ν (x) = C x d J ν 2 √ E x ≈ x→∞ C Γ(ν + 1) E 2 ν/2 x d−ν .(32)
The boundary condition on ψ
(+) consider the functions, f (x) = √ x and h(x) = − √ x, then the lim x−>0 √ x = 0 makes lim x−>0 √ x cos 2 √ E x − π 2 ν + 1 2 = 0.
• Hence, for the values of d < 0, the solutions ψ (+)
ν (x) are not well defined near zero. It restricts that d ≥ 0.
• But we have d − ν < 0 which fixes the lower bound of ν. To consider the lower bound value of ν as the least of the value of ν, we consider d = 0.
Hence, the eigenfunction, Eq. (29) becomes
ψ (+) ν (x) = CJ ν 2 √ E x , x ∈ (0, ∞).(34)
Similarly, the eigenfunction, Eq. (30) takes the form,
ψ (−) ν (x) =CJ ν 2 √ E x , x ∈ (−∞, 0).(35)
We also consider that ν > 0 from the fact that the Bessel functions J ν (0) are not well defined at ν = 0.
Parity
Now we use the parity condition on J ν . The solution (35), defined in the region x ∈ (−∞, 0), may be symmetric or anti-symmetric with ψ (+) ν (x). Consider a point near x = 0, then we havẽ
Cψ (−) ν (x) x=− = Cψ (+) ν (x) x= ,(36)
and so
C − (−1) νC J ν 2 √ E = 0.(37)
The odd parity determines ν = 1, 3, 5, ..., odd integers, and soC = −C, whereas even parity leads to ν = 2, 4, ..., even integers, so thatC = C. Hence, the parity condition fixes ν = n, n = 1, 2, 3, ....
As a result, we find that the coupling parameter (27) is now related with the quantum number 'n' as λ = n 2 − 2α 1 + 2γ 1 + 3 2
and so it is quantized which has also been confirmed by the semiclassical quantization method, vide Eq. (14).
Hence, the bound states from (34) and (35) become
ψ (+) n (x) = CJ n 2 √ E x , x ∈ (0, ∞) n = 1, 2, 3, .... (40) ψ (−) n (x) = C(−1) n J n 2 √ E |x| , x ∈ (−∞, 0) n = 1, 2, 3, ....(41)
The parity nature of the eigenfunctions (40) and (41) restricts the coupling parameter to take discrete values, that is expressed in terms of quantum number n in (39). Subsequently we analyze the energy eigenvalues in the following subsection. n (x) (vide Eq. (40) and Eq.(41)) are restricted to be zero at that point x = 0, that is
lim x−>0 ψ (±) n (x) = 0.(42)
Consequently, we have
lim x−>0 x π √ E cos 2 √ E x − π 2 n + 1 2 = 0.(43)
The above relation establishes that the energy eigenvalues are continuous, while the coupling parameter λ is quantized as in Eq. (39).
Normalizability condition of the states (40) and (41):
As the non-Hermitian ordered form of the Hamiltonian can be related with the Hermitian ordered form through similarity transformation, one can express the normalization condition for non-Hermitian ordered Hamiltonian as [23],
1 = ψ (±) n m 2η |ψ (±) n ,(44)
where η = γ 1 −α 1 2 . On substituting (40) in (44), we can get
1 = C 2 2 γ 1 −α 1 ∞ 0 1 x 4γ 1 −4α 1 J n 2 √ E x J n 2 √ E x dx.(45)
As d = 0, we have γ 1 − α 1 = 3 4 . By applying a simple transformation ρ = 1 x to (45), we can get
1 = C 2 2 3/4 ∞ 0 ρ J n 2 √ E ρ J n 2 √ E ρ dρ.(46)
On using the identity,
we can obtain the condition
1 = C 2 2 3/4 2 √ E δ 2 √ E − 2 √ E (48)
where δ(a − b) is the Dirac delta function which becomes infinity when a = b, otherwise it has zero value.
We now obtain,
C = √ 2 E δ 2 √ E − 2 √ E 1/2 .(49)
As the energy eigenvalue of the system is arbitrary and continuous, we have obtained the normalization constant in terms of Dirac delta function. This is analogous to the quantization of a free particle on a cone studied recently by Kowalski et al. [28]. Hence, we obtained the bounded states (29) in both the regions, x ∈ (0, ∞) and x ∈ (−∞, 0), as
ψ (±) n (x) = CJ n 2 √ E x , n = 1, 2, 3, ....(50)
The first two states (unnormalized) are plotted in the figure 3. One can reinterpret the normalization condition,
1 = ∞ −∞ ψ * n (x)ψ n (x) dx,(51)
by omitting the singular region (− , ) and reconsidering the integral (44) by
1 = 2C 2 ∞ 1 x 3 J n 2 √ E x J n 2 √ E x dx,(52)
in which we considered (50).
Let 1 x = ρ.
The integral (52) becomes
1 = 2C 2 1/ 0 ρ J n 2 √ E ρ J n 2 √ E ρ dρ.(53)
Now we use the identity [29] a 0 ρJ ν α νm ρ a
J ν α νm ρ a dρ = a 2 2 [J ν+1 (α νm )] 2 δ nm ,(54)
where δ nm is Kronecker delta function that takes the value 1 when n = m otherwise it takes zero. Here, α νm , m = 1, 2, 3, ...∞, is the m th zero of the Bessel function J ν , that is J ν (α νm ) = 0. The integral (53) now becomes
1 = C 2 2 J n+1 2 E N n 2(55)
which makes the energy eigenvalues to take the values,
E N n = 2 4 j N 2 n 2 , = 0,(56)
where j N n , N = 1, 2, 3, ...∞, n = 1, 2, 3, ... are zeroes of the Bessel function, J n . The normalization constant reads as
C N n = J n+1 2 √ E N n .(57)
The normalized eigenstates, vide (56) and (57), can be written as We have observed that one can possibly obtain the normalized eigenfucntions with the corresponding eigenvalues by restricting the motion of the particle around a point near to the origin ( = 0).
Hermitian ordering
In the previous section, we considered non-Hermitian ordered Hamiltonian (17) and solved the corresponding generalized Schrödinger equation that resulted in the general solution (50). In this sub-section, we discuss about the solution of the Hermitian ordered form of the Hamiltonian (19).
H her = 1 2 m γ 1 +α 1 2p m β 1p m γ 1 +α 1 2 + V (x).(19)
Instead of solving the Schrödinger equation corresponding to the Hermitian ordered Hamiltonian (19), we can obtain the solution from the relation (18) that relates the non-Hermitian ordered form (16) with the Hermitian ordered form through similarity transformation.
Hψ = m −ηĤ her m η ψ, 2η = γ 1 − α 1 .
Let m η ψ = φ. As we have 2η = γ 1 − α 1 = 3 2 from d = 0, we can write down the solution for (19) from (50),
φ n (x) = Cm η J n 2 √ E x = C x −3/2 J n 2 √ E x , n = 1, 2, 3, ...,(60)
where the normalization constant C is the same as obtained in (49). The solution (60) is singular at x = 0. Hence, for the system (1), the non-Hermitian ordered form (17) only yields bounded solutions (50).
Conclusion
In this work, we considered a nonlinear system of the quadratic Liénard type which admits temporally localized solutions at the classical level. Depending upon the positive and negative values of the coupling parameter λ, the solution is well defined or has a singular value in its domain. To start with, we implemented the WKB quantization condition which ensures that the coupling parameter λ would be quantized. While studying the quantum dynamics of the system, we considered a single term of the general ordered position-dependent mass Hamiltonian as the mass function which is singular at the origin and solved the underlying Schrödinger equation. We observed that the quantum system admits bounded solutions. Specifically, we find that the coupling parameter of the system gets quantized. We believe that such an observation is quite new to the literature as far as the quantization is concerned. The position dependent mass with δ-type mass profile considered in this paper may find application in the field of semiconductor physics, as in the case of Thomas-Fermi potential with δ−doped semiconductor [19]. We believe that our study widens the scope of quantizing other solvable classical nonlinear oscillators exhibiting novel dynamical features in a broader sense.
Figure 1 .
1(i) The plot of x(t) in Eq. (6) for C 1 = 1, C 2 = −5 and λ = 1 and (ii) the contour plot of x(t) given in Eq. (6) for various values of λ with C 1 = 1, and C 2 = −5 , where the blue shaded region denotes the possible values of λ for which the solutions are well defined and the white region denotes the values of λ for which the solutions are singular.
Figure 2 .
2The phase portrait of Hamiltonian (1) for different values of energy E = 0.5, 0.7, 0.8, 1 with λ = 0.5.
n = 1, 2, 3, ...
.
Energy: As m(x) = 2 x 4 is singular at x = 0, the eigenfunctions ψ (±)
n (ka)J n (kb)dk = 1 a δ(b − a), n = 0, 1, 2, ...,
Figure 3 .
3The plots of (i) ψ 1 (x) and (ii) ψ 2 (x) in Eq. (50).
.... N = 1, 2, 3, ..., = 0.(58)
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arxiv |
A New Paradigm Unifying the Concepts in Particle Abrasion and Breakage
Priya Tripathi
Dept. of Civil & Environmental Engineering
Florida International University
MiamiFL
Ph.DSeung Jae Lee
Dept. of Civil & Environmental Engineering
Florida International University
MiamiFL
Ph.DMoochul Shin
Dept. of Civil & Environmental Engineering
Western New England University
SpringfieldMA
Ph.DChang Hoon Lee
Dept. of Civil & Environmental Engineering
Western New England University
SpringfieldMA
A New Paradigm Unifying the Concepts in Particle Abrasion and Breakage
Tripathi et al. (2023) -1 -
This study introduces a new paradigm that unifies abrasion and breakage concepts, allowing for a holistic understanding of the comminution process. The significance of this paradigm lies in its ability to present both abrasion and breakage in a single big picture because both processes can co-occur under loading as particles are subjected to friction as well as collision. A comprehensive descriptive framework is employed to this end, which operates in a log-transformed surface-areato-volume ratio (A/V) and volume (V) space. This space facilitates a holistic characterization of the four-dimensional particle geometry features, i.e., volume (V), surface area (A), size (D), and shape (β). Consequently, this approach enables to systematically relate the co-occurring abrasion and breakage process to co-evolving particle shape and size. Transformative concepts including the breakage line, sphere line, and average shape-conserving line are introduced to describe the limit states and a special comminution process. This approach also uncovers a self-similar nature in evolving particle geometry during comminution, which will be a significant discovery for the granular materials research community given the most fundamental properties observed in natural phenomena.
INTRODUCTION
Comminution is an important phenomenon in the geotechnical engineering, and abrasion and breakage are the two most common phenomena in the comminution process. Soil particles experience both friction and collision when subjected to loading, and therefore abrasion and breakage may co-occur (Qian et al. 2014). If not, the comminution mode can transition from a state of predominant breakage to one of dominant abrasion (Xiao et al. 2022). Significant research efforts have been made to study the comminution, but which have focused on either abrasion (Domokos, Sipos, and Vá rkonyi 2009;Qian et al. 2014;Miller et al. 2014;Deiros Quintanilla et al. 2017;Sipos, Domokos, and Török 2021) or breakage (Hardin 1985;Altuhafi and Coop 2011;Zhang et al. 2015;Zheng et al. 2019;Buscarnera and Einav 2021;Einav 2007 Harmon, Arthur, and Andrade 2020;Wang and Arson 2016;Seo and Buscarnera 2022), without holistically considering both processes in a bigger framework. The comminution process necessarily involves particle geometry changes. While both abrasion and breakage reduce particle mass, abrasion tends to produce rounder particles, whereas breakage (crushing) often generates angular particles with sharp edges and corners. Additionally, the comminution process is influenced by factors such as particle mineralogy. For example, Bowman et al. (2001) conducted uniaxial compression tests on two types of sands, revealing that compression can lead to either abrasion or breakage depending on the particle mineralogy. The objective of this study is to provide a new paradigm that can integrate the concepts in abrasion and breakage, facilitating a holistic understanding of particle geometry evolution in comminution. To achieve this goal, a log-log space is employed, which presents the geometry in terms of particle surface-area-to-volume ratio (A/V) and particle volume (V). This approach allows for presenting change of particle geometry (including co-evolving shape and size) by both abrasion and breakage.
METHODOLOGY
Phenotypic Trait of Particle Geometries
For a group of particles from the same geological origin and loading history, the particle geometries exhibit a phenotypic trait that manifests as a power-law relation between the particles' surface-area-to-volume ratio (A/V) and volume (V) (Lee et al. 2022). We hypothesize that this power-law relationship continues to hold true in the process of comminution. It is worth noting that the phenotypic trait is a concept encompassing shape and size. As analogy, Koreans may exhibit variations in their physical appearances, they share a phenotypic trait due to their common genetic origin. Similarly, a family of mineral particles may exhibit different shapes and sizes but share a phenotypic trait due to the common geological origin. Graphically, when the data points are plotted in a log-log A/V and V space, a power regression unveils a line pattern if the particles are from the same family. The phenotypic trait then can be identified using the intercept and slope of power-regression line, which indicates the average shape angularity (β*) and the relation between particle shape and size (α), respectively. Average shape angularity: The shape angularity of an 'individual particle' can be quantitively determined by β which is equal to A 3 /V 2 . The value of β cannot be smaller than 36π (=113.09) which represents a sphere and is higher for a more angular shape. As shown in Figure 1, the value of β corresponds to the intercept at A/V = 1 in the log-transformed A/V and V space when a line is drawn with a slope of -3 from the data point (depicted as a yellow dot in the figure). The β* is similarly defined for a 'group of particles.' The β* value can be analytically obtained from the mean of β values as in Eq. (1). Graphically, it is also shown as the intercept at A/V = 1, obtained by evaluating a slope of -3 in the log-transformed A/V and V space ( Particle shape-size relation: The slope α of the power regression line informs the relation between particle shape and size. There are three cases: (a) Case 1: there is tendency that smaller particles (having a smaller V) are more angular than larger particles. In this case, the smaller particles have a higher β values, so the α value is greater than -3 (|α| < 3) as in Figure 1a; (b) Case 2: there is tendency that larger particles (having a larger V) are more angular than smaller particles. In this case, the α value is smaller than -3 (|α| > 3) as in Figure 1b; (c) Case 3: there is no particular trend between shape and size. In this case, all particles have similar angularity regardless of size. Then, α is equal to -3 (|α| = 3) as in Figure 1c. For example, for spheres of varying sizes, the A/V and V relation will exhibit |α| = 3. The size D can be estimated as the diameter of a sphere having the same volume V as the particle. The vertical axis, therefore, also represents D as in Figure 1. Therefore, this approach enables the representation of the four-dimensional particle geometry features (i.e., volume V, surface area A, size D, shape β) within a single space. Interested readers can refer to Lee et al. (2022). Upon further examination, Case 2 is uncommon for mineral particles from the reported data (Lee et al. 2022;Tripathi et al. 2023). Therefore, this study will focus on Cases 1 and 3 with |α| ≤ 3. Figure 1. Phenotypic trait identified in terms of α and β* for a group of particles (and the individual particle's shape angularity is presented by β1, β2, and β3): (a) Case 1: slope |α| < 3 in case smaller particles (smaller V) are more angular (higher β); (b) Case 2: slope |α| > 3 in case larger particles being more angular; (c) Case 3: slope |α| = 3 in case there is no particular shape-size relation.
The Framework Three axioms. The discussed log-transformed A/V and V space is utilized in this study to present a complete picture of particle geometry evolution in terms of V, A, D, and β caused by abrasion and breakage. The framework is built on the following three axioms (ax. 1 -3) considered as true:
▪ ax.1 -comminution reduces particle size (regardless of abrasion or breakage). ▪ ax.2 -abrasion results in the gradual rounding of particle shapes. ▪ ax.3 -breakage produces particles with angular shapes.
The three axioms are schematically shown in Figure 2a, where {PO} represents an initial set of freshly crushed particles. With ax.1, {PO} always moves down to a lower V due to decrease in size of particles. With ax.2, {PO} will move towards the left, because abrasion makes particle shape rounder, thus β* will reduce. On the other hand, with ax.3, {PO} moves to the right, increasing β*.
β* A < β* O < β* B,
showing the angularity change depending on the comminution process. If {PO} is not freshly crushed particles and underwent significant abrasion, a more dramatic shape change would occur after breakage. Therefore, the child particles will appear on the far-right of {PO} as shown as {PD} in Figure 2b, which will realize a much higher β* compared to β* O. Then, what would happen if both abrasion and breakage occur in an equal measure? We consider this as a 'special' comminution process because the equally occurring abrasion and breakage would conserve the initial shape angularity β* O. {PC} is the particle set after the special process and will be on the α = -3 line from {PO}. The β* is graphically evaluated as the intercept at A/V = 1 with the slope of α = -3 line, so moving along the α = -3 line is the only possible path for β* O = β* C. In this study, the α = -3 line is referred to as the average shape-conserving line. Abrasion: The regression line with initially |α| < 3 gradually converges towards |α| = 3. Angular particles typically undergo abrasion faster than round particles (Krumbein 1941;Janoo 1998). As all particles get rounded, shapes get more similar to one another, thus |α| → 3. In an extreme abrasion, the data points will cluster near the sphere line which represents a case where all particles are spheres. This sphere line is plotted with |α| = 3 and β* = 36π (sphere's angularity). Figure 2c schematically shows the concept. RO is the power regression line obtained for {PO} and RA corresponds to {PA}. The RO is initially |α| < 3, but with abrasion, RA approaches |α| = 3 (thus, the slope gets steeper). Breakage: The child (broken) particles inherit the phenotypic trait of the parent particles, so both sets will realize the same power regression line. See Figure 2c. RB is the power regression line that corresponds to {PB}, realizing RB = RO (where the blue dotted line indicates the power regression line for the child particles). The same power regression line will be realized with even further breakage; {PB(k)} is the particle set after k-th continuous breakage, which will manifest RB(k) = RO, as the child particles will inherit the phenotypic trait. In this study, this unique line not changing in breakage is called breakage line. A point O is at the intersection of breakage line and sphere line, which is called the geometric origin of comminution. When crushed particles undergo abrasion as well as breakage, the resulting data points will fall between the two limit lines. Equally occurring abrasion and breakage: While {PO}→{PC} occurs along α = -3 line (conserving β*), the slope |α| of RC (power regression line evaluated for {PC}) will become steeper than that of RB (= RO) due to the effect of abrasion. Breakage of initially abraded particles: In case {PO} is not freshly crushed particles and experienced significant abrasion, then |α| of RO will be high, and in extreme abrasion, it will be close to 3. Therefore, upon breakage, a new breakage line RD will emerge, and the slope will correspond to the slope |α| that could have been obtained from the freshly crushed parent particles prior to undergoing abrasion.
Five zones in the A/V and V space.
With consideration of the distinct comminution paths for abrasion, breakage, and average shape-conserving process (by equally occurring abrasion and breakage) as in Figure 2d, five in-between zones can be defined as the effects of abrasion and breakage are mixed up. See Figure 2e: (1) Zone A (in green) on the left of the average shapeconserving line informs more abrasion because the data points in this zone realize a lower β* than β* O; (2) Zone B1 (in red) is located on the right side of the average shape-conserving line and informs more breakage. The data points in this zone realize a higher β* than β* O; (3) Zone B2 (in yellow) is on the right side of the initial power regression line RO. If the particles data show up in this zone after breakage, it informs the parent particles underwent some significant abrasion prior to breakage; (4) Zone I1 (in blue) is underneath the sphere line. This is an impossible zone where data cannot exist because there is no rounder shaper than sphere; (5) Zone I2 (in white) is located above {PO}. This is another impossible zone because particles do not get larger after comminution.
Ultimate shape angularity. The shape angularity in extreme comminution will be determined at the ultimate grading, where no further evolution of the particle size distribution is expected (Einav 2007;Altuhafi and Coop 2011). The particle sets in the ultimate grading are denoted as {PA(ult)}, {PB(ult)}, and {PC(ult)} in Figure 2f. These sets are obtained after undergoing extreme abrasion, breakage, and an equally occurring abrasion and breakage process, respectively. The data points are located at the lowest possible V according to the ultimate grading. The average angularity then can be evaluated as β* A(ult) < β* C(ult) (= β* O) < β* B(ult). In an extreme abrasion, all particles in {PA(ult)} will be a sphere, and data points eventually will land on the sphere line, thereby β* A(ult) = 36π.
EXPERIMENTAL VALIDATION
The
Abrasion
Paixã o and Fortunato (2021) conducted micro-Deval abrasion tests on 30 granite particles. The particles are 3D scanned after 0 (i.e., before the test), 2000, and 14000 revolutions to capture the abrasion-induced geometry changes. The geometry data is obtained from the original paper and plotted as in Figure 3. The initial |α| is 1.8 (< 3), indicating smaller particles are more angular than the larger particles. After 14000 revolutions, |α| increases to 1.92 (closer to 3 than before), because angular particles are abraded more than round particles, resulting in relatively similar shape angularity overall compared to the initial condition at 0 revolution. The data points are migrating towards the lower left, decreasing β* value from 260.98 to 226.72, approaching the sphere line with β* = 36π (= 113.10). This experiment data demonstrates the {PO}→{PA} path.
Breakage
A sample of 10 particles collected from a batch of crushed granite particles is used for this study. The batch was obtained from a quarry located in Richmond, Virginia. The images of 10 Virginia granite (VG) particles are shown in Figure 4. Drop hammer test is performed on each particle to induce breakage to avoid potential abrasion that could occur as in uniaxial compression test of particles. A 5.5 lbs (2.5 kgf) standard Proctor hammer is dropped from a height of 2.5 inches (63.5 mm). The broken (child) particles larger than #10 sieve size (2 mm) are sifted and shown in Figure 4. Loss of finer particles is about 12%. The particles are 3D scanned before and after the test, from which 3D digital particles are developed. Polyga C504 structured light 3D scanner is used to capture the 3D particle geometry. The particle surface area (A) and volume (V) are obtained from the 3D digital particles. The geometry changes are plotted in Figure 5. The child particles inherit the phenotypic trait of the parent particles, thus both data sets realize the same power regression line as shown in Figure 5a. The average angularity β* increases from 285.90 to 389.11. This experiment data demonstrates the {PO}→{PB} path.
(a) (b) Figure 3. Particle geometry evolution by abrasion: (a) The power regression's slope |α| increases, implying overall particle shapes getting similar with abrasion; (b) The intercept β* decreases with revolutions, indicating the data points approaching the sphere line.
CONCLUDING REMARKS
This study presents a novel paradigm that integrates the concepts in abrasion and breakage. A descriptive framework is introduced to facilitate a holistic understanding of these phenomena in a log-log space, which presents the particle geometry in terms of surface-area-to-volume ratio (A/V) and volume (V). This approach enables to systematically explore the co-evolving particle shape and size in relation to both abrasion and breakage. This study proposes a power-law relationship continues to hold true in the comminution process and evidences the self-similarity (manifesting itself in a power-law) emerges from co-evolving particle shape and size. The self-similarity is found in many natural phenomena (Amitrano 2012;Barabasi 2003;Watts 1999) and it has been also reported in the context of particle crushing. However, the focus was primarily on the changes in particle size (Einav 2007). On the other hand, this study relates the self-similarity (characterized by a power-law) to the overall particle geometry changes (encompassing shape and size), which will be an exciting discovery for the research community. The proposed approach will be able to serve as a new conceptual foundation, enabling a comprehensive and coherent understanding of the complex comminution processes.
DATA AVAILABILITY
The dataset of digital particles generated from this study is available in the NSF DesignSafe-CI repository (Tripathi et al. 2023). Please see the Virginia granite group B particles in the dataset.
Figure 1 )
1. * = 10 log( ) ̅̅̅̅̅̅̅̅̅ = 10 log( )+ 3 × log( / ) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (1) where [… ] ̅̅̅̅̅ indicates the arithmetic mean of [… ].
Figure 2 .
2Schematic of the A/V and V space-based descriptive framework to present the particle geometry change caused by abrasion and breakage.Change in the intercept β* during comminution. An illustration is shown in Figure 2b, where {PA} and {PB} represent the particle sets after abrasion and breakage, respectively. With abrasion (ax.1 & 2), {PA} is located at the lower left of {PO}. With breakage (ax.1 & 3), {PB} is at the lower right of {PO}. It's worth noting that the number of data points will change as {PO}→{PB}, producing 'child' particles, unlike the abrasion as {PO}→{PA} without major fragmentations. The β* O is the average angularity of {PO}, β* A corresponds to {PA}, and β* B corresponds to {PB}, whereby
Change in the power regression's slope α during comminution.
Figure 4 .
4Virginia granite (VG) Particles before and after breakage (a) (b) Figure 5. Particle geometry evolution by abrasion: (a) The child particles inherit the phenotypic trait of parent particles, realizing the same regression line; (b) The intercept β* increases with the breakage.
; Cil and AlshibliarXiv:2306.04635v2
[cond-mat.soft] 8 Jun 2023
2014;
ACKNOWLEDGEMENTSThis work is partly sponsored by the US National Science Foundation (NSF) under the awards CMMI #1938431 and #1938285. The support is greatly appreciated. The authors extend their appreciation to Dr. Eric Koehler in Titan America LLC for providing the Virginia granite particles used in the breakage study.
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Characterization of Two-and Three-Dimensional Morphological Properties of Fragmented Sand Grains. Wenbo Zheng, Xinli Hu, Dwayne D Tannant, Kai Zhang, Cong Xu, 10.1016/j.enggeo.2019.105358Engineering Geology. 263105358Zheng, Wenbo, Xinli Hu, Dwayne D. Tannant, Kai Zhang, and Cong Xu. 2019. "Characterization of Two-and Three-Dimensional Morphological Properties of Fragmented Sand Grains." Engineering Geology 263 (December): 105358. https://doi.org/10.1016/j.enggeo.2019.105358.
| {'fraction_non_alphanumeric': 0.06259926800635315, 'fraction_numerical': 0.04747600303846419, 'mean_word_length': 4.617339022498061, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 2, 'https://': 21, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This study introduces a new paradigm that unifies abrasion and breakage concepts, allowing for a holistic understanding of the comminution process. The significance of this paradigm lies in its ability to present both abrasion and breakage in a single big picture because both processes can co-occur under loading as particles are subjected to friction as well as collision. A comprehensive descriptive framework is employed to this end, which operates in a log-transformed surface-areato-volume ratio (A/V) and volume (V) space. This space facilitates a holistic characterization of the four-dimensional particle geometry features, i.e., volume (V), surface area (A), size (D), and shape (β). Consequently, this approach enables to systematically relate the co-occurring abrasion and breakage process to co-evolving particle shape and size. Transformative concepts including the breakage line, sphere line, and average shape-conserving line are introduced to describe the limit states and a special comminution process. This approach also uncovers a self-similar nature in evolving particle geometry during comminution, which will be a significant discovery for the granular materials research community given the most fundamental properties observed in natural phenomena.', 'arxivid': '2306.04635', 'author': ['Priya Tripathi \nDept. of Civil & Environmental Engineering\nFlorida International University\nMiamiFL\n', 'Ph.DSeung Jae Lee \nDept. of Civil & Environmental Engineering\nFlorida International University\nMiamiFL\n', 'Ph.DMoochul Shin \nDept. of Civil & Environmental Engineering\nWestern New England University\nSpringfieldMA\n', 'Ph.DChang Hoon Lee \nDept. of Civil & Environmental Engineering\nWestern New England University\nSpringfieldMA\n'], 'authoraffiliation': ['Dept. of Civil & Environmental Engineering\nFlorida International University\nMiamiFL', 'Dept. of Civil & Environmental Engineering\nFlorida International University\nMiamiFL', 'Dept. of Civil & Environmental Engineering\nWestern New England University\nSpringfieldMA', 'Dept. of Civil & Environmental Engineering\nWestern New England University\nSpringfieldMA'], 'corpusid': 259095908, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9332, 'n_tokens_neox': 7844, 'n_words': 4220, 'pdfsha': '1292eb81946cf5f0015d6b4f0ea793afad497108', 'pdfurls': ['https://export.arxiv.org/pdf/2306.04635v2.pdf'], 'title': ['A New Paradigm Unifying the Concepts in Particle Abrasion and Breakage', 'A New Paradigm Unifying the Concepts in Particle Abrasion and Breakage'], 'venue': []} |
arxiv |
Probing horizon scale quantum effects with Love
Sayak Datta
Inter-University Centre for Astronomy and Astrophysics
Post Bag 4411 007Ganeshkhind, PuneIndia
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut)
D-30167HannoverGermany
Leibniz Universität Hannover
D-30167HannoverGermany
Probing horizon scale quantum effects with Love
(Dated: February 16, 2023)
Future gravitational wave detectors have been projected to be able to probe the nature of compact objects in great detail. In this work, we study the potential observability of the small-length scale physics near the black hole horizon with the tidal deformability of the compact objects in an inspiraling binary. We find that it is possible to probe them with extreme mass ratio inspirals. We discuss how the quantum effects can affect gravitational wave observables. This as a consequence is bound to shape our understanding of the quantum scale near the horizon.
I. INTRODUCTION
The discovery of gravitational waves(GWs) [1,2] paved the way towards probing fundamental physics. These observations provided a fillip to tests of General Relativity (GR) in the strong-field regime [3,4]; e.g., stringent bounds on the mass of the graviton and violations of Lorentz invariance have been placed [5][6][7]. As a result, GWs have become very important in the context of fundamental physics. Various possible distinctions between black holes (BHs) and other exotic compact objects (ECOs) based on tidal deformability [8][9][10][11], tidal heating [10,[12][13][14][15][16][17][18][19][20][21][22], multipole moments [12,[23][24][25][26][27], echoes in postmerger [28][29][30][31][32][33][34][35][36] and electromagnetic observations [34,[37][38][39][40] has been proposed in the literature.
One of the very intriguing questions in fundamental physics is how gravity behaves in the quantum regime. Since GWs bring information from the very close vicinity of BHs, it is expected that GWs may shed some light on this mystery [15,16,18,[41][42][43][44]. The idea behind such expectations follows from the fact that the Planck scale physics may affect the tidal Love numbers (TLNs) of the compact objects [10,11,45,46]. As compact objects coalesce, the information of the TLNs gets imprinted on the emitted GWs.
We study the challenges in achieving this due to the statistical error and the quantum noise. We will demonstrate for the first time that despite the quantum noise, it is possible to probe the near-horizon quantum scale physics with extreme mass ratio inspirals. As a result, not only do the small quantum corrections to the values of TLNs become measurable, but also inferring quantum noise will be possible. This will inevitably bring information from the quantum world near the horizon, shaping our understanding of the quantum nature of gravity.
In Sec. II we will discuss the basics of tidal deformability. Then in Sec. III the δ − k relation will be discussed. In Sec. IV the impact of quantum noise will be investigated. In V we will investigate the observability of small Love numbers with LISA. In Sec. VI limitation of the * [email protected] δ − k relation will be discussed. Then in Sec. VII a formalism will be constructed that is applicable for computing quantum contribution to the Love numbers. Finally, in Sec. VIII we will discuss the implication of our work and also its limitations.
II. TIDAL DEFORMABILITY
Consider a binary with the mass of the ith component to be m i in the inspiral phase. We can model these systems using the post-Newtonian (PN) theory, which is a weak-field/slow-velocity expansion of the field equations. The emitted GWs from such systems can be modeled in the frequency domain as [10,47],
h(f ) = A(f )e i(ψ P P (f )+ψ T H (f )+ψ T D (f )) ,(1)
where f is the GW frequency, A(f ) is the amplitude in the frequency domain. ψ P P (f ) is the contribution to the GW Fourier phase while treating the objects as spinning point particles, ψ T H (f ) is the contribution due to tidal heating, and ψ T D (f ) is the contribution due to their tidal deformability. In several works, it has been argued that ψ T H (f ) and ψ T D (f ) can be used to probe the nature of compact objects. As a result, it can be used as a distinguisher between black holes (BHs) and exotic compact objects (ECOs). In this work, we will focus only on ψ T D (f ). To leading PN order, this contribution for circular equatorial orbits is [48] ψ T D (f ) = − 117 8
(1 + q) 2 qΛ m 5 v 5 ,(2)
where v = (πmf ) 1/3 is the velocity, with m = m 1 + m 2 the total mass, and
26Λ = (1 + 12/q)λ 1 + (1 + 12q)λ 2 ,(3)
where, λ i = 2 3 k i m 5 i with k i the ( = 2, electric-type) TLNs and q = m 1 /m 2 is the mass ratio.
III. δ − k RELATION
TLNs are the response of a body to an external tidal field. It explicitly depends on the details of the internal structure of the compact object. It has been argued that for the BHs of GR, the TLN vanishes [49][50][51][52] 1 . Other compact objects unlike BHs, have a non-zero TLN. According to their equation of state, matter anisotropy, and fluid nature, neutron stars can have TLNs of O(10 2 ) [58][59][60][61][62][63][64][65][66][67] and similarly for the boson stars [8,9]. TLNs of some highly compact ECOs scales as ∼ 1/| log( )|, where δ ≡ r s −r H ≡ r H , where r s is the actual surface position of the ECO, and r H is the surface position of the horizon if it were a BH [8].
Motivated by this finding, it was argued in Ref. [10] that this logarithmic behavior can be used to possibly probe the Planck scale physics near the horizon (surface) of a BH (ECO). This logarithmic behaviour translates to the δ − k relation as follows (caveats are discussed later) [10],
δ = r s − r H = r H e −1/k(4)
Deviation of Planckian order (δ = pl ∼ O(10 −35 )meters) corresponds to k ∼ 10 −2 for masses of the BH ranging in the range (10 5 − 10 7 )M [10,68]. 2 From this it was proposed that by measuring small k, Planck scale physics can be probed.
IV. MEASURING QUANTUM NOISE
In such a case, it would seem that the only limitation disallowing us from such achievement is the sensitivity of the detectors. However, in [68] it has been argued that it is unlikely to be the case, as quantum noise of δ will populate at that level. As a result, the error in δ will get modified as [68],
σ Tot δδ = σ Stat r H r H 2 + 1 k 2 σ Stat kk 2 + a 2 2 pl δ 2 ≡ σ Stat δδ 2 + σ Sys δ δ 2(5)
where,δ andr H is the estimated value of δ and r H from the observation, and σ Sys δ = a 2 2 pl . σ Stat r H , σ Stat k , σ Stat δ are the statistical error in r H , k and δ respectively. Stat is the shorthand for statistical error. The error induced by quantum noise is a pl . Where pl is the Planck length and a is a number ∼ O (1).
Assuming this behavior of error, we can estimate a, which will help us in measuring the quantum noise. It is the first key observation of the current work. This will be possible to do since other parameters can be measured independently. From the observation we will have σ Stat M , σ Stat χ ,M ,χ. This can be used to estimate σ Stat r H ,r H . From the observation, the inferred value of TLNk will also be available. Therefore, if we can have an estimation of σ Stat k then we can estimate the a 2 . This can be done by performing simulations with injected synthetic signal in detectors withk and other observed parameters. Running a Bayesian estimation on that we can have an estimation of the statistical error, which is an artifact of the observation. With sufficiently sensitive detector σ Stat k can be reduced to very small values. By estimating those values from simulations we can estimate the systematic error, which is arising from the quantum nature. Having an estimation of a 2 can lead us to understand the quantum states near the horizon. For this purpose, in the next section, we will investigate if it is possible to reduce the statistical error sufficiently in future detectors.
V. OBSERVABILITY
Extreme mass ratio inspirals (EMRIs) are one of the promising sources of GW which will be observed with the future space-based Laser Interferometer Space Antenna (LISA) [69]. The emitted GW from these systems can stay in the detector band from months to a year. As a result, despite being small, with LISA we will be able to measure the TLNs of supermassive BHs in EMRI, quite precisely. Although the rates of EMRIs are not well understood it is expected that several such sources will be detected with LISA [70][71][72].
To estimate the effect of the TLN of these supermassive bodies in EMRI, we calculate dephasing as a function of k. We ignore the contribution of the secondary body. The primary body's mass is considered to be m 1 = M and the dimensionless spin is χ. A useful estimator to describe the effects of k in the phase is the total number of GW cycles (≡ N ) that accumulates within a given frequency band of the detectors. In terms of the frequencydomain phase ψ T D (f ) it is expressed as,
δφ = 2πN ≡ fISCO(M,χ) .4 mHz f df d 2 ψ T D (f ) df 2 ,(6)
where f ISCO is the GW frequency at the innermost stable circular orbit. In Fig. 1 and Fig. 2 we show the magnitude of the dephasing (δφ) in radian, as a function of k. The results are consistent with the expectations discussed in Ref. [73]. The black dashed horizontal line represents δφ = 1 radian. Dephasing δφ > 1 represents a strong effect [74][75][76][77][78][79]. In reality δφ > 1/ρ is a much more pertinent condition for an effect to be detectable, where ρ is signal to noise ratio (SNR) of the signal 3 . For this purpose, in Fig. 3 we plot the SNR of several sources situated at 1GPc, computed between .4 mHz and ISCO frequency. To compute the SNR the considered LISA sensitivity curve has been taken from [80]. As can be seen the SNR 1. If these same sources are nearby then the SNR will increase. For the sources considered the lowest value of SNR is ∼ 1. Therefore for such sources required dephasing would be ∼ 1. From the dephasing plot, it can be seen that it is possible to achieve. Note, k ∼ 10 −2 corresponds to Planck scale, assuming the δ − k relation in Eq. 4. Therefore, the smaller values correspond to the sub-Planckian scale that should be dominated by Planck scale noise. The result implies that the EMRIs can be the potential sources that will be sensitive to small-scale physics. However, considering δφ > 1 as an observational threshold has limitations. Although for very high SNR, this threshold can act as a sufficient condition to be detectable, it might not be good enough for low SNR sources. In such case, the statistical uncertainty on the phase could eventually overreach δφ, if k values are very small.
To be sensitive to the Planck scale physics it is atleast necessary that σ Stat δ <δ, where σ Stat δ is the statistical error of δ. In EMRIs the statistical error in δ will be dominated by the statistical error in k since a fractional error on mass and spin will be very less in LISA [81,82]. Hence, Assuming Eq. (4) to be valid, fork ∼ .005 (.01) to probe subPlanckian effects it is required that σ Stat k < 2.5 × 10 −5 (10 −4 ). From Fig. 1 and Fig. 2 it can be observed that such sensitivity can be reached with EM-RIs. Hence, statistical error is low enough in EMRIs. This does not mean that the Planck scale physics can be probed with this accuracy. It means that the dominating error will be just the quantum noise described in Eq.(5). As discussed before it can be used to estimate the quantum error itself, assuming the δ − k relation to be true. But in later sections, we will discuss why it is not just to assume the δ − k relation apriori. Rather we should use this opportunity to do an accurate measurement of the k to probe quantum correction or alternate theories of gravity. As well as we should try to investigate if there is any quantum error associated with k. The measurement of quantum error in k does not require δ −k relation to be valid apriori as it can arise from near horizon quantum effects. Note, the primary difference between the current work and Ref. [68] is that the considered sources are different. In the present work, the considered sources are EMRIs whereas, the sources considered in Ref. [68] and Ref. [10] are supermassive binaries.
σ Stat δδ ∼ σ Stat k k 2 .(7)
VI. INVALIDITY OF δ − k RELATION
In this section, we will argue that Eq. (4) is unlikely to hold in the context of GW observation. It is not justified to assume that k → 1/| log( )| scaling will be valid on a very small scale where quantum effects become important. This result has been derived assuming classical gravity. To probe small-scale physics, it is necessary for to be of that order. The conventional matter should collapse if it is distributed in such close proximity. The origin of such values of must be therefore exotic matter or quantum effects. In the above figure we demonstrate the SNR. The signal from .4mHz to ISCO frequency is considered. The SNR is lesser for total mass 10 7 M compared to 10 6 M . This is both due to a higher mass ratio and shorter duration of signal in the observable band. The sources are considered to be at 1GPc.
Hence, these systems are not "classical" systems to begin with. Consequently, it will become necessary to take into account the quantum properties of the states of the system to find the sub-leading contribution to the leading order classical results. This sub-leading "quantum corrections" most likely will be the interaction between the quantum observables at the quantum scale and the classical fields (discussed later). In such a case, the δ − k relationship is likely to get modified by k ∼ 1/| log( )| n + k q ( ), with n being a real number [11]. Therefore even though the first term starts to go to zero for very small , the second term survives and captures the details of the quantum nature. For BH as k = 0 classically, quantum effects can introduce nonzero k q , resulting in k = k q ( ).
It is important to ask, from which value of = q this behavior becomes important. If the compact objects are not sufficiently compact i.e. ECO q , then these quantum corrections (k q ) will not be important. However if ECO q , they can be used to probe the quantum scale near the horizon that is larger than the Planck scale. Another key issue is if any kind of δ − k relation seizes to exist then relations like Eq.(5) become invalid, making δ immeasurable from the measurement of k. But there will exist k q and non-zero systematic quantum noise in k, which will be discussed in the next section. Therefore, precise measurement of k and its error can help us probe the quantum nature near the horizon scale. As has already been demonstrated, EMRIs has such potential.
VII. LOVE IN THE QUANTUM WORLD
Due to the presence of an external tidal field, a nonzero quadrupole moment Q (multipole moment) gets induced on the bodies. In a linear regime, it is proportional to the external tidal field E, where the proportionality constant is the TLN (k). In the δ pl limit, a semiclassical quantum gravity approach can be applied to find the corrections to the classical contribution to the k.
Throughout our calculations, we will suppress the indices, and any non-scalar tensor will be represented by boldface. Therefore the tidal deformability can be defined as,
Q = −λE(8)
where, λ = 2 3 km 5 , with k and m being the TLN and the mass of the body (note m is not the total mass of a binary as was assumed before).
To find the contribution of the quantum effects we will consider quantum operators for all physical observables. We will assume none of the operators have zero eigenvalues, hence they are invertible 4 . We will separate the classical contribution and quantum fluctuation as,
λ →λ + λ cÎ Q →Q + Q cÎ E →Ê + E cÎ (9)
where λ c , Q c , E c are the classical contribution to the observables, andÎ (Î) is the tensor (scalar) identity operator. We will also assume that Eq. (8) is valid in this regime but in the sense of quantum operators 5 . Hence, it can be expressed as,
Q + Q cÎ = −λ c E cÎ −Êλ c −λE c −λÊ.(10)
Using this relation it is possible to identify the expressions of the classical contributions as well as the quantum contributions as,
λ c = − Q c E c ,λ = − Ê λ c +Q E c +Ê ≈ −Q E c + O(Ê). (11)
Note,Ê represents quantum corrections to the classical value of the external tidal field. Hence, this quantum correction represents the quantum correction of the external body's mass and the separation. In the right-most equation contribution ofÊ has been ignored. This result is equivalent to the expressions used in Ref. [11,45] (Check [46,83]). We will assume that the state of the system is |Ψ and we will suppress the Ψ while 4 In reality this stringent condition may not be required as all the required operators are scaled by a classical value. 5 It is likely that there will be some modification due to quantum effects. But for the current work, we will ignore such contributions.
writing the expectation value with respect to |Ψ . As a result, deformability gets modified as,
λ = λ c + λ ≡ λ c + λ q ,(12)
where represents expectation value, and,
λ q = − Ê λ c +Q E c +Ê ≈ − Q E c .(13)
Using this expression the systematic error in λ arising from the quantum nature can be expressed as,
σ Sys λ = Ê λ c +Q E c +Ê + λ q 2(14)
As the statistical error in k for EMRIs will be lower, observability of quantum noise solely will depend on the value of the standard deviation of the fluctuation ofk.
To find the corresponding result in k, we separate out each observable into its classical and quantum parts as,
k →k +Îk c , m →m +Îm c(15)
Using λ = 2 3 km 5 we find,
λ c = 2 3 m 5 c k c , k = 3λ 2m 5 c − 5k cm m c − 15λm 2m 6 c + O(m 2 ) λ q = 2 3 m 5 c k q + 5k c m m c + 5 km m c ,(16)
where, k q ≡ k . These expressions can be used to find the mean values of the macroscopic variables. If we separate out the mean value fromk aŝ k =x +Îk q then the error takes the simplified following form,
σ Sys k k = x 2 k ,(17)
Note, a knowledge of the quantum state of the body will not only allow estimating k q but also σ Sys k . Therefore if the systems do have quantum corrections, to measure its effect we have two observables to measure, namely the k q and σ Sys k . Since in EMRIs statistical error will be low, it can help us infer the systematic error.
It is important to point out that Eq.(4) is a modeldependent result found in Ref. [8]. However, other models have found different scaling relations, such as Ref. [11] found k ∼ 1/| ln | 2 . Therefore, approaching the problem of probing quantum scales assuming a particular δ − k relation is not just. Rather, measuring k q and it's quantum systematic error can shed some light on the nearhorizon quantum nature in a model-independent manner. It means that with EMRIs we can probe near horizon quantum scale larger than the Planck scale, making EMRIs the true GW microscopes.
Note, there is a degeneracy in the definition of k [8,49,58]. Therefore depending on the definition of k, λ ∝ k CF M P R m 5 [8] or λ ∝ k HBP R 5 [49,58]. In our work we considered the definition in Ref. [8], as connection with Planck scale physics is evident in this definition. However, most of the discussions in this work do not depend on one of the definitions. Therefore, while defininĝ k this issue needs to be resolved. If the definition in Ref. [49,58] is considered thenm will be replaced byR in the equations.
Using the prescription in this section we connect them with the observables. We have already argued that rather than focusing on any model dependent δ − k relation it is better to approach it in a model-agnostic manner. For that purpose, one should focus on measuring k q and σ Sys k . During parameter estimation, the measurement of k in this prescription will have both statistical and systematic error in a similar fashion as Eq. (5). Hence we can express it as follows:
σ Tot k k = σ Stat kk 2 + σ Sys k k 2(18)
where,k is the estimated value of k from the observation. Stat is the shorthand for statistical error just like before. As with EMRIs, the first term will become very small the error will be dominated by σ Sys k if σ Stat k σ Sys k . In the context of δ − k relation, this was precisely the case as a 2 1. Hence this can be used to infer σ Sys k or at least can be used to put some constraint on it.
This can be done by performing simulations with injected synthetic signal in detectors withk. Running a Bayesian estimation on that we can have an estimation of σ Stat k , which is an artifact of the observation. By estimating this value from simulations we can estimate the σ Sys k , which is arising from the quantum nature. As there will be other sources of systematic error also, i.e. incomplete noise realization, and post-Newtonian truncation error to mention a few, we will only be able to put some upper bound on the quantum noise. This can lead us to understand the quantum states near the horizon.
VIII. DISCUSSION
We have explored the resolving power of the EMRIs as gravitational microscope which can be used to probe near horizon physics with TLN k. The presence of the environmental effects could impact the GW signal [84][85][86][87] and exclusion of them may lead to erroneous measurements of TLNs [88]. Similarly, other competing effects can also mimic the effect of tidal deformability [13,14,20,28,29]. These should be taken into account to properly assess the potential of LISA. It is also required to study in detail from the theoretical standpoint the possible origin of these systems and their stability [89].
We have explicitly shown for the first time that very small values of k can add large dephasing in EMRIs. Our result suggests that it is possible for EMRIs to bring information regarding the quantum nature near the horizon scale. This paper also discusses the limitations of using the ECO relation between k and δ. We have also constructed a semi-classical formalism to take into account of the quantum effects. From the constructed formalism, it is evident that even if Eq. (4) is not valid, there will be quantum signatures on the observables, at least in principle. We discussed how it should be estimated. To achieve our conclusions we have assumed the binary to be in an equatorial circular orbit, which is unlikely to be true for EMRIs. This should be investigated in the future.
Quantum effects for large astrophysical BHs are usually considered to be negligibly small. This conclusion arises from the expectation that the strength of quantum effects is governed by the ratio 2 pl /r 2 s . However, it was argued in Ref. [11] that the strength of quantum effects can be much larger because they can be governed by the ratio of pl to the length scale of the fundamental theory of quantum gravity. In string theory, this is the string scale l s . As a result, the quantum effects are governed by the ratio g 2 s = 2 pl /l 2 s . Since g 2 s can be ∼ 0.1, it can have a larger contribution to the quantum effects [11]. This definitely requires further exploration.
Therefore it is high time to explore these avenues from the quantum gravity side. Finding possible effects of quantum gravity, as well as detailed numerical studies of coalescence of compact objects that has quantum contributions near their surfaces. This as a result will lead to proper quantification of quantum gravity effects on the GW observables.
Acknowledgement-I would like to thank Bhaskar Biswas, Richard Brito, Sumanta Chakraborty, Paolo Pani, Niels Warburton, and Nicolás Yunes for useful comments and also suggesting changes for the betterment of the article. I also thank Andrea Maselli, Swagat Mishra, Gabriel Andres Piovano, Karthik Rajeev, Shabbir Shaikh, and Yotam Sherf for useful discussions. I would like to thank University Grants Commission (UGC), India, for financial support for a senior research fellowship.
Appendix A: Dephasing-Mismatch
To assess the strength of an effect to be measurable in a GW detector with noise power spectral density S n (f ), the overlap O between two waveforms h 1 (t) and h 2 (t) are usually computed:
O(h 1 |h 2 ) = h 1 |h 2 h 1 |h 1 h 2 |h 2 ,(A1)
where, the inner product h 1 |h 2 is defined as,
h 1 |h 2 = 4 ∞ 0h 1h * 2 S n (f ) df . (A2)
The quantities with tilde stand for the Fourier transform and the star for complex conjugation. As the waveforms are defined up to an arbitrary time and phase shift, it is required to maximize the overlap (A1) over these quantities. This can be done by computing [90] O(h 1 |h 2 ) = 4 Two waveforms are considered to be indistinguishable for parameter estimation purposes if mismatch M 1/(2ρ 2 ) [74,75], where ρ is the SNR of the true signal. For an EMRI with an SNR ρ ≈ 20 (resp., ρ ≈ 100) one has M 10 −3 (resp., M 5×10 −5 ). For a large number of parameters, say D, this relation gets slightly modified as M D/(2ρ 2 ) [91].
h 1 |h 1 h 2 |h 2 max t0 F −1 h 1h * 2 S n (f ) (t 0 ) ,(A3)
Dephasing contribution (δφ) of an effect is indistinguishable from the absence of the effect in the context of scientific measurement if δφ 2 1/ρ 2 ∼ M. This condition is usually considered optimal in the sense that smaller dephasing than this is not measurable but not considering dephasing larger than this have distinguishable consequence [75]. The strongest LISA EM-RIs may have SNR of up to ρ ∼ 100 after matched filtering [75][76][77], so phase differences on the order of 1/ρ radians should be just detectable in matched filtering [75,78,79]. Keeping this in mind templates are constructed with δφ ≤ 1/ρ. Therefore for SNR ρ ≈ 20(100) M ∼ 10 −3 (resp., ∼ 5 × 10 −5 ) gets translated to dephasing δφ ≈ .05(.01). This implies that for any reasonable SNR, dephasing δφ > 1 would eventually be detectable. In light of this, usually, it is conventional wisdom to consider δφ ∼ 1 radian as the detection threshold.
FIG. 1 .
1We show the magnitude of dephasing (δφ) in radian, as a function of k. We varied M and q while keeping χ = .8.
FIG. 2 .
2We show the magnitude of the dephasing (δφ) in radian, as a function of k. We varied χ while keeping primary and secondary mass fixed at 10 7 M and 2M respectively.
FIG. 3. In the above figure we demonstrate the SNR. The signal from .4mHz to ISCO frequency is considered. The SNR is lesser for total mass 10 7 M compared to 10 6 M . This is both due to a higher mass ratio and shorter duration of signal in the observable band. The sources are considered to be at 1GPc.
where F −1 [g(f )](t) = +∞ −∞ g(f )e −2πif t df is the inverse Fourier transform.The overlap is defined in such a manner that O = 1 indicates a perfect agreement between the two waveforms. The mismatch (M) is defined as follows:M ≡ 1 − O (A4)
arXiv:2107.07258v3 [gr-qc] 15 Feb 2023
Recently in several works it has been demonstrated that the origin of the vanishing Love number is connected with the so-called Ladder symmetry[53][54][55][56][57]. In the presence of quantum hair this may break down.2 For an invariant definition of δ check[68].
For further details on the connection between dephasing, mismatch, and SNR, check appendix A.
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arxiv |
Integrable deformations of AdS/CFT
16 Mar 2022
Marius De Leeuw [email protected]
School of Mathematics
Hamilton Mathematics Institute Trinity College Dublin Dublin
Ireland
Anton Pribytok [email protected]
School of Mathematics
Hamilton Mathematics Institute Trinity College Dublin Dublin
Ireland
Ana L Retore [email protected]
School of Mathematics
Hamilton Mathematics Institute Trinity College Dublin Dublin
Ireland
Paul Ryan [email protected]
School of Mathematics
Hamilton Mathematics Institute Trinity College Dublin Dublin
Ireland
Integrable deformations of AdS/CFT
16 Mar 2022
In this paper we study in detail the deformations introduced in [1] of the integrable structures of the AdS 2,3 integrable models. We do this by embedding the corresponding scattering matrices into the most general solutions of the Yang-Baxter equation. We show that there are several non-trivial embeddings and corresponding deformations. We work out crossing symmetry for these models and study their symmetry algebras and representations. In particular, we identify a new elliptic deformation of the AdS 3 × S 3 × M 4 string sigma model.
Introduction
The main hallmark of integrable field theories is the factorisation of scattering events into sequences of two-body scattering processes [2]. This is due to the presence of a tower of conserved charges, which severely restrict possible scattering events. In particular, scattering is purely elastic and there is no particle production or annihilation. The property that three-body scattering factorises into a sequence of two-body scattering events in a consistent way imposes the following constraint on the two-body S-matrix S 23 S 12 S 23 = S 12 S 23 S 12 .
(1.1) This is the celebrated Yang-Baxter equation [3].
Integrable field theories have made numerous appearances in the context of the AdS/CFT correspondence [4] and on various string backgrounds the 1 + 1 dimensional theory on the string worldsheet defines an integrable QFT, see for example [5,6]. This is achieved by fixing the uniform light-cone gauge and decompactifying the worldsheet to a plane where the notion of asymptotic states and hence an S-matrix can be defined. From here, one can study scattering in perturbation theory for example.
It is clearly not feasible to explicitly compute a scattering process to all orders in perturbation theory. Instead one can make use of symmetry considerations. While the fixing of the light-cone gauge breaks some of the isometries of the initial background the residual symmetry is often highly constraining. We will focus our attention in this work on the AdS 3 × S 3 × T 4 , AdS 3 × S 3 × S 3 × S 1 and AdS 2 × S 2 × T 6 backgrounds which are known to lead to integrable QFTs, see for example [7][8][9]. At one loop the residual symmetry algebras can be determined by explicit calculation. From here one can conjecture the exact all-loop symmetry algebra. The symmetry algebras together with integrability (imposing the Yang-Baxter equation) completely determine the S-matrix up to the dressing phase, which is constrained by crossing symmetry instead.
An interesting question is the study of deformations of integrable QFTs which preserve integrablity. Numerous such deformations of the models described above have been constructed, for example η and λ deformations [10][11][12]. In these cases the isometry algebra is known to undergo a q-deformation. For instance, in the case of the AdS 5 × S 5 background whose lightcone gauge fixed model has su(2|2) ce symmetry, the deformed model has U q (psu(2|2) ce ) symmetry [13]. These deformations have also been worked out for the AdS 3 models and various generalisations and other deformations of our mentioned backgrounds have been constructed, see e.g. [11,[14][15][16][17][18][19][20].
The deformations just discussed are usually constructed at the level of the superstring action and it is not always clear how these deformations lift to the level of the worldsheet S-matrix and the corresponding symmetry algebra. In [1] we classified all possible S-matrices of integrable systems of so-called 6-and 8-vertex type. Physically, these correspond to S-matrices which preserve fermion number. We can embed the superstring S-matrices in these models by fixing various free functions and parameters. By varying these parameters we subsequently obtain integrable deformations of the S-matrices. Hence, we actually find a complete classification of possible integrable deformations to the aforementioned superstring S-matrices. In this way we therefore identify the possible integrable deformations of these holographic models. We find an interesting possible deformation structure, summarised in Figure 1. The massive scattering matrix of the AdS 2 superstring only admits one integrable deformation. It is an elliptic deformation, parameterized by the elliptic modulus. It exhibits the same Lie algebra, but in a different representation. Remarkably, the deformation does seem to affect the higher Yangian symmetry generators and hence might define a new type of quantum algebra.
More interestingly the S-matrix of the AdS 3 × S 3 × M 4 string sigma model admits two distinct deformations. The first deformation of the S-matrix is related to the quantum deformation found in [15,20] and the only extra degrees of freedom stem from the fact our functions are unconstrained and do not need to satisfy relations involving the physical constants such as the mass. This is why we refer to such deformations as functional deformations.
The second deformation is an elliptic deformation and it is a novel deformation. We show that it satisfies all the usual physical requirements such as crossing symmetry and braiding unitarity. Remarkably, for this model part of the symmetry algebra seems to be broken for non-zero values of the deformation parameter. It would be very interesting to see if the corresponding sigma model can be found. In particular, we find that the elliptic deformation is not a further deformation of the q-deformed model [15].
Finally, it is also interesting to notice that all the deformations of AdS 2,3 S-matrices satisfy the free fermion condition [21]. This allows for a great simplification in known results for lower-dimensional instances of AdS/CFT, including backgrounds supported by various fluxes. For 6-vertex like AdS 3 models and its deformations, the transfer matrix was rewritten for an arbitrary number of sites in a free-fermion form using Bogoliubov transformations.
Outline of this paper In this paper we examine the structure of these integrable deformations in more detail. In Section 2 we review the 6-and 8-vertex R-matrices which the conventional AdS 2,3 integrable systems can be embedded in. In Section 3 we discuss the AdS 2 deformations while in Section 4 we discuss the AdS 3 deformations and extend our deformed R-matrices for same-chirality scattering in order to account for scattering processes between particles with opposite chirality. In Section 5 we give the explicit map to recover known AdS 3 S-matrices, as well as its q-deformations. The deformed symmetry algebras are presented in Section 6. Finally, in Section 7 we discuss crossing symmetry and provide the crossing equations for all the deformations explicitly. We end with discussion and conclusions.
Review and notation
In order to preserve fermion number, for boson φ and fermion ψ the allowed scattering processes are φφ → φφ + ψψ ψψ → φφ + ψψ φψ → φψ + ψφ ψφ → ψφ + φψ with some weighting associated to each process, or in matrix form we have
R(u, v) = r 1 0 0 r 8 0 r 2 r 6 0 0 r 5 r 3 0 r 7 0 0 r 4 (2.1)
where u, v are the spectral parameters and each of the functions r j is assumed to depend on them. We will also use the (p, q) instead of (u, v) throughout the text to emphasize that these parameters correspond to the momenta of the particles being scattered. The R-matrices classified in [1] are the most general R-matrices of the above form.
The R-matrices of this type can be separated into two categories -so-called 6-vertex and 8-vertex models. Physically, 6-vertex models are those for which spin is conserved in scattering processes and so a boson pair cannot scatter to produce a fermion pair and vice-versa and as a result r 7 = r 8 = 0. In [1] both 6-and 8-vertex were further divided into two subcategories, dubbed A and B. These categories are described by the free fermion condition [21]. For applications to AdS integrable systems it is only the 6-vertex B and 8-vertex B which are relevant. Hence, for notation simplicity we will simply refer to them as 6-vertex or 6vB and 8-vertex or 8vB.
6-vertex
The R-matrix for the 6-vertex case has r 7 = r 8 = 0 and can be written as
r 1 (p, q) = h 2 (q) − h 1 (p) h 2 (p) − h 1 (p) , r 2 (p, q) = (h 2 (p) − h 2 (q))X(p)Y (p), r 3 (p, q) = h 1 (p) − h 1 (q) (h 2 (p) − h 1 (p))(h 2 (q) − h 1 (q)) 1 X(q)Y (q)
,
r 4 (p, q) = h 2 (p) − h 1 (q) h 2 (q) − h 1 (q) X(p)Y (p) X(q)Y (q) , r 5 (p, q) = Y (p) Y (q) , r 6 (p, q) = X(p) X(q) . (2.2)
h 1 , h 2 , X, Y are free functions. The Yang-Baxter equation is satisfied for any choice of them. Compared to [1] we have some extra functions in our R-matrix, namely X and Y . This is due to the fact that the R-matrix in [1] was solved by fixing some functions in the Hamiltonian using identifications such as normalization and local basis transformations. The new functions X, Y simply correspond to a twist and are important for the identification with the AdS 3 models and in order to obtain the crossing equations presented in Section 7.
8-vertex
The R-matrix is most conveniently written using Jacobi elliptic functions and we use the shorthand notation
sn = sn(u − v, k 2 ), cn = cn(u − v, k 2 ), dn = dn(u − v, k 2 ),(2.3)
to denote the elliptic functions of modulus k. To avoid potential ambiguities in conventions let us stress that we follow the convention that the elliptic functions written above is how they would be entered in Mathematica, and so we have, for example
dn 2 + k 2 sn 2 = 1 . (2.4)
The entries of the R-matrix are given by
r 1 = 1 sin η(u) sin η(v) sin η + cn dn − cos η + sn , r 2 = −1 sin η(u) sin η(v) cos η − sn + sin η − cn dn , r 3 = −1 sin η(u) sin η(v) cos η − sn − sin η − cn dn , r 4 = 1 sin η(u) sin η(v)
sin η + cn dn + cos η + sn , r 5 = r 6 = 1,
r 7 = r 8 = k sn cn dn , (2.5) where η ± = η(u)±η(v) 2 .
Deforming the AdS 2 S-matrix
The massive S-matrix for the AdS 2 integrable model [8] can only be embedded in the model 8vB due to the presence of the components r 7,8 . In order to construct the embedding we need to first perform various integrability-preserving transformations on the R-matrix which can be found in [22]. The main issue to be overcome is that the spectral parameters appearing in both models are different despite being denoted by the same letters u and v. To get around this we need to transform (u, v) → (G(u), G(v)) in one of the R-matrices and we take this to be in R 8vB . Notice, also that the R-matrix as in equation (2.5) is the boson-boson one, so in order to compare it with [8] one needs to do the appropriate modifications.
In [1] we compared the corresponding Hamiltonians to show that R AdS 2 is a special case of R 8vB , but it is also instructive to compare the R-matrices directly. We start by considering the (1, 4) component of both R-matrices which are [8], for R 8vB and R AdS 2 respectively,
(R 8vB ) 14 = k sn(G(u) − G(v)) cn(G(u) − G(v)) dn(G(u) − G(v)) , (3.1) (R AdS 2 ) 14 = 1 x + u x − u x + v x − v x − u − 1 x + u x + u x − u − x − v − 1 x + v x + v x − v 1 − 1 x + u x − u x + v x − v (3.2)
where x ± are the Zhukovski variables. Clearly, the (1, 4) component of R 8vB is of difference form, that is it only depends on the difference G(u) − G(v) of the spectral parameters. Let us expand the (1, 4) component of the AdS 2 R-matrix in u around v. We find
(x + x − ) 2 √ x − √ x + (x + x − − 1) (u − v) + O (u − v) 2 . (3.3)
In order to be purely of difference form we must have that the coefficient of u − v is a constant which we denote A:
(x + x − ) 2 √ x − √ x + (x + x − − 1) = A. (3.4)
Hence, after reinstating the G dependence, we solve to obtain
x + (v) = Tanh AG(v) + c 1 2 x − (v) . (3.5)
This completely fixes G in terms of x ± . After substituting (3.5) back into the (1, 4) component of the AdS 2 R-matrix we find that it reduces to simply 1
(R AdS 2 ) 14 = −Tanh (A(G(u) − G(v))) .
(3.6)
A comparison with the (1, 4) component of the 8vB R-matrix then tells us that we should take the limit k → ∞ in order to have this entry reduce to Tanh and furthermore the precise agreement requires that A = −i and we can take c 1 = 0, and so we find that 2
x + (u) = − Tan 2 (G(u)) x − (u) . (3.7)
Next, we make the substitution η(u) → arccot (kF (u)) and expand the 8vB R-matrix around k → ∞. By subsequently expanding around u = v we find that setting
F (u) = − 1 2 csc(G(u)) sec(G(u)) cot(G(u)) x − + i cot(G(u)) x − − i (3.8)
indeed reproduces the AdS 2 R-matrix. Notice that x − is in principle defined implicitly via the shortening condition
x + + 1 x + − x − − 1 x − = 2im h , (3.9)
where m is the mass, and the h is the coupling constant. But this relation is not needed for the mapping between the two models. Reversely, we can define m in terms of F, G in this way.
Deformation In order to embed R AdS 2 into R 8vB we sent the elliptic modulus k to infinity. We then immediately see a source of deformation comes from flowing away from infinity to finite values of k. This is the only available non-trivial source of deformation. There are also functional deformations corresponding to shifts of the form
η(u) → η(u) + λη(u) (3.10)
whereη(u) is an arbitrary function and λ is a deformation parameter. These deformations correspond to making the mass depend on the spectral parameter m → m(u). This completes the embedding of R AdS 2 into R 8vB . Later, we will discuss the deformed symmetry algebra in Section 6.
Deforming the AdS 3 S-matrix
In [1,22] we derived the most general 4 × 4 integrable R-matrices of 8-vertex type. These are exactly the R-matrices that are compatible with the graded structure that arises for the holographic integrable models. Hence, by embedding the R-matrices of AdS 3 in these R-matrices, we find the most general way in which they can be deformed compatible with the splitting in chiral blocks.
In this section we show that the massive S-matrix for the AdS 3 [6,24,9] integrable model can be embedded in both 6-vertex B and 8-vertex B extended models.
Let us explain how to construct the full 16×16 R-matrix given we have the 4×4 regular R-matrix, and then we apply this procedure to lift the models described in (2.1)-(2.5).
General procedure
Following the construction in [6], we decompose the full R-matrix R as
R = r LL 1 0 0 0 0 r LL 8 0 0 0 0 0 0 0 0 0 0 0 r LL 2 0 0 r LL 6 0 0 0 0 0 0 0 0 0 0 0 0 0 r LR 1 0 0 0 0 r LR 8 0 0 0 0 0 0 0 0 0 0 0 r LR 2 0 0 r LR 6 0 0 0 0 0 0 0 0 0 0 r LL 5 0 0 r LL 3 0 0 0 0 0 0 0 0 0 0 0 r LL 7 0 0 0 0 r LL 4 0 0 0 0 0 0 0 0 0 0 0 0 0 r LR 5 0 0 r LR 3 0 0 0 0 0 0 0 0 0 0 0 r LR 7 0 0 0 0 r LR 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r RL 1 0 0 0 0 r RL 8 0 0 0 0 0 0 0 0 0 0 0 r RL 2 0 0 r RL 6 0 0 0 0 0 0 0 0 0 0 0 0 0 r RR 1 0 0 0 0 r RR 8 0 0 0 0 0 0 0 0 0 0 0 r RR 2 0 0 r RR 6 0 0 0 0 0 0 0 0 0 0 r RL 5 0 0 r RL 3 0 0 0 0 0 0 0 0 0 0 0 r RL 7 0 0 0 0 r RL 4 0 0 0 0 0 0 0 0 0 0 0 0 0 r RR 5 0 0 r RR 3 0 0 0 0 0 0 0 0 0 0 0 r RR 7 0 0 0 0 r RR 4 (4.1)
where R ≡ R(u, v) and r AB i ≡ r AB i (u, v) and A, B ∈ {L, R}. The full R-matrix R satisfies the usual Yang-Baxter equation
R 12 (u, v)R 13 (u, w)R 23 (v, w) = R 23 (v, w)R 13 (u, w)R 12 (u, v) (4.2)
and R ij act on three vector spaces V of dimension four V ⊗ V ⊗ V. Let us allow for arbitrary scalar factors σ LR , then the functions r AB i (u, v) are matrix elements of the 4 × 4 R-matrices
R AB (u, v) = σ AB r AB 1 0 0 r AB 8 0 r AB 2 r AB 6 0 0 r AB 5 r AB 3 0 r AB 7 0 0 r AB 4 (4.3)
where R LL and R RR are regular, i.e.,
R LL (u, u) ∼ P and R RR (u, u) ∼ P,(4.4)
and P is the permutation operator for a Hilbert space of dimension two. The functions σ AB (u, v) are at this point arbitrary, but they will have to satisfy certain properties in order for both the blocks and the full R-matrix to satisfy crossing symmetry and braiding unitarity.
The fact that YBE (4.2) for the full 16×16 is satisfied is equivalent to the fact that the blocks {R LL (u, v), R RL (u, v), R LR (u, v), R RR (u, v)} satisfy eight Yang-Baxter equations given by all possible ways to distribute two chiralities into three Hilbert spaces:
YBE(A, B, C) = R AB 12 (u, v)R AC 13 (u, w)R BC 23 (v, w) − R BC 23 (v, w)R AC 13 (u, w)R AB 12 (u, v) = 0. (4.5) Each R AB ij (u, v) acts on V (A) ⊗ V (B) ⊗ V (C) , where each V (A)
is a vector space of dimension two and A, B, C ∈ {L, R}. So, each two dimensional vector space has a chirality associated to itself.
For example, for A = L, B = R and C = L we have
YBE(L, R, L) = R LR 12 (u, v)R LL 13 (u, w)R RL 23 (v, w) − R RL 23 (v, w)R LL 13 (u, w)R LR 12 (u, v) = 0.
(4.6) So, in other words, if R RR , R LL , R RL and R LR satisfy all the eight YBE's defined in (4.5) then the full R-matrix (4.1) will satisfy the equation (4.2).
At this point, it is important to remember that the method developed in [1] only allows one to compute regular R-matrices. So, our starting point is to put those matrices in the left-left and right-right blocks and use the six remaining YBE in (4.5) to fix the left-right and right-left blocks.
The complete procedure is the following:
1. We start by assuming that R LL and R RR are given by a regular 4 × 4 R-matrix as for example in equation (2.1). We will assume that both of these blocks can be deformed independently and so we use different names for the functions in the LL and RR blocks.
Notice that with this YBE(L, L, L) and YBE(R, R, R) are already satisfied.
We assume
R LR (u, v) and R RL (u, v) of the form (4.3). Then, we substitute R LR (u, v) together with R LL (u, v) in YBE(L, L, R) and solve them for r LR i . Similarly, solving in YBE(R, R, L) we construct r RL i .
3. The previous step fixes R LR (u, v) and R RL (u, v) apart from functions of one variable. We use the remaining four YBE(A, B, C) to fix these functions.
The last step is to substitute all r A B
i in the full R-matrix (4.1) and check that it indeed satisfies YBE (4.2).
For the final result, one of course needs to check braiding unitarity and to find the crossing relations. Braiding unitarity is checked in the next two sections, while crossing symmetry is discussed on a case by case basis in Section 7.
A deformation of AdS 3 : 6-vertex B
Now, for the 6 vertex B model, we present the four blocks obtained by applying the procedure above. For the left-left sector, (A = L and B = L) we have
r LL 1 = h L 2 (q) − h L 1 (p) h L 2 (p) − h L 1 (p) , r LL 7 = 0 = r LL 8 , (4.7) r LL 2 = (h L 2 (p) − h L 2 (q))X L (p)Y L (p), (4.8) r LL 3 = h L 1 (p) − h L 1 (q) (h L 2 (p) − h L 1 (p)) (h L 2 (q) − h L 1 (q)) 1 X L (q)Y L (q)
, (4.9)
r LL 4 = h L 2 (p) − h L 1 (q) h L 2 (q) − h L 1 (q) X L (p)Y L (p) X L (q)Y L (q) , (4.10) r LL 5 = Y L (p) Y L (q) , r LL 6 = X L (p) X L (q) . (4.11)
This block satisfies YBE(L, L, L) = 0. Notice that R LL looks slightly different from the R-matrix introduced in [1]. This is due to a twist 3 performed in order to make it satisfy crossing symmetry and compare it with AdS 3 R-matrix. For the right-right sector (A = R and B = R) we have
r RR 1 = h R 2 (q) − h R 1 (p) h R 2 (p) − h R 1 (p) , r RR 7 = 0 = r RR 8 (4.12) r RR 2 = h R 2 (p) − h R 2 (q) h R 2 (p) 2 X R (p)Y R (p), (4.13) r RR 3 = h R 2 (q) 2 h R 1 (p) − h R 1 (q) (h R 2 (p) − h R 1 (p)) (h R 2 (q) − h R 1 (q)) 1 X R (q)Y R (q)
, (4.14)
r RR 4 = h R 2 (q) 2 h R 2 (p) 2 h R 2 (p) − h R 1 (q) h R 2 (q) − h R 1 (q) X R (p)Y R (p) X R (q)Y R (q) , (4.15) r RR 5 = h R 2 (q) h R 2 (p) X R (p) X R (q) , r RR 6 = h R 2 (q) h R 2 (p) Y R (p) Y R (q) . (4.16)
This block is independent of R LL block and from the R-matrix introduced in [1] again due to a twist. Also, R RR satisfies YBE(R, R, R) = 0. So, the functions in R LL and R RR are independent and what connects them are the blocks of opposite chirality introduced now.
The right-left sector (A = R and B = L) is given by
r RL 1 = 1, r RL 5 = 0 = r RL 6 (4.17) r RL 2 = − h R 2 (p) 2 h R 2 (p) − h R 1 (p) 1 + h L 1 (q)h R 1 (p) 1 + h L 1 (q)h R 2 (p) 1 X R (p)Y R (p) , (4.18) r RL 3 = − h L 2 (q) − h L 1 (q) 1 + h L 2 (q)h R 2 (p) 1 + h L 1 (q)h R 2 (p) X L (q)Y L (q), (4.19) r RL 4 = −h R 2 (p) 2 h L 2 (q) − h L 1 (q) h R 2 (p) − h R 1 (p) 1 + h L 2 (q)h R 1 (p) 1 + h L 1 (q)h R 2 (p)) X L (q)Y L (q) X R (p)Y R (p) , (4.20) r RL 7 = i h R 2 (p) h L 2 (q) − h L 1 (q) 1 + h L 1 (q)h R 2 (p) Y L (q) Y R (p) , (4.21) r RL 8 = −i h R 2 (p) h L 2 (q) − h L 1 (q) 1 + h L 1 (q)h R 2 (p) X L (q) X R (p) . (4.22)
Finally, the left-right sector (A = L and B = R) is given by
r LR 1 = 1, r LR 5 = 0 = r LR 6 (4.23) r LR 2 = − 1 h L 2 (p) − h L 1 (p) 1 + h L 1 (p)h R 1 (q) 1 + h L 2 (p)h R 1 (q) 1 X L (p)Y L (p) , (4.24) r LR 3 = − h R 2 (q) − h R 1 (q) h R 2 (q) 2 1 + h L 2 (p)h R 2 (q) 1 + h L 2 (p)h R 1 (q) X R (q)Y R (q), (4.25) r LR 4 = − 1 h R 2 (q) 2 h R 2 (q) − h R 1 (q) h L 2 (p) − h L 1 (p) 1 + h L 1 (p)h R 2 (q) 1 + h L 2 (p)h R 1 (q)) X R (p)Y R (p) X L (p)Y L (p) , (4.26) r LR 7 = i h R 2 (q) h R 2 (q) − h R 1 (q) 1 + h L 2 (p)h R 1 (q) X R (q) X L (p) , (4.27) r LR 8 = − i h R 2 (q) h R 2 (q) − h R 1 (q) 1 + h L 2 (p)h R 1 (q) Y R (q) Y L (p) . (4.28)
With these four blocks satisfying all the eight YBE(A, B, C), the full R-matrix R is given by (4.1), and it satisfies (4.2). It is remarkable that it is possible to deform the left-left and right-right blocks independently and still obtain meaningful right-left and left-right blocks.
Because of this independence of the diagonal blocks, 6-vertex B is both a deformation of AdS 3 × S 3 × T 4 [6,24] and AdS 3 × S 3 × S 3 × S 1 [9]. More details about the comparison between these models and the undeformed ones are given in section 5. We find braiding unitarity in each of the four blocks
R RR (p, q)PR RR (q, p)P = B RR (p, q)I, (4.29) R LL (p, q)PR LL (q, p)P = B LL (p, q)I, (4.30) R RL (p, q)PR LR (q, p)P = B RL (p, q)I, (4.31) R LR (p, q)PR RL (q, p)P = B LR (p, q)I (4.32) where B RR (p, q) = h R 2 (p) − h R 1 (q) h R 2 (p) − h R 1 (p) h R 2 (q) − h R 1 (p) h R 2 (q) − h R 1 (q) σ RR (p, q)σ RR (q, p), (4.33) B LL (p, q) = h L 2 (p) − h L 1 (q) h L 2 (p) − h L 1 (p) h L 2 (q) − h L 1 (p) h L 2 (q) − h L 1 (q) σ LL (p, q)σ LL (q, p), (4.34) B RL (p, q) = 1 + h L 2 (q)h R 2 (p) 1 + h L 1 (q)h R 2 (p) 1 + h R 1 (q)h L 1 (p) 1 + h R 1 (p)h L 2 (q) σ RL (p, q)σ LR (q, p), (4.35) B LR (p, q) = 1 + h L 2 (p)h R 2 (q) 1 + h L 1 (p)h R 2 (q) 1 + h R 1 (q)h L 1 (p) 1 + h R 1 (q)h L 2 (p) σ LR (p, q)σ RL (q, p). (4.36)
With the above expressions, if
B(p, q) ≡ B RR (p, q) = B LL (p, q) = B RL (p, q) = B LR (p, q) (4.37)
then the full R-matrix R (4.1) automatically satisfies braiding unitarity
R(p, q)PR(q, p)P = B(p, q) 1. (4.38)
A deformation of AdS 3 : 8-vertex B
In this section we present the four 4 × 4 blocks (and consequently the full R-matrix) for the 8-vertex model introduced in [1]. This model can be seen as a deformation of 24,9]. The following notation will be used
AdS 3 × S 3 × M 4 R-matrix introduced in [6,η ± = η(u) ± η(v) 2 , sn AB ± = sn(G A (u) ± G B (v), k 2 ), (4.39) cn AB ± = cn(G A (u) ± G B (v), k 2 ), dn AB ± = dn(G A (u) ± G B (v), k 2 ). (4.40)
The explicit form of each matrix element r AB i is presented below, starting by the left-left block
r LL 1 = 1 sin(η(u)) sin(η(v)) − cos η + sn LL − + cn LL − dn LL − sin η + , r LL 2 = − 1 sin(η(u)) sin(η(v)) cos η − sn LL − − cn LL − dn LL − sin η − , r LL 3 = − 1 sin(η(u)) sin(η(v)) cos η − sn LL − − cn LL − dn LL − sin η − , r LL 4 = 1 sin(η(u)) sin(η(v)) cos η + sn LL − + cn LL − dn LL − sin η + , r LL 5 = g L (v) g L (u) , r LL 6 = g L (u) g L (v) , r LL 7 = kα g L (u)g L (v) cn LL − sn LL − dn LL − , r LL 8 = k g L (u)g L (v) α cn LL − sn LL − dn LL − . (4.41)
This is simply the 8-vertex B R-matrix where we added a diagonal local basis transformation with component g L . This block satisfies the YBE(L, L, L) (equation (4.5)).
Similarly, let us now introduce the right-right block
r RR 1 = 1 sin(η(u)) sin(η(v)) − cos η + sn RR − + cn RR − dn RR − sin η + , r RR 2 = − 1 sin(η(u)) sin(η(v)) cos η − sn RR − + cn RR − dn RR − sin η − , r RR 3 = − 1 sin(η(u)) sin(η(v)) cos η − sn RR − − cn RR − dn RR − sin η − , r RR 4 = 1 sin(η(u)) sin(η(v)) cos η + sn RR − + cn RR − dn RR − sin η + , r RR 5 = g R (u) g R (v) , r LL 6 = g R (v) g R (u) , r RR 7 = k g R (u)g R (v) α cn RR − sn RR − dn RR − , r RR 8 = kα g R (u)g R (v) cn RR − sn RR − dn RR − , (4.42)
that satisfies the YBE(R, R, R). Notice that a priori R LL (u, v) and R RR (u, v) can have different elliptic moduli k L , k R and different functions η L,R . However, by computing the LR and RL blocks we find that the parameters need to be related and the same is true for the function η. Indeed, by using different η L and η R in each block we found that they are related by η R = π − η L . Using the method described in Section 4.1 we can then construct the following right-left block
r RL 1 = 1 sin(η(u)) sin(η(v)) g R (u) g L (v) cos η − sn RL + + cn RL + dn RL + sin η − , r RL 2 = 1 sin(η(u)) sin(η(v)) g R (u) g L (v) cos η + sn RL + − cn RL + dn RL + sin η + , r RL 3 = 1 sin(η(u)) sin(η(v)) g R (u) g L (v) cos η + sn RL + + cn RL + dn RL + sin η + , r RL 4 = 1 sin(η(u)) sin(η(v)) g R (u) g L (v) − cos η − sn RL + + cn RL + dn RL + sin η − , r RL 5 = − kg R (u) α cn RL + sn RL + dn RL + , r RL 6 = kα g L (v) cn RL + sn RL + dn RL + , r RL 7 = − g R (u) g L (v) , r RL 8 = 1,(4.43)
and finally the left-right block
r LR 1 = 1 sin(η(u)) sin(η(v)) g R (v) g L (u) cos η − sn LR + + cn LR + dn LR + sin η − , r LR 2 = 1 sin(η(u)) sin(η(v)) g R (v) g L (u) cos η + sn LR + − cn LR + dn LR + sin η + , r LR 3 = 1 sin(η(u)) sin(η(v)) g R (v) g L (u) cos η + sn LR + + cn LR + dn LR + sin η + , r LR 4 = 1 sin(η(u)) sin(η(v)) g R (v) g L (u) − cos η − sn LR + + cn LR + dn LR + sin η − , r LR 5 = − kα g L (u) cn LR + sn LR + dn LR + , r LR 6 = kg R (v) α cn LR + sn LR + dn LR + , r LR 7 = − g R (u) g L (v) , r LR 8 = 1. (4.44)
We can now immediately construct the full R-matrix R (4.1) and check that it indeed satisfies the Yang-Baxter equation (4.2). Now let us discuss braiding unitarity. For the four blocks just presented we have
R RL (u, v)PR LR (v, u)P = B RL (u, v)I, (4.45) R LR (u, v)PR RL (v, u)P = B LR (u, v)I, (4.46) R RR (u, v)PR RR (v, u)P = B RR (u, v)I, (4.47) R LL (u, v)PR LL (v, u)P = B LL (u, v)I. (4.48) where B LL (u, v) σ LL (u, v)σ LL (v, u) = cn 2 L,L,− dn 2 L,L,− − sin 2 η + sin η(u) sin η(v) + k 2 sn 2 L,L,− − sn 2 L,L,− cos 2 η + sin η(u) sin η(v) (4.49) B RR (u, v) σ RR (u, v)σ RR (v, u) = cn 2 R,R,− dn 2 R,R,− − sin 2 η + sin η(u) sin η(v) + k 2 sn 2 R,R,− − sn 2 R,R,− cos 2 η + sin η(u) sin η(v) (4.50) g R (v) g L (u) B LR (u, v) σ LR (u, v)σ RL (v, u) = − cn 2 L,R,+ dn 2 L,R,+ sin 2 η − sin η(u) sin η(v) + sn 2 L,R,+ cos 2 η − sin η(u) sin η(v) − 1 (4.51) g R (u) g L (v) B RL (u, v) σ RL (u, v)σ LR (v, u) = − cn 2 R,L,+ dn 2 R,L,+ sin 2 η − sin η(u) sin η(v) + sn 2 R,L,+ cos 2 η − sin η(u) sin η(v) − 1,(4.52)
where cn A,B,± = cn(G A (u) ± G B (v), k 2 ), and similarly for dn and sn.
In order for the full R-matrix R(u, v) to satisfy braiding unitarity
R(u, v)PR(v, u)P = B(u, v)I (4.53)
it is necessary that
B(u, v) ≡ B LL (u, v) = B RR (u, v) = B RL (u, v) = B LR (u, v). (4.54)
This imposes additional constraints on σ AB .
Embeddings of AdS 3 × S 3 × M 4
Let us now show how to precisely embed the various AdS 3 R-matrices into the general ones that we derived in the previous section.
Recovering AdS
3 × S 3 × S 3 × S 1 R-matrix from 6-vertex B
Now we compare the 6 vertex B full R-matrix R(u, v) with the R-matrix for AdS 3 × S 3 × S 3 × S 1 [9] given in section A.1 and we obtain
h R 1 (p) = − x − R (p) β h L 1 (p) = β x − L (p), (5.1) h R 2 (p) = − x + R (p) β h L 2 (p) = β x + L (p),(5.2)
where β is an arbitrary constant. Also
X L (p) = ρ γ L (p) , Y L (p) = γ L (p) βρ x − L (p) − x + L (p) x − L (p) x + L (p) ,(5.
3)
X R (p) = −i ρ x + R (p) γ R (p) , Y R (p) = −i γ L (p) β ρ x − R (p)x + R (p) x − R (p) − x + R (p) ,(5.4)
where ρ is an arbitrary constant; and
σ LL (p, q) = x + L (p) − x − L (p) x + L (q) − x − L (p) (5.5) σ RR (p, q) = x + R (p) − x − R (p) x + R (q) − x − R (p) (5.6) σ LR (p, q) = x − L (p) x + L (p) 1 − x + L (p)x − R (q) 1 − x − L (p)x − R (q) ζ LR (p, q), (5.7) σ RL (p, q) = x − R (p) x + R (p) 1 − x + R (p)x − L (q) 1 − x − R (p)x − L (q) ζ RL (p, q). (5.8)
Notice that the left and right sectors have their own Zhukovsky variables. As explained in Appendix A, the AdS 3 × S 3 × T 4 is a special case of this where the x ± variables in both sectors coincide.
Recovering the AdS 3 q-deformation from 6-vertex B
The R-matrix in [15,20] can be obtained from the full 6vB R-matrix by making the following identifications:
h R 1 (p) = − x − R (p) β , h L 1 (p) = β x − L (p), (5.9) h R 2 (p) = − x + R (p) β , h L 2 (p) = β x + L (p),(5.10)
where β is an arbitrary constant. Also,
X L (p) = ρ γ L (p) , Y L (p) = 1 β ρ γ L (p) U L (p)V L (p)W L (p) 1 x − L (p) − x + L (p) , (5.11) Y R (p) = 1 β ρ x + R (p) γ R (p) , X R (p) = − ρ γ R (p) U R (p)V R (p)W R (p) x + R (p) x − R (p) − x + R (p) ,(5.12)
where ρ is an arbitrary constant; and
σ LL (p, q) = − U L (p)V L (p)W L (p) U L (q)V L (q)W L (q) x − L (p) − x + L (p) x − L (q) − x + L (p) , (5.13) σ RR (p, q) = − U R (p)V R (p)W R (p) U R (q)V R (q)W R (q) x − R (p) − x + R (p) x − R (q) − x + R (p) , (5.14) σ LR (p, q) = U R (q)V R (q)W R (q) (1 − x + L (p)x − R (q)) (1 − x + L (p)x + R (q)) , (5.15) σ RL (p, q) = U L (q)V L (q)W L (q) (1 − x + R (p)x − L (q)) (1 − x + R (p)x + L (q))
.
(5.16)
In particular, notice that the identifications (5.9)-(5.12) are invertible. This means that this q-deformation is the most general deformation possible. The only source of further deformations comes from the fact that our functions are completely unconstraint. This basically translates in making the constants in the q-deformed model, such as the mass, dependent on the spectral parameter. This is what we call a functional deformation.
Recovering AdS
3 × S 3 × S 3 × S 1 R-matrix from 8-vertex B
In order to recover the AdS 3 × S 3 × S 3 × S 1 R-matrix we need to take the limit k → 0 in the full 8-vertex R-matrix as presented in section 4.3 and compare it with [6,24] which is given in Appendix A.1 . After the limit k → 0, the map to recover AdS 3 ×S 3 ×S 3 ×S 1 R-matrix is the following:
σ LR (p, q) = x − R (q) x + R (q) x − L (p) − x + L (p) 1 − x − L (p)x − R (q) γ R (q) γ L (p) , (5.17) σ RL (p, q) = x − R (p) x + R (p) x − L (p) − x + L (p) 1 − x − L (q)x − R (p) γ R (p) γ L (q) , (5.18) σ LL (p, q) = g L (q) g L (p) x − L (p) − x + L (p) x − L (p) − x + L (q) γ L (q) γ L (p) , (5.19) σ RR (p, q) = g R (q) g R (p) x − R (p) − x + R (p) x − R (p) − x + R (q) γ R (q) γ R (p) . (5.20) Moreover, g L (p) = τ x − L (p) − x + L (p) γ L (p) 2 x + L (p) x − L (p) and g R (p) = τ x − R (p) − x + R (p) γ R (p) 2 x + R (p) x − R (p) ,(5.21)
and
G L (p) = π − i 4 log x − L (p)x + L (p) (5.22) G R (p) = π − i 4 log x − R (p)x + R (p) (5.23) η(p) = − i 2 log x + L (p) x − L (p) ,(5.24)
where τ = ±1. This identification only works since
x + L x − L = x + R x − R .
Deformed symmetry algebras
Let us now try to get some insight into the physical interpretation of the deformations that we have derived. The natural starting point for this is to consider the symmetry algebras that these models exhibit. We will determine the algebra generated by generators x such that quasi-cocommutativity is satisfied ∆ op (x)R = R∆(x) . (6.1)
General procedure
For a given R-matrix, the symmetry algebra is generated by the so-called RTT-relations
R 12 (u, v)T 1 (u)T 2 (v) = T 2 (v)T 1 (u)R 12 (u, v).(6.
2)
The T -matrices are matrices whose entries are the formal generators of the algebra and the above relations describe the fundamental commutation relations between them. Usually these relations describe algebras such as Yangian or quantum affine algebras corresponding to some underlying Lie algebra [25]. The procedure for dealing with the RTT-realization was first formulated in [26] for the AdS 5 × S 5 S-matrix and the Hubbard model and was later applied to AdS 2,3 [27][28][29].
The symmetry relations can be determined by expanding T around some point where R becomes (almost) the identity operator. For clarity, let us assume that this point is at
u = ∞. We write R(u, v) = R i 1 i 2 j 1 j 2 E i 1 j 1 ⊗ E i 2 j 2 (6.3)
and
T a (u) = E ij ⊗ 1 ⊗ T i j (u), T b (u) = E ij ⊗ T i j (u) ⊗ 1,(6.4)
then write (6.2) in component form. Next, we write an expansion of T as
T a b (u) = T a (−1)b + ∞ s=0 T a (s)b u −1−s . (6.5)
Then, it is easy to see that the RTT-relations reduce to regular commutation relations of some Lie algebra generated by the generators T a (0)b . The component T a (−1)b ≡ δ ab U a gives rise to the braiding element [26] of the underlying coproduct which is also completely fixed. Now, remarkably, the R-matrix does not only inherently contain the symmetry algebra, but also (part of) the representation theory of this algebra. Indeed, by considering the map ρ F : T a (u) → R ab (θ, u) (6.6) and using the Yang-Baxter equation we see that the components of the R-matrix provides the defining representation of the symmetry algebra. By a fusion procedure, more general representations can then also be constructed [30]. Conversely, the R-matrix is the unique object that intertwines the coproduct and opposite coproduct in the defining representation of the symmetry algebra (including higher-order generators). This is actually the method that is used in holographic integrable models. The symmetry algebra is determined by direct computation and then the scattering matrix is derived from this. In our situation, however, we already have the R-matrix and can now construct the corresponding symmetry algebra and compare this again with the symmetry algebras and representations of the integrable models coming from AdS 2,3 superstring sigma models.
Alternative direct calculation
The standard procedure of expanding the R-matrix around a point where it becomes diagonal crucially relies on the existence of such a point. In fact, for the specific case of the 8-vertex deformation of AdS 3 it is not possible to find such a point, despite the fact that the R-matrix is related to that of AdS 2 , for which it does work.
In light of this, we will extract the (defining representation of the) symmetry algebra directly from the 8vB R-matrix in a manner which does not depend on the specific model at hand (i.e. on the choice of free functions) nor on the existence of a point where the R-matrix becomes diagonal. We can then fine tune the obtained algebra to the model at hand.
Consider some symmetry generator Q. We putŘ(u, v) = PR(u, v) where P is the permutation operator and define
Q 12 (u, v) = Q(u) ⊗ 1 + U(u) ⊗ Q(v),(6.7)
where the tensor product is graded when we have both bosonic and fermionic degrees of freedom in our representation. Then the symmetry algebra relation (6.1) is that
Q 12 (v, u)Ř(u, v) =Ř(u, v)Q 12 (u, v). (6.8)
We can then differentiate this equation with respect to u and subsequently put u → v, which leaves us with a set of ODEs
[Q ⊗ 1 + U ⊗ Q, H] = Q ⊗ 1 + U ⊗ Q − U ⊗ Q . (6.9)
which can be solved directly and we have introduced the Hamiltonian density H(v) = ∂ uŘ (u, v) u→v . Notice that this equation greatly resembles the Sutherland equation that is crucial in the boost operator formalism of [1]. Moreover, we also see here that the symmetry algebra is fixed by the Hamiltonian density H of the system which nicely ties in to the bottom-up approach of our classification method [1]. Our approach will be to simply solve (6.9) for the derivatives of the remaining functions and plug them back in, obtaining a set of algebraic equations for the functions themselves. This approach is especially useful if the the R-matrix does not have a nice asymptotic behaviour.
6-vertex deformation of AdS 3
Let us now consider the 6-vertex deformation of the full AdS 3 R-matrices. We will start by considering two supercharges Q ± which we assume to have the form
Q ± = Q L ± 0 0 Q R ± (6.10) with Q L + = 0 0 a + 0 , Q L − = 0 a − 0 0 (6.11) Q R + = 0 b + 0 0 , Q R − = 0 0 b − 0 . (6.12)
We will equip these charges with coproducts ∆(Q ± ) which we assume to have the form together with coproducts of the following form
∆(Q + ) = Q + ⊗ 1 + U −1 ⊗ Q + , ∆(Q − ) = Q − ⊗ 1 + V −1 ⊗ Q − (6.13)
where U and V are block diagonal matrices acting as
U = U L 0 0 U R , V = V L 0 0 V R (6.14)
where each of U L,R and V L,R are scalar multiples of the identity operator.
By imposing the required commutation relations between the supercharges and the R-matrix we easily obtain
a + = 1 X L 1 t − h L 2 , a − = X L h L 1 − h L 2 s − h L 1 b + = i Y R h R 2 h R 1 − h R 2 1 + t h R 1 , b − = i h R 2 Y R 1 1 + s h R 2 (6.15)
where s and t are arbitrary constants and
U L = X L Y L (h L 1 − h L 2 ) t − h L 2 t − h L 1 , U R = h R 2 X R h R 2 Y R 1 h R 1 − h R 2 1 + t h R 1 1 + t h R 2 V L = 1 X L 1 Y L 1 h L 1 − h L 2 s − h L 1 s − h L 2 , V R = X R h R 2 Y R h R 2 (h R 1 − h R 2 ) 1 + s h R 2 1 + s h R 1 . (6.16)
At first glance it seems we have two one-parameter families of conserved charges Q + (t) and Q − (s). However only two such charges from each family are actually independent since for example for any T we can write
Q + (T ) = α Q + (t 1 ) + β Q − (t 2 ), t 1 = t 2 (6.17)
and as a result there are four independent supercharges Q + (t 1 ),
Q + (t 2 ), Q − (s 1 ), Q − (s 2 ).
The logic can then be reversed -starting from these supercharges the R-matrix is completely constrained up to normalisation in each of the four blocks by the requirement ∆ op (Q)R = R∆(Q) . (6.18)
We will now examine the resulting algebra more closely. We have the following commutation relations
{Q + (t 1 ), Q + (t 2 )} = 0, {Q − (s 1 ), Q − (s 2 )} = 0 . (6.19)
We introduce a two-parameter family of operators P(t, s) defined by
P(t, s) = {Q + (t), Q − (s)} . (6.20)
Clearly there are four independent such charges and all of them are central elements of the algebra. Their coproducts are given by
∆(P) = P ⊗ 1 + V −1 U −1 ⊗ P . (6.21)
The coproducts of the braiding factors are given by
∆(U) = U ⊗ U and ∆(V) = V ⊗ V.
Finally, the algebra can be extended by an additional C factor B with the commutation relations [B, Q ± ] = ±2Q ± (6.22) and with trivial coproduct.
Symmetries of deformed AdS 2
Let us now work out the symmetries of the different deformed models.
Direct approach and defining representation
As with the AdS 3 model above we will start our considerations by considering a supercharge Q of the form
Q = 0 b a 0 (6.23)
with coproduct ∆(Q) = Q ⊗ 1 + U ⊗ Q. Imposing as usual that ∆ op (Q)R = R∆(Q) leads to the following constraints
a = b ω 1 ω 2 ω 1 dn(2v) − 1 sn(2v) U = ω 1 ω 2 cos η sn(2v) + ω 2 sin η cn(2v) sin η + ω 1 ω 2 sn(2v) (6.24)
where ω 1 , ω 2 ∈ {−1, 1}. From the condition ∆ op (Q)R = R∆(Q) it is also possible to extract an ODE for b. However, it is very difficult to express the solution in a useful form. To this end, we will follow a different approach. Namely P = 1 2 {Q, Q} is central and as a result we have that ∆ op (P) = ∆(P) which implies P = ρ(1 − U 2 ) where ρ is some irrelevant constant. This relation provides an algebraic equation linking a, b and U allowing the symmetry generators to be completely determined. Note that the mentioned equation has two solutions. However, the two solutions differ only by a sign which in turn, along with ρ, only affects the normalisation of Q. Hence we can set ρ = 1 and choose whichever sign we like.
We now comment on the constructed supercharges corresponding to the four possible pairs (ω 1 , ω 2 ). We first note the following inversion property
U| ω 1 →−ω 1 = U −1 (6.25)
for fixed ω 2 easily verified by direct calculation. Next, we notice that although we seem to have four supercharges only two are actually independent since there are only two independent matrices of the form (6.23). For definiteness we will choose the two independent supercharges to be given by (ω 1 , ω 2 ) = (±1, −1) and denote the charges as
Q ± = Q| ω 1 =±1 , ω 2 = −1 (6.26)
and hence the coproducts are given by
∆(Q ± ) = Q ± ⊗ 1 + U ±1 ⊗ Q . (6.27)
The explicit form of the supercharges is in general quite complicated involving square roots of Jacobi elliptic functions. Huge simplifications occur at k = 1 where the elliptic functions degenerate into hyperbolic functions. The explicit solution is then given by U = sech(2v) sin η + tanh(2v) cos η tanh(2v) − sin η (6.28)
a = coth(v) 1−ω 1 2 (cos( η 2 ) − ω 1 sin( η 2 ) tanh(v)) coth(2v) sin η − ω 1 (6.29) b = tanh(v) 1−ω 1 2 cosh(v) cosh(v) sin( η 2 ) − ω 1 sinh(v) cos( η 2 )
. (6.30)
General approach and higher symmetries
We see that the underlying Lie algebra is the same for the deformed AdS 2 model as for the undeformed model. Let us now look at the higher symmetry generators by working out the RTT relations similar to [28].
RTT Let us assume that there is a point at which the 8vB R-matrix becomes diagonal. Without loss of generality, let us set this point to be at u = 0. We expand η(u) = a 0 + a 1 u + a 2 u 2 + . . . (6.31) It is easy to see that R only becomes diagonal if a 0 = a 1 = 0, so we assume this from now on. Next, we expand our monodromy matrix T in terms of symmetry generators as
T (u) = 1 0 0 U + H+B 2 UQ − Q + U B−H 2 u + . . . (6.32)
and we work out the fundamental commutation relations (6.2) expanded around u = 0.
To first order we find that U is central, i.e.
[U, T (v)] = 0. (6.33)
At second order we indeed recover the centrally extended psu(1|1) ce algebra together with the extra central element B.
Remarkably, we find that the k dependence drops out of the RTT relations in the first two orders, so the commutation relations of the first level Yangian generators are also unaffected and we therefore reproduce the algebra from [28], including the secret symmetry.
Deformed Yangian Since the first two levels of our algebra are exactly the same, the question arises if the deformation is purely realized in terms of the representation. However, starting from the second level, the deformation parameter appears in the structure constants. It is not possible to absorb this k dependence into redefinitions of generators of the full algebra. This seems to indicate that the algebra of this deformed model is not a standard Yangian. It would be interesting to find out exactly what the structure for the higher generators is.
Deformed Representation The fact that the Lie algebra of symmetries remains unchanged and only the representation is modified might suggest that the deformation of the AdS 2 model could be trivially absorbed into a redefinition of the physical parameters
e → e k , m → m k , p → p k . (6.34)
There are two ways to see that this is not the case and the deformation is non-trivial. The first comes from the fact that the AdS 2 R-matrix satisfies a certain linear constraint on the diagonal elements
r 1 − r 2 − r 3 − r 4 = 0 (6.35)
It can be easily checked that this condition is not satisfied for the general 8vB model except at k → ∞.
The second comes from the evaluation representation structure of the model. It can be shown that the fundamental representation of the Yangian does not admit an evaluation representation. This is in particular clear by looking at the central extensions. Since the k dependence only manifests itself at second level, we see that both the central extension P and its first level Yangian counterpartP 2P = {Q, Q} and 2P = {Q, Q} (6.36)
do not depend on k. However we find thatP = 1 2 {Q,Q} does depend on k. This prohibits the existence of an evaluation representation unless we take the k → ∞ limit. And indeed the AdS 2 model allows evaluation representations [28].
8-vertex deformation of AdS 3
The LL and RR blocks of the full R-matrix correspond to the 4 × 4 R-matrices which appeared above in the AdS 2 model. As such we have already classified the symmetry generators for these blocks. From the above discussion we know now that both the LL and RR scattering matrices exhibit a centrally extended psu(1|1) symmetry, whose representations depend on the elliptic parameter k. The question is now if this extends to a symmetry algebra of the full 16 × 16 S-matrix.
Let us write a supercharge Q as
Q = Q L 0 0 Q R . (6.37)
The matrices Q L are precisely of the form given for the AdS 2 deformation -we only need to make the replacement v → G L (v) in the argument of the elliptic functions. The matrices Q R can then be found almost for free -a quick check of the equations coming from ∆ op (Q)R = R∆(Q) yields that the equations involving different blocks do not mix and further the equations involving the entries of the RR block are identical to those for the LL block provided we simply swap replace
G L → G R , U L → U R , a L → b R and b L → a R .
The only other modification is that for Q R we must also include the normalisation factor ρ since we are only free to set the normalisation of one of the blocks. A straightforward calculation then yields that there are only two supercharges which survive for the full R-matrix, namely
Q L + 0 0 Q R − , Q L − 0 0 Q R + . (6.38)
As a result the symmetry algebra is given by a single copy of psu(1|1) ce .
The same statements made about the AdS 2 model also carry over to the current setting, namely for generic k the model does not admit evaluation representations and the higher order Yangian generators become deformed.
Crossing symmetry
In this section we study the presence of crossing symmetry in the deformed models. We find that the usual crossing type relations hold for both the AdS 3 and AdS 2 deformations, although the explicit form of the charge conjugation matrix C for the AdS 2 one is very unusual.
6-vertex deformation of AdS 3
The R-matrix described in Section 4.2 satisfies the crossing equations
C 1 R(p + ω, q) t 1 C −1 1 = R(p, q) −1 , C 2 R(p, q − ω) t 2 C −1 2 = R(p, q) −1 , (7.1)
where the superscript t 1 and t 2 denote transposition in the first and second vector space, respectively; ω is the crossing parameter and the conjugation matrix C is given by
C = 0 0 1 0 0 0 0 i 1 0 0 0 0 i 0 0 ,(7.2)
if and only if the following conditions are satisfied:
Condition 1: It is necessary that h L/R i (p ± ω) satisfy h R i (p ± ω) = − 1 h L i (p) , h L i (p ± ω) = − 1 h R i (p) , (7.3) for i = 1, 2 which implies that h A i (p ± 2ω) = h A i (p). (7.4)
Condition 2:
The functions X, Y need to satisfy
X A (p ± 2ω) = −X A (p) and Y A (p ± 2ω) = −Y A (p),(7.5)
and X R (p) = X L (p + ω) and Y R (p) = Y L (p + ω). (7.6)
Condition 3: Finally we find conditions on the scalar factors σ
σ AB (p + ω, q)σ BB (p, q) = h B 2 (p) − h B 1 (p) h B 2 (q) − h B 1 (p) , (7.7) σ AA (p + ω, q)σ BA (p, q) = h B 2 (p) − h B 1 (p) h B 2 (p) 1 + h A 1 (q)h B 2 (p) (1 + h B 1 (p)h A 1 (q)) (1 + h B 2 (p)h A 2 (q)) , (7.8) σ AA (p, q − ω) = −h A 2 (p)h B 2 (q)σ BB (p + ω, q), (7.9) σ BA (p, q − ω) = σ AB (p + ω, q) (7.10)
where A, B = {L, R} with A = B. In the last two equations we combined both crossing equations.
Notice that the charge conjugation matrix C (7.2) matches with the one in [9], while corresponds the complex conjugate of the one appearing in [6]. If one would like the exactly matching between C in (7.2) and [6] it is enough to perform a local basis transformation of the form
R → (W ⊗ W) R (W ⊗ W) −1 , W = diag(1, 1, 1, −1),(7.11)
on our 6-vertex B full R-matrix R.
8-vertex deformation of AdS 3
The full R-matrix for 8-vertex B, deformation of AdS 3 , described in Section 4.3 satisfies crossing symmetry (7.1) for
C = 0 0 1 0 0 0 0 i 1 0 0 0 0 i 0 0 ,(7.12)
given that its functions satisfy
G L (p + ω) = 2K − G R (p) = G L (p − ω), (7.13) G R (p + ω) = 2K − G L (p) = G R (p − ω), (7.14) η(p + ω) = 2π − η(p) = η(p − ω), (7.15) g R/L (p + ω) = g L/R (p) = − g R/L (p − ω) (7.16)
where K is the Elliptic Integral of the first kind. 4 . Moreover, the overall scalar factors σ AB of the different blocks need to satisfy the following crossing equations 5 and σ LL (p, q − ω) = −σ RR (p + ω, q) (7.21)
σ LL (p + ω, q)σ RL (p, q) = 1 i g R (p) g L (q) cn 2 R,L,+ dn 2 R,L,+ sin 2 η − sin η(p) sin η(q) − sn 2 R,L,+ cos 2 η − sin η(p) sin η(q) + 1 , (7.17) σ RR (p + ω, q)σ LR (p, q) = 1 i g R (q) g L (p)σ RR (p, q − ω) = −σ LL (p + ω, q) (7.22) σ RL (p, q − ω) = σ LR (p + ω, q) g R (q) g R (p) (7.23) σ LR (p, q − ω) = σ RL (p + ω, q) g L (p) g L (q) . (7.24)
As expected the crossing matrix (7.12) matches with the one in [9]. 4 One needs to be particularly careful with branch cuts for this calculation. This is the reason why we are giving the form of √ g R/L instead of g R/L itself. 5 To avoid cluttered expressions we denote Jacobi elliptic functions as cn R,L,+ etc in contrast to cn R,L + which was used earlier in the text.
Deformed AdS 2
In this section we address the question of whether the AdS 2 deformation given by the 8-vertex B R-matrix (2.5) satisfies crossing symmetry for any value of the deformation parameter k. The answer is particularly interesting. If we consider the boson-fermion R-matrix we do have crossing symmetry for any value of k as it will be described below.
In order to satisfy crossing symmetry the R-matrix has to satisfy
C 1 R st 1 12 (u + ω, v)C −1 1 = R 12 (u, v) −1 , C 2 R st 2 12 (u, v − ω)C −1 2 = R 12 (u, v) −1 (7.25)
where C is the charge conjugation matrix, ω is the crossing parameter and st i indicates super transposition in the space i. The R-matrix in equations (7.25) is the boson-fermion version 6 of (2.5), but with a dressing phase σ(u, v) and also {u, v} → {G(u), G(v)}. For C given by 7
C = 0 1 −i 0 (7.26) equations (7.25) are satisfied for σ(u + ω)σ(u, v) = −i cn 2 dn 2 sin 2 η − sin η(u) sin η(v) − sn 2 cos 2 η − sin η(u) sin η(v) + 1 = σ(u − ω)σ(u, v) (7.27)
and η(u + ω) = −η(u) + 2πn, η(u − ω) = −η(u) + 2πm, G(u + ω) = G(u) + 2 n K(k 2 ), G(u − ω) = G(u) + 2 m K(k 2 ), m, n ∈ Z, (7.28)
where K(k 2 ) is the Elliptic Integral of the first kind. Notice, however, that although this R-matrix is a deformation of the AdS 2 R-matrix, the charge conjugation matrix C is not similar to the one of AdS 2 . The undeformed AdS 2 R-matrix ( [8]) is diagonal and therefore bosons and fermions are transformed into their antiparticles by crossing symmetry. In our deformation, however, the C is off-diagonal and therefore bosons are transformed into fermions and vice-versa, which is highly unusual. It is important to highlight, that in the limit k → ∞ (which lead us back to AdS 2 ) both (7.26) and the diagonal C given in [8] satisfy the equations (7.25), but the diagonal one is the correct one on that case. It would be very important to try to construct the sigma model for this deformation and try to make sense of the off-diagonal C we found for arbitrary k.
For the boson-boson R-matrix there is crossing for any value of k except k = 1. The C is given by 8
C = 0 1 −1 0 (7.29)
and therefore also off-diagonal. The expression for σ(u ± ω) is basically the same as the boson-fermion one, just replacing the overall factor i by 1. Also,
η(u + ω) = −η(u) + πn, η(u − ω) = −η(u) − πm, G(u + ω) = G(u) + n K(k 2 ), G(u − ω) = G(u) + m K(k 2 )
, m, n are odd. (7.30) By comparing expressions (7.28) and (7.30) it is possible to see why the boson-fermion k = 1 case has crossing symmetry while the boson-boson one has not. The reason is that K(k 2 ) diverges for k = 1, but in the boson-fermion case we still can have m = n = 0 as a solution so K(k 2 ) does not appear, while for the boson-boson case m and n are odd, and therefore there is no way to have a finite expression when k = 1.
Conclusions
In this paper we classified regular deformations of the S-matrices of the integrable AdS 2,3 string sigma models. We found that the AdS 2 × S 2 × T 6 only admits one type of deformation. But interestingly, the AdS 3 × S 3 × M 4 scattering matrix admits two deformations, an elliptic and a functional one. We showed that all the deformations can be made crossing symmetric. It would also be important to know how many of our deformations correspond to moduli of the sigma models and how many would go out of the string theory [31].
There are numerous interesting directions for future research. The most pressing one would be to find a deformed string sigma model which gives rise to the new elliptic deformed S-matrix. Since the usual way of deforming by means of the modified classical Yang-Baxter equations gives rise to quantum deformations, a more general approach might be needed, e.g. through screening charge formalism for deformed sigma models and study screening parameter limits [32]. A first hint at what needs to be done is the fact that half of the symmetry seems to be broken, as it is only compatible with two instead of four supersymmetry generators. It would also be interesting to carry out the Bethe Ansatz for this model. This paper also sheds some light on the three parameter deformation [16,18]. In particular, we find that the quantum deformed model can only be further deformed functionally. Hence, it would be good if one could define a meaningful classical limit so that these models can be compared and gain some insight in the existence and form of the full quantum scattering matrix of the three parameter model. It is also very interesting to look if with the present approach one can identify 4-parametric deformation of AdS 3 × S 3 × M 4 , which could potentially follow from higher vertex models.
In this work we have also classified the possible integrable deformations of the AdS 2 model by considering the deformations of the massive modes. We were able to write a charge conjugation matrix C satisfying the crossing relations, but it still necessary to understand better such C, since it is of off-diagonal form. We also found a potential new quantum algebra that underlies this scattering matrix and it would be interesting to figure out its exact structure and if associated Yangian symmetries lead to known integrable deformations or create a new class. Even more interesting would be possible extensions of our deformed scattering matrix to massless modes. symmetry relations for it. MdL was supported by SFI, the Royal Society and the EP-SRC for funding under grants UF160578, RGF\R1\181011, RGF\EA\180167 and 18/EP-SRC/3590. Furthermore, AP. and ALR. were supported by the grants RGF\EA\180167 and 18/EPSRC/3590, respectively. P.R. was supported in part by a Nordita Visiting PhD Fellowship and by SFI and the Royal Society grant UF160578.
A AdS 3 R-matrices
In order to the comparison between the deformed and underformed R-matrices be performed it was necessary to rewrite the R-matrices for AdS 3 × S 3 × T 4 [6,24] and AdS 3 × S 3 × S 3 × S 1 [9] in a way that they satisfy YBE for any for arbitrary γ(p). Also, what we call γ(p) here was called η(p) in the referred papers in order to avoid confusion with our notation.
In this appendix, we present the exact form we used in the comparisons.
A.1 AdS 3 × S 3 × S 3 × S 1 For AdS 3 × S 3 × S 3 × S 1 [9]
the full R-matrix is composed of four blocks written according to chirality of the particles where R AB ≡ R AB (p, q), r AB i ≡ r AB i (p, q) and χ AB ≡ χ AB (p, q); and A and B are their chiralities i.e. A, B = R, L. Below we present the explicit form of the four blocks starting with the ones with same chirality: for RR
R AB = f AB χ AB r RR 1 = 1, r RR 2 = x − R (p) x + R (p) x + R (p) − x + R (q) x − R (p) − x − R (q) , r RR 3 = x + R (q) x − R (q) x − R (p) − x − R (q) x − R (p) − x + R (q) , r RR 4 = x − R (p) x + R (p) x + R (q) x − R (q) x − R (q) − x + R (p) x − R (p) − x + R (q) , r RR 5 = x − R (p) − x + R (p) x − R (p) − x + R (q) γ R (q) γ R (p) , r RR 6 = x − R (p) x + R (p) x + R (q) x − R (q) x − R (q) − x + R (q) x − R (p) − x + R (q) γ R (p) γ R (q) , r RR 7 = 0, r RR 8 = 0, (A.2)
the LL one given by
r LL 1 = 1, r LL 2 = x − L (p) x + L (p) x + L (p) − x + L (q) x − L (p) − x − L (q) , r LL 3 = x + L (q) x − L (q) x − L (p) − x − L (q) x − L (p) − x + L (q) , r LL 4 = x − L (p) x + L (p) x + L (q) x − L (q) x − L (q) − x + L (p) x − L (p) − x + L (q) , r LL 5 = x − L (p) x + L (p) x + L (q) x − L (q) x − L (q) − x + L (q) x − L (p) − x + L (q) γ L (p) γ L (q) , r LL 6 = x − L (p) − x + L (p) x − L (p) − x + L (q) γ L (q) γ L (p) , r LL 7 = 0, r LL 8 = 0, (A.3)
while LR is
r LR 1 = x − L (p) x + L (p) 1 − x + L (p)x − R (q) 1 − x − L (p)x − R (q)
, r LR 2 = 1,
r LR 3 = x − L (p) x + L (p) x − R (q) x + R (q) 1 − x + L (p)x + R (q) 1 − x − L (p)x − R (q) , r LR 4 = − x − R (q) x + R (q) 1 − x − L (p)x + R (q) 1 − x − L (p)x − R (q) , r LR 7 = x − L (p) x + L (p) x + R (q) − x − R (q) 1 − x − L (p)x − R (q) γ L (p) γ R (q) , r LR 5 = 0, r LR 8 = − x − R (q) x + R (q) x + L (p) − x − L (p) 1 − x − L (p)x − R (q) γ R (q) γ L (p) , r LR 6 = 0, (A.4)
and RL is given by
r RL 1 = x − R (p) x + R (p) 1 − x + R (p)x − L (q) 1 − x − R (p)x − L (q)
, r RL 2 = 1,
r RL 3 = x − R (p) x + R (p) x − L (q) x + L (q) 1 − x + R (p)x + L (q) 1 − x − R (p)x − L (q) , r RL 4 = − x − L (q) x + L (q) 1 − x − R (p)x + L (q) 1 − x − R (p)x − L (q) , r RL 7 = x − L (q) x + L (q) x + R (p) − x − R (p) 1 − x − R (p)x − L (q) γ L (q) γ R (p) , r RL 5 = 0, r RL 8 = − x − R (p) x + R (p) x + L (q) − x − L (q) 1 − x − R (p)x − L (q) γ R (p) γ L (q) , r RL 6 = 0. (A.5)
Also, the χ AB that appear in (A.1) are given by
χ LR (p, q) = x + L (p) x − L (p) −1/4 x + R (q) x − R (q) −1/4 1 − 1 x − L (p)x − R (q) 1 − 1 x + L (p)x + R (q) , (A.6)
where χ RL (p, q) can be obtained from χ LR (p, q) by doing {L ↔ R} while
χ RR = 1 = χ LL . (A.7)
Also, notice that γ L (p + ω) = i γ R (p)
x + R (p)
, and γ R (p + ω) = i γ L (p)
x + L (p)
, (A.8) and
x ± L (p + ω) = 1 x ± R (p)
, and x ± R (p + ω) = 1 x ± L (p)
.
(A.9)
Comparing with [9] notice that theirp is equal top = p + ω.
and then for RL
r RL 1 = x − (p) x + (p) x − (q) x + (q) 1 − x + (p)x − (q) 1 − x − (p)x − (q) r RL 3 = x − (q) x + (q) , r RL 2 = x − (q) x + (q) x − (p) x + (p) 1 − x + (p)x + (q) 1 − x − (p)x − (q) , r RL 4 = − x − (q) x + (q) 1 − x + (q)x − (p) 1 − x − (p)x − (q) , r RL 7 = x − (q) x + (q) 3/4 x + (p) x − (p) 1/4 x + (p) − x − (p) 1 − x − (p)x − (q)
γ(q) γ(p) , r RL 5 = 0,
r RL 8 = − x − (p) x + (p) x − (q) x + (q) 3/4 x + (q) − x − (q) 1 − x − (p)x − (q)
γ(p) γ(q) , r RL 6 = 0.
(A.14)
Notice that although this is the same R-matrix from [6], as mentioned above, we did two small modifications to it: the first was to write the blocks exclusively in terms of the Zhakowski variables by using
e i p = x + (p) x − (p) ; (A.15)
while the second was to rewrite the blocks R AB in a way that they satisfy YBE without the need to specify an expression for γ(p). Notice that by assuming 16) we can recover the R-matrices in [6].
γ(p) = e i p 4 i h 2 (x − (p) − x + (p)) (A.
A.3 Obtaining the S-matrix of AdS 3 × S 3 × T 4 from AdS 3 × S 3 × S 3 × S 1
One can actually obtain the R-matrix for AdS 3 × S 3 × T 4 starting from the one of AdS 3 × S 3 × S 3 × S 1 . In order to do that we need the following changes for x + R,L and x − R,L with a being a constant, while f AB (in (A.1)) are given by
x − L → x − , x − R → x − , x + L → x + and x − R → x + ,f LL (p, q) = − x + (p) x − (p) x − (q) x + (q) x − (p) − x + (q) x − (q) − x + (p) ζ LL (p, q),
f RR (p, q) = − x + (p)
x − (p)
x − (q) x + (q)
x − (p) − x + (q)
x − (q) − x + (p) ζ RR (p, q), f LR (p, q) = x + (p)
x − (p)
x + (q) x − (q) 1 − x − (p)x − (q) 1 − x + (p)x + (q)
ζ LR (p, q) χ LR (p, q) , f RL (p, q) =
x − (q) x + (q) ζ RL (p, q) χ RL (p, q) .
(A. 19)
Figure 1 :
1New deformations of AdS 3 and AdS 2 R-matrices.
σσ
LR (p + ω, q)σ RR (p, RL (p + ω, q)σ LL (p,
Working in an appropriate region such that we avoid branch cut issues.2 This relation is somewhat reminiscent of the procedure to arrive at the massless gamma variable[23].
A twist in the R-matrix is allowed because it keeps YBE invariant. It is also necessary in order to 6-vertex B R-matrix be a deformation of the R-matrices in[6,24,9,15].
Just apply the following transformation: r 4 (u, v) → −r 4 (u, v), r 7 (u, v) → i r 7 (u, v) and r 8 (u, v) → i r 8 (u, v), keeping the remaining r i (u, v) invariant.7 A similar solution exists for i → −i in C 8 A similar solution exists for replacing −1 → 1 in the matrix element 21 of C
Acknowledgments We would like to thank B. Hoare, S. Lukyanov, M. Alfimov, F. Seibold, C. Paletta, A. Torrielli, V. Korepin, S. J. van Tongeren and A. Sfondrini for useful discussions. We would like to thank B. Hoare for several interesting discussions about the AdS 2 deformation and for pointing out a problem that made possible writing the crossingFollowing[6,24], the four blocks can be written as followswhere R AB ≡ R AB (p, q), r AB i ≡ r AB i (p, q) and ζ AB ≡ ζ AB (p, q); and A and B are their chiralities i.e. A, B = R, L. Below we present the explicit form of the four blocks starting with the ones with same chirality: for RRand then for LLFor the blocks of opposite chirality we have for LR1 − x + (p)x + (q) γ(p) γ(q) , r LR 5 = 0,x − (q) x + (q)1/4x + (p)x − (p)
Classifying nearestneighbour interactions and deformations of AdS. M De Leeuw, C Paletta, A Pribytok, A L Retore, & P Ryan, 10.1103/PhysRevLett.125.031604arXiv:2003.04332Phys. Rev. Lett. 12531604M. de Leeuw, C. Paletta, A. Pribytok, A.L. Retore & P. Ryan, "Classifying nearest- neighbour interactions and deformations of AdS", Phys. Rev. Lett. 125, 031604 (2020), arXiv:2003.04332.
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arxiv |
SHAPE INVARIANCE AND EQUIVALENCE RELATIONS FOR PSEUDOWRONSKIANS OF LAGUERRE AND JACOBI POLYNOMIALS
David Gómez-Ullate
Yves Grandati
Robert Milson
SHAPE INVARIANCE AND EQUIVALENCE RELATIONS FOR PSEUDOWRONSKIANS OF LAGUERRE AND JACOBI POLYNOMIALS
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Pöschl-Teller potential.2010 Mathematics Subject Classification. 33C45, 81Q80, 42C05.
Introduction
Wronskian determinants arise in a natural fashion when iterating Darboux transformations, [1]. These ideas have been extensively used in the theory of integrable systems generally known as the dressing method (see for instance [2,3] and references therein).
In the context of classical orthogonal polynomials, Darboux transformations are essentially a factorization of the differential operator, and lead directly to exceptional orthogonal polynomials, a new class of Sturm-Liouville polynomial families originally introduced in [4,5], and developed over the past decade by many authors [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
From a physical point of view, exceptional orthogonal polynomials are (up to a gauge factor) the eigenfunctions of exactly solvable quantum potentials obtained by applying rational Darboux transformations to the harmonic oscillator (Hermite), isotonic oscillator (Laguerre) and trigonomeric Darboux-Pöschl-Teller potential (Jacobi). The new exactly solvable potentials are known are rational extensions [20][21][22], since the original potentials are modified by the addition of rational terms in a suitable variable.
In the mathematical context, Wronskian determinants whose entries are classical polynomials were considered by Karlin and Szegő [23], who provide expressions for the number of real zeros of the Wronskian of a sequence of consecutive polynomials. Sequences for which the Wronskian has constant sign were characterized by Adler in [24], and the result for arbitrary sequences has been completed by García-Ferrero et al. [25]. The complex zeros of these Wronskians form regular patterns and have also been the subject of intense study, [26,27] These Wronskian determinants of classical polynomials play an important role in the construction of rational solutions to nonlinear integrable equations of Painlevé type [28][29][30][31]. All known rational solutions to PIV, PV, and their higher order generalizations known as A N -Painlevé or Noumi-Yamada equations seem to be expressible in terms of suitable Wronskians whose entries are Hermite and Laguerre polynomials, although a rigorous proof of this fact has not been given yet (see [32] and references therein for a review of the symmetry approach to Painlevé equations).
In this paper we are concerned with describing the existence of an infinite number of identities among Wronskian determinants whose entries are Laguerre and Jacobi polynomials. Some of these equivalences were already noted in [33,34] at the Schrödinger's picture, but it was not until [35] that a full description was given for the Hermite case. From the quantum mechanical perspective, the identities reflect a notion of equivalence among different sets of Darboux transformations that end up in the same transformed potential, up to an additive shift. The term pseudo-Wronskian was introduced in [35] to describe the set of rules to construct the determinants with polynomial entries in terms of a Maya diagram, a type of diagram originally introduced by Sato in the theory of integrable systems [36] and first applied to exceptional orthogonal polynomials by Takemura, [37]. A Hermite pseudo-Wronskian is indexed by a single Maya diagram (or equivalently, a single partition), corresponding to the fact that a rational Darboux transformation of the harmonic oscillator is indexed by a finite sequence of integers specifying the seed functions for the transformation. The equivalance at the level of Maya diagrams corresponds to shifting the origin of the diagram.
In the Laguerre and Jacobi cases discussed here, the situation is richer as two independent families of seed functions can be considered for rational Darboux transformations, and thus two Maya diagrams must be given to specify the set of seed funtions in each family. Instead of a partition, these rational Darboux transformations are indexed by a Universal Character, which is essentially a pair of partitions as defined by Koike [38]. Universal characters generalize Schur polynomials, and they have been used in the construction of rational solutions to A N -Painlevé equations, [39,40].
Although we do not pursue this direction in this paper, it is worth noting that the pseudo-Wronskian determinants discussed here can be regarded as an extension of Jacobi-Trudi formulas in the theory of symmetric functions, [41]. Original Jacobi-Trudi formulas express a Schur polynomial s λ associated to a given partition λ as a determinant whose entries are complete homogeneous symmetric polynomials. Jacobi-Trudi formulas were extended to classical orthogonal polynomials in [42] and they are connected to exceptional orthogonal polynomials, [43]. The pseudo-Wronskian determinants in this paper can be regarded as a generalization of Jacobi-Trudi formulas that involve not just the partition λ or its conjugate partition λ , but also intermediate representations described by Durfee symbols, [44].
The paper is organized as follows. In Section 2 we introduce all the notions in combinatorics needed to formulate our results: Maya diagrams, Durfee symbols, etc. together with some basic operations on them. In Sections 3 and 4 we review the necessary concepts on Darboux transformations and shape invariant potentials. It includes a description of the discrete Γ symmetries acting on the parameters of the potentials that allow to define the notions of extended and shadow spectra, i.e. the families of possible seed functions for a rational Darboux transformation. The action of iterated Darboux transformations on the isotonic oscillator and Darboux-Pöschl-Teller potential are described in Section 5, and they are indexed by punctured universal characters. Finally, we introduce the Laguerre and Jacobi pseudo-Wronskians in Sections 6 and 7, together with the equivalence formulas and some explicit examples.
We would like to note that some of the equivalence formulas reported in this paper for the Laguerre pseudo-Wronskians have been obtained independently by Bonneux and Kuijlaars,[45], albeit with a different choice of indexing between seed functions and Maya diagrams.
2. Punctured Young diagrams and Durfee rectangles 2.1. Maya diagrams. We define a Maya diagram as an infinite row of boxes labelled by relative integers and which can be empty or filled by at most one "particle" (graphically represented by a filled dot). All the boxes sufficiently far away to the left are filled and all the boxes sufficiently far away to the right are empty.
A tuple of relative integers, N m = n 1 , ..., n k , n k , ..., n 1 , with n i > n i+1 ≥ 0 and 0 > n i+1 > n i (m = k +k), can be represented in an unique way by a Maya diagram, as depicted in Figure 1. Positive integers in the tuple correspond to filled boxes to the right of the origin and negative integers in the tuple to empty boxes to the left of the origin.
... ... In the following we use the term Maya diagram to designate both the diagram in graphical form and for the associated tuple. If the set of n i (respectively n i ) is empty, then the Maya diagram is said to be positive (respectively, negative). In the case of a positive Maya diagram, if all the entries are non-zero, then the Maya diagram is said to be strictly positive.
2.2.
Punctured Young diagrams. The correspondence between Maya and Young diagrams is well known [32,46]. Progressing from the left to the right, to each filled box we associate a vertical step upward and to each empty box an horizontal step to the right. The obtained broken line then serves as the lower and right boundary of the corresponding Young diagram.
In accordance with our definition, we associate to every Maya diagram a punctured Young diagram, which we define to be a Young diagram equiped with an origin. This last should be regarded as marked point on the right border of the Young diagram that separates the segments labelled by positive integers from the segments labelled by strictly negative integers. An example can be seen in Figure 2.
Supressing the origin, we recover a standard Young diagram, which we view as an equivalence class of punctured Young diagrams and which we name a shape class. As a canonical representative in every shape class we take the punctured Young diagram where the origin is located at the leftmost point of the topmost horizontal half-line. The corresponding Maya diagram is strictly positive. (9,8,5,4,2) is the canonical representative of the shape class of previous diagram.
2.3.
Maya diagrams and partitions. Every positive Maya diagram, i.e. every tuple of natural numbers N m = (n 1 , ..., n m ), is in one-to-one correspondence with an integer partition [20,42] (1)
|λ| = λ 1 + ... + λ m ,
where the l(λ) = m parts of the partition λ = (λ 1 , ..., λ m ) are given by
(2) λ i = n i − m + i, and (3) |λ| = m i=1 n i − m(m − 1)/2.
For the example in Figure 2, we have N 5 = (9,8,5,4,2), the corresponding partition is λ = 5 2 , 3 2 , 2 and |λ| = 18 (as usual we let λ k i denote λ i repeated k times). The λ i are the lengths of the columns of the corresponding Young diagram starting from the lower left corner and λ is a partition of the integer |λ|.
Note that, contrary to the usual convention, we permit a partition to terminate in a string of zeros. The reduced form λ of such a partition is obtained by suppressing this last string of zeros. This corresponds to fixing the origin at the endpoint of the last vertical segment on the boundary of the corresponding Young diagram. The tuple corresponding to λ consists of strictly positive integers; this last property characterizes the class of reduced partitions. A reduced partition is therefore characteristic of the shape class and its corresponding Young diagram.
To every reduced partition we can associate a conjugate partition λ [47,48] of the same integer |λ|, whose elements λ j are the lengths of the rows (starting from the top) of the Young diagram associated to λ. The conjugate partition can be visualized as the partition corresponding to the transpose (with respect to the main diagonal starting from the upperleft corner) of the original Young diagram. Formally, we have (4) l(λ) = λ 1 , l(λ) = λ 1 and (5)
λ j = # {i : λ i ≥ j} = # {i : n i ≥ j − i + l(λ)} .
An alternative description of an irreductible partition, i.e. a shape class, is via its Durfee symbol [47,48]. For that, we define the Durfee square of the partition to be the largest square contained in the Young diagram whose diagonal coincides with the main diagonal of the diagram.
Letting r × r denote the size of the Durfee square, we define the Durfee symbol to be the pair of partitions (6) λ | λ = λ 1 , ..., λ r | λ 1 , ..., λ r ,
where (7) λ i = λ i − r, λ i = λ i − r.
Thus, the parts of λ are the length of the columns below the Durfee square and the parts of λ are length of the rows to the right of the Durfee square. In this way the partition λ is a truncation of the original and λ is a truncation the conjugate partition.
×λ 1 = 2 λ 2 = 2 λ 3 = 1 λ 1 λ 2 λ 3 = = = 2 2 0
Durfee square Durfee symbol: 2.4. Punctured Young diagrams and Durfee symbols. The concept of Durfee square and Durfee symbol can be generalized to describe an arbitrary punctured Young diagram, and correspondingly, every Maya diagram. A Durfee rectangle is the k × k rectangle whose opposite vertices are the origin of a given punctured Young diagram and the upper left corner of the diagram, and whose left side of length k and upper side of length k lie on the left and upper border of the corresponding Young diagram [44]. In this way we are able to completely characterize a punctured Young diagram as the corresponding k × k Durfee symbol, which we define as
[2 2 , 0|2 2 , 1](8) d | d k×k = d 1 , ..., d k | d 1 , ..., d k ,
where the
(9) d i = λ i − k = n i − k + i,
stand for the heights of the truncated columns below the Durfee square and where the
(10) d j = λ j − k = − (n j + 1) − k + j,
represent the lengths of the truncated rows at the right of the Durfee rectangle. Note that
(11) l(λ) = k − k − n 1 = d 1 + k l(λ) = n 1 + 1 − k − k = d 1 + k.
The Durfee rectangle can be considered as the rectangular Young diagram corresponding to the partition
(12) R = k k .
From this point of view, d and d are respectively the partition and conjugate partition associated to each of the two connected components of the diagram λ − R, (see [47,48]).
We define an irreducible Durfee symbol to be one for which k ≤ l(λ) and k ≤ l(λ), i.e. for which the punctured Young diagram has an origin between its upper-right and lower left corners. With the exception of the two extremal cases of k = l(λ), k = 0 and k = 0, k = l(λ), the irreducible Durfee symbols are associated to non-flat rectangles. Thus, the total number of such symbols is (13) l(λ) + l(λ) + 1 = n 1 − n 1 + 2 = d 1 + d 1 + k + k + 1.
λ 1 = 1 λ 2 = 1 λ 1 λ 2 λ 3 λ 4 = = = = 3 3 1 1
Durfee rectangle Durfee symbol: If we fix the origin at the upper right corner of the Young diagram we have the representative of the shape class. The associated Durfee rectangle is horizontally flat and the corresponding generalized
[3 2 , 1 2 |1 2 ]Durfee symbol is (k = m) [λ 1 , ..., λ m | ∅].
If we fix the origin at the lower left corner of the Young diagram, the associated Durfee rectangle is vertically flat and the corresponding generalized Durfee symbol reduces to ∅ | λ 1 , ..., λ m (k = m).
2.5.
Shifting rules. Starting from the punctured Young diagram corresponding to d | d , a one-step displacement of the origin along the border of the Young diagram changes the generalized Durfee symbol in a simple way. The shifting rules are the following
Left-descending shift
We move of one step in the left (descending) direction. We denote by 1 d | d the resulting Durfee symbol
d k > 0 : 1 d | d = d 1 − 1, ..., d k − 1 | d 1 , ..., d k , 0 . (14) d k > 0 : 1 d | d = d 1 , ..., d k−1 | d 1 + 1, ..., d k + 1 .(15)
From equations (9) and (10), we deduce the corresponding updating rules for the associated Maya diagram N m :
n k > 0 : N m → 1 (N m ) = n 1 − 1, ..., n k − 1, (−1) , n 1 − 1, ..., n k − 1 . (16) n k > 0 : N m → 1 (N m ) = n 1 − 1, ..., n k−1 − 1, n 1 − 1, ..., n k − 1 .(17)
Right-ascending shift
Starting from the punctured Young diagram corresponding to d | d , a one-step displacement of the origin along the border of the Young diagram changes the generalized Durfee symbol in a simple way. The unit left and right shifting rules are, respectively, the following
1 d | d = d 1 − 1, ..., d k − 1 | d 1 , ..., d k , 0 if d k > 0 d 1 , ..., d k−1 | d 1 + 1, ..., d k + 1 if d k = 0. (18) d | d 1 = d 1 , ..., d k , 0 | d 1 − 1, ..., d k − 1 if d k > 0 d 1 + 1, ..., d k + 1 | d 1 , ..., d k−1 if d k = 0.(19)
At the level of the Maya diagram, the above transformation corresponds to a displacement of the origin one unit to the left and right, respectively. Let N m = n 1 , ..., n k , n k , ..., n 1 be the corresponding Maya diagram. From Eq (9) and Eq (10), we deduce the corresponding changes for the associated Maya diagram :
1 (N m ) = n 1 − 1, ..., n k − 1, −1, n 1 − 1, ..., n k − 1 if n k > 0 n 1 − 1, ..., n k−1 − 1, n 1 − 1, ..., n k − 1 if n k = 0. (20) (N m ) 1 = n 1 + 1, ..., n k + 1, 0, n 1 + 1, ..., n k + 1 if n k < −1 n 1 + 1, ..., n k + 1, n 1 + 1, ..., n k−1 + 1 if n k = −1.(21)
In the considered example, d | d = 3 2 , 1 2 | 1 2 , we have 11 distinct irreducible Durfee symbols in the associated shape class which are related by the following shift connections (the right and left shifts are represented by right and left arrows respectively)
5 2 , 3 2 , 2 | ∅ 4 2 , 2 2 , 1 | 0 3 2 , 1 2 , 0 | 0 2 3 2 , 1 2 | 1 2 (22) 2 2 , 0 2 | 1 2 , 0 2 2 , 0 | 2 2 , 1 2 2 | 3 2 , 2 1 2 | 3 2 , 2, 0 0 2 | 3 2 , 2, 0 2 0 | 4 2 , 3, 1 2 ∅ | 5 2 , 4, 2 2 .
Darboux Transformations
In this Section, and throughout the paper, the symbol ∝ will be heavily used to denote proportionality up to a constant factor, i.e.
f (x) ∝ g(x) ⇔ f (x) = C · g(x), C ∈ R.
3.1. One-step Darboux transformations. We consider a one dimensional Hamiltonian H = −d 2 /dx 2 + V (x), x ∈ I ⊂ R and the associated Schrödinger equation
(23) ψ (x) + (E − V (x)) ψ(x) = 0.
For every value of the eigenvalue E, there is a two dimensional space of formal eigenfunctions ψ(x) that satisfy the above equation. In the following we also suppose that, with Dirichlet boundary conditions on I, H admits a discrete spectrum of energies and eigenstates of the (E n , ψ n ) n∈{0,...,nmax} ⊆N where, without loss of generality, we can always suppose that the ground level of H is zero: E 0 = 0. We consider E as depending on an auxiliary spectral parameter λ, which identifies to the quantum number n for the energy levels. For a general value of λ, ψ λ means any solution of (23) for the eigenvalue E λ and for λ = n, it identifies the eigenstate ψ n .
Any formal eigenfunction ψ ν (x) of V (x) (i.e. H) can be used as a seed function for a Darboux transformation (DT) A(ψ ν ) which associates to the potential V (x) a modified potential (24) V (x)
A(ψν ) −→ V (ν) (x) = V (x) − 2 (log ψ ν (x)) ,
that we call an extension of V (x). The formal eigenfunction of V (ν) associated to the spectral parameter E λ is given by the Darboux-Crum formulas
(25) ψ (ν) λ (x) ∝ W (ψν ,ψ λ |x) ψν (x) if λ = ν, 1 ψν (x) , otherwise,
where W (y 1 , ..., y m | x) denotes the Wronskian of the family of functions y 1 , ..., y m
(26) W (y 1 , ..., y m | x) = y 1 (x) ... y m (x) ... . . . ... y (m−1) 1 (x) ... y (m−1) m (x) .
Chains of Darboux transformations.
At the formal level, the procedure can be straightforwardly iterated and a chain of m Darboux transformations can be completely characterized by the tuple (ν 1 , ..., ν m ) of spectral indices of the successive seed functions used in the chain. We note such a m-tuple by a capital letter N m , where the index m indicates the length of the chain N m = (ν 1 , ..., ν m ). ψ (Nm) λ is the formal eigenfunction associated to the eigenvalue E λ of the potential V (Nm) (x). A chain is non-degenerate if all the spectral indices ν i of the chain N m are distinct and is degenerate if some of them are repeated. In the rest of the paper, we use the following notation (27) (N m , λ) = (ν 1 , ..., ν m , λ) (N m + k) = (ν 1 + k, ..., ν m + k) .
For non-degenerate chains, the extended potentials and their eigenfunctions can be expressed via Wronskians of eigenfunctions of the initial potential [1,3], known as Crum's formulas. Assuming that all ν j and λ are distinct, we have
(28) ψ (Nm) λ (x) = W (Nm,λ) (x) W (Nm) (x) ,(29)V (Nm) (x) = V (x) − 2 log W (Nm) (x) , where W (Nm) (x) = W (ψ ν1 , ..., ψ νm | x).
We have shown [20] that these formulas can be extended to degenerate chains if we adopt the convention to suppress any pair of repeated index in the set of spectral indices associated to the chain, namely
(30) ψ (Nm) νi (x) = W (ν1,...,νi−1,νi+1,...,νm) (x) W (Nm) (x) ,
which extends the second equality in (25) and
(31) V (Nm,νi) (x) = V (x) − 2 log W (ν1,...,νi−1,νi+1,...,νm) (x) .
Translationally Shape Invariant Potentials
Consider a potential V (x; α) which depends upon a (multi)parameter α = α (1) , ..., α (M ) ∈ R M and with a (finite or infinite) bound state spectrum (E n , ψ n ) n≥0 , the ground level being supposed to be zero: E 0 (α) = 0.
Any potential of the same functional form as V (x; α), i.e. of the form V (x; β) + K, where K is a constant, will be said of the same shape class as V (x; α). This potential will be said shape invariant if its image by the Darboux transformation A(ψ 0 ) with seed function equal to the ground state [49][50][51], is in the same shape class as the initial potential, i.e. if (32)
V (0) (x; α) = V (x; f (α)) + R(α),
R (α) ∈ R and f (α) ∈ R M being two given functions of α. The corresponding Darboux transformation is usually called a supersymmetric quantum mechanical (SUSY QM) partnership. The bound state spectrum of such a shape invariant potential is then given by (n ≥ 1)
(33) E n (α) = n−1 k=0 R(α k ) = n−1 k=0 E 1 (α k ) ψ n (x; α) ∝ A + (α)... A + (α n−1 )ψ 0 (x; α n ), where α k = f (k) (α) = k times f • ... • f (α) and A + (α) = − d dx − (log (ψ 0 (x; α))) . When f is a simple translation f (α) = α + ε, ε = ε (1) , ..., ε (M ) , ε (k) = ±1
or 0, V is said to be translationally shape invariant (TSI) and we call it a TSI Potential. For all the known TSI potentials, we have M ≤ 2.
For the TSI potentials, the dispersion relation (energy as a function of the quantum number) is explicitly known and it has the general form
(34) E n (α) = g(α n ) − g(α),
where g(α) is a polynomial at most quadratic in α. The dispersion relation can be extended analytically to non integer values of the quantum number and we have in particular
(35) E −(n+1) (α) = n k=0 E −1 (α k ), n ≥ 0.
A function is said to be quasi-polynomial if it can be expressed, up to a gauge factor, as an orthogonal polynomial in a suitable variable its logarithmic derivative being rational in this variable. The eigenstates ψ n of a TSI potential are always quasi-polynomial.
The set of TSI potentials contains in particular all the potentials classicaly known to be exactly solvable, i.e. for which we know explicitly the dispersion relation and whose the eigenfunctions can be expressed in closed analytical form in terms of elementary transcendental functions, namely the harmonic, isotonic, Morse, Kepler-Coulomb, Eckart, Darboux-Pöschl-Teller hyperbolic and trigonometric and Rosen-Morse hyperbolic and trigonometric potentials.
In the rest of the paper, we concentrate on the isotonic oscillator and trigonometric Darboux-Pöschl-Teller (tDPT) potentials, whose eigenfunctions are expressible in terms of Laguerre and Jacobi polynomials respectively. The dispersion relation g(α) is linear for the isotonic oscillator and quadratic for the tDPT potential.
A common feature of these confining TSI potentials is the existence of a set of symmetries Γ i which act on the parameters and which are covariance transformations for the considered potential (36) α
Γi → Γ i (α) V (x; α) Γi → V (x; Γ i (α)) = V (x; α) + δ i (α) .
In the space of parameters, Γ i correspond to the reflections with respect to the coordinate axis. For the isotonic oscillator and tDPT potentials, the parameter space is two dimensional and we have three distinct symmetries Γ 1 , Γ 2 and Γ 3 = Γ 1 • Γ 2 .
A formal eigenfunction diverging at both endpoints of the definition interval will be said of (∞, ∞) type. A formal eigenfuntion diverging at one or the other extremity of the definition interval will be said of (0, ∞) or (∞, 0) type and a formal eigenfuntion tending to zero at both endpoints of the definition interval will be said of (0, 0) type, which includes the L 2 eigenfunctions. 4.1. Pseudo spectra of the trigonometric Darboux-Pöschl-Teller potential. The trigonometric Darboux-Pöschl-Teller (tDPT) potential is defined on the interval (0, π/2) by
(37) V (x; α, β) = (α + 1/2)(α − 1/2) sin 2 x + (β + 1/2)(β − 1/2) cos 2 x − (α + β + 1) 2 , |α| , |β| > 1/2.
With Dirichlet boundary conditions at 0 and π/2 and in the case α, β > 1/2, it has the following spectrum
(38) E n (α, β) = (α n + β n + 1) 2 − (α + β + 1) 2 = 4n(α + β + 1 + n) ψ n (x; α, β) = (1 − z) (α+1/2)/2 (1 + z) (β+1/2)/2 P (α,β) n (z) , n ∈ N,
where z = cos 2x (ψ n is of (0, 0) type on [0, π/2]). The P (α,β) n (z) are the usual Jacobi polynomials [52,53] and (α n , β n ) = (α + n, β + n).
Note that V (x; α, β) is invariant under the parity reflection x → π/2 − x ( which corresponds to z → −z) combined with the permutation of the parameters (α ↔ β). This implies in a direct way the standard reflection property of the Jacobi polynomials [52,53]:
(39) P (α,β) n (−z) = (−1) n P (β,α) n (z) .
The tDPT potential is a TSI potential, with (α, β) ∈ R 2 and ε = (+1, +1)
(40) V (0) (x; α, β) = V (x; α 1 , β 1 ) + 4(α + β + 2) ψ (0) n (x; α, β) ∝ ψ n−1 (x; α 1 , β 1 )
.
It possesses three discrete parametric symmetries:
(1) The Γ 1 symmetry which acts as
(41) (α, β) Γ1 → (α, −β) V (x; α, β) Γ1 → V (x; α, −β) = V (x; α, β) + 4β(α + 1),
and generates the conjugate shadow spectrum of V (x; α, β) :
(42) E n (α, β) ψ n (x; α, β) Γ1 → E n−β (α, β) = 4 (n − β) (n + α + 1) = E −(n+1)−α (α, β) φ −(n+1) (x; α, β) = ψ n (x; α, −β) = (1 − z) (α+1/2)/2 (1 + z) (−β+1/2)/2 P (−α,β) n (z) , where φ −(n+1) (x; α, β) = ψ n−β (x; α, β) = ψ −(n+1)−α (x; α, β) is of (0, ∞) type on [0, π/2]. Note also that Γ 1 (V (x; α k , β l )) = V (x; α k , β −l ) + 4β −l (α k + 1)(43)Γ 1 (ψ n (x; α k , β l )) = φ n (x; α k , β −l ), n ∈ Z(44)
(2) The Γ 2 symmetry which acts as
(45) (α, β) Γ2 → (−α, β) , (α, β) Γ2 → (−α, β) V (x; α, β) Γ2 → V (x; −α, β) = V (x; α, β) + 4α(β + 1)
and generates the shadow spectrum of V (x; α, β):
(46) E n (α, β) ψ n (x; α, β) Γ2 → E n−α (α, β) = 4 (n − α) (n + β + 1) φ n (x; α, β) = ψ n (x; −α, β) = (1 − z) (−α+1/2)/2 (1 + z) (β+1/2)/2 P (−α,β) n (z) ,
where φ n (x; α, β) = ψ n−α (x; α, β) is of (∞, 0) type on [0, π/2]. We have also
(47) Γ 2 (V (x; α k , β l )) = V (x; α −k , β l ) + 4α −k (β l + 1) Γ 2 (ψ n (x; α k , β l )) = φ −(n+1) (x; α −k , β l ), n ∈ Z.
(3) The Γ 3 = Γ 1 • Γ 2 symmetry which acts as
(48) (α, β) Γ3 → (−α, −β) , V (x; α, β) Γ3 → V (x; −α, −β) = V (x; α, β) + 4(α + β) ψ n (x; α, β) Γ3 → ψ −(n+1) (x; α, β) ,
and generates the conjugate spectrum of V (x; α, β).
(49) E n (α, β) ψ n (x; α, β) Γ3 → E −(n+1) (α, β) = −4 (n + 1) (α + β − n) ψ −(n+1) (x; α, β) = ψ n (x; −α, −β) = (1 − z) (−α+1/2)/2 (1 + z) (−β+1/2)/2 P (−α,−β) n (z) , where ψ −(n+1) (x; α, β) is of (∞, ∞) type. Note also that (50) Γ 3 (V (x; α k , β l )) = V (x; α −k , β −l ) + 4 (α −k + β −l ) Γ 3 (ψ n (x; α k , β l )) = ψ −(n+1) (x; α −k , β −l ), n ∈ Z.
The conjugate spectrum is obtained by a mirror symmetry (center at ν = −1/2) on the values of the spectral parameter: ν → −(ν + 1). The tDPT potential presents in addition to the true spectrum, the three pseudo-spectra defined above: conjugate (E −(n+1) ), shadow (E n−α ) and conjugate shadow (E n−β ) spectra. The union of the quasi-spectrum and of its conjugate will be called the extended spectrum. The union of the shadow spectrum and of its conjugate will be called the extended shadow spectrum.
The extended shadow spectrum is thus obtained by shifting the integer values of the spectral parameter associated to the spectrum, by β and α. Due to the structure of the dispersion relation, which induces the identity E n−β = E −(n+1)−α , it can also be obtained by shifting the values of the extended and conjugate spectra by α. Under this aspect, the conjugate shadow spectrum E −(n+1)−α appears to be the shadow of the conjugate spectrum E −(n+1) . Starting from the usual eigenstates, the Γ i symmetries generate all the quasi-polynomial eigenfunctions of the tDPT potential which are all contained in the extended or extended shadow spectra.
If α = β, and α and β are not integers, the four pseudo-spectra are distinct and in the rest of the paper we limit our study to this non-degenerate case. For integer values of α or β the pseudo-spectra can coalesce, giving rise to interesting but specific phenomena [22].
4.2.
Pseudo spectra of the Isotonic oscillator. The isotonic oscillator potential (with zero ground level E 0 = 0) is defined on the positive half line (0, +∞) by (51) V (x; ω, α) = ω 2 4
x 2 + (α + 1/2) (α − 1/2) x 2 − ω (α + 1) , |α| > 1/2.
If we impose Dirichlet boundary conditions at 0 and infinity and assume α > 1/2 and α / ∈ N, it has the following spectrum (z = ωx 2 /2)
(52) E n (ω) = 2nω ψ n (x; ω, α) = z (α+1/2)/2 e −z/2 L α n (z) , n ≥ 0,
where ψ n is of (0, 0) type on [0, +∞). The isotonic oscillator is a TSI potential, with (ω, α) ∈ R 2 and ε = (0, +1), since
(53) V (0) (x; ω, α) = V (x; ω, α 1 ) + 2ω ψ (0) n (x; ω, α) ∝ ψ n−1 (x; ω, α 1 )
, with α n = α+n. This potential also possesses three discrete parametric symmetries, which we describe below.
(1) The Γ 1 symmetry which acts as
(54) (ω, α) Γ1 → (−ω, α) V (x; ω, α) Γ1 → V (x; −ω, α) = V (x; ω, α) + 2ω (α + 1) ,
and generates the conjugate shadow spectrum of V (x; ω, α) :
(55) E n (ω) ψ n (x; ω, α) Γ1 → E −(n+1)−α (ω) = −2 (n + 1 + α) ω < 0 φ −(n+1) (x; ω, α) = ψ n (x; −ω, α) = z (α+1/2)/2 e z/2 L α n (−z) , n ≥ 0, where φ −(n+1) (x; ω, α) = ψ −(n+1)−α (x; ω, α) is of (0, ∞) type on [0, +∞). Note also that (56) Γ 1 (V (x; ω, α k )) = V (x; ω, α k ) + 2ω (α k + 1) Γ 1 (ψ n (x; ω, α k )) = φ −(n+1) (x; ω, α k ), n ∈ Z.
(2) The Γ 2 symmetry which acts as
(57) (ω, α) Γ2 → (ω, −α) V (x; ω, α) Γ2 → V (x; ω, −α) = V (x; ω, α) + 2ωα,
and generates the shadow spectrum of V (x; ω, α) :
(58) E n (ω) ψ n (x; ω, α) Γ2 → E n−α (ω) = 2 (n − α) ω φ n (x; ω, α) = ψ n (x; ω, −α) = z (−α+1/2)/2 e −z/2 L −α n (z) , n ≥ 0,
where φ n (x; ω, α) is of (∞, 0) type on [0, +∞). We have in particular
(59) φ −1 (x; ω, α) = z (α+1/2)/2 e z/2 = 1/φ 0 (x; ω, α 1 ).
and more generally
(60) Γ 2 (V (x; ω, α k )) = V (x; ω, α −k ) + 2ωα −k Γ 2 (ψ n (x; ω, α k )) = φ n (x; ω, α −k ), n ∈ Z.
(3) The Γ 3 = Γ 1 • Γ 2 symmetry which acts as
(61) (ω, α) Γ3 → (−ω, −α) V (x; ω, α) Γ3 → V (x; −ω, −α) = V (x; ω, α) + 2ω,
and generates the conjugate spectrum of V (x; ω, α) :
(62) E n (ω) ψ n (x; ω, α) Γ3 → E −(n+1) (ω) = −2 (n + 1) ω < 0 ψ −(n+1) (x; ω, α) = ψ n (x; −ω, −α) = z (−α+1/2)/2 e z/2 L −α n (−z) , n ≥ 0,
where ψ −(n+1) (x; ω, α) is of (∞, ∞) type on [0, +∞). In particular, we have (63) ψ −1 (x; ω, α) = z (−α+1/2)/2 e z/2 = 1/ψ 0 (x; ω, α −1 ). and more generally
(64) Γ 3 (V (x; ω, α k )) = V (x; ω, α −k ) + 2ω Γ 3 (ψ n (x; ω, α k )) = ψ −(n+1) (x; ω, α −k ), n ∈ Z.
Here again, the presence of the infinite centrifugal barrier at the origin induces the existence of an extended shadow spectra, (E n−α ) n∈Z . It is obtained by shifting the integer values of the spectral parameter associated to the extended spectrum (E n ) n∈Z by α. Note that, the range −1/2 < α < 1/2 that we have excluded previously is nevertheless interesting because then the potential is weakly attractive at z = 0. This attraction is indeed sufficiently weak that we still get a self-adjoint eigenvalue problem but for which one has to use non-Dirichlet boundary conditions.
Mixed Darboux Chains for TSI potentials.
To produce the rational extensions of the considered potentials we have at our disposal all the quasipolynomial eigenfunctions, that can be used as seed functions in Darboux transformations:
• the eigenstates and their conjugates which can be gathered into one family ψ n , labelled by positive and negative integer values of the quantum number n ∈ Z and which form the extended spectrum. • the shadow and conjugate shadow eigenfunctions which can be gathered into one family φ n , labelled by positive and negative integer values of the quantum number n ∈ Z and which form the extended shadow spectrum. Using ψ n and φ n , n ∈ Z as seed functions for chains of Darboux transformations, the obtained potentials are rational functions in the adapted variable. In the general case, the tuple of spectral indices associated to the chain contains indices associated to the extended spectrum (n 1 , ..., n m ) and indices associated to the extended shadow spectrum (l 1 , ..., l r ) and the corresponding chain of Darboux transformations will be called a mixed chain. The chain is then characterized by a bi-tuple of spectral indices If either m = 0 or r = 0, the bi-tuple reduces to a standard tuple L r or N m and the chain is called a single type chain. A reducible chain is a single type chain whose associated tuple contains consecutive integers starting from 0 in increasing order or starting from −1 in decreasing order. The bi-tuple N m ⊗ L r can be associated to a pair of Maya diagrams, one for N m and the other one for L r .
In the rest of the paper we will use the following notation:
N m = n 1 , ..., n k , n k , ..., n 1 with k + k = m. n i > n i+1 ≥ 0, 0 > n i+1 > n i ,
L r = l 1 , ..., l κ , l κ , ..., l 1 , with κ + κ = r, l i > l i+1 ≥ 0, 0 > l i+1 > l i .
The bi-tuple N m ⊗ L r can be equivalently associated to a pair of punctured Young diagrams or to a direct product Durfee symbols (see equations (9) and (10))
(66) d | d ⊗ δ | δ = d 1 , ..., d k | d 1 , ..., d k ⊗ δ 1 , ..., δ κ | δ 1 , ..., δ κ , where d i = n i − k + i, δ i = l i − κ + i and d j = − (n i + 1) − k + i, δ j = − l i + 1 − κ + i.
Refering to Koike [38] and Tsuda [39], we call d | d ⊗ δ | δ a punctured universal character, which can be thought of as the direct product of two punctured Young diagrams.
In [34], using shape invariance arguments, in the case of mixed chains containing only seed functions of the extended spectrum, we have proven that every mixed chain is equivalent to a single type chain N m = (n 1 , ..., n m ) ∈ N m of seed functions of the spectrum. In other words these type of extensions can be gathered into shape classes of potentials whose representative element is an extension with an associated Durfee symbol of the form [λ 1 , ..., λ m | ∅]. Such a shape class is in one to one correspondence with the Young diagram of the partition λ which represents a shape class of punctured Young diagrams. Using different approaches, Odake [33] has obtained equivalent results for chains containing only seed functions of the extended shadow spectrum and Takemura [37] has extended this results for general mixed chains for the isotonic oscillator and tDPT potentials.
As we will see, all these results are direct consequences of the shape invariance and parametric symmetries of the primary potential. The resulting equivalence formulas are just obtained by applying shape invariance for its rationally extended descendants. In fact, all the rational extensions of the three primary confining TSI potentials and all the associated exceptional orthogonal polynomials, can be put in one to one correspondence with a punctured universal character.
The equivalence must be understood in the following manner at the level of potentials and Durfee symbols: shifting the origin in a punctured Young diagram modifies the chain of Darboux transformations but produces a potential in the same shape class. Equivalently, the pseudowronskians with orthogonal polynomial entries (which are, up to gauge factors, the Wronskians of the seed functions associated to the chains) corresponding to equivalent symbols will be identical up to an overall factor. 5. Shape invariance and formulas for the derivatives of Laguerre and Jacobi polynomials 5.1. Isotonic oscillator and Laguerre polynomials. In terms of punctured universal characters, the shape invariance property (53) for the isotonic oscillator becomes
(67) V [0|∅]⊗[∅|∅] (x; ω, α) = V (0)⊗(∅) (x; ω, α) = V (x; ω, α 1 ) + E 1 (ω) ψ [0|∅]⊗[∅|∅] n (x; ω, α) = ψ (0)⊗(∅) n (x; ω, α) ∝ ψ n−1 (x; ω, α 1 ) , n ∈ Z.
In the particular case n = 0, we have by (62):
ψ [0|∅]⊗[∅|∅] 0 (x; ω, α) = 1/ψ 0 (x; ω, α) ∝ ψ −1 (x; ω, α 1 ).
Observe that φ x; ω, α) for the formal eigenvalue E n−α (ω), that is, a formal eigenfunction of V (x; ω, α 1 ) for the formal eigenvalue E n−α (ω) − E 1 (ω) = E n−α1 (ω). Since it is also quasi-polynomial we necessarily have
(68) φ [0|∅]⊗[∅|∅]
n (x; ω, α) ∝ φ n (x; ω, α 1 ), n ∈ Z.
We can now combine the results above with the Γ symmetries. As an example consider first the case of Γ 3 . Combining (67) and (68) with (62) and (64), we have
V [∅|0]⊗[∅|∅] (x; ω, α) = V (−1)⊗(∅) (x; ω, α) = V (x; ω, α) − 2 (log ψ −1 (x; ω, α)) (69) = Γ 3 V (x; ω, α) − 2 (log ψ 0 (x; ω, α)) − 2ω = Γ 3 (V (x; ω, α 1 ) + 2ω) − 2ω, that is, (70) V [∅|0]⊗[∅|∅] (x; ω, α) = V (x; ω, α −1 ) − 2ω.
Moreover
ψ [∅|0]⊗[∅|∅] n (x; ω, α) = W (ψ −1 , ψ n | x; ω, α) ψ −1 (x; ω, α) = Γ 3 W ψ 0 , ψ −(n+1) | x; ω, α ψ 0 (x; ω, α) (71) ∝ Γ 3 ψ −(n+1)−1 (x; ω, α 1 ) ∝ ψ n+1 (x; ω, α −1 ), n ∈ Z, and φ [∅|0]⊗[∅|∅] n (x; ω, α) = W (ψ −1 , φ n | x; ω, α) ψ −1 (x; ω, α) = Γ 3 W ψ 0 , φ −(n+1) | x; ω, α ψ 0 (x; ω, α) (72) ∝ Γ 3 φ −(n+1) (x; ω, α 1 ) ∝ φ n (x; ω, α −1 ), n ∈ Z.
We can repeat the same elementary algebraic manipulation with the whole set of Γ i symmetries and then, starting from (67) and (68), obtain in a very direct way the complete set of basic shape invariance formulas of the isotonic oscillator. For the potential, they are given by
(73) V [0|∅]⊗[∅|∅] (x; ω, α) = V (x; ω, α 1 ) + E 1 (ω) V [∅|0]⊗[∅|∅] (x; ω, α) = V (x; ω, α −1 ) − 2ω V [∅|∅]⊗[0|∅] (x; ω, α) = V (x; ω, α −1 ) V [∅|∅]⊗[∅|0] (x; ω, α) = V (x; ω, α 1 ),
for the eigenfunctions of the extended spectrum by
(74) ψ [0|∅]⊗[∅|∅] n (x; ω, α) ∝ ψ n−1 (x; ω, α 1 ) ψ [∅|0]⊗[∅|∅] n (x; ω, α) ∝ ψ n+1 (x; ω, α −1 ) ψ [∅|∅]⊗[0|∅] n (x; ω, α) ∝ ψ n (x; ω, α −1 ) ψ [∅|∅]⊗[∅|0] n (x; ω, α) ∝ ψ n (x; ω, α 1 ), n ∈ Z,
and for the eigenfunctions of the extended shadow spectrum by
(75) φ [0|∅]⊗[∅|∅] n (x; ω, α) ∝ φ n (x; ω, α 1 ) φ [∅|0]⊗[∅|∅] n (x; ω, α) ∝ φ n (x; ω, α −1 ) φ [∅|∅]⊗[0|∅] n (x; ω, α) ∝ φ n−1 (x; ω, α −1 ) φ [∅|∅]⊗[∅|0] n (x; ω, α) ∝ φ n+1 (x; ω, α 1 ), n ∈ Z.
Remarkably, the classical formulas for the derivative Laguerre polynomials [52,53], are a direct transcription of the shape invariance formulas when expressed in a explicit form. Indeed, the Darboux-Crum formula (25) allows to write (74) as
(76) W (ψ 0 , ψ n | x; ω, α) ∝ ψ 0 (x; ω, α)ψ n−1 (x; ω, α 1 ) W (ψ −1 , ψ n | x; ω, α) ∝ ψ −1 (x; ω, α)ψ n+1 (x; ω, α −1 ) W (φ 0 , ψ n | x; ω, α) ∝ φ 0 (x; ω, α)ψ n (x; ω, α −1 ) W (φ 0 , ψ n | x; ω, α) ∝ φ −1 (x; ω, α)ψ n (x; ω, α 1 ), n ∈ Z.
Combined with the following standard properties [54] of Wronskians
(L α n (z)) ∝ L α+1 n−1 (z) (z −α e z L −α n (−z)) ∝ z −α−1 e z L −α−1 n+1 (−z) (z −α L −α n (z)) ∝ z −α−1 L −α−1 n (z) (e z L α n (−z)) ∝ e z L α+1 n (−z)(78)
.
Taking the initial condition in z = 0 (79) L α n (0) = (n + α) (n + α − 1) ... (α + 1) n! , (L α n (0)) = − n α + 1 L α n (0) = −L α+1 n−1 (0), this leads to the well known formulas for the derivatives [52,53] (80)
(L α n (z)) = −L α+1 n−1 (z) (z −α e z L −α n (−z)) = (n + 1) z −α−1 e z L −α−1 n+1 (−z) (z −α L −α n (z)) = (n − α) z −α−1 L −α−1 n (z) (e z L α n (−z)) = e z L α+1 n (−z)
, and more generally for the k-th derivative
(81) (L α n (z)) (k) = (−1) k L α+k n−k (z) (z −α e z L −α n (−z)) (k) = (n + 1) k z −α−k e z L −α−k n+k (−z) (z −α L −α n (z)) (k) = (n − α) k z −α−k L −α−k n (z) (e z L α n (−z)) (k) = e z L α+k n (−z) ,
where (x) m and (x) m are the usual ascending and falling factorials
(82) (x) m = x(x + 1)...(x + m − 1), (x) m = x(x − 1)...(x − m + 1).
5.2.
Trigonometric Darboux-Pöschl-Teller potential and Jacobi polynomials. In terms of punctured universal characters, the shape invariance property (40) for the tDPT potential can be written as
(83) V [0|∅]⊗[∅|∅] (x; α, β) ≡ V (0)⊗(∅) (x; α, β) = V (x; α 1 , β 1 ) + E 1 (α, β) ψ [0|∅]⊗[∅|∅] n (x; α, β) ≡ ψ (0)⊗(∅) n (x; α, β) ∝ ψ n−1 (x; α 1 , β 1 ) , n ∈ Z.
In the particular case n = 0, we have by (42) (84) ψ
[∅|0]⊗[∅|∅] 0 (x; α, β) = 1/ψ 0 (x; α, β) = (1 − z) (−α−1/2)/2 (1 + z) (−β−1/2)/2 ∝ ψ −1 (x; α 1 , β 1 ) . Observe that φ [0|∅]⊗[∅|∅] n (x; α, β) is a formal eigenfunction of V [0|∅]⊗[∅|∅] (x; α, β) for the formal eigenvalue E n−α (α, β), i.e. (see above) φ [0|∅]⊗[∅|∅] n (x; α, β) is a formal eigenfunction of V (x; α 1 , β 1 ) for the formal eigenvalue E n−α (α, β) − E 1 (α, β) = E n−α1 (α, β).
Since it is also quasi-polynomial we necessarily have
(85) φ [0|∅]⊗[∅|∅] n (x; α, β) ∝ φ n (x; α 1 , β 1 ), n ∈ Z.
Proceeding as in the case of the isotonic oscillator, by combining the Γ i , i = 1, 2, 3 symmetries (49) with the results above, we obtain the following transformed potentials:
V [0|∅]⊗[∅|∅] (x; α, β) = V (x; α 1 , β 1 ) + E 1 (α, β) V [∅|0]⊗[∅|∅] (x; α, β) = V (x; α −1 , β −1 ) + E −1 (α, β) V [∅|∅]⊗[0|∅] (x; α, β) = V (x; α −1 , β 1 ) V [∅|∅]⊗[∅|0] (x; α, β) = V (x; α 1 , β −1 )
, whose eigenfunctions of the extended spectrum are
(86) ψ [0|∅]⊗[∅|∅] n (x; α, β) ∝ ψ n−1 (x; α 1 , β 1 ) ψ [∅|0]⊗[∅|∅] n (x; α, β) ∝ ψ n−1 (x; α −1 , β −1 ) ψ [∅|∅]⊗[0|∅] n (x; α, β) ∝ ψ n (x; α −1 , β 1 ) ψ [∅|∅]⊗[∅|0] n (x; α, β) ∝ ψ n (x; α 1 , β −1 )
, and whose eigenfunctions of the extended shadow spectrum are
(87) φ [0|∅]⊗[∅|∅] n (x; α, β) ∝ φ n (x; α 1 , β 1 ) φ [∅|0]⊗[∅|∅] n (x; α, β) ∝ φ n (x; α −1 , β −1 ) φ [∅|∅]⊗[0|∅] n (x; α, β) ∝ φ n−1 (x; α −1 , β 1 ) φ [∅|∅]⊗[∅|0] n (x; α, β) ∝ φ n+1 (x; α 1 , β −1 )
. This exhausts the complete set of shape invariance formulas for the quasi-polynomial eigenfunctions of the tDPT potential. Combining the shape invariance formulas established above with the Darboux-Crum formula, we can deduce in a direct way some identities for the Jacobi polynomials. In the same way as for the isotonic oscillator, we first have
(88) W (ψ 0 , ψ n | x; α, β) ∝ ψ 0 (x; α, β)ψ n−1 (x; α 1 , β 1 ) W (ψ −1 , ψ n | x; α, β) ∝ ψ −1 (x; α, β)ψ n+1 (x; α −1 , β −1 ) W (φ 0 , ψ n | x; α, β) ∝ φ 0 (x; α, β)ψ n (x; α −1 , β 1 ) W (φ 0 , ψ n | x; α, β) ∝ φ −1 (x; α, β)ψ n (x; α 1 , β −1 ), n ∈ Z.
From equations (38), (49), (46) and (42), we deduce immediately
(89) P (α,β) n (z) ∝ P (α+1,β+1) n−1 (z) (1 − z) α (1 + z) β P (α,β) n (z) ∝ (1 − z) α−1 (1 + z) β−1 P (α−1,β−1) n+1 (z) (1 − z) α P (α,β) n (z) ∝ (1 − z) α−1 P (α−1,β+1) n (z) (1 + z) β P (α,β) n (z) ∝ (1 + z) β−1 P (α+1,β−1) n (z) ,
Evaluating the above expressions in z = 1, we have (90) P (α,β) n (1) = (n + α) (n + α − 1) ... (α + 1) n! , P (α,β) n (1) = n + α + β + 1 2 P (α+1,β+1) n−1 (1) .
and making use of the reflection formula for Jacobi polynomials [52,53]
P (β,α) n (−z) = (−1) n P (α,β) n (z) ,
we obtain
(91) P (α,β) n (z) = n+α+β+1 2 P (α+1,β+1) n−1 (z) (1 − z) α (1 + z) β P (α,β) n (z) = −2 (n + 1) (1 − z) α−1 (1 + z) β−1 P (α−1,β−1) n+1 (z) (1 − z) α P (α,β) n (z) = − (n + α) (1 − z) α−1 P (α−1,β+1) n (z) (1 + z) β P (α,β) n (z) = (n + β) (1 + z) β−1 P (α+1,β−1) n (z)
.
More generally, for the k-th derivative we have (82)
(92) P (α,β) n (z) (k) = 1 2 k (n + α + β + 1) k P (α+k,β+k) n−k (z) (1 − z) α (1 + z) β P (α,β) n (z) (k) = (−2) k (n + 1) k (1 − z) α−k (1 + z) β−k P (α−k,β−k) n+k (z) (1 − z) α P (α,β) n (z) (k) = (−1) k (n + α) k (1 − z) α−k P (α−k,β+k) n (z) (1 + z) β P (α,β) n (z) (k) = (n + β) k (1 + z) β−k P (α+k,β−k) n (z)
.
We see thus that the classical formulas for the derivatives of Jacobi polynomials [52,53], are just an explicit transcription of the complete set of shape invariance formulas for the eigenfunctions of the tDPT potential.
Equivalence of Laguerre pseudowronskians
In this section we shall derive many identities among determinants whose entries are associated Laguerre polynomials, by exploiting the notion of equivalence of punctured universal characters explained in Section 2.
Consider a mixed chain associated to the bi-tuple N m ⊗ L r = n 1 , ..., n k , n k , ..., n 1 ⊗ (l 1 , ..., l r ).
We consider first the case where n i > n i+1 > 0 and −1 ≥ n i+1 > n i . We add to the first Maya diagram the index 0 (ie we add to the preceding chain a Darboux transformation associated to the seed function ψ 0 ) obtaining the bi-tuple (0, N m ) ⊗ L r .
Considering V (0,Nm)⊗Lr as an extension of V (0)⊗∅ , the Crum formula (28) applied to the successive seed functions along the chain gives, after telescopic cancellation: On the other hand, if we consider V (0,Nm)⊗Lr as an extension of V , the same use of Crum's formulas (28) leads after telescopic cancellation to W (ψ 0 , ψ n1 , ..., ψ n1 , φ l1 , ..., φ lr | x; ω, α) = ψ 0 (x; ω, α) ψ (0)⊗∅ n1 (x; ω, α) ...ψ Comparing the last two identities and making use of shape invariance (74) and Eq(75), we obtain immediately
(93) W (0,Nm)⊗Lr (x; ω, α) ∝ ψ 0 (x; ω, α) W (Nm−1)⊗Lr (x; ω, α 1 ) ,
or, using (63) and redefining α → α −1
(94) W (Nm−1)⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (0,Nm)⊗∅ (x; ω, α −1 ) .
Suppose n k < −1. By shifting the indices n i → n i + 1, n i → n i + 1, (94) can be rewritten as (see (20))
(95) W Nm⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (Nm) 1 ⊗Lr (x; ω, α −1 ) .
Suppose now that in N m we have n k = 0 (in which case n k−1 > 0). (93) is still valid provided we change k into k − 1. Then (96) W (0,Nm)⊗Lr (x; ω, α) ∝ ψ 0 (x; ω, α) W (Nm−1)⊗Lr (x; ω, α 1 ) ,
i.e. with the shifting rule (17) (97) W Nm⊗Lr (x; ω, α) ∝ ψ 0 (x; ω, α) W 1 (Nm)⊗Lr (x; ω, α 1 ) .
Consider in a second step, a mixed chain associated to the bi-tuple of relative integers N m ⊗ L r to which we add to the Maya diagram N m the index (−1) (i.e. we add a DT associated to the seed function ψ −1 to the corresponding chain) obtaining (−1, N m ) ⊗ L r . Proceeding as above, using the Darboux-Crum and Crum formulas as well as telescopic cancellation and the shape invariance properties (74) and (75), we obtain
(98) W (−1,Nm)⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (Nm+1)⊗Lr (x; ω, α −1 ) .
If n k > 0, shifting the indices (n i → n i − 1, n i → n i − 1) and using (63) we have equivalently
(99) W Nm⊗Lr (x; ω, α) ∝ ψ 0 (x; ω, α) W 1 (Nm)⊗Lr (x; ω, α 1 ) . If n k = −1 (in which case n k−1 < −1), (98) gives (100) W (−1,Nm)⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (Nm+1)⊗Lr (x; ω, α −1 ) ,
i.e. (see Eq (17))
(101) W Nm⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (Nm) 1 ⊗Lr (x; ω, α −1 ) .
Summarizing, we have for every bi-tuple N m ⊗ L r (102) W Nm⊗Lr (x; ω, α) ∝ ψ 0 (x; ω, α) W 1(Nm)⊗Lr (x; ω, α 1 ) W Nm⊗Lr (x; ω, α) ∝ ψ −1 (x; ω, α) W (Nm) 1 ⊗Lr (x; ω, α −1 ) .
We can repeat exactly the same lines by acting on the Maya diagram L r rather than on N m . We obtain in the same way
(103) W Nm⊗Lr (x; ω, α) ∝ φ 0 (x; ω, α) W Nm⊗ 1(Lr ) (x; ω, α 1 ) W Nm⊗Lr (x; ω, α) ∝ φ −1 (x; ω, α) W Nm⊗(Lr) 1 (x; ω, α 1 ) .
Note that the identities (76) and the formulas for the derivatives of Laguerre polynomials (76) constitute in fact the lowest order case of (102) and (103).
If we look at the consequences of the relations (102) and (103) at the level of the extended potentials, we obtain first, using (73)
V Nm⊗Lr (x; ω, α) = V (x; ω, α) − 2 log ψ 0 (x; ω, α) W 1(Nm)⊗Lr (x; ω, α 1 ) (104) = V (0)⊗∅ (x; ω, α) − 2 log W 1(Nm)⊗Lr (x; ω, α 1 ) = V (x; ω, α 1 ) + E 1 (ω) − 2 log W 1 (Nm)⊗Lr (x; ω, α 1 ) = V 1 (Nm)⊗Lr (x; ω, α 1 ) + E 1 (ω).
More generally, we have
(105) V Nm⊗Lr (x; ω, α) = V 1(Nm )⊗Lr (x; ω, α 1 ) + E 1 (ω) V Nm⊗Lr (x; ω, α) = V (Nm) 1 ⊗Lr (x; ω, α −1 ) + E −1 (ω) V Nm⊗Lr (x; ω, α) = V Nm⊗ 1(Lr ) (x; ω, α 1 ) V Nm⊗Lr (x; ω, α) = V Nm⊗(Lr) 1 (x; ω, α −1 ).
In terms of punctured universal characters the preceding results (102), (103) and (105) can be rewritten as
(106) W [d|d]⊗[δ|δ] (x; ω, α) ∝ ψ 0 (x; ω, α) W 1[d|d]⊗[δ|δ] (x; ω, α 1 ) W [d|d]⊗[δ|δ] (x; ω, α) ∝ ψ −1 (x; ω, α) W [d|d] 1 ⊗[δ|δ] (x; ω, α −1 ) W [d|d]⊗[δ|δ] (x; ω, α) ∝ φ 0 (x; ω, α) W [d|d]⊗ 1[δ|δ] (x; ω, α −1 ) W [d|d]⊗[δ|δ] (x; ω, α) ∝ φ −1 (x; ω, α) W [d|d]⊗[δ|δ] 1 (x; ω, α 1 ) and (107) V [d|d]⊗[δ|δ] (x; ω, α) = V 1[d|d]⊗[δ|δ] (x; ω, α 1 ) + E 1 (ω) V [d|d]⊗[δ|δ] (x; ω, α) = V [d|d] 1 ⊗[δ|δ] (x; ω, α −1 ) + E −1 (ω) V [d|d]⊗[δ|δ] (x; ω, α) = V [d|d]⊗ 1[δ|δ] (x; ω, α −1 ) V [d|d]⊗[δ|δ] (x; ω, α) = V [d|d]⊗[δ|δ] 1 (x; ω, α 1 ), where d | d ⊗ δ | δ ( d | d = d 1 , ..., d k | d 1 , .
.., d k with k+k = m, and δ | δ = δ 1 , ..., δ κ | δ 1 , ..., δ κ with κ + κ = r) is the punctured universal character corresponding to the bi-tuple N m ⊗ L r .
By repeating the procedure we can push the origins in both punctured Young diagrams in the upper right corner.
Indeed, by doing −n 1 = d 1 + k successive shifts in the right-ascending direction in the first Young diagram and l 1 + 1 = δ 1 + κ shifts in the same direction in the second one, we obtain (108)
W [d|d]⊗[δ|δ] (x; ω, α) ∝ d1+k−1 i=0 ψ −1 (x; ω, α −i ) δ1+κ−1 j=0 φ −1 x; ω, α j−(d1+k) W [λ|∅]⊗[µ|∅] x; ω, α δ1+κ−d1−k .
Equivalently we can choose to push the origin on each punctured Young diagram in the lower left corner by doing n 1 + 1 = d 1 + k successive shifts in the left-descending direction in the first Young diagram and l 1 + 1 = δ 1 + κ shifts in the same direction in the second one, obtain then (109)
W [d|d]⊗[δ|δ] (x; ω, α) ∝ d1+k−1 i=0 ψ 0 (x; ω, α i ) δ1+κ−1 j=0 φ 0 (x; ω, α d1+k−j ) W [∅|λ]⊗[∅|µ] x; ω, α d1+k−(δ1+κ) .
In the same manner we arrive at
(110) W [d|d]⊗[δ|δ] (x; ω, α) ∝ d1+k−1 i=0 ψ −1 (x; ω, α −i ) δ1+κ−1 j=0 φ 0 x; ω, α −(d1+k+j) W [λ|∅]⊗[∅|µ] x; ω, α −(d1+k+δ1+κ) W [d|d]⊗[δ|δ] (x; ω, α) ∝ d1+k−1 i=0 ψ 0 (x; ω, α i ) δ1+κ−1 j=0 φ −1 (x; ω, α d1+k+j ) W [∅|λ]⊗[∅|µ] x; ω, α d1+k+δ1+κ .
For the potentials, this gives
(111) V [d|d]⊗[δ|δ] (x; ω, α) = V [λ|∅]⊗[µ|∅] x; ω, α δ1+κ−d1−k + E −(d1+k) (ω) V [d|d]⊗[δ|δ] (x; ω, α) = V [∅|λ]⊗[∅|µ] (x; ω, α d1+k−δ1−κ ) + E d1+k (ω).
These formulas constitute the general transcription of the shape invariance property of the isotonic potential at the level of its rationally extended descendants.
In the following we choose as canonical representative of a given shape class the punctured universal character [λ | ∅] ⊗ [µ | ∅] and then express all the extensions of this class in terms of W [λ|∅]⊗ [µ|∅] . Using (55) and (62), we obtain a more explicit form for (108)
W [d|d]⊗[δ|δ] (x; ω, α) ∝ z −α(d1+k−δ1−κ)/2 × e z(d1+k+δ1+κ)/2 (112) ×z (d1+k−δ1−κ) 2 /4 × W [λ|∅]⊗[µ|∅] x; ω, α −(d1+k)+δ1+κ , where z = ωx 2 /2.
From (62), (58) and (55) we have
(113) ψ −(n+1) (x; ω, α) = z −α e z ψ 0 (x; ω, α) L −α n (−z) φ n (x; ω, α) = z −α ψ 0 (x; ω, α) L −α n (z) φ −(n+1) (x; ω, α) = e z ψ 0 (x; ω, α) L α n (−z) .
Then using (109), (108) and (81), and noting that m = k + k, r = κ + κ, we deduce that
W [d|d]⊗[δ|δ] (x; ω, α) ∝ z α(k−k−κ+κ)/2 e −z(k−k+κ−κ)/2 (114) ×z (k−k−κ+κ) 2 /4 z −(k+κ)(k+κ−1) × L [L [d|d]⊗[δ|δ] (z; α) = det − → L (j) (z, α) 1≤j≤k , − → L (j) (z, α) 1≤j≤k , − → Λ (j) (z, α) 1≤j≤κ , − → Λ (j) (z, α) 1≤j≤κ , where(116) L (j) i (z, α) = (−1) i−1 L α+i−1 nj −i+1 (z) L (j) i (z, α) = (−n j ) i−1 × z m+r−i L −α−i+1 −nj +i−2 (−z) Λ (j) i (z, α) = (l j + α) i−1 × z m+r−i L −α−i+1 lj (z) Λ (j) i (z, α) = L α+i−1 −lj −1 (−z)
, i = 1, ..., m + r,
with n j = d j + k − j ≥ 0, l j = δ j + κ − j ≥ 0, −n j − 1 = d j + k − j ≥ 0 and −l j − 1 = δ j + κ − j ≥ 0. If [λ | ∅]
is in the same shape class as d | d = d 1 , ..., d k | d 1 , ..., d k , we can use (114) replacing k and κ by 0 and k and κ by l(λ) = d 1 + k and l(µ) = δ 1 + κ respectively. It gives
W [λ|∅]⊗[µ|∅] (x; ω, α) ∝ z α(d1+k−δ1−κ)/2 × e −z(d1+k+δ1+κ)/2 (117) ×z (d1+k−δ1−κ) 2 /4 × z −(δ1+κ)(δ1+κ−1) × L [λ|∅]⊗[µ|∅] (z; α) .
The equivalence relation (112) can then be rewritten as
W [d|d]⊗[δ|δ] (x; ω, α) ∝ z α(k−k−κ+κ)/2 × e −z(k−k+κ−κ)/2 ×z (k−k−κ+κ) 2 /4 × z −(δ1+κ)(δ1+κ−1) × L [λ|∅]⊗[µ|∅] z; α −(d1+k)+δ1+κ ,
and finally, using Eq(114) we arrive at
(118) L [d|d]⊗[δ|δ] (z; α) ∝ z (k+κ)(k+κ−1)−(δ1+κ)(δ1+κ−1) × L [λ|∅]⊗[µ|∅] z; α −(d1+k)+δ1+κ ,
which is the equivalence relation for Laguerre pseudowronskians.
L i,j = (−1) i L (α+i) j (z) (119)L i,j = (j + i) i L (−α−i) j (−z) (120) Λ i,j = (j − α) i L (−α−i) j (z) (121)Λ i,j = L (α+i) j (−z),(122)
the pseudo-Wronskians in question have the explicit determinantal form
L [2,1|1]⊗[3|2] (z; α) = det L 0,1 −1 0 0 0 L 0,3 L 1,2 L 2,1 −1 0 z 4L 0,3 z 3L 1,3 z 2L 2,4 zL 3,5L4,6 z 4 Λ 0,3 z 3 Λ 1,3 z 2 Λ 2,3 zΛ 3,3 Λ 4,3 Λ 0,2Λ1,2Λ2,2Λ3,2Λ4,2 ,(123)L [3,2,1|∅]⊗[4,1 2 |∅] (z; α) = det L 0,1 −1 0 0 0 0 L 0,3 L 1,2 L 2,1 −1 0 0 L 0,5
L 1,4 L 2,3 L 3,2 L 4,1 L 5,0 z 5 Λ 0,1 z 4 Λ 1,1 z 3 Λ 2,1 z 2 Λ 3,1 zΛ 4,1 Λ 5,1 z 5 Λ 0,2 z 4 Λ 1,2 z 3 Λ 2,2 z 2 Λ 3,2 zΛ 4,2 Λ 5,2 z 5 Λ 0,6 z 4 Λ 1,6 z 3 Λ 2,6 z 2 Λ 3,6 zΛ 4,6 Λ 5,6 Figure 5. Graphical description of the equivalence formula (125) between two Laguerre pseudo-Wronskians.
.(124)⊗ ∼ z −4 ⊗
Equivalence relations for the Jacobi pseudowronskians
Proceeding exactly as in the case of the isotonic oscillator of the previous section, we obtain the following shifted Wronskians
(126) W [d|d]⊗[δ|δ] (x; α, β) ∝ ψ 0 (x; α, β) W 1[d|d]⊗[δ|δ] (x; α 1 , β 1 ) W [d|d]⊗[δ|δ] (x; α, β) ∝ ψ −1 (x; α, β) W [d|d] 1 ⊗[δ|δ] (x; α −1 , β −1 ) W [d|d]⊗[δ|δ] (x; α, β) ∝ φ 0 (x; α, β) W [d|d]⊗1[δ|δ] (x; α −1 , β 1 ) W [d|d]⊗[δ|δ] (x; α, β) ∝ φ −1 (x; α, β) W [d|d]⊗[δ|δ] 1 (x; α 1 , β −1 )
and consequently
W [d|d]⊗[δ|δ] (x; α, β) ∝ d1+k−1 i=0 ψ 0 (x; α i , β i ) δ1+κ−1 j=0 φ 0 (x; α −j+d1+k , β j+d1+k ) (127) ×W [∅|λ]⊗[∅|µ] x; α d1+k−(δ1+κ) , β δ1+κ+d1+k , W [d|d]⊗[δ|δ] (x; α, β) ∝ d1+k−1 i=0 ψ −1 (x; α −i , β −i ) δ1+κ−1 j=0 φ −1 x; α j−(d1+k) , β −j−(d1+k) (128) ×W [λ|∅]⊗[µ|∅] x; α δ1+κ−(d1+k) , β −(d1+k+δ1+κ) and W [d|d]⊗[δ|δ] (x; α, β) ∝ d1+k−1 i=0 ψ −1 (x; α −i , β −i ) δ1+κ−1 j=0 φ 0 x; α −j−(d1+k) , β j−(d1+k) (129) ×W [λ|∅]⊗[∅|µ] x; α −(d1+k+δ1+κ) , β −(d1+k)+δ1+κ .
For the extended potentials, we have
(130) V [d|d]⊗[δ|δ] (x; α, β) = V [λ|∅]⊗[µ|∅] x; α d1+k−(δ1+κ) , β d1+k+δ1+κ + E −(d1+k) (α, β) V [d|d]⊗[δ|δ] (x; α, β) = V [∅|λ]⊗[∅|µ] (x; α δ1+κ−(d1+k) , β −(δ1+κ+d1+k) ) + E d1+k (α, β) V [d|d]⊗[δ|δ] (x; α, β) = V [λ|∅]⊗[∅|µ] (x; α d1+k+δ1+κ , β d1+k−(δ1+κ) ) + E −(d1+k) (α, β) .
Using (42) and (49), expression (129) becomes
W [d|d]⊗[δ|δ] (x; α, β) ∝ (1 − z) (d1+k−δ1−κ) 2 /4−α(d1+k−δ1−κ)/2 · (1 + z) (d1+k+δ1+κ) 2 /4−β(d1+k+δ1+κ)/2 × W [λ|∅]⊗[µ|∅] x; α −(d1+k)+δ1+κ , β −(d1+k+δ1+κ) ,
where z = cos 2x.
Using (77) and the identities (49), (46) and (42), we can write
(131) ψ −(n+1) (x; α, β) = (1 − z) −α (1 + z) −β ψ 0 (x; α, β) P (−α,−β) n (z) φ n (x; α, β) = (1 − z) −α ψ 0 (x; α, β) P (−α,β) n (z) φ −(n+1) (x; α, β) = (1 + z) −β ψ 0 (x; α, β) P (α,−β) n (z)
, which combined with the properties of the derivatives of Jacobi polynomials (92), and noting that m = k + k, r = κ + κ, gives ,
− → P (j) 1≤j≤k , − → Π (j) 1≤j≤κ , − → Π (j) 1≤j≤κ , where (133) P (j) i (z; α, β) = (nj +α+β+1)i−1 2 i−1 P (α+i−1,β+i−1) nj −i+1 (z) P (j) i (z; α, β) = (−2) i−1 (−n j ) i−1 × 1 − z 2 m+r−i P (−α−i+1,−β−i+1) −nj +i−2 (z) Π (j) i (z; α, β) = (−1) i−1 (l j − α) i−1 (1 − z) m+r−i P (−α−i+1,β) lj (z) Π (j) i (z; α, β) = −l j − 1 − β i−1 (1 + z) m+r−i P (α,−β−i+1) −lj −1 (z) , with n j = d j + k − j ≥ 0, l j = δ j + κ − j ≥ 0, −n j − 1 = d j + k − j ≥ 0 and −l j − 1 = δ j + κ − j ≥ 0. If [λ | ∅] is P 0,1 P 1,0 0 0 0 P 0,3 P 1,2 P 2,1 P 3,0 0 (1 − z 2 ) 4P 0,3 (1 − z 2 ) 3P 1,3 (1 − z 2 ) 2P 2,4 (1 − z 2 )P 3,5P4,6 (1 − z) 4 Π 0,3 (1 − z) 3 Π 1,3 (1 − z) 2 Π 2,3 (1 − z)Π 3,3 Π 4,3 (1 + z) 4Π 0,2 (1 + z) 3Π 1,2 (1 + z) 2Π 2,2 (1 + z)Π 3,2Π4,2 ,(141)P [3,2,1|∅]⊗[4,1 2 |∅] (z; α, β) = det P 0,1 P 1,0 0 0 0 0 P 0,3 P 1,2 P 2,1 P 3,0 0 0 P 0,5 P 1,4 P 2,3 P 3,2 P 4,1 P 5,0 (1 − z) 5 Π 0,1 (1 − z) 4 Π 1,1 (1 − z) 3 Π 2,1 (1 − z) 2 Π 3,1 (1 − z)Π 4,1 Π 5,1 (1 − z) 5 Π 0,2 (1 − z) 4 Π 1,2 (1 − z) 3 Π 2,2 (1 − z) 2 Π 3,2 (1 − z)Π 4,2 Π 5,2 (1 − z) 5 Π 0,6 (1 − z) 4 Π 1,6 (1 − z) 3 Π 2,6 (1 − z) 2 Π 3,6 (1 − z)Π 4,6 Π 5,6 .(142)
An explicit calculation applied to equation (136) now gives where K(α, β) = 23040(β − 1)(β + 1)(β + 2) (α − 4)α (α + 2) 4 (α − β − 3) 2 (α − β + 2)(α + β + 1)(α + β + 3)
. Figure 6. Graphical description of the equivalence formula (143) between two Jacobi pseudo-Wronskians.
⊗ ∼ (1 − z) −4 (1 + z) 2 ⊗
Summary and Outlook
In [35] we introduced the concept of Hermite pseudo-Wronskians determinants and proved many equivalence formulas among them. In this paper we have extended the analysis to Laguerre and Jacobi pseudo-Wronskians. Equivalence formulas express different but equivalent manners to perform rational Darboux transformations on the isotonic oscillator and tDPT potential such that the final potential is the same up to an additive shift.
In the Laguerre and Jacobi setting, the situation is richer, as rational Darboux transformations can be state-adding, state-deleting or isospectral, i.e. we can choose seed functions for rational Darboux transformations out of four families, instead of just two families in the Hermite case. Thus, equivalences are obtained by shifting the origin in two independent Maya diagrams. The equivalence relations can be viewed as a generalization of the usual shape invariance property at a multi-step level, combined with the discrete Γ symmetries of the primary potentials (isotonic and tDPT).
As discussed in [35] for the generalized Hermite and Okamoto classes, the existence of these equivalence relations provides optimized determinantal representations for the polynomials associated to rational solutions of (higher order) Painlevé equations of A N -type. This will be the subject of further investigation, aimed at providing a complete classification of the rational solutions of A N -Painlevé equations, together with their optimized (pseudo)-Wronskian representation.
Figure 1 .
1Unique Maya diagrama associated to the tuple (6, 5, 2, 1, −2 − 3).
Figure 2 .
2Left: Maya diagram (6, 5, 2, 1, −2, −3) and its corresponding punctured young diagram. Right: The Maya diagram
Figure 3 .
3Durfee square of the partition λ = (5 2 , 3 2 , 2) and corresponding Durfee symbol 2 2 , 0 | 2 2 , 1 .
Figure 4 .
4Punctured Young diagram associated to the Maya diagram (6, 5, 2, 1, −2, −3), corresponding to a 2 × 4 Durfee rectangle and Durfee symbol [3 2 , 1 2 |1 2 ].
( 65 )
65N m ⊗ L r = (n 1 , ..., n m ) ⊗ (l 1 , ..., l r ) .
; ω, α) is a formal eigenfunction of V [0|∅]⊗[∅|∅] (
( 77 )
77W (uy 1 , ..., uy m | x) = u m W (y 1 , ..., y m | x) W (y 1 , ..., y m | x) = dz dx m(m−1)/2 W (y 1 , ..., y m | z) , equation (76) can be rewritten as (see (52), (62), (58) and (55))
Example 6. 1 .
1As an example of an equivalence relation, consider the case where d | d = [2, 1 | 1], N 3 = (3, 1, −2) and δ | δ = [3 | 2], L 2 = (3, −3). Setting
××P
in the same shape class as d | d = d 1 , ..., d k | d 1 , ..., d k , we can use (132) replacing k and κ by 0 and k and κ by l(λ) = d 1 + k and l(µ) = δ 1 + κ respectively. This givesW [λ|∅]⊗[µ|∅] (x; α, β) ∝ (1 − z) (d1+k−δ1−κ) 2 /4−(δ1+κ)(δ1+κ−1) (1 + z) P [λ|∅]⊗[µ|∅] (z; α, β) .The equivalence relation can then be rewritten asW [d|d]⊗[δ|δ] (x; α, β) ∝ (1 − z) (k−k−κ+κ) 2 /4−(δ1+κ)(δ1+κ−1) (1 + z) [λ|∅]⊗[µ|∅] z; α −(d1+k)+δ1+κ , β −(d1+k+δ1+κ)By making use of (132), the previous formula can be finally expressed asP [d|d]⊗[δ|δ] (z; α, β) ∝ (1 − z) (k+κ)(k+κ−1)−(δ1+κ)(δ1+κ−1) (1 + z) (k+κ)(k+κ−1) (136) ×P [λ|∅]⊗[µ|∅] z; α −(d1+k)+δ1+κ , β −(d1+k+δ1+κ) ,which is the equivalence relation between Jacobi pseudowronskians.Example 7.1. As an example of an equivalence relation, consider the case where d | d = [2, 1 | 1], N 3 = (3, 1, −2) and δ | δ = [3 | 2], L 2 = (3, −3). We then have λ = (3, 2, 1) and µ = 4, 1 2 . -Wronskians in question have the explicit determinantal form P [2,1|1]⊗[3|2] (z; α, β) = det
( 143 )
143K(α, β)P [2,1|1]⊗[3|2] (z; α, β) = (1 − z) −4 (1 + z) 2 × P [3,2,1|∅]⊗[4,1 2 |∅] (z; α 1 , β −5 ) ,
d|d]⊗[δ|δ] (z; α) , where L [d|d]⊗[δ|δ] (z; α) is a polynomial which we shall denote as a Laguerre pseudowronskian. A Laguerre pseudowronskian is defined by the following determinantal expression(115)
An explicit calculation applied to equation (118) now gives L [2,1|1]⊗[3|2] (z; α) = z −4 L [3,2,1|∅]⊗[4,1 2 |∅] (z; α + 1) .(125)
1
90
(α − 4)α (α + 2) 4
W [d|d]⊗[δ|δ] (x; α, β) ∝ (1 − z) (k−k−κ+κ) 2 /4−(k+κ)(k+κ−1) (1 + z) (k−k+κ−κ) k+κ−κ)/2 × P [d|d]⊗[δ|δ] (z; α, β) ,where P [d|d]⊗[δ|δ] (z; α, β) is a polynomial which we shall denote as a Jacobi pseudowronskian. A Jacobi pseudowronskian is defined by the following determinantal expression2 /4−(k+κ)(k+κ−1)
× (1 − z)
α(k−k−κ+κ)/2 (1 + z)
β(k−(132)
P [d|d]⊗[δ|δ] (z; α, β) = det
−
→
P (j)
1≤j≤k
AcknowledgementsThe research of D.G.U. has been supported in part by Spanish MINECO-FEDER Grant MTM2015-65888-C4-3. He also acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554). The research of R.M. was supported in part by NSERC grant RGPIN-228057-2009. They both would like to thank Université de Lorraine for its hospitality during their visit in the summer of 2016 where this work was initiated. The draft was finalized during the thematic trimester on "Orthogonal polynomials and Special functions in approximation theory and mathematical physics", held in Madrid at the Institute of Mathematical Sciences in the Fall of 2017, whose financial support is also acknolwedged.
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Supersymmetry, shape invariance and exactly solvable potentials. R Dutt, A Khare, U P Sukhatme, Am. J. Phys. 6R. Dutt, A. Khare and U. P. Sukhatme, Supersymmetry, shape invariance and exactly solvable potentials, Am. J. Phys. 6 (1988) 163-168.
Derivation of exact spectra of the Schrodinger equation by means of supersymmetry. L Gendenshtein, JETP Lett. 38L. Gendenshtein, Derivation of exact spectra of the Schrodinger equation by means of supersymmetry, JETP Lett. 38 (1983) 356-359.
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Higher transcendental functions. A Erdélyi, W Magnus, F Oberhettinger, F G Tricomi, Mc Graw-HillNew YorkA. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Mc Graw-Hill, New York, 1953.
A treatise on the theory of determinants. T Muir, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera. New York; Madrid, SpainDover152804028049 Madrid, Spain. Departamento de Física Teórica, Universidad Complutense de MadridT. Muir (revised and enlarged by W.H. Metzler), A treatise on the theory of determinants, Dover, New York, 1960. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera 15, 28049 Madrid, Spain. Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain.
Laboratoire de Physique et Chimie Théoriques. UMR CNRS 7019Université de Lorraine-Site de MetzLaboratoire de Physique et Chimie Théoriques, UMR CNRS 7019, Université de Lorraine-Site de Metz,
E-mail address: [email protected], [email protected]. D F Bvd, Arago, France Metz, [email protected], NS, B3H 3J5, CanadaDepartment of Mathematics and Statistics, Dalhousie UniversityBvd D. F. Arago, F-57070, Metz, France. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada. E-mail address: [email protected], [email protected], [email protected]
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arxiv |
Sensor-less Angle and Stiffness Control of Antagonistic PAM Actuator Using Reference Set
May 7, 2021
Takaya Shin
Department of Mechanical and Intelligent Systems Engineering
The University of Electro-Communications
TokyoJapan
Kiminao Kogiso
Department of Mechanical and Intelligent Systems Engineering
The University of Electro-Communications
TokyoJapan
Sensor-less Angle and Stiffness Control of Antagonistic PAM Actuator Using Reference Set
May 7, 2021pneumatic artificial muscleangle-and-stiffness controlsensor-less controlunscented kalman filter
This paper proposes a simultaneous control method for the angle and stiffness of the joint in an antagonistic pneumatic artificial muscle (PAM) actuator system using only pressure measurements, and clarifies the allowable references for the PAM actuator system. To achieve a sensor-less control, the proposed method estimates the joint angle and contraction forces using an unscented Kalman filter that employs a detailed model of the actuator system. Unlike previous control methods, the proposed method does not require any encoder and force sensor to achieve angle and stiffness control of the PAM actuator system. Experimental validations using three control scenarios confirm that the proposed method can control the joint angle and stiffness simultaneously and independently. Moreover, it is shown that a reference admissible set can be used as an indicator to establish reference values by demonstrating that the reference set covers the experimentally obtained trajectories of the angle and stiffness. Ugurlu et al.[18] realized simultaneous control of position and stiffness in an antagonistically driven PAM actuator. These studies experimentally determined a reference in joint stiffness, while they did not clarify an admissible reference set for joint stiffness and angle. Clarification of the admissible information is essential for the practical implementation and systematic design of PAM actuators for applications such as antagonistic force maximization[19]. The use of sensors such as an encoder and a force sensor can provide efficient tracking performance of a PAM actuator. However, a force sensor is relatively expensive and heavy. As a result, a sensor-less approach have been proposed. For example, a static force map was used in [20] to estimate the joint torque of a PAM actuator. A static force map was also used in[21]to control an antagonistic PAM joint actuator system. If the joint angle can be estimated, the cost of designing and producing a PAM actuator could be further reduced. Our previous study of[22]proposed a detailed mathematical model of an antagonistic PAM actuator system. The feature of the study is that a pressure sensor is used to estimate PAM's joint angle and torque. An accurate PAM model is likely to be helpful for estimating joint stiffness as well as angle because both are related to the working pressure.The purpose of this paper, therefore, is to propose the simultaneous sensor-less control method for the joint angle and stiffness of an antagonistic PAM actuator system, and to develop a procedure to obtain a set of admissible references, defined as pairs of stiffnesses and joint angles. To realize sensor-less control using only pressure measurements, this study uses an unscented Kalman filter (UKF) to estimate the joint angle and contraction forces by employing the detailed model in[22]. The contribution of this study is that the proposed sensor-less angle/stiffness control method using a UKF represents a novel approach in robotics. Moreover, it does not require any encoder, which previous relevant studies[4,[9][10][11][12][13][14][15][16][17][18]have relied upon, to achieve simultaneous control. This sensor-less angle/stiffness control approach can realize a low-cost, lightweight actuator that ensures safe contact with humans and environments. Furthermore, this paper also presents experimental results of the proposed control method using a previously developed antagonistic PAM actuator system testbed. The results indicate that the reference admissible set obtained using the model is useful in choosing allowable references and confirm that the reference set covers the experimentally obtained trajectories for the stiffness and angle. Three scenarios are chosen from the characterized reference admissible set, and it is confirmed that the proposed method can independently control the stiffness and angle.The remainder of this paper is organized as follows: Section 2 presents the antagonistic PAM actuator system and its mathematical model and derives the expressions for the joint stiffness. Section 3 proposes the angle/stiffness control system and describes its details. Section 4 presents the experimental results of sensor-less angle/stiffness control. Finally, Section 5 concludes this paper.Antagonistic PAM Actuator System and Its Joint Stiffness ModelThis section briefly introduces a practical antagonistic PAM actuator system and its mathematical model, as presented in our previous study[22], and then mathematically describes the joint stiffness using this model.
Introduction
The McKibben pneumatic artificial muscle (PAM) is an actuator that provides a high forceto-weight ratio, and it is lightweight and has excellent flexibility. These properties enable the construction of a lightweight, highly backdrivable, and direct-drive actuator. Therefore, a PAM is a suitable actuator for devices that often contact human beings, such as assistant robots, nursing care robots, and rehabilitation orthoses [1]. The PAM comprises an internal rubber bladder surrounded by inextensible threads that are braided in a spiral. When compressed air is supplied to a PAM, the diameter of the rubber bladder increases, causing the long axis of the PAM to contract. Though the PAM generates a contraction force when compressed air is supplied, it does not generate any force when the compressed air is released. An antagonistic structure consisting of two PAMs arranged in parallel, with one PAM connected to the other via a joint, is often used to realize rotational motion and perform behaviors similar to those of human muscles [2][3][4][5][6][7][8]. However, the PAM exhibits high nonlinearity owing to its pressure dynamics, valve characteristics, and friction, making its modeling and control quite challenging.
The PAM is a variable-stiffness actuator with adjustable rigidity or hardness. Variable stiffness can enhance contact and collision safety between robots and humans or environments and improve the comfort of wearing rehabilitation orthotics. In the case of rehabilitation, the stiffness prescribed by physiotherapists changes according to the treatment phase; thus, it must be adjusted step-by-step. There have been several studies on a model-based stiffness or compliance control using a PAM actuator [4,[9][10][11][12][13][14][15][16][17][18], and some of the studies are as follows. Cao et al. [16] proposed model-based angle-compliance control to develop a robotic gait rehabilitation device. pressure tank stores compressed air and is connected to the PDCVs and PAMs by air tubes. Air flows adjusted by the PDCVs are used to drive the PAMs and rotates the joint part. The encoder and torque meter respectively measure the joint angle and torque, and the pressure sensors measure the inner pressure of the PAMs. The voltage signals to the two PDCVs (u 1 and u 2 ) are the inputs of the system, the joint angle ψ, inner pressure of the PAMs (P 1 and P 2 ), torque τ , and joint stiffness K P are the outputs of the system. The values of ψ, P 1 , P 2 , and τ are obtained by the sensors, and K P is obtained using the equation derived below. The PC has a 3.2 GHz CPU and 8 GB of RAM; the operating system is Ubuntu 12.04 and the the preemption-patched Xenomai 2.6.2.1 is installed. The sampling period, T stp , was set to 1 ms. The range of the joint angle is ± 25 deg, and the range of the output torque is ± 3.0 Nm. Figure 2. Geometric structure of antagonistic PAM joint.
PAM2
Brief Introduction of PAM Actuator Model
A state-space model that considers the noises in the antagonistic PAM system is given as follows:
x(t) = f σ (x(t), u(t)) + v(t) if x(t) ∈ X σ ,(1a)y(t) = h(x(t)) + w(t),(1b)
where t ∈ R ≥0 is the time; R ≥0 is a nonnegative real number set; u := [u 1 u 2 ] T ∈ U ⊂ R 2 is the input voltages to the two PDCVs, in which U := [0, 10] 2 is a set of allowable inputs determined by the characteristics of the PDCV; the state variable is x := [ψψ P 1 P 2 ] T ∈ R 4 , and the output variable is y := [ψ P 1 P 2 τ K P ] T ∈ R 5 , in which P := [200, 750] is defined as a set of allowable pressures determined by the specification of the PAMs, i.e., P 1 , P 2 ∈ P; v and w are the process noise and observation noise, respectively; f σ : R 4 → R 4 is a nonlinear function with 18 subsystems, and it switches according to if-then rules; X σ := {x ∈ R 4 |Ψ σ (x) > 0} is the state set, where σ ∈ Σ := {1, 2, · · · , 18} is the subsystem's index; Ψ σ (x) is a function derived from the modes in the form of if-then rules; finally, the function h : R 4 → R 5 is an observation equation. Tables 1 and 2 respectively show the model parameters and their identified values for the antagonistic PAM actuator. Details and parameter settings of the PAM actuator model can be found in [22].
Joint Stiffness
The joint stiffness equation describing the antagonistic PAM actuator system is based on the derivation process, discussed in [18]. However, this study derives a different-form of equation for joint stiffness because the mechanical structures of the actuator and PAM force model differ.
Considering the geometric relationship shown in Figure. 2, the lengths of the two PAMs, l 1 and l 2 , are respectively given by
l 1 (t) = L 0 − ∆L(t), l 2 (t) = L 0 + ∆L(t),(2)
where the horizontal displacement of the two PAMs due to rotation is so small that it can be neglected, that is, ∆L(t) ≈ r sin ψ(t); r is the radius of the seesaw; and L 0 is the PAM length when the seesaw is at the horizontal position.
The PAM contraction force has static characteristics that relate inner pressure under the joint fixed [23]. The contraction force F can therefore be described as a function of the PAM inner pressure and length as follows:
F i (l i (t), P i (t)) = v i (l i (t))P i (t) + w i (l i (t)),(3)
where
i ∈ I := {1, 2} and v i (l i (t)) = p v1i l i (t) + p v2i ,(4a)w i (l i (t)) = p w1i l i (t) + p w2i . (4b)
The joint torque τ can be described as follows:
τ (t) = r cos ψ(t) (F 1 (t) − F 2 (t)) .(5)
Joint stiffness is used in this study as an index to describe the joint resistance to an applied moment. As the joint stiffness decreases, the joint becomes more flexible in response to external forces; thus, the interaction between robots and people or environments becomes safer. The joint stiffness, denoted as K P , is defined as the partial differentiation of the joint torque (5) as follows:
K P (t) = − ∂τ (t) ∂ψ(t) = r sin ψ(t)(F 1 (t) − F 2 (t)) − r cos ψ(t) ∂F 1 (t) ∂l 1 (t) ∂l 1 (t) ∂ψ(t) − ∂F 2 (t) ∂ψ(t) ∂l 2 (t) ∂ψ(t) = r sin ψ(t)(F 1 (t) − F 2 (t)) + r 2 cos 2 ψ(t) ∂F 1 (t) ∂l 1 (t) + ∂F 2 (t) ∂l 2 (t) .(6)
Applying (4) to (3), the PAM contraction forces and the partial differentiation of the PAM length can be obtained:
F i (t) = (p v1i l i (t) + p v2i )P i (t) + (p w1i l i (t) + p w2i ), ∂F i (t) ∂l i (t) = p v1i P i (t) + (p v1i l i (t) + p v2i ) ∂P i (t) ∂l i (t) + p w1i .
Note that ∂P i /∂l i is negligible when the volume of the PAM is much smaller than that of the air tank [18]; asa result, (6) can be written as follows:
K P = r sin ψ(t)(F 1 (t) − F 2 (t)) + r 2 cos 2 ψ(t)(p v11 P 1 (t) + p w11 + p v12 P 2 (t) + p w12 ).
By defining α i (t) := p v2i P i (t) + p w2i , K P can be described as
K P = r sin ψ(t)(F 1 (t) − F 2 (t))(7)+ r 2 cos 2 ψ(t) F 1 (t) − α 1 (t) l 1 (t) + F 2 (t) − α 2 (t) l 2 (t) .D 1 , D 2 , D 3 : coefficients of polynomial (m, m 2 , m 3 ) p v1i , p v2i , p w1i , p w2i : coefficient of force for PAMi (-) A 1i , A 2i : orifice area of PDCVi (m 2 ) k 1 , k 2 : polytropic indexes (-) T p : Coulomb friction coefficient of PAM (-) µs : Coulomb friction coefficient of shaft (-) EstimatedD 1 (m) −2.440 × 10 −2 r (m) 0.0365 D 2 (m 2 ) 6.824 × 10 −3 L 0 (m) 0.165 D 3 (m 3 ) −4.254 × 10 −4 M (kg) 0.256 p v11 (-) 7.045 × 10 −3 g (m/s 2 ) 9.80 p v21 (-) −1.017 × 10 −3 P tank (Pa) 0.7100 × 10 6 p w11 (-) −5.568 × 10 2 Pout (Pa) 0.1013 × 10 6 p w21 (-) 72.86 k (-) 1.40 p v12 (-) 6.423 × 10 −3 R (J/kg·K) 287 p v22 (-) −9.184 × 10 −4 T (K) 293 p w12 (-) −197.8 J (kg·m 2 ) 4.263 × 10 −4 p w22 (-) −15.75 ks (N·m/rad) 4.117 × 10 −4 A 11 (m 2 ) 5.184 ×10 −8 cs (N·s) 2.256 × 10 −3 A 12 (m 2 ) 7.776 ×10 −8 k 1 (-) 1.100 T p (-) 4 × 10 8 k 2 (-) 0.4545 µs (-) 0.2
Design of Sensor-less Control System with Admissible References
This section describes the proposed simultaneous sensor-less control method for the angle and stiffness and discusses a procedure for depicting a reference admissible set on an angle-stiffness plane.
Sensor-less Angle/Stiffness Control
The sensor-less angle/stiffness control system proposed in this study consists of a reference generator, PI controllers, and a UKF. A block diagram of the proposed control system is shown in Figure. 3; a similar control system configuration was applied in [18] using force sensors for feedback control. The detailed structure of the control system is described below.
Reference generator
The reference generator provides adequate contraction force reference signals to each of the PAMs (F 1 andF 2 ) using a computed torque command τ c and a given reference joint stiffness K P . The relation amongF 1 ,F 2 , τ c , andK P can be derived algebraically from the obtained stiffness equation. UsingF 1 ,F 2 , τ c , andK P , (5), and (7) can be rewritten as follows: Figure 3. Block diagram of sensor-less angle/stiffness control system.
τ c (t) = r cos ψ(t) F 1 (t) −F 2 (t) ,K P = r sin ψ(t)(F 1 (t) −F 2 (t)) + r 2 cos 2 ψ(t) F 1 (t) − α 1 (t) l 1 (t) +F 2 (t) − α 2 (t) l 2 (t) .
Then,F 1 andF 2 can be obtained from the above two equations as follows:
F 1 (t) = 1 r 2 cos 2 ψ(t) l 1 (t)l 2 (t) l 1 (t) + l 2 (t) K P (t) + r cos ψ(t) l 2 (t) − tan(ψ(t)) τ c −r 2 cos 2 ψ(t) α 1 l 1 (t) + α 2 l 2 (t) ,(8)F 2 (t) =F 1 (t) − τ c r cos ψ(t) .(9)
The error between the reference angleψ and the measured angleψ is then converted into a command torque τ c by the feedback PI controller according to:
x ψ (k + 1) = x ψ (k) + T stp e ψ (k), τ c (k) = G ψ I x ψ (k) + G ψ P e ψ (k),
where e ψ :=ψ −ψ; x ψ ∈ R is the controller state; G ψ P and G ψ I are respectively the proportional and integral gains, which were set to 15 and 10, respectively. Based on the command torque τ c and reference joint stiffnessK P , force signals (F 1 andF 2 ) are generated by the reference generator according to (8) and (9), respectively. The resulting values ofF 1 andF 2 are then compared with the estimated forcesF 1 andF 2 , respectively, and the resulting errors are fed to the PI controllers to generate control input voltages u 1 and u 2 , respectively. These controllers are governed as follows:
x F i (k + 1) = x F i (k) + T stp e F i (k), u i (k) = G F I x F i (k) + G F P e F i (k),
where i ∈ I, e F i :=F i −F i and x F i ∈ R is the controller state. The proportional gain G F P and the integral gain G F I of the force controllers were respectively set to 0.08 and 0.15 using trial and error.
State estimator
This study uses the UKF [24,25] to estimate the state of the antagonistic PAM actuator system based on the pressure sensor information; the rotary encoder is used only to evaluate the resulting estimation. The UKF applies the pressure sensor information and the actuator model from [22] to estimate the joint angle and torque, and the torque is used to compute the stiffness.
A discrete-time representation of the actuator model (1) used in the UKF can be given as follows by the fourth-order Runge-Kutta method:
x(k + 1) = f σ (x(k), u(k)) + v(k) if x(k) ∈ X σ , y(k) = g(x(k)) + w(k), with g(x(k)) = 0 0 1 0 0 0 0 1 x(k),
where v ∈ R 4 and w ∈ R 2 are the process and observation noise of the system, respectively; v and w are zero-man white noises with the covariance matrices Q ∈ R 4 × R 4 and R ∈ R 2 × R 2 , respectively. Table 3 lists the UKF's parameters applied in this study. The estimated PAM forces (F 1 andF 2 ) are obtained by substituting the estimated state variables into (3).
Reference Admissible Set
When considering a tracking control problem, prior information of an allowable reference helps an operator to set reference values. Thus, this study introduces a reference admissible set W ⊂ R 2 , which is defined as follows,
W := [ ψ K P ] T ∈ R 2 | ∀P i (∞) ∈ P, ∀i ∈ I, ∃κ u (P 1 (∞), P 2 (∞)) ∈ U, 0 = f σ (x(∞), κ u (P 1 (∞), P 2 (∞)), y(∞) = h(x(∞))} ,
where x(∞) and y(∞) denote a state and corresponding output of the antagonistic PAM system (1) in a steady state, and κ u is a function that yields control inputs corresponding to pressures P i in a steady state. This reference admissible set can be computed as explained below. In a steady state, the derivative term in the equation of the seesaw motion (Eq. (8) in [22]) becomes zero so that the equation can be written as follows:
ψ(∞) = τ − T f k s .(11)
where T f is the resistant torque due to friction, and it holds T f = T 0 ∧ |T 0 | ≤ T s + T p in a steady Procedure 1
Computation of a reference admissible set W .
Step 0: Identify the model parameters listed in TABLE 1.
Step 1: Obtain the static relationships between the two pressures P 1 , P 2 , and the joint angle ψ using (12a) by giving P 1 , P 2 ∈ P.
Step 2: Calculate two joint stiffness values corresponding to the maximum and minimum stiffness at a specific joint angle using (7) and the static relationships obtained in Step 1.
Step 3: Apply Step 2 to the entire driving angle range of the actuator system.
Step 4: Plot the relationships between the joint angle and stiffness obtained in Steps 2 and 3 on an anglestiffness plane.
Step 5: Apply the same procedure as in Steps 1 to 4 to (12b).
Step 6: The area enclosed by both of the rhombic areas in the angle-stiffness plane obtained in Steps 4 and 5 is the reference admissible set W .
state, in which T 0 is an external torque except for the frictional term, that is, T 0 := τ − k s ψ, and
T s = r p µ s |F 1 (∞) + F 2 (∞) − M g|, T p = µ p 1 (P 1 (∞) − P out ) 2 + 1 (P 2 (∞) − P out ) 2 .
The friction torque T f takes a positive or negative value depending on the direction of the external torque. Considering the maximum static friction, that is, T f = T s + T p , (11) holds as follows:
ψ(∞) = τ − (T s + T p ) k s , (12a) ψ(∞) = τ + (T s + T p ) k s .(12b)
By giving P 1 , P 2 ∈ P and using (12), the static relationship between the PAM inner pressures and the joint angle is illustrated in Figure. 4(a), where the colored solid lines and dashed lines correspond to (12a) and (12b), respectively. As shown in the figure, different combinations of pressures correspond to achieve different joint angles. Moreover, for a given particular joint angle, there exist certain combinations of pressures corresponding to the maximum and minimum stiffness. For example, for a joint angle of 10 deg (the green-dashed line), the point marked in (a) corresponds to the minimum joint stiffness (P 1 = 310 (kPa) and P 2 = 200 (kPa)), and the point marked in (b) corresponds to the maximum joint stiffness (P 1 = 750 (kPa) and P 2 = 450 (kPa)).
Using the points (a) and (b) enables the computation of the corresponding stiffness by (7), and these data are reported on an angle-stiffness plane in Figure. 4 (b). Finally, applying the same procedure to the operating joint angle range of the actuator, the reference admissible stiffnesses are represented by the light yellow-colored area in Figure. 4 (b). In this figure, the solid and dashed lines have the same meanings as those in Figure. 4(a). It should be noted that it is generally necessary to generate a pressure difference between two PAMs to obtain the desired angle; thus the smaller the amplitude of the joint angle is, the larger the range of the joint stiffnesses that can be set is. In summary, the procedure applied in this study to compute a reference admissible set is described in Procedure 1.
Sensor-less Angle/Stiffness Control Experiments
This section demonstrates that the reference admissible set helps to determine a reference joint angle and stiffness, and the proposed control method is verified by conducting control experiments of the antagonistic PAM actuator system.
Applicability of Reference Set
The reference set serves as a guide to set the reference values so that the experimentally obtained steady-state responses of a joint angle and stiffness remain within the reference set. The following voltage signals (u 1 and u 2 ) are input to the PDCVs to cover the entire driving pressure range of the PAM (200 to 750 kPa) and obtain the time response of the joint angle and stiffness.
u 1 (t) = 6 0 ≤ t ≤ 25 4.7 25 < t ≤ 55 , u 2 (t) = 6
0 ≤ t ≤ 10 4.5 10 < t ≤ 25 6 25 < t ≤ 40 4.5 40 < t ≤ 55
.
The time responses of the inner pressures of the two PAMs are shown in Figure. 5(a), the time responses of the joint angle and stiffness are shown in Figure. 5 (b), and the joint angle and stiffness trajectories on the admissible set are shown in Figure. 5 (c). Focusing on the steadystate values of the joint angle and stiffness at 0 to 10, 25, 40, and 55 s, marked by black circles in Figure. 5 (c), all of the steady-state values of the joint angle and stiffness are clearly within the reference set. Thus, this set can be used as an indicator for setting the reference joint angle and stiffness.
Sensor-less Angle/Stiffness Control
Three experimental results for the proposed sensor-less angle/stiffness control are shown in Figures. 6 to 8. In each of the figures, (a) shows the time response of the pressure, (b) shows the time response of the joint angle and stiffness, and (c) shows the trajectory of the joint angle and stiffness on the computed admissible reference set. In (b) and (c), the black dashed line is the reference, the solid red line represents the estimation based on the pressure sensor measurements, and the solid green line represents the actual value, computed by (7) based on measurements using the pressure sensors and the encoder.
A reference of a joint angle was defined in each of the experiments in Figures. 6, 7, and 8 by a sinusoidal function with a period of 10 s and amplitudes set to 15, 10, and 5 deg, respectively. To serve as a reference stiffness, three step-like signals within the computed set in Figure. 4 (b) were chosen with ranges of 7.2 to 6.5, 8 to 5.5, and 9 to 4 Nm/rad, respectively. The proposed control method can be observed to accurately track the angle and stiffness to the references in all three figures. Furthermore, it can be clearly observed in (c) of each figure that the angle/stiffness trajectories are within the reference admissible set. These results confirm that the proposed method is capable of controlling the joint angle and stiffness independently without an encoder.
Conclusion
This study proposed a sensor-less angle/stiffness control method for an antagonistic PAM actuator system and provided a procedure to obtain a set of admissible references, defined as pairs of stiffnesses and joint angles. In order to realize the sensor-less control using only pressure measurements, this study applied a UKF within a detailed model to estimate the joint angle and contraction forces. It was then demonstrated that the reference admissible set obtained using the model helps to choose an allowable reference. Three experiments were conducted using the characterized reference admissible set, and it was confirmed that the proposed method can control the stiffness and angle simultaneously and independently based only on the measured PAM pressures. These experimental results indicate that the antagonistic PAM actuator is applicable to various devices that must be lightweight, low-cost, and interact safely with humans, such as nursing care robots, rehabilitation orthoses, and power-assist orthoses.
In future work, a control method for PAM-actuated devices will be developed considering the presence of disturbances such as reaction torque from the human arm. This could realize safer operation of human-assisting robots by combining stiffness control with a disturbance observer.
Schematic of antagonistic PAM system.
Figure 1 .
1Antagonistic PAM system[22].
relationships between two PAM pressures and joint angles.
Figure 4 .
4Computation of a reference admissible set using the presented procedure.
Time response of joint angle and stiffness.(c) Trajectyories on the computed reference admissible set.
Figure 5 .
5Experimental results for angle and stiffness response.
Table 2 .
2Identified parameters of antagonistic PAM system
parameter
value parameter
value
rp (m)
0.006
A survey on applications of pneumatic artificial muscles. G Andrikopoulos, G Nikolakopoulos, S Manesis, Proceedings of the 19th mediterranean conference on control automation. the 19th mediterranean conference on control automationAndrikopoulos G, Nikolakopoulos G, Manesis S. A survey on applications of pneumatic artificial muscles. In: Proceedings of the 19th mediterranean conference on control automation. 2011. p. 1439- 1446.
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Safe and compliant guidance by a powered knee exoskeleton for robot-assisted rehabilitation of gait. P Beyl, K Knaepen, S Duerinck, M V Damme, B Vanderborght, R Meeusen, D Lefeber, Advanced Robotics. 255Beyl P, Knaepen K, Duerinck S, Damme MV, Vanderborght B, Meeusen R, Lefeber D. Safe and compliant guidance by a powered knee exoskeleton for robot-assisted rehabilitation of gait. Advanced Robotics. 2011;25(5):513-535.
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Novel considerations on static force modeling of pneumatic muscle actuators. G Andrikopoulos, G Nikolakopoulos, S Manesis, ASME Transactions on Mechatronics. 216Andrikopoulos G, Nikolakopoulos G, Manesis S. Novel considerations on static force modeling of pneumatic muscle actuators. in proceedings of the IEEE/ASME Transactions on Mechatronics. 2016; 21(6):2647-2659.
Musculoskeletal model for path generation and modification of an ankle rehabilitation robot. P K Jamwal, S Hussain, Y H Tsoi, S Q Xie, IEEE Transactions on Human-Machine Systems. 505Jamwal PK, Hussain S, Tsoi YH, Xie SQ. Musculoskeletal model for path generation and modification of an ankle rehabilitation robot. IEEE Transactions on Human-Machine Systems. 2020;50(5):373- 383.
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Torque and variable stiffness control for antagonistically driven pneumatic muscle actuators via a stable force feedback controller. B Ugurlu, P Forni, C Doppmann, J Morimoto, Ieee/rsj international conference on intelligent robots and systems. Ugurlu B, Forni P, Doppmann C, Morimoto J. Torque and variable stiffness control for antag- onistically driven pneumatic muscle actuators via a stable force feedback controller. In: Ieee/rsj international conference on intelligent robots and systems. 2015. p. 1633-1639.
Position control considering passive stiffness of rubberless artificial muscle antagonistic drive system. N Saito, Y Yamadaira, T Satoh, Proceedings of the ieee conference on systems. the ieee conference on systemsSaito N, Yamadaira Y, Satoh T. Position control considering passive stiffness of rubberless artificial muscle antagonistic drive system. In: Proceedings of the ieee conference on systems, process and control. 2015. p. 61-66.
Robust variable stiffness control of McKibben type pneumatic artificial muscle arm by using multiple model error compensators. In: 17th international conference on control, automation and systems. Y Yamamoto, N Matsunaga, H Okajima, Yamamoto Y, Matsunaga N, Okajima H. Robust variable stiffness control of McKibben type pneu- matic artificial muscle arm by using multiple model error compensators. In: 17th international con- ference on control, automation and systems. 2017. p. 957-962.
Stability and joint stiffness analysis of legged robot's periodic motion driven by mckibben pneumatic actuator. Y Sugimoto, D Nakanishi, M Nakanishi, K Osuka, Advanced Robotics. 318Sugimoto Y, Nakanishi D, Nakanishi M, Osuka K. Stability and joint stiffness analysis of legged robot's periodic motion driven by mckibben pneumatic actuator. Advanced Robotics. 2017; 31(8):441-452.
MIMO sliding mode controller for gait exoskeleton driven by pneumatic muscles. J Cao, S Q Xie, R Das, IEEE Transactions on Control Systems Technology. 261Cao J, Xie SQ, Das R. MIMO sliding mode controller for gait exoskeleton driven by pneumatic muscles. IEEE Transactions on Control Systems Technology. 2018;26(1):274-281.
Independent stiffness and force control of antagonistic pneumatic artificial muscles joint. X Zhao, H Ma, D Ye, D Zhang, Proceedings of the 2nd international conference on advanced robotics and mechatronics. the 2nd international conference on advanced robotics and mechatronicsZhao X, Ma H, Ye D, Zhang D. Independent stiffness and force control of antagonistic pneumatic artificial muscles joint. In: Proceedings of the 2nd international conference on advanced robotics and mechatronics. 2017. p. 734-739.
Stable control of force, position, and stiffness for robot joints powered via pneumatic muscles. B Ugurlu, P Forni, C Doppmann, E Sariyildiz, J Morimoto, IEEE Transactions on Industrial Informatics. 1512Ugurlu B, Forni P, Doppmann C, Sariyildiz E, Morimoto J. Stable control of force, position, and stiff- ness for robot joints powered via pneumatic muscles. IEEE Transactions on Industrial Informatics. 2019;15(12):6270-6279.
A systematic design strategy for antagonistic joints actuated by artificial muscles. S Dirven, A Mcdaid, ASME Transactions on Mechatronics. 226Dirven S, McDaid A. A systematic design strategy for antagonistic joints actuated by artificial muscles. in proceedings of the IEEE/ASME Transactions on Mechatronics. 2017;22(6):2524-2531.
Modeling, identification and control of an antagonistically actuated joint for telerobotic systems. G Bauer, B Grossmann, T Schlegl, H Kobayashi, Proceedings of the 41st annual conference of the ieee industrial electronics society. the 41st annual conference of the ieee industrial electronics societyBauer G, Grossmann B, Schlegl T, Kobayashi H. Modeling, identification and control of an antago- nistically actuated joint for telerobotic systems. In: Proceedings of the 41st annual conference of the ieee industrial electronics society. 2015. p. 3696-3701.
A sensor-less approach of a torque controller for pneumatic muscle actuator driven joints. M Martens, A Passon, I Boblan, Proceedings of the 3rd international conference on control, automation & robotics. the 3rd international conference on control, automation & roboticsMartens M, Passon A, Boblan I. A sensor-less approach of a torque controller for pneumatic muscle actuator driven joints. In: Proceedings of the 3rd international conference on control, automation & robotics. 2017. p. 477-482.
Detailed dynamic model of antagonistic pam system and its experimental validation: Sensor-less angle and torque control with UKF. submitted to IEEE/ASME Transactions on Mechatronics. T Shin, T Ibayashi, K Kogiso, Shin T, Ibayashi T, Kogiso K. Detailed dynamic model of antagonistic pam system and its ex- perimental validation: Sensor-less angle and torque control with UKF. submitted to IEEE/ASME Transactions on Mechatronics. 2020. [Online]. Available: https://arxiv.org/abs/2009.09229. May 7, 2021
A new approach to modeling hysteresis in pneumatic artificial muscle using the Maxwell-slip model. T Vo-Minh, T Tjahjowidodo, H Ramon, H V Brussel, IEEE/ASME Transactions on Mechatronics. 161Vo-Minh T, Tjahjowidodo T, Ramon H, Brussel HV. A new approach to modeling hysteresis in pneu- matic artificial muscle using the Maxwell-slip model. IEEE/ASME Transactions on Mechatronics. 2011;16(1):177-186.
A new method for the nonlinear transformation of means and covariances in filters and estimators. S Julier, J Uhlmann, H F Durrant-Whyte, IEEE Transactions on Automatic Control. 453Julier S, Uhlmann J, Durrant-Whyte HF. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control. 2000;45(3):477- 482.
A systematization of the Unscented Kalman Filter theory. Hmt Menegaz, J Y Ishihara, G A Borges, A N Vargas, IEEE Transactions on Automatic Control. 6010Menegaz HMT, Ishihara JY, Borges GA, Vargas AN. A systematization of the Unscented Kalman Filter theory. IEEE Transactions on Automatic Control. 2015;60(10):2583-2598.
a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower. May 7, 2021 (a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower).
Time responses of joint angle (upper) and stiffness (lower). Time responses of joint angle (upper) and stiffness (lower).
Trajectories on the reference admissible set. Trajectories on the reference admissible set.
Figure 6. Angle/stiffness control results (amplitude of reference angle: 15deg). Figure 6. Angle/stiffness control results (amplitude of reference angle: 15deg).
a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower. May 7, 2021 (a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower).
Time responses of joint angle (upper) and stiffness (lower). Time responses of joint angle (upper) and stiffness (lower).
Trajectories on the reference admissible set. Trajectories on the reference admissible set.
Figure 7. Angle/stiffness control results (amplitude of reference angle: 10deg). Figure 7. Angle/stiffness control results (amplitude of reference angle: 10deg).
a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower. May 7, 2021 (a) Time responses of actual inner pressure of PAM1 (upper) and PAM2 (lower).
Time responses of joint angle (upper) and stiffness (lower). Time responses of joint angle (upper) and stiffness (lower).
Trajectories on the reference admissible set. Trajectories on the reference admissible set.
Figure 8. Angle/stiffness control results (amplitude of reference angle: 5deg). Figure 8. Angle/stiffness control results (amplitude of reference angle: 5deg).
| {'fraction_non_alphanumeric': 0.057310662947865264, 'fraction_numerical': 0.030948294357434028, 'mean_word_length': 4.289970208540218, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 1, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "This paper proposes a simultaneous control method for the angle and stiffness of the joint in an antagonistic pneumatic artificial muscle (PAM) actuator system using only pressure measurements, and clarifies the allowable references for the PAM actuator system. To achieve a sensor-less control, the proposed method estimates the joint angle and contraction forces using an unscented Kalman filter that employs a detailed model of the actuator system. Unlike previous control methods, the proposed method does not require any encoder and force sensor to achieve angle and stiffness control of the PAM actuator system. Experimental validations using three control scenarios confirm that the proposed method can control the joint angle and stiffness simultaneously and independently. Moreover, it is shown that a reference admissible set can be used as an indicator to establish reference values by demonstrating that the reference set covers the experimentally obtained trajectories of the angle and stiffness. Ugurlu et al.[18] realized simultaneous control of position and stiffness in an antagonistically driven PAM actuator. These studies experimentally determined a reference in joint stiffness, while they did not clarify an admissible reference set for joint stiffness and angle. Clarification of the admissible information is essential for the practical implementation and systematic design of PAM actuators for applications such as antagonistic force maximization[19]. The use of sensors such as an encoder and a force sensor can provide efficient tracking performance of a PAM actuator. However, a force sensor is relatively expensive and heavy. As a result, a sensor-less approach have been proposed. For example, a static force map was used in [20] to estimate the joint torque of a PAM actuator. A static force map was also used in[21]to control an antagonistic PAM joint actuator system. If the joint angle can be estimated, the cost of designing and producing a PAM actuator could be further reduced. Our previous study of[22]proposed a detailed mathematical model of an antagonistic PAM actuator system. The feature of the study is that a pressure sensor is used to estimate PAM's joint angle and torque. An accurate PAM model is likely to be helpful for estimating joint stiffness as well as angle because both are related to the working pressure.The purpose of this paper, therefore, is to propose the simultaneous sensor-less control method for the joint angle and stiffness of an antagonistic PAM actuator system, and to develop a procedure to obtain a set of admissible references, defined as pairs of stiffnesses and joint angles. To realize sensor-less control using only pressure measurements, this study uses an unscented Kalman filter (UKF) to estimate the joint angle and contraction forces by employing the detailed model in[22]. The contribution of this study is that the proposed sensor-less angle/stiffness control method using a UKF represents a novel approach in robotics. Moreover, it does not require any encoder, which previous relevant studies[4,[9][10][11][12][13][14][15][16][17][18]have relied upon, to achieve simultaneous control. This sensor-less angle/stiffness control approach can realize a low-cost, lightweight actuator that ensures safe contact with humans and environments. Furthermore, this paper also presents experimental results of the proposed control method using a previously developed antagonistic PAM actuator system testbed. The results indicate that the reference admissible set obtained using the model is useful in choosing allowable references and confirm that the reference set covers the experimentally obtained trajectories for the stiffness and angle. Three scenarios are chosen from the characterized reference admissible set, and it is confirmed that the proposed method can independently control the stiffness and angle.The remainder of this paper is organized as follows: Section 2 presents the antagonistic PAM actuator system and its mathematical model and derives the expressions for the joint stiffness. Section 3 proposes the angle/stiffness control system and describes its details. Section 4 presents the experimental results of sensor-less angle/stiffness control. Finally, Section 5 concludes this paper.Antagonistic PAM Actuator System and Its Joint Stiffness ModelThis section briefly introduces a practical antagonistic PAM actuator system and its mathematical model, as presented in our previous study[22], and then mathematically describes the joint stiffness using this model.", 'arxivid': '2105.02462', 'author': ['Takaya Shin \nDepartment of Mechanical and Intelligent Systems Engineering\nThe University of Electro-Communications\nTokyoJapan\n', 'Kiminao Kogiso \nDepartment of Mechanical and Intelligent Systems Engineering\nThe University of Electro-Communications\nTokyoJapan\n'], 'authoraffiliation': ['Department of Mechanical and Intelligent Systems Engineering\nThe University of Electro-Communications\nTokyoJapan', 'Department of Mechanical and Intelligent Systems Engineering\nThe University of Electro-Communications\nTokyoJapan'], 'corpusid': 233864515, 'doi': '10.1080/01691864.2022.2046502', 'github_urls': [], 'n_tokens_mistral': 11524, 'n_tokens_neox': 9821, 'n_words': 6140, 'pdfsha': 'eea8fd0b15785e5118f2ac54dcf6a61f3cd12d12', 'pdfurls': ['https://arxiv.org/pdf/2105.02462v1.pdf'], 'title': ['Sensor-less Angle and Stiffness Control of Antagonistic PAM Actuator Using Reference Set', 'Sensor-less Angle and Stiffness Control of Antagonistic PAM Actuator Using Reference Set'], 'venue': []} |
arxiv |
SPLIT INVOLUTION COUPLED TO ACTUAL GAUGE SYMMETRY
August 1994
I A Batalin
) P.N.Lebedev Physical Institute
117924MoscowRussia † (
S L Lyakhovich
Tomsk State University
634050TomskRussia
I V Tyutin
) P.N.Lebedev Physical Institute
117924MoscowRussia † (
SPLIT INVOLUTION COUPLED TO ACTUAL GAUGE SYMMETRY
August 1994arXiv:hep-th/9408090v1 16 Aug 1994 P. N. Lebedev Physical Institute Preprint FIAN/TD/8-94 I. E. Tamm Theory Department † E-mail address: [email protected],[email protected] 2
The split involution quantization scheme, proposed previously for pure secondclass constraints only, is extended to cover the case of the presence of irreducible first-class constraints. The explicit Sp(2)-symmetry property of the formalism is retained to hold. The constraint algebra generating equations are formulated and the Unitarizing Hamiltonian is constructed. Physical operators and states are defined in the sense of the new equivalence criterion that is a natural counterpart to the Dirac's weak equality concept as applied to the first-class quantities. † E-mail address: [email protected],[email protected] 1 It goes without saying that arbitrary second-class constraints (whose Fermionic component number is divisible by 4) and Hamiltonian can be transformed locally to the polarized basis subjected to eqs. (1.1), (1.2). What is not so evident that there exists a valuable set of relativistic dynamical systems such that the Dirac's hamiltonianization procedure, being applied directly to the original relativistic Lagrangian, just produces the polarized constraint basis.
Introduction
In previous paper [1] of the present authors the split involution formalism has been proposed for canonical quantization of dynamical systems with pure second-class constraints.
The formalism implies no extra variables to be introduced with the purpose of converting original constraints into effective ones of the first-class.
On the other hand, the total set of original second-class constraints is supposed to be polarized by splitting into two interchangeable subsets, T a µ , a = 1, 2, to satisfy the so-called "split involution" relations
(ıh) −1 [T {a µ , T b} ν ] = U {aρ µν T b} ρ ,(1.H complete = H + ε ab (ıh) −2 [Q b , [Q a , B]] (1.5)
where B is a "gauge-fixing" Bosonic operator. Being the physical quantities defined in an appropriate way, they do not depend on a particular choice of a "gauge" operator B. This independence is quite a nontrivial feature of the split involution scheme, because pure second-class constraints do not generate an actual gauge symmetry. The algebra generating equations (1.3) as well as the Hamiltonian (1.5) possess the Sp(2)covariant form which is characteristic to the formalism developed in Refs. [2,3] to quantize gauge-invariant theories in a ghost-antighost symmetric fashion. However, the number of ghosts (and antighosts) introduced in the formalism [2,3] is twice as compared with the corresponding number in the split involution theory. Moreover, the ghost numbers of the generating operators (Q 1 , Q 2 ) are (+1, +1) in the split involution scheme, while in the ghost-antighost symmetric theory these numbers are (+1, −1).
In the present paper we generalize the split involution formalism by including original first-class constraints into it. When doing this we retain the explicit Sp(2)-symmetry property of the method to hold.
We assign ghost canonical pairs to constraints of both the classes and require the ghost number operators G ′ and G ′′ of the first and second class, respectively, to be conserved separately. In accordance with this requirement, a pair of the ghost number values, denoted by gh ′ and gh ′′ , is assigned to each admitted operator of the theory.
Then we formulate the extended version of the gauge algebra generating equations. We require the generating operator of the first-class constraint algebra to be nilpotent modulo contributions similar to the gauge-fixing term in r.h.s. of (1.5). Thereby we define the equivalence criterion that is a natural counterpart to the Dirac's weak equality concept as applied to the first-class quantities. The conservation property of the first-class generating operator is also formulated in the sense of the new equivalence criterion proposed.
The constraint algebra generating equations are shown to possess the group of automorphisms that enables one to make the first (resp. second)-class constraints be a set of momenta (resp. a set of canonical pairs). The maximal group of automorphisms is given by semidirect product of three groups that are: ghost-dependent canonical transformations,cnumerical symplectomorphisms, and exact shifts initiated by the new equivalence criterion.
In terms of the constraint algebra generating operators we construct the complete Unitarizing Hamiltonian of the theory. We modify the definition (1.5) by adding the genuine gauge-fixing term required by the presence of original first-class constraints.
Finally, we formulate the definitions of physical operators and physical states in the sense of the new equivalence criterion.
Notations and Conventions. As usual, ε(A) represents the Grassmann parity of the quantity A.
If n = n + + n − is the total number of some superobjects, then n + (n − ) indicates the number of Bosons (Fermions) among them.
The standard supercommutator of the operators A, B is defined by the formula
[A, B] ≡ AB − BA(−1) ε(A)ε(B) . (1.6)
By ε ab we denote the constant Sp(2)-invariant tensor
ε ab = 0 1 −1 0 , (1.7)
while its inverse is denoted as ε ab :
ε ab ε bc = δ a c . (1.8)
We also use the standard notations for symmetrization
A {ab} ≡ A ab + A ba ,(1.9)
and antisymmetrization
A [ab] ≡ A ab − A ba . (1.10)
Greek indices of first(second)-class constraints are taken from the first(second),half of the Greek alphabet, α, . . . , λ(µ, . . . , ω). The same convention holds for related quantities.
By gh ′ (A) (gh ′′ (A)) we denote the first(second)-class ghost number of quantity A. The other notation is clear from the context.
Constraint Algebra
Let (q i , p i ), i = 1, . . . , n = n + + n − ,
(2.1) ε(q i ) = ε(p i ) ≡ ε i , gh ′ (q i ) = −gh ′ (p i ) = 0, gh ′′ (q i ) = −gh ′′ (p i ) = 0, (2.2) (q i ) † = q i , (p i ) † = p i (−1) ε i , (2.3)
be a set of the original phase variable operators whose equal-time nonzero super-commutation relations are
(ıh) −1 [q i .p j ] = δ i j .(2.T α = T α (p, q), ε(T α ) ≡ε α , (2.6) α = 1, . . . , m ′ = m ′ + + m ′ − , (2.7) T a µ = T a µ (p, q), ε(T a µ ) ≡ ε µ , (2.8) a = 1, 2; µ = 1, . . . , m ′′ = m ′′ + + m ′′ − , (2.9) m ′′ − = 2k, m ± ≡ m ′ ± + m ′′ ± < n ± ,(2.10)
to satisfy the following involution relations
(ıh) −1 [T {a µ , T b} ν ] = U {aρ µν T b} ρ , (2.11) (ıh) −1 [T a µ , T α ] =Ũ aβ µα T β + U ν µα T a ν , (2.12) (ıh) −1 [T α , T β ] =Ũ γ αβ T γ + 1 2 ε ab W µν αβ (T b ν δ ρ µ − T b µ δ ρ ν (−1) εµεν − ıhU bρ νµ )T a ρ , (2.13) (ıh) −1 [H, T a µ ] = V ν µ T a ν , (2.14) (ıh) −1 [H, T α ] =Ṽ β α T β + 1 2 ε ab W µν α (T b ν δ ρ µ − T b µ δ ρ ν (−1) εµεν − ıhU bρ νµ )T a ρ . (2.15)
where the structure coefficient operators are some functions of the original phase variables (2.1), and the following antisymmetry properties are supposed to hold:
U aρ µν = −U aρ νµ (−1) εµεν ,Ũ γ αβ = −Ũ γ βα (−1)ε αεβ , (2.16) W µν αβ = −W νµ αβ (−1) εµεν = −W µν βα (−1)ε αεβ , (2.17) W µν α = −W νµ α (−1) εµεν . (2.18)
Let us also require the supercommutators
∆ ab µν ≡ (ıh) −1 [T [a µ , T b] ν ],(2.19)
enumerated by collective indices (a, µ), (b, ν), to form an invertible operator-valued matrix:
∆ ⇒ ∃ ∆ −1 (2.20)
This condition implies the constraints (2.8) to be of the second-class. In their own turn the involution relations (2.12),(2.13) imply the constraints (2.6) to be of the first class. Let us require for these constraints to commute with the operators (2.1) to give an operator-valued supermatrix whose invertible Bose-Bose and Fermi-Fermi blocks are of the maximal possible sizes m ′ + × m ′ + and m ′ − × m ′ − , respectively, which requirement is an operator version to the irreducibility condition.
As for the second-class constraints, they are irreducible due to the condition (2.20). The irreducibility property determines the quantum rules of "dividing by constraints", i.e. characteristic form of the most general operator solution to the basic set of homogeneous linear equations
Z µ T a µ +Z aα T α = 0, (2.21) Z {aµ T b} µ +Z abα T α = 0,Z [ab]α = 0, (2.22) Z abµ T c µ + cycle(a, b, c) = 0, Z [ab]µ = 0, (2.23) Z µν 1 2 ε ab (T b ν δ ρ µ − T b µ δ ρ ν (−1) εµεν − ıhU bρ νµ )T a ρ +Z α T α = 0, Z µν = −Z νµ (−1) εµεν ,(2.24)
which are obtained by applying the Jacobi identity to all the involution relations (2.11) -(2.15). In the Appendix these equations will be considered in more details. It would be just desirable to avoid imposing further restrictions on the constraint algebra (2.11) -(2.15). Unfortunately, we are unable to prevent such restrictions for the present. Therefore we have to impose the following extra condition on the structure coefficientsŨ aβ µα entering the cross-sector relation (2.12) that involves constraints of the both classes:
(ıh) −1 [T {a µ ,Ũ b}β να ] − (ıh) −1 [T {a ν ,Ũ b}β µα ](−1) εµεν −Ũ {aγ µαŨ b}β νγ (−1) εν (εα+εγ ) + +Ũ {aγ ναŨ b}β µγ (−1) εµ(εα+εγ+εν ) − U {aρ µνŨ b}β ρα =Ũ {aργ µνα (T b} ρ δ β γ − ıhŨ b}β ργ )(−1) εµεα , (2.25)
where the new structure coefficient operatorsŨ aγρ µνα are supposed to possess the antisymmetry propertyŨ
aγρ µνα = −Ũ aγρ νµα (−1) εµεν +ενεα+εαεµ . (2.26)
Let us consider the status of the restriction (2.25). By applying the Jacoby identity to the constraint algebra (2.11) -(2.15) and then making use of the above mentioned quantum "rules of dividing by constraints", one can show the operatorsŨ aβ µα to satisfy the relation that differs from the one (2.25) by the extra contributioñ
U abγλ µα (T λ δ β γ − T γ δ β λ (−1)ε γελ − ıhŨ β γλ ) (2.27)
to r.h.s. Thus, in fact, the condition (2.25) is equivalent to the requirement for the contribution (2.27) to vanish. On the other hand, one can consider the cross-sector relation (2.12) to be the covariant constancy property of the constraints, being the structure coefficientsŨ aβ µα , U aρ µν treated to serve as the connection components. From this viewpoint, l.h.s. of (2.25) is nothing else but the corresponding curvature components. The condition (2.25), being treated classically, requires for the curvature to vanish on the second-class constraint surface, while the algebra (2.11) -(2.13) itself implies a weaker condition to be satisfied that the curvature components should vanish on the surface of all the constraints. Now let us comment in brief the most characteristic features of the involution relations (2.11) -(2.15).
First of all we observe that the split involution relations (2.11), (2.14) retain their original form [1] specific to the pure second-class constraint case. Further, the cross-sector constraint supercommutators are actually restricted in two respects: the operatorsŨ aβ µν are subordinated to the relations (2.25), and the operators U ν µα do not possess their own Sp(2)-indices. Finally, let us turn to the first-class constraint involution relations (2.13), (2.15). Being these relations treated classically, second-class constraints are allowed to contribute only quadratically, which assertion is a consequence of the Jacoby identity. Such quadratic contributions are just represented by the second and third terms in r.h.s. of (2.13), (2.15), and these terms possess the specific structure characterized by the antisymmetry property of the coefficients ε ab W νµ in their indices a, b and µ, ν. However, ath = 0 second-class constraints appear to be allowed quantum-mechanically to contribute to (2.13), (2.15) linearly with the effective coefficients − 1 2 ıhε ab W νµ U bρ µν . These linear quantum contributions, represented by the fourth terms in r.h.s. of (2.13), (2.15), are necessary in order to provide the operator compatibility of the formal constraint algebra.
Given the initial operators (2.5), (2.6), (2.8), the involution relations (2.11) -(2.15) serve to determine the lowest structure coefficient operators
U aρ µν ,Ũ aβ µα , U ν µα ,Ũ γ αβ , W µν αβ , V ν µ ,Ṽ β α , W µν α (2.28)
up to a natural arbitrariness. By making use of the Jacoby identity together with the irreducibility property of the constraints, one can derive the necessary compatibility conditions to the involution relations (2.11) -(2.15). These new conditions, including the one (2.25), contain new structure coefficient operators to be determined at this level. On the other hand, these relations reduce to an admissible extent the arbitrariness in the preceding-level structure coefficient operators. Continuing this procedure, one generates, step by step, an infinite gauge algebra initiated by the operators (2.5), (2.6), (2.8).
In the next Section we formulate the generating equations that give automatically an infinite set of structure relations of the constraint gauge algebra.
Constraint algebra generating equations
As a next step let us introduce the ghost phase variable operators. We assign a ghost canonical pair to each first-class constraint operator:
T α → (C ′α ,P ′ α ), α = 1, . . . , m ′ , (3.1) ε(C ′α ) = ε(P ′ α ) =ε α + 1, (3.2) gh ′ (C ′α ) = −gh ′ (P ′ α ) = 1, gh ′′ (C ′α ) = −gh ′′ (P ′ α ) = 0. (3.3) (C ′α ) † = C ′α , (P ′ α ) † = −P ′ α (−1)ε α . (3.4)
In the same way we assign a ghost canonical pair to each (a = 1, 2)-pair of the second-class constraint operators (2.8),
T a µ → (C ′′µ ,P ′′ µ ), µ = 1, . . . , m ′′ , (3.5) ε(C ′′µ ) = ε(P ′′ µ ) = ε µ + 1, (3.6) gh ′ (C ′′µ ) = −gh ′ (P ′′ µ ) = 0, gh ′′ (C ′′µ ) = −gh ′′ (P ′′ µ ) = 1 (3.7) (C ′′µ ) † = C ′′µ , (P ′′ µ ) † = −P ′′ µ (−1) εµ . (3.8)
The equal-time nonzero supercommutators of the ghost operators introduced are
(ıh) −1 [C ′α ,P ′ β ] = δ α β , (ıh) −1 [C ′′µ ,P ′′ ν ] = δ µ ν . (3.9)
Further, introduce the generating operators Ω a (q, p, C ′ ,P ′ , C ′′ ,P ′′ ), ε(Ω a ) = 1, (3.10)
gh ′ (Ω a ) = 0, gh ′′ (Ω a ) = 1, (3.11) Ω(q, p, C ′ ,P ′ , C ′′ ,P ′′ ), ε(Ω) = 1, (3.12) gh ′ (Ω) = 1, gh ′′ (Ω) = 0, (3.13) K(q, p, C ′ ,P ′ , C ′′ ,P ′′ ), ε(K) = 0, (3.14) gh ′ (K) = 2, gh ′′ (K) = −2, (3.15) H(q, p, C ′ ,P ′ , C ′′ ,P ′′ ), ε(H) = 0, (3.16) gh ′ (H) = 0, gh ′′ (H) = 0, (3.17) Λ(q, p, C ′ ,P ′ , C ′′ ,P ′′ ), ε(Λ) = 1, (3.18) gh ′ (Λ) = 1, gh ′′ (Λ) = −2, (3.19)
and subordinate them to the following generating equations:
[Ω a , Ω b ] = 0, (Ω a ) † = Ω a ,(3.20)
[Ω a , Ω] = 0, (Ω) † = Ω, (3.21) [
Ω, Ω] = ε ab (ıh) −1 [Ω b , [Ω a , K]], (K) † = K, (3.22) [Ω a , H] = 0, (H) † = H,(3.23)
[
Ω, H] = ε ab (ıh) −1 [Ω b , [Ω a , Λ]], (Λ) † = Λ. (3.24)
Let us seek for a solution to these equations in the form of CP-ordered series expansion in ghost powers: The algebra generating equations (3.20) -(3.24) admit the following group of automorphisms:
Ω a = C ′′µ T a µ + 1 2 (−1) εν C ′′ν C ′′µ U aρ µνP ′′ ρ (−1) ερ + (−1)ε α C ′α C ′′µŨ aβ µαP ′ β (−1)ε β + . . . , (3.25) Ω = C ′α T α + 1 2 (−1)ε β C ′β C ′αŨ γ αβP ′ γ (−1)ε γ + (−1)ε α C ′α C ′′µ U ν µαP ′′ ν (−1) εν + . . . , (3.26) K = 1 2 (−1)ε β C ′β C ′α W µν αβP ′′ νP ′′ µ (−1) εν + . . . , (3.27) H = H − C ′′µ V ν µP ′′ ν (−1) εν − C ′αṼ β αP ′ β (−1)ε β + . . . , (3.28) Λ = 1 2 C ′α W µν αP ′′ νP ′′ µ (−1) εν + . . . .A = A 1 · A 2 · A 3 (3.30) where A 1 is the standard unitary group Ω a → U −1 Ω a U,(3.1 2 (−1) (εν +εµεα) C ′α C ′′ν C ′′µŨ aβρ µναP ′′ ρP ′ β (−1) ερ . Ω → U −1 ΩU, K → U −1 KU, (3.32) H → U −1 HU, Λ → U −1 ΛU, (3.33) A 2 = GL(2, R) is the group of c-numerical nondegenerate linear transformations Ω a → S a b Ω b , Ω → Ω, H → H, (3.34) K → λ −1 K, Λ → λ −1 Λ, λ ≡ det(S a b ),(3.
35)
A 3 is the group of exact shifts
Ω a → Ω a , (3.36) Ω → Ω + ε ab (ıh) −2 [Ω b , [Ω a , Ξ]],(3.
Unitarizing Hamiltonian
Introduce now the following new canonical variable operators which are the antighosts:
(P ′α ,C ′ α ), α = 1, . . . , m ′ (4.1) ε(P ′α ) = ε(C ′ α ) =ε α + 1, (4.2) gh ′ (P ′α ) = −gh ′ (C ′ α ) = 1, gh ′′ (P ′α ) = −gh ′′ (C ′ α ) = 0, (4.3) (P ′α ) † = P ′α , (C ′ α ) † = −C ′ α (−1)ε α , (4.4) (P ′′µ ,C ′′ µ ), µ = 1, . . . , m ′′ , (4.5) ε(P ′′µ ) = ε(C ′′ µ ) = ε µ + 1, (4.6) gh ′ (P ′′µ ) = −gh ′ (C ′′ µ ) = 0, gh ′′ (P ′′ µ ) = −gh ′′ (C ′′ µ ) = 1, (4.7) (P ′′µ ) † = P ′′µ , (C ′′ µ ) † = −C ′′ µ (−1) εµ ,(4.8)
and dynamically-active Lagrange multipliers:
(λ α , π α ), α = 1, . . . , m ′ , (4.9) ε(λ α ) = ε(π α ) =ε α , (4.10)
gh ′ (λ α ) = −gh ′ (π α ) = 0, gh ′′ (λ α ) = −gh ′′ (π α ) = 0, (4.11) (λ α ) † = λ α (−1)ε α , (π α ) † = π α ,(4.12)
(λ a µ ), a = 1, 2, µ = 1, . . . , m ′′ , (4.13) ε(λ a µ ) = ε µ , (4.14)
gh ′ (λ a µ ) = gh ′′ (λ a µ ) = 0. (4.15) (λ a µ ) † = λ a µ .
(4.16)
The equal-time nonzero supercommutators of the new operators introduced are
(ıh) −1 [P ′α ,C ′ β ] = δ α β , (ıh) −1 [P ′′µ ,C ′′ ν ] = δ µ ν , (4.17) (ıh) −1 [λ α , π β ] = δ α β , (ıh) −1 [λ a µ , λ b ν ] = ε ab d µν , (4.18)
where a constant matrix d µν is supposed to be invertible and possesses the following symmetry properties
d νµ = d µν (−1) εµεν , d * νµ = d µν . (4.19)
Let us extend the generating operators (3.10), (3.12) by including the phase variable operators (4.1), (4.5), (4.9), (4.14) via the formulae
Q = Ω + P ′α π α ,
(4.20)
Q a = Ω a + P ′′µ λ a µ , a = 1, 2, (4.26)
(F ) † = −F, (B) † = −B. (4.27)
The gauge-fixing operators F and B may depend on the total set of phase variables of the extended phase space. In the simplest case these gauge operators can be chosen in the form
F = λ αP ′ α + (χ α + C ′′µ V να µP ′′ ν (−1) εν +εα )C ′ α + . . . , (ıh) −1 [T a µ , χ α ] = V να µ T a ν , (4.28) B =P ′′ µC ′′ ν d νµ , d µν d νρ = δ ρ µ . (4.29)
Further, let us introduce the ghost number operators
G ′ = 1 2 (C ′αP ′ α (−1)ε α −P ′ α C ′α ) + 1 2 (P ′αC ′ α (−1)ε α −C ′ α P ′α ), (4.30) G ′′ = 1 2 (C ′′µP ′′ µ (−1) εµ −P ′′ µ C ′′µ ) + 1 2 (P ′′µC ′′ µ (−1) εµ −C ′′ µ P ′′µ ). (4.31)
Then we have The total ghost number operator is naturally defined as
(ıh) −1 [G ′ , A] = gh ′ (A)A, G ′ |Φ = gh ′ (|Φ )|Φ ,G = G ′ + G ′′ . (4.34)
As a next step, let us define the physical operators and physical states. An operator O is called the physical one iff Let Γ be the total set of phase variable operators of the extended phase space, and let Γ(t) satisfies the Heisenberg equations governed by the Unitarizing Hamiltonian (4.24). Then the physical matrix elements Φ|O(Γ(t))|Φ 1 do not depend on a particular choice of gauge-fixing operators F and B.
gh ′ (O) = gh ′′ (O) = 0, (4.35) [Q a , O] = 0, [Q, O] = ε ab (ıh) −1 [Q b , [Q a , E]].
Further Generalization and Geometric Interpretation
It has been implied in the above considerations that the second-class constraints themselves retain their algebraic properties to be the same as they are in the pure second-class case. In particular, no first-class constraints enter the split involution relations (1.1), (1.2) actually.
In this section we intend to generalize the set of constraint algebra generating equations in order to make it possible for the first-class constraints contribute explicitly to the modified split involution relations.
The main idea can be explained as follows. Let the original second-class constraints T a µ are allowed to contain the first-class admixture. Let us suppose that the corresponding admixture to the generating operators Ω a is representable in the form and Ω is the first-class generating operator to be determined selfconsistently. It is quite natural to require for the pure second-class generating operators
Ω a − (ıh) −1 [A a , Ω] (5.3)
to satisfy the equations similar to the above-given ones (3.20), (3.21): Ω], Ω] = 0, (5.5)
[Ω a − (ıh) −1 [A a , Ω], Ω b − (ıh) −1 [A b , Ω]] = 0, (5.4) [Ω a − (ıh) −1 [A a ,
Besides, we have to subordinate the first-class generating operator Ω to the equation similar the one (3.22):
[Ω, Ω] = ε ab (ıh) −1 [Ω b − (ıh) −1 [A b , Ω], [Ω a − (ıh) −1 [A a , Ω], K]].
(5.6)
In the same way we formulate the equations similar to the above-given ones (3.23), (3.24):
[
Ω b − (ıh) −1 [A b
, Ω], H] = 0, (5.7)
[
Ω, H] = ε ab (ıh) −1 [Ω b − (ıh) −1 [A b , Ω], [Ω a − (ıh) −1 [A a , Ω], Λ]]. (5.8)
The generating operators Ω a , Ω, K, H, Λ are searched in the form of the corresponding series expansions (3.25) -(3.29), whereas the new operators A a are expanded in ghost powers as
A a = C ′′µX aβ µP ′ β (−1)ε β + 1 2 (−1)ε α C ′α C ′′µX aβγ µαP ′ γP ′ β (−1)ε γ + + 1 2 (−1) εν C ′′ν C ′′µX aαρ µνP ′′ ρP ′ α (−1) ερ + . . . . (5.9)
Here we refrain from considering in details the explicit form of a constraint algebra generated by eqs. (5.4) -(5.8) to the lowest order in ghosts. The only comment to be given here concerns the modified cross-sector relations. Instead of (2.12) we have:
(ıh) −1 [T a µ , T α ] =Ũ aβ µα T β + U ν µα T a ν + + 1 2 (δ γ αX aβ µ − δ β αX aγ µ (−1)ε βεγ + ıhX aγβ µα )((ıh) −1 [T β , T γ ] −Ũ δ βγ T δ ).
(5.10) Being treated at the classical level, these relations determine the quantitiesX aβ µ to serve as coefficients of a linear dependence between the cross-sector supercommutators {T a µ , T α } and the pure first-class-sector ones {T α , T β }.
The following Existence Theorem apparently holds for the generating equations (5.4) -(5.8): if these equations are satisfied to the lowest order in ghosts and, besides, the equations (5.4) themselves are satisfied to the C ′ (C ′′ ) 2P ′ -order, then there exists a formal solution for the generating operators Ω a , Ω, K, H, Λ to all orders in ghosts.
It is an interesting circumstance that l.h.s. of eqs.
∇ aH = 0, ∇ aΛ = 0, (5.17) where ∇ a ≡ ∂ a − (ıh) −1 adĀ a , ∂ a ≡ ∂ ∂ξ a (5.18)
stand for the covariant ξ-derivative components, so that for arbitrary E(ξ) we have
∇ a E = ∂ a E − (ıh) −1 [Ā a , E]. (5.19)
We suppose the connectionĀ a to be flat:
∂ [aĀb] − (ıh) −1 [Ā a ,Ā b ] =
Conclusion
So, we have extended the split involution formalism to cover the case of the presence of irreducible first-class constraints. Thereby the miraculous supersymmetry yielded by the split involution relations is coupled to the actual gauge symmetry initiated by the original first-class constraints.
The most characteristic feature of the formalism proposed is the appearance of the new equivalence criterion explicitly-quadratic in second-class constraints that is a natural counterpart to the Dirac's weak equality concept as applied to the first-class quantities.
It is quite evident from this viewpoint that all the double-supercommutator contributions in (3.22), (3.24), (3.39), (4.24), (4.36) as well as the quadratic operator in r.h.s. of (4.38) are of the same origin.
All the main results are extendable in a straightforward way to cover the case of finitestage reducibility of the first and second-class constraints included.
Appendix. Quantum Rules of Dividing by Constraints
In this Appendix we represent the general solution to the equation (2.21) -(2.24).
First of all, let us introduce the following remarkable operators :
W n ≡ n−1 m=0 Ω m (C,P)| C→−ıh ∂r ∂P (−1) ε(P ) , (A.1) W a n ≡ n−1 m=0 Ω a m (C,P)| C→−ıh ∂r ∂P (−1) ε(P ) , (A.2)
where C ≡ (C ′ , C ′′ ),P ≡ (P ′ ,P ′′ ) is a condensed notation for ghost operators, and
Ω m ∼ (C) m+1 (P) m , Ω a m ∼ (C) m+1 (P) m (A.3)
are the corresponding homogeneous monomials entering the ghost power series expansions to the generating operators Ω, Ω a ,
Ω = ∞ m=0 Ω m , Ω a = ∞ m=0 Ω a m . (A.4)
In particular we have
Ω 0 = C ′α T α , (A.5) Ω 1 = 1 2 (−1)ε β C ′β C ′αŨ γ αβP ′ γ (−1)ε γ + (−1)ε α C ′α C ′′µ U ν µαP ′′ ν (−1) εν , (A.6) Ω 2 = 1 12 (−1) (ε β +εαεγ ) C ′γ C ′β C ′αŨ δλ αβγP ′ λP ′ δ (−1)ε λ + + 1 2 (−1) (εα+ε β εµ) C ′β C ′α C ′′µŨ γν µαβP ′′ νP ′ γ (−1) εν + 1 4 (−1) (εν +εαεµ) C ′α C ′′ν C ′′µ U ρσ µναP ′′ σP ′′ ρ (−1) εσ , (A.7) Ω a 0 = C ′′µ T a µ , (A.8) Ω a 1 = 1 2 (−1) εν C ′′ν C ′′µ U aρ µνP ′′ ρ (−1) ερ + (−1)ε α C ′α C ′′µŨ aβ µαP ′ β (−1)ε β , (A.9) Ω a 2 = 1 12 (−1) (εν +εµερ) C ′′ρ C ′′ν C ′′µ U aνστ µνρP ′′ τP ′′ σ (−1) ετ + + 1 2 (−1) (εν +εαεµ) C ′α C ′′ν C ′′µŨ aβρ µναP ′′ ρP ′ β (−1) ερ + 1 4 (−1) (εα+ε β εµ) C ′β C ′α C ′′µŨ aγδ µαβP ′ δP ′ γ (−1)ε δ .
(A.10)
As applied from the right to arbitrary CP-ordered polinomial of the highest power n in ghost momentaP, the operators (A.1), (A.2) possess the important formal properties W n W a n−1 + W a n W n−1 = 0, W {a n W where
Z 1 ≡ Z µP ′′ µ (−1) εµ ,Z a 1 ≡Z aαP ′ α (−1)ε α . (A.16)
It can be shown that the general solution for Z 1 , Z a 1 is
Z 1 = (E 3 W b 3 +Ẽ b2W
their superscripts a, b. Besides, the Hamiltonian H is supposed to satisfy the relations 1(ıh) −1 [H, T a µ ] = V ν µ T a ν .(1.2) One generates the "gauge" algebra, initiated by the relations (1.1), (1.2) by solving the equations [Q a , Q b ] = 0, [Q a , H] = 0, (1.3) for the Fermions Q a and Boson H in the form of a series expansion in ghost powers Q a = C µ T a µ + . . . , H = H + . . . . (1.4) Then one constructs the complete Unitarizing Hamiltonian of the theory in the following Sp(2)-symmetric form
, we have chosen the CP-ordering only for the sake of convenience of the general analysis. Depending on a particular representation of constraints some other choice of ghost ordering may appear to be more relevant, such as the Weyl-or Wick-ordering in field-theory case.Byinserting the expansions (3.25) -(3.29) into the left generating equations in (3.20) -(3.24), one obtains to the second order in ghosts just the constraint involution relations (2.11) -(2.15), whereas to higher orders in ghosts we obtain all the higher structure relations 2 of the gauge algebra initiated by the given operators (2.5), (2.6), (2.8). On the other hand, the right equations in (3.20) -(3.24) determine the properties of the constraints and higher structure coefficients with respect to the Hermitian conjugation. Thus the equations (3.20) -(3.24) describe the gauge algebra generating mechanism comprehensively. The following Existence Theorem holds for the proposed generating equations (3.20) -(3.24): if the constraint involution relations (2.11) -(2.5) are satisfied together with the conditions (2.20), (2.25) and the ones requiring for the first-class constraints T α to be irreducible in the above formulated sense, then there also exist all the higher structure coefficients in the expansions (3.25) -(3.29) and, thus, there exists a formal solution of the algebra generating equations. Besides, it can be shown that all the Hermiticity properties in (3.20) -(3.24) can also be satisfied by the solution obtained.
31) 2
2In particular, the relation (2.25) is generated by the left equation(3.20) to the C ′ (C ′′ ) 2P ′ -order, whereas the corresponding contribution to Ω a is of the form (see also eq. (A.10) of the Appendix)
37 )K
37→ K + 2(ıh) −1 [Ω, Ξ] + ε ab (ıh) −3 [[Ω b , Ξ], [Ω a , Ξ]] + (ıh) −1 [Ω a , X a ], (3.38)H → H + (ıh) −1 [Ω, Ψ] + ε ab (ıh) −1 [Ω b , [Ω a , Φ]], [Ω a , Ψ] = 0, (3.39) Λ → Λ + (ıh) −1 [Ξ, H] + (ıh) −1 [Ω, Φ] + 1 2 (ıh) −1 [K, Ψ]+ +(ıh) −2 [Ξ, [Ω, Ψ]] + ε ab (ıh) −3 [[Ξ, Ω b ], [Ω a , Φ]] + (ıh) −1 [Ω a , Y a ].(3.40) Under the premises of the Existence Theorem the group of automorphisms (3.30) is the maximal possible one and, thus, describes the natural arbitrariness of a solution to the algebra generating equations (3.20) -(3.24) comprehensively. The exact shift transformations (3.37) -(3.40) enable one to make the new operatorsK andΛ vanish. Then one can apply the ghost-dependent canonical transformations (3.31) -(3.33) to make the generating operators take the Abelian form Ω a abelian = C ′′µ t µ , Ω abelian = C ′α t α , (3.41) [t {a µ , t b} ν ] = 0, [t a µ , t α ] = 0, [t a , t β ] = 0, (3.42) [H abelian , t a µ ] = 0, [H abelian , t α ] = 0. (3.43)
Q) = 1, gh ′ (Q) = 1, gh ′′ (Q) = 0, (4.22) ε(Q a ) = 1, gh ′ (Q a ) = 0, gh ′′ (Q a ) = 1.(4.23)The extended operators Q, Q a satisfy the same equations (3.20) -(3.24) as their minimalsector counterparts Ω, Ω a do.The complete Unitarizing Hamiltonian of the theory readsH complete = H + (ıh) −1 [Q, F ] + ε ab (ıh) −2 [Q b , [Q a ,B]], [Q a , F ] = 0, (4.24) where ε(F ) = 1, gh ′ (F ) = −1, gh ′′ (F ) = 0, (4.25) ε(B) = 0, gh ′ (B) = 0, gh ′′ (B) = −2.
) −1 [G ′′ , A] = gh ′′ (A)A, G ′′ |Φ = gh ′′ (|Φ )|Φ . (4.33)
, the Hamiltonian (4.24) is a physical operator just in the sense of this definition. A state |Φ is called the physical one iff gh ′ (|Φ ) = gh ′′ (|Φ ) = 0, (4.37)Q a |Φ = 0, Q|Φ = ε ab (ıh) −1 Q b Q a |E .(4.38) The physical matrix elements Φ|O|Φ 1 depend neither on the arbitrariness (3.30) in determining the generating operators Ω a , Ω, K, H, Λ, nor on the arbitrariness of r.h.s. of eqs. (4.36), (4.38).
ghost-dependent operators A a are introduced, ε(A a ) = 0, gh ′ (A a ) = −1, gh ′′ (A a ) = 1, (A a ) † = A a , (5.2)
(5. 4 )
4-(5.8) possess the structure of a natural first-class counterpart of the well-known Dirac's bracket, being eq.the "Lagrange multiplier" operators A a to an admissible extent. It is just the form that generalizes in the most natural way the lowest-order relations (5.10). Now, let us consider an interesting geometric extension to the set of eqs.(5.4) -(5.8). First of all, introduce a pair of real Bosonic ghost parameters ξ a , ε(ξ a ) = 0, gh ′ (ξ a ) = 1, gh ′′ (ξ a ) = −1, ξ * a = ξ a . (5.12) Next, let us define the ξ-,Ω] = ε ab (ıh) −1 [∇ bΩ , [∇ aΩ ,K]], (5.14) ∇ a ∇ bΩ = 0, ∇ aK = 0, (5.15) [Ω,H] = ε ab (ıh) −1 [∇ bΩ , [∇ aΩ ,Λ]], (5.16)
|
immediately from eqs. (5.14) -(5.17), (5.20) that∇ a [Ω,Ω] ≡ 2[∇ aΩ ,Ω] ξ=0 ≡ Ω, ∂ aΩ | ξ=0 ≡ Ω a ,(5.24)A a | ξ=0 ≡ A a ,K| ξ=0 ≡ K, (5.25) H| ξ=0 ≡ H,Λ| ξ=0 ≡ Λ, the flatness condition (5.20) there exists a ξ-dependent canonical transformation that results for the connection componentsĀ a in their vanishing. Thus one returns naturally to the case considered in previous Sections.Further, let us extend the operatorΩ via the formulāQ ≡Ω + P ′α π α + P ′′µ λ a µ ξ a . (5.27)Then the ξ-dependent Unitarizing Hamiltonian readsH complete ≡H + (ıh) −1 [Q,F ] + ε ab (ıh) −2 [∇ bQ , [∇ aQ ,B]], (5.28)where ξ-dependent gauge-fixing operatorsF ,B should satisfy the conditions∇ aF = 0 [F , ∇ aQ ] = 0, ∇ aB = 0. (5.29)Finally, let us define the ξ-dependent physical operators and states. An operatorŌ is called the physical one iff:gh ′ (Ō) = gh ′′ (Ō) = 0, (5.30) [Q,Ō] = ε ab (ıh) −1 [∇ bQ , [∇ aQ ,Ē]], (5.31) ∇ aŌ = 0, ∇ aĒ = 0. (5.32) A state |Φ is called the physical one iff gh ′ (|Φ ) = gh ′′ (|Φ ) = 0, (5.33) Q|Φ = ε ab (ıh) −1 (∇ bQ )(∇ aQ )|Ē , (5.34) ∇ a |Φ = 0, ∇ a |Ē = 0, (5.35) where the covariant derivative operators ∇ a are applied to arbitrary state | . . . via the formula: ∇ a | . . . ≡ (∂ a − (ıh) −1 A a )| . . . . By construction, the physical matrix elements are ξ-independent: Φ |Ō|Φ 1 = Φ|O|Φ 1 (5.36) where unbared operators and states in r.h.s. coincide with the corresponding bared ones taken at ξ = 0. (Of course, one can choose another fixed point ξ 0 instead of ξ = 0.) The physical matrix elements (5.36) are also independent of a particular choice of operators F , B, E and states |E entering eqs. (5.28) -(5.35) taken at ξ = 0.
| C→−ıh ∂r ∂P (−1) ε(P ) , (A.12) where ∆ ≡ ε ab (ıh) −1 [Ω b , [Ω a , K]] = ∞ n=2 ∆ n , ∆ n ∼ (C) n (P) n−2 , (A.13) ∆ 2 =W 2 W b 2 W a 1 ε ab ,W 2 = (ıh) −1 K 2 | C→−ıh ∂r ∂P (−1) ε(P ) , (A.14)and K m is the (P) m -order in the expansion (3.27). Now, let us consider the equation (2.21) to represent it in the form
where E 3 ∼ (P ′′ ) 3 ,Ẽ 2 ∼P ′′P ′ ,Ẽ a ∼ (P ′ ) 2 (A.19)are arbitrary operators. By eliminating ghost operators from the representations (A.17), (A.18), one decodes the general solution for Z µ ,Z aα in the formZ µ =Ẽ αν Π µ να + (E τ σρ Π ξνa ρστ + ıhẼ aβα W ξν αβ )Π µb νξ ε ba ,(A.20) Next, let us consider the equation (2.22) to represent it in the form where E 2 ∼ (P ′′ ) 2 ,Ẽ a ∼P ′′P ′ ,Ẽ ab 2 ∼ (P ′ ) 2 (A.32) are arbitrary operators. By decoding the representations (A.30), (A.31) one obtains the general solution for Z aµ , Z abα : Z aµ = (E ρν δ a c + Further, let us represent the equation (2.23) in the formThe general solution is given by the formula2 )W a
2 ε ab +Ẽ 2 W 2 ,
(A.17)
Z a
1 =Ẽ 2 W a
2 +Ẽ a
2 W 2 ,
(A.18)
2
Z aα =Ẽ βµΠαa
µβ +Ẽ aγβΠα
βγ ,
(A.21)
where
Π ν
µα ≡ −T α δ ν
µ (−1)ε αεµ − ıhU ν
µα ,
(A.22)
Π στ a
µνρ ≡ [(δ σ
µΠ
τ a
νρ − δ τ
µΠ
σa
νρ (−1) εσετ )(−1) εµεν + cycle(ρ, µ, ν)] + (ıh) 2 U aστ
µνρ ,
(A.23)
Π ρa
µν ≡
1
2
(T a
µ δ ρ
ν − T a
ν δ ρ
µ (−1) εµεν ) − ıhU aρ
µν ,
(A.24)
Π ρa
µν ≡ T a
µ δ ρ
ν − T a
ν δ ρ
µ (−1) εµεν − ıhU aρ
µν ,
(A.25)
Π βa
µα ≡ T a
µ δ β
α − ıhŨ aβ
µα ,
(A.26)
Π γ
αβ ≡ T α δ γ
β − T β δ γ
α (−1)ε αεβ − ıhŨ γ
αβ .
(A.27)
Z
{a
1 W
b}
1 +Z ab
1 W 1 = 0,Z [ab] = 0,
(A.28)
where
Z a
1 ≡ Z aµP ′′
µ (−1) εµ ,Z ab
1 ≡Z abαP ′
α (−1)ε α .
(A.29)
The general solution for Z a
1 ,Z ab
1 is given by the formulae
Z a
1 = E 2 W a
2 +Ẽ a
2 W 2 +
1
2Ẽ
ac
2W 2 W b
2 ε bc ,
(A.30)
Z ab
1 =Ẽ
{a
2 W
b}
2 +Ẽ ab
2 W 2 ,Ẽ
[ab]
2
= 0,
(A.31)
2
1
2
ıhẼ abβα W ρν
αβ ε cb )Π µc
νρ +Ẽ aαν Π µ
να ,
(A.33)
Z abα =Ẽ {aβµΠ αb}
µβ +Ẽ abγβΠα
βγ .
(A.34)
Z ab
1 W c
1 + cycle(a, b, c) = 0, Z
[ab]
1
= 0,
(A.35)
where
Z ab
1 ≡ Z abµP ′′
µ (−1) εµ .
(A.36)
Z ab
1 = E
{a
2 W
b}
2
(A.37)
where
E a
2
∼ (P ′′ ) 2
(A.38)
are arbitrary operators.
It follows from (A.37) that the general solution for Z abµ is of the formFinally, let us turn to the equation (2.24) as represented in the formThe general solution is given by the formulaeare arbitrary operators. By decoding the representations (A.42), (A.43), one obtains the general solution for Z µν , Z α in the formΠ βρa µνα ≡ (Π ρa µν δ β α +Π βa µα δ ρ ν (−1) εν (εα+ε β ) − −Π βa να δ ρ µ (−1) εµ(εα+ε β +εν ) )(−1) εµεα + (ıh) 2Ũ aβρ µνα ,
. I A Batalin, S L Lyakhovich, I V Tyutin, Mod.Phys. Lett. 71931I.A.Batalin, S.L.Lyakhovich, I.V.Tyutin, Mod.Phys. Lett. A7 (1992) 1931.
. I A Batalin, P M Lavrov, I V Tyutin, J.Math. Phys. 316I.A.Batalin, P.M.Lavrov, I.V.Tyutin, J.Math. Phys. 31 (1990) 6
. I A Batalin, P M Lavrov, I V Tyutin, J.Math. Phys. 312708I.A.Batalin, P.M.Lavrov, I.V.Tyutin, J.Math. Phys. 31 (1990) 2708
| {'fraction_non_alphanumeric': 0.12929552012591447, 'fraction_numerical': 0.04109708822757877, 'mean_word_length': 3.071437047585143, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 27, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "The split involution quantization scheme, proposed previously for pure secondclass constraints only, is extended to cover the case of the presence of irreducible first-class constraints. The explicit Sp(2)-symmetry property of the formalism is retained to hold. The constraint algebra generating equations are formulated and the Unitarizing Hamiltonian is constructed. Physical operators and states are defined in the sense of the new equivalence criterion that is a natural counterpart to the Dirac's weak equality concept as applied to the first-class quantities. † E-mail address: [email protected],[email protected] 1 It goes without saying that arbitrary second-class constraints (whose Fermionic component number is divisible by 4) and Hamiltonian can be transformed locally to the polarized basis subjected to eqs. (1.1), (1.2). What is not so evident that there exists a valuable set of relativistic dynamical systems such that the Dirac's hamiltonianization procedure, being applied directly to the original relativistic Lagrangian, just produces the polarized constraint basis.", 'arxivid': 'hep-th/9408090', 'author': ['I A Batalin \n) P.N.Lebedev Physical Institute\n117924MoscowRussia † (\n', 'S L Lyakhovich \nTomsk State University\n634050TomskRussia\n', 'I V Tyutin \n) P.N.Lebedev Physical Institute\n117924MoscowRussia † (\n'], 'authoraffiliation': [') P.N.Lebedev Physical Institute\n117924MoscowRussia † (', 'Tomsk State University\n634050TomskRussia', ') P.N.Lebedev Physical Institute\n117924MoscowRussia † ('], 'corpusid': 14744943, 'doi': '10.1142/s0217751x95000930', 'github_urls': [], 'n_tokens_mistral': 15071, 'n_tokens_neox': 13195, 'n_words': 6588, 'pdfsha': 'b7aa51de0530953d73c5c29b7b96e4c3720d172f', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9408090v1.pdf'], 'title': ['SPLIT INVOLUTION COUPLED TO ACTUAL GAUGE SYMMETRY', 'SPLIT INVOLUTION COUPLED TO ACTUAL GAUGE SYMMETRY'], 'venue': []} |
arxiv |
1986. 2011. 1995. 2019. 2021. 2022. 2021. 2013. 2014. 2020. 2021. 2010. 2019. 2016. 1992. 2018. 2011. 2019. 2016. 2020
K P Hertzog
R K Honeycutt
J W Robertson
S Kafka
R K Honeycutt
J W Robertson
G W Turner
T Kato
T Kato
T Kato
T Kato
F.-J Hambsch
T Kato
VSOLJ Variable Star Bull
106101986. 2011. 1995. 2019. 2021. 2022. 2021. 2013. 2014. 2020. 2021. 2010. 2019. 2016. 1992. 2018. 2011. 2019. 2016. 202010.1086/1758411 V476 Cyg (Nova Cyg 1920) is currently a dwarf nova -first such an object in the period gap?
V476 Cyg (Nova Cyg 1920) is a bright, fast nova reaching a photographic magnitude of 2.0. Using the Zwicky Transient Facility (ZTF) public database, I found that this nova is currently a dwarf nova with a cycle length of ∼24 d. Compared to other classical novae currently in dwarf nova-type states, outbursts of V476 Cyg are rapidly rising and short with durations of a few days. Based on the AAVSO observations, this nova was probably already in the dwarf nova-type phase in 2016, 96 years after the nova eruption. I found a possible orbital period of 0.1018002(6) d using the ZTF data, which would place the object in the period gap. This supposed short orbital period appears to explain the features and faint absolute magnitudes of the observed dwarf nova outbursts. If this period is confirmed, V476 Cyg is a classical nova with the shortest orbital period with distinct dwarf nova outbursts and in which a nova eruption was recorded in the modern era. I also compared with the outburst properties with V446 Her (Nova Her 1960), which currently shows SS Cyg-type outbursts. The transition to the dwarf nova-phase in V476 Cyg occurred much earlier (∼100 yr) than what has been supposed (∼1000 yr) for classical novae below the period gap. V476 Cyg would not only provide an ideal laboratory of the behavior of an irradiated accretion disk in which tidal instability is expected to work, but also an ideal laboratory of the effect of a massive white dwarf on dwarf nova outbursts.
Introduction
V476 Cyg was discovered as a bright Galactic nova by . The visual peak magnitude by was 2.2 on 1920 August 23. Photographic observations showed relatively slow rise from 7.0 mag to the peak (2.0 mag) which took place in 7 d . This nova was a fast nova with t 2 =16.5 d or t 2 =6 d . The nova was classified as a D-class one with a weak dust dip in the light curve by . Leslie Peltier described that the post-nova could be sometimes glimpsed in his autobiographical Starlight Nights ) 1 . He indeed followed this nova since its maximum in 1920 (when he was at an age of 20) and saw it around 16 mag or slightly below it between 1961 and 1972 according to the AAVSO International Database 2 . reported a low-resolution spectrum at V =17.33. reported a magnitude of V ∼18.7 and suspected that either there were significant flux errors in or the nova was variable on a short time-scale. discussed that wiggles in the spectrum of V476 Cyg might be a signature of a dwarf nova.
V476 Cyg as a dwarf nova
Using the Zwicky Transient Facility (ZTF: Masci et al. 2019) public data 3 , I found that this object is currently a dwarf nova (T. Kato on 2020 March 5, vsnet-chat 8457 4 ) [for general information of cataclysmic variables and dwarf novae, see e.g. ]. Here I report on this object using the ZTF data up to the end of 2021. The light curve is shown in figure 1. I must note, however, neither all outbursts were detected nor all outbursts were detected at their peaks by ZTF. The quiescent brightness varied relatively strongly, and the object was bright in 2019 August-September (BJD 2458700-2458760). During this bright phase, there was a outburst starting on BJD 2458718 (2019 August 22; figure 2), which had a shoulder [or referred to as an embedded precursor by ] and the peak brightness (r ∼16.5 and g ∼16.6) was brighter than the other outbursts. There were equally bright outbursts in 2018 September, peaking on BJD 2458386 (first panel of figure 1) and in 2020 June, peaking on BJD 2459010 (third panel of 1). The former outburst apparently had a shoulder as in the 2019 August one.
The color was g − r=+0.1 at outburst peak, while it was redder (g − r ∼ +0.5) in quiescence. This was probably due to the presence of a close, physically unrelated, companion star Gaia EDR3 2089624258068065152 with a Gaia magnitude G=19.07 (Gaia Collaboration et al. 2021).
There were also CCD observations in the AAVSO International Database between 2016 and 2019. Short outbursts can be recognized by comparing with the ZTF data (figure 3). The AAVSO observations were unfiltered CCD ones obtained by HKEB (K. Hills, UK). At least a few outbursts recorded by ZTF were also recorded by AAVSO CCD observations. There was a bright outburst on 2017 February 14 (unfiltered CCD magnitude 15.3). The dwarf nova state should have started before 2016. Although there were some CCD observations with significant variations in 2007, the data were not sufficient to identify them as dwarf nova outbursts.
3 Nova in the period gap?
The mean outburst interval derived from the best recorded part (BJD 2459300-2459510) was 24.1(1.4) d. The durations of most these outbursts were short (2-3 d), suggesting that V476 Cyg has a relatively short orbital period. There was time-resolved photometry by ZTF on one night (figure 4). This run suggests a period of ∼0.10 d. With the help of this candidate period, I analyzed the ZTF data in quiescence (figure 5) using phase dispersion minimization (PDM: analysis after removing the global trends by locally-weighted polynomial regression (LOWESS: . The error was estimated by methods of and Kato et al. (2010). Although this period looks like the orbital period, it might come from the physically unrelated companion star and needs to be confirmed by further observations. If this period is the orbital period of V476 Cyg, this object is in the period gap. This period appears to be consistent with the outburst behavior mostly showing short outbursts. The brightest dwarf nova outburst in the ZTF data had M V =+5.5 using A V =0.7 ) and the Gaia parallax (Gaia Collaboration et al. 2021). This is relatively faint among dwarf novae (see e.g. ) and appears to be consistent with a short orbital period. Kato (2022) showed that WZ Sge stars start showing superhumps at M V =+5.4. The present result of V476 Cyg is comparable to this value.
The borders of the period gap is somewhat variable depending the authors. I use the range 0.090-0.13 d based on equation (17) in Knigge et al. (2011). Well-established novae in the period gap include IM Nor (recurrent nova) , V Per , QU Vul , V597 Pup , and some more borderline or less established cases. None of these object shows dwarf nova-type outbursts.
Novae showing dwarf nova outbursts after the eruption
There are well-established classical novae which currently show dwarf nova-type outbursts. I summarized them in table 1. There have been many references for GK Per and I only listed a few of them. It might be worth noting that already reported dwarf nova-like outbursts for V446 Her before the nova eruption. This phenomenon may have been similar to the reported case in Mróz et al. (2016). V446 Her currently shows dwarf nova-type outburst typical for an SS Cyg star with long and short outbursts in the ZTF data (in contrast to the statement in : figure 6, see also the light curve in 1994 in Honeycutt et al. 1995). V392 Per was also a dwarf nova ) before the 2018 nova eruption (e.g. Munari et al. 2020), whose most recent slowly rising outburst was observed in 2016 February-April [detected by the VSOLJ observer Mitsutaka Hiraga and the AAVSO observer Carey Chiselbrook (cvnet-outburst message on 2016 February 28)]. BC Cas is currently in IW And-type state (Kato and Kojiguchi 2020) [see e.g. Simonsen (2011); Kato (2019) for IW And-type stars]. A recent light curve for X Ser is also present in Kimura et al. (2018). A discussion on V1017 Sgr can be also found in . The most recent dwarf nova-type outburst occurred in 2007. A outburst of V2109 Oph was detected by the Gaia satellite as Gaia21dza 5 . This outburst was a slowly rising one and the orbital period was suspected to be long (T. Kato, vsnet-alert 26178 6 ).
BK Lyn was suggested to be the counterpart of the Chinese "guest star" in 101 A.D. (Hertzog 1986; Although WY Sge (nova eruption in 1783) was once considered to be a dwarf nova , Naylor et al. (1992); pointed out that it is just an ordinary old nova. Modern ZTF observations do not show any sign of dwarf nova outbursts contrary to the expectation by the hibernation scenario . See also for modern observations of WY Sge.
Although listed old novae showing low-amplitude outburst (they referred to as stunted outbursts) in V841 Oph, V728 Sco, V1059 Sgr, V849 Oph, V363 Sgr, HS Pup and V2572 Sgr, the dwarf nova-type nature is not apparent from their light curves for most objects. I included only V728 Sco, which showed recurrent outbursts similar to dwarf novae by more than 1 mag, in the table.
Three of the objects in the table are long-period systems (orbital periods more than 1 d) and have evolved secondaries. It is understandable that a considerable fraction of this table is composed of such objects, since these objects have a large accretion disk and it is unstable to thermal instability even under mass-transfer rates typical for ordinary (short-period) novalike systems (Kim et al. 1992). Kim et al. (1992) predicted that outbursts in such systems are inside-out-type, approximately symmetric ones, which agree with the observations of these post-novae. This is apparently not the case for V476 Cyg. The outbursts in V476 Cyg rise rapidly and they are apparently outside-in outbursts. If the suspected orbital period is correct, this behavior is consistent with the short-period nature. Among the table, the only confirmed short-period object is BK Lyn, whose dwarf nova-type phase was likely a transient phenomenon and the suspected nova eruption occurred nearly 2000 years ago. In this regard, the case of V476 Cyg with a long-lasting dwarf nova-type phase would be unique. estimated that novae below the period gap show dwarf nova outbursts after the nova eruption when the white dwarf cools sufficiently after ∼1000 years. If the suspected orbital period of V476 Cyg is correct, this object can be an exception. The case of V476 Cyg may reflect the rapid evolution (with t 2 =16.5 d or 6 d) of the nova eruption and rapid subsequent cooling.
Shoulder or failed superoutburst?
The nature of the shoulder in the dwarf nova-type outburst is not still clear. considered it to be similar to precursor outbursts in SU UMa-type superoutbursts. Kato and Hambsch (2021) suggested that it originates when the disk reaches the tidal truncation radius. In the special case of V363 Lyr, the outburst accompanied by a shoulder was 0.3-0.4 mag brighter than other outbursts and showed periodic modulations with a period slightly longer than the orbital period (Kato 2021). The nature of this variation is still unclear (Kato 2021). Compared to the light curves by , such as that of SS Cyg, the case of V476 Cyg looks more similar to that of V363 Lyr. It would be worth performing time-resolved photometry during such outbursts to detect possible periodic signals as in V363 Lyr. Other shorter outbursts in V476 Cyg have variable peak brightness, although it was more constant at 17.0 mag in the late 2020 to the 2021 seasons (later part of the third panel and the fourth panel of figure 1). Considering the suspected orbital period in section 3, these outbursts with shoulders may be analogous to SU UMa-type superoutbursts [a "failed superoutburst" is also known in SU UMa stars, during which tidal instability is not sufficiently strong to produce a full superoutburst ], although the durations were much shorter. Determination of the orbital period by radial-velocity studies is desired. Considering that many dwarf novae in the period gap have been identified as SU UMa stars [e.g. V1006 Cyg (Kato et al. 2016); MN Dra nature of dwarf nova outbursts in V476 Cyg. Since the object appears to be still declining from the 1920 nova eruption, this object would provide an ideal laboratory of the behavior of an irradiated accretion disk in which tidal instability is expected to work. This object would also be an ideal laboratory of the effect of a massive white dwarf on dwarf nova outbursts.
Figure 1 :Figure 2 :
12and showed a transient ER UMa-type phase in 2011-2012 (Patterson et al. 2013; Kato et al. 2013, ZTF light curve of V476 Cyg. ZTF light curve of V476 Cyg. Enlargement of the bright state in 2019 August-September. The tick represents a shoulder in the bright outburst. The symbols are the same as in figure 1. 2014). This state had apparently started as early as in 2005. The object is currently in novalike state and no dwarf nova outbursts are observed.
Figure 3 :Figure 4 :
34Combined ZTF and AAVSO light curve of V476 Cyg. The AAVSO observations were unfiltered CCD ones obtained by HKEB (K. Hills, UK). At least a few outbursts recorded by ZTF were also recorded by AAVSO CCD observations. Short-term variation recorded in r-band time-resolved photometry by ZTF.
Figure 5 :Figure 6 :
56); NY Ser], superoutbursts may be expected in V476 Cyg. Continued observations and timely time-resolved photometry would clarify the PDM analysis of V476 Cyg using the ZTF data in quiescence. (Upper): PDM analysis. A sharp signal at 0.1018002(6) d was detected. (Lower): mean profile. ZTF light curve of V446 Her. The current behavior is indistinguishable from that of ordinary SS Cygtype dwarf novae. The BJD scale is the same as in figure 1 (V476 Cyg). One can easily see the shortness of outbursts in V476 Cyg.
Table 1 :
1Novae showing dwarf nova outbursts after the eruptionObject
Eruption Orbital Period (d) References
V728 Sco
1862
-
Vogt et al. (2018)
V606 Aql
1899
-
Kato and Kojiguchi (2021)
GK Per
1901
1.996803
Crampton et al. (1986); Bianchini et al. (1986); Šimon (2002)
X Ser
1903
1.478
Thorstensen and Taylor (2000); Šimon (2018)
V476 Cyg
1920
0.101800?
this paper
BC Cas
1929
-
Kato and Kojiguchi (2020)
V446 Her
1960
0.2070
Thorstensen and Taylor (2000); Honeycutt et al. (2011)
V2109 Oph
1969
-
vsnet-alert 26178
V1017 Sgr
1919
5.78629
Sekiguchi (1992); Webbink et al. (1987)
BK Lyn
101?
0.07498
Ringwald et al. (1996b); Patterson et al. (2013)
I read this story in the book translated to Japanese. 2 <http://www.aavso.org/data-download>.3 The ZTF data can be obtained from IRSA <https://irsa.ipac.caltech.edu/Missions/ztf.html> using the interface <https://irsa.ipac.caltech.edu/docs/program_interface/ztf_api.html> or using a wrapper of the above IRSA API <https://github.com/MickaelRigault/ztfquery>. 4 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-chat/8457>.
<http://gsaweb.ast.cam.ac.uk/alerts/alert/Gaia21dza/>. 6 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26178>.
AcknowledgementsThis work was supported by JSPS KAKENHI Grant Number 21K03616. The author is grateful to the ZTF team for making their data available to the public. We are grateful to Naoto Kojiguchi for helping downloading the ZTF data. This research has made use of the AAVSO Variable Star IndexReferencesWe provide two forms of the references section (for ADS and as published) so that the references can be easily incorporated into ADS.
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arxiv |
Integrating out heavy scalars with modified EOMs: matching computation of dimension-eight SMEFT coefficients
5 Feb 2023
Upalaparna Banerjee
Joydeep Chakrabortty [email protected]
Indian Institute of Technology Kanpur
208016KalyanpurKanpur
Uttar Pradesh
India
Christoph Englert [email protected]
School of Physics & Astronomy
University of Glasgow
G12 8QQGlasgowU.K
Shakeel Ur Rahaman
Indian Institute of Technology Kanpur
208016KalyanpurKanpur
Uttar Pradesh
India
Michael Spannowsky [email protected]
Institute for Particle Physics Phenomenology
Department of Physics
Durham University
DH1 3LEDurhamU.K
Integrating out heavy scalars with modified EOMs: matching computation of dimension-eight SMEFT coefficients
5 Feb 2023
The shift in focus towards searches for physics beyond the Standard Model (SM) employing model-independent Effective Field Theory (EFT) methods necessitates a rigorous approach to matching to guarantee the validity of the obtained results and constraints. The limits on the leading dimension-six EFT effects can be rather inaccurate for LHC searches that suffer from large uncertainties while exploring an extensive energy range. Similarly, precise measurements can, in principle, test the subleading effects of the operator expansion. In this work, we present an algorithmic approach to automatise matching computations for dimension-eight operators for generic scalar extensions with proper implementation of equations of motion. We devise a step-by-step procedure to obtain the dimension-eight Wilson coefficients (WCs) in a non-redundant basis to arrive at complete matching results. We apply this formalism to a range of scalar extensions of the SM and provide tree-level and loop-suppressed results. Finally, we discuss the relevance of the dimension-eight operators for a range of phenomenological analyses, particularly focusing on Higgs and electroweak physics.
Introduction
Searches for physics beyond the Standard Model (BSM) chiefly performed at the Large Hadron Collider (LHC) have, so far, not revealed any significant deviation from the Standard Model (SM) predictions. This is puzzling, on the one hand, given the SM's plethora of known flaws and shortcomings. On the other hand, these findings have motivated the application of model-independent techniques employing Effective Field Theory (EFT) [1] to LHC data. The EFT approach breaks away from the assumption of concrete model-dependent correlations, thus opening up the possibility of revealing new (and perhaps non-canonical) BSM interactions through a holistic approach to data correlation interpretation. The inherent assumption of such an approach is that there is a significant mass gap between the BSM spectrum and the (process-dependent) characteristic energy scale at which the LHC operates to "integrate out" BSM states to obtain a low energy effective description that is determined by the SM's particle and symmetry content.
Efforts to apply EFT to the multi-scale processes of the LHC environment have received considerable interest recently, reaching from theory-led proof-of-principle fits to LHC data [2][3][4][5][6][7] (with a history of almost a decade) over the adoption of these techniques by the LHC experiments (e.g. [8,9] for recent examples), to perturbative improvements of the formalism [10][11][12][13][14][15][16][17][18][19][20][21]. In doing so, most attention has been devoted to SMEFT at dimension-six level [22] L
= L SM + i c i Λ 2 O i . (1.1)
While EFT is a formidable tool to put correlations at the forefront of BSM searches, the significant energy coverage of the LHC can lead to blurred sensitivity estimates even in instances when Eq. (1.1) is a sufficiently accurate expansion. When pushing the cut-off scale Λ to large values, the experimental sensitivity to deviations from the SM can be too small to yield perturbatively meaningful or relevant constraints when matched to concrete UV scenarios (see e.g. [23]). In contrast, dimension-eight contributions can be sizeable when the new physics cut-off Λ is comparably low in the case of more significant BSM signals at the LHC. To understand the ramifications for concrete UV models, it is then important to (i) have a flexible approach to mapping out the dimension-eight interactions and (ii) gauge the importance of dimension-eight contributions relative to those of dimension-six to quantitatively assess the error of the (potential) dimension-eight truncation.
A common bottleneck in constructing EFT interactions is removing redundancies. This is historically evidenced by the emergence of the so-called Warsaw basis [22]. Equations of motion are typically considered in eliminating redundant operators. Still, they are not identical to general (non-linear) field redefinitions, which are the actual redundant parameters of the field theory [24][25][26][27][28]. When truncating a given operator dimension, this can be viewed as a scheme-dependence not too unfamiliar from renormalisable theories, however, with less controlled side-effects when the new physics scale is comparably low. Additional operator structures need to be included to elevate classical equations of motion to field re-definitions [28,29] to achieve a consistent classification at the dimension-eight level.
In this work, we devise a generic approach to this issue which enables us to provide a complete framework to match any dimension-eight structure that emerges in the process of integrating-out a heavy non-SM scalar and obtaining the form of the WCs. Along the way of systematically re-organising the operators into a non-redundant basis, resembling the one discussed in Refs. [30,31], we show that removing the higher-derivative operators produced at the dimension-six level itself can induce a non-negligible effect on dimensioneight matching coefficients along with the direct contribution to the same which can be computed following the familiar methodologies of the Covariant Derivative Expansion (CDE) [32,33] of the path integral [34][35][36], or diagrammatic approach [37,38]. Finally, it is worth mentioning that, even though the one-loop effective action at dimension eight is yet to be formulated, it is possible to receive equally suppressed, loop-induced corrections from the dimension-six coefficients computed precisely at one-loop. These can present themselves as the leading order contributions for the WCs, which generally appear at one-loop. This paper is organised as follows: in Sec. 2, we discuss the implementation of the Higgs field equation of motion and study its equivalence with field redefinitions. This gives rise to the desired dimension-eight operator structures after removing redundancies (Sec. 2.3). Our approach is tested and validated against available results for the real triplet scalar extension in Sec. 3. In Sec. 4, the matching coefficients are presented explicitly considering a range of scalar extensions of the SM. Finally, the significance of the dimension-eight operators is analysed based on observables in a model-dependent manner in Sec. 5. We conclude in Sec. 6.
Complete matching at dimension-eight
We start by studying the structures of the higher-dimensional operators that can arise from heavy scalar extensions of the SM generically once the heavy field (Φ) is integrated out. The most generalised structure of the renormalised Lagrangian involving heavy scalars can be written as [32,39]:
L Φ ⊃ Φ † (P 2 − m 2 − U (x)) Φ + (Φ † B(x) + h.c.) + 1 4 λ Φ (Φ † Φ) 2 . (2.1)
Here, U (x) and B(x) contain the interactions that are quadratic and linear in Φ, respectively, and only involve the lighter degrees of freedom. Once Φ is integrated out, we obtain a tower of operators that can be arranged according to their canonical dimension. It is important to note that the operators generated in this process might not be independent. Depending on phenomenological considerations, several sets of operators are defined in the literature. A set of dimension-six operators was prescribed in Ref. [40]. It was improved by systematically removing the redundant structures and promoting it to form a complete non-redundant basis in Ref. [22] 1 , popularly known as the Warsaw basis. There is another set of operators known as the Green's set [42][43][44], which is over-complete. The operators here are independent under the Fierz identities and integration by parts but otherwise redundant on account of equations of motion 2 . This source of redundancy contributes to higher dimensional operators. In this paper, we use the Mathematica package CoDEx [45] to generate WCs of the operators in the SILH set [46,47] up to one-loop, including the relevant redundant terms. Since we are interested in the corrections to the dimension-eight coefficients resulting from the dimension-six redundant structures, we recast the SILH operators into Green's set-like structures 3 to single out redundant and non-redundant operators using the following equations:
Q H ≡ 1 2 ∂ µ (H † H)∂ µ (H † H) = − 1 2 (H † H) (H † H), Q T ≡ 1 2 H † ← → D µ H H † ← → D µ H = −2 (D µ H † H)(H † D µ H) − 1 2 (H † H) (H † H), Q R ≡ (H † H)(D µ H † D µ H) = 1 2 (H † H) (H † H) − (H † H)(D 2 H † H + H † D 2 H) , Q D ≡ D 2 H † D 2 H = − 1 2 (Y pq (ψ p ψ q ) D 2 H + h.c.) − λ (H † H)(D 2 H † H + H † D 2 H), Q 2W ≡ − 1 2 D µ W I µν 2 = − g 2 W 32 Y −1 pq (ψ p ψ q ) D 2 H + h.c. − ig 2 W 4 ψ p γ µ τ I ψ p (H † i ← → D I µ H) + g 2 W 4 λ 2 Y −1 pq Y −1 qp (H † H) 3 + g 2 W 8 (D µ H † H)(H † D µ H) − g 2 W 16 Q R + g 2 W 32 (1 + 2λ Y −1 pq Y −1 qp ) (H † H)(D 2 H † H + H † D 2 H), Q 2B ≡ − 1 2 ∂ µ B µν 2 = − g 2 Y 4 Y −1 pq (ψ p ψ q ) D 2 H + h.c. − ig 2 Y ψ p γ µ ψ p (H † i ← → D µ H) +g 2 Y Q R + 2g 2 Y λ 2 Y −1 pq Y −1 qp (H † H) 3 + g 2 Y 2 (1 + λ Y −1 pq Y −1 qp ) (H † H)(D 2 H † H + H † D 2 H), Q 2G ≡ − 1 2 D µ G a µν 2 = − g 2 G 3 Y −1 pq (ψ p ψ q ) D 2 H + h.c. + 4g 2 G 3 λ 2 Y −1 pq Y −1 qp (H † H) 3 + 2g 2 G 3 λ Y −1 pq Y −1 qp (H † H)(D 2 H † H + H † D 2 H), Q W ≡ ig W (H † τ I ← → D µ H)D ν W I µν = ig 2 W 2 (H † i ← → D I µ H)(ψγ µ τ I ψ) − g 2 W 8 (D µ H † H)(H † D µ H) + g 2 W 16 Q R − g 2 W 32 (H † H)(D 2 H † H + H † D 2 H), Q B ≡ ig Y (H † ← → D µ H)∂ ν B µν = ig 2 Y (H † ← → D µ H)(ψγ µ ψ) − 2g 2 Y Q R −g 2 Y (H † H)(D 2 H † H + H † D 2 H), Q W W ≡ g 2 W (H † H)W I µν W I,µν , Q BB ≡ g 2 Y (H † H)B µν B µν , Q W B ≡ 2g W g Y (H † τ I H)W I µν B µν , Q GG ≡ g 2 G (H † H)G a µν G a µν ,(2.2)
where Y pq denotes the SM Yukawa coupling matrix, {p, q} ∈ (1, 2, 3) are the flavour indices. We denote the Wilson coefficients of the SILH operators as C i with i labelling the operators in Eqs. (2.2). Taking into account all the H-involved structures that can appear from a scalar extension of the SM, one can write:
L = L (4) SM + λ (H † H) 2 + ζ(6)1 (H † H) 3 + ζ(6)2 (H † H) (H † H) + ζ(6)3 (D µ H † H)(H † D µ H) + ζ (6) 4 (H † H)(B µν B µν ) + ζ (6) 5 (H † H)(W I µν W Iµν ) + ζ (6) 6 (H † τ I H)(B µν W Iµν ) + ζ (6) 7 (H † H)(G a µν G aµν ) + ζ (6) 8,1 (H † i ← → D µ H)(ψ γ µ ψ) + ζ (6) 8,2 (H † i ← → D I µ H)(ψ τ I γ µ ψ) + ξ (6) 1 (H † H)(D 2 H † H + H † D 2 H) + ξ (6) 2 (ψ ψ) D 2 H + h.c. . (2.3)
We highlight the redundant terms in Eq. (2.3) in bold font; they need to be removed. We now discuss our approach to properly implement the Higgs equation of motion (EOM) to compute the corrections to dimension-eight coefficients.
C 6 + g 2 W 4 λ 2 Y −1 pq Y −1 qp C 2W + 2g 2 Y λ 2 Y −1 pq Y −1 qp C 2B + 4g 2 G 3 λ 2 Y −1 pq Y −1 qp C 2G ζ (6) 2 − 1 2 C H − 1 2 C T + 1 2 C R − g 2 W 32 C 2W + g 2 Y 2 C 2B + g 2 W 32 C W − g 2 Y C B ζ (6) 3 − 2C T + g 2 W 8 C 2W − g 2 W 8 C W ζ (6) 4 g 2 Y C BB ζ (6) 5 g 2 W C W W ζ (6) 6 2g W g Y C W B ζ (6) 7 g G 2 C GG ζ (6) 8,1 − ig 2 Y C 2B + ig 2 Y C B ζ (6) 8,2 − ig 2 W 4 C 2W + ig 2 W 2 C W ξ (6) 1 − 1 2 C R + g 2 W 32 (1 + 2λ Y −1 pq Y −1 qp )C 2W + g 2 Y 2 (1 + λ Y −1 pq Y −1 qp )C 2B + 2g 2 G 3 λ Y −1 pq Y −1 qp C 2G − g 2 W 16 C W − λ C D ξ (6) 2 −Y −1 pq g 2 W 32 C 2W + g 2 Y 4 C 2B + g 2 G 3 C 2G − 1 2 Y pq C D
Removing redundancies: Field Redefinition & Higgs Field EOM
In QFT, the experimentally observable quantities are related to the S-matrix elements, which remain invariant under field redefinition. Naively, this can be inferred from the fact that when calculating correlation functions using the path integral formalism, the field is just an integration variable. The correlation functions and S-matrix elements can be connected by the LSZ-reduction formula [48,49]. In the case of a renormalisable Lagrangian, we exploit this freedom and rewrite the Lagrangian in the canonical form. In an effective theory, we can perform non-linear field redefinitions due to the presence of higher dimensional operators. This invariance gives rise to a rule to remove redundant terms from the effective Lagrangian and leads to the construction of "on-shell" effective theory [24].
One way of removing the redundancies with higher derivatives of operators involving the Higgs field H is to redefine the field in a perturbative manner [24,28,29,50]. For example, to remove the term ξ
1 ) will be removed. Subsequently, it will give rise to higher dimensional operator at O (ξ (6) 1 ) 2 . Now the same outcome can be achieved by employing the EOM judiciously. We compute the classical EOM for the Higgs considering all possible structures up to dimension-six from Eq. (2.3)
D 2 H = − 2λ (H † H)H − Y + 3 ζ (6) 1 (H † H) 2 H + ζ (6) 3 (D µ H † H)D µ H + ξ (6) 1 (D 2 H † H + H † D 2 H)H + ξ (6) 1 (H † H) D 2 H − ζ (6) 3 D µ (H † D µ H)H + 2ζ (6) 2 H (H † H) + ξ (6) 1 D 2 [(H † H)H] + ζ (6) 4 H(B µν B µν ) + ζ (6) 5 H(W I µν W I,µν ) + ζ (6) 6 (τ I H)(B µν W I,µν ) + ζ (6) 7 H(G a µν G aµν ) + i ζ (6) 8,1 D µ H (ψγ µ ψ) + i ζ (6) 8,1 (D µ H)(ψγ µ ψ) + i ζ (6) 8,2 D I µ H (ψγ µ τ I ψ) + i ζ (6) 8,2 (D I µ H)(ψγ µ τ I ψ) + ξ (6) 2 D 2 (ψ ψ) . (2.4)
Here, λ = λ − λ with λ being the SM Higgs quartic coupling in the renormalisable Lagrangian. λ is the direct contribution to the former, obtained from integrating out the heavy field as shown in Eq. (2.3). The underlined part on the right-hand side of the Eq. (2.4) denotes the contribution from the renormalisable part of the Lagrangian (L
SM ), which is considered as the first-order term in the EOM. The remainder arises from the effective operators at dimension-six and is considered second-order terms [49]. We can think of Eq. (2.4) as some special field redefinition and directly employ the first-order terms to remove the redundancies in Eq. (2.3), but this will not generate any higher dimensional structures. This is the common practice to obtain the complete basis at dimension six. Working with the second-order terms is a non-trivial task, substituting it directly into Eq. (2.3) to obtain a contribution to higher dimension operators could lead to incompleteness as pointed out in [29]: The missing contributions can be encapsulated by including a term, (1/2) ξ
L = ξ (6) 1 2 D µ (H † H)H † D µ (H † H)H − 6 ξ (6) 1 2 λ (H † H) 4 = −2 ξ (6) 1 2 λ (H † H) 4 + ξ (6) 1 2 (H † H) 2 (H † Y + Y † H) − ξ (6) 1 2 (H † H) 2 (D µ H † D µ H).
(2.5)
Since our primary concern is the redundant operators involving the Higgs field, the boxed structures in the Eq. (2.4) containing the derivative of fermion fields can be reduced to other structures by applying the first-order fermionic EOM. We are now ready to implement the methodology discussed above to compute the dimension-eight coefficients from dimension-six operators.
1 − ζ (6) 3 ξ (6) 1 O (8) H 6 D 2 ,2 4ζ (6) 3 ξ (6) 1 O (8) ψ 2 H 4 D,1 i ζ (6) 8,1 ξ (6) 1 O (8) ψ 2 H 4 D,2 i ζ (6) 8,2 ξ (6) 1 O (8) ψ 4 DH,1 i ζ (6) 8,1 ξ (6) 2 O (8) ψ 4 DH,2 i ζ (6) 8,2 ξ (6) 2 O (8) H 4 B 2 2ζ (6) 4 ξ (6) 1 O (8) H 4 W 2 2ζ (6) 5 ξ (6) 1 O (8) H 4 W B 2ζ (6) 6 ξ (6) 1 O (8) H 4 G 2 2ζ (6) 7 ξ (6) 1 O (8) ψ 2 B 2 H ζ (6) 4 ξ (6) 2 O (8) ψ 2 W 2 H ζ (6) 5 ξ (6) 2 O (8) ψ 2 W BH ζ (6) 6 ξ (6) 2 O (8) ψ 2 G 2 H ζ (6) 7 ξ (6) 2 O (8) ψ 2 H 5 − 4 ξ (6) 1 2 − 4ζ (6) 2 ξ (6) 1 + 1 2 ζ (6) 3 ξ (6) 1 Y SM O (8) H 6ζ (6) 1 ξ (6) 1 − 6 ξ (6) 1 2 λ − 4λ (4(ξ (6) 1 ) 2 +3ζ (6) 1 ξ (6) 2 − 12λ ξ (6) 1 ξ (6) 2 + 2λ ξ (6) 2 ζ (6) 3 +4ζ (6) 2 ξ (6) 1 − 1 2 ζ (6) 3 ξ (6) 1 ) −8λ ξ (6) 2 ζ (6) 2 O (8) ψ 2 H 3 D 2 ,1 +4ξ (6) 2 ζ (6) 2 + 2ξ (6) 1 ξ (6) 2 O (8) ψ 2 H 3 D 2 ,2 −ξ (6) 2 ζ (6) 3 O (8) ψ 4 H 2 ,1 (−4ξ (6) 1 ξ (6) 2 − 4ζ (6) 2 ξ (6) 2 + 2ζ (6) 3 ξ (6) 2 ) Y SM O (8) ψ 4 H 2 ,2 (−3ξ (6) 1 ξ (6) 2 − 2ξ (6) 2 ζ (6) 2 ) Y SM
Impact of dimension-six structures on dimension-eight coefficients
It is a common practice to employ the first-order EOM, i.e., the classical equation of motion obtained from the renormalisable Lagrangian, to transform the operators from one basis to another at a given mass dimension. Here we extend this strategy to generate higher-order terms in the effective Lagrangian. The contribution to the WCs arising from the EOM substitution considering second-order terms will be important. In Table 2 we provide the contribution to dimension-eight operators coming from the dimension-six Lagrangian. The operator structures are shown in appendix A.
As the process of integrating out heavy fields becomes more complicated at higher operator dimensions, our method of generating WCs from lower dimension ones becomes economical. Following the expressions shown in Table 2, one can quickly work out the dimension-eight contribution without explicitly performing the matching at that order. In the following subsection, we compute the complete basis at dimension eight.
Removing redundancies at dimension-eight
We consider all (redundant and non-redundant) structures that only involve H and its derivatives at dimension-eight that can arise directly after integrating out at tree-level.
L (8) eff = ζ (8) 1 (H † H) 4 + ζ (8) 2 (H † H)(H † D µ H D µ H † H) + ζ (8) 3 (H † H) 2 (D µ H † D µ H) + ζ (8) 4 (D µ H † D ν H)(D ν H † D µ H) + ζ (8) 5 (D µ H † D ν H)(D µ H † D ν H) + ζ (8) 6 (D µ H † D µ H)(D ν H † D ν H)+ + ξ (8) 2 (D µ H † D µ H)(D 2 H † H + H † D 2 H) + ξ (8) 3 ( D µ H † H)(D 2 H † D µ H) + h.c. + ξ (8) 4 (D 2 H † H)(D 2 H † H) + h.c. + ξ (8) 5 (D µ H † H)(D µ H † D 2 H) + h.c. + ξ (8) 6 (D 2 H † D 2 H)(H † H) + ξ (8) 7 (H † D 2 H)(D 2 H † H). (2.6)
The redundant structures, written in bold in Eq. (2.6), can be expressed in terms of the non-redundant basis structures in the following manner 4 :
ξ (8) 1 (H † H) 2 (D 2 H † H + H † D 2 H) = − 4λ ξ (8) 1 (H † H) 4 − ξ (8) 1 (H † H) 2 (Y † H) + h.c. ξ (8) 2 (D µ H † D µ H)(D 2 H † H + H † D 2 H) = − 4λ ξ (8) 2 (H † H) 2 (D µ H † D µ H) − ξ (8) 2 (H † Y)(D µ H † D µ H) + h.c. , ξ (8) 3 ( D µ H † H)(D 2 H † D µ H) + h.c. = − 4λ ξ (8) 3 (H † H)(H † D µ H)(D µ H † H) − ξ (8) 3 (H † Y)(D µ H † D µ H) + h.c. , ξ (8) 4 (D 2 H † H)(D 2 H † H) + h.c. = 8λ 2 ξ (8) 4 (H † H) 4 + 4λ ξ (8) 4 (H † H) 2 (Y † H) + h.c. + ξ (8) 4 (Y † H)(Y † H) + h.c. , ξ (8) 5 (D µ H † H)(D µ H † D 2 H) + h.c. = − 4λ ξ (8) 5 (H † H)(H † D µ H)(D µ H † H) − ξ (8) 5 (D µ H † H)(D µ H † Y) + h.c. , ξ (8) 6 (D 2 H † D 2 H)(H † H) =4λ 2 ξ (8) 6 (H † H) 4 + 2λ ξ (8) 6 (H † H) 2 (Y † H) + h.c. + ξ (8) 6 (H † H)(Y † Y), ξ (8) 7 (H † D 2 H)(D 2 H † H) =4λ 2 ξ (8) 7 (H † H) 4 + 2λ ξ (8) 7 (H † H) 2 (Y † H) + h.c. + ξ (8) 7 (H † H)(Y † Y).
(2.7)
In Table 3, we present the coefficients in the non-redundant basis. Before cross-checking the proposed method in the next section, we summarise our framework in the flowchart depicted in Fig. 1. This work considers SM extensions of only a single heavy scalar. CoDEx [45] has been used to generate the operators and the WCs in the SILH set up to one-loop at dimension-six. We compute the EOM (only for the Higgs field), including contributions from dimension-six operators; we substitute the EOM in the redundant structures. The first-order terms transform the redundant structure of dimensionsix to non-redundant structures, while the second-order terms generate dimension-eight
Operator
Wilson coefficients
Operator Wilson coefficients Table 3. Matching contributions to the non-redundant dimension-eight operators from the dimensioneight structures after implementing the EOM. Here Y SM denotes the SM Yukawa coupling. We have suppressed the flavour indices without any loss of generality and continued throughout the rest of the paper.
O (8) H ζ (8) 1 − 4λ ξ (8) 1 + 8λ 2 ξ (8) 4 O (8) ψ 2 H 5 (−ξ (8) 1 + 4λ ξ (8) 4 + 2λ ξ (8) 6 +4λ 2 ξ (8) 6 + 4λ 2 ξ (8) 7 +2λ ξ (8) 7 ) Y SM O (8) H 6 D 2 ,1 ζ (8) 3 − 4λ ξ (8) 2 O (8) H 6 D 2 ,2 ζ (8) 2 − 4λ ξ (8) 3 − 4λ ξ (8) 5 O (8) ψ 2 H 3 D 2 ,1 (−ξ (8) 2 − ξ (8) 3 ) Y SM O (8) ψ 2 H 3 D 2 ,2 (−ξ (8) 5 ) Y SM O (8) ψ 4 H 2 ,1 (ξ (8) 6 + ξ (8) 7 ) Y 2 SM O (8) ψ 4 H 2 ,2 (ξ (8) 4 ) Y 2 SM
operators. To compensate for the missing contribution that renders the EOM equivalent to a field redefinition, we need to add a term proportional to the second-order derivative of the effective action. Furthermore, we calculate dimension-eight operators by integrating out the heavy field at the tree level, which gives rise to the leading effects at dimension eight. We then substitute the first-order terms in the EOM to convert them into a complete basis and combine all these contributions to obtain the complete matching result.
The following section applies this to reproduce the known results for the real triplet extension of the SM to validate our methodology.
Cross-validation of the method
To cross check our approach, we first turn to the example case of a real triplet scalar (Φ) SM extension. The BSM part of the Lagrangian reads
L Φ = 1 2 (D µ Φ a )(D µ Φ a ) − 1 2 m 2 Φ Φ a Φ a + 2kH † τ a HΦ a − η(H † H)Φ a Φ a − 1 4 λ Φ (Φ a Φ a ) 2 . (3.1)
Integrating out the heavy scalar leads to some correction to the renormalisable term (H † H) 2 as discussed in Eq. (2.3), the coefficient λ for this case is: λ = k 2 /(2m 2 Φ ). The SILH dimension-six coefficients have been tabulated in Table 4. The one-loop contribution to the matching can be categorised into two different classes: the contribution arising from integrating out scalars from purely heavy loops has been highlighted in blue, and terms from heavy-light mixed loops are shown in red.
We compute the Green's set-like coefficients first following the relations provided in Table 1 shown in Table 2. These contributions are computed considering only the tree-level part of the SILH coefficients. Thus, they are on the same footing as the direct dimensioneight contributions; they are tabulated separately in Table 5. The direct contributions at dimension-eight after "integrating out" can be captured by a total of seven coefficients as specified in Eq. (2.6). The values for the coefficients are
ζ (8) 1 = 2η 2 k 2 m 6 Φ − k 4 4m 8 Φ λ Φ , ζ(8)2 = 8ηk 2 m 6 Φ , ζ(8)3 = − 4ηk 2 m 6 Φ , ζ (8) 4 = 4k 2 m 6 Φ , ζ(8)5 = 0, ζ(8)6 = − 2k 2 m 6 Φ , ξ(8)1 = 2ηk 2 m 6 Φ , ξ(8)2 = − 2k 2 m 6 Φ , ξ(8)3 = 4k 2 m 6 Φ , ξ(8)4 = k 2 2m 6 Φ , ξ(8)5 = 0, ξ(8)6 = 2k 2 m 6 Φ , ξ(8)7 = − k 2 m 6 Φ . (3.2)
In Table 6, we show the direct contributions to dimension-eight structures removing redundancies at dimension-eight following the relations given in Table 3. Lastly, in Table 7, we provide the complete tree-level matching results at dimension-eight. Here, for comparison, we focus exclusively on those operators whose coefficients were previously derived in Ref. [50] employing the field redefinition of the Higgs field. We can connect the structure of O (8) H 6 D 2 ,2 given in Ref. [50] with O Table 7. Total tree-level matching of the dimension-eight coefficients for the real-triplet scalar extension of SM of Eq. (3.1). explicit structures of the operators), in the following way:
H 6 D 2 ,2 (see appendix A for the SILH Op. Wilson Coefficients SILH Op. Wilson Coefficients O 6 − ηk 2 m 4 Φ − η 3 8m 2 Φ π 2 − 5ηk 2 λ Φ 8m 4 Φ π 2 + 13η 2 k 2 8m 4 Φ π 2 O H η 2 16m 2 Φ π 2 − 3ηk 2 8m 4 Φ π 2 − 9k 4 32m 6 Φ π 2 + 47ηk 4 16m 6 Φ π 2 + 19k 6 16m 8 Φ π 2 − 2ηk 2 λ m 4 Φ π 2 − 2k 4 λ m 6 Φ π 2 + 5k 2 λ 16m 4 Φ π 2 + 11k 2 λ 2 16m 4 Φ π 2 − 5k 4 λ Φ 16m 6 Φ π 2 O R 2k 2 m 4 Φ + 5k 2 λ Φ 4m 4 Φ π 2 − 21ηk 2 16m 4 Φ π 2 O T k 2 m 4 Φ + 5k 2 λ Φ 8m 4 Φ π 2 − ηk 2 2m 4 Φ π 2 − 21k 4 32m 6 Φ π 2 + 25k 2 λ 32m 4 Φ π + k 4 32m 6 Φ π 2 + 3k 2 λ 32m 4 Φ π 2 O W W η 96m 2 Φ π 2 + 25k 2 768m 4 Φ π 2 O 2W g 2 W 480m 2 Φ π 2 O W B − k 2 128m 4 Φ π 2 O BB 3k 2 256m 4 Φ π 2 O W − k 2 288m 4 Φ π 2 O B − 7k 2 96m 4 Φ π 2O (8) H 5k 6 m 10 Φ + 6k 4 η m 8 Φ − 10k 4 λ m 8 Φ O (8) ψ 2 H 5 − k 4 m 8 Φ Y SM O (8) H 6 D 2 ,1 − 5k 4 m 8 Φ O (8) H 6 D 2 ,2 8k 4 m 8 ΦO (8) H 2k 6 m 10 Φ + 4ηk 4 m 8 Φ − 8λk 4 m 8 Φ − k 4 λ Φ 4m 8 Φ O (8) ψ 2 H 5 − 2k 4 m 8 Φ − 2ηk 2 m 6 Φ + 4λk 2 m 6 Φ Y SM + 2η 2 k 2 m 6 Φ − 8ηλk 2 m 6 Φ + 8λ 2 k 2 m 6 Φ O (8) H 6 D 2 ,1 − 4k 4 m 8 Φ − 4ηk 2 m 6 Φ + 8λk 2 m 6 Φ O (8) H 6 D 2 ,2 8k 4 m 8 Φ + 8ηk 2 m 6 Φ − 16λk 2 m 6 ΦO (8) H 7k 6 m 10 Φ + 2k 2 (2λ−η) 2 m 6 Φ + (40η−72λ−λ Φ )k 4 4m 8 Φ O (8) H 6 D 2 ,1 − k 4 m 8 Φ O (8) H 6 D 2 ,2 (4η−8λ)k 2 m 6 Φ + 8k 4 m 8 Φ O (8) ψ 2 H 5 Y SM −3k 4 −2m 2 k 2 (η−2λ) m 8 ΦO (8) ψ 4 H 2 ,1 k 2 m 6 Φ Y 2 SM O (8) ψ 4 H 2 ,2 k 2 2m 6 Φ Y 2 SM O (8) ψ 2 H 3 D 2 ,1 − 2k 2 m 6 Φ Y SM O (8) H 4 D 4 ,1 4k 2 m 6 Φ O (8) H 4 D 4 ,3 − 2k 2 m 6 Φ(H † H)(H † σ I H)(D µ H † σ I D µ H) = 2 (H † H)(D µ H † H)(H † D µ H)−(H † H) 2 (D µ H † D µ H). (3.3)
These results are in agreement with the expressions provided for the tree-level matching of the dimension-eight coefficients in Table 10 of Ref. [50]. The remaining structures that arise after integrating out the heavy triplet scalar at dimension-eight, mainly at tree-level, including two and four-fermionic operators are shown in Table 8.
Example models
This section applies the formalism described above to several example models to generate dimension-eight operators. We use CoDEx [45] to obtain the operators and associated WCs in the SILH set at dimension six up to one loop, which we tabulate for each model. The coefficients are passed through Eqs. (2.2) to (2.5) that yield the contribution of dimension-six operators to dimension-eight operators. For clarity, we only present the leading contribution from the dimension-six tree-level generated operators and the direct integrated-out contribution at dimension eight. Subleading (yet non-negligible) corrections to the coefficients that arise from loop-generated operators can be obtained accordingly 5 , and the complete list of contributions can be obtained from a Mathematica notebook provided here. The models (apart from the leptoquark one) we discuss below are chosen as they generate operators at the tree level (see, e.g., the discussion in Refs. [53,54]). We can classify the contributions to WCs into the following two categories:
• Tree-level contributions: In this category, we only consider the contribution from those WCs generated at the tree level at dimension six. When the EOM is applied, they contribute on a par with the tree-level generated dimension-eight operators. Their combined effects are then considered to be the leading order contributions at dimension eight. We will mainly focus on this type of contribution and tabulate results for each model. It should be noted that these operators also receive subleading loop-induced contributions. The complete expressions for the coefficients can be found in the Mathematica notebook.
• Loop-induced and/or higher order contributions: If the redundant dimension-six operators are generated at one-loop-level or the coefficients introduced through the application of the EOM appear with loop-level contributions or both, the WCs contain (1/16π 2 ) or (1/16π 2 ) 2 suppressions, depending on interference between tree and loop parts. We will not list these types of contributions here, except for the leptoquark model for demonstration purposes. However, as mentioned above, the provided Mathematica notebook contains all contributions, and interested readers are referred to the here documented results.
Complex Triplet Scalar
The SM can be extended with an electroweak complex triplet scalar (∆) to explain the generation of neutrino masses through type-II seesaw mechanism [55,56]. This model also offers interesting collider signatures comprising rare lepton number and flavour violating processes [57]. In addition to contemplating the phenomenological significance of this model, a consistent effort has been made in the recent past to explore the effective theory of such an extension, see Refs. [58][59][60]. The BSM part of the Lagrangian reads:
L ∆ ⊃ (D µ ∆ † )(D µ ∆) − m 2 ∆ (∆ † ∆) − λ ∆ H † σ I H∆ † + h.c. − λ 1 (∆ † ∆) 2 − λ 2 (∆ † T I ∆)(∆ † T I ∆) − λ 3 (H † H)(∆ † ∆) − λ 4 (H † σ I H)(∆ † T I ∆), (4.1)
(we neglect the interaction with the fermion fields in the following). As the Lagrangian contains a linear interaction for the field ∆, the renormalisable structure (H † H) 2 gets an extra contribution proportional to the coupling λ = 2λ 2 ∆ /m 2 ∆ . Table 9 contains the complete matching at one-loop-order for the dimension-six SILH coefficients. After the tree-level integrating we obtain the following WCs at dimension-eight:
ζ (8) 1 = 2(λ 3 − λ 4 ) 2 λ 2 ∆ m 6 ∆ − 8(λ 1 + λ 2 )λ 4 ∆ m 8 ∆ ; ζ (8) 2 = 8(λ 3 − λ 4 )λ 2 ∆ m 6 ∆ ; ζ(8)3 = 4(λ 3 − λ 4 )λ 2 ∆ m 6 ∆ ; ζ (8) 4 = 8λ 2 ∆ m 6 ∆ ; ξ(8)1 = − 2(λ 3 − λ 4 )λ 2 ∆ m 6 ∆ ; ξ(8)3 = 8λ 2 ∆ m 6 ∆ ; ξ (8) 6 = 4λ 2 ∆ m 6 ∆ ; ξ (8) 7 = 4λ 2 ∆ m 6 ∆ . (4.2)
As mentioned before, the total contribution to the dimension-eight operators is categorised into two categories depending on how they contribute. Below we write down the operators in their respective categories.
• Tree-level contribution: O Table 9. WCs of dimension-six SMEFT operators in the SILH set after integrating out the Complex Triplet Scalar Eq. (4.1). The terms within braces ({}) denote the contribution from pure heavy loops, whereas the brackets ([ ]) mark the contribution from light-heavy mixed loops. We only use the uncoloured coefficients for further calculation here. The complete calculation can be found in the provided Mathematica notebook.
H 6 D 2 ,1 , O(8)H 6 D 2 ,2 , O(8)H , O(8)ψ 2 H 5 , O(8)ψ 2 H 3 D 2 ,1 , O(8)λ 4 λ 2 ∆ 2m 4 ∆ − 2λ 3 λ 2 ∆ m 4 ∆ − λ 3 3 32π 2 m 2 ∆ − λ 3 λ 2 4 64π 2 m 2 ∆ O H 2λ 2 ∆ m 4 ∆ + λ 2 3 32π 2 m 2 ∆ − 3λ 3 λ 2 ∆ 4π 2 m 4 ∆ + λ 4 λ 2 ∆ 8π 2 m 4 ∆ − 2λ 1 λ 3 λ 2 ∆ π 2 m 4 ∆ − λ 2 λ 3 λ 2 ∆ π 2 m 4 ∆ − λ 1 λ 4 λ 2 ∆ 2π 2 m 4 ∆ − λ 2 λ 4 λ 2 ∆ 4π 2 m 4 ∆ + 23λλ 2 ∆ 8π 2 m 4 ∆ − 11λ 4 ∆ 2π 2 m 6 ∆ + 13λ 2 3 λ 2 ∆ 8π 2 m 4 ∆ − λ 3 λ 4 λ 2 ∆ 2π 2 m 4 ∆ − 8λ 3 λλ 2 ∆ π 2 m 4 ∆ + 3λ 2 4 λ 2 ∆ 64π 2 m 4 ∆ O T − 2λ 2 ∆ m 4 ∆ − 2λ 1 λ 2 ∆ π 2 m 4 ∆ − λ 2 λ 2 ∆ π 2 m 4 ∆ − 5λ 1 λ 4 ∆ π 2 m 6 ∆ − 5λ 2 λ 4 ∆ π 2 m 6 ∆ + 24λ 3 λ 4 ∆ π 2 m 6 ∆ + 7λ 4 λλ∆ 2 4π 2 m 4 ∆ + λ 2 4 768π 2 m 2 ∆ + λ 3 λ 2 ∆ 2π 2 m 4 ∆ − 11λ 4 λ 2 ∆ 24π 2 m 4 ∆ + 11λ 2 λ 2 ∆ π 2 m 4 ∆ − 41λ 4 λ 4 ∆ 8π 2 m 6 ∆ − 62λλ 4 ∆ π 2 m 6 ∆ + 74λ 6 ∆ π 2 m 8 ∆ − 3λλ 2 ∆ 8π 2 m 4 ∆ + 4λ 4 ∆ 3π 2 m 6 ∆ O R 4λ 2 ∆ m 4 ∆ + 4λ 1 λ 2 ∆ π 2 m 4 ∆ + 2λ 2 λ 2 ∆ π 2 m 4 ∆ + λ 2 4 384π 2 m 2 ∆ O W − λ 2 ∆ 72π 2 m 4 ∆ − 13λ 3 λ 2 ∆ 8π 2 m 4 ∆ + λ 4 λ 2 ∆ 6π 2 m 4 ∆ + 11λλ 2 ∆ 2π 2 m 4 ∆ − 37λ 4 ∆ 3π 2 m 6 ∆ O B 11λ 2 ∆ 24π 2 m 4 ∆ O D λ 2 ∆ 8π 2 m 4 ∆ O W W λ 3 96π 2 m 2 ∆ + 25λ 2 ∆ 192π 2 m 4 ∆ O W B λ 4 384π 2 m 2 ∆ − 13λ 2 ∆ 96π 2 m 4 ∆ O BB λ 3 64π 2 m 2 ∆ + 11λ 2 ∆ 64π 2 m 4 ∆ O 2W g 2 W 240π 2 m 2 ∆ O 2B g 2 Y 160π 2 m 2 ∆
• Loop-induced and/or higher order contribution: O
ψ 2 H 3 D 2 ,2 , O(8)H 4 X 2 , O (8) ψ 2 X 2 H , O(8)
ψ 4 H 2 ,2 . The WCs corresponding to these operators are listed in the Mathematica notebook.
General Two Higgs Doublet Model
One of the simplest and well-motivated extensions of the SM Higgs sector is the inclusion of an additional SU (2) L scalar doublet (H) with hypercharge Y = 1/2, the well-known two-Higgs doublet model (2HDM) [61,62]. Many UV complete theories contain a 2HDM in their minimal versions. This model has also been well discussed within SMEFT framework by integrating out the additional heavy Higgs doublet leading to dimension-six effective operators at one-loop [63][64][65]. The relevant part of the BSM Lagrangian reads The Wilson coefficients of dimension-six operators in the SILH set are presented in Table 11. After integrating out at tree-level the non-zero dimension-eight coefficients are given by
L H ⊃ (D µ H † )(D µ H) − m 2 H H † H − λ H 4 (H † H) 2 + η H ( H † H) + η H (H † H) H † H + H † H − λ 1 H † H (H † H) 2 − λ 2 (H † H)( H † H) − λ 3 ( H † H) 2 + (H † H) 2 . (4.3) Operator Wilson coefficients Operator Wilson coefficients O (8) H 6 D 2 ,1 8(λ 3 −λ 4 )λ 2 ∆ m 6 ∆ − 20λ 4 ∆ m 8 ∆ O (8) H 6 D 2 ,2 − 8(4λ−λ 3 +λ 4 )λ 2 ∆ m 6 ∆ + 32λ 4 ∆ m 8 ∆ O (8) H 2 16λ 2 +4λ(λ 3 −λ 4 )+(λ 3 −λ 4 ) 2 λ 2 ∆ m 6 ∆ O (8) ψ 2 H 5 2(8λ+λ 3 −λ 4 )λ 2 ∆ m 6 ∆ − 36λ 4 ∆ m 8 ∆ Y SM − 8(21λ+λ 1 +λ 2 −λ 3 +λ 4 )λ 4 ∆ m 8 ∆ + 208λ 6 ∆ m 10 ∆ O (8) H 4 D 4 ,1 8λ 2 ∆ m 6 ∆ O (8) ψ 2 H 3 D 2 ,1 − 8λ 2 ∆ m 6 ∆ Y SM O (8) ψ 4 H 2 ,1 8λ 2 ∆ m 6 ∆ Y 2 SM+ 3η 2 H λ H 32m 2 H π 2 + 3η H η H λ 2 8m 2 H π 2 − λ 3 1 48m 2 H π 2 O H − 3η H η H 8π 2 m 2 H + λ 2 1 48π 2 m 2 H + λ 1 λ 2 48π 2 m 2 H + 3η H η H λ 2 8m 2 H π 2 − λ 2 1 λ 2 32m 2 H π 2 − λ 3 2 96m 2 H π 2 − λ 1 λ 2 3 8m 2 H π 2 + λ 2 2 192π 2 m 2 H + λ 2 3 48π 2 m 2 H + 5η 2 H 16π 2 m 2 H − λ 2 λ 2 3 8m 2 H π 2 + 15η 2 H λ 8m 2 H π 2 − 3η 2 H λ 1 4m 2 H π 2 − 13η 2 H λ 2 16H 2 π 2 − 7η 2 H λ 3 4H 2 π 2 O 2B g 2 W 960m 2 H π 2 O R − 3η H η H 8m 2 H π 2 + λ 2 2 96m 2 H π 2 + λ 2 3 24m 2 H π 2 + η 2 H 8m 2 H π 2 O T λ 2 2 192m 2 H π 2 − λ 2 3 48m 2 H π 2 O W W λ 1 384m 2 H π 2 + λ 2 768m 2 H π 2 O BB λ 1 384m 2 H π 2 + λ 2 768m 2 H π 2 O W B λ 2 384m 2 H π 2 O 2W g 2 W 960m 2 H π 2ζ (8) 1 = − η 2 H m 4 H (λ 1 + λ 2 + 2λ 3 ); ζ (8) 3 = − η 2 H m 4 H ; ξ (8) 1 = − η 2 H m 4 H . (4.4)
We split the operators into our categories:
• Tree-level contribution: O (8)
H 6 D 2 ,1 , O (8) H , O(8)
ψ 2 H 5 . The WCs corresponding to these operators are listed in Table 12.
• Loop-induced and/or higher order contribution: O (8) The WCs corresponding to these operators are provided in the Mathematica notebook.
H 6 D 2 ,2 , O (8) ψ 2 H 4 D,1 , O (8) ψ 2 H 4 D,2 , O(8)ψ 4 DH,1 , O(8)ψ 4 DH,2 , O(8)ψ 2 H 3 D 2 ,1 , O(8)ψ 2 H 3 D 2 ,2 , O(8)H 4 X 2 , O(8)ψ 2 X 2 H , O(8)ψ 4 H 2 ,1 , O(8O (8) H η 2 H (4λ−λ 1 −λ 2 −2λ 3 ) m 4 H O (8) ψ 2 H 5 η 2 H m 4 H Y SM O (8) H 6 D 2 ,1 − η 2 H m 4 H
Complex quartet Scalar (hypercharge Y = 3/2)
To generate neutrino masses through higher dimensional operators, an SU (2) L quartet (Σ) with hypercharge Y = 3/2 can be added to the SM [66][67][68]. Focusing on this part of the BSM Lagrangian,
L Σ ⊃ (D µ Σ † )(D µ Σ) − M 2 Σ Σ † Σ + (η Σ Σ † jkl H j H k H l + h.c.) − k Σ 1 (H † H)(Σ † Σ) −k Σ 2 (H † m H n )(Σ † jkn Σ jkm ) − λ Σ 1 (Σ † Σ) 2 − λ Σ 2 (Σ † T a Σ) 2 , (4.5)
we can integrate out the heavy scalar and match to the dimension-six SMEFT operators (see also [30,64,65]). After matching we obtain the effective operators and associated WCs in terms of the UV parameters, shown in Table 13. Tree-level matching generates the following dimension-eight operators coefficients:
ζ (8) 1 = − |η Σ | 2 (k Σ 1 + k Σ 2 ) M 4 Σ ; ζ (8) 2 = − 6|η Σ | 2 M 4 Σ ; ζ (8) 3 = − 6|η Σ | 2 M 4 Σ . (4.6)
Again, we split the operators into the two categories: • Tree-level contribution: O
SILH Op. Wilson Coefficients SILH Op. Wilson Coefficients O6 η 2 Σ M 2 Σ − k 2 Σ 1 k Σ 2 16π 2 M 2 Σ − k 3 Σ 1 24π 2 M 2 Σ − 7k Σ 1 k 2 Σ 2 144π 2 M 2 Σ − 9η 2 Σ k Σ 1 8π 2 M 2 Σ O H k 2 Σ 1 24π 2 M 2 Σ + k Σ 1 k Σ 2 24π 2 M 2 Σ + k 2 Σ 2 96π 2 M 2 Σ + 3η 2 Σ 8π 2 M 2 Σ − k 3 Σ 2 72π 2 M 2 Σ + 5η 2 Σ λ Σ 1 8π 2 M 2 Σ + 15η 2 Σ λ Σ 2 32π 2 M 2 Σ − 19η 2 Σ k Σ 2 16π 2 M 2 Σ O 2B 3g 2 Y 160π 2 M 2 Σ O R 5k 2 Σ 2 432π 2 M 2 Σ + 3η 2 Σ 4π 2 M 2 Σ O T 5k 2 Σ 2 864π 2 M 2 Σ − 3η 2 Σ 8π 2 M 2 Σ O W W 5k Σ 1 192π 2 M 2 Σ + 5k Σ 2 384π 2 M 2 Σ O BB 3k Σ 1 64π 2 M 2 Σ + 3k Σ 2 128π 2 M 2 Σ O W B 5k Σ 2 192π 2 M 2 Σ O 2W g 2 W 96π 2 M 2 ΣO (8) H 6 D 2 ,1 − 6η 2 Σ m 4 Σ O (8) H 6 D 2 ,2 − 6η 2 Σ m 4 Σ O (8) H − η 2 Σ k Σ1 m 4 Σ − η 2 Σ k Σ2 m 4 ΣO 6 − η 2 S k S M 4 S − η 2 S k S λ S 16π 2 M 4 S − k 3 S 24π 2 M 2 S O H η 2 S M 4 S + η 2 S λ S 16π 2 M 4 S + k 2 S 48π 2 M 2 S + 11η 2 S k 2 S 8π 2 M 4 S + 37η 4 S k S 16π 2 M 6 S − 3η 4 S λ 2π 2 M 6 S + 43η 6 S 48π 2 M 8 S − 17ηS 2 k S 24π 2 M 4 S + 9η 2 S λ 32π 2 M 4 S − 5η 4 S 12π 2 M 6 S − 3η 2 S k S λ 2π 2 M 4 S + 9η 2 S λ 2 16π 2 M 4 S − η 4 S λ S 32π 2 M 6 S O W − 7η 2 S 288π 2 M 4 S O B − 7η 2 S 288π 2 M 4 S O D η 2 S 96π 2 M 4 S O W B η 2 S 128π 2 M 4 S O W W η 2 S 256π 2 M 4 S O BB η 2 S 256π 2 M 4 SH 6 D 2 ,1 , O(8)H , O(8)H 6 D 2 ,2 .(8)
The WCs corresponding to these operators are listed in Table 14.
• Loop-induced and/or higher order contribution: O (8)
ψ 2 H 5 , O (8) ψ 2 H 4 D,1 , O (8) ψ 2 H 4 D,2 , O (8) ψ 4 DH,1 , O (8) ψ 4 DH,2 , O (8) ψ 2 H 3 D 2 ,1 , O (8) ψ 2 H 3 D 2 ,2 , O(8)H 4 X 2 , O (8) ψ 2 X 2 H , O(8)ψ 4 H 2 ,1 , O(8)
ψ 4 H 2 ,2 . The WCs corresponding to these operators are shown in the Mathematica notebook.
Real Singlet Scalar Model
The addition of a real singlet scalar to the SM is motivated by a range of SM shortcomings, related to dark matter, baryogenesis, and the electroweak hierarchy problem [69][70][71]. This model has been discussed extensively within the EFT framework through a complete one-loop matching to the SMEFT up to dimension-six [36,38,58,64,65,72]. Here, we systematically extend these results. The Lagrangian involving the Real Singlet Scalar field (S) is given by:
L S ⊃ 1 2 (∂ µ S) 2 − 1 2 M 2 S S 2 − η S (H † H)S − k S (H † H)S 2 − 1 4! λ S S 4 . (4.7)
After integrating out the heavy field S, we obtain an additional contribution to the quartic coupling of Higgs:λ = −η 2 S /M 2 S , along with the dimension-six SILH set operators as shown in Table 15. Since no redundant operator at dimension-six is generated at tree-level, there is no tree-level contribution to dimension-eight operators from dimension-six. Thus the dominant contribution arises solely from removing redundancies at dimension-eight level itself. The non-zero coefficients of dimension-eight operators generated via integrating out
O (8) H 2η 2 S k 2 S m 6 S − 8η 2 S k S λ m 6 S O (8) H 6 D 2 ,1 4η 2 S k S m 6 S − 8λ η 2 S k S m 6 S + 8η 2 S k S λ 2 m 6 S + 16η 4 S k S λ m 8 S O (8) H 4 D 4 ,3 2η 2 S m 6 S − η 4 S λ S 24m 8 S + 8η 6 S k S m 10 S O (8) ψ 2 H 5 − 2η 2 S k S Y SM (1−2λ) m 6 S + 4η 4 S k S Y SM m 8 S O (8) ψ 2 H 3 D 2 ,1 − 2η 2 S k S Y SM m 6 S O (8) ψ 4 H 2 ,2 η 2 S k S Y 2 SM 2m 6 S O (8) ψ 4 H 2 ,1 η 2 S k S Y 2 SM m 6 S1 = 2η 2 S k 2 S M 6 S − λ S η 4 S 24M 8 S ; ζ (8) 3 = 4η 2 S k S M 6 S ; ζ (8) 6 = 2η 2 S M 6 S ; ξ (8) 1 = 2η 2 S k S M 6 S ; ξ (8) 2 = 2η 2 S k S M 6 S ; ξ (8) 4 = η 2 S k S 2M 6 S ; ξ (8) 7 = η 2 S k S M 6 S . (4.8)
We use Eq. (2.7) to remove the redundancies from the above equation and rewrite them in the complete basis of Table 3; the coefficients of non-redundant SMEFT dimension-eight operators are shown in Table 16. Expressed in the categories detailed above we arrive at
• Tree-level contribution: O
H 6 D 2 ,1 , O(8)H , O(8)ψ 2 H 3 D 2 ,1 , O(8)ψ 4 H 2 ,1 , O(8)ψ 2 H 5 , O(8)ψ 4 H 2 ,2 .(8)
The WCs corresponding to these operators are listed in Table 14.
• Loop-induced and/or higher order contribution: There is no redundancy at the dimension-six level, and no loop induced operators can be generated by the equation of motion. We can generate dimension-eight operators at one-loop-level itself by integrating out the heavy degree of freedom. This is beyond the scope of this paper, and we will leave this for future work.
Scalar Leptoquark
Next, we consider the BSM model where the SM is extended by a scalar leptoquark, having quantum numbers (3, 2, 1/6) under the SM gauge group SU (3) C × SU (2) L × U (1) Y . This scenario has recently received lots of attention as it can potentially address observed anomalies in B-meson decays [73,74]. This model has also been analysed within the EFT framework, see Refs. [58,64]. We focus on the scalar interaction part of the Lagrangian for our discussion, which reads
L Θ ⊃ (D µ Θ † )(D µ Θ) − M 2 Θ (Θ † Θ) − η Θ 1 (Θ † Θ)(H † H) − η Θ 2 (Θ † τ I Θ)(H † τ I H) − λ Θ 1 (Θ † Θ) 2 − λ Θ 2 (Θ † τ I Θ)(Θ † τ I Θ). (4.9)
No linear coupling of the Higgs field is present, and we do not obtain effective operators at tree-level. Note that although dimension-eight one-loop-level operators are beyond the scope of this work, we can still capture contributions to dimension-eight operators using our formalism. Table 17 shows the one-loop generated operators and the corresponding WCs. Table 18 contains the WCs contribution coming from the loop induced operators at dimension-six only. The two operators quoted there is not an exhaustive list but serves the purpose of demonstrating that we can still obtain non-zero WCs from dimension-six operators without dimension-eight one-loop-level matching. Categorising the WCs as above we find
O 6 − η 3 Θ 1 16π 2 M 2 Θ − 3η Θ 1 η 2 Θ2 256π 2 M 2 Θ O H η 2 Θ 1 16π 2 M 2 Θ O R η 2 Θ 2 128π 2 M 2 Θ O T η 2 Θ 2 256π 2 M 2 Θ O W W η Θ 1 128π 2 M 2 Θ O BB η Θ 1 1152π 2 M 2 Θ O W B η Θ 2 768π 2 M 2 Θ O 2W g 2 W 320π 2 M 2 Θ O 2B g 2 Y 2880π 2 M 2 ΘO (8) H 6 D 2 ,1 g 4 W g 4 Y λ 2 14745600π 4 m 4 Θ Y 2 SM + g 4 W g 4 Y λ 7372800π 4 m 4 Θ Y SM + g 4 W g 4 Y 3686400π 4 m 4 Θ − η 2 Θ1 g 4 W 20480π 4 m 4 Θ1 + η 2 Θ2 g 4 W 131072π 4 m 4 Θ − η 2 Θ1 g 4 W λ 20480π 4 m 4 Θ Y SM + η 2 Θ2 g 4 W λ 327680π 4 m 4 Θ Y SM + g 8 W λ 2 26214400π 4 m 4 Θ Y 2 SM − g 8 W λ 6553600π 4 m 4 Θ Y SM − g 8 W 5242880π 4 m 4 Θ − η 2 Θ2 g 4 Y 184320π 4 m 4 Θ − η 2 Θ1 g 4 Y λ 23040π 4 m 4 Θ Y SM + η 2 Θ2 g 4 Y λ 368640π 4 m 4 Θ Y SM + g 8 Y λ 2 33177600π 4 m 4 Θ Y 2 SM + g 8 Y λ 4147200π 4 m 4 Θ Y SM + η 2 Θ1 η 2 Θ2 1024π 4 m 4 Θ − 5η 4 Θ2 65536π 4 m 4 Θ O (8) H − g 4 W g 4 Y λ 1843200π 4 m 4 Θ − g 4 W g 4 Y λ 2 1228800π 4 m 4 Θ Y SM − 11g 4 W g 4 Y λ 3 7372800π 4 m 4 Θ Y 2 SM + η 2 Θ g 4 W λ 10240π 4 m 4 Θ − 3η 3 Θ1 g 4 W 40960π 4 m 4 Θ − 9η Θ1 η 2 Θ2 g 4 W 655360π 4 m 4 Θ + η 2 Θ2 g 4 W λ 65536π 4 m 4 Θ − g 8 W λ 2621440π 4 m 4 Θ + η 2 Θ1 g 4 W λ 2 10240π 4 m 4 Θ Y SM − 3η 3 Θ1 g 4 W λ 40960π 4 m 4 Θ Y SM − 9η Θ1 η 2 Θ2 g 4 W λ 655360π 4 m 4 Θ Y SM + 7η 2 Θ2 g 4 W λ 2 163840π 4 m 4 Θ Y SM − 7g 8 W λ 2 3276800π 4 m 4 Θ Y SM − 23g 8 W λ 3 13107200π 4 m 4 Θ Y 2 SM + η 2 Θ2 g 4 Y λ 92160π 4 m 4 Θ + η 2 Θ1 g 4 Y λ 2 11520π 4 m 4 Θ Y SM − η 3 Θ1 g 4 Y λ 15360π 4 m 4 Θ Y SM − η Θ1 η 2 Θ2 g 4 Y λ 81920π 4 m 4 Θ Y SM + η 2 Θ2 g 4 Y λ 2 184320π 4 m 4 Θ Y SM − g 8 Y λ 2 2073600π 4 m 4 Θ1 Y SM + g 8 Y λ 3 16588800π 4 m 4 Θ1 Y 2 SM − η 2 Θ1 η 2 Θ2 λ 512π 4 m 4 Θ + 3η 3 Θ1 η 2 Θ2 2048π 4 m 4 Θ1 + 9η Θ1 η 4 Θ2 32768π 4 m 4 Θ − 5η 4 Θ2 λ 32768π 4 m 4 Θ
• Tree-level contribution: No tree-level contribution.
• Loop-induced and/or higher order contribution: There is only loop induced contribution in this case.
H 6 D 2 ,2 , O(8)ψ 2 H 5 , O(8)ψ 2 H 3 D 2 ,1 , O (8) ψ 2 H 3 D 2 ,2 , O (8) ψ 4 H 2 ,1 , O (8) ψ 4 H 2 ,2 , O (8) ψ 2 H 4 D,1 , O (8) ψ 2 H 4 D,2 , O (8) ψ 4 DH,1 , O (8) ψ 4 DH,2 O (8) H 4 X 2 , O(8)
ψ 2 X 2 H . The full contribution can be accessed from the Mathematica notebook.
Impact of dimension-eight operators on BSM scenarios
Given the plethora of data available after LHC Run-II and Run-III, we broadly classify the above UV theories by investigating their low-energy phenomenology in this section, emphasising the relevance of the dimension-eight operators.
Different observables and precision measurements provide strong discriminators between UV scenarios when using matched EFT results. In this sense, the dimension-eight effects provide quantitatively crucial additional information. To gain a qualitative understanding of UV discrimination employing the results above, we consider three categories of experimental observables for guidance: (i) electroweak precision observables (EWPO), (ii) Higgs signal strength (HSS) measurements, and (iii) vector boson scattering (VBS) measurements. We analyse the cases discussed in Sec. 4, reviewing the interplay of (i)-(iii) as shown in Fig. 2.
The characteristics of different models can be analysed by adjudging their responses to the following questions. First, one needs to note which effective operators emerge from each model under consideration. As the observables can be parametrized in terms of the effective operators, the observable-model correspondence can be set up directly. Based on that, one can classify different UV models by carefully scrutinising the overlapping sets of operators contributing to a set of observables. The second question relates to the order of the perturbative expansion at which the operators are being produced. That will give a hint of their possible sensitivity towards the observables. Keeping all these points in mind, we have clubbed those models that show degenerate sensitivity and prepared the different classes. Each class contains degenerate models with respect to their response to that particular observable in consideration.
The first question points to the need for new measurements with distinct features to correctly classify a wide range of complete models. The second one emphasises the ability of existing measurements to constrain the underlying UV parameter spaces determines the measurements' BSM UV sensitivity.
Such an observable-based categorisation has been studied recently (see e.g. [75]) for dimension-six operators up to one loop, mainly in the Warsaw basis. Bringing dimensioneight operators into the picture helps resolve model degeneracies at the dimension-six level even without introducing new measurements. However, since the number of independent structures increases rapidly beyond dimension six, we confine our discussion only to the structures emerging for the six scalar extensions discussed in Sec. 3 and Sec. 4.
Electroweak precision observables
The precise measurements of the electroweak observables naturally calls for improvements on the theoretical side. This can be achieved by performing theoretical computations at next-to-leading order (NLO) and by extending the effective series expansion. As described in Refs. [19,50,76], we organise the operators below into lists based on how they contribute to EWPO:
Dimension-six LO : {O HD , O HW B , O He , O Hu , O Hd , O (1) Hq , O (3) Hq , O (1) Hl , O(3)
Hl , O ll };
(5.1)
Dimension-six NLO : {O 2 , O HB , O HW , O W , O uB , O uW , O ed , O ee , O eu , O lu , O ld , O le , O (1) lq , O (3) lq , O qe , O (1) qd , O (3) qq , O (1) qq , O (1) qu , O (1) ud , O uu , O dd }; (5.2) Dimension-eight LO : {O ψ 2 H 5 , O(1)HD,2 , O(2)HD,2 , O(1)ψ 2 H 4 D , O(2)ψ 2 H 4 D , O H 4 W B , O(1)ψ 4 H 2 , O (2) ψ 4 H 2 }. (5.3)
Contrary to the Φ and ∆ extensions, wherein {O HD } and {O HD , O ll }, respectively, are produced at tree-level (see e.g. Refs. [65,75]) and therefore provide the dominant contribution to the observable, other operators in Eq. (5.1) are generally sourced by heavy one-loop insertions. Their subleading contributions are comparable to the ones resulting from the operators produced at the tree-level and contribute to the observable at NLO (noted in Eq. (5.2)). This implies that the BSM parameter space of such an extension is less sensitive to EWPO than other observables when only dimension-six is considered, which includes the models: {H, Σ, S}. The situation can be remedied by taking dimension-eight operators shown in Eq. (5.3) into account. In this case, if these operators appear at the tree level, the constraints can be improved significantly. Hence based on sensitivity towards EWPO, we can divide all the models into two broad categories:
Higgs signal strength measurements
Higgs signal strengths (HSSs) are inherently connected to the interplay of fundamental mass generation in the SM and electroweak symmetry breaking. Therefore, the analysis of the HSSs is a relevant discriminator in the space of Higgs sector extensions. Certain dimensioneight operators imply non-negligible effects when constraining the BSM parameter space through HSS measurements, for instance, when the new physics occurs at a relatively low scale or if new couplings occur at tree-level after BSM states have been integrated out. According to the Refs. [65,77,78], the operators that affect the HSS measurements are listed below: [75]), these models are seemingly less sensitive to HSS measurements at dimension-six. Following a similar approach as the one described in Sec. 5.1, we can infer that the impact of the operators given in Eq. (5.5) should be considered to properly explore the parameter space of {Σ, Θ}. The fact that Σ generates O
Dimension-six : {O H , O H2 , O HD , O HB , O HW , O HW B , O eH , O uH , O dH , O He , O Hu , O Hd , O(1)Hq , O(3)Hq , O(1)Hl , O(3)Hl }; (5.4) Dimension-eight : {O (8) H , O(8)H 6 D 2 ,1 , O(8)H 6 D 2 ,2 , O ψ 2 H 5 , O H 4 B 2 , O H 4 W 2 ,
Vector boson scattering measurements
Vector boson scattering measurements have been very crucial in the study of the electroweak sector, particularly in constraining anomalous gauge couplings, which have been discussed in detail in SMEFT at dimension-six [79][80][81][82]. Individual bounds on dimension-eight couplings have been derived from VBS data as well [83][84][85]. At dimension-six, a total of nine operators contribute to the modification of observables, through gauge-self couplings (
Hq , O(1)
Hq }). Among these, O HD and the two-fermionic operators are mainly constrained from EWPO observables [86], while the operators {O W , O HW , O HB , O HW B } remain to be constrained by VBS measurements [87]. These are typically produced at one-loop level (see Ref. [75]). For a low enough cut-off scale, these can be comparable to dimension-eight tree-level contributions
H 4 D 4 , O(1)H 4 D 4 , O(2)H 4 D 4 }. (5.7)(3)
We note that, {O
HD,2 , O
HD,2 } contribute to the EW sector through the modification of gauge boson masses (see Appendix D of Ref. [78] for more details). Thus they are mostly constrained by EWPO. Models that produce the full or a subset of the rest of the mentioned dimension-eight operators (i.e., {Φ,∆,S}), can be efficiently constrained by VBS measurements:
Class
Conclusions
The indirect search for new physics using EFT, while providing an ingenious way to uncover the physics that might lie just beyond our reach, faces several critical challenges when tracing constraints to possible complete and renormalisable UV scenarios. Moreover, since one encounters new signatures at dimension eight that may unravel the microscopic nature of new interactions, including their effects can become a vital question when looking for new physics in a model-independent way. However, performing a global analysis of the entire parameter space of dimensions six and eight SMEFT is unrealistic. Broad modeldependent correlations can then help to hone the sensitivity to new interactions. This requires a transparent and effective way to perform matching to new physics scenarios beyond dimension six. In this work, we have explored these two issues in detail.
H 4 D 4 , O(2)
H 4 D 4 } are mainly constrained with VBS data, and filter out {H, Σ} which do not produce these operators. We highlight the operators produced by {Θ} that contribute to all observables with a box, these are loop-suppressed.
We present an easy-to-implement approach to compute the dimension-eight matching coefficients, capturing loop effects consistently. Furthermore, the method employs EOMs instead of the traditional field-redefinition formalism. The "missing piece" of the EOM that elevates it to a similar footing with field redefinition is included in a model-independent manner. It must be stressed that, while removing the redundant structures at dimension six, we obtain one-loop or two-loop equivalent contributions to dimension-eight structures due to the interference among the pieces generated at the tree and one-loop order.
We have applied this method to six different scalar extensions of the SM at one-loop order of the dimension-six coefficients considering both heavy-heavy and heavy-light loop propagators, validating our approach against results documented in the literature. Finally, we have clarified the relevance of dimension-eight operators for classifying UV-complete models given Higgs and electroweak measurements and VBS data.
A Relevant operator structures
Here we discuss the operator structure that differs from the Green's set as defined in Ref. [42]. At dimension-six there are four structures in the Φ 4 D 2 operator class. The operators are the following: Among the above structures, the first two (O H , O HD ) are considered to be the independent and part of the complete Warsaw basis. We can ignore the last one, O HD , which is CPviolating and does not appear in our analysis. The redundant operator O HD is the important structure for our analysis. In order to remove this redundancy, we derive the contribution to higher dimension i.e., dimension-eight in our case. Instead of using this exact structure, we use the following relation to convert it to a suitable form
(H † H)(D µ H † D µ H) = 1 2 (H † H) (H † H) − (H † H)(D 2 H † H + H † D 2 H) (A.2)
Since we are replacing one redundant structure with another that is related to the former by the integration by parts, we term it a Green's set-like structure. We collect all relevant dimension-six operators in Table 19. Dimension-eight can be found in Table 20.
B Renormalisable SM Lagrangian and EOM
The renormalisable SM Lagrangian is where y i denotes the U (1) Y hypercharges of the fermions. We have also used the following the notation [13,22],
L SM = − 1 4 G A µν G Aµν −Q R (H † H)(D µ H † D µ H) Q T 1 2 H † ← → D µ H H † ← → D µ H Q D (D 2 H † )(D 2 H) Q 2W − 1 2 D µ W I µν 2 Q 2B − 1 2 ∂ µ B µν 2 Q W ig W (H † τ a ← → D µ H)D ν W a µν Q B ig Y (H † ← → D µ H)∂ ν B µν Q W W g 2 W (H † H)W a µν W a,µν Q BB g 2 Y (H † H)B µν B µν Q W B 2g W g Y (H † τ a H)W a µν B µνH † i ← → D β H = iH † (D β H) − i(D β H † )H, (B.5) H † i ← → D I β H = iH † τ I (D β H) − i(D β H † )τ I H ,
to write the operators in the well-known compact forms.
H † H)(D 2 H † H + H † D 2 H) in Eq. (2.3), we can use the redefinition H → H + ξ
H † H)H, in which case the redundancy at the level of O(ξ
H † H) 2 δ 2 L/δH † δH.Calculating this term from Eq. (2.3) we obtain the following contribution:
Figure 1 .
1Flow chart depicting the algorithmic approach considered to compute matching coefficients for both dimension-six and dimension-eight operators. Here, "First order EOM" and "Second order EOM" are formulated from the renormalizable and the dimension-six parts of the SM Lagrangian respectively.
Class-A : {Φ, ∆, H, Σ, S}, Class-B : {Θ}.
D 2 ,2 at tree-level, as shown in
{O W }), gauge-Higgs couplings ({O HD , O HW , O HB , O HW B }) and fermion-
Dimension-six : {O W , O HB , O HW , O HW B };
-A : {Φ, ∆, S}, Class-B : {H, Σ, Θ}. (5.8)
Figure 2 .
2Interplay of different observables for the categorisation of complete models based on their sensitivity towards specific observable(s). {Φ, ∆, H, Σ, S} produce any one or both from the set {O D 2 ,2 } at tree-level. They contribute to all observables and are therefore severely constrained by EWPO. {O
O
H = (H † H) (H † H), O HD = |H † D µ H| 2 O HD = (H † H)(D µ H † D µ H), O HD = (H † H)D µ (H † i ← → D µ H). (A.1)
The coefficients of the Green's set-like structures in Eq. (2.3) can be expressed in terms of SILH coefficients through the relations given inTable 1.Coefficients of
Green's set-like
operators
Relation in terms of SILH coefficients
ζ
(6)
1
Table 1 .
1Translation of SILH coefficients into the Green's set-like form. Here, Y pq denotes the SM Yuakawa coupling, {p, q} ∈ (1, 2, 3) are the flavour indices.
Table 2 .
2Contributions to the dimension-eight operators from the dimension-six structures after implementing the EOM. Here Y SM denotes the SM Yukawa coupling. We have suppressed the flavour indices without any loss of generality.
and derive the corrections to dimension-eight from dimension-six structures asSM + heavy
scalar
Dimension-6 operators
Non-redundant
structures
Redundant
structures
First order
EOM
Using CoDEx
up to one-loop
Dimension-8 operators
Non-redundant
structures
Redundant
structures
First order
EOM
Second order
EOM
Integrating out the
heavy scalar
up to tree level
Table 4 .
4Dimension-six SILH Wilson coefficients relevant for integrating out the real-triplet scalar of Eq. (3.1) at one-loop. The terms within braces ({}) denote the contribution from pure heavy loops, whereas the brackets ([ ]) mark the contribution from light-heavy mixed loops.Operator
Wilson coefficients
Operator
Wilson coefficients
Table 5 .
5Contributions to dimension-eight operators from dimension-six structures when integrating out the heavy real triplet scalar of Eq. (3.1). Here only the tree-level matching of the dimension-six structures have been considered while computing the results.Operator
Wilson coefficients
Operator
Wilson coefficients
Table 6 .
6Contributions to dimension-eight operators from dimension-eight structures when integrating out the heavy real triplet scalar of Eq. (3.1).Operator
Wilson coefficients
Operator
Wilson coefficients
Table 8 .
8Dimension-eight matching coefficients for the real-triplet scalar extension of SM,
Eq. (3.1).
ψ 4 H 2 ,1 .The WCs corresponding to these operators are listed inTable 10.SILH Op.
Wilson Coefficients
SILH Op.
Wilson Coefficients
O 6
Table 10 .
10Total dimension-eight tree-level contribution after integrating out the Complex Triplet Scalar of Eq. (4.1). Complete results at one-loop with additional operators are presented in the Mathematica notebook.SILH Op.
Wilson Coefficients
SILH Op.
Wilson Coefficients
O 6
η 2
H
m 2
H
Table 11 .
11WCs of dimension-six SMEFT operators in the SILH set after integrating out an additional Higgs Doublet, Eq. (4.3). The terms within braces ({}) denote the contribution from pure heavy loops, whereas the brackets ([ ]) mark the contribution from light-heavy mixed loops. We only use the uncoloured coefficients for further calculation here. The complete calculation can be found in the Mathematica notebook.
) ψ 4
ψH 2 ,2 .Operator
Wilson coefficients
Operator
Wilson coefficients
Operator
Wilson coefficients
Table 12 .
12Totaldimension-eight tree-level contribution after integrating out additional an Higgs
Doublet, Eq. (4.3). Complete contributions at one-loop including additional operators are presented
in the Mathematica notebook.
Table 13 .
13WCs of dimension-six SMEFT operators in the SILH set after integrating out the Quartet Scalar of Eq. (4.5). The terms within braces ({}) denote the contribution from pure heavy loops, whereas the brackets ([ ]) mark the contribution from light-heavy mixed loops. We only use the uncoloured coefficients for further calculation here. The complete calculation can be found in the Mathematica notebook.Operator
Wilson coefficients
Operator
Wilson coefficients
Operator
Wilson coefficients
Table 14 .
14Total dimension-eight tree-level contribution after integrating out the Quartet Scalarof Eq. (4.5). Complete one-loop results including additional operators are available from the
Mathematica notebook.
Table 15 .
15WCsof dimension-six SMEFT operators in the SILH set after integrating out the Real
Singlet Scalar of Eq. (4.7). The terms within braces ({}) denote the contribution from pure heavy
loops, whereas the brackets ([ ]) mark the contribution from light-heavy mixed loops.
Table 16 .
16Non-redundant SMEFT dimension-eight operators and their corresponding coefficients
after integrating out the Real Singlet Scalar of Eq. (4.7).
are
ζ
(8)
Table 17 .
17WCsof dimension-six SMEFT operators in the SILH set for the Scalar Leptoquark of
Eq. (4.9). The terms within braces ({}) denote the contribution from pure heavy loops.
Operator
Wilson coefficients
Table 18 .
18Contribution to dimension-eight operators from dimension-six operators. This table does not capture the full contribution. Full results are available from our Mathematica notebook.
Table 18
18captures only a subset of operators which gets contribution
from the lower dimension operators. The others are : O
(8)
O H 4 W B }. (5.5) Since the models {Φ, ∆, H, S} produce subsets of these operators {O H , O H2 , O HD , O uH , O dH , O eH } at tree-level, while for {Σ, Θ} they are generated at one-loop (see Ref.
Table 14 ,
14leads to similar a priori sensitivity of Higgs measurements as for {Φ, ∆, H, S}. HSSs therefore discriminate: Class-A : {Φ, ∆, H, Σ, S}, Class-B : {Θ}.
µν B µν + (D µ H † )(D µ H) +1
4
W I
µν W Iµν −
1
4
B ψ=q,u,d,l,e
ψ i /
D ψ
Operator
Op. Structure
Operator
Op. Structure
Q H
1
2 ∂ µ (H † H)∂ µ (H † H)
Q 6
|H| 6
Table 19 .
19Dimension-six operator structures in the SILH set.[D α , B αβ ] = g Y ψ=u,d,q,e,l ψ y i γ β ψ + H † i ← → D β H ,
These operators can be related to the structures shown in Ref.[44].
In Refs.[51,52] some of the dimension-eight operators upto one-loop order for a few models have been computed.
AcknowledgementsThe authors thank Wrishik Naskar for comments after carefully reading this work during its draft stage. We also acknowledge helpful comments from Dave Sutherland.Here, we have ignored the (negative) mass term for the Higgs field which is not relevant for our analysis. We can calculate the EOMs for various fields using Eq. (B.1). These EOMs constitute the first-order approximation of the full EOM, and they can be used to transform one set to another at a given mass dimension. The first-order EOM for Higgs field is[13,28]the EOMs for the fermions are[13,28]i /with Yukawa couplings Y i ≡ Y SM,i . The EOMs for the gauge fields are[13,28][D α , G αβ ] A = g G ψ=u,d,q ψ T A γ β ψ,[D α , W αβ ] I = g W 1 2 q τ I γ β q + 1 2 l τ I γ β l + 1 2
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arxiv |
Towards a holographic marginal Fermi liquid
27 Jun 2011
Kristan Jensen
Department of Physics
University of Victoria
V8W 3P6VictoriaBCCanada
Shamit Kachru
Department of Physics
Stanford University and SLAC
94305StanfordCAUSA
Andreas Karch
Department of Physics
University of Washington
98195SeattleWAUSA
Joseph Polchinski
KITP and Department of Physics
UCSB
93106Santa BarbaraCAUSA
Eva Silverstein
Department of Physics
Stanford University and SLAC
94305StanfordCAUSA
Towards a holographic marginal Fermi liquid
27 Jun 2011
We present an infinite class of 2+1 dimensional field theories which, after coupling to semi-holographic fermions, exhibit strange metallic behavior in a suitable large N limit. These theories describe lattices of hypermultiplet defects interacting with parity-preserving supersymmetric Chern-Simons theories with U (N ) × U (N ) gauge groups at levels ±k. They have dual gravitational descriptions in terms of lattices of probe M2 branes in AdS4 × S 7 /Z k (for N ≫ 1, N ≫ k 5 ) or probe D2 branes in AdS4 × CP 3 (for N ≫ k ≫ 1, N ≪ k 5 ). We discuss several challenges one faces in maintaining the success of these models at finite N , including backreaction of the probes in the gravity solutions and radiative corrections in the weakly coupled field theory limit.
Introduction Local quantum criticality, an invariance under rescaling of energies that leaves the spatial momenta fixed, has been invoked as a potential explanation of interesting phases seen in a variety of condensed matter systems [1]. One leading approach for explaining the anomalous transport properties of the strange metallic phase, the marginal Fermi liquid (MFL) [2], involves a locally critical sector of spin and charge fluctuations, coupled to a Fermi sea.
In general, the theory of non-Fermi liquids is still in its infancy. One recently developed method of obtaining controlled models of non-Fermi liquids uses holography. The study of fermion probes in black brane backgrounds with AdS 2 ×R 2 near-horizon geometries [3][4][5][6], or equivalently the semi-holographic prescription of [7], readily gives rise to non-Fermi liquid behavior. In the latter approach, free fermions are mixed with fermionic operators from a large-N locally critical sector, dual to fermions living in AdS 2 . A distinct holographic mechanism realizing non-Fermi liquid transport arises on probe branes in Lifshitz backgrounds [8].
Much of the work on the holographic approach to non-Fermi liquids has so far been at the level of 4d effective AdS gravity theories, with the scaling dimensions of operators in the dual field theory appearing as free parameters (masses of bulk fields). It would be useful to have microscopic dual pairs where the field theory dynamics giving rise to local criticality is visible in a conventional field theoretic Lagrangian, and the scaling properties of the non-Fermi liquid can be predicted by the concrete dual field theory instead of being parameterized as unknowns. [28] One goal of our work is to provide an infinite class of such theories where it is natural to obtain precisely the scaling dimensions required for marginal Fermi liquid behavior.
A second goal has been to remedy one of the residual defects in the models of [4]; there, the precise nature of the non-Fermi liquid depends sensitively on the Fermi momentum k F (since the dimensions of the relevant fermionic operators depend on k F ). In the models we describe here, the relevant scaling dimension ∆ which (with the right value) gives rise to marginal Fermi liquid behavior, is independent of k F . This allows an arbitrary shape of the Fermi surface, a useful feature since this is not protected from renormalization group flow.
A third goal has been to clarify when and how locally critical behavior can occur in a higher-dimensional (D ≥ 2 dimensional) quantum field theory. Local criticality is a rather exotic property, which needs to be better understood. By definition, it entails quantum mechanical degrees of freedom propagating independently at every point in space, not suppressed by gradient terms. On the other hand, in higherdimensional quantum field theories, the ultraviolet physics contains itinerant fields which propagate in all directions, with gradient terms in their Lagrangian. Even if one begins with a sector of localized degrees of freedom (like the defects we study), which in itself exhibits local criticality, this sector generically mixes with the itinerant fields through interaction terms. These can, and generally would be expected to, induce gradients. Yet surprisingly, among holographic gravity systems dual to very strongly coupled field theories, one often finds solutions with AdS 2 symmetry (using either the AdS-Reissner-Nordström (RN) black brane, or the world-volumes of appropriate probe branes [11] as we shall do here). These solutions are common because they are not terribly hard to obtain, whether by the relatively prosaic matter of stabilizing the extra dimensions of string theory or by stably embedding a probe brane along an AdS 2 slice. However even in the large-N approximation of a gauge theory with N colors, strong effects of the itinerant fields are included, so this is a nontrivial result of gauge/gravity duality. Therefore, we wish to begin an analysis of whether this emergence of local criticality is only an artifact of the extreme strongly coupled limit where the gravity description is appropriate, or whether instead a similar mechanism exists also at weaker coupling and finite N . In the second part of this note we discuss the interaction between impurities, which is a finite N effect but becomes important at low energies. In some cases this spoils the local criticality, but in others this may survive to the IR.
The brane system Instead of obtaining AdS 2 in the nearhorizon limit of an AdS-RN black brane, a setup which incurs various instabilities, we choose to obtain the AdS 2 regions on the worldvolumes of lattice defects, as in [11,12]. A variety of field theoretic toy-models suggest that lattices of defects interacting with itinerant electrons could be a reasonable starting point for strange metal phenomenology (see e.g. [13][14][15]).
Such lattices can be implemented in various ways, differing in their symmetries and in the quantum numbers of the operators in the theory. The model of [11] involves a lattice of defect fermions interacting with the 4d N = 4 supersymmetric Yang-Mills theory, and is engineered by intersecting D3 and D5-branes (with the D5-branes wrapping AdS 2 × S 4 regions in the near-horizon AdS 5 × S 5 geometry of the D3-branes). The supersymmetry preserved by that lattice model is somewhat unconventional (allowing e.g. purely fermionic defect representations); therefore we will mostly focus on a different lattice system which is 2+1 dimensional and enjoys a more powerful supersymmetry algebra for some values of our discrete parameters. This, however, entails extraneous bosonic degrees of freedom at the lattice sites, and the examples containing only fermions on the defects can be analyzed similarly.
In the most symmetric case, the brane configuration we study is given, in M-theory, by M2 and M2 ′ branes: Here, an x denotes a dimension wrapped by the given brane stack, blanks denotes dimensions where the given branes are localized at a common point, and :: denotes dimensions in which the given branes are individually localized but form a lattice. In this configuration, the two stacks intersect along a lattice in the 1-2 plane.
Our family of theories will depend on two parameters: N and k. N denotes the number of M2 branes in the stack above; the M2 ′ branes are equally spaced in a square lattice, and the lattice spacing is the only scale in the problem (so it doesn't constitute a new parameter). The second parameter k arises as follows. We consider a Z k orbifold which acts as follows on the four complex coordinates transverse to the M2s:
g k : z i = x 2i+1 + ix 2i+2 , z i → e i2π k z i , i = 1...4 . (2)
The set of M2 ′ branes wrap the locus [16]
z 1 = z 2 = 0, z 3 =z 4 .(3)
and their orbifold images under (2). For k = 1 this embedding is equivalent to the one in (1). We treat even and odd k symmetrically, defining the orbifold action to identify points on different, mirror branes (rather than taking the g k/2 k element to identify points on the same brane in the case k even).
The global symmetry of the M2-brane theory is partially broken by the orbifolding and the presence of the M2 ′ probes; from SO(8) × SO(2) to SO(6) × U (1) × Z 4 for k = 1, and down to SU (2) × U (1) 2 × Z 4 for k > 1. The Z 4 factor here represents the symmetry of the lattice. At large k (such that k 5 ≫ N ≫ 1), it follows from the analysis in [17] that the near-horizon region of the system of M2 and M2 ′ branes is described more accurately using different variables in terms of type IIA string theory with D2 and D2 ′ branes on a nontrivial geometry with background 2-form gauge flux.
The field theory The field theory on the M2 branes in these geometries has been studied in great detail [17]. A general 3d supersymmetric Chern-Simons theory with at least N = 2 supersymmetry has an action including the terms [18]:
S = d 3 x k 4π Tr(A∧dA+ 2 3 A 3 )+D µφi D µ φ i +iψ i γ µ D µ ψ i − 16π 2 k 2 (φ i T a Ri φ i )(φ j T b Rj φ j )(φ k T a R k T b R k φ k ) − 4π k (φ i T a Ri φ i )(ψ j T a Rj ψ j ) − 8π k (ψ i T a Ri φ i )(φ j T a Rj ψ j ) .(4)
Here T a R are the generators of the gauge group in representation R, and the scalars φ i and fermions ψ i are superpartners in a chiral multiplet. These terms arise from integrating out the scalars and fermions of the massive vector multiplet and flowing to the deep infrared limit of the theory.
The field theory on our M2 branes is a special case of this theory, with gauge groups U (N ) × U (N ) appearing at levels ±k. The 't Hooft coupling of this theory is N/k and so is large in the holographic limits. The matter fields φ i are four bi-fundamental fields A 1,2 and B 1,2 , in the (N,N ) and (N , N ) representations respectively. In addition to the basic supersymmetric action written above for these fields, we add an N = 3 superpotential
W = 2π k ǫ ab ǫȧ˙bT r(A a BȧA b B˙b) .(5)
Here a, b = 1, 2 and the superpotential has been written in a manifestly SU (2) × SU (2) symmetric manner. The full symmetry of the field theory is in fact enhanced to an SO(6)× U (1) b (with the baryonic U (1) b acting with charge ±1 on the A and B fields), and the theory with these choices enjoys an enhanced N = 6 supersymmetry [17]. [29] The probe M2 ′ branes give rise to localized degrees of freedom; in the type IIA string theory limit of the brane construction these arise from strings stretching between the D2 branes and a lattice of probe D2 ′ branes. In the simplest case of k = 1, these are hypermultiplets, with the fermions transforming as spinors in the dimensions transverse to both branes (and the bosons transforming as spinors along 1234). The infrared Chern-Simons theory is more difficult to analyze directly, since the appropriate type IIB brane construction involves non-perturbative ingredients. However, by generalizing the methods of [17] one can obtain a plausible hypothesis for the spectrum [16], in which defect hypermultiplets are added to both gauge groups. One reason that this is plausible is that the dual probe branes respect parity, which in the field theory exchanges the gauge group factors. The bosonic quantum mechanical degrees of freedom Q 1,2 andQ 1,2 at each site transform as follows. Q i transforms in the N of the ith U(N) gauge group (and is a singlet under the other), whileQ i transforms in the conjugate manner; these also transform as spinors under the Lorentz group in the 1234 directions. Each boson is accompanied by a fermion partner so there are also defect fermions χ 1,2 ,χ 1,2 ; these do not transform as spinors in the 1234 directions, but do in the remaining directions. Starting from the ABJM theory, the defect probe branes preserve 8 supercharges in the special case of k = 1, and more generally they preserve 4 supercharges [16]. We expect a similar spectrum of localized degrees of freedom on the defects for all k.
While the overall system preserves at least 4 supercharges in all cases, the superspace structure is unconventional and we have not been able to find a packaging in the standard superspace arising in 4d N = 1 supersymmetry. (For instance, from the IIB brane configuration used to obtain the N = 6 theories in [17], supplemented by our defects as in [16], it is clear that there are no spatial directions along which one could T-dualize to obtain a higher-dimensional theory with a conventional superspace; either the probe branes or the ABJM configuration itself breaks the needed higherdimensional translation symmetries). However, the couplings of the A i , B j fields to the Qs andQs can be inferred by the following logic. Under translations of the M2 branes along the 34 directions, the Q,Q degrees of freedom should remain massless, while other motions should separate the M2s and M2 ′ s and give Q,Q a mass. In a standard way, one can identify motion in the transverse space to the M2 branes with (eigenvalues of) appropriate gauge-invariant composites of the A, B fields. First, we identify motion in the 34 directions with A 1 B 1 + A 2 B 2 . Then, we expect component couplings localized at the defects depending on the other bilinears in A i , B i ; these are of the form
∆S = dt i |(A 1 B 1 −A 2 B 2 )Q i | 2 +|(A 1 B 2 −A 2 B 1 )Q i | 2 + |(A 1 B 2 + A 2 B 1 )Q i | 2 (6)
with similar couplings toQ i . For the fermions, there are related couplings
∆S = dtχ α Γ M αβ X M χ β(7)
with X M corresponding to the real and imaginary parts of Gaugeinvariant composite operators can be formed from these fields. We will shortly compute the dimensions of low-lying defect operators at strong 't Hooft coupling and large N using the gravity side of the correspondence, and then comment on the field theory description of these operators.
A 1 B 1 −A 2 B 2 , A 1 B 2 ±A 2 B 1
Computation of operator dimensions using holography A standard extension of the holographic dictionary relates the dimensions ∆ of scalar operators localized at the lattice sites in our construction, to the masses of scalar KK modes arising in the M2 ′ brane world-volume action, via the formula
m 2 localized = ∆(∆ − 1) .(8)
The fermionic spectrum may be inferred by supersymmetry. We briefly discuss the calculation in the simplest case, k = 1. The fluctuations of the transverse scalars to a given M2 ′ brane (the x I = x 5 , x 6 , .., x 10 directions in space) are all related by an SO(6) symmetry, so we may focus on a single scalar. The M2 ′ brane wraps an AdS 2 × S 1 geometry. The fluctuations can be expanded in Fourier modes on the S 1 . If we let r denote the radial coordinate in AdS 2 and focus on static fluctuations, then
δx I (r, φ) = l δx I,l (r)e ilφ(9)
with φ the angular coordinate on the wrapped S 1 . The resulting Laplace equation for δx I,l (r) reveals that
m 2 l = − 1 4 + l 2 4(10)
which corresponds to scalar operators of dimension
∆ l = 1 2 + l 2 .(11)
The lowest operator in the tower, with l = 0, gives a sextet of scalar primaries with ∆ = 1/2; its Fermi partner is a quartet of ∆ = 1 fermionic defect operators. We will see in the next subsection that this ∆ = 1 multiplet of fermionic operators plays an important role in obtaining semi-holographic descriptions of marginal Fermi liquids. There is also a second tower of operators, arising from fluctuations of the M2 ′ branes along the two transverse spatial directions to their worldvolume in AdS 4 , i.e. the x 1,2 directions in (1). The tower arising from these fluctuations is distinguished from the tower above by global quantum numbers. For example, the fluctuations in the AdS directions transform under the SO(2) rotation symmetry of the x 1,2 plane (which is broken to Z 4 by the lattice), and are singlets under the SO(6) global symmetry discussed above, while the fluctuations in the x 5,···10 directions transform non-trivially under SO(6) but are Z 4 invariant. While this second tower contains some fermionic operators of ∆ = 1/2 which would be dangerous if they coupled to the semi-holographic fermions, such couplings can be forbidden by the SO(6) × Z 4 symmetry in a "natural" way (in the sense of the renormalization group).
The spectrum for higher k may be most easily inferred from the k = 1 case by the following logic. We can obtain the higher k brane configurations by Z k orbifolds of appropriate lattice configurations on AdS 4 × S 7 . The orbifold action is free on the S 7 (the fixed point at z i = 0 in C 4 is removed in the near-horizon limit), and therefore, all of the low-lying modes in the orbifold theory are Z k invariant modes in the original k = 1 theory. Correlation functions of the dual operators will enjoy large N inheritance from the parent k = 1 theory, similarly to the theories discussed in [19]. (New degrees of freedom that might be introduced by the orbifolding, analogous to twisted states in string theory, are very massive in the supergravity regime, due to the free orbifold action). A simple analysis following this logic implies that the spectrum is the same for all k > 1; so in particular, ∆ = 1 fermionic operators arise in these theories (and any lower ∆ fermionic operators from the second tower can rendered safe as above, by using global quantum numbers). A careful discussion of the KK spectra of these theories, and the matching with operators in the dual defect field theories, will appear in [20].
Coupling to semi-holographic fermions The theory we have constructed above is locally critical in the large N limit. That is, because the probe M2 ′ branes wrap AdS 2 slices of the AdS 4 geometry, the excitations of the bulk fields localized on the probe branes can be classified by the quantum numbers of a locally critical quantum theory, and the correlation functions of the operators dual to localized bulk excitations (computed using the standard AdS/CFT dictionary) obey the constraints following from local criticality. These are precisely correlation functions of operators involving defect fields in the dual field theory. Now, we couple the defect field theory we have constructed to semi-holographic fermions, following [7]. Namely, if we call the full action of the lattice system above (including both the bulk gauge theory and the defect fields) S LC , we consider the theory with
S total = S LC (A, B, Q,Q)+ J,J ′ dt c † J (iδ J,J ′ ∂ t + µδ J,J ′ + t J,J ′ )c J ′ + g J dt (c † J O F J + Hermitian conjugate) .(12)
In (12), we are coupling a normal theory of a weakly coupled Fermi surface (governing the excitations of the c fermion) to the strongly coupled locally critical sector, through the coupling constant g mixing c with (in any natural theory) the lowest dimension fermionic operator O F that has the right quantum numbers to couple to c. Using large N factorization, it is then easy to show that the g = 0 Green's function of the c fermion
G 0 (k, ω) ∼ 1 ω − v|k − k F (k)|(13)
is modified to
G g (k, ω) ∼ 1 ω − v|k − k F (k)| − g 2 G(k, ω) ,(14)
where
G(ω) = dt e iωt O F J (t)O F † J (0) .(15)
This two-point function is fixed by the scaling symmetry of the LC theory to be G(ω) = c ∆ ω 2∆−1 where ∆ is the dimension of O F (and, importantly, G(ω) ∼ c ω log(ω) in the degenerate case ∆ = 1). The correction term in the denominator of G g will dominate the low-frequency behavior if ∆ ≤ 1. Unitarity allows any ∆ ≥ 1 2 and this scaling dimension is a free parameter in the general approaches of [4,7]. The marginal Fermi liquid behavior of [2] appears in the case that the dimension of O F is precisely 1. Therefore, the question is, are there natural circumstances in which the theory S LC (A, B, Q,Q) has a leading fermionic operator of ∆ = 1 which can couple to c?
The theories we have constructed above naturally come with defect operators of ∆ = 1, as indicated by our calculation of the KK spectrum on the probe M2 ′ branes. It is interesting to consider where these come from in field theory language. The field theory has gauge-invariant operators of the form
∂ tQ1 Aχ 2 , ∂ tQ2 Bχ 1 , ∂ t Q 1 Bχ 2 , ∂ t Q 2 Aχ 1 .(16)
(as well as related quartets of operators of the schematic form χ 1 ψ A χ 2 , · · · andχ 1 A∂ t Q 2 , · · · ). These have ∆ = 1 at weak coupling, and are good candidates for the duals of the probe defect operators we computed on the gravity side (arising in the tower of fluctutations of the M2 ′ branes along x 5,··· ,10 ). Suppose that upon extrapolating to strong coupling (at large N), the weak-coupling dimensions of these operators are indeed protected, i.e. that the weak-coupling engineering dimensions of the fields correspond to their scaling dimensions under the locally critical scaling governing the defect sector in the probe limit. Then, assigning appropriate global quantum numbers to c, one can choose one of these as the lowest dimension fermionic operator that c can couple to in the localized sector.
Returning to the dual gravitational description, we can see that the idea above does work at least in the probe approximation. By appropriate choice of global quantum numbers (under the Z 4 lattice symmetry and the (subgroup of) SO(6) preserved by the brane configuration), one can guarantee that no lower ∆ operators from the second tower of fluctuations in the previous subsection infect the leading-order c-fermion correlators (14) after coupling to the large N sector. We conclude that we can work directly in the probe limit and obtain a marginal Fermi liquid by identifying O F with the lowest fermionic operator in the first tower of defect fields computed above. This has ∆ = 1, and as emphasized in the introduction, this dimension is independent of momentum.
Backreaction Up until now we have ignored the backreaction of the impurities on the itinerant fields, and therefore on each other. Thus we have been studying the dynamics of a single impurity interacting strongly with itinerant fields. The gravity side exhibits the successes it does because the probe branes each wrap an AdS 2 region, and the symmetries of local quantum criticality are manifest, even including the highly nontrivial field theory interactions that are re-summed by the tree-level gravity solution.
At scales of order the lattice spacing the backreaction is a 1/N effect, but at lower energies it must become important. The scale symmetry of the itinerant fields, which the impurity system inherits, acts on the spatial coordinates. At energies of order N −1/2 times the fundamental scale the number of impurities in a scaling volume is of order N , and the effect of the impurities on the itinerant fields and on each other can no longer be neglected. Do these effects inevitably generate corrections to the action which destroy the locally critical behavior -is the behavior seen in the gravity regime a peculiarity of very strongly coupled large N theories, which would not extrapolate to any more realistic systems -or can it be robust in some circumstances? And, if locally critical behavior survives to the far IR, how do the operator dimensions there relate to those we have found at higher energy?
Staying in the limit of strong 't Hooft coupling, gauge/gravity duality transforms this field theory question into the problem of finding the supergravity solution with backreaction. This can still be a challenging problem, but one can get insight from a simple energetics argument. We start with the M theory brane configuration (1). We are looking for an IR geometry AdS 2 × R 2 × X, which we will for convenience compactify to AdS 2 × T 2 × X. We study this with the Ansatz X = S 7 , averaging the energy density of the impurity 2 ′ branes over the compact dimensions. Let A, T , and S be the respective radii of the three factors AdS 2 × T 2 × S 7 . The effective action dimensionally reduced to 1+1 dimensions is of the form
S = d 2 x −T 2 S 7 + A 2 T 2 S 5 − N ′ 2 A 2 S − N 2 2 A 2 T 2 S 7 .
(17) We work in units where the M theory scale is one, and ignore order one coefficients. The respective terms come from the curvatures of AdS 2 and S 7 , the 2 ′ -brane tensions, and the 7form flux from the 2-branes. In other situations it would be natural to Weyl transform to an effective potential, but this is not possible for AdS 2 ; instead we directly extremize with respect to A in addition to T and S.
One finds that there is an extremum (with physically acceptable positive values for the moduli) such that
A ∼ S ∼ N 1/6 2 , T ∼ N ′1/2 2 /N 1/3 2 .(18)
The radius S is parametrically the same as for the pure M2 system. The density of defects is N ′
2 /T 2 = N 2/3 2 .
What is happening is that the lattice defects provide a force acting against the contraction of the two spatial dimensions, hence helping to drive the system towards a fixed point where the bulk modes are locally critical. In the probe approximation, the itinerant fields retained their relativistic scaling, and each independent impurity was invariant under a scale transformation leaving its position fixed. Here there is a common locally critical scaling of the whole geometry. This result is encouraging, but we should improve the Ansatz. We have averaged the action of the 2 ′ branes over the S 7 , but in fact they are wrapped on a circle and we should consider moduli corresponding to the contraction of this circle. Thus we represent S 7 as a circle over CP 3 , with radius F for the fiber circle and B for the base. The action becomes
S = d 2 x −T 2 F B 6 + A 2 T 2 F B 4 − A 2 T 2 F 3 B 2 −N ′ 2 A 2 F − N 2 2 A 2 T 2 F B 6 .(19)
One now finds that there is no physical extremum; the contraction of the fiber is not stabilized. Nevertheless, there are brane systems that realize the solution (18). Consider a system with several kinds of impurity brane, with different orientations in the transverse spacetime. If the configuration of M2 ′ branes is sufficiently uniform and isotropic, the spherical Ansatz will be a good approximation. [30] Such a configuration will necessarily break supersymmetry (for supersymmetric configurations, at least with N ≥ 2, there will always be an unstable fiber circle). It is also necessary to stabilize the angular configuration, for example by taking a sufficiently symmetric configuration, and by keeping relatively nonsupersymmetric branes far enough apart to avoid tachyons. With the scaling (18) the typical transverse distance between the branes is larger than the M theory scale, so one expects that the latter difficulty may be avoided. Although with a symmetric distribution there should be a solution of the equations of motion, it may be an unstable saddle point; with the lack of supersymmetry there is no a priori guarantee against disallowed tachyons. Without having addressed all the possible instabilities, something that might benefit from further model building, we simply take from this construction the lesson already noted that lattice flavors contribute to producing local criticality on the gravity side.
As an aside, the absence of supersymmetric solutions could also be anticipated from another point of view. We are looking for solutions where the color branes remain localized in the 3-4 directions in which the impurity branes are extended. In Refs. [22] it is shown that these do not exist for brane intersections of spatial dimension 0 (as here) or 1. The interpretation was that the scalar fields Q on the intersection are spread out on their moduli space due to low-dimensional quantum effects, which implies that the brane intersection delocalizes and the AdS IR region disappears. In nonsupersymmetric systems, masses will generically be generated for these scalars. In the appendix we analyze an impurity system that has no such impurity scalars.
Orbifolding by Z k does not affect the energetics, and so the discussion above can be applied with N 2 → N k, giving in M theory units A ∼ S ∼ N 1/6 k 1/6 , R 11 ∼ N 1/6 /k 5/6 ,
T ∼ N ′1/2 2 /N 1/3 k 1/3(20)
and in string units
A ∼ S ∼ N 1/4 k 1/4 , g s ∼ N 1/4 /k 5/4 , T ∼ N ′1/2 2 /N 1/4 k 1/4 .(21)
The same applies if the orbifold action (2) is replaced by one acting only on two complex coordinates z 3,4 , generating the brane configuration 0 1 2 3 4 5 6 7 8 9 D2 x x x D6 x x x x x x x D2 ′ x :: :: x x (22) with N color D2-branes and k D6-branes. This is a nice example, having a weakly coupled conformal point for N 2 ≪ N 6 (as in Refs. [23]) and an AdS 4 dual description for N 2 ≫ N 6 [24]. The radius S and coupling g s are parametrically the same as for the pure D2-D6 system. In particular one sees that the condition that the radius be large (in string units) is N 2 ≫ N 6 , and that there then is a weakly coupled IIA dual for N 2 ≪ N 5 6 and an M-theory dual for N 2 ≫ N 5 6 . The density of defects is N ′
2 /T 2 = N 1/2 2 N 1/2
6 . Even if we find a supergravity solution, there is a general argument that suggests that the local critical scaling cannot persist indefinitely into the IR. The scaling would imply a density of states
ρ(E) = Aδ(E) + B/E(23)
per energy (and exponential in the volume). The first term is the widely noted zero-temperature entropy. If only this term is present, the Hamiltonian in the critical sector is zero: there is no dynamics (e.g. a dimension 1 operator would have a correlator δ ′ (t) rather than 1/t 2 ). So the B term is necessary, but its integral diverges, so local criticality must always break down at sufficiently low energy. In the gravity description, the density B comes from bulk states, and so is of order 1/N 2 . Thus the breakdown takes place at exponentially small scales, which seems more promising than the N −1/2 breakdown scale of the probe approximation.
Ref. [8] identified a specific breakdown mechanism, whereby the scaling exponents of the spatial directions were shifted (at all scales) from 0 to O(1/N ), thus rendering the density of states convergent. This is a rather special property of the system studied there. More generally, local criticality might persist until the finite density of states per volume forces it to break down.
Backreaction at weak coupling It is encouraging that we have found possible stable systems with the desired IR properties, but the gravity methods are still only controlled in a peculiar limit, from the field theory perspective. Here we discuss some related issues in direct analysis of the dual field theory. We start with the field theory corresponding to the brane system (22). This is an N = 8 supersymmetric 3d Yang-Mills theory, with defect hypermultiplets. In such theories, with a Maxwell action, the conformal symmetry that will emerge in the IR is far from manifest. A second approach, via the Chern-Simons theories of [17], has been the one we've followed in the bulk of the paper. The IR conformal behavior of the bulk theory is much clearer here, as the gauge fields do not appear with a dimensionful coupling, and the starting (bulk) Lagrangian has no dimensionful parameters. It is interesting to contrast our expectations for radiative corrections arising from the two approaches.
Starting from the 3d N = 8 Yang-Mills theory with hypermultiplet defects, and following the techniques of [25], it is easy to write a superspace Lagrangian. The problems with finding a 4d N = 1 superspace do not arise in this perspective; the additional complications of the ABJM brane construction [17] are not present, and one can straightforwardly T-dualize to find an N = 1 presentation. In terms of the brane construction with D2 branes wrapping x 1,2 and D2 ′ branes wrapping x 3,4 , it is convenient to perform the T-duality is along the 7, 8, 9 directions and to treat those as the spatial directions of the N = 1 field theory, with x 1,2 being internal dimensions. The bulk action is
S = 1 g 2 3 dtd 2 x T r[ d 2 θ 1 2 W α W α + ǫ ijk φ i (∂ j φ k − [φ j , φ k ]/3 √ 2) + h.c. + 2 d 4 θ( √ 2∂ i +φ i )e −V (− √ 2∂ i + φ i )e V +∂ i e −V ∂ i e V ] + WZW term .(24)
Here, The index n runs over the lattice sites, and n subscripts on a bulk field simply indicate that the field is to be evaluated at position of the nth site. This has the intuitively expected features; for instance, motions of the D2 branes along x 5,6,7,8,9 , given the correspondence with fields above, can be seen to mass up the defect hypermultiplets.
∂ 1 = ∂ x 1 + i∂ x 2 , while ∂ 2,3 → 0, and (φ i ) † =φ i . W α is an SU (N )
Integrating out the auxiliary D-field in the gauge multiplet generates inter-defect interactions. For simplicity we focus on the Abelian (N = 1) case; defect hypermultiplet scalars are denoted by η. Then the couplings of the auxiliary field are:
S D = 1 g 2 3 dt d 2 x ( 1 2 D 2 − 2 √ 2(φ 1∂ 1 D +φ 1 ∂ 1 D) +φ 1φ 1 ) + 1 2 n D n (|η c n | 2 − |η n | 2 ) . (26)
Integrating out D, the action becomes:
S D = 1 g 2 3 dt d 2 x (−2[∂ 1 Z 1 + ∂ 1Z 1 ] 2 + |Ż 1 −ζ| 2 ) (27)
where we've defined
ζ(z 1 ) = 1 8π √ 2 n (|η c n | 2 − |η n | 2 ) z 1 − z 1n(28)
and
φ 1 = Z 1 − ζ .(29)
The |ζ| 2 term in (27) exhibits cross-couplings between the η hypermultiplet fields that would naively ruin local criticality. One would also get similar terms by integrating out A 0 and φ 3 . The generation of inter-defect interactions is not tied to supersymmetry, but these terms sum to a cross-coupling term in the Kähler potential for the defect hypermultiplets. [31] This makes it seem unlikely that the local criticality of the gravity regime can survive to finite N and coupling, where a field theory analysis should be reliable. However, it is important to remember that our starting point here has been the 3d N = 8 Yang-Mills theory, and this UV Lagrangian is valid only far from the IR fixed point which we know governs the physics on the N M2 branes (even at finite N ).
To get an alternate perspective, we can also try to compute the inter-defect corrections arising from coupling the defect hypermultiplets to the doubled Chern-Simons theory which captures the fixed-point physics. In fact, a simple toymodel already illustrates the important difference between the Chern-Simons defect theories and the Yang-Mills defect theories. An Abelian Chern-Simons gauge field coupled to defect fermions χ n would be governed by an action
S = dt d 2 z [A 0 (∂ z Az − ∂zA z ) − A z (∂ 0 Az − ∂zA 0 ) + Az(∂ 0 A z − ∂ z A 0 ) + n δ (2) (z − z n )χ † n A 0 χ n ] .(30)
One can see directly that integrating out A 0 will not generate a dangerous inter-defect coupling here, as it is a nonpropagating field. The A and B fields do propagate, but these couple to the defect fields only quadratically as in Eqs. (6,7) and so do not generate tree level corrections. A full field-theoretic analysis of the radiative corrections to the ABJM theory coupled to hypermultiplet defects is beyond the scope of our work. It will be interesting to see to what extent the absence of induced inter-defect couplings applies in the full model; the simple computation above suggests that at least the most obvious dangerous cross-couplings visible from the Yang-Mills perspective, do not characterize the physics of the IR fixed point theory coupled to hypermultiplet defects. Especially in the cases k = 1, 2, where the full model enjoys enhanced supersymmetry, non-renormalization theorems strongly constrain the possible generation of fourfermion cross-coupling terms (see for instance [26]); constraints on higher multi-fermion terms are less obvious. It would be most interesting to push this analysis further, and construct systems of defect fermions interacting with itinerant fields where local criticality can be seen robustly directly from field theoretic arguments. acknowledge the hospitality of the Aspen Center for Physics while this work was in progress. S.K. also acknowledges the hospitality of the organisers of the 5th Asian Winter School at Jeju Island, and thanks the participants for asking many interesting questions about related subjects. This research was supported in part by the National Science Foundation under grants PHY-02-44728, PHY05-51164 and PHY07-57035, and by the DOE under contracts DE-AC03-76SF00515 and DE-FG02-96ER40956. KJ was supported by NSERC, Canada.
Appendix: The 3.5 system To begin let us consider a variant of the construction of [11], who studied the brane configuration 0 1 2 3 4 5 6 7 8 9 D3
x x x x D5(5) x :: :: :: x x x x x (31) As before, an x indicates a direction in which the given branes are extended, and a :: indicates a direction in which they are in a lattice configuration. The 3-5 intersections are 0 + 1 dimensional, representing defects in the dual gauge theory. For this system, with 8 ND directions, only fermions live on the intersections, which is very natural for the intended applications.
In the limit that the 5-branes are probes, the D3-branes generate an AdS 5 × S 5 spacetime, with each 5-brane wrapped on an AdS 2 × S 4 subspace. However, the spatial directions contract in the IR of the AdS 5 geometry, so the 5-brane density diverges there and their backreaction cannot be neglected. At large N , the backreaction becomes a large effect at energies which are parametrically small compared to the lattice scale (as noted in [11]). [32] We are looking for an IR geometry AdS 2 × R 3 × X, which we will for convenience compactify to AdS 2 × T 3 × X. We study this with the Ansatz X = S 5 , averaging the energy density of the 5-branes over the compact dimensions. Let A, T , and S be the respective radii of the three factors AdS 2 × T 3 × S 5 . The effective action dimensionally reduced to 1+1 dimensions is of the form
S = d 2 x − T 3 S 5 g 2 s + A 2 T 3 S 3 g 2 s − N 5 A 2 S 4 g s − N 2 3 A 2 T 3 S 5 .
(32) We work in units where the string length is one, and ignore order one coefficients. The respective terms come from the curvatures of AdS 2 and S 5 , the 5-brane tensions, and the RR 5-form flux, and do not distinguish between pure D5-branes and a mix of D5s and D5s. In other situations it would be natural to Weyl transform to an effective potential, but this is not possible for AdS 2 ; instead we directly extremize with respect to A. One readily verifies that the action has no stationary points for finite values of the moduli A, T, S, g s . This analysis precludes an AdS 2 × T 3 × S 5 solution in the case that the 5-branes are oriented in many directions on the S 5 , averaging to a symmetric source.
One way of understanding the absence of an AdS 2 solution in the infrared in this case is that the N = 4 super Yang-Mills sector has a line of fixed points, parameterized by the string coupling g s . The additional lattice branes source this mode and altogether there are not enough independent forces to fix g s , T, S, and A. If we include electric and magnetic flavors, these can fix g s . Having done this, an AdS 2 solution fixing the other moduli does arise.
and α, β spinor indices running over the directions transverse to both the M2s and the M2 ′ s. The dimensions of the fields determined from their kinetic terms at weak coupling are ∆(Q) = ∆(Q) = − 1 2 , ∆(χ) = ∆(χ) = 0, and ∆(A) = ∆(B) = 1 2 .
gauge field strength superfield, while V is the vector superfield. In 3d N = 4 language, one should think of φ 1,2 as the scalars in a hypermultiplet and φ 3 as the complex adjoint scalar in the vector multiplet. In Wess-Zumino gauge, the WZW term vanishes. The fields in the above action can be interpreted as follows: D2 gauge field Wilson lines along x 1,2 and D2 motions alongx 3,4 are packaged in φ 1,2 ; D2 motions along x 5,6 are contained in φ 3 ; and the vector multiplet V has θθ components consisting of A 0 and x 7,8,9 . The hypermultiplets H, which transform in the fundamental of SU (N ), have localized actions n dt d 4 θ (H c n e VnH c n +H n e −Vn H n ) − d 2 θ H c n φ 3,n H n − h.c. . (25)
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Other approaches to a microscopic construction of AdS2 duals include Refs. 10Other approaches to a microscopic construction of AdS2 duals include Refs. [9, 10].
In the special cases k = 1, 2, the supersymmetry is further enhanced to N = 8 and the global symmetry to. In the special cases k = 1, 2, the supersymmetry is further enhanced to N = 8 and the global symmetry to SO(8).
Such smeared sources have been studied in Ref. 21Such smeared sources have been studied in Ref. [21].
There is also a log-divergent same-site kinetic term which we believe can be cancelled by a renormalization of this Kähler potential. There is also a log-divergent same-site kinetic term which we believe can be cancelled by a renormalization of this Kähler potential.
Note that the lattice breaks all the conformal symmetries of AdS5: the embedding geometry of each defect is invariant under a different SO(2, 2). The fully backreacted solutions for a single stack of D5 defects will be discussed in. 27Note that the lattice breaks all the conformal symmetries of AdS5: the embedding geometry of each defect is invariant under a different SO(2, 2). The fully backreacted solutions for a single stack of D5 defects will be discussed in [27].
| {'fraction_non_alphanumeric': 0.0534716996078742, 'fraction_numerical': 0.03537053907394938, 'mean_word_length': 4.097415707651928, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 36, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We present an infinite class of 2+1 dimensional field theories which, after coupling to semi-holographic fermions, exhibit strange metallic behavior in a suitable large N limit. These theories describe lattices of hypermultiplet defects interacting with parity-preserving supersymmetric Chern-Simons theories with U (N ) × U (N ) gauge groups at levels ±k. They have dual gravitational descriptions in terms of lattices of probe M2 branes in AdS4 × S 7 /Z k (for N ≫ 1, N ≫ k 5 ) or probe D2 branes in AdS4 × CP 3 (for N ≫ k ≫ 1, N ≪ k 5 ). We discuss several challenges one faces in maintaining the success of these models at finite N , including backreaction of the probes in the gravity solutions and radiative corrections in the weakly coupled field theory limit.', 'arxivid': '1105.1772', 'author': ['Kristan Jensen \nDepartment of Physics\nUniversity of Victoria\nV8W 3P6VictoriaBCCanada\n', 'Shamit Kachru \nDepartment of Physics\nStanford University and SLAC\n94305StanfordCAUSA\n', 'Andreas Karch \nDepartment of Physics\nUniversity of Washington\n98195SeattleWAUSA\n', 'Joseph Polchinski \nKITP and Department of Physics\nUCSB\n93106Santa BarbaraCAUSA\n', 'Eva Silverstein \nDepartment of Physics\nStanford University and SLAC\n94305StanfordCAUSA\n'], 'authoraffiliation': ['Department of Physics\nUniversity of Victoria\nV8W 3P6VictoriaBCCanada', 'Department of Physics\nStanford University and SLAC\n94305StanfordCAUSA', 'Department of Physics\nUniversity of Washington\n98195SeattleWAUSA', 'KITP and Department of Physics\nUCSB\n93106Santa BarbaraCAUSA', 'Department of Physics\nStanford University and SLAC\n94305StanfordCAUSA'], 'corpusid': 54018109, 'doi': '10.1103/physrevd.84.126002', 'github_urls': [], 'n_tokens_mistral': 15618, 'n_tokens_neox': 13280, 'n_words': 8665, 'pdfsha': 'c54069f5a40a59abec451a7f67343c6a5e1cf38a', 'pdfurls': ['https://arxiv.org/pdf/1105.1772v2.pdf'], 'title': ['Towards a holographic marginal Fermi liquid', 'Towards a holographic marginal Fermi liquid'], 'venue': []} |
arxiv |
ANCIENT SOLUTION OF MEAN CURVATURE FLOW IN SPACE FORMS
28 Nov 2019
L I Lei
ANDHongwei Xu
Entao Zhao
ANCIENT SOLUTION OF MEAN CURVATURE FLOW IN SPACE FORMS
28 Nov 2019
In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.|H| 4 + 4(n − 1)c|H| 2 . It is easy to check that α(n, |H|, c) is strictly bigger 2010 Mathematics Subject Classification. 53C44, 53C40. Key words and phrases. Ancient solution, mean curvature flow of submanifolds, pinching theorem, second fundamental form.
Introduction
Let F : M n × (T 1 , T 2 ) → N n+p be a smooth family of immersions into a Riemannian manifold that satisfies
(1) ∂ ∂t F (x, t) = H(x, t),
where H(x, t) is the mean curvature vector of the submanifold M t = F (M, t).
(1) is the negative gradient flow of the volume functional on the submanifolds and F (x, t) is called the solution of the mean curvature flow. The mean curvature flow of hypersurfaces has been investigated extensively, see for instance the fundamental papers of Huisken [15,16,17] on the smooth convergence theory of the mean curvature flow. For the mean curvature flow of submanifolds with higher codimensions, fruitful results were obtained. For example, following the work of Huisken [15,16,17], Andrews and Baker [1,3], Liu, Xu, Ye and Zhao [24,26] proved several smooth convergence theorems for the mean curvature flow of submanifolds of arbitrary codimensions under pointwise curvature pinching conditions. Motivated by the rigidity theorem for closed submanifolds with parallel mean curvature vector and the topological sphere theorem for complete submanifolds in space forms [32,36,37], it was proposed in [25,28] that the mean curvature flow of closed submanifolds satisfying initial curvature condition |h| 2 α(n, |H|, c) and |H| 2 + n 2 c > 0 in a complete simply connected space form F n+p (c) with constant curvature c = 0 would converges smoothly to a round point in finite time or to a totally geodesic sphere in F n+p (c) with c > 0. Here h denotes the second fundamental form of the submanifold and α(n, |H|, c) = nc + n|H| 2 2(n−1) − n−2 2(n−1) than 1 n−1 |H| 2 + 2c. Lei-Xu [20] proved the smooth convergence theorem of mean curvature flow in F n+p (c) with c < 0 and n 6 under the curvature condition |h| 2 α(n, |H|, c) and |H| 2 + n 2 c > 0. For the case c > 0, Lei-Xu [21] proved the smooth convergence theorem of mean curvature flow in F n+p (c) (n 6) under a sharp curvature condition |h| 2 γ(n, |H|, c). Meanwhile, some smooth convergence theorems for the mean curvature flow of arbitrary codimension in space forms under integral curvature pinching conditions were proved in [25,27]. See [22,28] for recent progress in the smooth convergence theory of mean curvature flow of arbitrary codimensions.
In the present paper, we focus on the ancient solution of the mean curvature flow, which is the solution of (1) on the time interval (−∞, T ) for some T < ∞. In recent years, various researchers have investigated ancient solutions of the mean curvature flow of codimension one. Closed convex ancient solutions of the curve shortening flow in the plane have been completely classified by Daskalopoulos-Hamilton-Sesum [12] to be either shrinking round circles or Angenent ovals. In higher dimension, a rigidity theorem was proved in [18] stating that a closed and convex ancient solution of the mean curvature flow on (−∞, 0) in the Euclidean space R n+1 with principal curvatures λ i and mean curvature function |H| of M t satisfying λ i ǫ|H| for a positive constant ǫ is a family of shrinking spheres. For the ancient solution in the sphere, Bryan-Louie [8] have proved that the only closed, embedded and convex ancient solutions to the curve shortening flow in the unit sphere S 2 are equators or shrinking circles, and for higher dimensional case some characterizations of spherical ancient solution in terms of curvature pinching have been given in [18]. In higher codimensional case, it was proved in [29,31] that the ancient solution to the mean curvature flow in the Euclidean space R n+p satisfying certain pointwise curvature pinching condition uniformly on (−∞, 0) is a family of shrinking spheres, and similar rigidity phenomenon holds for the ancient solutions in the unit sphere S n+p under the pinching condition |h| 2 1 n−1 |H| 2 + 2 for n 4 and similar but stronger pinching condition for n = 2, 3. It should be remarked that the curvature pinching conditions in [29,31] are proposed according to the convergence theorems for the mean curvature flow in [1,3]. On the other hand, Brendle-Choi [4] proved that the rotationally symmetric bowl soliton is the only complete noncompact ancient solution of mean curvature flow in R 3 which is strictly convex and noncollapsed, and they [5] also generalized this result to higher dimensional case assuming additionally that the solution is uniformly two-convex. For more results about ancient solutions of the mean curvature flow and other geometric flows, see [2,6,7,11,13,14,35], et.al.
In the late 1960's, Simons [34], Lawson [19], and Chern-do Carmo-Kobayashi [10] proved a famous rigidity theorem for n-dimensional compact minimal submanifolds in the unit sphere S n+p under the pinching condition |h| 2 n/ 2− 1 p . In [23], Li-Li improved the pinching constant to n/ 1 + 1 2 sgn(p − 1) . Inspired by the Simons-Lawson-Chern-do Carmo-Kobayashi-Li-Li rigidity theorem, we prove the following theorem for ancient solutions of mean curvature flow in the sphere. The term "spherical cap" means a family of shrinking totally umbilical spheres. Our pinching condition in Theorem 1 is sharp. In fact, the Clifford minimal hypersurfaces in S n+1 satisfy |h| 2 = n, and the Veronese minimal surface in S 4 satisfies |h| 2 = 4 3 . The Clifford hypersurfaces and Veronese surface are obviously static ancient solutions of the mean curvature flow, so the pinching condition for the case p = 1 or the case n = 2 and p = 2 in Theorem 1 is optimal.
Theorem 1 is a direct consequence of the following
(i) lim t→−∞ max Mt |h| 2 − min 3 n+2 , 4(n−1) n(n+2) |H| 2 < n, for p = 1, (ii) lim t→−∞ max Mt |h| 2 − 4 3n |H| 2 < 2n 3 , for p = 2, (iii) lim t→−∞ max Mt |h| 2 − 4 3n |H| 2 < 3n 5 or lim t→−∞ max Mt |h| 2 − 1 n |H| 2 < 2n 3 , for p 3.
Then M t is either a shrinking spherical cap or a totally geodesic sphere.
The curvature condition in Theorem 2 is obviously weaker than those in [29,31]. It should be mentioned that the pinching constant in Theorem 2 is sharp.
If the ambient space is the hyperbolic space H n+p with constant curvature −1, we prove the following rigidity theorem. In Theorem 3,h is the tracefree second fundamental form of a submanifold. As far as we know, Theorem 3 is the first result in the literature about the rigidity of ancient solutions of mean curvature flow in the hyperbolic space. Now we give more details on the shrinking spheres in the hyperbolic space. A geodesic sphere of radius r in the hyperbolic space has mean curvature n coth r. Thus, the family of geodesic spheres whose radii are given by r(t) = arccosh e −nt is an ancient solution of mean curvature flow. Furthermore, the mean curvature functions of the geodesic spheres are |H|(t) = ne −nt / √ e −2nt − 1. We see that |H|(t) are always larger than n, and tend to n as t → −∞.
The rigidity of submanifolds under integral curvature pinching condition is also an attractive topic in submanifold theory. For instance, Shiomaha, Xu and Gu [33,38,39] proved several geometric and topological rigidity theorems for submanifolds in F n+p (c) with c 0 under the pinching condition of the curvature integral under conformal transformations of the ambient space and is a higher dimensional generalization of the classical Willmore functional for 2-dimensional surfaces. Inspired by these global rigidity theorems, we prove the following
Theorem 4. Let F : M n × (−∞, 0) → F n+p (c) be a compact ancient solution of mean curvature flow in a space form F n+p (c) with c 0. Suppose there holds lim t→−∞ Mt |h| n dµ t < C(n),
where C(n) is a positive constant explicitly depending on n. Then M t is either a shrinking sphere or a totally geodesic sphere in F n+p (c) with c > 0.
Theorem 4 is also the first result as far as we know in the literature about the rigidity of ancient solutions of mean curvature flow under integral curvature pinching condition.
The paper is organized as follows. In Section 2, we recall the evolution equations along the mean curvature flow in space forms and derive several curvature inequalities which will be used in the proofs of theorems. In Section 3, we first prove several rigidity theorems for ancient solutions of mean curvature flow in the sphere, which combined together verify Theorems 1 and 2, then we give the proof of Theorem 3. Theorem 4 will be proved in Section 4.
Preliminaries
Let M n be an n-dimensional Riemannian submanifold immersed in an (n + p)dimensional simply connected space form F n+p (c) with constant curvature c. We denote by ∇ and ∇ the Levi-Civita connections of the submanifold M and the ambient space F n+p (c), respectively. The second fundamental form of M is defined as
h(u, v) = ∇ u v − ∇ u v for tangent vector fields u, v on M .
Let {e i |1 i n} be a local orthonormal frame for the tangent bundle and {ν α |1 α p} be a local orthonormal frame for the normal bundle. In a local frame, we set h(e i , e j ) = α h α ij ν α . The mean curvature vector is given by
H = α H α ν α , H α = i h α ii . Leth = h − 1 n H ⊗ g be the tracefree second fundamental form. Its norm has the relation |h| 2 = |h| 2 − 1 n |H| 2 . Let F : M n × I → F n+p (c) be a mean curvature flow. Let M t = F (M, t)
be the submanifold at t. Andrews and Baker derived the following evolution equations along the mean curvature flow [1,3].
Lemma 1. For the mean curvature flow F : M n × I → F n+p (c), we have (i) ∂ ∂t |h| 2 = ∆|h| 2 − 2|∇h| 2 + 2R 1 + 4c|H| 2 − 2nc|h| 2 , (ii) ∂ ∂t |H| 2 = ∆|H| 2 − 2|∇H| 2 + 2R 2 + 2nc|H| 2 , (iii) ∂ ∂t |h| 2 = ∆|h| 2 − 2|∇h| 2 + 2R 1 − 2 n R 2 − 2nc|h| 2 , where R 1 = α,β i,j h α ij h β ij 2 + i,j,α,β k (h α ik h β jk − h β ik h α jk ) 2 , R 2 = i,j α H α h α ij 2 .
In particular, if the codimension is 1, then
R 1 = |h| 4 , R 2 = |H| 2 |h| 2 .
Huisken [15] obtained the following estimate for the gradient terms
(2) |∇h| 2 3 n + 2 |∇H| 2 , which implies (3) |∇h| 2 = |∇h| 2 − 1 n |∇H| 2 2(n − 1) n(n + 2) |∇H| 2 .
We will need the following matrix inequality due to Li-Li [23].
Proposition 1. Let A 1 , · · · , A p be p( 2) symmetric matrices. Then α,β |A α A β − A β A α | 2 + α,β [tr(A α A β )] 2 3 2 α |A α | 2 2 .
Using Proposition 1, we have
R 1 = α,β i,jh α ijh β ij 2 + i,j,α,β k (h α ikh β jk −h β ikh α jk ) 2 + 2 n i,j α H αhα ij 2 + 1 n 2 |H| 4 3 2 |h| 4 + 2 n i,j α H αhα ij 2 + 1 n 2 |H| 4 (4) = 3 2 |h| 4 + 2 n R 2 − 1 n 2 |H| 4 .
At a point on M , we choose an orthonormal frame {ν α } for the normal space, such that H = |H|ν 1 , and an orthonormal frame {e i } for the tangent space, such
that h 1 ij = λ i δ ij . Then h1 ij is also diagonal, we denote its diagonal elements bẙ λ i . Thusλ i = λ i − 1 n |H| andh α ij = h α ij for α > 1. With the special frame, R 1 becomes R 1 = i λ 2 i 2 + 2 α>1 iλ ih α ii 2 + α,β>1 i,jh α ijh β ij 2 +2 α>1 i =j λ i −λ j h α ij 2 + α,β>1 i,j k h α ikh β jk −h α jkh β ik 2 .
We use Cauchy-Schwarz inequality to get
α>1 iλ ih α ii 2 α>1 iλ 2 i i h α ii 2 .
We also have
α>1 i =j λ i −λ j h α ij 2 α>1 i =j 2 λ 2 i +λ 2 j h α ij 2 2 α>1 iλ 2 i i =j h α ij 2 .
Thus we get
α>1 iλ ih α ii 2 + α>1 i =j λ i −λ j h α ij 2 2 iλ 2 i α>1 i,j h α ij 2 .
Using Proposition 1, we obtain
α,β>1 i,jh α ijh β ij 2 + α,β>1 i,j k h α ikh β jk −h α jkh β ik 2 ξ α>1 i,j h α ij 2 2 , whereξ = 1 + 1 2 sgn(p − 2)
. Therefore, we obtain
R 1 i λ 2 i 2 + 4 iλ 2 i α>1 i,j h α ij 2 +ξ α>1 i,j h α ij 2 2 . We set P = α>1 i,j h α ij 2 .
This gives i λ 2 i = |h| 2 − P and iλ 2 i = |h| 2 − P . Then we obtain
(5) R 1 |h| 4 + 2|h| 2 − 2 n |H| 2 P − ξ −1 P 2 , where ξ = 2 4−sgn(p−2)
. We also have
(6) R 2 = i (|H|λ i ) 2 = |H| 2 (|h| 2 − P ).
Pointwisely pinched ancient solutions in curved space forms
In this section we consider the rigidity of ancient solutions of mean curvature flow in curved space forms under pointwise curvature pinching conditions.
Ancient solutions in spheres.
We first consider codimension 1 ancient solutions and prove the following Proof. Set γ = 2(n − 1) n(n + 2) and k = min γ, 2γ − 1 n .
From the pinching condition, there exists a positive constant a, such that for all t < 0 there holds |h| 2 − k|H| 2 < a < n.
We study the function
f = |h| 2 γ|H| 2 + a .
Notice that f is always less than 1.
We have
∂ ∂t f = 1 γ|H| 2 + a ∂ ∂t |h| 2 − γf ∂ ∂t |H| 2 and ∆f = 1 γ|H| 2 + a ∆|h| 2 − γf ∆|H| 2 − 2γ ∇f, ∇|H| 2 .
Then we get from the evolution equations that
∂ ∂t − ∆ f = 2γ γ|H| 2 + a ∇f, ∇|H| 2 − 2 γ|H| 2 + a (|∇h| 2 − γf |∇H| 2 ) +2f |h| 2 − n − γ γ|H| 2 + a (|H| 2 |h| 2 + n|H| 2 ) .(7)
Firstly, we estimate the gradient terms in (7). From (3) we have
|∇h| 2 γ|∇H| 2 γf |∇H| 2 .
Secondly, we estimate the reaction terms.
|h| 2 − n − γ γ|H| 2 + a (|H| 2 |h| 2 + n|H| 2 ) = a|h| 2 − n(2γ|H| 2 + a) γ|H| 2 + a a(2γ|H| 2 + a) − n(2γ|H| 2 + a) γ|H| 2 + a = (a − n)(2γ|H| 2 + a) γ|H| 2 + a a − n. Letting δ = n − a, we obtain ∂ ∂t − ∆ f 2γ γ|H| 2 + a ∇f, ∇|H| 2 − 2δf.
Applying the maximum principle, we get
∀s < t, max Mt f e −2δ(t−s) max Ms f.
Letting s → −∞, we obtain max Mt f = 0, i.e., M t is a totally umbilical sphere.
To derive the rigidity theorem in higher codimensions, we need the following inequality.
Lemma 2. Let b and ξ be positive numbers. Define
G(x, y) = 2b 4 3 x + b + x(y − 1) x + 2b − xy + ξ −1 y 2 1 3 x + b + 2y − 1. If 1 2 ξ < b −1 − 1, then sup x,y∈[0,+∞) G(x, y) < 0.
Proof. We write G(x, y) as a quadratic function of y.
G(x, y) = 2b 4 3 x + b − x x + 2b − 1 + b(7x + 12b) (x + 2b)(x + 3b) y − 3ξ −1 x + 3b y 2 .
So, for x 0, we have
sup y∈R G(x, y) = 2b 4 3 x + b − x x + 2b − 1 + ξ 12(x + 3b) b(7x + 12b) x + 2b 2 = [12(x + 2b) 2 (x + 3b)] −1 × (8) 144b 3 (bξ + b − 1) + 24b 2 (7bξ + 13b − 11)x +b(49bξ + 184b − 144)x 2 + 8(4b − 3)x 3 . From 1 2 ξ < b −1 − 1, we have bξ + b < 1, 7bξ + 13b < 11, 49bξ + 184b < 144 and b < 3 4 .
So the coefficients of x on the right hand side of (8) are all negative. Therefore, we obtain sup x,y∈[0,+∞) G(x, y) < 0. Proof. From the pinching condition, there exists a positive constant b, such that for all t < 0 there holds (9) |h| 2 − 1 3n |H| 2 < bn < n ξ + 1 .
We study the function f = |h| 2 |H| 2 + 2bn 2 .
It follows from (9) that f < 1 2n . We have
∂ ∂t f = 1 |H| 2 + 2bn 2 ∂ ∂t |h| 2 − f ∂ ∂t |H| 2 and ∆f = 1 |H| 2 + 2bn 2 ∆|h| 2 − f ∆|H| 2 − 2 ∇f, ∇|H| 2 .
From the evolution equations, we get
∂ ∂t − ∆ f = 2 |H| 2 + 2bn 2 ∇f, ∇|H| 2 − 2 |H| 2 + 2bn 2 (|∇h| 2 − f |∇H| 2 ) + 2 |H| 2 + 2bn 2 R 1 − 1 n R 2 − 2f n + 1 |H| 2 + 2bn 2 (R 2 + n|H| 2 ) .(10)
Firstly, we estimate the gradient terms in (10). From (3) we have
|∇h| 2 1 2n |∇H| 2 f |∇H| 2 .
Secondly, we estimate the reaction terms. From (5), (6) and the pinching condition, we have
R 1 − 1 n R 2 |h| 2 |h| 2 + 2|h| 2 P − 1 n |H| 2 P − ξ −1 P 2 |h| 2 |h| 2 + 2P − 1 n |H| 2 P + ξ −1 P 2 1 3n |H| 2 + bn .
Hence we get
2 |H| 2 + 2bn 2 R 1 − 1 n R 2 − 2f n + 1 |H| 2 + 2bn 2 (R 2 + n|H| 2 ) (11) 2f |h| 2 + 2P − 1 n |H| 2 P + ξ −1 P 2 1 3n |H| 2 + bn − n − |H| 2 (|h| 2 − P + n) |H| 2 + 2bn 2 = 2f 2bn 2 |h| 2 + |H| 2 (P − n) |H| 2 + 2bn 2 − 1 n |H| 2 P + ξ −1 P 2 1 3n |H| 2 + bn + 2P − n 2f 2bn 2 4 3n |H| 2 + bn + |H| 2 (P − n) |H| 2 + 2bn 2 − 1 n |H| 2 P + ξ −1 P 2 1 3n |H| 2 + bn + 2P − n = 2nf G 1 n 2 |H| 2 , 1 n P .
Here G is the function defined in Lemma 2. Since 1 2 ξ < b −1 − 1, G 1 n 2 |H| 2 , 1 n P has a negative upper bound −δ.
Inserting (11) into (10), we obtain
∂ ∂t − ∆ f 2 |H| 2 + 2bn 2 ∇f, ∇|H| 2 − 2nδf.
Applying the maximum principle, we get
∀s < t, max Mt f e −2nδ(t−s) max Ms f.
Letting s → −∞, we obtain max Mt f = 0, i.e., M t is a totally umbilical sphere.
If p 3, we verify the following theorem with different pinching coefficients.
|h| 2 < 2n 3 ,
then M t is either a shrinking spherical cap or a totally geodesic sphere.
Proof. Let
f = |h| 2 |H| 2 + 5 3 n 2 .
Then we have f < 2 5n . Similarly to (10), we get
∂ ∂t − ∆ f = 2 |H| 2 + 5 3 n 2 ∇f, ∇|H| 2 − 2 |H| 2 + 5 3 n 2 (|∇h| 2 − f |∇H| 2 ) + 2 |H| 2 + 5 3 n 2 R 1 − 1 n R 2 − 2f n + 1 |H| 2 + 5 3 n 2 (R 2 + n|H| 2 ) . From (3) we have |∇h| 2 2 5n |∇H| 2 f |∇H| 2 .
From (4) and (6), we have
2 |H| 2 + 5 3 n 2 R 1 − 1 n R 2 2 |H| 2 + 5 3 n 2 3 2 |h| 4 + 1 n |H| 2 |h| 2 − 1 n |H| 2 P = 2f 1 2 |h| 2 + |h| 2 − 2 n · |H| 2 P |H| 2 + 5 3 n 2 and 2f n + 1 |H| 2 + 5 3 n 2 (R 2 + n|H| 2 ) = 2f n + 1 |H| 2 + 5 3 n 2 |H| 2 (|h| 2 + n − P ) = 2f |h| 2 + 2n − 5 3 n 2 |H| 2 + 5 3 n 2 (|h| 2 + n) − 2f |H| 2 P |H| 2 + 5 3 n 2 2f |h| 2 + 2n − 5 3 n 2 |H| 2 + 5 3 n 2 1 n |H| 2 + 5n 3 − 2 n · |H| 2 P |H| 2 + 5 3 n 2 = 2f |h| 2 + n 3 − 2 n · |H| 2 P |H| 2 + 5 3 n 2 . Therefore, we get ∂ ∂t − ∆ f 2 |H| 2 + 5 3 n 2 ∇f, ∇|H| 2 + |h| 2 − 2n 3 f. Set δ = − sup M ×(−∞,0) |h| 2 − 2n 3 .
It follows from the maximum principle that
∀s < t, max Mt f e −δ(t−s) max Ms f.
Hence, we obtain max Mt f = 0, i.e., M t is a totally umbilical sphere.
Now we give the proof of Theorem 2. Proof of Theorem 2. By the assumption, there exists t 0 < 0, such that
sup M×(−∞,t0) (|h| 2 − κ|H| 2 ) < α,
where (κ, α) = min{ 3 n+2 , 4(n−1) n(n+2) }, n for p = 1, (κ, α) = ( 4 3n , 2n 3 ) for p = 2, (κ, α) = ( 4 3n , 3n 5 ) or ( 1 n , 2n 3 ) for p 3. Then combining the results of Theorems 5, 6 and 7, we complete the proof of Theorem 2.
Ancient solutions in hyperbolic spaces.
Let 2n(n+2) , otherwise. For positive numbers ε and n, we define a function ϕ : (n 2 , +∞) → R by (12) ϕ
(x) = x 1 − n 2 x 2+ε .
Then it has the following properties.
Lemma 3. The function ϕ(x) satisfies (i) ϕ(x) < x, ϕ ′ (x) < 1, ϕ ′′ (x) > 0, (ii) 1 − xϕ ′ (x) ϕ(x) = − (2+ε)n 2 x−n 2 , (iii) ϕ ′ (x) + 2xϕ ′′ (x) < 4.
Proof. By the definition of ϕ, we get ϕ(x) < x.
Differentiating ϕ, we get
ϕ ′ (x) = x + n 2 (1 + ε) x 1 − n 2 x 1+ε , ϕ ′′ (x) = n 4 (1 + ε)(2 + ε) x 3 1 − n 2 x ε .
Since ϕ ′′ (x) > 0, we have ϕ ′ (x) < ϕ ′ (+∞) = 1. Now we can check
1 − xϕ ′ (x) ϕ(x) = − (2 + ε)n 2 x − n 2 .
By a direct computation, we have
ϕ ′ (x) + 2xϕ ′′ (x) = n 4 (1 + ε)(3 + 2ε) + n 2 εx + x 2 x 2 1 − n 2 x ε .
Replacing x by n 2 y , we set
ψ(y) = [(1 + ε)(3 + 2ε)y 2 + εy + 1](1 − y) ε for 0 < y < 1. Since ψ ′ (y) = (1 + ε)(2 + ε)y(1 − y) ε−1 [3 − (3 + 2ε)y], we have sup 0<y<1 ψ(y) = ψ 3 3 + 2ε = 2(7ε + 6) 2ε + 3 2ε 2ε + 3 ε .
Taking the logarithm, we set θ(ε) = log 2(7ε + 6) 2ε + 3 + ε log 2ε 2ε + 3 .
Then we have
θ ′ (ε) = 3(7ε + 9) (2ε + 3)(7ε + 6) + log 2ε 2ε + 3 , θ ′′ (ε) = 27(7ε 2 + 17ε + 12) ε(2ε + 3) 2 (7ε + 6) 2 .
Since θ ′′ (ε) > 0, we have θ ′ (ε) < θ ′ (+∞) = 0. This implies θ(ε) < θ(0+) = log 4. Thus we obtain ψ(y) < 4 for 0 < y < 1.
Now we give the proof of Theorem 3. Proof of Theorem 3. We study the function
f = |h| 2 ϕ(|H| 2 )
.
The pinching condition implies that f k. First, we derive the evolution equation of f . By a direct computation, we have
∂ ∂t f = 1 ϕ(|H| 2 ) ∂ ∂t |h| 2 − f ∂ ∂t ϕ(|H| 2 ) and ∆f = 1 ϕ(|H| 2 ) ∆|h| 2 − f ∆ϕ(|H| 2 ) − 2 ∇f, ∇ϕ(|H| 2 ) . From Lemma 1 (ii), we have ∂ ∂t − ∆ ϕ(|H| 2 ) = −2ϕ ′ (|H| 2 )(|∇H| 2 − R 2 + n|H| 2 ) − ϕ ′′ (|H| 2 ) ∇|H| 2 2 −2(ϕ ′ (|H| 2 ) + 2|H| 2 ϕ ′′ (|H| 2 ))|∇H| 2 +2ϕ ′ (|H| 2 )(R 2 − n|H| 2 ).
Therefore we obtain
∂ ∂t − ∆ f 2 ϕ(|H| 2 ) ∇f, ∇ϕ(|H| 2 ) (13) − 2 ϕ(|H| 2 ) [|∇h| 2 − f (ϕ ′ (|H| 2 ) + 2|H| 2 ϕ ′′ (|H| 2 ))|∇H| 2 ] + 2 ϕ(|H| 2 ) R 1 − 1 n R 2 + 2f n − ϕ ′ (|H| 2 ) ϕ(|H| 2 ) (R 2 − n|H| 2 ) .
From (3) and Lemma 3 (iii), we have
f (ϕ ′ (|H| 2 ) + 2|H| 2 ϕ ′′ (|H| 2 ))|∇H| 2 4k|∇H| 2 2(n − 1) n(n + 2) |∇H| 2 |∇h| 2 .
From (5), (6), we get
2 ϕ(|H| 2 ) R 1 − 1 n R 2 + 2f n − ϕ ′ (|H| 2 ) ϕ(|H| 2 ) (R 2 − n|H| 2 ) 2 ϕ(|H| 2 ) |h| 2 |h| 2 + 2P |h| 2 − 1 n P |H| 2 +2f n − ϕ ′ (|H| 2 )|H| 2 ϕ(|H| 2 ) (|h| 2 − P − n) = 2f |h| 2 + n − ϕ ′ (|H| 2 )|H| 2 ϕ(|H| 2 ) (|h| 2 − n) + 2P ϕ(|H| 2 ) 2|h| 2 − 1 n |H| 2 + f ϕ ′ (|H| 2 )|H| 2 .
If p = 1, then P is identically zero. If p 2, it follows from the pinching condition and Lemma 3 (i) that
2|h| 2 − 1 n |H| 2 + f ϕ ′ (|H| 2 )|H| 2 2kϕ(|H| 2 ) − 1 n |H| 2 + f |H| 2 2k|H| 2 − 1 n |H| 2 + k|H| 2 = 3k − 1 n |H| 2 0.
By Lemma 3 (ii), we have
|h| 2 + n − ϕ ′ (|H| 2 )|H| 2 ϕ(|H| 2 ) (|h| 2 − n) = 1 − ϕ ′ (|H| 2 )|H| 2 ϕ(|H| 2 ) (|h| 2 − n) + 2n = − (2 + ε)n |H| 2 − n 2 (n|h| 2 − n 2 ) + 2n −(2 + ε)n + 2n = −εn.
Hence, we obtain
∂ ∂t − ∆ f 2 ϕ(|H| 2 ) ∇f, ∇ϕ(|H| 2 ) − 2εnf.
This implies f ≡ 0, i.e., M t is a totally umbilical sphere.
Integral pinched ancient solutions
Ancient solutions in the Euclidean space.
We need the following Sobolev type inequality on submanifolds [30]. Noting that ∂ ∂t dµ t = −|H| 2 dµ t , we have d dt Mt
|h| 2 dµ t = d dt Mt 3 2 |h| 2 − 1 4 |H| 2 dµ t = Mt ∂ ∂t 3 2 |h| 2 − 1 4 |H| 2 − |H| 2 3 2 |h| 2 − 1 4 |H| 2 dµ t = Mt −3|∇h| 2 + 1 2 |∇H| 2 + 3R 1 − 2R 2 −|H| 2 3 2 |h| 2 − 1 4 |H| 2 dµ t .
By (3), we have
(15) − 3|∇h| 2 + 1 2 |∇H| 2 −|∇h| 2 .
By (5) and (6), we have
3R 1 − 2R 2 3|h| 4 + 3(2|h| 2 − |H| 2 )P − 2|H| 2 (|h| 2 − P ) 3|h| 4 + 6|h| 4 − 2|H| 2 |h| 2 (16) = 9|h| 4 + |H| 2 |h| 2 − 1 4 |H| 4 . So we get d dt Mt |h| 2 dµ t Mt 9|h| 4 − |∇h| 2 − 1 2 |H| 2 |h| 2 dµ t .
From Proposition 2, we have
Mt |h| 4 dµ t 1 2 B(2) Mt (2|h||∇h| + |h| 2 |H|)dµ t B(2) Mt |h| 2 dµ t 1 2 Mt (2|∇h| + |h||H|) 2 dµ t 1 2 B(2) √ C Mt (6|∇h| 2 + 3|h| 2 |H| 2 )dµ t 1 2 . Setting C = 1 60B(2) 2 , we get Mt |h| 4 dµ t 1 10 Mt |∇h| 2 + 1 2 |h| 2 |H| 2 dµ t .
Hence we obtain
(17) d dt Mt |h| 2 dµ t − Mt |h| 4 dµ t − Mt |h| 2 dµ t 2 Mt 1dµ t .
Let I(t) = Mt |h| 2 dµ t , vol (t) = Mt 1dµ t . We prove I(t) is always zero by contradiction. Assume there exists t 0 , such that I(t 0 ) > 0. Thus, for all t < t 0 , we get from (17) that d dt
I −1 (t) vol −1 (t).
Integrating, we get
(18) I −1 (t 0 ) − I −1 (t) t0 t vol −1 (τ )dτ.
Then we estimate the volume of M t . By (14), we have d dt
vol (t) = − Mt |H| 2 dµ t −C, whereC = 2C + 16π. This yields vol (t) vol (t 0 ) +C(t 0 − t).
Thus, we get from (18) that
I −1 (t 0 ) I −1 (t) + 1 C log 1 +C vol (t 0 ) (t 0 − t) .
The right hand side of the above inequality tends to +∞ as t → −∞, which leads to a contradiction. Therefore, we obtain I(t) = 0 for all t.
Denote by f + the positive part of a function f . Now we derive the following integral inequality.
Letting ε → 0, we obtain
M U q−1 + ∆U dµ − q − 1 4q 2 1 2B(n) 2 M U qn n−2 + dµ n−2 n − M |H| 2 U q + dµ .
Let n 3. Consider the compact ancient solution F :
M n × (−∞, 0) → R n+p . Set U = |h| 2 − 1 n 2 |H| 2 .
Then we have the following estimate.
Lemma 5. For any number q > 1, the following inequality holds along the flow.
d dt Mt U q + dµ t −A 1 (n, q) + A 2 (n, q) Mt |h| n dµ t 2 n Mt U qn n−2 + dµ t n−2 n ,
where A 1 (n, q), A 2 (n, q) are positive constants depending only on n and p.
Proof. From the evolution equations, we get
∂ ∂t − ∆ U = −2 |∇h| 2 − 1 n 2 |∇H| 2 + 2 R 1 − n + 1 n 2 R 2 .
By (3) we have
(21) |∇h| 2 − 1 n 2 |∇H| 2 1 3n |∇H| 2 .
From (5), (6), we get
R 1 − n + 1 n 2 R 2 |h| 4 + 2|h| 2 − 2 n |H| 2 P − n + 1 n 2 |H| 2 (|h| 2 − P ) U (|h| 2 + 2P ) (22) U + (|h| 2 + 2|h| 2 ) = U + 3|h| 2 + 1 n |H| 2 .
Hence we obtain
(23) ∂ ∂t U ∆U − 2 3n |∇H| 2 + 2U + 3|h| 2 + 1 n |H| 2 .
It follows from (23) and Lemma 4 that
d dt Mt U q + dµ t Mt qU q−1 + ∂ ∂t U dµ t q Mt U q−1 + ∆U dµ t + 2q Mt U q + 3|h| 2 + 1 n |H| 2 dµ t − q − 1 4q · 1 2B(n) 2 Mt U qn n−2 + dµ t n−2 n + Mt q − 1 4q |H| 2 + 2q 3|h| 2 + 1 n |H| 2 U q + dµ t .
Note that |H| 2 n 2 |h| 2 on the support of U + . We use Hölder's inequality to get
Mt q − 1 4q |H| 2 + 2q 3|h| 2 + 1 n |H| 2 U q + dµ t n 2 4 + 4qn Mt |h| 2 U q + dµ t n 2 4 + 4qn Mt |h| n dµ t 2 n Mt U qn n−2 + dµ t n−2 n .
Thus, we complete the proof. Proof. We choose C(n) such that
C(n) < min q∈{ n 2 , n 2 2(n−2) } A 1 (n, q) A 2 (n, q) n 2 .
Here A 1 (n, q), A 2 (n, q) are constants in Lemma 5. Letting J r (t) = Mt U r/2
+ dµ t , we get d dt J n (t) −C(n)(J n 2 /(n−2) (t)) n−2 n ,
whereC(n) is a positive constant. Hence, for any t 1 < t 2 < 0, we have
J n (t 1 ) − J n (t 2 ) C (n) t2 t1
(J n 2 /(n−2) (t)) n−2 n dt.
It follows from Lemma 5 that J n 2 /(n−2) (t) is decreasing. Thus t2 t1 (J n 2 /(n−2) (t)) n−2 n dt (t 2 − t 1 )(J n 2 /(n−2) (t 2 )) n−2 n .
On the other hand, we have
J n (t 1 ) − J n (t 2 ) J n (t 1 ) Mt 1 |h| n dµ t1 < C(n).
Hence we get
(t 2 − t 1 )(J n 2 /(n−2) (t 2 )) n−2 n < C(n) C(n) .
Letting t 1 → −∞, we obtain J n 2 /(n−2) (t 2 ) = 0. Thus U 0 for all t.
From (23) we get ∂ ∂t U ∆U − 2 3n |∇H| 2 .
Assume |H| attains 0 at (x 0 , t 0 ). Then U also attains 0 at this point. The strong maximum principle implies that U ≡ 0. Then we get |∇H| ≡ 0. Thus, M t0 is minimal, which is not possible. So H is non-vanishing along the flow. Applying the rigidity theorem for ancient solutions in [29], we obtain the conclusion. Noting that ∂ ∂t dµ t = −|H| 2 dµ t , we have d dt Mt |h| 2 dµ t = d dt Mt
3 2 |h| 2 − 1 4 |H| 2 − 1 dµ t = Mt ∂ ∂t 3 2 |h| 2 − 1 4 |H| 2 − |H| 2 3 2 |h| 2 − 1 4 |H| 2 − 1 dµ t = Mt −3|∇h| 2 + 1 2 |∇H| 2 + 3R 1 − 2R 2 −6|h| 2 − |H| 2 3 2 |h| 2 − 1 4 |H| 2 dµ t .
Using (15) and (16) Through the composition of immersions M n → S n+p → R n+p+1 , M n can be regarded as a submanifold in R n+p+1 , whose mean curvature is |H| 2 + n 2 . Thus Proposition 2 implies Mt |h| 4 dµ t Therefore, we obtain Mt |h| 2 dµ t = 0 for all t. This implies Mt U n 2 + dµ t ≡ 0. Therefore, M t satisfies |h| 2 1 n 2 |H| 2 for all t < 0. Applying the rigidity theorem in the previous section, we obtain the conclusion.
Proof of Theorem 4. By the assumption, there exists t 0 < 0, such that Mt |h| n dµ t < C(n) for all t ∈ (−∞, t 0 ). Then combining the results of the present section, we complete the proof of Theorem 4.
Theorem 1 .
1Let F : M n × (−∞, 0) → S n+p be a compact ancient solution of mean curvature flow in the unit sphere. 2 = 0 and M t is either a shrinking spherical cap or a totally geodesic sphere.
Theorem 2 .
2Let F : M n × (−∞, 0) → S n+p be a compact ancient solution of mean curvature flow in the unit sphere. Suppose F satisfies one of the following conditions:
Theorem 3 .
3Let F : M n × (−∞, 0) → H n+p be a compact ancient solution of mean curvature flow in the hyperbolic space. Suppose there exists a positive number ε, such that for all t < 0, M t satisfies |H| > n and |h| 2 k|H| 2 1 − n+2) , otherwise. Then M t is a family of shrinking spheres.
Theorem 5 .
5Let F : M n × (−∞, 0) → S n+1 be a compact ancient solution of mean curvature flow in the unit sphere. If sup M×(−∞,0) |h| 2 − min 3 n+2 , 4(n−1) n(n+2) |H| 2 < n, then M t is either a shrinking spherical cap or a totally geodesic sphere.
Theorem 6 .
6Let F : M n × (−∞, 0) → S n+p (p 2) be a compact ancient solution of mean curvature flow in the unit sphere. sgn(p−2) . Then M t is either a shrinking spherical cap or a totally geodesic sphere.
Theorem 7 .
7Let F : M n × (−∞, 0) → S n+p (p 3) be a compact ancient solution of mean curvature flow in the unit sphere. If sup M×(−∞,0)
F
: M n × (−∞, 0) → H n+p be a compact ancient solution of mean curvature flow in the hyperbolic space with constant curvature −1. Suppose there exists a positive number ε, such that for all t < 0, M t satisfies |H| > n and |h| 2 k|H| 2 1 −
Proposition 2 .
2Let M be an n-dimensional closed submanifold in the Euclidean space. For any nonnegative C 1 -function f on M , we have | + f |H|)dµ, where B(n) is a positive constant depending only on n. Now we investigate the integral pinched ancient solution in dimension two. Theorem 8. Let F : M 2 × (−∞, 0) → R 2+p be a compact ancient solution of mean curvature flow. Suppose for all t < 0, there holds Mt |h| 2 dµ t < C, where C is an explicit positive constant. Then M t is a shrinking sphere. Proof. Let χ(M ) be the Euler characteristic of M . The Gauss-2 dµ t = 2πχ(M ).
Theorem 9 .
9Let F : M n × (−∞, 0) → R n+p (n 3) be a compact ancient solution of mean curvature flow. Suppose for all t < 0, there holds Mt |h| n dµ t < C(n), where C(n) is a positive constant explicitly depending on n. Then M t is a shrinking sphere.
4. 2 .
2Ancient solutions in the sphere. First we investigate the integral pinched ancient solution in dimension two. Theorem 10. Let F : M 2 × (−∞, 0) → S 2+p be a compact ancient solution of mean curvature flow. Suppose for all t < 0, there holds Mt |h| 2 dµ t < C, where C is an explicit positive constant. Then M t is either a shrinking spherical cap or a totally geodesic sphere.Proof. The Gauss-
.+
Let F : M n ×(−∞, 0) → S n+p (n 3) be a compact ancient solution of mean curvature flow. Suppose for all t < 0, there holds Mt |h| n dµ t < C(n), where C(n) is a positive constant explicitly depending on n. Then M t is either a shrinking spherical cap or a totally geodesic sphere. Proof. By choosing C(n) = D2(dµ t .
again, we get |H| 2 |h| 2 − 6|h| 2 dµ t .d
dt Mt
|h| 2 dµ t
Mt
9|h| 4 − |∇h| 2 −
1
2
Mt 2|h||∇h| + |h| 2 |H| 2 + 4 dµ t |H| 2 + 2|h| 2 dµ t . Mt |h| 2 dµ t .1
2
B(2)
B(2)
Mt
|h| 2 dµ t
1
2
Mt
2|∇h| + |h| |H| 2 + 4
2
dµ t
1
2
B(2)
√
C
Mt
(6|∇h| 2 + 3|h| 2 (|H| 2 + 4))dµ t
1
2
.
Setting C =
1
54B(2) 2 , we get
Mt
|h| 4 dµ t
1
9 Mt
|∇h| 2 +
1
2
|h| 2 Hence we have
d
dt Mt
|h| 2 dµ t −4
M |h| n dµ. This curvature integral is called the Willmore functional. It is invariant
Lemma 4. Suppose M is an n( 3)-dimensional closed submanifold in the Euclidean space, and U is a C 2 -function on M . For any q > 1, we haveProof. Set U ε = U 2 + + ε for ε > 0. Then U ε is C 1 -differentiable, and ∇U ε = U+ Uε ∇U . By the divergence theorem, we haveApplying Proposition 2 to U q(n−1)/(n−2) ε , we haveCombining(19)and(20), we get Proof. Using(21)and(22)again, we haveHence we get d dt Mt U n 2Regarding M n as a submanifold in R n+p+1 with mean curvature |H| 2 + n 2 , we get from Lemma 4 thatThus we get
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A general gap theorem for submanifolds with parallel mean curvature in R n+p. H W Xu, J R Gu, Comm. Anal. Geom. 15H. W. Xu and J. R. Gu, A general gap theorem for submanifolds with parallel mean curvature in R n+p . Comm. Anal. Geom. 15 (2007), 175-193.
| {'fraction_non_alphanumeric': 0.1112071947423037, 'fraction_numerical': 0.05971734940949745, 'mean_word_length': 3.029367844698855, 'pattern_counts': {'":': 0, '<': 64, '<?xml version=': 0, '>': 33, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 45, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.|H| 4 + 4(n − 1)c|H| 2 . It is easy to check that α(n, |H|, c) is strictly bigger 2010 Mathematics Subject Classification. 53C44, 53C40. Key words and phrases. Ancient solution, mean curvature flow of submanifolds, pinching theorem, second fundamental form.', 'arxivid': '1910.05496', 'author': ['L I Lei ', 'ANDHongwei Xu ', 'Entao Zhao '], 'authoraffiliation': [], 'corpusid': 204509589, 'doi': '10.1090/tran/8267', 'github_urls': [], 'n_tokens_mistral': 17910, 'n_tokens_neox': 14726, 'n_words': 8358, 'pdfsha': 'a502111e725e8bac9f8ca72922c09030bc6ddd1e', 'pdfurls': ['https://arxiv.org/pdf/1910.05496v2.pdf'], 'title': ['ANCIENT SOLUTION OF MEAN CURVATURE FLOW IN SPACE FORMS', 'ANCIENT SOLUTION OF MEAN CURVATURE FLOW IN SPACE FORMS'], 'venue': []} |
arxiv |
A local potential for the Weyl tensor in all dimensions
20 Aug 2004 March 24, 2022
S Brian Edgar
Department of Mathematics
Linköpings universitet LinköpingS-581 83Sweden
José M M Senovilla
Física Teórica
Universidad del País Vasco
Apartado 64448080BilbaoSpain
A local potential for the Weyl tensor in all dimensions
20 Aug 2004 March 24, 2022
In all dimensions n ≥ 4 and arbitrary signature, we demonstrate the existence of a new local potential -a double (2, 3)-form, P ab cde -for the Weyl curvature tensor C abcd , and more generally for all tensors W abcd with the symmetry properties of the Weyl tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -a double (2, 1)-form, H ab c -is proven to be a particular case of the new potential: its double dual.
PACS Numbers: 02.40. Ky,04.20Cv In n dimensions, the existence of a 1-form potential A a for the 2-form electromagnetic field F ab enables the electromagnetic field equations to be written as a wave equation (in Lorentz signature) for the potential, which is particularly simple in the differential gauge 1 A a ;a = 0 [12]:
∇ 2 A a = F ab ;b .
In four dimensions, Lanczos [11] proposed the existence of a double (2, 1)-form potential H ab c = H [ab] c for the double (2, 2)-form Weyl tensor C ab cd (see e.g. [14] for the definition and properties of r-fold forms), and this result was confirmed in [2] (for any double (2,2)-form with the algebraic properties of the Weyl conformal curvature tensor) and equivalently for any symmetric spinor φ ABCD in [10]; see also [1], [5].
It is straightforward to conjecture a direct n-dimensional analogue for the Lanczos-Weyl equation (see (13) below) but, unfortunately, it has been shown that such a potential cannot exist, in general, in dimensions n > 4 [6]. As a consequence, interest in the existence of potentials for the Weyl tensor in dimensions n > 4 has diminished.
However there are a number of reasons for a continuing interest in a potential for the Weyl tensor, especially one which is defined in arbitrary dimensions. In particular, one attractive property of any possible potential for the Weyl tensor is that it has units L −1 , the same as the connection (or the first derivative of the metric). This means that any of its squares (such as its superenergy tensor [14]) would have units L −2 which are precisely the units we would expect for quantities related to gravitational 'energies', in contrast with the familiar Bel-Robinson superenergy tensor [3,14] whose units are L −4 (Roberts [13] pointed this out for the Lanczos potential). As a matter of fact, another related attractive property of these potentials is that they are at a level similar to the connection. As the potential is an 'integral' of the Weyl tensor, which itself is a second derivative of the metric, the potential and the connection are necessarily connected. The advantage is that the potential is a tensorial object. Then, of course, the impossibility of constructing tensors from the metric and its first derivatives manifest itself -as it must-in the lack of uniqueness of the potentials. Thus, already at this stage we realize that any potential must be affected by a gauge freedom.
In this letter we shall show, in arbitrary dimension and signature, that all (2, 2)-forms with the algebraic properties of the Weyl tensor, locally have a double (2,3)-form potential P ab cde = P [ab]
[cde] . We will further show that, in four dimensions, the double Hodge dual of the new potential, which is a double (2,1)-form, is exactly the classical Lanczos potential. Hence a very natural and very general potential for the Weyl tensor is this new double (2, 3)-form potential.
We begin by considering an arbitrary n-dimensional pseudo-Riemannian manifold with metric g ab of any signature. We call a Weyl candidate any double (2,2)-form X abcd = X [ab]cd = X ab [cd] with the algebraic properties of the Weyl conformal curvature tensor:
X a bca = 0, X a[bcd] = 0 (=⇒ X abcd = X cdab ),(1)
so that X abcd is a traceless and symmetric double (2,2)-form. We will now exploit these properties and rearrange as follows,
∇ 2 X abcd = 1 2 (X abcd;e e + X;e b] + 2 R e dn[a X n b]ec − R e cn[a X n b]ed + R n[c X n d]ab + R n[a X n b]cd − 1 2 (R cden X ab en + R aben X cd en ) (2)
where R ab is the Ricci tensor. Using the shorthand notation {R ⊗ X} abcd for all the terms linear in the curvature tensors, that is to say, the second line of (2), we can rewrite by using (1) repeatedly
∇ 2 X abcd = 1 2 3X ab[
Now define the double (2, 3)-form
P abcde = P [ab][cde] ≡ 3 2 X ab[cd;e](4)
which will inherit from X abcd the properties
P a[bcde] = 0, P ab abc = 0.(5)
Immediate consequences from the first of these are the following useful properties
P e [bcd]e = 0, P [abcd]e = 0, P abcde = 3P [cde]ab = 3P a[cde]b , P a[bc]de = −P a[de]bc .(6)
Hence (3)
It is easily confirmed that this expression constructed from P abcde , ignoring the {R⊗X} abcd terms, has the necessary Weyl candidate index symmetries. Now consider any Weyl candidate W abcd . We can always find a Weyl candidate 'superpotential' X abcd locally for W abcd by appealing to the Cauchy-Kowalewsky theorem [4] which guarantees a local solution of the linear second order equation
∇ 2 X abcd − {R ⊗ X} abcd = W abcd(8)
in a given analytic background space. From the superpotential X abcd we can then construct the potential P abcde using (4), and obtain our main result:
Theorem 1 Any Weyl candidate tensor field W abcd has a double (2, 3)-form local potential P abcde with the properties (5) such that
W ab cd = P ab cdi ;i + P cd abi ;i − 2P i[c abi ;d] − 2P i[a cdi ;b] .(9)
The potential itself can be given in terms of a Weyl candidate superpotential X abcd by (4) so that W abcd is given in terms of this superpotential by (8).
Of course the above theorem has a direct application to the Weyl tensor C ab cd of any pseudo-Riemannian manifold.
The number of independent components of the potential can be computed easily by using the properties (5) and the result is (n + 2)n(n − 3)(n 2 − n + 4)/24 (16 if n = 4, 70 if n = 5). This is larger than the number of independent components of a Weyl candidate, which is known to be (n + 2)(n + 1)n(n − 3)/12 (that is, 10 if n = 4, 35 if n = 5). It is also larger (equal, in the case n = 4) than the number of independent Ricci rotation coefficients, or of independent components of the connection in a given basis. Although this large number of components may seem unsatisfactory, one must bear in mind that this number can be substantially reduced in any particular case by means of the gauge differential freedom as we show in [7].
The choice of P abcde is not unique. As is usual with potentials -see, e.g., [12] one can redefine significant parts of P abcde;f without altering the combination in (9) which produces a particular W abcd . From known results about p-forms, and standard arguments on the independence of the exterior differential versus the divergence, we can easily identify and exploit some of the gauge freedom for the new potential. A detailed discussion on gauge, with explicit formulas for the gauge redefinition of P abcde , will be given in [7].
Consider now the special case of n = 4. It has already been noted that if n = 4 the Weyl tensor, and more generally all Weyl candidates, have a so-called Lanzcos potential [11], [2], [10], [1]. This is a double (
What is the relation, if any, between the new potential P abcde and the Lanczos potential?
To answer this question, we first of all remark that a double (2,3)-form is equivalent, via dualization with the Hodge * operator, to a double (2,1)-form in n = 4. For the formulas and conventions about the Hodge dual operator we refer the reader to [14]allowing for an extra sign depending on the signature of the space. In particular, for any traceless double (2,2)-form, and denoting the canonical volume element 4-form by η abcd = η [abcd] , we have [14] ( * W * ) abcd ≡ 1 4 η abef η cdgh W ef gh =⇒ ( * W * ) abcd = ǫ W abcd where ǫ = ±1 = sign(det(g ab )) is a sign depending on the signature (ǫ = 1 in positivedefinite metrics, and ǫ = −1 in the Lorentzian case). Similarly, for any double (2,3)-form P abcde we can write [14] ( * P * ) abc ≡ 1 12
η abef η dghc P ef dgh =⇒ P ab cde = 6ǫ ( * P * ) [a [cd δ b]
e] , P i cabi = ǫ ( * P * ) abc .
Observe that the properties (5) translate for the double dual into, respectively, ( * P * ) [abc] = 0, ( * P * ) ab b = 0 which are the Lanczos potential properties (10) exactly. Hence, by taking the double dual of (9), using the previous formulas and after a little bit of algebra we can prove that, in four dimensions
ǫ W ab cd = 2( * P * ) ab [c;d] + 2( * P * ) cd [a;b] − 2δ
which coincides with (11) by identifying H abc = ǫ( * P * ) abc .
For completeness, we give also the inverse formula: P abcde = ǫ ( * H * ) abcde . Therefore, we have recovered the classical Lanczos potential for the Weyl tensor in four dimensions as the double dual of the new potential P abcde . Note that the familiar differential gauge of the Lanczos potential H abc ;c , [11], [2], [10], [1] becomes H abc ;c = ǫ ( * P * ) abc ;c , and so the differential gauge for the new potential resides in P ab [cde;f ] , via double dualization.
In dimensions greater than four, a natural generalization of (11) to arbitrary dimension is to keep the double (2,1)-form H abc with properties (10) and consider the appropriate formula analogous to (11)
However, as already noted, this Lanczos potential exists exclusively in n = 4 dimensions [6]. On the other hand, we now know that there is a different counterpart to the Lanczos potential in n > 4 dimensions; it is the double dual of P ab cde , i.e., a double (2, n − 3)-form H ab c 1 c 2 ...c n−3 , and an expression -giving any Weyl candidate in terms of such a potential -can be obtained in any dimension by the same method that led to (12). Clearly such a dimensionally dependent result is much less natural and convenient that our defining formula (9), which is independent of dimension. Therefore, in particular, we have proved that the natural and proper way to consider a potential for a double (2,2)-form is as a double (2,3)-form; this version carries over to all dimensions by means of (9). The traditional double (2,1)-form Lanczos potential H ab c in n = 4 dimensions is nothing but the double dual of the general potential P ab cde .
In electromagnetic theory the local existence of the potential A a for the field tensor F ab is a direct consequence of applying Poincare's Lemma to one of Maxwell's equations, F [ab;c] = 0. However, it is important to note that the local existence of the new potential P abcde for the Weyl tensor does not require any such 'field equations'. Our result is actually analogous to the result that any 2-form always has a pair of local potentials, a 1-form A a and a 3-form B abc such that
F ab = 2A [a;b] − B abc ;c .
If the 2-form F ab is closed, then B abc can be chosen to be zero, while if it is divergence-free, then A a can be set to zero, [12], [10]. Our result for Weyl candidates is the generalization of this fact but, given the special structure of traceless and symmetric double (2,2)-forms, we achieve dealing with just one potential. A detailed discussion will be presented in [7]. Finally we emphasise that all our results are local and depend on the analiticity of the pseudo-Riemannian metric, [4]. However, as is usually the case, we expect that these results can be generalized, by using appropriate techniques of existence and uniqueness of solutions to differential equations, to the smooth case and even to spaces of low differentiability; from the point of view of general relativity, we can appeal to stronger theorems [8], [9] when we specialise to spaces with Lorentz signature, and the second order differential equations in the theorems become wave equations.
can be rewritten as ;b] + {R ⊗ X} abcd = P abcde ;e + P cdabe ;e − 2P e [c|abe|;d] − 2P e [a|cde|;b] + {R ⊗ X} abcd .∇ 2 X abcd = P abcde
;e + P cdabe
;e + 4P e[ab][c
e
;d] + 4P e[cd][a
e
which keeps the Weyl candidate symmetries. This formula is unique and readsW ab
cd = 2H ab
[c;d] + 2H cd
[a;b] −
4
n − 2
δ
[a
[c H b]e
d];e + H d]e
b];e .
A semicolon indicates covariant derivative with respect to the canonical connection. As usual, we use round and square brackets to denote symmetrization and antisymmetrization of indices, respectively. Our convention for the Riemann tensor follows from the Ricci identity: 2v a;[bc] = R d abc v d .
AcknowledgementsJMMS gratefully acknowledges financial support from the Wenner-Gren Foundations, Sweden, and from grants BFM2000-0018 of the Spanish CICyT and no. 9/UPV 00172.310-14456/2002 of the University of the Basque Country. JMMS thanks the Applied Mathematics Department at Linköping University, where this work was carried out, for hospitality.
Existence of Lanczos potentials and superpotentials for the Weyl spinor/tensor. F Andersson, S B Edgar, Class. Quantum Gravity. 182297Andersson, F. and Edgar S.B. (2001). Existence of Lanczos potentials and superpo- tentials for the Weyl spinor/tensor. Class. Quantum Gravity, 18, 2297.
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The Lanczos potential for the Weyl curvature tensor: existence, wave equations and algorithms. S B Edgar, A Höglund, Proc. Roy. Soc. A. 453835Edgar, S. B. and Höglund, A. (1997). The Lanczos potential for the Weyl curvature tensor: existence, wave equations and algorithms. Proc. Roy. Soc. A, 453, 835.
The Lanczos potential for Weyl candidates exists only in four dimensions. S B Edgar, A Höglund, Gen. Rel. Grav. 322307Edgar, S.B. and Höglund A.(2000).The Lanczos potential for Weyl candidates exists only in four dimensions. Gen. Rel. Grav., 32,2307.
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Gauge freedom for general potentials and new potentials for curvature tensors. S B Edgar, J M M Senovilla, In preparationEdgar, S.B. and Senovilla, J.M.M., Gauge freedom for general potentials and new potentials for curvature tensors. In preparation.
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arxiv |
Distribution of Global Defensive k-Alliances over some Graph Products
Mostafa Tavakoli [email protected]
Department of Applied Mathematics
Faculty of Mathematical Sciences
Ferdowsi University of Mashhad
P.O. Box 115991775MashhadIran
Sandi Klavžar [email protected]
Faculty of Mathematics and Physics
University of Ljubljana
Slovenia
Faculty of Natural Sciences and Mathematics
University of Maribor
Slovenia
Institute of Mathematics, Physics and Mechanics
LjubljanaSlovenia
Distribution of Global Defensive k-Alliances over some Graph Products
global allianceglobal defensive k-alliance, hierarchical product of graphslexicographic product of graphs AMS Subj Class: 05C69, 05C76
If G = (V G , E G ) is a graph, then S ⊆ V G is a global defensive k-alliance in G if (i) each vertex not in S has a neighbor in S and (ii) each vertex of S has at least k more neighbors inside S than outside of it. The global defensive k-alliance number of G is the minimum cardinality among all global defensive k-alliance in G. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds are also discussed.
Introduction
Alliances form a phenomena in many ways, say in politics, in relations between people (alliances such as common friendship), in social networks (say, Twitter users following each other), in natural sciences (say, animals from the same group), to list just a very small sample of examples. Since graphs are standard mathematical models for several of such phenomena, there is a strong need for a theory of alliances in graphs. The foundation for this theory was set up in [13], where alliances were classified into defensive, offensive, and powerful.
In this paper we are interested in defensive alliances, the theory of which has been recently surveyed in [23]. More precisely, we are interested in global defensive k-alliances, a concept that widely generalizes global defensive alliances and goes back to the paper [16] published two years before [13]. We refer to [15,20] for several mathematical properties of global defensive k-alliances and to [10,13,16] for applications of alliances in as different areas as national defence, studies of RNA structures, and fault-tolerant computing.
Since determining an optimal (global) defensive (k-)alliance is NP-hard, a way to approach the problem is via dynamic programming: decompose a given graph into smaller parts, solve the problem on these smaller graphs, and deduce a solution or an approximate the original problem from the obtained partial solutions. Graph products and similar operations are natural candidates for such an approach. In [6], the global defensive alliance was investigated on the join, the corona, and the composition (alias lexicographic product) of graphs. Very recently, the global defensive k-alliance was studied on the Cartesian product, the strong product, and the direct product of graphs [21,22]. In this paper we continue this direction of research by investigating the global defensive k-alliance on additional graph products and graph operations, in particular extending some previous results.
In the rest of the introduction concepts and notation needed are introduced, in particular the global defensive k-alliance is formally defined. In the subsequent section we study the generalized hierarchical product, in Section 3 we proceed with the lexicographic product, while in Section 4 we consider the corona and the edge corona product.
Throughout this article, G = (V G , E G ) stands for a simple graph of order n(G) = |V G | and size m(G) = |E G |. The degree of v ∈ V G is denoted by deg G (v), and the minimum and the maximum degree of G by δ G and ∆ G , respectively. If X ⊆ V G , then the subgraph induced by X is denoted by
G V . If S ⊆ V G and v ∈ V G , then N S (v) is the set of neighbors of v in S, that is, N S (v) := {u ∈ S | uv ∈ E G }. The complement of S in V G is denoted byS. Let X ⊆ V G . Then D ⊆ V G dominates X,
if every vertex from X \ D has a neighbor in D. When X = V G we say that D is a dominating set of G. The domination number γ(G) is the cardinality of a smallest dominating set of G. Let G = (V G , E G ) be a graph and k ∈ {−δ G , . . . , δ G }. Then a nonempty set S ⊆ V G is a global defensive k-alliance in G if the following conditions hold:
1. S is a dominating set of G and 2. for every v ∈ S, |N S (v)| ≥ |NS(v)| + k.
When S fulfils Condition 2 it is a defensive k-alliance. To shorten the presentation, we will abbreviate global defensive k-alliance as GDk-A and defensive k-alliance as Dk-A. Note that GDk-A is a Dk-A which is also a dominating set.
The global defensive k-alliance number γ d k (G) of G (abbreviated as GDk-A number) is the smallest order of a GDk-A in G. If G admits not a single GDk-A, we set γ d k (G) = ∞. Similarly, The defensive k-alliance number γ k (G) of G (abbreviated as Dk-A number) is the minimum cardinality among all Dk-As in G. If G does not contain a Dk-A set then we set γ k (G) = ∞. Finally, we use the notation [n] = {1, . . . , n}.
Generalized hierarchical products
A graph G together with a fixed vertex subset U ⊆ V G will be denoted by G(U ). If G and H are graphs, and U ⊆ V G , then the generalized hierarchical product G(U ) H is the graph with the vertex set V G ×V H , vertices (g, h) and (g , h ) being adjacent if and only if either g = g ∈ U and hh ∈ E H , or gg ∈ E G and h = h . Note that when U = V G , then the generalized hierarchical product is just the classical Cartesian product of graphs [8], that is, G(V G ) H = G H. The generalized hierarchical product was introduced for the first time in [4], we also refer to [1,2,3,17] for additional results on it as well as on its applications.
Consider
G(U ) H and let h ∈ V (H). Then the set of vertices {(g, h) : g ∈ V G } is called a G-layer over h.
Similarly an H-layer over g ∈ V G is defined. Note that a G-layer over h induces a subgraph of G(U ) H isomorphic to G and an H-layer over g ∈ U induces a subgraph of G(U ) H isomorphic to H. Theorem 1. If G and H are graphs and U ⊆ V G , then
γ d k (G(U ) H) ≤ γ d k (G)n(H) . Proof. Let U ⊆ V G , let S G be a GDk-A in G with |S G | = γ d k (G)
, and set S = S G ×V H . Note first that S is a domination set of G(U ) H. Indeed, since S G is a dominating set of G, every G-layer over h ∈ V H is dominated by the intersection of S with the layer. So all G-layers are dominated by S and therefore G(U ) H is dominated by S.
To show that S is a Dk-A in G(U ) H consider an arbitrary vertex (g, h) from S. If g ∈ U , then we have
|N S ((g, h))| − |NS((g, h))| = (|N S G (g)| − |NS G (g)|) + |N V H (h)| ≥ k + deg H (h) ≥ k , and if g / ∈ U , then |N S ((g, h))| − |NS((g, h))| = (|N S G (g)| − |NS G (g)|) + 0 ≥ k .
We have thus seen that S is a GDk-A. Since |S| = γ d k (G)n(H), the argument is complete.
Let Γ be the graph of the truncated cube, see the right-hand side of Fig. 1. Then Γ can be represented as the hierarchical product G(U ) P 2 , where G is the graph of order 12 on the left-hand side of Fig. 1 and U = {g 1 , g 4 , g 9 , g 12 }.
Figure 1: The graph Γ = G(U ) P 2 , where U = {g 1 , g 4 , g 9 , g 12 }.
The set S G = {g 5 , g 6 , g 7 , g 8 } (drawn in black in Fig. 1) is a global defensive (−1)-alliance in G. Since γ(G) = 4 (which follows for instance from the fact that the vertices g 1 , g 4 , g 9 , g 12 are pairwise at distance at least 3), we have γ d −1 (G) = 4. Hence Theorem 1 yields γ d −1 (G(U ) P 2 ) ≤ γ d −1 (G)n(P 2 ) = 8. Now, let S Γ be a minimal global defensive (−1)-alliance in Γ. Since Γ is a 3regular graph and for every v of V Γ we have |N S Γ (g)| − |NS Γ (g)| ≥ −1, each vertex u of S Γ must have at least one neighbor in S Γ . On the other hand, S Γ dominates vertices of Γ and so γ d −1 (Γ) ≥ 8; hence the the inequality of Theorem 1 is sharp. As already mentioned, G(V G ) H = G H. Hence Theorem 1 for the case of the Cartesian product reads as:
γ d k (G H) ≤ γ d k (G)n(H) .
By the well known commutativity property of the Cartesian product operation, this bound further implies that
γ d k (G H) ≤ min{γ d k (H)n(G), γ d k (G)n(H)}.
The special case of the latter result for k = 1 has been recently obtained in [21].
Lexicographic products
The lexicographic product G[H] of graphs G and H has V (G[H]) = V G ×V H , vertices (g 1 , h 1 ) and (g 2 , h 2 ) being adjacent if either g 1 g 2 ∈ E G , or g 1 = g 2 and h 1 h 2 ∈ E H . Let S be a GDk-A in G with k ≥ 0 and set S = S G × V H . Since |N S G (g)| − |NS G (g)| ≥ k, we get that
|N S G (g)|n(H) − |NS G (g)|n(H) ≥ kn(H) ,
which in turn implies that for any vertex h ∈ V H , h)| and |NS G (g)|n(H) = |NS(g, h)| this means that S is a global defensive (kn(H) + δ H )-alliance. As S is also a dominating set, we conclude that
|N S G (g)|n(H) + deg H (h) − |NS G (g)|n(H) ≥ kn(H) + δ H . Since |N S G (g)|n(H) + deg H (h) = |N S (g,γ d kn(H)+δ H (G[H]) ≤ n(H)γ d k (G) .
For k > 0 this is a better result than Proof. First, suppose that k > 2. Let S G be a GDk-A in G and D be a 1-perfect code in G S G . Then |D| = γ(G S G ). (This fact is well known and has been independently established several times, see [9,Theorem 9].) which also holds for every k ≥ 2.
Set S = (S G \ D) × V H ∪ D × {v} where v is a vertex of minimum degree in H. It is easy to check that S is a dominating set in G[H]. We claim that S is a Dk-A in G[H]. To prove this claim, let (g, h) ∈ S. If (g, h) ∈ (S G \ D) × V H , then |N S ((g, h))| − |NS((g, h))| = (|N S (g)| − 1)n(H) − (|NS(g)| + 1)n(H) + deg H (h) + 2 = n(H)(|N S (g)| − |NS(g)|) − 2n(H) + deg H (h) + 2 ≥ (k − 2)n(H) + deg H (h) + 2 ≥ k,
We have thus proved that S is a Dk-A in G[H]. Moreover, since S contains the copy of S G in the G-layer over v, by the definition of the lexicographic product we also infer that S is a dominating set. We conclude that S is a GDk-A in G[H]. Since clearly |S| = n(H)(γ d k (G) − γ(G S )) + γ(G S ), the proof is complete.
The proof of the bound from Theorem 2 does not work for k = 1. The reason is that the inequality (k − 2)n(H) + deg H (h) + 2 ≥ k for the case k = 1 reduces to −n(H) + deg H (h) + 2 ≥ 1 which clearly does not hold in general.
Corona and edge corona products
The corona product G • H of graphs G and H is the graph obtained from the disjoint union of G and n(G) copies of H bijectively assigned to the vertices of G, where each vertex v ∈ V G is adjacent to all the vertices of the assigned copy of H. This product was introduced in [7], see also [18,19]. Proof. Let G be the subgraph of G • H isomorphic to G and let H i , i ∈ [n(G )], be the the isomorphic copy of H corresponding to the ith vertex of G . Note that γ(G • H) = n(G) and consequently γ d
k (G • H) ≥ n(G). Hence if δ G − n(H) ≥ k, then V G is a Dk-A set in G • H so that γ d k (G • H) = n(G) holds in this case.
For the general case, consider an arbitrary defensive (k − 1)-alliance S H in H. Since each vertex of H i has exactly one neighbor outside H i , the set (∪
n(G) i=1 S H i )∪V G is a GDk-A in G • H, where S H i is the copy of S H in H i . Therefore, γ d k (G • H) ≤ n(G) + n(G)γ k−1 (H) = n(G)(1 + γ k−1 (H)).
Also, if S H is a global defensive (k +1)-alliance in H, then from the same reasons as above, the set ∪
n(G) i=1 S H i is a GDk-A in G • H. Hence γ d k (G • H) ≤ n(G)γ d k+1 (H) and therefore, γ d k (G • H) ≤ min{n(G) + n(G)γ k−1 (H), n(G)γ d k+1 (H)}.
Consider the corona products G • K m , m ≥ 2. Since γ d 1 (K m ) = (m + 2)/2 and γ d −1 (G) = (m + 1)/2 (cf. [23]), Theorem 3, yields Another corona-like product was recently introduced as follows. The edge corona G♦H of graphs G and H is obtained by taking one copy of G and m(G) disjoint copies of H associated to the edges of G, and for every edge uv ∈ E G joining u and v to every vertex of the copy of H associated to uv, see [11,14]. For the statement of the next result recall that if S G is a subset of vertices of a graph G, then its complement is denoted withS G .
γ d 0 (G • K m ) ≤ min{n(G)(1 + γ d −1 (K m )),
Theorem 4. Let G and H be two graphs. Then
γ d k (G♦H) ≤ min{m(G)γ d k+2 (H), γ d k+n(H)∆ G (G) + γ d k+2 (H)|E G S G |, γ d k n(H)+1 (G) + n(H)|E G S G | + γ d k+2 (H)|E G S G | I(k)},
where S G is a global defensive (k + n(H)∆ G )-alliance in G, S G is a global defensive Proof. Let G denote the copy of G in G♦H, and let H i be the copy of H corresponding to an edge e i ∈ E G .
If S H is a global defensive (k + 2)-alliance in H, then since each vertex of H i has exactly two neighbors outside H i , the set ∪
m(G) i=1 S H i , is a GDk-A in G♦H (again, S H i is the copy of S H in H i ). Thus γ d k (G♦H) ≤ n(G)γ d k+2 (H)
. Let next S G be a global defensive (k +n(H)∆ G )-alliance in G, and S H is a global defensive (k + 2)-alliance in H. Since each vertex of H i has exactly two neighbors outside H i and as each vertex of G has at most ∆ G n(H) neighbors outside G , the copy of S G in G together with the copies of S H is each of the copies of H corresponding to the edges from G S G form a global defensive k-alliance in G♦H.
So, γ d k (G♦H) ≤ γ d k+n(H)∆ G (G) + γ d k+2 (H)|E G S G |.
Suppose now that S G is a global defensive k n(H)+1 -alliance in G. Let S 1 be the set of vertices of G♦H that lie in the copies of H corresponding to the edges of G S G . In addition, in every copy of H corresponding to the edges from S G select a global defensive (k + 2)-alliance in H. Then set S = S G ∪ S 1 ∪ S 2 . We claim that S is a GDk-A in G♦H.
S is a dominating set in G♦H, because S G dominates G and all the copies of H above its edges as well as above copies of H above edges with one endpoint in S G , while the other copies of H are dominated by S 2 . To show that S is a Dk-A in G♦H consider u ∈ S G . Then The sun graph S n is obtained by replacing every edge of a cycle C n by a triangle C 3 , cf. [5]. See Fig. 2 for S 3 . From our point of view note that S n = C n ♦K 1 . Since γ d 2 (K 1 ) = ∞, γ d 2 (C 3 ) = 3, γ d 0 (C 3 = 2, |E C 3 S G | = 1, |E C 3 S G | = 0, and I(0) = 1, Theorem 4 implies that γ d 0 (S 3 ) = γ d 0 (C 3 ♦P 1 ) ≤ 3. Actually, γ d 0 (S 3 ) = 3 (in Fig. 2 elements of a global defensive alliance in C 3 ♦P 1 are colored black), we have the inequality in Theorem 4.
such a case k < kn(H) + δ H , and so γ d k (G[H]) ≤ γ d kn(H)+δ H (G[H]). Eballe et al. [6] obtained some upper bounds of γ d k (G[H]) for the case k = 0 and H = K m . In the next theorem, we present an upper bound on γ d k (G[H]) for the case when there exists a GDk-A in G with some special structure. For this sake recall that an r-perfect code in G = (V G , E G ) is a subset D of V G for which the balls of radius r centered at the vertices of D form a partition of V G , cf.[12].
Theorem 2 .
2Let k > 0, let S be a smallest GDk-A set in G, and suppose that G S has a 1-perfect code. If H is a graph with more than one vertex, then G[H] has a GDk-A. Moreover, if k ≥ 2, thenγ d k (G[H]) ≤ n(H)(γ d k (G) − γ(G S )) + γ(G S ) .
which holds true because k ≥ 2.Consider next a vertex (g, h) ∈ D × {v}. Then|N S ((g, h))| − |NS((g, h))| = |N S (g)|n(H) − |NS(g)|n(H) − δ H = n(H)(|N S (g)| − |NS(g)|) − δ H ≥ kn(H) − δ H ≥ k ,
Theorem 3 .
3If G and H are graphs, thenγ d k (G • H) ≤ min{n(G)(1 + γ d k−1 (H)), n(G)γ d k+1 (H)} .Moreover, if δ G − n(H) ≥ k, then γ d k (G • H) = n(G).
n(G)γ d 1 (K m )} = min{n(G)(1 + (m + 1)/2 , n(G) (m + 2)/2 } = n(G) (m + 2)/2 . On the other hand, it was proved in [6, Corollary 3.7] that if m ≥ 2 and ∆(G) < m − 1, then γ d 0 (G • K m ) = n(G) (m + 1)/2 . It follows that the bound of Theorem 3 is sharp for G • K m for all even m.
if k > −(n(H) + 1), ∞; otherwise .
|N S (u)| − |NS(u)| = (|N S G (u)The same conclusion is clear when u ∈ S \ S G . Therefore, S is a k-alliance in G♦H and so γ dk (G♦H) ≤ γ d k n(H)+1 (G) + n(H)|E G S G | + γ d k+2 (H)|E G S G |.
Figure 2 :
2The sun graph S 3
Acknowledgements. S.K. acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-7110, J1-9109, N1-0043). The authors are indebted to the referees for helpful remarks which leaded us to correct and improve the paper.
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Alliances in graphs. P Kristiansen, S M Hedetniemi, S T Hedetniemi, J. Combin. Math. Combin. Comput. 48P. Kristiansen, S.M. Hedetniemi, S.T. Hedetniemi, Alliances in graphs, J. Com- bin. Math. Combin. Comput. 48 (2004) 157-177.
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Global defensive k-alliances in graphs. J A Rodríguez-Velázquez, J M Sigarreta, Discrete Appl. Math. 157J.A. Rodríguez-Velázquez, J.M. Sigarreta, Global defensive k-alliances in graphs, Discrete Appl. Math. 157 (2009) 211-218.
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| {'fraction_non_alphanumeric': 0.09437988015253314, 'fraction_numerical': 0.025649173778826948, 'mean_word_length': 3.25516708518447, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'If G = (V G , E G ) is a graph, then S ⊆ V G is a global defensive k-alliance in G if (i) each vertex not in S has a neighbor in S and (ii) each vertex of S has at least k more neighbors inside S than outside of it. The global defensive k-alliance number of G is the minimum cardinality among all global defensive k-alliance in G. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds are also discussed.', 'arxivid': '1812.02992', 'author': ['Mostafa Tavakoli [email protected] \nDepartment of Applied Mathematics\nFaculty of Mathematical Sciences\nFerdowsi University of Mashhad\nP.O. Box 115991775MashhadIran\n', 'Sandi Klavžar [email protected] \nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nSlovenia\n\nFaculty of Natural Sciences and Mathematics\nUniversity of Maribor\nSlovenia\n\nInstitute of Mathematics, Physics and Mechanics\nLjubljanaSlovenia\n'], 'authoraffiliation': ['Department of Applied Mathematics\nFaculty of Mathematical Sciences\nFerdowsi University of Mashhad\nP.O. Box 115991775MashhadIran', 'Faculty of Mathematics and Physics\nUniversity of Ljubljana\nSlovenia', 'Faculty of Natural Sciences and Mathematics\nUniversity of Maribor\nSlovenia', 'Institute of Mathematics, Physics and Mechanics\nLjubljanaSlovenia'], 'corpusid': 119714651, 'doi': '10.1007/s10100-018-00605-w', 'github_urls': [], 'n_tokens_mistral': 8048, 'n_tokens_neox': 7289, 'n_words': 4288, 'pdfsha': 'bc40e6a8271becabbf161f2035c26608b910259b', 'pdfurls': ['https://arxiv.org/pdf/1812.02992v1.pdf'], 'title': ['Distribution of Global Defensive k-Alliances over some Graph Products', 'Distribution of Global Defensive k-Alliances over some Graph Products'], 'venue': []} |
arxiv |
Unified Optimal Analysis of the (Stochastic) Gradient Method
23 Dec 2019
Sebastian U Stich
EPFL
Unified Optimal Analysis of the (Stochastic) Gradient Method
23 Dec 2019
In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on µ-convex functions under a (milder than standard) L-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after T iterations aswhere σ 2 measures the variance in the stochastic noise. For deterministic gradient descent (GD) and SGD in the interpolation setting we have σ 2 = 0 and we recover the exponential convergence rate. The bound matches with the best known iteration complexity of GD and SGD, up to constants. * [email protected], Machine Learning and Optimization Lab (MLO), EPFL, Switzerland.
Introduction
We consider the unconstrained optimization problem f ⋆ := min x∈R n f (x) , where f : R n → R is a convex continuously differentiable function. We consider a stochastic approximation scenario-comprising the classic deterministic setting-where only unbiased estimates of the gradient of f are available and study the convergence rate of stochastic gradient descent (SGD).
Formally, we assume that we have an increasing sequence of σ-fields {F t } t≥0 , such that x 0 ∈ R n is F 0 measurable and such that for all t ≥ 0 the iterates of SGD are given as:
x t+1 = x t − γ t g t ,(1)
where {γ t } t≥0 denotes a sequence of (positive) stepsizes and g t ∈ R n is a stochastic gradient of f , satisfying the following three assumptions.
Assumption 1 (Unbiased gradient oracles). Almost surely,
E [g t | F t ] = ∇f (x t ) , ∀t ≥ 0 ,(2)
where here ∇f (x t ) denotes the gradient of f at x t .
Assumption 2 ((L, σ)-smoothness). There exists two constants L, σ 2 ≥ 0, s.t.
E g t 2 | F t ≤ 2L(f (x t ) − f ⋆ ) + σ 2 , ∀t ≥ 0 .(3)
This assumption generalizes the standard smoothness assumption as we will explain in Section 2 below. We further assume that f is µ-convex (with respect to the optimum x ⋆ -a slight relaxation of the standard assumption, see e.g. (Necoara et al., 2019)) and denote by x ⋆ a minimizer of f in R n .
Assumption 3 (µ-convexity). There exists x ⋆ ∈ arg min x∈R n f (x) and a constant µ ≥ 0 with
µ 2 x − x ⋆ 2 + f (x) − f ⋆ ≤ ∇f (x), x − x ⋆ , ∀x ∈ R n .(4)
Note that this assumption is weaker than the standard strong convexity (for µ > 0) or convexity (for µ = 0) assumptions that require such an inequality to hold for arbitrary pairs x, y ∈ R d and not only for y = x ⋆ .
Contribution
Let {x t } t≥0 denote the iterates of (1). We show that for appropriate stepsizes γ t and an appropriately defined average iterate after T iterations,x T := 1 WT T t=0 w t x t for weights w t ≥ 0 and W T := T t=0 w t , it holds for R = x 0 − x ⋆ :
Ef (x T ) − f ⋆ + µE x T +1 − x ⋆ 2 = O min LR 2 exp − µT 4L + σ 2 µT , LR 2 T + σR √ T .
We further also give a simpler proof that shows, up to polylogarithmic factors 1 ,
Ef (x T ) − f ⋆ + µE x T +1 − x ⋆ 2 =Õ LR 2 exp − µT 2L + σ 2 µT ,
and that only relies on constant stepsizes in (1). This analysis unifies the analyses of gradient descent and SGD for smooth functions. In the deterministic case and in the iterpolation setting (where σ 2 = 0), we recover the exponential convergence rates of these algorithms (up to a factor 4 in the exponent) when µ > 0. Furthermore, the result for convergence in function values is tight up to absolute (non-problem specific) constants (Nesterov, 2004). Similarly, in the stochastic setting we recover the best known rates not only for the function values but also for the squared distance of the last iterate to the optimum (Nemirovski et al., 2009;Shamir and Zhang, 2013).
Related Work
Whilst the first analyses of stochastic gradient descent (SGD) (Robbins and Monro, 1951) focused on asymptotic results (Chung, 1954), the focus shifted to non-asymptotic results in recent years.
For µ > 0, Bach and Moulines (2011) give a bound E x T − x ⋆ 2 =Õ ( L µ ) 2 R 2 exp − µ L T + σ 2 µ 2 T , this was later improved by Needell et al. (2016)
to E x T − x ⋆ 2 =Õ L µ R 2 exp − µ L T + σ 2 µ 2 T .
Up to polylogarithmic factors this is the same rate as we show here in a slightly more general setting-however, their result only covers the distance x T − x ⋆ 2 of the iterates. Deducing from this result a rate for the function values via the smoothness inequality f (x T ) − f ⋆ ≤ L 2 x T − x ⋆ 2 introduces a superflous condition number factor L µ . In the quest of deriving optimal rates-up to constant factors-in function suboptimality, different averaging schemes have been studied (Ruppert, 1988;Polyak, 1990;Rakhlin et al., 2012;Shamir and Zhang, 2013). Lacoste-Julien et al. (2012) give a simple proof for f (x T ) − f ⋆ = O G 2 µT , where here G 2 ≥ σ 2 is an upper bound on the gradient norms, E g t 2 | F t ≤ G 2 . Analyses under this assumption are not optimal in the deterministic setting where σ 2 = 0, but G 2 > 0 in general.
The (L, σ 2 )-smoothness assumption appeared in this form recently in e.g. (Grimmer, 2019), though very similar conditions have been studied in the literature (Bertsekas and Tsitsiklis, 1996;Schmidt and Roux, 2013;Needell et al., 2016;Bottou et al., 2018;Gower et al., 2019). We will discuss a few of these in Section 2 below. In contrast to the bounded gradient assumption, these assumption allow to chose σ 2 = 0 in nontrivial situations and thus allow to recover faster rates in general. However, adapting the proof technique from (Lacoste-Julien et al., 2012) to the relaxed assumptions considered here (cf. Lemma 7 below, or (Stich et al., 2018;Grimmer, 2019)
) gives f (x T )−f ⋆ = O L 2 R 2 µT 2 + σ 2 µT ,
where the dependence on the initial distance R is not optimal, i.e. not exponentially decreasing as in (Bach and Moulines, 2011). Gower et al. (2019) generalize the results of (Needell et al., 2016) for the convergence of the distance x T − x ⋆ 2 to the setting considered here and obtain the same rate as stated earlier in this subsection.
To keep our focus, we do not discuss obvious generalizations of our bounds to other settings here. For instance convergence under average smoothness or importance sampling (Bach and Moulines, 2011;Needell et al., 2016) or expected smoothness conditions (Gower et al., 2018).
Motivating Examples
In this section we give a few examples that motivate Assumption 2.
Example 1 (Gradient Descent). In the non-stochastic setting, we have g t = ∇f (x t ), ∀t ≥ 0. If f is L-smooth, then f is also (L, 0)-smooth, as seen by the choice y = x ⋆ in the following inequality that holds for convex L-smooth functions (Nesterov, 2004, Theorem 2.1.5):
1 2L ∇f (x) − ∇f (y) 2 ≤ f (x) − f (y) − ∇f (y), x − y , ∀x, y ∈ R n .(5)Hence, we recover the O L x 0 − x ⋆ 2 exp − µT L
convergence rate which coincides with the best known rate (Nesterov, 2004, Theorem 2.1.15) for the function value convergence in this setting. 2 Example 2 (Stochastic Gradient Descent). In the stochastic setting, we have g t = ∇f (x t ) + ξ t , where {ξ t } t≥0 are independent, zero-mean noise terms, with uniformly bounded second moment E ξ t 2 ≤ σ 2 for a constant σ 2 ≥ 0. Again, by relying on (5), we see that Assumption 2 is satisfied for L-smooth functions:
E g t 2 | F t = ∇f (x t ) 2 + E ξ t 2 | F t ≤ (2L(f (x t ) − f ⋆ ) + σ 2 .
Hence, we recover the O σ 2 µT convergence rate of SGD for the function values and the O σ 2 µ 2 T rate for the last iterate-which are the best known rates (Rakhlin et al., 2012;Shamir and Zhang, 2013). We like to point out that we here do not need to rely on the frequently used bounded-gradient assumption, as e.g. in (Lacoste-Julien et al., 2012;Rakhlin et al., 2012;Shamir and Zhang, 2013).
SGD has also been analyzed under various similar growths conditions, for instance assumptions of the form E g t 2 | F t ≤ ν 1 + ν 2 ∇f (x t ) 2 , for two constants ν 1 , ν 2 ≥ 0, see e.g. (Schmidt and Roux, 2013;Bertsekas and Tsitsiklis, 1996;Bottou et al., 2018;Nguyen et al., 2018). By virtue of (5), we see that these settings are also comprised in Assumption 2 and covered here. 2 )-smooth gradient oracle , as can be seen from (cf. (Needell et al., 2016)):
E g t 2 | F t = E i ∇f i (x t ) − ∇f i (x ⋆ ) + ∇f i (x ⋆ ) 2 ≤ 2E i ∇f i (x t ) − ∇f i (x ⋆ ) 2 + 2E i ∇f i (x ⋆ ) 2 (5) ≤ 4L(f (x t ) − f ⋆ ) + 2E i ∇f i (x ⋆ ) 2 .
When the loss at the optimum vanishes, i.e. ∇f i (x ⋆ ) = 0, ∀i ∈ [m]-the so called interpolation setting-we have σ 2 = 0 and we recover linear convergence of SGD, as e.g. in (Schmidt and Roux, 2013;Needell et al., 2016;Ma et al., 2018). The above observation also holds in more general settings, such as e.g. under expected smoothness or weak growth conditions (cf. (Gower et al., 2018(Gower et al., , 2019).
2 The constant L is tight here (Nesterov, 2004). However-as a side remark-we like to point out that our proof reveals the
improved bound Ef (x T ) − f ⋆ + µE x T +1 − x ⋆ 2 = O µ x 0 − x ⋆ 2 exp − µT L + σ 2 µT if T = Ω µ
L is sufficiently large (an assumption that appears sometimes in the literature-though does not improve the worst-case complexity for arbitrary T ).
Convergence Analysis Part I-Deriving a Recursion
Following standard techniques, we prove the following lemma:
Lemma 1. For x 0 ∈ R d , let {x t } t≥0 denote
the iterates of SGD (1) generated on a function f under Assumptions 1-3 for stepsizes
γ t ≤ 1 2L , ∀t ≥ 0. Then E x t+1 − x ⋆ 2 ≤ (1 − µγ t )E x t − x ⋆ 2 − γ t (Ef (x t ) − f ⋆ ) + γ 2 t σ 2 .(6)
Proof. By definition,
E x t+1 − x ⋆ 2 | F t = E x t − x ⋆ 2 − 2γ t g t , x t − x ⋆ + γ 2 t g t 2 | F t (2) = x t − x ⋆ 2 − 2γ t ∇f (x t ), x t − x ⋆ + γ 2 t E g t 2 | F t (3),(4) ≤ x t − x ⋆ 2 − 2γ t µ 2 x t − x ⋆ 2 + f (x t ) − f ⋆ + γ 2 t 2L(f (x t ) − f ⋆ ) + σ 2 ,
where we also used µ-convexity in the last inequality. By re-arranging and taking expectation on both sides, we get:
E x t+1 − x ⋆ 2 ≤ (1 − µγ t )E x t − x ⋆ 2 − 2γ t (1 − Lγ t )(Ef (x t ) − f ⋆ ) + γ 2 t σ 2 ,
and the claim follows by observing (1 − Lγ t ) ≥ 1 2 for γ t ≤ 1 2L .
A classic convergence result (cf. Section 1.2). Lemma 6 is a key tool that allows to derive convergence results for SGD. To exemplify, we show there a first result for µ-convex functions with µ > 0. By choosing constant stepsizes γ t ≡ γ ≤ 1 2L (to be specified below) and relaxing (6)
to E x t+1 − x ⋆ 2 ≤ (1 − µγ)E x t − x ⋆ 2 + γ 2 σ 2 we obtain after unrolling the recurrence, E x T +1 − x ⋆ 2 ≤ (1 − µγ) T x 0 − x ⋆ 2 + γσ 2 µ .(7)
This intermediate results shows that SGD with constant stepsizes reduces the initial error term x 0 − x ⋆ 2 linearly, but only converges towards a γσ 2 µ -neighborhood of x ⋆ (cf. discussions in e.g. (Bach and Moulines, 2011;Bottou et al., 2018)). To obtain a convergence guarantee that holds for arbitrary accuracy, we need to choose the stepsize γ carefully:
• If 1 2L ≥ ln(max{2,µ 2 x0−x ⋆ 2 T /σ 2 }) µT then we choose γ = ln(max{2,µ 2 x0−x ⋆ 2 T /σ 2 }) µT . • If otherwise 1 2L < ln(max{2,µ 2 x0−x ⋆ 2 T /σ 2 }) µT then we pick γ = 1 2L .
With these choices of γ, we can show 3
E x T +1 − x ⋆ 2 =Õ x 0 − x ⋆ 2 exp − µT 2L + σ 2 µ 2 T .
This result does not show convergence of the function values f (x t ) − f ⋆ and theÕ(·) notation hides logarithmic factors. In the next section, we show how we can address both these issues.
Convergence Analysis Part II-Solving the Recursion
In this section, we consider two non-negative sequences {r t } t≥0 , {s t } t≥0 , that satisfy the relation
r t+1 ≤ (1 − aγ t )r t − bγ t s t + cγ 2 t ,(8)
for all t ≥ 0 and for parameters b > 0, a, c ≥ 0 and non-negative stepsizes {γ t } t≥0 with γ t ≤ 1 d , ∀t ≥ 0, for a parameter d ≥ a, d > 0.
By considering the special case r t = x t − x ⋆ 2 , s t = (Ef (x t ) − f ⋆ ), a = µ, b = 1, c = σ 2 and d = 2L, we see that (8) comprises the setting of Lemma 2, and thus the three lemmas that follow below will prove the claims from Section 1.1.
Constant
Stepsizes (with Log Terms). First, we derive a suboptimal (up to polylogarithmic factors) solution of (8).
Lemma 2. Let {r t } t≥0 , {s t } t≥0 as in (8) and a > 0. Then there exists a constant stepsize γ t ≡ γ ≤ 1 d such that for weights w t := (1 − aγ) −(t+1) and W T :
= T t=0 w t it holds: b W T T t=0 s t w t + ar T +1 =Õ dr 0 exp − aT d + c aT .
Decreasing Stepsizes (Avoiding Log Terms). In Lemma 2 above we collected suboptimal logarithmic terms. The averaging scheme with exponentially decreasing weights has a too short effective window to reduce the variance at the optimal O 1 T rate. In contrast, averaging schemes with polynomial weights can in general achieve the optimal O 1 T decrease of the statistical term, but do not decrease the optimization term exponentially fast (see e.g. (Lacoste-Julien et al., 2012), (Shamir and Zhang, 2013)). This suggests that a combination of these averaging strategies might yield the best results. We analyze a simple two-phase scheme, that first performs T 2 iterations without averaging and then switches to suffix averaging scheme for the remaining iterations (this analysis could be generalized to α-suffix averaging as in (Rakhlin et al., 2012)).
Lemma 3. Let {r t } t≥0 , {s t } t≥0 as in (8), a > 0. Then there exists stepsizes γ t ≤ 1 d and weighs w t ≥ 0,
W T := T t=0 w t , such that: b W T T t=0 s t w t + ar T +1 ≤ 32dr 0 exp − aT 2d + 36c aT .
Sublinear Rate (a = 0). The previous lemma allows to derive the main result presented in this note. For completeness, we also recite a lemma that solves the recursion in the special case a = 0.
Lemma 4 ((Karimireddy et al., 2019)).
Let {r t } t≥0 , {s t } t≥0 as in (8) for a ≥ 0. Then there exists a constant stepsize γ t ≡ γ ≤ 1 d such that b T + 1 T t=0 s t ≤ dr 0 T + 1 + 2 √ cr 0 √ T + 1 .
SGD convergence rates. To conclude this section, we now briefly summarize our main result that follows by replacing the variables in Lemmas 2-4 by the values stated at the beginning of this section, and observing
f (x T ) ≤ 1 WT T t=0 w t f (x t ) for convex f .
Theorem 5. For x 0 ∈ R d , let {x t } t≥0 denote the iterates of SGD (1) generated on a function f under Assumptions 1-3 for stepsizes γ t ≤ 1 2L , ∀t ≥ 0. Then there exists stepsizes γ t ≤ 1 2L and weights w t ≥ 0 such that it holds for all T ≥ 0:
Ef (x T ) − f ⋆ + µE x T +1 − x ⋆ 2 ≤ min 64LR 2 exp − µT 4L + 36σ 2 µT , 2LR 2 T + 2σR √ T ,
where here R := x 0 − x ⋆ , and again W T := T t=0 w t andx T := T t=0 w t x t . Proof. The theorem follows from the decrease Lemma 6 and Lemmas 3 and 4 with a = µ, b = 1, c = σ 2 and d = 2L. For this we observe that all stepsizes γ t ≤ 1 d ≡ 1 2L , ∀t ≥ 0. We here did not explicitly state the (suboptimal) convergence result for tuned constant stepsizes that follows directly from Lemma 2.
Discussion
We study the iteration complexity of the (stochastic) gradient descent method and recover-simultaneouslythe best known rates for the function value suboptmality for an average iterate of SGD and the distance to the optimal solution of the last iterate of SGD. Our analysis focuses on the general stochastic setting, but-as a special case-we also recover the exponential convergence rates in the deterministic setting. This unified analysis address several shortcomings of previous works.
Whilst we only consider (strongly) convex functions here, further extension of the framework to larger function classes would obviously be an interesting future direction. We would like to remark that Assumption 2 potentially also covers a much larger class of functions than the few examples discussed in Section 2. For instance, the approximate gradient oracles introduced in (Devolder et al., 2014) satisfy this assumption as well (cf. (Devolder et al., 2014, Theorem 1)) and, interestingly, Hölder continuous functions (which are in general not continuously differentiable) still admit approximate gradient oracles. However, as these oracles are not unbiased in general, Assumption 1 prevents the immediate application our framework in this extended setting (though, extension of our results under mild relaxations of the unbiasedness Assumption 1, similar as e.g. in (Bottou et al., 2018), are immediately possible).
A small drawback our results is that one needs knowledge of T and the problem parameters µ, L, σ to implement the schemes presented here (to decide on the stepsize, and for switching to the suffix averaging). In practice, some of these limitations can be remedied by the doubling trick. Further, just knowing T up to some constant factor approximation is sufficient to recover the optimal rate up to constant factors.
Example 3 (
3Empirical Risk Minimization). In machine learning applications the objective function has often a known sum structure, f (x) := 1 m m i=1 f i (x) for f i : R n → R convex and L-smooth. By picking one index i ∼ u.a.r. [m], uniformly at random, g t := ∇f i (x t ) is an unbiased, (2L, 2 m m i=1 ∇f i (x ⋆ )
Here we follow the standard convention that O hides constants andÕ hides constants and factors polylogarithmic in the problem parameters. For the ease of exposition, we sometimes ignore absolute constants ν in the exponent when discussing results and write O(exp[−T ]) instead of O(exp[−νT ]). We diligently report these constants when stating new results.
We refer the readers to the proof of Lemma 2 below for detailed computations in a very similar setting.
AcknowledgmentsWe would like to thank Martin Jaggi for his support and his helpful comments and Praneeth Karimireddy for his suggestions to improve the first version of this manuscript.A Technical LemmasA.1 Constant Stepsizes (with Log Terms)Proof of Lemma 2. We start by re-arranging (8) and multiplying both sides with w t :By summing from t = 0 to t = T , we obtain a telescoping sum:With the estimateswe can further simplify the left and right hand sides:Now the lemma follows by carefully tuning γ. Consider the two cases:• If 1 d ≥ ln(max{2,a 2 r0T 2 /c}) aT then we choose γ = ln(max{2,a 2 r0T 2 /c}) aT and get that Equation(9)isas in case 2 ≥ a 2 r 0 T 2 /c it holds ar 0 T ≤ 2c aT .• If otherwise 1 d < ln(max{2,a 2 r0T 2 /c}) aT then we pick γ = 1 d and get that Equation(9)isCollecting these two cases concludes the proof.A.2 Decreasing Stepsizes (Avoiding Log Terms)For the proof of Lemma 3 we need auxiliary results, for both of which we do not claim much novelty here.Lemma 6. Let {r t } t≥0 , {s t } t≥0 be as in (8) for a > 0 and for constant stepsizes γ t ≡ γ := 1 d , ∀t ≥ 0. Then it holds for all T ≥ 0:Proof. This follows by relaxing(8), and unrolling:The next lemma is similar to the result derived in(Lacoste-Julien et al., 2012, Sec. 3.2), except that we cannot chose the stepsizes γ t arbitrarily large and hence need to take care of the initial conditions. Similar results were presented e.g. in(Stich et al., 2018;Grimmer, 2019).Lemma 7. Let {r t } t≥0 , {s t } t≥0 as in(8)for a > 0 and for decreasing stepsizes γ t := 2 a(κ+t) , ∀t ≥ 0, with parameter κ := 2d a , and weights w t := (κ + t). Thenwhere here again W T := T t=0 w t .Proof. We start as in the proof of Lemma 2,where the equality follows from the definition of γ t and w t and the inequality from (κ +• and W T = (2κ+T )(T +1) 2 ≤ 2(κ+T )(1+T ) 2 ≤ (κ + T ) 2 for κ = 2d a ≥ 1. By applying these two estimates we conclude the proof.We can now combine the findings of these two lemmas.Proof of Lemma 3. For integer T ≥ 0, we choose stepsizes and weights as follows:for κ = 2d a and t 0 = T 2 . We will now show that these choices imply the claimed result.We start with the case T ≤ d a . This case is similar to the proof of Lemma 2 and it suffices to consider Equation 9 for the choice γ = 1 d . We observe that Equation 9 simplifies toIf T > d a , then we obtain from Lemma 6 thatFrom Lemma 7 we have for the second half of the iterates:aT .Now we observe that the restart condition r t0 satisfies:These inequalities show the claim.A.3 Sub-linear Convergence rateProof of Lemma 4. We start by re-arranging (8) and summing from t = 0 to t = T :Now we carefully select γ.• If 1 d 2 ≤ r0 c(T +1) , then we pick γ = 1 d with γ ≤ √ r0 √ c(T +1) and verify r 0 γ(T + 1) + cγ ≤ dr 0 T + 1 + √ cr 0 √ T + 1 .• If otherwise 1 d 2 > r0 c(T +1) then we pick γ = √ r0 √ c(T +1) to obtain r 0 γ(T + 1) + cγ ≤ 2 √ cr 0 √ T + 1 .
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| {'fraction_non_alphanumeric': 0.07052835676199709, 'fraction_numerical': 0.048888581123190916, 'mean_word_length': 3.8026272031925505, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 35, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on µ-convex functions under a (milder than standard) L-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after T iterations aswhere σ 2 measures the variance in the stochastic noise. For deterministic gradient descent (GD) and SGD in the interpolation setting we have σ 2 = 0 and we recover the exponential convergence rate. The bound matches with the best known iteration complexity of GD and SGD, up to constants. * [email protected], Machine Learning and Optimization Lab (MLO), EPFL, Switzerland.', 'arxivid': '1907.04232', 'author': ['Sebastian U Stich \nEPFL\n\n'], 'authoraffiliation': ['EPFL\n'], 'corpusid': 195848212, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10283, 'n_tokens_neox': 8495, 'n_words': 5083, 'pdfsha': '195cd45ceb3dce621c591f032976e63d08640e36', 'pdfurls': ['https://export.arxiv.org/pdf/1907.04232v2.pdf'], 'title': ['Unified Optimal Analysis of the (Stochastic) Gradient Method', 'Unified Optimal Analysis of the (Stochastic) Gradient Method'], 'venue': []} |
arxiv |
Cheeger-like inequalities for the largest eigenvalue of the graph Laplace operator
2021
Jürgen Jost
Max Planck Institute for Mathematics in the Sciences
D-04103Leipzig, LeipzigGermany, Germany
Raffaella Mulas [email protected]
Max Planck Institute for Mathematics in the Sciences
D-04103Leipzig, LeipzigGermany, Germany
Cheeger-like inequalities for the largest eigenvalue of the graph Laplace operator
J Graph Theory
202110.1002/jgt.22664Received: 20 January 2021 | Accepted: 8 February 2021A R T I C L E Correspondence Raffaella Mulas, Max Planck Institute for Mathematics in the Sciences,
We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.K E Y W O R D SCheeger-like constant, largest eigenvalue, normalized Laplacian, spectral theory ≥ λ n n − 1 n This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
| INTRODUCTION
The (normalized) Laplace operator is a very powerful tool for the study of graphs, as its spectrum encodes important information [8,4,15,2]. Here we consider unweighted and undirected (but oriented) graphs without loops, multiple edges, and isolated vertices. For a fixed such graph on n vertices, let's arrange the n eigenvalues of the Laplace operator, counted with multiplicity, as ≤ ⋯ ≤ λ λ . n 1
We have λ = 0 1 , and the multiplicity of the eigenvalue 0 equals the number of the connected components of the graph. Thus, (1) if and only if the graph is connected. Henceforth, we shall only consider connected graphs. There is also a quantitative aspect. As we shall explain in more detail below, λ 2 estimates the coherence of the graph, that is, how different it is from a disconnected one.
λ > 0 2
The largest eigenvalue, which is the main object of interest of this article, satisfies with equality if and only if the graph is complete. For noncomplete gaphs ≥ λ n n + 1 − 1 , n with equality if and only if the graph either is obtained from a complete graph by removing a single edge or consists of two complete graphs of size n + 1 2 that share a single vertex [16]. In the other direction
≤ λ 2 n(2)
with equality if and only if at least one connected component of the graph is bipartite. For connected graphs, the first nonzero eigenvalue λ 2 is controlled both above and below by the Cheeger constant h, a quantity that measures how difficult it is to partition the vertex set into two disjoint sets V 1 and V 2 such that the number of edges between V 1 and V 2 is as small as possible and such that the volume of both V 1 and V 2 , that is, the sum of the degrees of their vertices, is as big as possible. In particular,
≤ ≤ h λ h 1 2 2 . 2 2(3)
Furthermore, there is an interesting characterization of h obtained by writing λ 2 using the Rayleigh quotient and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator, as we shall see in Section 2.
In this paper, we want to explore an analogue of this for λ n . In the same sense that by (3), λ 2 estimates how different a connected graph is from being disconnected, by (2), λ 2 − n should quantify how different the graph is from being bipartite. One might therefore try to find the best (in a suitable sense) bipartite subgraph of our graph, because for a bipartite graph, the Rayleigh quotient that we shall discuss below is 2, the maximal possible value. In fact, as it turns out, that subgraph can be quite small. More precisely, we shall introduce a new constant that is an analogue of the Cheeger constant in the sense that it can be characterized by writing λ n using the Rayleigh quotient and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator. This constant is very simple,
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≔ Q v w max 1 deg + 1 deg . v w edges Ã
nalogously to the Cheeger estimate (3), we shall prove that it controls the largest eigenvalue λ n both above and below. Therefore, Q is an analogue of the Cheeger constant for the largest eigenvalue. As we had explained above, λ 2 controls how different the graph is from a connected. Analogously, in view of (2), one should expect that λ 2 − n measures the difference from a bipartite graph.
Throughout the paper we shall also prove new general results of spectral graph theory that are useful to prove or discuss our main result.
| Structure of the paper
In Section 2 we discuss the Laplace operator, the Cheeger constant, the dual Cheeger constant and the edge-Laplacian, as preliminaries to our work. In Section 3, and in particular in JOST AND MULAS | 409 Theorem 3, we present our main results and we prove them in Section 4. In Section 5 we motivate the choice of Q, in Section 6 we discuss the precision of our lower bound for λ n , and finally in Section 7 we discuss the precision of our upper bound.
| PRELIMINARIES
In this section, we present some well-known results of spectral graph theory as preliminaries to our work; a general reference is [8].
From here on we fix a graph V E Γ = ( , ) on n vertices. We also fix an arbitrary orientation on Γ, that is, we see each edge as an arbitrarily ordered pair of its endpoints. Given ∈ e v w E = ( , )
, we say that v is the input of e and w is its output. The fixed orientation is needed to do the computations when considering a function → γ E :
. However, the results are independent of the chosen orientation because, if one reverses the orientation of some edges, changing the sign of γ on these edges leads to the same results. Therefore, the oriented edges considered here should not be confused with directed edges. Moreover, we shall use the notation v w for indicating (oriented) edges when input and output do not need to be distinguished.
| Laplace operator and its eigenvalues
Let Id be the n n × identity matrix, let A be the adjacency matrix of Γ, let D be the diagonal degree matrix and let by the min-max principle, the eigenfunctions of the other eigenvalues must be orthogonal to them with respect to the scalar product
≔ ∑ ⋅ ⋅ f g v f v g v ( , ) deg ( ) ( ) v .
The orthogonality to the constants is satisfied also by the eigenfunctions of λ n , but in this case we do not need to specify it.
| Cheeger constant
For a connected graph V E Γ = ( , ), the Cheeger constant is defined as
≔ | | h E S S S S min ( ,¯) min{vol( ), vol(¯)} , S where, given ∅ ≠ ⊊ ≔ ⧹ | | S V S
V S E S S ,¯, ( ,¯) denotes the number of edges with one endpoint in S and the other in S, and
≔ ∑ ∈ S v vol( ) deg( ) v S
. The following theorem [1,12] gives two important bounds for λ 2 in terms of h. Theorem 1. For every connected graph,
≤ ≤ h λ h 1 − 1 − 2 . 2 2(4)
Also, the following theorem [8, Theorem 2.8 and Corollary 2.9] shows the interesting relation between h and λ 2 when, in the characterizations of λ 2 via the Rayleigh quotient, we replace the L 2 -norm by the L 1 -norm both in the numerator and denominator.
Theorem 2. For every connected graph,
∑ ∑ ⋅ → ∈ ∈ h f v f w v f v t = min max~( ) − ( ) deg ( ) − f V t v w v V : non constant and ≤ ∑ ∑ ⋅ ≤ ∑ → ≠ ⋅ ∈ ∈ h f v f w v f v h 1 2 min~( ) − ( ) deg ( ) . f V f v f v v w v V : s . t . 0 , d e g ( )=0 v V Remark 3. Interestingly, the quantity ∑ ∑ ⋅ → ∈ ∈ f v f w v f v t min max~( ) − ( ) deg ( ) − f V t v w v V :
non constant that characterizes h in Theorem 2 is equal to the second smallest eigenvalue of the 1-Laplacian [6,13,14,7,9].
Our Theorem 3 is an analogue of Theorems 1 and 2 for the largest eigenvalue λ n in terms of our new constant Q. Before stating it, we shall discuss the dual Cheeger constant and the edge-Laplacian.
| Dual Cheeger constant
In literature, there is already a Cheeger-like constant that bounds the largest eigenvalue [3,5]. It is defined as
JOST AND MULAS | 411 ≔ | | ⊔ ⊔ h E V V V V max ( , ) vol( ) + vol( ) , V V V V partitions = 1 2 1 2 1 2 3
it is called the dual Cheeger constant and it satisfies an analogue of (4),
≤ ≤ h λ h 2¯1 + 1 − (1 −¯) . n 2
The two constants h and h are actually related to each other [5]. For the dual Cheeger constant, however, there is no result analogous to Theorem 2 [10]. This motivates the definition of the new constant Q that again bounds λ n and, additionally, satisfies an analogue of Theorem 2.
| Edge-laplacian
Associated to the Laplace operator there is also the edge-Laplacian, defined as
≔ L D , E T −1 where is the | | | | V E × incidence matrix of Γ.
Instead on acting on functions defined on the vertex sets, L E acts on functions defined on the edge set. It has the same nonzero spectrum of L (i.e., the nonzero eigenvalues are the same, counted with multiplicity) and the multiplicity of the eigenvalue 0 for L E equals the number of cycles of Γ [15]. We can therefore write the largest eigenvalue (that coincides for L and L E ) also in terms of the Rayleigh quotient for functions on the edge set, by applying the min-max principle to L E :
∑ ∑ ∑ ∑ ∑ ∑ → ≠ ⋅ → ≠ ⋅ ∈ ∈ ∈ ( ) λ = m a x = max . n f V f f v f w v f v γ E γ γ e γ e γ e :
, 0 In Section 3 we shall present an analogue of Theorems 1 and 2, where:
( ( ) − ( )) deg ( ) : , 0 ( )− ( ) ( ) v w v V v V v e v e v e E
• We look at λ n instead of λ 2 .
• We use Q instead of h.
• We use the point of view of the edge-Laplacian for considering the Rayleigh quotient and characterize Q.
| MAIN RESULTS
Before stating our main theorem, let's recall that for a graph Γ we have defined the new Cheeger-like constant
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≔ Q v w max 1 deg + 1 deg . v w
Let's also define the constant
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≔ ⋅ ≥ τ w v n v v w max (deg − deg + ) deg deg + deg . v w w ṽ : deg deg Theorem 3. For every graph, ∑ ⋅ ∑ ∑ ∑ → ≠ ∈ ∈ Q γ e γ e γ e = max ( ) − ( ) ( ) γ E γ v V v e v e vand ≤ ≤ ⋅ Q λ Q τ. n
Observe that the characterization of Q appearing in Theorem 3 equals the Rayleigh quotient we have used for writing λ n from the point of view of the edge-Laplacian, replacing the L 2 -norm by the L 1 -norm. Therefore, such a characterization is analogous to the one of h in Theorem 2. We prove Theorem 3 in Section 4. Also, in Section 5 we motivate the choice of Q, in Section 6 we discuss whether the lower bound appearing in Theorem 3 is sharp, and in Section 7 we discuss the sharpness of the upper bound.
| PROOF OF THE MAIN RESULTS
We split the statement of Theorem 3 into three parts. The first part, Lemma 4, contains the characterization of Q. The second part, Lemma Proof. To prove that
≤ ∑ ⋅ ∑ ∑ ∑ → ≠ ∈ ∈ Q γ e γ e γ e max ( ) − ( ) ( ) , γ E γ v V v e v e v e E :
, 0
1 deg : input in : output out in out fix an edge v v ( , ) 1 2 that maximizes + v w 1 deg 1 deg over all ∈ v w E ( , )
and let → γ E ′: be 1 on v v ( , ) 1 2 and 0 otherwise. Then, . Then, This proves the claim. □ As a corollary of Lemma 4, we get another characterization of Q. . Then, by taking ⊂ Γ Γ as the bipartite graph containing only the edge v v ( , ) 1 2 , we get that If there is no such vertex, then
∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ⋅ | | ≤ ⋅ | | ∈ ∈ → ≠ ∈ ∈ Q v v γ e γ e γ e γ e γ e γ e = 1 deg + 1 deg = ′( ) − ′ ( ) ′( ) max ( ) − ( ) ( ) . v V v e v e v e E γ E γ v V v e v e v e ELet → ≠ γ E γ: ,ˆ0 be a maximizer for ∑ ⋅ ∑ ∑ ∑ ∈ ∈ γ e γ e γ e ( ) − ( ) ( ) v V v e v e v e E ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ⋅ | | ≥ | |⋅ ⋅ | | ≥ ⋅ ⋅ | | ∈ ∈ ∈ → ≠ ∈ ∈ Q v w v w γ e γ e v w v γ e v γ e γ e γ e γ e γ e = max 1 deg + 1 deg = max 1 deg + 1 deg ˆ( ) ( ) 1 deg + 1 deg = 1 deg ˆ( ) 1 deg ˆ( ) −ˆ( ) = max ( ) − ( ) ( ) . v w v w e E v w v V e v v V e v e v γ E γ v V v e v e vCorollary 5. ∑ | | ⊂ ∈ Q E = max (Γ ) . v V v v Γ Γ bipartite deg ( ) deg ( ) Γ Γ Proof. Let's fix ⊂ Γ′ Γ that maximizes ∑ | | ∈ E (Γ ) . v V v v deg ( )deg′( ) − ′ ( ) ′( ) = (Γ′) = max (Γ ) . γ E γ v V v e v e v e E v V v e v e v e E v V v v v V v v⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∑ | | ≥ ⊂ ∈ E v v v w Q max (Γ ) 1 deg + 1 deg = max 1 deg + 1 deg = . v V v v v w Γ Γ bipartite deg ( ) deg ( ) 1 2Γ Γ □
| Lower bound for the largest eigenvalue
′( ) − ′ ( ) ′( ) = 1 deg + 1 deg = . n γ E γ v V v e v e v e E v V v e v e v≤ ≤ Q n n 1 2 + 1 2 = 1 − 1 .
Therefore, the bound in Lemma 6 is better than the usual bound ≤ λ n n n − 1 only for a small class of graphs. However, the aim of our work is not to find the best possible bounds forλ n but the best possible bounds for λ n involving Q, to show that Q is a Cheeger-like constant. We shall see, in Section 6, that the bound in Lemma 6 is actually the best possible lower bound for λ n involving only Q.
| Upper bound for the largest eigenvalue
Lemma 7. For every graph,
≤ ⋅ λ Q τ. n
Proof. We apply [18,Theorem 5] to obtain
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ | ∩ | ≤ ⋅ ⋅ ≤ ⋅ ≥ ≥ ≥ λ v w v w v w n w w v n w v w w v n v v w Q τ 2 − min ( ) ( ) max{deg , deg } 2 − min deg + deg − deg = max deg − deg + deg = max 1 deg + 1 deg (deg − deg + ) deg deg + deg . n v w v w w v v w w v v w w ṽ : deg deg ~: deg deg ~: deg deg □
Observe that the bound in Lemma 7 is not a better upper bound for λ n than the one in [18,Theorem 5]. Nevertheless, it is a good upper bound for λ n involving Q, as we shall see in Section 7.
| CHOICE OF Q
Let us motivate the choice of Q. As we have discussed in Section 2,
∑ ∑ ⋅ → ≠ ∈ λ f v f w v f v = max~( ( ) − ( )) deg ( ) n f V f v w v V : , 0 2 2 (5) ∑ ⋅ ∑ ∑ ∑ → ≠ ∈ ∈ ( ) γ e γ e γ e = max ( ) − ( ) ( ) . γ E γ v V v e v e v
We have chosen Q to be the constant that can be written as (6) by replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator. We could have chosen to work on the constant that can be written as (5) by replacing the L 2 -norm by the L 1 -norm, but such a constant is actually equal to 1 for all graphs, as shown by the following lemma. Furthermore, while the characterization of the Cheeger constant is interesting also because it is equal to the second smallest eigenvalue of the 1-Laplacian, one cannot get an analogous constant in this sense because the largest eigenvalue of the 1-Laplacian equals 1 for every graph, as shown in [8,Theorem 5.1]. For completeness, we shall provide a proof.
Lemma 8. For every graph,
∑ ∑ ⋅ → ≠ ∈ f v f w v f v max~( ) − ( ) deg ( ) = 1. f V f v w v V : , 0 Proof. Let → f V : be a maximizer of ∑ ∑ ⋅ ∈ f v f w v f ṽ ( ) − ( ) deg ( ) v w v V
and assume, without loss of generality, that
∑ ⋅ | | ∈ v f v deg ˆ( ) = 1 v V . Then, ∑ | | ≤ ∑ | | | | ∑ ⋅ | | ∑ ∑ → ≠ ⋅ ∈ ∈ f v f w f v f w v f v max =ˆ( ) −ˆ( ) ( ) +ˆ( ) = deg ˆ( ) = 1. f V f f v f w v f v v w v w v V : , 0 ( ) − ( ) deg ( )ṽ w v V
To see the inverse inequality, let → f Ṽ : that is 1 on a fixed vertex and 0 on all other vertices. Then,
∑ ∑ ⋅ ≥ ∑ ∑ ⋅ → ≠ ∈ ∈ f v f w v f v f v f w v f v max~( ) − ( ) deg ( )~~( ) −~( ) deg ~( ) = 1. f V f v w v V v w v V :
, 0 □ 6 | HOW GOOD IS THE LOWER BOUND?
To see that ≤ Q λ n is a sharp lower bound, consider the case of For further motivating our upper bound, we shall:
(1) Prove that, for each graph on n nodes, ⋅ τ n < 0.54 and 0.54 is the best ε with a precision of two decimal places such that
≤ ⋅ ⋅ λ Q ε n. n (2) Prove that there is no bound of the form ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ≤ ⋅ λ Q n c 2 + , n
if c is a constant that does not depend on n, as we might be tempted to do by looking at the example of regular graphs.
To prove these two points, we shall first discuss one-sided bipartite graphs, a new big class of graphs that includes among others petal graphs, complete graphs, and complete bipartite graphs.
| One-sided bipartite graphs
Definition Fix n and k such that
≤ k n 0 < − 2. Let V E Γ = ( , ) be a graph on n vertices such that ⊔ | | V V V V k = , = 1 2 2 therefore | | V n k v v = − ,1 1 2 for each ∈ v V 1 1 and ∈ v V v n k , deg = − 2 2 2 for each ∈ v V 2 2 and v d deg = 1 for each ∈ v V 1 1 , for some ≥ d k.
Call such a graph a k d ( , )-one-sided bipartite graph.
Remark 5. In a k d ( , )-one-sided bipartite graph, the vertex set is divided into two sets V 1 and V 2 . All possible edges between V 1 and V 2 are there, the k vertices in V 2 are not connected to each other and the vertices in V 1 all have degree d, therefore there are edges between vertices of V 1 if and only if d k > (Figure 1). In particular, a k d ( , )-one-sided bipartite graph is:
• The petal graph if k = 1 and d = 2.
• The complete graph K n if k = 1 and d n = − 1.
• The graph Proof. This follows easily by definition of one-sided bipartite graphs and by [11,Theorem 2.6], which states that a d-regular graph on n nodes exists if and only if at least one of d and n is even. □ In Theorem 11 we shall prove that for a one-sided bipartite graph with ≥ d n k − , λ d k d = + n and for a k d ( , )-one-sided bipartite graph with d n k < − ,
≤ ≤ d k d λ n d + .
n Let's prove a preliminary lemma first.
Definition [4]. Given a vertex v 1 , let ⊂ v V ( ) 1 be the set of neighbors of v 1 . We say that v 1 and v 2 are duplicate vertices if v v ( ) = ( ) 1 2 .
Observe that, in particular, duplicate vertices have the same degree and they cannot be neighbors of each other.
, the matrices L and are similar, therefore they have the same spectrum, including multiplicities, although the eigenfunctions can be different.By the Courant-Fischer-Weyl min-max principle, we can write the eigenvalues≤ ⋯ ≤ λ λ n 1of L in terms of the Rayleigh quotient[8, pp. 4 and 5]. In particular,
⊂ Γ Γ bipartite. Let's fix an orientation and let → γ E ′: (Γ) be 1 on each oriented edge in E (Γ′) and 0 otherwise. Then,
Lemma 6 .
6For every graph, ≤ Q λ . n Proof. As in the proof of Lemma 4, fix an edge v v
□
Remark 4. Observe that ≥ Q n n − 1 if and only if there exists a vertex of degree 1. In fact, if there exists such a vertex, then
k = 2 and d n = − 1. • The complete bipartite graph K d n k , − if d k = . • Not bipartite if d k > . • A d-regular graph if d n k = − .Lemma 9. Given n k , , and d such that ≥ ≤ n k n 3, 0 < − 2, and ≤ ≤ k d n − 1, there exists a k d ( , )-one-sided bipartite graph on n nodes if and only if at least one of d k − and n k − is even.
F
I G U R E 1 A k d ( , )-one-sided bipartite graph on 7 nodes, with k = 3 and d = 5. The black nodes are the ones of degree d
and, since the graph is nonbipartite, λ < 2 n . Therefore, if we look for a bound of the form Hence ≤ Q λ n is actually a good lower bound involving only Q for each n.7 | HOW GOOD IS THE UPPER BOUND?To see that the bound ⋅ Q τ is actually a good upper bound for λ n , let us first construct an example for which the bound Example 1. For d-regular graphs, it is easy to see that Q = d 2 and τ = , that is, the inequality in Lemma 7 becomes an equality.K 2 : here, Q λ
= = 2
2
. Also, for
n > 2, consider a nonbipartite graph such that there exists an edge v w
( , ) with
v
deg = 1 and
w
deg = 2. Then, clearly
Q = 1 +
1
2
=
3
2
⋅ ≤
Q ν λ ,
n
we must have
≤
≃
ν
λ
Q
<
4
3
1.33.
n
≤ ⋅
λ
Q τ
n
is sharp.
n
2 , therefore
≤ ⋅
λ
Q τ
n
is equivalent to
≤
λ
n
d
.
n
In the particular case of the complete graph K d n
, = − 1
n
and λ =
n
n
n − 1 [8] therefore
⋅
λ
Q τ
=
n
Lemma 10.If v 1 and v 2 are duplicate vertices and f is an eigenfunction for an eigenvalue ≠ λ 1 of L, f v f v ( ) = ( ).1 2Proof. An eigenvalue λ of L with eigenfunction f satisfies for each vertex v,n For a k d ( , )-one-sided bipartite graph with d n k < − ,n Proof. For any fixed k d ( , )-one-sided bipartite graph, let ≠ λ 0, 1 be an eigenvalue for L with eigenfunction f . By construction, in a k d ( , )-one-sided bipartite graph all k vertices in V 2 of degree n k − are duplicate vertices. Therefore, by Lemma 10,In particular, since we are assuming ≠ λ 1, this implies that ≠ α 0 n k − , hence we can writewhich is a contradiction. Therefore we must haveThis proves that d k d + is an eigenvalue, thereforen Now, in the particular case of ≥ d n k − , we can prove also the inverse inequality by applying[18,Theorem 5], which states thatLet's consider the possible cases.• Case 1:1. In this case, v w d deg = deg = . Also, v and w have k neighbors in common in V 2 and at least d k n k 2( − ) − ( − ) neighbors in common in V 1 . Therefore,where the last inequality follows from the assumption that ≥ d n k − . Therefore,and by[18,Theorem 5]this implies thatn therefore that the equality holds in this case. It remains to prove that, for d n k < − ,Let again ≠ λ 0, 1 be an eigenvalue for L with eigenfunction f . We know that f v ( ) 2 must be constant for each ∈ v V Therefore, since we are assuming d n k < − , we have that λ n d < .Let's now consider the case f v ( ) = 0 2 . We have thatIn fact, To prove(7), it suffices to show that, for d−regular graphs on n nodes, the largest eigenvalue of the nonnormalized Laplace operator is at most n. This is actually true for every graph, because for the nonnormalized Laplacian the complete graph has largest eigenvalue equal to n and, if an edge is added into a graph, then none of its Laplacian eigenvalues can decrease[17]. Therefore,n ′ This proves that any eigenvalue of L, in the case when d n k < − , is at most ∕ n d. Therefore, in particular, ≤ ∕ λ n d n . □ Remark 6. Observe also that, for k d ( , )-one-sided bipartite graphs with ≥ d n k − ,| ConclusionsAs a consequence of Theorem 11, we can prove the following corollary that further motivates the upper bound in Lemma 7.Corollary 12.(1) For each graph on n nodes, ⋅ τ n < 0.54 and 0.54 is the best ε with a precision of two decimal places such thatif c is a constant that does not depend on n.Proof.(1) By writing in WolframAlpha[19]:By writing in WolframAlpha[19]: Therefore, if we look for an upper bound of λ n of the form ⋅ Q g n ( ), we must have
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The dual Cheeger constant and spectra of infinite graphs. F Bauer, B Hua, J Jost, Adv. Math. 251F. Bauer, B. Hua, and J. Jost, The dual Cheeger constant and spectra of infinite graphs, Adv. Math. 251 (2014), 147-194.
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The 1-Laplacian Cheeger cut: Theory and algorithms. K C Chang, S Shao, D Zhang, J. Comput. Math. 5K. C. Chang, S. Shao, and D. Zhang, The 1-Laplacian Cheeger cut: Theory and algorithms, J. Comput. Math. (2015), no. 5, 443-467.
K C Chang, S Shao, D Zhang, arXiv:1607.00489Spectrum of the signless 1-Laplacian and the dual Cheeger constant on graphs. K. C. Chang, S. Shao, and D. Zhang, Spectrum of the signless 1-Laplacian and the dual Cheeger constant on graphs, arXiv:1607.00489, 2016.
A first course in graph theory. G Chartrand, P Zhang, Dover PublicationsMineola, New York, USG. Chartrand, and P. Zhang, A first course in graph theory, Mineola, New York, US: Dover Publications, 2012.
Difference equations, isoperimetric inequality and transience of certain random walks. J Dodziuk, Trans. Am. Math. Soc. 2842J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Am. Math. Soc. 284 (1984), no. 2, 787-794.
An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. M Hein, T Bühler, NIPS. M. Hein, and T. Bühler, An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA, NIPS, 2010, pp. 847-855.
Beyond spectral clustering-tight relaxations of balanced graph cuts. M Hein, S Setzer, Adv. Neural Inform. Process. Syst. 24M. Hein, and S. Setzer, Beyond spectral clustering-tight relaxations of balanced graph cuts, Adv. Neural Inform. Process. Syst. 24 (2011), 2366-2374.
Hypergraph Laplace operators for chemical reaction networks. J Jost, R Mulas, Adva. Math. 351J. Jost, and R. Mulas, Hypergraph Laplace operators for chemical reaction networks, Adva. Math. 351 (2019), 870-896.
Spectral gap of the largest eigenvalue of the normalized graph Laplacian. J Jost, R Mulas, F Münch, 10.1007/s40304-020-00222-7Comm. Math. Stat. to appearJ. Jost, R. Mulas, and F. Münch, Spectral gap of the largest eigenvalue of the normalized graph Laplacian, Comm. Math. Stat., 2021, to appear. https://doi.org/10.1007/s40304-020-00222-7
Completion of Laplacian integral graphs via edge addition. S Kirkland, Discrete Math. 295S. Kirkland, Completion of Laplacian integral graphs via edge addition, Discrete Math. 295 (2005), 75-90.
O Rojo, R L Soto, A new upper bound on the largest normalized Laplacian eigenvalue. 7O. Rojo, and R. L. Soto, A new upper bound on the largest normalized Laplacian eigenvalue, Oper. Matrices 7 (2013), 323-332.
Cheeger-like inequalities for the largest eigenvalue of the graph Laplace Operator. Wolfram Alpha, Llc Wolframalpha, ; J Jost, R Mulas, 10.1002/jgt.22664J Graph Theory. 97How to cite this articleWolfram Alpha LLC, WolframAlpha, available at https://www.wolframalpha.com How to cite this article: J. Jost, and R. Mulas, Cheeger-like inequalities for the largest eigenvalue of the graph Laplace Operator, J Graph Theory. (2021), 97, 408-425. https://doi.org/10.1002/jgt.22664
| {'fraction_non_alphanumeric': 0.07683603032703407, 'fraction_numerical': 0.028478744672022934, 'mean_word_length': 3.0476335877862595, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 4, 'https://': 3, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 144, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.K E Y W O R D SCheeger-like constant, largest eigenvalue, normalized Laplacian, spectral theory ≥ λ n n − 1 n This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.', 'arxivid': '1910.12233', 'author': ['Jürgen Jost \nMax Planck Institute for Mathematics in the Sciences\nD-04103Leipzig, LeipzigGermany, Germany\n', 'Raffaella Mulas [email protected] \nMax Planck Institute for Mathematics in the Sciences\nD-04103Leipzig, LeipzigGermany, Germany\n'], 'authoraffiliation': ['Max Planck Institute for Mathematics in the Sciences\nD-04103Leipzig, LeipzigGermany, Germany', 'Max Planck Institute for Mathematics in the Sciences\nD-04103Leipzig, LeipzigGermany, Germany'], 'corpusid': 204976431, 'doi': '10.1002/jgt.22664', 'github_urls': [], 'n_tokens_mistral': 9731, 'n_tokens_neox': 8349, 'n_words': 5385, 'pdfsha': '577eeb71a5ab32b9e10e9da8109093d818b2be99', 'pdfurls': None, 'title': ['Cheeger-like inequalities for the largest eigenvalue of the graph Laplace operator', 'Cheeger-like inequalities for the largest eigenvalue of the graph Laplace operator'], 'venue': ['J Graph Theory']} |
arxiv |
Analytical investigations on non-minimally coupled scalar fields outside neutral reflecting shells
8 Oct 2022
Yan Peng
School of Mathematical Sciences
Qufu Normal University
273165QufuShandongChina
Analytical investigations on non-minimally coupled scalar fields outside neutral reflecting shells
8 Oct 2022numbers: 1125Tq0470Bw7420-z * yanpengphy@163com 2
We study the existence of scalar fields outside neutral reflecting shells. We consider static massive scalar fields non-minimally coupled to the Gauss-Bonnet invariant. We analytically investigated properties of scalar fields through the scalar field equation. In the small scalar field mass regime, we derive a compact resonance formula for the allowed masses of scalar fields in the composed scalar field and shell configurations.
I. INTRODUCTION
The famous black hole no hair theorem has attracted a lot of attention from physicists and mathematicians for decades. If true, it states that asymptotically flat black holes cannot support static scalar field hairs outside horizons, for progress see references [1]- [19] and reviews [20,21]. This no hair property was usually attributed to the existence of absorbing boundary conditions at black hole horizons. So it is of great interest to examine whether such no hair behavior can appear in horizonless spacetimes.
In the horizonless spacetime, Hod recently proved a new type no hair theorem that static scalar field hairs cannot exist outside asymptotically flat neutral reflecting stars [22,23]. Furthermore, such no hair theorem also holds for systems constructed with static scalar fields and neutral horizonless reflecting shells [24,25].
In the asymptotically dS spacetimes, the static scalar field hair also cannot exist outside neutral horizonless reflecting stars [26]. And further studies showed that such no scalar hair theorem is a general property for horizonless objects with reflecting boundary conditions [27]- [32].
However, asymptotically flat black holes can support static scalar fields non-minimally coupled to electromagnetic Maxwell fields, which violates the spirit of no hair theorems [33,34]. Similarly, Hod proved that scalar fields can exist outside charged horizonless reflecting shells when considering non-minimally couplings between static scalar fields and electromagnetic Maxwell fields [35]. In particular, Hod derived a remarkably compact resonance formula for the allowed masses of the spatially regular scalar fields supported by charged shell configurations. In the background of neutral horizonless reflecting shells, exterior scalar fields also can exist when including non-minimally couplings between scalar fields and the Gauss-Bonnet invariant [36]. Inspired by analysis in [35], we plan to carry out an analytical investigation on configurations composed of scalar fields and neutral reflecting shells in the scalar-Gauss-Bonnet gravity.
This work is organized as follows. We firstly introduce the system with scalar fields non-minimally coupled to the Gauss-Bonnet invariant outside neutral horizonless reflecting shells. Then we shall derive a remarkably compact resonance formula for the allowed masses of scalar fields supported by neutral reflecting shells. We will summarize our main results in the last section.
II. SCALAR FIELD EQUATIONS IN THE REFLECTING SHELL BACKGROUND
We consider scalar fields non-minimally coupled to the Gauss-Bonnet invariant. The Lagrangian density of the scalar-Gauss-Bonnet gravity is given by [37][38][39][40][41][42][43]
L = R − |∇ α Ψ| 2 − µ 2 Ψ 2 + f (Ψ)R 2 GB .(1)
Here R is the Ricci scalar curvature, Ψ is the scalar field with mass µ and f (Ψ)R 2 GB describes the coupling,
where R 2 GB is the Gauss-Bonnet invariant [39, 40] R 2 GB = R µνρσ R µνρσ − 4R µν R µν + R 2(2)
and f (Ψ) is a general function of Ψ. When neglecting matter fields' backreaction on the metric, the Gauss-
Bonnet invariant term is R 2 GB = 48M 2 r 6 .
Without loss of generality, the function f (Ψ) can be putted in a simple form f (Ψ) = ηΨ 2 in the linear limit, where η is the coupling strength parameter.
The spherically symmetric spacetime is characterized by the curved line element [40]
ds 2 = −g(r)dt 2 + dr 2 g(r) + r 2 (dθ 2 + sin 2 θdφ 2 ).(3)
The radial coordinate depending function g(r) is the metric solution. The angular coordinates are labeled as θ and φ respectively. And the radius of the shell is defined as r s .
With variation methods, we obtain the scalar field equation [37]- [43] [
(∇ α − iqA α )(∇ α − iqA α ) − µ 2 + ηR 2 GB ]Ψ = 0.(4)
We choose to study a stationary massive scalar field in the form
Ψ(r, θ, φ) = lm ψ lm (r) r S lm (θ)e imφ(5)
with l, m representing the integer harmonic parameters. S lm (θ) is the angular scalar eigenfunction with the eigenvalue l(l + 1), where l is the spherical harmonic index [35]. For simplicity, we label the characteristic radial function ψ lm (r) as ψ(r).
The equation (4) and field decomposition (5) yield the ordinary differential equation
[ d 2 dr 2 − µ 2 − l(l + 1) r 2 + 48ηM 2 r 6 ]ψ = 0.(6)
Here we take the metric outside shells in the flat spacetime limit with g = 1 and g ′ = 1.
At the shell radius, the scalar field satisfies reflecting surface conditions ψ(r s ) = 0.
At the infinity, the scalar field is spatially regular, leading to the vanishing condition
ψ(∞) = 0.(8)
III. ANALYTICAL FORMULA FOR THE ALLOWED MASSES OF SCALAR FIELDS
We investigate on properties of scaler fields outside reflecting shells through the ordinary differential equation (6). The equation can be analytically studied in two regions r ≪ 1/µ and r ≫ ( √ ηM ) 1/2 . In the small scalar field mass regime µ ≪ 1, we can analyze the system in the overlapping region ( √ ηM ) 1/2 ≪ r ≪ 1/µ [35]. In the overlapping region, we apply matching methods to obtain a formula of the masses of the non-minimally coupled scalar fields outside reflecting shells.
In the limit r ≪ 1/µ, the ordinary differential equation (6) can be expressed as
[ d 2 dr 2 − l(l + 1) r 2 + 48ηM 2 r 6 ]ψ = 0.(9)
The general mathematical solution of the equation (9) is
ψ(r) = A 1 r 1 2 J 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 ) + A 2 r 1 2 Y 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 ),(10)
where J ν (z) and Y ν (z) are Bessel functions of the first and second kinds respectively. And the coefficients A 1 and A 2 are normalization constants.
In the limit z → 0, the Bessel functions asymptotically behave as s (see Eqs. 9.1.7 and 9.1.9 of [44])
J ν (z) = (z/2) ν Γ(1 + ν) · [1 + O(z 2 )],(11)Y ν (z) = − Γ(ν) π(z/2) ν · [1 + O(z 2 )].(12)
In the radial overlap region ( √ ηM ) 1/2 ≪ r ≪ 1/µ, there is the relation √ ηM r 2 ≪ 1 and solution (10) of the scalar field equation (9) can be expressed as
ψ(r) = A 1 ( √ 3 √ ηM ) 1 2 l+ 1 4 Γ( 1 2 l + 5 4 ) r −l − A 2 Γ( 1 2 l + 1 4 ) π( √ 3 √ ηM ) 1 2 l+ 1 4 r l+1 .(13)
In the limit r ≫ ( √ ηM ) 1/2 , the ordinary differential equation (6) can be approximated by
[ d 2 dr 2 − µ 2 − l(l + 1) r 2 ]ψ = 0.(14)
The general mathematical solution of (14) can be expressed by the Bessel functions in the form
ψ(r) = B 1 r 1 2 J l+ 1 2 (iµr) + B 2 r 1 2 Y l+ 1 2 (iµr),(15)
where the coefficients B 1 and B 2 are normalization constants.
According to behaviors (11) and (12), equation (15) can be mathematically expressed as
ψ(r) = B 1 (iµ/2) l+ 1 2 Γ(l + 3 2 ) · r l+1 − B 2 Γ(l + 1 2 ) π(iµ/2) l+ 1 2 · r −l(16)
for the scalar field in the overlap region ( √ ηM ) 1/2 ≪ r ≪ 1/µ. Therefore, two analytically derived mathematical expressions (13) and (16) for the scalar field ψ(r) are both valid in the intermediate radial region
( √ ηM ) 1/2 ≪ r ≪ 1/µ.(17)
This fact allows us to match the solutions (13) and (16) in the region (17) and obtain relations between coefficients A i and B i as
B 1 = −A 2 Γ( 1 2 l + 1 4 )Γ(l + 3 2 ) π · (− 4 √ 3 √ ηµ 2 M ) 1 2 l+ 1 4 ,(18)B 2 = −A 1 π Γ( 1 2 l + 5 4 )Γ(l + 1 2 ) · (− √ 3 √ ηµ 2 M 4 ) 1 2 l+ 1 4 .(19)
In the large-argument (z → ∞), the Bessel functions behave as (see Eqs. 9.2.1 and 9.2.2 of [44])
J ν (z) = 2 πz · [cos(z − 1 2 νπ − 1 4 π)] · [1 + O(z −1 )],(20)Y ν (z) = 2 πz · [sin(z − 1 2 νπ − 1 4 π)] · [1 + O(z −1 )],(21)
and the solution (15) has the asymptotic functional behavior
ψ(r → ∞) = B 1 2 iπµ cos(iµr − 1 2 lπ − 1 2 π) + B 2 2 iπµ sin(iµr − 1 2 lπ − 1 2 π).(22)
The asymptotic behavior (22) and the boundary condition (8) yield the relation
B 2 = iB 1 .(23)
With relations (18), (19) and (23), we derive the relation between A i in the form
( 3ηµ 4 M 2 16 ) 1 2 l+ 1 4 = i[ Γ(l + 1 2 )Γ(l + 3 2 )Γ( 1 2 l + 1 4 )Γ( 1 2 l + 5 4 ) π 2 ] A 2 A 1 .(24)
We investigate on properties of scalar fields in the regime µ ≪ 1. So there is the relation µr s ≪ 1. The expression (10) holds at the shell surface r s satisfying µr s ≪ 1. The boundary condition (7) and the solution (10) yield expression of A 2 /A 1 as
A 2 A 1 = − J 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) Y 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) .(25)
Substituting the ratio A 2 /A 1 of (25) into relation (24), we arrive at the resonance equation
√ ηµ 2 M = 4 √ 3 (−1) 1 2l+1 [ Γ(l + 1 2 )Γ(l + 3 2 )Γ( 1 2 l + 1 4 )Γ( 1 2 l + 5 4 )J 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) π 2 Y 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) ] 2 2l+1 .(26)
The relation (17) yields the relation
√ ηµ 2 M ≪ 1.(27)
According to equation (26), the relation (27)
implies 2 √ 3 √ ηM r 2 s ≈ j 1 4 + 1 2 l,n ,(28)
where {j ν.n } n=∞ n=1 are positive zeros of the Bessel function J ν (x).
The Bessel functions can be expanded around the positive zeros in the form (see Eq. 9.1.27d of [44])
J 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) = −J 5 4 + 1 2 l (j 1 4 + 1 2 l,n ) · ∆ n · [1 + O(∆ n )](29)
and
Y 1 4 + 1 2 l ( 2 √ 3 √ ηM r 2 s ) = Y 1 4 + 1 2 l (j 1 4 + 1 2 l,n ) · [1 + O(∆ n )],(30)
where ∆ n is the small quantity defined as
∆ n = 2 √ 3 √ ηM r 2 s − j 1 4 + 1 2 l,n ≪ 1.(31)
Substituting the expansions (29) and (30) into (26), one obtains the formula
√ ηµ 2 M = 4 √ 3 (−1) 1 2l+1 [ Γ(l + 1 2 )Γ(l + 3 2 )Γ( 1 4 l + 1 4 )Γ( 1 4 l + 5 4 )J 5 4 + 1 2 l (j 1 4 + 1 2 l,n ) π 2 Y 1 4 + 1 2 l (j 1 4 + 1 2 l,n ) ∆ n ] 2 2l+1 ,(32)
which shows that the allowed values of scalar field masses are discrete.
IV. CONCLUSIONS
We studied the existence of scalar fields outside neutral reflecting shells in the flat spacetime limit. We considered static massive scalar fields non-minimally coupled to the Gauss-Bonnet invariant. We investigated on properties of scaler fields outside reflecting shells through the ordinary differential equation (6). The system is amenable to an analytical analysis in two regions r ≪ 1/µ and r ≫ ( √ ηM ) 1/2 , where η is the coupling parameter, M is the shell mass and µ is the scalar field mass. In the small scalar field mass regime µ ≪ 1, we can analyze the system in the overlapping region ( √ ηM ) 1/2 ≪ r ≪ 1/µ. In the overlapping region, we applied matching methods to derive a remarkably compact resonance formula (32) for the allowed masses of the supported spatially regular scalar fields outside neutral horizonless reflecting shells.
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An analytic effective model for hairy black holes. Yves Brihaye, Thomas Delplace, Carlos Herdeiro, Eugen Radu, Phys. Lett. B. 782Yves Brihaye, Thomas Delplace, Carlos Herdeiro, Eugen Radu, An analytic effective model for hairy black holes, Phys. Lett. B 782(2018)124-130.
Uniqueness of higher-dimensional phantom field wormholes. Marek Rogatko, Phys. Rev. D. 97224001Marek Rogatko,Uniqueness of higher-dimensional phantom field wormholes, Phys. Rev. D 97(2018)no.2,024001.
Stationary scalar configurations around extremal charged black holes. J C Degollado, C A R Herdeiro, Gen. Rel. Grav. 452483J. C. Degollado and C. A. R. Herdeiro, Stationary scalar configurations around extremal charged black holes, Gen. Rel. Grav. 45, 2483 (2013).
Shadows of Kerr black holes with scalar hair. P V P Cunha, C A R Herdeiro, E Radu, H F Rúnarsson, Phys. Rev. Lett. 115211102P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Rúnarsson, Shadows of Kerr black holes with scalar hair, Phys. Rev. Lett. 115, 211102(2015).
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J D Bekenstein, arXiv:gr-qc/9605059Black hole hair: 25-years after. J. D. Bekenstein, Black hole hair: 25-years after, arXiv:gr-qc/9605059.
Asymptotically flat black holes with scalar hair: a review. A R Carlos, Eugen Herdeiro, Radu, Int.J.Mod.Phys.D. 241542014Carlos A. R. Herdeiro, Eugen Radu,Asymptotically flat black holes with scalar hair: a review, Int.J.Mod.Phys.D 24(2015)09,1542014.
No-scalar-hair theorem for spherically symmetric reflecting stars. S Hod, Physical Review D. 94104073S.Hod, No-scalar-hair theorem for spherically symmetric reflecting stars, Physical Review D 94, 104073 (2016).
No nonminimally coupled massless scalar hair for spherically symmetric neutral reflecting stars. S Hod, Physical Review D. 9624019S.Hod, No nonminimally coupled massless scalar hair for spherically symmetric neutral reflecting stars,Physical Review D 96, 024019 (2017).
Charged massive scalar field configurations supported by a spherically symmetric charged reflecting shell. S Hod, Physics Letters B. 763275S.Hod, Charged massive scalar field configurations supported by a spherically symmetric charged reflecting shell,Physics Letters B 763, 275 (2016).
Marginally bound resonances of charged massive scalar fields in the background of a charged reflecting shell. S Hod, Physics Letters B. 768S.Hod, Marginally bound resonances of charged massive scalar fields in the background of a charged reflecting shell, Physics Letters B 768(2017)97-102.
No-hair theorems for a static and stationary reflecting star. Srijit Bhattacharjee, Sudipta Sarkar, Physical Review D. 9584027Srijit Bhattacharjee, Sudipta Sarkar, No-hair theorems for a static and stationary reflecting star, Physical Review D 95, 084027 (2017).
Scalar field condensation behaviors around reflecting shells in Anti-de Sitter spacetimes. Yan Peng, Bin Wang, Yunqi Liu, Eur.Phys.J. C. 788680Yan Peng, Bin Wang, Yunqi Liu, Scalar field condensation behaviors around reflecting shells in Anti-de Sitter spacetimes, Eur.Phys.J. C 78 (2018) no.8, 680.
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Non-minimally coupled massive scalar field configurations supported by charged reflecting shells: Analytic treatment in the weak coupling regime. Shahar Hod, Physics Letters B. 826136926Shahar Hod, Non-minimally coupled massive scalar field configurations supported by charged reflecting shells: Analytic treatment in the weak coupling regime, Physics Letters B 826(2022)136926.
Analytical investigations on formations of hairy neutral reflecting shells in the scalar-Gauss-Bonnet gravity. Yan Peng, Eur.Phys.J.C. 80202Yan Peng, Analytical investigations on formations of hairy neutral reflecting shells in the scalar-Gauss-Bonnet gravity, Eur.Phys.J.C 80(2020)3,202.
Black Hole Hair in Generalized Scalar-Tensor Gravity. P Thomas, Shuang-Yong Sotiriou, Zhou, Phys. Rev. Lett. 112251102Thomas P. Sotiriou and Shuang-Yong Zhou, Black Hole Hair in Generalized Scalar-Tensor Gravity, Phys. Rev. Lett. 112(2014)251102.
New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories. Daniela D Doneva, Stoytcho S Yazadjiev, Phys. Rev. Lett. 120131103Daniela D. Doneva and Stoytcho S. Yazadjiev, New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories, Phys. Rev. Lett. 120(2018)131103.
Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling. Hector O Silva, Jeremy Sakstein, Leonardo Gualtieri, Thomas P Sotiriou, Emanuele Berti, Phys. Rev. Lett. 120131104Hector O. Silva, Jeremy Sakstein, Leonardo Gualtieri, Thomas P. Sotiriou, Emanuele Berti, Spontaneous scalar- ization of black holes and compact stars from a Gauss-Bonnet coupling, Phys. Rev. Lett. 120(2018)131104.
Spontaneous scalarization of Gauss-Bonnet black holes: Analytic treatment in the linearized regime. Shahar Hod, Phys. Rev. D. 10064039Shahar Hod, Spontaneous scalarization of Gauss-Bonnet black holes: Analytic treatment in the linearized regime, Phys. Rev. D 100(2019)064039.
Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories. G Antoniou, A Bakopoulos, P Kanti, Phys. Rev. Lett. 120131102G. Antoniou, A. Bakopoulos, P. Kanti, Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss- Bonnet Theories, Phys. Rev. Lett. 120(2018)131102.
Spontaneous scalarisation of charged black holes. A R Carlos, Eugen Herdeiro, Nicolas Radu, José A Sanchis-Gual, Font, Phys. Rev. Lett. 121101102Carlos A. R. Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, José A. Font, Spontaneous scalarisation of charged black holes, Phys. Rev. Lett. 121(2018)101102.
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arxiv |
arXiv:physics/0007086v1 [physics.gen-ph] HOW HIGH THE TEMPERATURE OF A LIQUID BE RAISED WITHOUT BOILING?
28 Jul 2000
Mala Das
Department of Physics
Bose Institute
93 / 1 A. P. C. Road700009CalcuttaIndia
B K Chatterjee
Department of Physics
Bose Institute
93 / 1 A. P. C. Road700009CalcuttaIndia
B Roy
Department of Physics
Bose Institute
93 / 1 A. P. C. Road700009CalcuttaIndia
S C Roy
Department of Physics
Bose Institute
93 / 1 A. P. C. Road700009CalcuttaIndia
arXiv:physics/0007086v1 [physics.gen-ph] HOW HIGH THE TEMPERATURE OF A LIQUID BE RAISED WITHOUT BOILING?
28 Jul 2000‡ Lines to be deleted from the manuscript for blindfold refereeingnumbers: 6460-i646470Fx6460Qb6460My 1
How high a temperature of a liquid be raised beyond its boiling point without vaporizing (known as the limit of superheat) is an interesting subject of investigation. A new method of finding the limit of superheat of liquids is presented here. The superheated liquids are taken in the form of drops suspended in a dust free gel. The temperature of the superheated liquid is increased very slowly from room temperature to the temperature at which the liquid nucleates to boiling. The nucleation is detected acoustically by a sensitive piezo-electric transducer, coupled to a multi channel scaler (MCS) and the nucleation rate is observed as a function of time. The limit of superheat measured by the present method supersedes other measurements and theoretical predictions in reaching the temperature closest to the critical temperature of the liquids.
Any fluid that exists in the liquid form above its boiling temperature is said to be superheated. These liquids are in a metastable state in the thermodynamic sense and can be nucleated to form vapor by homogeneous nucleation or by the presence of heterogeneous nucleation sites such as gas pockets, vapor bubbles, solid impurities etc. or by the radiation interactions caused by charged particles, neutrons etc. Vapor embryos of different sizes, which are responsible for homogeneous nucleation, are produced at thermal equilibrium in the superheated liquid. The superheated state owes its existence to an energy barrier which causes the vapor embryo to collapse, rather than lead to nucleation, if it is less than a critical size.
A liquid can not be superheated up to the critical temperature, there is a limit to the maximum attainable temperature for any given liquid without boiling. This limit is called the 'limit of superheat of the liquid' (T sl ), where the height of the energy barrier which maintains the superheated state is of the order of kT and this temperature is a characteristic of any liquid. In addition to its importance in basic science, the knowledge of T sl is important in a number of industrial operations where a hot, nonvolatile liquid comes in contact with a cold volatile liquid. If the temperature of the hot liquid reaches to the limit of superheat of the cold liquid, explosive boiling would result. This explosive boiling is a potential hazard in damaging equipment and injure personnel in the vicinity of the blast [1]. The study of T sl has another importance since the discovery of bubble chamber by Glaser [2] and superheated drop detector [3]. The operation of this detector depends on the degree of superheat of the liquid, more the liquid is superheated more sensitive is the detector to lower energy radiations [4]. The minimum energy detectable by such detector is therefore limited by the limit of superheat of the detecting liquid. The limit of superheat of liquids can be estimated from the theory and can be measured experimentally. Theoretical calculations are performed either from the pure thermodynamic considerations or using the statistical mechanics. Very good and comprehensive reviews on homogeneous nucleation of lquid and on the limit of superheat are available in literature [5,6,7].
One has to note that theoretical calculations are performed for 'pure' homogeneous nucleation where the chance of heterogeneous nucleation arising out of various interfaces with different surface energies e.g. gas-liquid, liquid-liquid, solid-gas etc. is completely excluded.
Experimental results reported so far are far below the critical temperature of the liquids. One of the reasons being that observing 'pure' homogeneous nucleation experimentally, without any chance of heterogeneous nucleation is difficult to achieve.
Hence, the goal is to reduce the chance of heterogeneous nucleation as far as possible and to use an improved method of quantitative detection of nucleation to see how close one can reach experimentally to the predicted limit of superheat. The present experiment is designed to achieve this goal. Superheated sample used in this investigation is a homogeneous suspension of superheated drops of three liquids (R-12 : CCl 2 F 2 , R-114 : C 2 Cl 2 F 4 and R-22 : CHClF 2 ) in a dust free, visco-elastic, degassed gel medium. Suspending the superheated liquid in another liquid (gel) reduces the chance of heterogeneous nucleation. Nucleation is detected acoustically by a piezoelectric transducer [8] and the pulses thus received are digitized and recorded as a function of time by a multichannel scaler. This improved method of determining T sl supersedes all other measured values in reaching closest to the critical temperatures.
Reviews on previous experimental techniques of measuring the limit of superheat of liquid have been described in detail by Avedisian [6]. As has been found from this literature, all previous experiments except one rely on the qualitative observation of the nucleation visually and therefore the present measurement constitutes the first quantitative measurement of T sl using digital electronics.
The limit of superheat can be estimated either from the thermodynamic stability theory or from the analysis of the dynamics of formation of the critical sized vapor embryos (statistical mechanical theory). The superheated state of a liquid is a metastable state and the limit of this metastable state is represented on the P-V diagram by the spinodal curves. For a pure liquid, the spinodal curve or the thermodynamic limit of superheat is defined by states for which
dP dV T = 0(1)
Temperley [9] calculated the value of maximum superheat temperature using van der Waals' equation of state. The maximum limit of superheat of a given liquid can be expressed as
t m = 27T c 32(2)
where t m is the limit of superheat of the liquid . For mathematical simplicity this has been calculated by considering the ambient pressure to be zero. At atmospheric pressure i.e. at P=1, t m will be slightly greater than the corresponding value at P=0. Other equations of state such as modified Bertholet equation and Redlich-Kwong equation have also been used to calculate the limit of superheat [5]. As has been observed by Blander and Katz [5], experimental values of thermodynamic limit clearly exceeded the Van der Waals limit at least for five liquids.
For most of the organic liquids the thermodynamic limit of superheat can be represented empirically [1] by
T sl = T c [0.11(P/P c ) + 0.89](3)
where T c is the critical temperature, P c is the critical pressure and P is the ambient pressure.
Another method of estimating T sl using statistical mechanics involves considerations of the rate processes of nucleation to form vapor embryos in a superheated liquid. This method does not yield an absolute value of T sl but it allows one to estimate the probable rate of formation of critical-sized vapor embryos in a superheated liquid at a given temperature. If the rate is very low within the time scale of the experiment, one considers no nucleation would occur, while if the rate is very high, then one assumes that T sl has been reached. The rate of homogeneous nucleation 5 (J) as given approximately by the Volumer-Doring formula is given by [
1] J = Nf exp − B kT(4)
where J is the expected rate of formation of critical sized vapor embryos per unit volume, f is a frequency factor which in general is of the order of 10 11 sec −1 , N is the number density of molecules in the superheated liquid and B is the minimum amount of energy needed to form a vapor bubble of critical size as given by Gibbs [10] from reversible thermodynamics is
B = 16πγ 3 (T )/3(p v − p o ) 2(5)
where γ(T ) is the liquid-vapor interfacial tension, P v is vapor pressure of the superheated liquid and P o is the ambient pressure. It is to be noted in this connection that which value of J is proper to calculate T sl is not defined and therefore one has to make some 'judicious choice' of a rate which would correspond to T sl . A J value of 10 6 nucleation/cm 3 .sec is often used to define the limit of superheat temperature.
It is to be noted that all the above discussions are related with the classical theory of nucleation. Effect of other factors like diffusion, viscosity and other hydrodynamical constraints are discussed by Blander and Katz [5]. As has been pointed out by them, contributions arising out of these effects in calculating T sl of pure liquids are not very significant. 0.89 [6] As could be seen from the table, the measured limit of superheat exceeded the predicted limit of superheat and other experimental values. It is to be noted in this connection that all theretical predictions are approximate as discussed before.
Therefore the present experimental measurements indicate the need of improved calculation of limit of superheat. That the Van der Waals' limit is exceeded was reported before by Blander and Katz [5]. This table also gives an useful insight about the nucleation process. As can be seen from the
The experiment is carried out with superheated liquids of R12 (b.p. -29.79 o C), R114 (b.p. 3.6 o C) and R22 (b.p. -40.5 o C). The superheated drops are suspended 6 in dust free, de-gassed visco-elastic gel. The gel is a mixture of 'aquasonic' gel available commercially and glycerine. A glass vial containing the superheated drops homogeneously suspended in gel is placed on the top of a thin layer of degassed gel taken in a beaker. The gel in the beaker improves the acoustic coupling between the superheated drops in the vial and the transducer. The beaker is placed on a piezoelectric transducer with a coupling gel. Some pure gel is placed on the top of the sample and a thermometer was inserted in the pure gel so as to avoid any contact with the superheated liquid sample, thus reducing the chance of heterogeneous nucleation from the liquid-glass interface. The nucleation in superheated drops is detected by the transducer, the output of the transducer is digitized and recorded by a multi channel scaler. The vial was wrapped with a heating coil covering the gel and sample. The temperature of the sample is increased slowly from room temperature and the count rate (dN/dt) is recorded in MCS. As nucleation proceeds, the number of superheated drops are depleted and hence the nucleation rate is normalized with respect to the number of drops present. What we expect ideally is ( 1 N ) dN dt is zero till the temperature reaches the limit of superheat where there will be a sudden increase in ( 1 N ) dN dt (entire liquid nucleates) and will be no nucleation beyond this temperature. Considering the experimental uncertainty, one may observe the similar behavior as presented in Fig. 1. The comparison of observed limit of superheat with other experimental resultsis presented in the table below. The reduced limit of superheat defined as T sl /T c (taken in o K) for these liquids is also presented in the
table, for liquids with lower boiling points it is harder to reach closer to the critical temperature. This is quite expected as the chances of heterogeneous nucleation increases in case of liquids with lower boiling points. Whether complete elimination of heterogeneous nucleation in experimental 8 measurement is possible or not is an open question. No other measurement have beenable to reach so close to the critical temperature. It is to be noted in this connection that the limit of superheat of only 14 liquids out of 56 liquids studied by Blander and Katz[5] hardly exceeded 90% of the critical temperature.Therefore, by reducing the chances of heterogeneous nucleation by suspending the superheated sample in another 'pure' liquid and using precise electronic measurement we have been able to reach closer to the critical temperature hitherto unattainable. Inspite of the fact that theoretical calculations are performed for 'pure' homogeneous nucleation, they fall below the experimental values indicate the inadequacy of the present method of calculation discussed here and warrants improved calculations.
‡ Lines to be deleted from the manuscript for blindfold refereeing. ‡ Lines to be deleted from the manuscript for blindfold refereeing
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arxiv |
Nonautonomous discrete rogue waves and interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients
18 May 2011
Zhenya Yan
Institute of Systems Science
Key Laboratory of Mathematics Mechanization
AMSS
Chinese Academy of Sciences
100190BeijingChina
Dongmei Jiang
Institute of Systems Science
Key Laboratory of Mathematics Mechanization
AMSS
Chinese Academy of Sciences
100190BeijingChina
Department of Mathematics
Qingdao University of Technology
266033QingdaoChina
W M Liu
Beijing National Laboratory for Condensed Matter Physics
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Nonautonomous discrete rogue waves and interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients
18 May 2011numbers: 0545Yv4265Tg4265Wi
We analytically investigate the nonautonomous discrete rogue wave solutions and their interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients, which possess complicated wave propagations in time and are beyond the usual discrete rogue waves. When the amplitude of the tunnel coupling coefficient between sites decreases, these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values. Moreover, we find that the interaction between nonautonomous discrete rogue waves is elastic. In particular, these results can reduce to the usual discrete rogue wave solutions when the gain or loss term is ignored.
I. INTRODUCTION
Rogue waves (alias as freak waves, monster waves, killer waves, giant waves or extreme waves), as an important physical phenomenon, are localized both in space and time and depict a unique event that 'appears from nowhere and disappears without a trace' [1]. Rogue waves (RWs) are also known as 'rogons' if they reappear virtually unaffected in size or shape shortly after their interactions [2]. The study of RWs has become a significant subject in many fields since they can signal catastrophic phenomena such as thunderstorms, earthquakes, and hurricanes. RWs have been found in the ocean [3,4], nonlinear optics [5][6][7], Bose-Einstein condensates [8], the atmosphere [9], and even the finance [10]. Moreover, some experimental observations have shown that optical RWs do exist and play a positive role in nonlinear optical fibres [5][6][7]. Such optical RWs differ from oceanic RWs that play a negative role and lead to many accounts of such waves hitting passenger ships, container ships, oil tankers, fishing boats, and offshore and coastal structures, sometimes with catastrophic consequences [4]. In particular, the analytical RWs have been obtained for the nonlinear Schrödinger (NLS) equation [1,11,12], as well as some of their extensions with the varying coefficients [2], the higher orders [13], or the higher dimensions [14].
The discrete NLS equation [15] and the Ablowitz-Ladik (AL) lattice [16,17], as two prototypical discretizations of the continuum NLS equation, have been studied extensively in the field of nonlinear science. The former is nonintegrable, but has some interesting applications of physics [18][19][20][21][22]. The latter is integrable and possesses an infinite number of conservation laws [16,17], as well as has been depicted as an effective lattice to study properties of the intrinsic localized modes [23]. Moreover the nonintegrable discrete NLS equation can also be regarded as a perturbation of the integrable AL lattice [24]. In addition, the Salerno model (SM) has also been presented, interpolating between the nonintegrable discrete NLS equation (µ = 0) and the intregable AL lattice (ǫ = 0) in the form [25][26][27][28][29][30] iψ n,t +(ψ n+1 +ψ n−1 )(1+µ|ψ n | 2 )+ ǫ|ψ n | 2 ψ n + vψ n = 0, (1) which can be derived on the basis of a variational principle δL SM /δψ * n = 0 from the Lagrangian density
L SM = n i(ψ * n ψ n,t −ψ n ψ * n,t )+ 4 Re (ψ * n ψ n+1 ) +µ(ψ n+1 + ψ n−1 )ψ * n |ψ n | 2 + ǫ|ψ n | 4 + 2v|ψ n | 2 ,(2)
where ψ n ≡ ψ n (t) stands for the complex field amplitude at the nth site of the lattice, the parameter µ is the intersite nonlinearity and corresponds to the nonlinear coupling between nearest neighbors, ǫ measures the intrinsic onsite nonlinearity, and v describes the inhomogeneous frequency shift. The SM has been applied in biology [25], and Bose-Einstein condensates [30]. Recently, the discrete RWs have also drawn much attention. The discrete NLS equation (i.e., for the case µ = 0, ǫ = 1, and v = −2 in Eq. (1)) has numerically been verified to support discrete RWs [31]. The SM has also been found to admit discrete RWs from the viewpoint of statistical analysis [32]. More recently, it has also been shown that exact discrete RWs [33,34] can exist in the AL lattice (i.e., for the case µ = 1, ǫ = 0, and v = −2 in Eq. (1)) on the basis of the limit cases of their multi-soliton solutions [35]. However, there is no reports to date about nonautonomous discrete RWs except for the AL lattice and the discrete Hirota equation [33,34].
In the present paper, we will explore exact nonautonomous discrete RW solutions and interaction of the generalized Ablowitz-Ladik-Hirota (ALH) lattice with variable coefficients given by Eq. (3), i.e., the generalized case of Eq. (1) without the intrinsic onsite nonlinearity, where the tunnel coupling coefficients and the intersite nonlinearity are the time-modulated, complex-valued and real-valued functions, respectively, the inhomogeneous frequency shift are space-and time-modulated, real-valued functions, and the time-dependent gain or loss term is added. To do so, we will make use of the differential-difference symmetry analysis that can connect this equation with variable coefficients with the simpler ones. We show that, for the attractive intersite nonlinearity, the generalized ALH lattice with variable coefficients can support nonautonomous discrete RWs in terms of the rogue wave solutions of the discrete Hirota equation. Moreover, we also exhibit wave propagations of nonautonomous discrete RW solutions and their interaction for some chosen parameters and functions.
The rest of this paper is organized as follows. In Sec. II, we introduce the generalized ALH lattice with variable coefficients, which contains some special lattice models, such as the AL lattice, the discrete Hirota equation, and the generalized AL lattice. In Sec. III, by analyzing the phase of the complex field amplitude, we systematically present a similarity transformation reducing the generalized ALH lattice with variable coefficients to the discrete Hirota equation. In Sec. IV, we determine the self-similar variables and constraints satisfied by the coefficients in Eq. (3). Moreover, we analyze the relations among these functions such that we find that the intersite nonlinearity (the inhomogeneous frequency shift) is related to the gain or loss term (the tunnel coupling coefficient). Sec. V mainly discusses nonautonomous discrete RW solutions and their interaction of Eq. (3) for some chosen parameters and functions. For the given periodic gain or loss term, when the amplitude of the tunnel coupling coefficient between sites decreases, these nonautonomous discrete rogue wave solutions are localized in space and keep the localization longer in time, which differ from the usual discrete rogue waves. Finally, we give some conclusions in Sec. VI.
II. THE GENERALIZED ABLOWITZ-LADIK-HIROTA LATTICE WITH VARIABLE COEFFICIENTS
We here address the generalized Ablowitz-Ladik-Hirota (ALH) lattice with variable coefficients modeled by the following lattice
iΨ n,t + Λ(t)Ψ n+1 + Λ * (t)Ψ n−1 1 + g(t)|Ψ n | 2 −2v n (t)Ψ n + iγ(t)Ψ n = 0,(3)
which can be derived in terms of a variational principle δL ALH /δΨ * n = 0 from the following Lagrangian density
L ALH = n i(Ψ * n Ψ n,t −Ψ n Ψ * n,t ) + 4 Re [Λ(t)Ψ * n Ψ n+1 ] +g(t)[Λ(t)Ψ n+1 +Λ * (t)Ψ n−1 ]Ψ * n |Ψ n | 2 −2[2v n (t)−iγ(t)]|Ψ n | 2 ,(4)
where Ψ n ≡ Ψ n (t) stands for the complex field amplitude at the nth site of the lattice, the complex-valued function Λ(t) is the coefficient of tunnel coupling between sites and can be rewritten as Λ(t) = α(t) + iβ(t) with α(t) and β(t) being differentiable, real-valued functions, g(t) stands for the time-modulated intersite nonlinearity, v n (t) is the space-and time-modulated inhomogeneous frequency shift, and γ(t) denotes the time-modulated effective gain or loss term.
In fact, this nonlinear lattice model (3) contains many special lattice models, such as the AL lattice for the case α(t) = const., β(t) = v n (t) = γ(t) = 0, and g(t) = const. [16,17], the AL equation with additional term accounding for dissaption for the case α(t) = const., β(t) = v n (t) = 0, γ(t) = const., and g(t) = const. [26], the discrete Hirota equation for the case α(t) = const., β(t) = const., v n (t) = γ(t) = 0, and g(t) = const. [35], the generalized AL lattice given by Eq. (1) for the case α(t) = const., β(t) = γ(t) = 0, and g(t) = const. [36], and the discrete modified KdV equation for the case α(t) = v n (t) = γ(t) = 0, β(t) = const., and g(t) = const. [37].
III. DIFFERENTIAL-DIFFERENCE SIMILARITY REDUCTIONS AND CONSTRAINT EQUATIONS
We consider the spatially localized solutions of Eq. (3), i.e., lim |n|→∞ |Ψ n (t)| = 0. To this aim, we search for a proper similarity transformation connecting solutions of Eq. (3) with those of the following discrete Hirota equation with constant coefficients [35], namely iΦ n,τ + λΦ n+1 +λ * Φ n−1 1+|Φ n | 2 −2Re (λ)Φ n = 0, (5) which can be derived in terms of a variational principle δL H /δΨ * n = 0 from the following Lagrangian density
L H = n i(Φ * n Φ n,τ −Φ n Φ * n,τ ) + 4 Re (λΦ * n Φ n+1 ) +(λΦ n+1 +λ * Φ n−1 )Φ * n |Φ n | 2 −4 Re (λ)|Φ n | 2 ,(6)
where Φ n ≡ Φ n (τ ) is a complex dynamical variable at the nth site of the lattice, τ ≡ τ (t) is a real-valued function of time to be determined, and the complex-valued parameter λ can be rewritten as λ = a + ib with a and b being real-valued parameters. The discrete Hirota model (5) contains some special physical models, such as the AL lattice for the case a = 1 and b = 0 [16,17] and the discrete mKdV equation for the case a = 0 and b = 1 [37]. It has been shown in Ref. [38] that the discrete Hirota equation (5) is in fact an integrable discretization of the three-order NLS equation (also known as the Hirota equation) [39] iq t + a(q xx + |q| 2 q) − ib(q xxx + 6|q| 2 q x ) = 0, (7) which plays an important role in nonlinear optics [40,41]. To show the above-mentioned aim, we here apply the similarity transformation in the form
Ψ n (t) = ρ(t)e iϕn(t) Φ n [τ (t)](8)
to Eq. (3), where the function ρ(t) and phase ϕ n (t) are both real-valued functions of indicated variables to be determined. To conveniently substitute ansatz (8) into Eq. (3) and to further balance the phases in every term in Eq. (3), i.e., Ψ n+1 (t), Ψ n (t), and Ψ n−1 (t), we should firstly know the explicit expression of the phase ϕ n (t) in transformation (8) in space.
Here we consider the case that the phase is expressed as a quadratic polynomial in space with coefficients being functions of time in the form ϕ n (t) = p 2 (t)n 2 + p 1 (t)n + p 0 (t) with p 0,1,2 (t) being functions of time, which is similar to the phases in the continuous NLS (or GP) equations with variable coefficients [42]. Based on the symmetry analysis, we balance the coefficients of these terms Ψ n+1 (t), Ψ n (t), and Ψ n−1 (t) such that we find that the phase in transformation (8) should be a first degree polynomial in space with coefficients being functions of time, namely
ϕ n (t) = p 1 (t)n + p 0 (t),(9)
where p 0,1 (t) are functions of time to be determined. Eq. (8) with the condition (9) allows us to reduce Eq. (3) to Eq. (5), variables in this reduction can be determined from the requirement for the new complex field amplitude Φ n (τ (t)) to satisfy Eq. (5). Thus, we substitute transformation (8) into Eq. (3) along with Eq. (9) and after relatively simple algebra obtain the following system of ordinary differential equationṡ
ρ(t) + γ(t)ρ(t) = 0, (10a) aτ (t) − α(t) cos p 1 (t) + β(t) sin p 1 (t) = 0, (10b) [bβ(t) + aα(t)] sin p 1 (t) + [aβ(t) − bα(t)] × cos p 1 (t) = 0, (10c) 2v n (t) +ṗ 1 (t)n +ṗ 0 (t) + 2[β(t) sin p 1 (t) −α(t) cos p 1 (t)] = 0, (10d) g(t)ρ 2 (t) = 1,(10e)
where the dot denotes the derivative with respect to time. Therefore, if system (10) Firstly, we solve Eqs. (10a)-(10c) to obtain the functions ρ(t), τ (t), and p 1 (t) in transformation (8).
And then we consider Eqs. (10d) and (10e) to determine the inhomogeneous frequency shift v n (t) and the intersite nonlinearity g(t) in Eq. (3) in terms of aboveobtained functions ρ(t), τ (t), and p 1 (t).
Thus, we have established a similarity transformation (8) connecting solutions of Eq. (5) and those of Eq. (3). In particular, we here exhibit our approach in terms of two lowest-order discrete rogue wave solutions of Eq. (5) as seeding solutions to find nonautonomous discrete rogue wave solutions of Eq. (3).
IV. DETERMINING THE SIMILARITY TRANSFORMATION AND COEFFICIENTS
It follows from Eqs. (10a)-(10c) that we can obtain the variables ρ(t), p 1 (t) and τ (t) in transformation (8) in the form
ρ(t) = ρ 0 exp − t 0 γ(s)ds ,(11a)p 1 (t) = tan −1 bα(t) − aβ(t) aα(t) + bβ(t) ,(11b)τ (t) = (a 2 + b 2 ) −1/2 t 0 [α 2 (s) + β 2 (s)] 1/2 ds, (11c)
where ρ 0 is an integration constant. Now it follows from Eqs. (10d) and (10e) along with Eqs. (11a) and (11b) that we further find the inhomogeneous frequency shift v n (t) and intersite nonlinearity g(t) in the form
g(t) = ρ −2 0 exp 2 t 0 γ(s) ds , (12a) v n (t) = v 1 (t)n + v 0 (t),(12b)
where we have introduced two functions in the inhomogeneous frequency shift v n (t) in the form
v 1 (t) = α(t)β(t) −α(t)β(t) 2[α 2 (t) + β 2 (t)] , (13a) v 0 (t) = a α 2 (t) + β 2 (t) a 2 + b 2 1/2 −ṗ 0 (t) 2 ,(13b)
where p 0 (t) is an arbitrary differentiable function of time. It follows from Eqs. (12a)-(13b) that, in these coefficients of Eq. (3), the intersite nonlinearity g(t) (the inhomogeneous frequency shift v n (t)) is related to the gain or loss term γ(t) (the tunnel coupling Λ(t) = α(t) + iβ(t)). This means that only two varying coefficients (e.g., γ(t) and Λ(t) = α(t)+iβ(t)) are left free. Moreover, it follows from Eq. (12a) that the intersite nonlinearity g(t) is always positive (i.e., the attractive intersite nonlinearity). In addition, it follows from Eq. (11a) that the gain or loss term γ(t) can also control the function ρ(t), which is used to modulate the amplitude of the complex field Ψ n (t).
For the inhomogeneous frequency shift v n (t) given by Eq. (12b), when α(t) = cβ(t) with c being a constant, the inhomogeneous frequency shift v n (t) is a linear function of space n with coefficients being functions of time. In the absence of the discrete space n in the inhomogeneous frequency shift, i.e., v n (t) ≡ v(t), which means thatṗ 1 (t) = 0 on the basis of Eq. (10d), there exist two cases to be discussed: i) If p 1 (t) = 0 in which we have p 1 (t) = p 1 = const = 0, then this means that ϕ n (t) is still a linear function of the discrete space n, i.e., ϕ n (t) = p 1 n+p 0 (t). In this case, the variable function τ (t) and the inhomogeneous frequency shift v n (t) are given by
τ (t) = t 0 α(s)ds a cos(p 1 ) + b sin(p 1 ) , v n (t) = aα(t) a cos(p 1 ) + b sin(p 1 ) −ṗ 0 (t) 2 ,
and ρ(t), g(t) are same as Eqs. (11a) and (12a); ii) If p 1 (t) = 0, i.e., tan −1 {[bα(t) − aβ(t)]/[aα(t) + bβ(t)]} = 0, which means that the relation for the coefficients in Eq. (3), α(t) = (a/b)β(t), is required and ϕ n (t) is only a functions of time, i.e., ϕ n (t) ≡ p 0 (t), then the variable function τ (t) and the inhomogeneous frequency shift v n (t) are given by the form
τ (t) = 1 a t 0 α(s)ds, v n (t) = α(t) −ṗ 0 (t) 2 ,
and ρ(t), g(t) are same as Eqs. (11a) and (12a), where γ(t), α(t) and p 0 (t) are free functions of time, and a, b, ρ 0 are all free parameters.
V. NONAUTONOMOUS DISCRETE ROGON SOLUTIONS AND INTERACTION
In general, we have a large degree of freedom in choosing the coefficients of similarity transformation (8) and Eq. (3). As a consequence, we can obtain an infinitely large family of exact solutions of the generalized ALH lattice with variable coefficients given by Eq. (3) in terms of exact solutions of the discrete Hirota equation (5) and transformation (8). In particular, if we consider discrete rogon solutions of Eq. (5) as seeding solutions, then we can obtain many types of nonautonomous (including arbitrary time-dependent functions) discrete rogon (rogue wave) solutions of Eq. (3). As two representative examples, we consider the lowest-order discrete rogon solutions of Eq. (5) as two examples [33] to study the dynamics of nonautonomous discrete rogon solutions of Eq. (3).
A. Nonautonomous discrete one-rogon solution
Firstly, based on the similarity transformation (8) and one-rogon solution of the discrete Hirota equation (5), we present the nonautonomous discrete one-rogon solution (also known as the first-order rational solution) of Eq. (3) in the form
Ψ (1) n (t) = ρ 0 √ µ exp − t 0 γ(s)ds + i[ϕ n (t) +φ n (τ )] × 1− 4(1 + µ) 1 + 4iµ √ a 2 + b 2 τ (t) 1 + 4µn 2 + 16µ 2 (1 + µ)(a 2 + b 2 )τ 2 (t) ,(15)
where the part phaseφ n (τ ) is defined bŷ
ϕ n (τ ) = 2τ (t) (1+µ) a 2 + b 2 −a −n tan −1 (b/a) ,(16)
µ is a positive parameter, the variable τ (t) is given by Eq. (11c), and the phase ϕ n (t) = p 1 (t)n + p 0 (t) with p 1 (t) given by Eq. (11b) and p 0 (t) being an arbitrary differentiable function of time.
To illustrate the wave propagations of the obtained nonautonomous discrete one-rogon solution (15), we can choose these free parameters in the form
α(t) = c 1 sin(2t), β(t) = c 2 cos(t),γ(t) = γ 0 sin(t) cos 2 (t), a = b = µ = ρ 0 = 1,(17)
where γ 0 , c 1,2 are constants. Figure 1 depicts the profiles of the coefficient v 1 (t) = [α(t)β(t) −α(t)β(t)]/{2[α 2 (t) + β 2 (t)]} of the first degree term of the inhomogeneous frequency shift v n (t) given by Eq. (13a), the attractive intersite nonlinearity g(t) given by Eq. (12a), and the gain or loss term γ(t) vs time for the parameters given by Eq. (17). The evolution of the intensity distribution for the one-rogon solution given by Eq. (15) is illustrated in Fig. 2 for parameters γ 0 = c 1 = c 2 = 1. The discrete rogue wave solution is localized both in space and in time, thus revealing the usual discrete 'rogue wave' features. However if we fix the coefficient γ 0 = 1 of the gain or loss term and adjust the coefficients c 1 = 0.01, c 2 = 0.02 of the tunnel couplings α(t) and β(t) given by Eq. (17), then the evolution of the intensity distribution for the one-rogon solution is changed (see Fig. 3), and it follows from Fig. 3 that the discrete one-rogon solution in this case is localized in space and keep the localization longer in time than usual rogue waves (see [33]). Moreover, it follows from Figs. 2(c) and 3(c) that the amplitude of the discrete one-rogon solution decreases as time increases, and the amplitude in Fig. 2(c) is decreased faster than one in Fig. 3(c) as time increases.
B. The interaction between nonautonomous discrete rogon solutions
Here we consider the interaction between nonautonomous discrete rogon solutions. To do so, we apply a second-order rational solution of the discrete Hirota equation (5) to the similarity transformation (8) such that we can obtain the nonautonomous discrete tworogon solution of Eq. (3) in the form
Ψ (2) n (t) = ρ 0 √ µ exp − t 0 γ(s)ds + i[ϕ n (t) +φ n (τ )] × 1− 12(1+µ) P (2) n (τ )+i T (τ ) 1+µ Q (2) n (τ ) H (2) n (τ ) ,(18)
which displays the interaction between nonautonomous discrete rogon solutions, where µ is a positive parameter, the functions P
(2) n (τ ), Q(2)
n (τ ) and H (2) n (τ ) are all polynomials of space and time given by P (2) n (τ ) = 5T 2 + 6(N + 2µ + 3)T + N 2
+(6 − 4µ)N − 3(4µ + 1), Q (2) n (τ ) = T 2 + 2(N + 1)T + N 2 −(16µ + 6)N − 3(8µ + 5), H (2) n (τ ) = T 3 +3(N +8µ + 9)T 2 +3(N 2 −6N −16µN +48µ 2 +72µ+33)T +N 3 +(3 − 8µ)N 2 +(27 + 24µ + 16µ 2 )N +9,
where we have introduced N = 4µn 2 and T = 16µ 2 (1 + µ)(a 2 + b 2 )τ 2 , τ ≡ τ (t) is given by Eq. (11c), the part phase ϕ n (t) = p 1 (t)n+p 0 (t) with p 1 (t) given by Eq. (11b) and p 0 (t) being an arbitrary differentiable function of time, and the part phaseφ n (τ ) is given by Eq. (16).
Similarly, we can choose these free parameters given by Eq. (17) for the nonautonomous discrete two-rogon solution given by Eq. (18) except for µ = 1/16. Figures 4 and 5 depict the evolution of intensity distribution for the interaction between nonautonomous discrete rogon solutions (discrete two-rogon solution) given by Eq. (18) for different parameters γ 0 , c 1 , and c 2 , respectively. Moreover, we find that the interaction between nonautonomous discrete rogue waves is elastic. Figure 4 shows that the nonautonomous discrete two-rogon solution is localized both in space and in time, thus revealing the usual discrete 'rogue wave' features for the chosen parameters γ 0 = 1, c 1 = 2, c 2 = 1, but if we fix the coefficient γ 0 = 1 of the gain or loss term and adjust the coefficients c 1 = 0.2, c 2 = 0.1 of the tunnel couplings α(t) and β(t) given by Eq. (17), then the evolution of the intensity distribution for the discrete two-rogon solution is changed (see Fig. 5), and it follows from Fig. 5 that the nonautonomous discrete two-rogon solution is localized in space and keep the localization longer in time than usual rogue waves. Moreover, it follows from Figs. 4(c) and 5(c) that the amplitude of the discrete two-rogon solution decreases as time increases, and the amplitude in Fig. 4(c) decreases faster than one in Fig. 5(c) as time increases.
VI. CONCLUSIONS
In conclusion, we have studied nonautonomous discrete rogon solutions and their interaction in the generalized Ablowitz-Ladik-Hirota lattice with the varying tunnel coupling, intersite nonlinearity, inhomogeneous frequency shift, and gain or loss term given by Eq. (3) on the basis of the differential-difference similarity reduction (8). We found its some nonautonomous discrete rogon solutions when the intersite nonlinearity g(t) (the inhomogeneous frequency shift v n (t)) is related to the gain or loss term γ(t) (the tunnel coupling Λ(t) = α(t) + iβ(t)) (see Eqs. (12a) and (12b)). This denotes that only two coefficients (i.e., γ(t) and Λ(t) = α(t) + iβ(t)) are left free.
In particular, we have studied wave propagations of nonautonomous discrete rogon solutions and interaction for some chosen parameters, which exhibit complicated rogue wave construes. For the given periodic gain or loss term, when the amplitude of the tunnel coupling coefficient between sites decreases, these nonautonomous discrete rogon solutions are localized in space and keep the localization longer in time, which differ from the usual discrete rogue waves of nonlinear discrete equations (e.g., the AL lattice and the discrete Hirota equation) [33].
Moreover, nonautonomous discrete rogon solutions and interaction may provide more documents to further understand the physical mechanism of discrete rogue wave phenomena. The approach may also be extended to other discrete nonlinear lattices with variable coefficients for studying their discrete rogue wave solutions and wave propagations.
is consistent, then we have constructed an algorithm generating nonautonomous solutions of Eq. (3) based on transformation (8) and solutions of Eq. (5):
FIG. 1 .
1(color online). Profiles of the coefficients of the generalized ALH lattice with variable coefficients given by Eq. (3) vs time for the parameters given by Eq.(17)with parameters γ0 = c1 = c2 = 1. (a) the coefficient v1(t) given by Eq. (13a) of the first degree term of the inhomogeneous frequency shift vn(t) given by Eq. (12b), (b) nonlinearity g(t) given by Eq. (12a), and (c) the gain or loss term γ(t).
FIG. 2 .
2(color online). Profiles of nonautonomous discrete one-rogon solution (15) of the generalized ALH lattice with variable coefficients given by Eq. (3) for the parameters given by Eq. (17) with γ0 = c1 = c2 = 1. (a) the intensity distribution |Ψ
n
(t)| 2 for t = 0, 0.2, which means that the amplitude decreases as time increases, and the peak falls the lower position after time exceeds about 1.
FIG. 3 .
3(color online). Profiles of nonautonomous discrete one-rogon solution (15) of the generalized ALH lattice with variable coefficients given by Eq. (3) for the parameters given by Eq. (17) with γ0 = 1, c1 = 0.01, and c2 = 0.02. (a) the intensity distribution |Ψ
n
(t)| 2 with max (n,t) |Ψ
n
(t)| 2 for t = 0, 3, which means that the amplitude decreases as time increases, and the peak falls the lower position after time exceeds about 70.
FIG. 4 .
4(color online). Profiles of the interaction between nonautonomous discrete rogon solution given by Eq. (18) of the generalized ALH lattice with variable coefficients given by Eq. (3) for the parameters given by Eq. (17) with γ0 = 1, c1 = 2, and c2 = 1. (a) the intensity distribution |Ψ
n
(t)| 2 with max (n,t) |Ψ
n
(t)| 2 , (c) the intensity distributions |Ψ
t = 0, 0.6, which means that the amplitude decreases as time increases, and the peak falls the lower position after time exceeds about 3.
FIG. 5 .
5(color online). Profiles of the interaction between nonautonomous discrete rogon solution given by Eq. (18) of the generalized ALH lattice with variable coefficients given by Eq. (3) for the parameters given by Eq. (17) with γ0 = 1, c1 = 0.2, and c2 = 0.1. (a) the intensity distribution |Ψ
n
(t)| 2 with max (n,t) |Ψ
n
(t)| 2 , (c) the intensity distributions |Ψ
t = 0, 1.8, which means that the amplitude decreases as time increases, and the peak falls the lower position after time exceeds about 65.
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| {'fraction_non_alphanumeric': 0.09729778202905483, 'fraction_numerical': 0.048723620493812396, 'mean_word_length': 3.6987359550561796, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We analytically investigate the nonautonomous discrete rogue wave solutions and their interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients, which possess complicated wave propagations in time and are beyond the usual discrete rogue waves. When the amplitude of the tunnel coupling coefficient between sites decreases, these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values. Moreover, we find that the interaction between nonautonomous discrete rogue waves is elastic. In particular, these results can reduce to the usual discrete rogue wave solutions when the gain or loss term is ignored.', 'arxivid': '1105.3511', 'author': ['Zhenya Yan \nInstitute of Systems Science\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina\n', 'Dongmei Jiang \nInstitute of Systems Science\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina\n\nDepartment of Mathematics\nQingdao University of Technology\n266033QingdaoChina\n', 'W M Liu \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n'], 'authoraffiliation': ['Institute of Systems Science\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina', 'Institute of Systems Science\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina', 'Department of Mathematics\nQingdao University of Technology\n266033QingdaoChina', 'Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina'], 'corpusid': 118451133, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12527, 'n_tokens_neox': 10802, 'n_words': 5721, 'pdfsha': '988411ac5443285dcebca2f3e456a27199ee3fb9', 'pdfurls': ['https://arxiv.org/pdf/1105.3511v1.pdf'], 'title': ['Nonautonomous discrete rogue waves and interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients', 'Nonautonomous discrete rogue waves and interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients'], 'venue': []} |
arxiv |
DOMAIN SPECIFIC WAV2VEC 2.0 FINE-TUNING FOR THE SE&R 2022 CHALLENGE
Alef Iury [email protected]
Institute of Informatics
Institute of Informatics Federal
Federal University of Goiás
Brazil
Siqueira Ferreira
Institute of Informatics
Institute of Informatics Federal
Federal University of Goiás
Brazil
Gustavo Dos
University of Goiás
Brazil
Reis Oliveira
University of Goiás
Brazil
DOMAIN SPECIFIC WAV2VEC 2.0 FINE-TUNING FOR THE SE&R 2022 CHALLENGE
speech recognition · portuguese · prepared speech · spontaneous speech · wild data
This paper presents our efforts to build a robust ASR model for the shared task Automatic Speech Recognition for spontaneous and prepared speech & Speech Emotion Recognition in Portuguese (SE&R 2022). The goal of the challenge is to advance the ASR research for the Portuguese language, considering prepared and spontaneous speech in different dialects. Our method consist on fine-tuning an ASR model in a domain-specific approach, applying gain normalization and selective noise insertion. The proposed method improved over the strong baseline provided on the test set in 3 of the 4 tracks available.
Introduction
The performance of Automatic Speech Recognition systems (ASRs) has increased significantly with the development of modern neural network topologies and the use of massive amount of data to train the models [1]. Although the accuracy of recent models improved for high-resource languages, such as English, the development of ASR models in other languages is still a difficult task using the same technologies [2,3]. In this scenario, Self-Supervised Learning (SSL), a method in which representations with semantic information are learned by using unlabelled data, emerged as an important advance, allowing the training of deeper models using less labelled data [4,5]. In this line of work, this paper explores the use of the Wav2vec 2.0 [6], a framework for self-supervised learning of discrete representations from raw audio data. Wav2vec 2.0 ( Figure 1) is inspired by previous works in unsupervised pre-training for speech recognition, that is, Wav2vec [7] and Vq-Wav2vec [4]. During pre-training the model learns speech representations solving a contrastive task which requires identifying the correct quantized latent speech representations of a masked time step among a set of distractors. After the self-supervised pre-training, the model can be fine-tuned on labeled data in a supervised task, like ASR, adding a randomly initialized linear projection on top of the context network with N classes and a loss function specific to the task at hand, like CTC, for instance.
The model shows important results among low resource languages. In Portuguese, for example, [8] and [9] demonstrated that the fine-tuning of the Wav2vec 2.0 model achieves state-of-the-art (SOTA) results only using publicly available datasets.
An important aspect to consider when training an ASR model is the quality and the domain of the data [10,11,12]. While most of the available public datasets are composed of prepared speech [9], mostly read sentences [13,14], the domain of real ASRs are far more complex, mainly because it is formed by spontaneous speech and different speech dialects. Quality is another issue: most of ASR use cases involve high noise environments or low recording equipment, which is not adressed in most of the public datasets available [9].
To stimulate research that can advance the present SOTA in ASR in Portuguese, for both prepared and spontaneous speech, the shared-task Automatic Speech Recognition for spontaneous and prepared speech & Speech Emotion Figure 1: Illustration of the Wav2vec 2.0 framework [6].
Recognition in Portuguese (SE&R 2022) introduces a new baseline for ASR and a new dataset in Portuguese [9]. The Corpus of Annotated Audios (CORAA ASR), a large corpus of spontaneous and prepared speech, is composed by various subsets in Portuguese with different characteristics. The baseline achieves a Word Error Rate (WER) of 24.18% on CORAA ASR test set, a difficult dataset containing samples with low quality, noise, and a variety of domains and dialects.
In this work, we investigate the fine-tuning of the baseline model [9] proposed by the shared task, a fine-tuned model based on the Wav2vec 2.0 XLSR-53 [15], using only public available Portuguese datasets, including the CORAA ASR dataset. We conducted several experiments in different domains for the challenge and explored the use of selective noise insertion and audio normalization during training. This work is organized as follows: Section 2 discuss the proposed methods and Section 3 shows and discusses the obtained results. Finally, Section 4 presents the conclusions of this work.
Methods
Datasets
We used several publicly available datasets in Portuguese. Besides CORAA ASR, most of them is composed by prepared speech. In general, we opted to use all the data in the gathered datasets for training, except the dev part of the CORAA ASR, as presented in Table 1. The datasets used in this work are:
• CETUC [13]: contains approximately 145 hours of Brazilian Portuguese speech distributed among 50 male and 50 female speakers, each pronouncing approximately 1,000 phonetically balanced sentences selected from the CETEN-Folha 1 corpus;
• Common Voice (CV) 7.0 [16]: is a project proposed by Mozilla Foundation with the goal to create a wide open dataset in different languages. In this project, volunteers donate and validate speech using the official site 2 ;
• Multilingual LibriSpeech (MLS) [14]: a massive dataset available in many languages. The MLS is based on audiobook recordings in public domain like LibriVox 3 . The dataset contains a total of 6k hours of transcribed data in many languages. The set in Portuguese used in this work 4 Table 1: Dataset splits used in this work.
Experiments
Our experiments consists on the fine-tuning of the baseline model of [9]. For each experiment, we trained the model for 5 epochs, using a batch size of 192, and using Adam [22] where the learning rate is 3 e−05 that is warmed up for the first 400 updates, then linearly decayed for the remained. For the experiments, we used a NVIDIA TESLA V100 32GB, a NVIDIA TESLA Tesla P100 16GB and a NVIDIA A100 80GB, depending on the type of audio pre-processing used. The code to replicate the results is available at https://github.com/alefiury/SE-R_2022_Challenge_Wav2vec2.
In total, we conducted five main experiments to test our methods:
• Experiment 1: Wav2vec 2.0 XLSR-53 -Base: For this experiment, the model was fine-tuned considering the whole train set, but did not receive neither normalization nor noise addition;
• Experiment 2: Wav2vec 2.0 XLSR-53 -Norm: The model was fine-tuned with the whole train set with normalization. For the normalization, the mean gain of all the audios in the train set was considered;
• Experiment 3: Wav2vec 2.0 XLSR-53 -Norm and SNA: The model was fine-tuned with gain normalization and selective noise addition. The audios were normalized considering the mean gain of all the audios in the train set, and those audios pertaining to datasets that were considered to have a low presence of noise, namely MLS and CETUC, received randomly one of the following 5 possible types of noises: additive noise, being music or nonspeech noises from the MUSAN Corpus [23], Room impulse responses [24], Addition or reduction of gain, Pitch shift and Gaussian noise;
• Experiment 4: Wav2vec 2.0 XLSR-53 -Norm + Prepared Speech: Model fine-tuned based on the final trained model of Experiment 2, but considering just the prepared speech data from the CORAA ASR dataset, trained for more 5 epochs;
• Experiment 5: Wav2vec 2.0 XLSR-53 -Norm + Spontaneous Speech: Model fine-tuned based on the final trained model of Experiment 2, but considering just the spontaneous speech data from the CORAA ASR dataset, trained for more 5 epochs.
Results and Discussion
The shared-task consists of 4 tracks. Each track has a domain specific scenario, that includes prepared speech and spontaneous speech. In this regard, we conducted a prior analysis (Section 3.1) using the dev set to select the best approaches based on 3 of the 4 tracks available: Mixed, Prepared Speech PT_BR and Spontaneous Speech. The best models were selected and then submitted for evaluation. Our final results are presented in Section 3.2.
Dev Set Analysis
Overall, our models did not show a huge improvement in performance when compared to the baseline model. Even though we fine-tuned a model that is considered the state-of-the-art in Brazilian Portuguese, we suspect that the number of training epochs might have been insufficient to obtain an increase in performance, or that the baseline model might have already reached a local optima.
Furthermore, as presented in Table 2, the model that was fine-tuned using prepared speech clearly improved the results in the Prepared Speech subset (and consequently the Mixed subset). The same phenomenon could not be seen in the Spontaneous Speech subset. A possible explanation to this fact is that most of the data of the datasets that were added to the train set are comprised of prepared speech, which might have contributed to the increase in performance in this particular domain. Another possible explanation is the low number of training epochs used to train the models. Table 3: Dev set analisys by dataset in CORAA ASR Table 4 compares the baseline with our selected models in the test set. The model Wav2vec 2.0 XLSR-53 -Norm + Prepared Speech surpassed the strong baseline in the Prepared Speech PT_BR, Prepared Speech PT_PT and the Mixed tracks. As seen in the results based on the dev set, the fact that most of the data of the datasets that were added to the train set are comprised of prepared speech, might have contributed to the increase in performance in this domain in both Portuguese variants. Lastly, even though we were not able to surpass the baseline model in the Spontaneous Speech track, we achieved competitive results with both submitted models.
Final Results
Additional Experiments
After selecting and submitting the best results, we performed some additional experiments to further explore our proposed methods using the dev set. We tried to use text correction in the outputs of the ASR models, and train the normalized models for a longer period of time using early stopping considering the prepared speech and spontaneous speech data from the CORAA ASR dataset.
The text correction was done with an additional post-processing step, using a KenLM [25] model. For the different tasks, we used 2 different KenLM models: one for spontaneous speech, which was built using subsets of the CORAA ASR dataset containing spontaneous speech phrases. And the other one was built considering wikipedia in portuguese texts, as proposed by [3]. Both were 4-grams. We found that this post-processing of the predictions from the ASR models did not improve the results from the dev set on the Prepared Speech PT_BR and Spontaneous Speech tracks, as can be seen in Table 5, in fact they were worse. One possible explanation is that some of the decoder hyper-parameters did not work well with our ASR models. Another possibility is that the 4-gram trained with spontaneous text was built with a small amount of text, which might have decreased the performance of the model.
Conclusions
In this work we presented our efforts to build a robust ASR model using multiple approaches, such as selective noise insertion and domain specific fine-tuning. In our experiments we found that fine-tuning a strong baseline with additional public available data in multiple domains and using normalization, even for a few epochs, can improve performance. With our results we were able to improve on the test set in 3 of the 4 tracks available over the strong baseline provided.
As future works, we plan to train a ASR model using a dynamic noise insertion approach that do not depend on choosing specific datasets previously.
Table 2 :
2Dev set analisys by subset Additionally, the noise insertion did not gave further improvement in performance. Nevertheless, the results of the SNA model in some more noisy subsets of the CORAA ASR dataset, like ALIP and NURC, for instance, showed some interesting and promising results when compared to the baseline. These results are shown inTable 3.ALIP
NURC-Recife
C-ORAL-BRASIL I
SP2010
Table 4 :
4Test set analisys by subset
However, the results on the Prepared Speech PT_PT track were much better compared to the previous experiments. This result suggests that the LM might improve results when there are few domain data used to train the Wav2vec model, since most of our training data was composed by Brazilian Portuguese audios. Furthermore, as we had suspected earlier, the model with gain normalization that were trained considering the spontaneous speech data from the CORAA ASR corpus and for a longer period of time performed better in its respective subtrack, strengthening our hypothesis that our results did not improve in the main experiments due to a low number of training epochs.Prepared Speech PT_BR
Prepared Speech PT_PT
Spontaneous Speech
Mixed
Model
CER
WER
CER
WER
CER
WER
CER
WER
Norm + Prepared Speech + Prepared KenLM
5.59%
14.11%
15.23%
32.36%
16.26%
30.66%
13.33%
26.95%
Norm + Prepared Speech + Spontaneous KenLM
5.83%
15.58%
16.52%
36.74%
16.73%
33.78%
13.95%
29.97%
Norm + Spontaneous Speech + Prepared KenLM
5.80%
13.94%
15.80%
32.58%
16.38%
30.75%
13.59%
27.00%
Norm + Spontaneous Speech + Spontaneous KenLM
6.05%
15.52%
16.58%
36.98%
16.91%
33.45%
14.11%
29.85%
Norm + Prepared Speech + Early Stopping
4.54%
12.96%
15.73%
35.97%
14.75%
30.16%
12.45%
27.31%
Norm + Spontaneous Speech + Early Stopping
4.59%
13.14%
15.97%
36.52%
14.66%
29.83%
12.47%
27.33%
Table 5 :
5Additional Experiments. All the experiments used the Wav2vec 2.0 XLSR-53 pre-trained model for finetuning.
https://www.linguateca.pt/cetenfolha/ 2 https://commonvoice.mozilla.org/pt 3 https://librivox.org/ 4 http://www.openslr.org/94/
AcknowledgmentsThis research was funded by CEIA with support by the Goiás State Foundation (FAPEG grant #201910267000527)5. We also would like to thank Cyberlabs Group 6 for the support for this work.
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An investigation of deep neural networks for noise robust speech recognition. L Michael, Dong Seltzer, Yongqiang Yu, Wang, 2013 IEEE international conference on acoustics, speech and signal processing. IEEEMichael L Seltzer, Dong Yu, and Yongqiang Wang. An investigation of deep neural networks for noise robust speech recognition. In 2013 IEEE international conference on acoustics, speech and signal processing, pages 7398-7402. IEEE, 2013.
Rethinking evaluation in asr: Are our models robust enough?. Tatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Paden Tomasello, Jacob Kahn, Gilad Avidov, Ronan Collobert, Gabriel Synnaeve, arXiv:2010.11745arXiv preprintTatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Paden Tomasello, Jacob Kahn, Gilad Avidov, Ronan Collobert, and Gabriel Synnaeve. Rethinking evaluation in asr: Are our models robust enough? arXiv preprint arXiv:2010.11745, 2020.
Robust wav2vec 2.0: Analyzing domain shift in self-supervised pre-training. Wei-Ning Hsu, Anuroop Sriram, Alexei Baevski, Tatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Jacob Kahn, Ann Lee, Ronan Collobert, Gabriel Synnaeve, arXiv:2104.01027arXiv preprintWei-Ning Hsu, Anuroop Sriram, Alexei Baevski, Tatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Jacob Kahn, Ann Lee, Ronan Collobert, Gabriel Synnaeve, et al. Robust wav2vec 2.0: Analyzing domain shift in self-supervised pre-training. arXiv preprint arXiv:2104.01027, 2021.
Lsf and lpc-derived features for large vocabulary distributed continuous speech recognition in brazilian portuguese. Abraham Vfs Alencar, Alcaim, IEEE42nd Asilomar conference on signals, systems and computersVFS Alencar and Abraham Alcaim. Lsf and lpc-derived features for large vocabulary distributed continuous speech recognition in brazilian portuguese. In 2008 42nd Asilomar conference on signals, systems and computers, pages 1237-1241. IEEE, 2008.
Mls: A large-scale multilingual dataset for speech research. Vineel Pratap, Qiantong Xu, Anuroop Sriram, Gabriel Synnaeve, Ronan Collobert, Interspeech 2020Vineel Pratap, Qiantong Xu, Anuroop Sriram, Gabriel Synnaeve, and Ronan Collobert. Mls: A large-scale multilingual dataset for speech research. Interspeech 2020, Oct 2020.
Unsupervised cross-lingual representation learning at scale. Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Édouard Grave, Myle Ott, Luke Zettlemoyer, Veselin Stoyanov, Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. the 58th Annual Meeting of the Association for Computational LinguisticsAlexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Édouard Grave, Myle Ott, Luke Zettlemoyer, and Veselin Stoyanov. Unsupervised cross-lingual representation learning at scale. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 8440-8451, 2020.
Common voice: A massively-multilingual speech corpus. Rosana Ardila, Megan Branson, Kelly Davis, Michael Kohler, Josh Meyer, Michael Henretty, Reuben Morais, Lindsay Saunders, Francis Tyers, Gregor Weber, Proceedings of the 12th Language Resources and Evaluation Conference. the 12th Language Resources and Evaluation ConferenceRosana Ardila, Megan Branson, Kelly Davis, Michael Kohler, Josh Meyer, Michael Henretty, Reuben Morais, Lindsay Saunders, Francis Tyers, and Gregor Weber. Common voice: A massively-multilingual speech corpus. In Proceedings of the 12th Language Resources and Evaluation Conference, pages 4218-4222, 2020.
The multilingual tedx corpus for speech recognition and translation. Elizabeth Salesky, Matthew Wiesner, Jacob Bremerman, Roldano Cattoni, Matteo Negri, Marco Turchi, W Douglas, Matt Oard, Post, arXiv:2102.01757arXiv preprintElizabeth Salesky, Matthew Wiesner, Jacob Bremerman, Roldano Cattoni, Matteo Negri, Marco Turchi, Douglas W Oard, and Matt Post. The multilingual tedx corpus for speech recognition and translation. arXiv preprint arXiv:2102.01757, 2021.
Projeto alip (amostra linguística do interior paulista) e banco de dados iboruna: 10 anos de contribuição com a descrição do português brasileiro. Sebastião Carlos Leite Gonçalves, Estudos Linguísticos. São Paulo48Sebastião Carlos Leite Gonçalves. Projeto alip (amostra linguística do interior paulista) e banco de dados iboruna: 10 anos de contribuição com a descrição do português brasileiro. Estudos Linguísticos (São Paulo. 1978), 48(1):276-297, abr. 2019.
C-oral -brasil i: Corpus de referência do portugues brasileiro falado informal. Tommaso Raso, Heliana Mello, 042012Tommaso Raso and Heliana Mello. C-oral -brasil i: Corpus de referência do portugues brasileiro falado informal. pages 362-367, 04 2012.
Nurc digital um protocolo para a digitalização, anotação, arquivamento e disseminação do material do projeto da norma urbana linguística culta (nurc). Miguel OliveiraJr, 3CHIMERA: Romance Corpora and Linguistic StudiesMiguel Oliveira, Jr. Nurc digital um protocolo para a digitalização, anotação, arquivamento e disseminação do material do projeto da norma urbana linguística culta (nurc). CHIMERA: Romance Corpora and Linguistic Studies, 3:149-174, 01 2016.
Mapping paulistano portuguese: the sp2010 project. R B Mendes, L Oushiro, Fizenze University Press012012Firenze, ItalyMendes R B and Oushiro L. Mapping paulistano portuguese: the sp2010 project. pages 459-463. Firenze, Italy: Fizenze University Press, 01 2012.
Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, 3rd International Conference on Learning Representations. San Diego, CA, USAConference Track ProceedingsDiederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.
Musan: A music, speech, and noise corpus. David Snyder, Guoguo Chen, Daniel Povey, David Snyder, Guoguo Chen, and Daniel Povey. Musan: A music, speech, and noise corpus, 2015.
A study on data augmentation of reverberant speech for robust speech recognition. Tom Ko, Vijayaditya Peddinti, Daniel Povey, Michael L Seltzer, Sanjeev Khudanpur, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Tom Ko, Vijayaditya Peddinti, Daniel Povey, Michael L. Seltzer, and Sanjeev Khudanpur. A study on data augmentation of reverberant speech for robust speech recognition. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5220-5224, 2017.
KenLM: Faster and smaller language model queries. Kenneth Heafield, Proceedings of the Sixth Workshop on Statistical Machine Translation. the Sixth Workshop on Statistical Machine TranslationEdinburgh, ScotlandAssociation for Computational LinguisticsKenneth Heafield. KenLM: Faster and smaller language model queries. In Proceedings of the Sixth Workshop on Statistical Machine Translation, pages 187-197, Edinburgh, Scotland, July 2011. Association for Computational Linguistics.
| {'fraction_non_alphanumeric': 0.04301243450379291, 'fraction_numerical': 0.031907405959177286, 'mean_word_length': 4.954598370197904, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 4, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This paper presents our efforts to build a robust ASR model for the shared task Automatic Speech Recognition for spontaneous and prepared speech & Speech Emotion Recognition in Portuguese (SE&R 2022). The goal of the challenge is to advance the ASR research for the Portuguese language, considering prepared and spontaneous speech in different dialects. Our method consist on fine-tuning an ASR model in a domain-specific approach, applying gain normalization and selective noise insertion. The proposed method improved over the strong baseline provided on the test set in 3 of the 4 tracks available.', 'arxivid': '2207.14418', 'author': ['Alef Iury [email protected] \nInstitute of Informatics\nInstitute of Informatics Federal\nFederal University of Goiás\nBrazil\n', 'Siqueira Ferreira \nInstitute of Informatics\nInstitute of Informatics Federal\nFederal University of Goiás\nBrazil\n', 'Gustavo Dos \nUniversity of Goiás\nBrazil\n', 'Reis Oliveira \nUniversity of Goiás\nBrazil\n'], 'authoraffiliation': ['Institute of Informatics\nInstitute of Informatics Federal\nFederal University of Goiás\nBrazil', 'Institute of Informatics\nInstitute of Informatics Federal\nFederal University of Goiás\nBrazil', 'University of Goiás\nBrazil', 'University of Goiás\nBrazil'], 'corpusid': 251196765, 'doi': '10.48550/arxiv.2207.14418', 'github_urls': ['https://github.com/alefiury/SE-R_2022_Challenge_Wav2vec2.'], 'n_tokens_mistral': 7698, 'n_tokens_neox': 6654, 'n_words': 3671, 'pdfsha': '8bda9efb50dc80d010bbb88263deab664b42806c', 'pdfurls': ['https://export.arxiv.org/pdf/2207.14418v1.pdf'], 'title': ['DOMAIN SPECIFIC WAV2VEC 2.0 FINE-TUNING FOR THE SE&R 2022 CHALLENGE', 'DOMAIN SPECIFIC WAV2VEC 2.0 FINE-TUNING FOR THE SE&R 2022 CHALLENGE'], 'venue': []} |
arxiv |
Heterogeneity-stabilized homogeneous states in driven media
Zachary G Nicolaou
SUPPLEMENTARY DISCUSSION 1
Daniel J Case
SUPPLEMENTARY DISCUSSION 1
Ernest B Van Der Wee
SUPPLEMENTARY DISCUSSION 1
Michelle M Driscoll
SUPPLEMENTARY DISCUSSION 1
Adilson E Motter
SUPPLEMENTARY DISCUSSION 1
Heterogeneity-stabilized homogeneous states in driven media
SUPPLEMENTARY INFORMATION
Band gaps in a pendulum array with heterogeneous massesHere, we consider an alternative pendulum model in which the masses of the pendula vary rather than the lengths. This case is of easy experimental implementation and illustrates the generality of band gap opening by periodic heterogeneities (see Methods). The equations of motion for the driven array are given bywhere 1 ≤ i ≤ N . In the absence of driving (a d = 0), linearization around θ i = 0 ∀i leads to an expression in the form of (9):where the heterogeneity is determined by the differing M i .Supplementary Fig. 1shows how wave modes split into distinct branches for a specific periodic heterogeneity with a three-particle unit cell,with L = 1, g = 1, and κ = 1. Similar to the pendulum array discussed in the main text, when this system is parametrically driven with driving frequencies around twice the frequencies in a band gap, there are no resonant wave modes that can be easily excited, and thus heterogeneity-stabilized homogeneous states emerge.
Here, we consider an alternative pendulum model in which the masses of the pendula vary rather than the lengths. This case is of easy experimental implementation and illustrates the generality of band gap opening by periodic heterogeneities (see Methods). The equations of motion for the driven array are given by
M i Lθ i = −ηLθ i − M i [g − a d ω 2 d cos(ω d t)] sin(θ i ) + κL[sin(θ i+1 − θ i ) + sin(θ i−1 − θ i )],(1)
where 1 ≤ i ≤ N . In the absence of driving (a d = 0), linearization around θ i = 0 ∀i leads to an expression in the form of (9):
M iθi + ηθ i + j (M i g/L + 2κ)δ i j − κ(δ i+1 j + δ i−1 j ) θ j = 0,(2)
where the heterogeneity is determined by the differing M i . Supplementary Fig. 1 shows how wave modes split into distinct branches for a specific periodic heterogeneity with a three-particle unit cell,
M i = 1 − λ i mod 3 = 0, 1 i mod 3 = 1, 1 + λ i mod 3 = 2,(3)
with L = 1, g = 1, and κ = 1. Similar to the pendulum array discussed in the main text, when this system is parametrically driven with driving frequencies around twice the frequencies in a band gap, there are no resonant wave modes that can be easily excited, and thus heterogeneity-stabilized homogeneous states emerge. In the main text, we established that the homogeneous states are stabilized in the band gaps when heterogeneity is introduced, in the sense that infinitesimal perturbations around them decay. However, other stable states may also emerge when the system is driven. If multiple stable states coexist, sufficiently large perturbations may induce transitions between states even when the initial state is stable against infinitesimal perturbations. Here, we consider heterogeneity-stabilized homogeneous states in the presence of finite-size perturbations. For concreteness, we focus on the driven pendulum array with periodic heterogeneity considered in the main text, which is defined by alternating pendulum length
L i = L + (−1) i ∆.
We find that alternative stable states do, in fact, emerge in the periodic pendulum array in parameter regions that exhibit heterogeneity-stabilized homogeneous states, as shown in Supplementary Fig. 2. This follows because the instability of any particular wave mode becomes subcritical when the driving frequency is smaller than the resonant frequency of that mode (left panel in Supplementary Fig. 2a), which should be contrasted with the supercritical form of the instabilities for larger driving frequencies (right panel in Supplementary Fig. 2a). The difference between the driving frequency and the resonant frequency of a particular mode is called the detuning parameter for that mode, so modes exhibit subcritical instabilities for negative detuning parameters. In the subcritical cases, an unstable swinging state vanishes at a secondary instability boundary of a new stable periodic swinging state that emerges for increasing driving amplitude. Since the detuning parameter for the mode is negative in this case, there exists a different, lower frequency mode that is resonant with the driving frequency. Thus, the secondary instability boundary for the swinging states may lie above or below the (primary) instability boundary for the uniform state, which consists of the envelope of the instability boundaries over all modes.
The central panel in Supplementary Fig. 2a shows the primary instability boundaries for the pendulum array in the N → ∞ limit with ∆ = 0 and ∆ = 0.35 (solid lines). The dashed lines show the instability boundary for the k = π/2 mode that is split by the bad gap, the dotted lines show the secondary instability boundaries for this mode in the subcritical cases. For the homogeneous array (orange lines), the secondary instability boundary for this mode lies above the primary instability boundary, so the system is not susceptible to finite-size instabilities below the primary instability boundary. For the heterogeneous array in this figure (blue lines), on the other hand, the secondary instability boundary for this mode lies below the primary instability boundary, and finite-size perturbations can therefore excite the array to the periodic swinging state below the primary instability boundary. In the outsets, the θ (a) i describe the swinging state amplitudes, defined by the value of the phases at the time points where i θ 2 i is maximized. Interestingly, we find that the periodic swinging states are not the only alternative stable states for the heterogeneous array. Supplementary Fig. 2b shows the swinging state amplitudes for a particularly interesting localized stable state that we observe for random initial conditions in an array of 32 pendula for the parameter values marked by the × (ω d = 3.5 and a d = 0.05) and heterogeneity in Supplementary Fig. 2a. An animation of this localized state is also available as part of Supplementary Movie 1. Since these localized states can coexist in a variety of spatial configurations, the heterogeneous pendulum array exhibits a high degree of multistability. Similar gap soliton solutions have been observed around band gaps in other media, emerging through a snaking bifurcation in pattern-formation models [S1]. While beyond the scope of this work, we expect that such gap solitons will also exist in Faraday instability systems with periodic substrates within regions of HSHS. We argue that this is expected because the bifurcations of instability modes have been shown to become subcritical for negative detuning parameters in Faraday wave experiments with homogeneous substrates [S2], in direct analogy with the orange lines in Supplementary Fig. 2a.
Finite-size perturbations can induce a transition between the homogeneous state and the stable swinging states below the instability boundary. To quantify the stability against finite-size perturbations, we simulate the system with a random initial perturbation given by an initial θ i uniformly distributed in [−απ, απ] and an initialθ i = 0. Here, α quantifies the size of the perturbation. The system is evolved until it approaches a stable state. Supplementary Fig. 2c shows the time-averaged value of the phases (after the decay of the initial transient) averaged over 1024 random perturbation realizations for arrays of various sizes for the parameter values shown by the × in Supplementary Fig. 2a. The stability transition sharpens as the number of pendula increases and, for large N , the homogeneous state is stable against almost all simulated finite-amplitude perturbations for α < 0.1. Thus, finite-size perturbations can destabilize heterogeneity-stabilized homogeneous states and lead to nontrivial dynamical states, but doing so requires large perturbations.
. 1. Band-gap opening and HSHS for the pendulum array with heterogeneous masses. a, Schematic of the pendulum array with a three-particle unit cell described by Supplementary Eqs. (1)-(3). b, Frequency ω vs. wavenumber k for λ = 0 (blue lines) and λ = 0.35 (red lines) in the N → ∞ limit for the system shown in a. Two gaps open between the three branches in the heterogeneous case. c, Instability boundaries for the systems shown in b, with green shading showing areas exhibiting HSHS associated with the respective band gaps in b.SUPPLEMENTARY DISCUSSION 2Finite-size perturbations and gap solitons in driven pendulum arrays
Fig. 2 .
2Multistability and finite-size perturbations in pendulum array model. a, Primary instability boundary for the homogeneous array (solid orange line) and heterogeneous array with ∆ = 0.35 (solid blue line) in the N → ∞ limit. The boundaries for the k = π/2 instability mode (dashed lines) and the corresponding secondary instability boundary (dotted lines) are also shown, along with the region of HSHS (green area). Outsets: driving amplitude vs. swinging amplitudes for negative detuning parameters (subcritical bifurcation, left panel) and positive detuning parameters (supercritical bifurcation, right panel). Solid lines show stable states, dashed lines show unstable states, and symbols (also marked in the central panel) indicate the instability boundary (filled circles and diamonds) and the secondary instability boundary (open circles) for a particular mode. b, Swinging amplitude vs. pendulum index for a localized gap soliton state in a heterogeneous array determined by the parameters marked by × in a, with short pendula shown as orange dots and long pendula shown as blue dots. c, Average (over time and perturbation realization) of the squared phases of final stable state θ 2 i vs. perturbation size α for heterogeneous arrays of various size with the other parameters corresponding to × in a.
Gap solitons and forced snaking. B C Ponedel, E Knobloch, Phys. Rev. E. 9862215Ponedel, B. C. & Knobloch, E. Gap solitons and forced snaking. Phys. Rev. E 98, 062215 (2018).
Pattern formation outside of equilibrium. M C Cross, P C Hohenberg, Rev. Mod. Phys. 65851Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993).
| {'fraction_non_alphanumeric': 0.03685220729366603, 'fraction_numerical': 0.01161228406909789, 'mean_word_length': 4.648238482384824, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Band gaps in a pendulum array with heterogeneous massesHere, we consider an alternative pendulum model in which the masses of the pendula vary rather than the lengths. This case is of easy experimental implementation and illustrates the generality of band gap opening by periodic heterogeneities (see Methods). The equations of motion for the driven array are given bywhere 1 ≤ i ≤ N . In the absence of driving (a d = 0), linearization around θ i = 0 ∀i leads to an expression in the form of (9):where the heterogeneity is determined by the differing M i .Supplementary Fig. 1shows how wave modes split into distinct branches for a specific periodic heterogeneity with a three-particle unit cell,with L = 1, g = 1, and κ = 1. Similar to the pendulum array discussed in the main text, when this system is parametrically driven with driving frequencies around twice the frequencies in a band gap, there are no resonant wave modes that can be easily excited, and thus heterogeneity-stabilized homogeneous states emerge.', 'arxivid': '2108.01087', 'author': ['Zachary G Nicolaou \nSUPPLEMENTARY DISCUSSION 1\n\n', 'Daniel J Case \nSUPPLEMENTARY DISCUSSION 1\n\n', 'Ernest B Van Der Wee \nSUPPLEMENTARY DISCUSSION 1\n\n', 'Michelle M Driscoll \nSUPPLEMENTARY DISCUSSION 1\n\n', 'Adilson E Motter \nSUPPLEMENTARY DISCUSSION 1\n\n'], 'authoraffiliation': ['SUPPLEMENTARY DISCUSSION 1\n', 'SUPPLEMENTARY DISCUSSION 1\n', 'SUPPLEMENTARY DISCUSSION 1\n', 'SUPPLEMENTARY DISCUSSION 1\n', 'SUPPLEMENTARY DISCUSSION 1\n'], 'corpusid': 236209579, 'doi': '10.1038/s41467-021-24459-0', 'github_urls': [], 'n_tokens_mistral': 2683, 'n_tokens_neox': 2298, 'n_words': 1641, 'pdfsha': '5c1c404db516eebe50dc62e48a744ab01b69e8b5', 'pdfurls': None, 'title': ['Heterogeneity-stabilized homogeneous states in driven media', 'Heterogeneity-stabilized homogeneous states in driven media'], 'venue': []} |
arxiv |
AN INFINITE LIE ALGEBRA ASSOCIATED WITH THE QUANTUM COULOMB FIELD
arXiv:hep-th/9810085v1 13 Oct 1998
Andrzej Staruszkiewicz
Department of Physics and Astronomy
Department of Theoretical Physics
University of South Carolina Columbia
29208South Carolina
Jagiellonian University
KrakowPoland
AN INFINITE LIE ALGEBRA ASSOCIATED WITH THE QUANTUM COULOMB FIELD
arXiv:hep-th/9810085v1 13 Oct 1998⋆ Address after September 15, 1992 :PACS numbers: 1220Ds
The theory of the quantum Coulomb field associates with each Lorentz frame, i. e., with each unit, future oriented time-like vector u, the operator of the number of transveral infrared photons N (u) and the phase S(u) which is the coordinate canonically conjugated with the total charge Q: [Q, S(u)] = ie, e being the elementary charge. It is shown that the operators N (u), Q e S(u) and Q 2 form an infinite Lie algebra. One can conclude from this algebra that ∆N (u) = (4/π)Q 2 , where ∆ is the Laplace operator in the Lobachevsky space of four-velocities u, thus relating the total charge Q with the number of infrared photons.
A charged particle, when scattered, produces an infrared electromagnetic field which behaves at the spatial infinity like the inverse of distance. Gervais and Zwanziger [1] gave a clean way to separate the infrared field from the rest : performing the rescaling lim λ→∞ λA µ (λx) one obtains an electromagnetic potential which is homogeneous of degree −1 and represents thus a pure infrared field free from high frequency contaminations. Now, from the physical point of view, the infrared electromagnetic field is a free dynamical system : scattered charges produce infrared fields but infrared fields do not scatter charges; there is no back reaction. This allows to treat the infrared field as a free field; such a treatment is physically rigorous. This means also that the infrared field can be quantized in the usual way [2] . The resulting field theory is remarkable as it contains the quantum analog of the classical Coulomb field and allows to make meaningful and meaningfully arrived at statements on the magnitude of the fine structure constant e 2 /hc . In particular the value e 2 /hc = π is seen to be critical as it separates two kinematically distinct regimes of the quantum Coulomb field [3] .
In the quantum theory of infrared fields the classical Coulomb field
A 0 = Q r , A = 0,(1)
is a global solution of the classical equation of motion, to be used, for the sake of completeness, in the procedure of quantization. The theory is summarized in [2] on page 364 in the form of all relevant canonical commutation relations, written in a fixed Lorentz reference frame which we imagine as being at rest. We repeat this summary here for the reader's convenience:
[Q, S 0 ] = ie, [Q, c lm ] = 0, [S 0 , c lm ] = 0,(2)[c lm , c † l ′ m ′ ] = 4πe 2 δ ll ′ δ mm ′ .(3)
Here Q is the total charge, e is the elementary charge, S 0 is the phase, a coordinate canonically conjugated with the operator Q e ; c † lm and c lm are, respectively, creation and annihilation operators for transversal infrared photons; they are numbered by the numbers l = 1, 2, ... and m = −l, −l+1, ..., l known from the theory of angular momentum.
The vacuum state |0 > is defined by the relations (Ref. [2], p. 364)
c lm |0 >= 0, < 0|c † lm = 0, Q|0 >= 0, < 0|Q = 0,(4)
which are standard for transversal photons but nonstandard for the total charge Q. The reason for this is that the total charge Q is a Lorentz invariant quantity while the phase S 0 is not; hence any definition of the vacuum state involving the phase S 0 would necessarily violate the Lorentz invariance.
The operator
N = 1 4πe 2 lm c † lm c lm (5)
gives the number of transversal infrared photons; its spectrum consists of nonnegative integers, N = 0, 1, 2, .... Now, the whole theory is Lorentz invariant as it may be represented as a quantum field theory of a massless scalar field "living" in the three-dimensional hyperboloid of unit space-like four-vectors. Therefore, all the previous relations may be written in exactly the same form in another Lorentz reference frame :
[Q, S ′ 0 ] = ie, N ′ = 1 4πe 2 lm c ′ † lm c ′ lm ,(6)
etc.
Prime over a quantity denotes the same quantity in the moving reference frame which, without loss of generality, will be assumed to move in the z-direction. There is no prime over Q since Q = Q ′ is a Lorentz scalar.
We can formulate now the main goal of this Letter: to relate primed quantities N ′ , S ′ 0 with the nonprimed ones N , S 0 . It will be seen that the appropriate relation has the form of an infinite Lie algebra.
In what follows it will be convenient to simplify notation and to replace the pair of indices (lm) by a single index α. Thus the quantum mechanics of charge is simplified to the following form
[Q, S 0 ] = ie, [Q, c α ] = 0, [S 0 , c α ] = 0,(7)
[c α , c † β ] = 4πe 2 δ αβ ,
and
c α |0 >= 0, < 0|c † α = 0, Q|0 >= 0, < 0|Q = 0.(9)
We shall prove two lemmas which together give the Lorentz transformation of the amplitudes c α and S 0 .
Lemma 1.
The amplitude c ′ α is a linear combination of the amplitudes c α and Q but does not contain the phase S 0
c ′ α = β A αβ c β + B α Q.(10)
Moreover, the matrix A αβ is unitary.
Indeed, since c ′ α |0 >= 0, c ′ α must be a linear combination of those unprimed amplitudes which annihilate the vacuum state, i. e., c α and Q. Unitarity of the matrix A αβ follows from the fact that the Lorentz transformation (10) must preserve the canonical commutation relations
[c ′ α , c ′ † β ] = 4πe 2 δ αβ .(11)
Lemma 2.
The amplitude S ′ 0 is a linear combination of the amplitudes S 0 , c α and c † α but does not contain the charge Q
S ′ 0 = S 0 − 1 4πie αβ B α A αβ c β − B α A αβ c † β .(12)
It is easy to check that this expression is indeed consistent with the Lorentz invariance of the commutation relations which involve the phase S 0 : calculating [Q, S ′ 0 ], [c ′ α , S ′ 0 ] and [c ′ † , S ′ 0 ] and using (10) one finds identity in each case. This does not show that the term proportional to the charge Q on the right hand side is absent. Using, however, the explicit expression of S ′ 0 through the solutions of the wave equation on the three-dimensional hyperboloid of space-like unit four-vectors given in Ref. [2], one finds indeed that the phase S ′ 0 does not contain the total charge Q.
To summarize, we have the following expressions for the Lorentz transformation of all the amplitudes :
c ′ α = β A αβ c β + B α Q,(13)c ′ † α = β A αβ c † β + B α Q,(14)S ′ 0 = S 0 − 1 4πie αβ B α A αβ c β − B α A αβ c † β ,(15)
where the matrix A αβ is unitary. We have
N ′ = 1 4πe 2 α c ′ † α c ′ α .(16)
Putting into (16) c ′ α and using unitarity of the matrix A αβ one obtains
N ′ = N + Q 4πe 2 αβ B α A αβ c β + B α A αβ c † β + Q 2 4πe 2 α |B α | 2 . (17) It is shown in Ref. [2] that α |B α | 2 = 8e 2 (λcothλ − 1),(18)
where λ is the hyperbolic angle between the time axis of the moving frame and that of the rest frame.
Let us take the commutator [N, S ′ 0 ]. Using (12) and the obvious relations [N,
c † α ] = c † α , [N, c α ] = −c α , one finds [N, S ′ 0 ] = 1 4πie αβ B α A αβ c β + B α A αβ c † β .(19)
Hence
N ′ = N + i Q e [N, S ′ 0 ] + 2 Q 2 π (λcothλ − 1).(20)
Since the total charge Q commutes with the operator N , the last equation (20) can be written in the form
[N, Q e S ′ 0 ] = i N − N ′ + 2 Q 2 π (λcothλ − 1) .(21)
This equation has already the form characteristic for the Lie algebras. To close the algebra we need further [N, N ′ ] and [ Q e S 0 , Q e S ′ 0 ]. One has immediately from the previous formula (21) and canonical commutation relations
[N, N ′ ] = i Q e (S ′ 0 − S 0 ), [ Q e S 0 , Q e S ′ 0 ] = i Q e (S 0 − S ′ 0 ).(22)
An objection may be raised that the operator ( Q e )S 0 is "ill defined" as it can be represented in the rest frame as the differential operator (i ∂ ∂S 0 S 0 ), S 0 being an angular variable with the period 2π. However, one sees that the right hand sides of all the commutators contain only differences of two phases which are perfectly well defined. Such differences may be also introduced in the left hand sides; for example, the commutator [N, Q e S ′ 0 ] can be written as [N, Q e (S ′ 0 − S 0 )] since [N, Q e S 0 ] = 0. The same can be done for all other commutators, which shows that all the commutators written above have perfectly well defined meaning.
It will be useful to change notation. We introduce the unit, timelike, future oriented vector u which indicates the time axis of the rest frame and a similar vector v which indicates the time axis of the moving frame. All such vectors together form the Lobachevsky space of four-velocities. Using this notation we have
N (u), N (v) = i Q e − S(u) + S(v) ,(23)N (u), Q e S(v) = i N (u) − N (v) + 2 Q 2 π (uv) ,(24)Q e S(u), Q e S(v) = i Q e S(u) − S(v) ,(25)
where (uv) = λcothλ − 1 , λ being the hyperbolic angle between u and v : g µν u µ v ν = coshλ. Commutators involving Q 2 are omitted because they are completely obvious.
The Lobachevsky space of four-velocities is a three-dimensional locally Euclidean space of constant negative curvature with the metric induced by the metric of the four-dimensional space-time in which it is immersed. One can execute in this space all invariant operations of tensor analysis. Let us apply the Laplace operator ∆ v (v indicates that it is to be applied at the point v) to both sides of the commutator
N (u), Q e S(v) = i N (u) − N (v) + 2 Q 2 π (uv) .(26)
One finds ∆ v (uv) = 2 by direct calculation. Thus
N (u), Q e ∆ v S(v) = i − ∆ v N (v) + 4 Q 2 π .(27)
But ∆ v S(v) = 0, as can be shown again from the explicit expression for S(v) given in Ref. [2]. Hence, dropping the index v which is not necessary anymore we have
∆N = 4 Q 2 π .(28)
This equation is remarkable as it connects the total charge Q with the operator N of the number of transversal infrared photons which has an absolute scale, its spectrum consisting of nonnegative integers.
. J.-L Gervais, D Zwanziger, Phys. Lett. 94389J.-L. Gervais and D. Zwanziger, Phys. Lett. B94, 389(1980)
. A Staruszkiewicz, Ann. Phys. (N.Y.). 190354A. Staruszkiewicz, Ann. Phys. (N.Y.) 190, 354(1989).
. A Staruszkiewicz, Acta Phys. Pol. 23591A. Staruszkiewicz, Acta Phys. Pol. B23, 591(1992).
| {'fraction_non_alphanumeric': 0.06987154534838459, 'fraction_numerical': 0.028513040093421566, 'mean_word_length': 3.491695804195804, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The theory of the quantum Coulomb field associates with each Lorentz frame, i. e., with each unit, future oriented time-like vector u, the operator of the number of transveral infrared photons N (u) and the phase S(u) which is the coordinate canonically conjugated with the total charge Q: [Q, S(u)] = ie, e being the elementary charge. It is shown that the operators N (u), Q e S(u) and Q 2 form an infinite Lie algebra. One can conclude from this algebra that ∆N (u) = (4/π)Q 2 , where ∆ is the Laplace operator in the Lobachevsky space of four-velocities u, thus relating the total charge Q with the number of infrared photons.', 'arxivid': 'hep-th/9810085', 'author': ['Andrzej Staruszkiewicz \nDepartment of Physics and Astronomy\nDepartment of Theoretical Physics\nUniversity of South Carolina Columbia\n29208South Carolina\n\nJagiellonian University\nKrakowPoland\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nDepartment of Theoretical Physics\nUniversity of South Carolina Columbia\n29208South Carolina', 'Jagiellonian University\nKrakowPoland'], 'corpusid': 10422764, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3519, 'n_tokens_neox': 3017, 'n_words': 1985, 'pdfsha': 'aa70c948d72e53d6048c5d10fcadb3564cfdcfb4', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9810085v1.pdf'], 'title': ['AN INFINITE LIE ALGEBRA ASSOCIATED WITH THE QUANTUM COULOMB FIELD', 'AN INFINITE LIE ALGEBRA ASSOCIATED WITH THE QUANTUM COULOMB FIELD'], 'venue': []} |
arxiv |
From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)
Shubham Ojha
Center of Excellence of Artificial Intelligence
Odisha University of Technology and Research Bhubaneswar
Siddharth Mohapatra
Center of Excellence of Artificial Intelligence
Odisha University of Technology and Research Bhubaneswar
Aditya Narendra
Center of Excellence of Artificial Intelligence
Odisha University of Technology and Research Bhubaneswar
Ipsit Misra
Center of Excellence of Artificial Intelligence
Odisha University of Technology and Research Bhubaneswar
From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)
Artificial Intelligence for Education(AIEd)Smart ApplicationsEducational Systems
The world around us has undergone a radical transformation due to rapid technological advancement in recent decades. The industry of the future generation is evolving, and artificial intelligence is the next change in the making popularly known as Industry 4.0. Indeed, experts predict that artificial intelligence (AI) will be the main force behind the following significant virtual shift in the way we stay, converse, study, live, communicate and conduct business. All facets of our social connection are being transformed by this growing technology. One of the newest areas of educational technology is Artificial Intelligence in the field of Education (AIEd). This study emphasis the different applications of Artificial Intelligence in education from both an industrial and academic standpoint. It highlights the most recent applications of AIEd, with some of its main areas being the reduction of instructors' burden and students' contextualized learning novel transformative evaluations, and advancements in sophisticated tutoring systems. It analyses the AIEd's ethical component and the influence of this transition on people, particularly students and instructors as well. Finally, the article touches on AIEd's potential future research and practices. The goal of this study is to introduce the present-day applications to its intended audience.
I. Introduction:
The emergence of artificial intelligence(AI) has created a tremendous technical change in recent years.AI as defined by Marvin Minsky, artificial intelligence(AI) is the science of making machines do things that would require intelligence if done by men. This topic began as a research area of computer science engineering, but due to its significant absorption of ideas from neurology, cognitive science, philosophy and other disciplines, it has become extremely interdisciplinary, making it difficult even for experts to find an agreeable definition of artificial intelligence. It is a system that has capabilities(such as language or perception) and intelligent conduct that were originally thought to be exclusive to mankind and carry out a certain task. In simpler terms, artificial intelligence (AI) is a discipline of computer science dealing with the emulation of human intelligence by behaving intelligently. This formidable technology has brought about a shift in the way we live in the world.
By incorporating AI-based solutions, sectors including manufacturing, healthcare, etc, are undergoing a sea of change in their operational methodologies. Around the world, the education sector exhibits a similar pattern. Given the largely advantageous digital changes that AI brings into the system, AI has undoubtedly created challenges to traditional ways of education. For the past 30 years, researchers have been studying integration of artificial intelligence in education(AIEd).AIEd has achieved significant success in strengthening connections between teachers and student where the connections were lacking or needed improvement. With the use of AI effective teaching techniques, evaluation systems, and feedback mechanisms can also be introduced. Additionally, weaknesses in the existing systems can be identified, and variety of student responses like boredom and concentration can be captured to make learning a interactive environment.
This essay provides a survey of the most recent advancements in artificial intelligence in education. It starts out by going over numerous fields of education and learning that have made us of AI, then shifts to the areas on which we see the industry concentrating, and it ends with a remark on further fields of development with AI in Education providing a succinct overview of the domain.
II. Methodology:
This multi-phase study conducts a comprehensive analysis of peer-reviewed articles on AI in education. To discover eligible publications for complete analysis, a multi-phase search and selection process was used.
A. Archival Databases
Using the proliferation of online journals and freely accessible resources, even with well-defined criteria, it's nearly difficult to do a thorough search. This study was methodically planned to concentrate on research articles gathered within one of the most commonly used web-based databases, Google Scholar (SCI/SSCI). The database used as the source was chosen as It compiles publications from the Social Science Citation Index (SSCI) and the Science Citation Index (SSCI). Besides that, Possibly not including more recent journals in one of the Science database, further searches were carried out to find newly published papers on newly published papers. Furthermore, given the interdisciplinary nature of AIED, relevant research is frequently published in more conferences on AI and learning science such as the Conference and Workshop on International Conference on Learning Representations (ICLR), International Conference on Machine Learning (ICML), International Conference of the Learning Sciences, and Neural Information Processing Systems (NEURIPS) (ICLS).
B. Criteria for Searches and Selection
The source database was searched several times and carried out employing a variety of essential crucial phrase assemblages and search tactics, such as "AI," "artificial intelligence," and "education.". Two non-English articles and 27 duplicates were eliminated during the first screening of the 307 items that were produced after 18 rounds of Web of Science searches. Furthermore, looks for supporting information to study screening were undertaken on the websites. To achieve the goals of the research, first step were taken to sort the papers according to their year of publication. Second step was taken to find subdomains that were mainly covered in the papers. It was also discovered that the most recent AIEd research can be separated into four major subdomains, as illustrated in fig 2. which were chosen as research and study subjects for the paper .In order to meet the research objectives, a set of inclusion and exclusion criteria were developed and applied. For subsequent analysis, only English language refereed journal papers providing empirical, evidence-based investigations were chosen. The methodology for the paper analysis and selection process is given below in With good reason, recent AIED research has prioritised teachers over other educational institution stakeholders.
No matter the type of learning environment-face-to-face or online-teachers are at the centre of it. Utilizing participatory design methodologies, new AIED technologies are developed with teachers, parents, and students in mind. In order to empower teachers and provide them the freedom to concentrate on teaching rather than other responsibilities, educators have long faced the difficulty of reducing teacher burden.
Teachers will need to make adjustments more frequently as a result of the widespread emphasis on online learning and the creation of new tools to support it. It is essential for teachers to upgrade and retrain themselves in order to adapt to this generation, particularly the new skills they must pick up in order to fully profit from AIED. In order to understand, evaluate, and adapt to new educational technological tools as they become available, people must first become digitally literate. They might or might not use these tools, but it's important to know what they offer and whether or not they help instructors to reduce their burden. Zoom video calling was frequently used to deliver classes remotely throughout the outbreak. In addition to setting up courses on Zoom, teachers also need to understand how to use the breakout rooms for group work and the whiteboard for freehand writing. In order to evaluate the data supplied by these ed-tech tools and to decide what kind of data and analytics tool they will need to better understand students, teachers will also need to develop their analytical skills. As a result, teachers will be able to buy only the ed-tech products they require, which will lighten their workload. Finally, in order to incorporate new tools into their daily practises, educators will need to master new collaborative, group, and management abilities. They will be responsible for efficiently managing these extra resources.
In recent studies, Talk Moves [1], an application designed to give teachers feedback and incite fruitful dialogues in their maths courses, was introduced. This application employs models such as BERT and LSTM paired with BOW with Glove to provide individualised feedback based on recordings. Another study introduced a Neural Geometric Solver [2] that is capable of automatically solving geometric problems trained using a dataset named GeoQA; the study also proposed its usage of jigsaw location prediction, geometric elements prediction, and knowledge points prediction using neural nets and LSTM as a decoder-encoder pair. The vast majority of work has also been completed for handwriting text validation and document conversion.
A study by Singh and Karayev [3] developed a model for automatic handwriting text identification from images and translation into text, all done in sequence using neural networks and transformers which may be used for easy document generation, including notes, and easy assignment verification. PREREQ [4] is a solution for Pre-Requisite Annotation and identifying proper background information when growing online education technology. It is a supervised learning strategy that efficiently learns the prerequisites for online educational courses from videos and other resources. OCR is a well-studied topic, but latex-based OCR is a new concept described in a recent study [5] that uses neural nets to directly output text from a latex-based document, which can help educators decode research papers and keynotes to text while saving time and effort. A brief overview of these works are given below in TABLE II.
B. Adaptive Curriculum
Based on their social background, level of financial security, mental stability, and prior knowledge of the subject, each learner has a unique learning environment. When instruction is tailored to these changing circumstances, it is most successful. The use of AIED can help identify a learner's specific learning gaps, suggest appropriate reading material based on those gaps, and offer detailed solutions to challenging issues. These tailored learning plans not only allow for simple development of students into higher level ideas, but also ensure that the fundamentals of each concept are carefully read and taken care of. Aside from that, tailor-made educational curriculum enables institutions to build more individualised programmes with greater student participation with the teacher, increasing the entire programme experience.
For instance, researchers created the open-source platform iTalk2Learn [6] to help students in grades 5 through 11 learn mathematics more effectively. This tutor spoke with the students, identifying when a student was having trouble with fractions and conducting the appropriate interviews.. Another latest study by Alkadi and Inkpen [7] established a model that classifies student reading materials based on their readability using multiple machine learning algorithms, which can help educators aid readers with different reading levels in selecting resources that fit their readability level. A different introduced making changes to general taxonomy books PDF2PreReq [8] introduced dynamic textbook algorithms that link necessary concepts with each topic and chapter, saving time and producing a better experience.
Student frustration can frequently lead to delays in concept formation and understanding. In order to address contemporary curriculum and interview settings, Betty's Brain [9], an interviewing agent, captures these irritation lapses. A brief overview of these works are given in below TABLE III. Analysis of various student frustration by use of virtual agent during studying
Self-Collected Dataset through student interviews
For Understanding perspective of student during study process and set curriculum accordingly
C. Intelligent Tutoring Technologies
A computer programme known as an intelligent tutoring system makes an effort to communicate with a human teacher in order to give students individualised instruction. The concept of ITS in AIED has existed for some time. High standards have always been set for ITS's capacity to support education. IT has been noticed a considerable gap between what ITS were promised to accomplish and what they've actually been instrumental in providing throughout the years.
The majority of ITS in recent years have been subject-and subject-focused, like ASSISTments [10]. Each intelligent tutoring system focuses on a certain subject, however results has shown that they are effective at giving students relevant information, interacting with students, and improving students' academic achievement.
An Intelligent Question Answering System [11] was proposed in a different study. It combines intelligent systems for answering questions based on knowledge graphs, incorporates big data technology, and uses them to quickly and accurately answer questions from high school students while connecting the knowledge points that are relevant to the questions. It also analyses students' questioning behaviour and anticipates student learning behaviour to offer information on the impact of instruction.
In a work that combined BASEBERT models with domain adaptive pre training, a generic language model [12] for question-answering was proposed that surpassed simultaneously with the most recent state-of-the-art model. A neural model [13] with visual attention was developed in a different study, and it can be trained to learn how to mark up a mathematical formula in LaTeX from its image. This paradigm can be used to teach computers various coding methods. A brief overview of these works are given in below TABLE IV. TABLE IV.
AN OVERVIEW OF WORKS COVERED UNDER INTELLIGENT TUTORING SYSTEMS
Name of the Work Approach Dataset Use Cases
ASSITments [10] Intelligent Ecosystem of teachers and students to provide assessments and assistance together
Self-made dataset of individual modules with Q/A, videos etc.
Provides Immediate feedback to student's on their work and analyses data to students.
The creation and study of an intelligent question-answering system. [11] Knowledge graph-based big data system for answering questions
D. Smart Assessments
Any assessment of a student's work or performance in a classroom (or any type of judgement or evaluation) is referred to as assessment . Assessments, together with curriculum, learning, and teaching, are described as one of the three pillars of schooling by Hill and Barber [28]. Today's generation exams are designed to evaluate students' knowledge, comprehension, and skills. The best tests assessment platforms would consider the whole spectrum of student skills and offer useful data on learning outcomes. However, each learner and their learning process are unique. One issue that is brought up in relation to more general notions of educational evaluation is how standardised assessments may be utilised to evaluate each student, who has unique capacities, passions, and expertise.
Researchers at UCL Knowledge Lab created an intelligent assessment tool called AI Assess as an illustration [14]. Three models-the knowledge model, the analytics model, and the student model-were used to assess students' math and science learning. Each topic's information was stored in the knowledge component, student progress was monitored using the student model, and interactions between students were examined using the analytics component. Similar to this, Samarakou et al. developed an AI assessment tool [15] that evaluates students qualitatively in order to relieve teachers of the burden of spending hours analysing each activity. Applying ML methods such as RL, semantic analysis, voice recognition, and NLP can raise the calibre of assessments conducted with these technologies.
Another study published by Huang and Li [16] introduced BERTEdu, a Chinese education-based pre-trained language model that discovers similar tasks based on input by capturing semantics, diverse texts, and formulas from the exercise. A recent study uses Knowledge Tracing to propose a broad framework [17] that takes into account student engagement, the amount of difficulty of instructional activities, and natural language processing embeddings of each concept's text and predicts future performance of a student using neural nets. A brief overview of these works are given in below TABLE V.
IV. DISCUSSION
The field of AI for education must continue to advance in order to benefit not only education (increasing inclusivity, helping students grasp complex ideas, and increasing accessibility), but also the development of reasoning-capable AI systems. The development of AI systems that support human capacities should be considered in addition to the development of autonomous AI systems. To construct robust AI systems in the field of education, study must be done in the following areas: (a)Similar to the computer vision field, we need better benchmarking datasets such as ImageNet[39], COCO [40]. Curation of consensus benchmark dataset will provide researchers with a consistent baseline for their model, allowing them to enhance their model further. (b) We must create AI systems whose thought processes are transparent, i.e., the systems must be able to articulate their thought processes and have those processes be understandable by humans. (c) Many STEM fields require the capacity to digest knowledge from various modalities, including texts and visuals. As a result, much development in multimodal DL is required. (d)In the past, symbolic approaches to stem fields produced positive outcomes. Their key strength is their ability to reason in an interpretable manner. However, they are not scalable. So much research is needed to figure out how to combine ML and symbolic methods to AI, i.e. near symbolic AI that can ensure better AI for Education systems.
In order to examine AIED over longer time periods and at the institutional, regional, and national levels, research must be broadened. Utilizing cutting-edge technology like text mining, learning analytics, and data visualisations is also required to progress AIED research. Emerging educational research approaches are ideal for investigations on revolutionary technology like AIED, in particular educational design research (EDR), are strongly advised because they enable educators to integrate their research inquiries as part of the technology development and implementation cycle in real-world settings. When educators help with the competition, development, or evaluation of AI technology for educational objectives, EDR can be particularly beneficial. Among the range of AIEDs accessible, some have drawn more attention from researchers than others, For example, the review discovered that while very few academic publications mentioned the use of chatbots or ML in educational field, the majority of studies concentrated on intelligent tutors or personalised learning environments. Therefore, more AIED technologies should be covered in future research, especially ones that have gotten less attention over time.
V. CONCLUSION
The motive of the paper was to highlight key technologies in AI that will create a great impact in the educational space by filling the gaps that persist in today's education space. The main highlights of the paper are to make readers cognizant of various problems that exist in today's education system, what technologies exist in today's AI that can fill those gaps, and in what manner today's AI technologies lack the capability to build nextgeneration AI educational tools. An in-depth analysis of the literature was used as part of a qualitative research study. AI in education has great potential and has the ability to completely revolutionize the educational space but its full potential has not been fully realized. Various stakeholders like technocrats, politicians, teachers, students, and others should collaborate on ideas to harness AI's full potential in the educational space. As AI in education can have both positive and negative side effects so the various stakeholders should correct calculations about its trade-off before deploying AI technologies in the real world. The main aim of AI is not to replace various stakeholders but to empower them. With proper policy making and careful implementation AIEd is going to prove an game changer.
VI References
Fig 1 .
1The Methodology of Paper Analysis and Selection Process
TABLE I .
ISELCTED SUB DOMAINS OF APPLICATIONS OF AI FOR EDUCATION CHOSEN FOR THIS STUDYSub Domains of Work
Purpose of Working
Teacher's / Instructor's Workload Decrement
To lessen the amount of work teachers have to do without
affecting student learning
Adaptive Curriculum
On the basis of students' contexts and learning histories, offer
them personalised and/or customised learning experiences.
Intelligent Tutoring Systems
Construct educational scenarios that engage students, offer
individualised feedback, and improve their comprehension of
particular subjects.
Smart Assessments
Increasing understanding of students which covers not just
what people are cognizant about, but also how they learn and
which pedagogies are most effective for them, thanks to the
use of adaptive assessment.
III. AI for Education Applications
Virtually endless options in education fields are made possible by AI technology. The following categories of
learning technology were examined by the twenty studies as part of a range of educational AI applications.
A. Teacher's/ Instructor Workload Decrement( [1] [2] [3] [4] [5] )
B. Adaptive Curriculum ( [6] [7] [8] [9])
C. Intelligent Tutoring Systems([10] [11] [12] [13] )
D. Smart Assessments([14] [15] [16] [17])
A. Teacher's / Instructor's Workload Decrement
TABLE II .
IIAN OVERVIEW OF WORKS COVERED UNDER INSTRUCTOR'S WORKLOAD DECREMENTName of the Work
Approach
Dataset
Use Cases
Talk Moves Application[1]
Uses SOA-NLP techniques to
provide feedback to teachers
Self-Collected Datasets with
175757 sentences of each type
Can be used for providing
new method for feedback
collection
GeoQA[2]
LSTM and ResNet-101 as
encoder-decoder for solving
geometric questions
GeoQA, a dataset with 4998
geometric issues, answers
questions about geometry.
Can be Used for doubt
clearing and reduce teacher's
workload
Full Page Handwriting
Recognition[3]
Neural Network based
Handwritten Text Recognition
model
IAM dataset-Benchmark for
handwriting recognition
Can be used for conversion of
handwritten textual
information into digital format
or vice-versa
PREREQ[4]
Supervised Learning Method
for finding concept wise pre-
requisite relations
University Course
Dataset(Benchmark Dataset) and a
new self-created NPTEL MOOC
Dataset
Can be used for teacher's to
easily find pre requisites for
each concept and create class
discourse accordingly
What you Get is What you see [5]
Deep Learning System for
OCR for latex based
documents
IM2LATEX-100K dataset with
rendered mathematical
expressions from published papers
Used by teachers to easily
create resources from latex
based documents.
TABLE III .
IIIAN OVERVIEW OF WORKS COVERED UNDER ADAPTIVE CURRICULUMName of the Work
Approach
Dataset
Use Cases
iTalk2Learn[6]
Smart Curriculum Based
Maths Tutoring
Not Available
For Adaptive Curriculum
setting based on student' need
and expertise to improve
interest and knowledge
Classifying Documents based on
Multiple Readability Levels[7]
Deep Learning based system
for Classification of articles
Newsela Dataset
For personalized article
recommendation based on
expertise
PDF2PreReq [8]
End to End pipeline for
generating dependency graphs
for existing curriculum
Not Available
For finding dependency on
underlying concepts and smart
curriculum setting
Affect-Targeted Interviews for
Understanding Student
Frustration[9]
TABLE V .
VAN OVERVIEW OF WORKS COVERED UNDER SMART ASSESMENTSName of the Work
Approach
Dataset
Use Cases
AIAssess[14]
AI based assessment system
which provides adaptive tasks
based on student's progress
Not Available
For Adaptive Curriculum
setting based on student' need
and expertise to improve
interest and knowledge
Implementation of AI
Assessments in Engg. Lab
Education[15]
Cognitive theory based
assessments with feedback
ensuring qualitative evaluation
Experimentative MATLAB course
based dataset
For providing improvised
evaluation and testing in
diverse coursework.
An Empirical Study of Finding
Similar Exercises [16]
BERT based model to find
similar exercises in various
fields of education
Self-Created Dataset involving
education based corpus
For finding similar exercises
and enabling easy assessments
setting
Deep Knowledge Tracing using
Temporal Convolutional
Networks[17]
Language and neural network
based model for tracing
knowledge of students in
assessments
Algebra 2007-2008 Dataset
For planning and setting
assessments based on
students' knowledge level and
improve setting process.
Using Transformers to Provide Teachers with Personalized Feedback on their Classroom Discourse: The TalkMoves Application. Abhijit & Suresh, Jennifer & Jacobs, Lai, & Vivian, Tan, & Chenhao, Ward, & Wayne, James & Martin, Tamara Sumner, 10.48550/arXiv.2105.07949Suresh, Abhijit & Jacobs, Jennifer & Lai, Vivian & Tan, Chenhao & Ward, Wayne & Martin, James & Sumner, Tamara. Using Transformers to Provide Teachers with Personalized Feedback on their Classroom Discourse: The TalkMoves Application.(2021). https://doi.org/10.48550/arXiv.2105.07949
GeoQA: A Geometric Question Answering Benchmark towards Multimodal Numerical Reasoning. Jiaqi Chen, Jianheng Tang, Jinghui Qin, Xiaodan Liang, Lingbo Liu, Eric P Xing, Liang Lin, 10.48550/arXiv.2105.14517Jiaqi Chen, Jianheng Tang, Jinghui Qin, Xiaodan Liang, Lingbo Liu, Eric P. Xing, Liang Lin.GeoQA: A Geometric Question Answering Benchmark towards Multimodal Numerical Reasoning. (2022) .https://doi.org/10.48550/arXiv.2105.14517
Full Page Handwriting Recognition via Image to Sequence Extraction. S Sumeet, Sergey Singh, Karayev, 10.48550/arXiv.2103.06450Sumeet S. Singh, Sergey Karayev,Full Page Handwriting Recognition via Image to Sequence Extraction, (2022), https://doi.org/10.48550/arXiv.2103.06450
Inferring Concept Prerequisite Relations from Online Educational Resources. S Roy, M Madhyastha, S Lawrence, V Rajan, AAAI. 3301S. Roy, M. Madhyastha, S. Lawrence, and V. Rajan, "Inferring Concept Prerequisite Relations from Online Educational Resources", AAAI, vol. 33, no. 01, pp. 9589-9594, (2019).
What You Get Is What You See: A Visual Markup Decompiler. Y Deng, A Kanervisto, A M Rush, abs/1609.04938ArXiv. Deng, Y., Kanervisto, A., & Rush, A.M.What You Get Is What You See: A Visual Markup Decompiler. (2016). ArXiv, abs/1609.04938.
Talk, tutor, explore, learn: intelligent tutoring and exploration for robust learning. B Grawemeyer, S Gutierrez-Santos, W Holmes, M Mavrikis, N Rummel, C Mazziotti, R Janning, AIEDMadridGrawemeyer, B., Gutierrez-Santos, S., Holmes, W., Mavrikis, M., Rummel, N., Mazziotti, C., Janning, R.: Talk, tutor, explore, learn: intelligent tutoring and exploration for robust learning, p. 2015. AIED, Madrid (2015)
Classifying Documents to Multiple Readability levels. Alkadi, Inkpen, AAAI2021 Spring Symposium on Artificial Intelligence for K-12 Education. Alkadi.,InkPen.,Classifying Documents to Multiple Readability levels., In AAAI2021 Spring Symposium on Artificial Intelligence for K-12 Education,(2021)
Pdf2PreReq: Automatic Extraction of Concept Dependency Graphs fromAcademic Textbooks. Rushil Thareja, V Venktesh, Mukesh Mohania, AAAI2022 Artificial Intelligence for Education. Rushil Thareja, Venktesh V and Mukesh Mohania. Pdf2PreReq: Automatic Extraction of Concept Dependency Graphs fromAcademic Textbooks. In AAAI2022 Artificial Intelligence for Education,(2022)
Affect-Targeted Interviews for Understanding Student Frustration. Ryan S Baker, Nidhi Nasiar, Jaclyn L Ocumpaugh, Stephen Hutt, Juliana M A L Andres, Stefan Slater, Matthew Schofield, Allison Moore, Luc Paquette, Artificial Intelligence in Education: 22nd International Conference. Anabil Munshi, and Gautam BiswasRyan S. Baker, Nidhi Nasiar, Jaclyn L. Ocumpaugh, Stephen Hutt, Juliana M. A. L. Andres, Stefan Slater, Matthew Schofield, Allison Moore, Luc Paquette, Anabil Munshi, and Gautam Biswas.Affect-Targeted Interviews for Understanding Student Frustration. In Artificial Intelligence in Education: 22nd International Conference(2021)
The ASSISTments Ecosystem: Building a Platform that Brings Scientists and Teachers Together for Minimally Invasive Research on Human Learning and Teaching. N T Heffernan, C L Heffernan, 10.1007/s40593-014-0024-xInt J Artif Intell Educ. 24Heffernan, N.T., Heffernan, C.L. The ASSISTments Ecosystem: Building a Platform that Brings Scientists and Teachers Together for Minimally Invasive Research on Human Learning and Teaching. Int J Artif Intell Educ 24, 470-497 (2014) https://doi.org/10.1007/s40593-014-0024-x
Design and Research of Intelligent Question-Answering(Q&A) System Based on High School Course Knowledge Graph. Zhijun Yang, Yang Wang, Jianhou Gan, Hang Li, Ning Lei, 10.1007/s11036-020-01726-wMob. Netw. Appl. 265Zhijun Yang, Yang Wang, Jianhou Gan, Hang Li, and Ning Lei. 2021. Design and Research of Intelligent Question- Answering(Q&A) System Based on High School Course Knowledge Graph. Mob. Netw. Appl. 26, 5 (Oct 2021), 1884-1890. https://doi.org/10.1007/s11036-020-01726-w
MathBERT: A Pretrained Language Model for General NLP Tasks in Mathematics Education. Jia Tracy Shen, Michiharu Yamashita, Ethan Prihar, Neil Heffernan, Xintao Wu, Ben Graff, Dongwon Lee, 10.48550/arXiv.2106.07340NeurIPS 2021 MATHAI4ED Workshop. Jia Tracy Shen, Michiharu Yamashita, Ethan Prihar, Neil Heffernan, Xintao Wu, Ben Graff, Dongwon Lee. MathBERT: A Pre- trained Language Model for General NLP Tasks in Mathematics Education,In NeurIPS 2021 MATHAI4ED Workshop.(2021). https://doi.org/10.48550/arXiv.2106.07340
Teaching Machines to Code: Neural Markup Generation with Visual Attention. Sumeet Singh, 10.48550/arXiv.1802.05415Singh, Sumeet.Teaching Machines to Code: Neural Markup Generation with Visual Attention. (2018). https://doi.org/10.48550/arXiv.1802.05415
Towards artificial intelligence-based assessment systems. Rosemary Luckin, 10.1038/s41562-016-0028https:/doi.org/10.48550/arXiv.1310.3174Nature Human Behaviour. Luckin, Rosemary. (2017). Towards artificial intelligence-based assessment systems. Nature Human Behaviour. 1. 0028.10.1038/s41562-016-0028https://doi.org/10.48550/arXiv.1310.3174
Implementation of artificial intelligence assessment in engineeringlaboratory education. M Samarakou, E Fylladitakis, P Prentakis, S Athineos, Samarakou, M., Fylladitakis, E., Prentakis, P., Athineos, S.: Implementation of artificial intelligence assessment in engineer- inglaboratory education. https://files.eric.ed.gov/fulltext/ED557 263.pdf (2014).
An Empirical Study of Finding Similar Exercises. Tongwen Huang, Xihua Li, 10.48550/arXiv.2111.08322Tongwen Huang, Xihua Li (2021) An Empirical Study of Finding Similar Exercises. https://doi.org/10.48550/arXiv.2111.08322
Deep Knowledge Tracing using Temporal Convolutional Networks. Ait Khayi, N , Proceedings of the Workshop ArtificialIntelligence for Education (IJCAI 2021. the Workshop ArtificialIntelligence for Education (IJCAI 2021Ait Khayi, N. Deep Knowledge Tracing using Temporal Convolutional Networks. Proceedings of the Workshop ArtificialIntelligence for Education (IJCAI 2021)), (2021). Retrieved from https://par.nsf.gov/biblio/10290861.
| {'fraction_non_alphanumeric': 0.03482102056359482, 'fraction_numerical': 0.019984767707539985, 'mean_word_length': 5.093558566920364, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 11, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "The world around us has undergone a radical transformation due to rapid technological advancement in recent decades. The industry of the future generation is evolving, and artificial intelligence is the next change in the making popularly known as Industry 4.0. Indeed, experts predict that artificial intelligence (AI) will be the main force behind the following significant virtual shift in the way we stay, converse, study, live, communicate and conduct business. All facets of our social connection are being transformed by this growing technology. One of the newest areas of educational technology is Artificial Intelligence in the field of Education (AIEd). This study emphasis the different applications of Artificial Intelligence in education from both an industrial and academic standpoint. It highlights the most recent applications of AIEd, with some of its main areas being the reduction of instructors' burden and students' contextualized learning novel transformative evaluations, and advancements in sophisticated tutoring systems. It analyses the AIEd's ethical component and the influence of this transition on people, particularly students and instructors as well. Finally, the article touches on AIEd's potential future research and practices. The goal of this study is to introduce the present-day applications to its intended audience.", 'arxivid': '2301.10026', 'author': ['Shubham Ojha \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Siddharth Mohapatra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Aditya Narendra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Ipsit Misra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Shubham Ojha \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Siddharth Mohapatra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Aditya Narendra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n', 'Ipsit Misra \nCenter of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n\n'], 'authoraffiliation': ['Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n', 'Center of Excellence of Artificial Intelligence\nOdisha University of Technology and Research Bhubaneswar\n'], 'corpusid': 256194605, 'doi': '10.48550/arxiv.2301.10026', 'github_urls': [], 'n_tokens_mistral': 8407, 'n_tokens_neox': 7387, 'n_words': 4560, 'pdfsha': '7ddebbf3e8a324061bc1ba6659d8247a39fd7133', 'pdfurls': ['https://export.arxiv.org/pdf/2301.10026v1.pdf'], 'title': ['From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)', 'From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)', 'From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)', 'From Robots to Books: An Introduction to Smart Applications of AI in Education (AIEd)'], 'venue': []} |
arxiv |
A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories
December 12, 2022 8 Dec 2022
Javier Fuentes-Martín
Departamento de Física Teórica y del Cosmos
Universidad de Granada
Campus de FuentenuevaE-18071GranadaSpain
PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics
Johannes Gutenberg University
D-55099MainzGermany
Matthias König
Physik Department T31
Technische Universität München
James-Franck-Str. 1D-85748GarchingGermany
Julie Pagès
Department of Physics
University of California at San Diego
9500 Gilman Drive, La Jolla92093-0319CAUSA
‡ Anders
Eller Thomsen §[email protected]¶[email protected]
Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics
University of Bern
CH-3012BernSwitzerland
§
Felix Wilsch
Physik-Institut
Universität Zürich
CH-8057ZürichSwitzerland
A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories
December 12, 2022 8 Dec 2022
Studying the impact of new-physics models on low-energy observables necessitates matching to effective field theories at the relevant mass thresholds. We introduce the first public version of Matchete, a computer tool for matching weakly-coupled models at one-loop order. It uses functional methods to directly compute all matching contributions in a manifestly gauge-covariant manner, while simplification methods eliminate redundant operators from the output. We sketch the workings of the program and provide examples of how to match simple Standard Model extensions. The package, documentation, and example notebooks are publicly available at https://
Introduction
The advent of the LHC heralded a new era for beyond-the-Standard-Model (BSM) physics.
With the discovery of the Higgs boson and no direct signs of new resonances, we see indications of a mass gap up to the scale of yet-to-be-discovered new physics (NP). The focus of the community is shifting to precision flavor and electroweak physics in order to search for indirect signs of new particles and potentially probe scales far beyond the reach of resonance searches. The result has been a renaissance of Effective Field Theories (EFTs) applied to BSM physics often using the Standard Model Effective Theory (SMEFT), whose basis was first determined after the LHC went into service [1]. The use of EFTs goes all the way back to Fermi's theory and has long since reached maturity within the Standard Model (SM), facilitating SM predictions for many precision observables. Now, new methods are rapidly being developed for BSM physics with an aspiration of reaching a similar level of maturity. The new challenge to achieving this goal is the need for a near-complete level of generality, as the nature of NP has yet to be revealed.
To determine the low-energy effects of high-scale NP, one typically has to perform sequential matching to consecutive EFTs at the relevant mass thresholds and renormalization group (RG) running between these scales. In the absence of any light new particles, the running and matching machinery is already available to handle computations below the NP mass threshold: the one-loop RG equations in the SMEFT [2][3][4][5], the matching to the Low-Energy Effective Theory (LEFT) at the weak scale [6][7][8], and the LEFT RG equations [9] have been determined and even implemented in computational tools [10][11][12]. Many tools are also available for phenomenological analyses of theories within the SMEFT and LEFT frameworks [13][14][15][16][17][18][19][20][21][22][23][24].
The sticking point for a long time has been performing the matching computation of BSM models to their EFTs. Although it is tempting to think of the target EFT as the SMEFT, we should bear in mind that realistic BSM constructions can contain a rich NP sector spanning large ranges of energy scales, calling for intermediate-scale EFTs. Alternatively, the presence of additional light fields, for example axion-like or dark-matter particles, demands extensions of the SMEFT (see, e.g., [25][26][27][28]). The unclear nature of both the UV model and the target EFT makes matching a formidable task. Functional methods promise a direct approach to the problem [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. They entirely circumvent the matching of individual amplitudes and produce the EFT Lagrangian directly, albeit unsimplified, without requiring any prior knowledge about its structure or symmetries. The method has produced general results in the form of the Universal One-Loop Effective Action [49][50][51][52][53], several tools to assist part of the matching computations [54][55][56][57], and has been used for a number of simple BSM models [58][59][60][61][62]. Nevertheless, the package we present here represents the first truly automated, end-to-end one-loop matching tool based on functional methods, making it competitive with the diagrammatic matchmakereft [63] but with the advantages of the functional approach. Thanks to these new tools, fast and competent matching requiring little more than the press of a button is finally becoming feasible. Not only that, matching tools can easily be repurposed to compute RG equations for other EFTs, as both types of computations require the evaluation of loop integrals in the hard region.
Here we introduce a first public, proof-of-concept version (v0.1.0) of the Mathematica package Matchete-Matching Effective Theories Efficiently-to solve the problem of matching weakly coupled UV models to their EFTs at the relevant mass thresholds. It uses functional methods [43][44][45], which facilitates direct matching without the need for specifying a target basis for the EFT. This feature is especially useful in theories that match into EFTs other than the SMEFT or when extending EFT matching beyond dimension-six operators. The automated application of these methods was previously demonstrated by the authors in the SuperTracer package [56], which Matchete supersedes. Furthermore, we make significant headway with the challenging task of automatically simplifying the EFT Lagrangian to an on-shell basis. The design of the package includes a simple and user-friendly interface that considerably simplifies the user input while still allowing for very generic implementations. In essence, the user can write down the Lagrangian in a Mathematica notebook, in manner that is very close to a pen-and-paper form, and leave the rest to the package. While there are still many features and capabilities that we would like to implement over the next years, this proof-of-concept release already represents a major leap in the development of (functional) matching tools and can greatly assist many matching computations, including those in multiple BSM scenarios. The limitations in this release are reflected in the discussion of the future prospects of the package in Section 4. This paper is meant as a short introduction to give a flavor of the first public version of Matchete. The paper contains a brief description of the underlying package structure and gives some hands-on examples of how to use it. It is not a comprehensive guide to the use of the program. For more detailed instructions, the user is encouraged to consult the documentation notebook included with the distribution. Section 2 presents the organization and use of the package in broad scopes, touching on the specific methods used in the computation. This The user has to specify the (gauge) groups, fields, and couplings of a UV model before writing down the Lagrangian. This can be passed to functions for EFT matching at tree-and one-loop level. Simplification methods with identities and field redefinitions can then reduce the EFT Lagrangian.
is followed with some concrete usage example in Section 3, including simple extensions of the SM, to give the reader a feel for the practical applications. We conclude the paper in Section 4 with a short discussion of the direction of future package developments.
Organization of the Package
Matchete is organized around Lagrangians and operators, which are the objects the user will interact with in the workflow illustrated in Figure 1. With standard functional methods, a UV Lagrangian is matched to an EFT Lagrangian at tree-and one-loop level. However, by nature of the functional approach, the EFT Lagrangian is not simplified and contains many redundant operators that need to be reduced to a basis. A core part of Matchete consists of powerful methods for simplifying the matching result to a (near-)basis to bridge the gap to a useful EFT Lagrangian. Various functions are also available for manipulating the output in various ways and identifying individual contributions. The simplification methods do not handle evanescent contributions yet, so the output is in a d-dimensional basis, which is redundant in a physical renormalization scheme [64].
Model setup and internal representations
Much of the user experience invariably concerns the input of models into the program. First, one must specify what (gauge) groups and representations are available for objects to transform under before one can specify what manner of fields and couplings are involved. Only then can a Lagrangian be written down. To achieve this, Matchete contains specific methods for Lorentz contractions and Dirac algebra.
Symmetry groups
All manner of group invariants show up in quantum-field-theory computations. These can be as simple as Kronecker deltas or generators of a representation, or they can be much more complicated once more exotic representations are involved. We will refer to all such invariant tensors as Clebsch-Gordan (CG) coefficients in line with the well-known SU(2) case. We are unaware of analytic rules for evaluating generic contractions of CG coefficients that apply to all cases, so a more constructive approach is used in Matchete.
Matchete contains a module for handling all things related to group and representation theory, to allow for generic gauge and/or symmetry groups. Upon specifying a simple Lie group, the module can determine the weights, dimensions, and other information regarding the representations with standard methods. 1 To determine CG coefficients, which describe how to combine weights from multiple representations in an invariant manner, we implemented the algorithm of Refs. [66,67], which casts the problem in terms of linear algebra. With this method, the program can explicitly construct the CG tensors.
When the user specifies a simple (gauge) group to be included in a model, Matchete automatically generates several common representations and CG coefficients and more can be initialized by the user with build-in routines. The CGs are referred to symbolically for all input and output purposes. However, when contracting CG objects, the symbols are replaced internaly with numerical tensors, contracted, and, finally, projected to a basis of CG coefficients. This procedure allows for efficient evaluation of CG products, with a minimum of inconvenience to the user.
Fields and couplings
All objects in a Lagrangian have properties associated with them that are necessary for determining what algebraic manipulations are possible. The field and coupling objects appearing in Matchete Lagrangians carry most of this information with them as they are passed along to various routines. Although concise, the amount of information contained in a Lagrangian, or even an operator, is considerable, which can be useful for careful manipulations of the output should the user so desire. In most common cases, the user will want to exploit the convenience of the notebook format to view the output in a more legible form, and build-in routines allow the user to print Matchete objects in the format of a regular textbook.
Despite all the information contained in field objects, users can simply refer to them with their label (name) and the indices they might have. All that is required for the user is to define the properties of a field: spin, mass, flavor and gauge representations, and whether if it self-conjugate (real for scalars, Majorana for fermions). Gauge fields are even easier to implement, as they are automatically defined with their gauge group. Couplings, similarly, need to be defined beforehand to specify their flavor indices and mass dimensions.
Matching step
The input UV Lagrangian is matched to an EFT under the assumption that all heavy masses are of the same order M a ∼ Λ (otherwise, the matching will have to be performed sequentially accounting for RG running), which sets the heavy scale of the problem. This allows for arranging the EFT as a double expansion in the heavy scale and the loop order:
L EFT = =0 n=4 (4π) 2 Λ n−4 L ( ,n) EFT .
(2.1)
Matchete features routines for computing L EFT . There is no fundamental obstacle preventing the evaluation of higher-dimensional terms with the current implementation, although limits of computing power make pushing beyond dimension-six for one-loop terms time intensive. In practice, the mass expansion is performed in terms of the light dimensions, counting the canonical dimension of light fields, covariant derivatives, and light masses/dimensional couplings.
Tree level
Matching at tree level comes down to solving the equations of motion (EOMs) of the heavy fields as it is commonly done by hand. This approach has also been applied to automated tree-level matching in MatchingTools [54]. Schematically, with heavy fields Φ a and light fields φ, the UV action is
S UV [Φ, φ] = x 1 2 Φ a ∆ −1 ab (D, M )Φ b +S int ,
where ∆ −1 is the appropriate kinetic operator for the heavy fields and S int is the interacting part of the action (including both heavy and light fields). The solution to the heavy-field EOMs in the presence of light fields readsΦ
a [φ] = −∆ ab δS int δΦ b Φ [φ], φ . (2.2)
Matchete is equipped with routines to take functional derivatives of the action after which this equation can be solved iteratively order by order in the mass expansion. With the solution in hand, the tree-level EFT is given by
S ( =0) EFT [φ] = S UV Φ [φ], φ . (2.3)
An efficient truncation of higher-order terms ensures excellent performance of this method.
One-loop level
One-loop contributions to the EFT encode the high-energy components of one-loop effects in the UV theory. In the functional formalism, there is but a single functional topology at one-loop order, which is captured by a supertrace-a generalization of the functional trace that accounts for the presence of mixed bosonic and fermionic objects. The key object to consider is the fluctuation operator
δ 2 S UV δη j δη i Φ [φ], φ = δ ij ∆ −1 i − X ij , η = (Φ, φ),(2.4)
where again ∆ −1 denotes the kinetic operator and X ij are interaction terms. The master formula for the one-loop matching in terms of these objects is
S ( =1) EFT = i 2 STr ln ∆ −1 hard − i 2 ∞ k=1 1 k STr (∆X) k hard . (2.5)
Here, hard indicates that loop integrands are expanded around loop momenta q ∼ Λ, following the method of expansion by regions [68,69]. This form allows for a straightforward counting of light mass dimensions, and the resulting series can be truncated at the relevant order in the mass expansion.
For the actual computation of Eq. (2.5), we follow the implementation outlined in Ref. [56] based on the developments of Refs. [43][44][45]. The procedure allows for simultaneous treatment of all particle spins and mixed heavy and light states in the loop. Furthermore, the traces can be evaluated in a manifestly gauge covariant manner using the Covariant Derivative Expansion (CDE) [29][30][31]. Altogether, the method allows for a very algorithmic and efficient approach to evaluating all loop contributions simultaneously. Moreover, it remains possible to pinpoint specific contributions based on the fields propagating in the loops by targeting specific supertraces.
Simplifications
Properly simplifying the output Lagrangian is a challenge related to the long-standing problem of finding a basis for the higher-dimensional operators of an EFT. We distinguish between simplification with exact identities (integration-by-parts and group identities, and commutation relations), taking the Lagrangian to the Green's basis, and using field redefinitions to produce a simplified Lagrangian with on-shell equivalence. The exact simplification relates operators linearly, and can be applied to individual operators as well as the full EFT Lagrangian. On the other hand, field redefinitions work non-linearly and make sense only when acting on the EFT Lagrangian as a whole.
Green's basis
To reduce EFT Lagrangians to a Green's basis, we use methods from linear algebra, as this allows for efficient and robust simplifications. One can think of L EFT as an element in the vector space O equipped with a basis {O a } consisting of all operators in the absence of any exact identities. That is, the elements of this basis span the complete set of gauge and Lorentz-invariant monomials of the fields, their covariant derivatives, and CG coefficients (including Dirac matrices). This vector space is redundant once the exact identities, relating the basis operators, are accounted for. Each identity relation can be represented as a vector that is equivalent to 0. Together, the identity vectors span a subspace I ⊆ O, and we can identify the coset O/I with the set of Green's basis Lagrangians. Simplifying L EFT then comes down to finding a convenient basis for O/I and determining the representative element of the equivalence class [L EFT ] defined by the coset.
To arrive at the Green's basis in Matchete, we employ the following strategy: For all basis elements (operators) we encounter, we generate the complete set of possible identities using integration by parts, Jacobi identities, commutation of covariant derivatives, and gamma matrix identities (such as γ µ γ ν = g µν − iσ µν ). Denoting the vectors in O corresponding to the resulting identities by I n , it follows that I = span {I n } . The operator basis of O allows for the decomposition
I n = a M na O a ,(2.6)
which in turn defines a matrix M with the coordinate vectors of the identities as its rows.
With standard methods, M is brought to reduced row echelon form M , and we observe that the non-zero rows describe a basis (in coordinate space) for I. Conveniently, the first "1" in each non-zero row of M effectively picks out a set of redundant operators {O r } r∈R , which can be eliminated in the EFT Lagrangian. The rows of M , thus, describe a set of identities for the equivalence classes:
O r + b / ∈R M rb O b = [0], ∀r ∈ R. (2.7)
Using these identities, all {O r } r∈R can be eliminated from the representative element of [L EFT ], that is, the Green's basis Lagrangian. By absorption of gauge couplings into gauge fields, all entries in M are numbers, allowing for efficient matrix manipulations of M . The main challenge to implementing the simplification procedure described above is that of identifying identical operators based on their internal representation. To this end, Matchete relies heavily on pattern matching to identify, e.g., different labeling of the dummy indices, permutations of indices on symmetric tensors, and orderings of terms in products. By choosing an ordering of the basis {O a }, it is possible to dictate a preference as to what operators are considered redundant, that is, in {O r } r∈R . While the choice is somewhat arbitrary, we can ensure that the maximal number of operators that can be removed with field redefinitions are kept in the basis. Additional requirements are enforced to ensure that the ouput Lagrangian is manisfestly Hermitian.
The main limitation of our current approach is the need to hard-code all possible identities in Matchete. However, additional identities can be added in a modular manner. The initial version notably does not include Fierz identities, as the proper handling of these necessitate the evaluation of evanescent contributions [64]. This is something we expect to address in future updates. In any event, the lack of implementation of identities does not result in an invalid result from the simplification method, merely a non-minimal operator basis for L EFT as the full identity space I is not found.
Field redefinitions
After the simplifications outlined in the previous section have been performed, we are left with a Lagrangian that contains redundant operators that can be removed by field redefinitions. To classify these operators, we first define for each field type the object D (ψ) corresponding to the kinetic piece of the field EOM and whose definition is given in Table 1. Operators with at least one occurrence of D (ψ) can be removed from the Lagrangian employing a field redefinition of the field ψ. For operators at the highest power (that is, dimension-six Field Type Objects Definition Table 1: Definitions of the operator D (ψ) for the various field types. In the scalar and vector cases, operators acting on a complex-conjugated field follow straightforwardly from replacing the field with its complex conjugate in the definition. In the last line, A µν denotes the usual field-strength tensor associated with the vector field A.
Scalar ϕ D (ϕ) D µ D µ ϕ Dirac fermion ψ D (ψ) γ µ D µ ψ D ψ (D µψ )γ µ Majorana fermion η D (η) γ µ D µ ψ D η T (D µ η T )γ T µ Vector A D (A ν ) D µ A µν
operators when one is working up to dimensions six), such field redefinitions are equivalent to replacing the field EOM at leading power in the corresponding operator. In the presence of effective operators of different power-counting orders, this procedure misses some of the power-corrections to the lower-order operators and yields an incorrect result [70][71][72][73][74][75][76][77]. We, therefore, employ field redefinitions for the sake of generality. The general procedure is as follows: First, one identifies all instances of D (ψ) for all fields ψ appearing in the Lagrangian. Then one reads off the coefficient of the operators with of these EOM objects and performs a redefinition of the fields with these coefficients. For illustration, consider a real scalar Lagrangian of the form
L = 1 2 (D µ ϕ)(D µ ϕ) − 1 2 m 2 ϕ 2 + c Λ 2 ϕ 3 D µ D µ ϕ = 1 2 (D µ ϕ)(D µ ϕ) − 1 2 m 2 ϕ 2 + c Λ 2 ϕ 3 D (ϕ) .
(2.8)
The field redefinition ϕ → ϕ + c Λ 2 ϕ 3 removes the redundant operator when inserted into the kinetic term and produces a quartic operator when inserted into the mass term:
L → L = 1 2 (D µ ϕ)(D µ ϕ) − 1 2 m 2 ϕ 2 − c m 2 Λ 2 ϕ 4 . (2.9)
For complex fields as well as Majorana fermions, one reads off the coefficients of the conjugated EOM objects and averages over them. The complete list of field redefinitions for each field type is given in Table 2.
When eliminating a redundant operator with D (ψ), the field redefinitions will generate contributions only at higher order in the EFT power counting or at the same dimension but with fewer derivatives. In practice, therefore, one proceeds in an iterative fashion, seeking out redundant operators at the lowest order in the EFT counting and removing them by field redefinitions. The lowest-order operators that can appear here are kinetic-mixing terms at dimension four. At higher powers, the procedure needs to be repeated since operators may contain more than one D (ψ) object, and the redefinition removes only one occurrence. Once all redundant terms are removed at a given order in power-counting, the procedure is repeated at the next order, until no more redundant operators remain.
Field Type
Redundant operators
Field redefinition Special care needs to be taken in the case of Abelian gauge fields. Since removing kineticmixing terms between gauge fields amounts to complicated redefinitions of the charges under the associated gauge groups, we choose to keep them explicit. 2 Hence, in the presence of kinetic mixing, field redefinitions should be modified as follows: Consider a set of vector fields A i µ that exhibit kinetic mixing parameterized by a mixing matrix Z and a redundant operator involving the vector fields:
Real scalar ϕ χD (ϕ) ϕ → ϕ + χ Complex scalar φ χD (φ) + D φ † ∆ φ → φ + 1 2 (χ † + ∆) Majorana fermion η χD (η) + D η T ∆ η → η + iC(χ T − ∆) Dirac fermion ψ χD (ψ) + D ψ ∆ ψ → ψ − i 2 (χ + ∆) Real vector field A D (A µ ) χ µ A µ → A µ − χ µ Complex vector field A D (A µ ) χ µ + D A † µ ∆ µ A µ → A µ − 1 2 (χ † µ + ∆ µ )L = − 1 4 A i µν Z ij A j,µν + χ ν i D µ A i µν ≡ − 1 4 A i µν Z ij A j,µν + χ ν i D A i ν . (2.10)
The appropriate field redefinition is
A i µ = A i µ − (Z −1 · χ µ ) i .
As long as the deviation of Z from identity is perturbative (in either loop counting or EFT power-counting), its inverse can be easily obtained.
An important subtlety arises from matching corrections to the couplings of operators of mass-dimension lower than four. That is, mass terms and cubic scalar interactions. For example-as is famously the case with the Higgs boson in the SM-if heavy degrees of freedom coupled to light scalars are integrated out at loop level, the latter receive mass corrections proportional to the hard scale. This upsets the power-counting of the effective theory, since a mass term for a light scalar of the form
δL = − c 1 2 Λ 2 ϕ 2 ,(2.11)
is formally of dimension two in the EFT counting even if c 1 is loop suppressed. In this case,
Matchete introduces an effective coupling m 2 ϕ,eff ,
− 1 2 m 2 ϕ + c 1 Λ 2 ϕ 2 → − 1 2 m 2 ϕ,eff ϕ 2 ,(2.12)
that is treated as EFT dimension two, such that the term is of dimension four again. In doing so, the program assumes a (fine-tuned) cancellation between the tree-level mass and the loop correction when the power enhancement from Λ 2 is large enough to overcome the loop suppression from c 1 .
Conventions
In this section we clearly state the overarching conventions used in Matchete to prevent unnecessary confusion on the matter. For the metric we use the "mostly-minus" signature: g µν = diag(+1, −1, −1, −1). Meanwhile, we take the antisymmetric Levi-Civita tensor to satisfy ε 0123 = −ε 0123 = +1 while the chiral spinor projectors are P L = 1 2 (1 − γ 5 ) and P R = 1 2 (1 + γ 5 ). The covariant derivatives of the gauge groups are automatically generated and used throughout the package. They are defined by D µ = ∂ µ − igT a A a µ (note the sign) for non-Abelian groups with gauge field A µ , with g being the coupling g and T a the Hermitian generators, which normalize as tr[T a T b ] = 1 2 δ ab for fundamental representations. For the Abelian gauge groups, T a is replaced by the charge.
All computations are performed in dimensional regularization (DR) with spacetime dimension d = 4 − 2 . The renormalization scheme is MS in line with most BSM computations. The treatment of γ 5 is a point of contention in DR and fraught with potential errors. In the initial release, we employ naive dimensional regularization (NDR). Namely, we use the anticommuting γ 5 and impose the four-dimensional identity tr[γ µ γ ν γ ρ γ σ γ 5 ] = −4iε µνρσ .
(2.13)
Trace cyclicity is lost in this manner, but as long as the EFT computations follow the Diractrace reading point of the matching computation, all ambiguities cancel [56]. The source of the few ambiguities in the matching stem from IR divergences in loops with heavy and light fermions. In these cases, the Matchete reading points can be inferred, since it will always read these supertraces starting with a heavy fermion propagator. This is not a particularly elegant solution, and we plan to explore other approaches for handling γ 5 in future updates.
Using Matchete
The Matchete package is free software under the terms of the GNU General Public License v3.0 and is publicly available in the GitLab repository https://gitlab.com/matchete/matchete
The package can be installed in one of two ways:
i ) Automatic installation: The simplest way to download and install Matchete is to run the following command in a Mathematica notebook:
In[1]:= Import["https://gitlab.com/matchete/matchete/-/raw/master/install.m"]
This will download and install Matchete in the Applications folder of Mathematica's base directory.
ii ) Manual installation: The user can also manually download the package from the GitLab repository. In this case, the user has to specify the location of the downloaded package with 3
In[1]:= PrependTo[$Path,"directory"];
where directory is the path to the Matchete folder.
Once installed, the user can load Matchete in a fresh Mathematica kernel by running:
Vector-like fermion toy-model
To illustrate the use of Matchete with a simple but comprehensive example, we consider a variation of the toy model of Ref. [56] with a U(1) gauge symmetry, two charged vector-like fermions ψ and Ψ, and a real scalar singlet φ. The Lagrangian is given by
L = − 1 4 F µν F µν + 1 2 (∂ µ φ) 2 − 1 2 m 2 φ φ 2 +ψ i / D ψ +Ψ(i / D − M )Ψ − yψ L φ Ψ R + h.c. , (3.1)
where D µ ψ = ∂ µ ψ − ie A µ ψ (and similarly for Ψ). We take ψ to be massless, φ to have a light mass m φ , and Ψ to have a heavy mass M . The low-energy EFT, describing physics at energies much lower than M , is obtained by integrating out Ψ. We proceed to show how the matching is performed in Matchete. 4 As a first step, the user has to define all (gauge) symmetries of the theory. We define the U(1) symmetry of the present example, labeled U1e, by where the PlusHc routine automatically adds the Hermitian conjugate of its argument. With the NiceForm formatting, we can then verify that the Lagrangian does in fact agree with our expectations:
In[7]:= LUV //NiceForm Out[7]= - 1 4 A µν 2 + 1 2 (D µ φ) 2 - 1 2 mφ 2 φ 2 + i(ψ · γ µ · D µ ψ) + i(Ψ · γ µ · D µ Ψ) -M(Ψ · Ψ)
-yφ (ψ · P R · Ψ) -yφ (Ψ· P R · ψ)
Next, we integrate out the heavy fermion Ψ with the Match routine: where the option EFTOrder -> 6 prescribes the EFT expansion is terminated at dimension-six operators, and LoopOrder -> 1 indicates that the matching is performed at one-loop order. The resulting EFT Lagrangian LEFT is given in a redundant, unsimplified form. It can be simplified to an off-shell Green's basis by calling More commonly, we wish to also use field redefinitions to achieve an even more simplified EFT that still reproduces the same on-shell physics.
µ 2 M 2 φ 4 + 1 3 y 3 y 3 1 M 2 φ 6 + 1 3 yy e 2 M 2 φ 2 A µν 2 - 2 15 e 4 M 2 (ψ · γ µ · ψ) 2 + 7 36 yy e 2 M 2 (ψ · γ µ · ψ)(ψ · γ µ P L · ψ)
where we set all the poles to zero, assuming that both the UV Lagrangian LUV and the EFT Lagrangian LEFTonShell are properly renormalized in the MS-scheme. In a slight abuse of notation, is used in the output to denote the loop factor and ensure consistent truncation of the loop expansion, i.e. for one-loop computations Matchete sets 2 = 0. For numerical values, one simply needs to replace → 1/(16π 2 ). We observe that there are no redundant operators left in this EFT Lagrangian. The simplified output has canonically normalized kinetic terms for the matter fields, leaving only the non-trivial factor on the gauge kinetic term in lieu of a coupling correction. The coupling cφφ is automatically introduced to account for the hard scale contribution to the mass correction of the scalar field, as described in Sec. 2.3.2. The user is notified when such replacements happen and can retrieve the definitions of the effective couplings in the resulting Lagrangian using the PrintEffectiveCouplings command: If desired, the effective couplings can be replaced by their definitions in terms of the original input couplings using the ReplaceEffectiveCouplings command.
Real singlet scalar BSM extension
Our first BSM example is the venerable singlet extension of the SM previously matched in Refs. [79,80]. A real, heavy scalar field φ, which is a singlet under the SM gauge group is added to the SM. The resulting Lagrangian for this UV model is
L = L SM + 1 2 (∂ µ φ) 2 − 1 2 M 2 φ − µ 3! φ 3 − λ φ 4! φ 4 − Aφ|H| 2 − κ 2 φ 2 |H| 2 . (3.2)
Assuming the mass of the scalar to be heavy compared to the EW scale, the singlet can be integrated out from the theory to arrive at the corresponding SMEFT Lagrangian. We have validated the full one-loop dimension-six result of this matching and obtained agreement with the calculation of Ref. [80]. 5 Here, we will show how this simple SM extension can be implemented in Matchete and how to select specific contributions from the matching computation.
Since we are dealing with a SM extension, the task of inputting the model is simpler. The first step is to load the SM Lagrangian, which is already predefined in Matchete, by running 6
In[1]:= LSM = LoadModel["SM", ModelParameters -> {"µ" -> mH, "λ" -> λh}];
where we rename the Higgs mass parameter to mH and the quartic Higgs coupling to λh. This command defines all SM symmetries, couplings and fields, and saves the SM Lagrangian into the LSM variable. For completeness, we also provide the full SM definition in Matchete in Appendix A. The implementation shown there agrees with the internal implementation that is loaded when using LoadModel["SM"].
Next, we have to define the BSM field φ with mass M by The matching to the SMEFT is again performed with the Match routine. For the tree-level matching, we find
1 2 A 2 1 M 2 H i H j H i H j + 1 6 A 2 1 M 6 -3κ M 2 + A µ H i H j H k H i H j H k -A 2 1 M 4 H i D µ H j H i D µ H j + - 1 2 A 2 1 M 4 H i H j H i D 2 H j + H.c.
where we only print the NP contributions in the EFT after applying off-shell operator simplification, such as integration-by-parts identities. The first operator is a modification of the Higgs quartic coupling, the second is Q H in the Warsaw basis, the third can be exchanged for the operator Q H of the Warsaw basis, 7 and the last term can be removed by applying appropriate field redefinitions or the Higgs EOM. This last simplification step can be performed by applying EOMSimplify [LEFT0]. For brevity, we do not show the result here, but it can be found in the example notebook Examples/Singlet Scalar Extension.nb, included in the public release of Matchete. This notebook contains the full matching of this model at one loop as well as the comparison to the results presented in Ref. [80]. The one-loop matching and the full simplification of the resulting EFT Lagrangian is performed similarly: Again, the resulting Lagrangian is too long to show here, but it can be found in the example notebook. In the following, we demonstrate how to extract a particular contribution from the EFT Lagrangian, using the SelectOperatorClass routine. As an example, we extract the fully leptonic four-fermion operator These operators are related by integration by parts, but their difference is an operator that can be removed by applying Higgs field redefinitions. Therefore, the choice between Q H or Q HD affects the matching conditions for a wide set of different operator classes. The examples shown here are, however, not affected by this. In the example notebook, we show how to manually match the results provided by Matchete to the Warsaw basis.
In
Vector-like lepton BSM extension
As our final example, we consider a vector-like lepton extension of the SM with the same quantum numbers as the SM lepton singlet, namely E ∼ (1, 1) −1 . The Lagrangian for this model is given by
L = L SM + E(i / D − M E )E − y p E¯ p H E R + h.c. ,(3.3)
where H is the SM Higgs, p is the SM SU(2) L lepton doublet, and the index p denotes SM flavor. The NP model parameters M E and y p E are a real scalar and a complex flavor vector, respectively. The matching result of this model was already presented as an example for the matchmakereft matching code [63], which uses a diagrammatic approach. We find full agreement with this result, hence providing essential validation for both implementations.
In what follows, we show how to input the Lagrangian (3.3) and illustrate the Matchete functions relevant to the cross check. As in the previous example, we are dealing with a SM extension and the first step is to load the SM Lagrangian, which is already predefined:
In[2]:= LSM = LoadModel["SM"];
This command defines all SM symmetries, couplings and fields, and saves the SM Lagrangian into the LSM variable. The next step is to define the NP field 8
In LEFT0 -LSM //NiceForm
Out[6]= i yE p yE r 1 ME 2 D µ H i H j (l r j · γ µ P L · l ip ) + i yE p yE r 1 ME 2 H i H j (l r j · γ µ P L · D µ l ip )
This result is not manifestly Hermitian but it can be recast into a manifestly Hermitian form using IBP identities via the GreensSimplify routine:
In[7]:= LEFT0 -LSM //GreensSimplify //NiceForm Out[7]= i 2 yE p yE r 1 ME 2 D µ H i H j (l r j · γ µ P L · l ip ) -H i D µ H j (l r j · γ µ P L · l ip ) + i 2 yE p yE r 1 ME 2 H i H j (l r j · γ µ P L · D µ l ip ) -H i H j (D µ l r j · γ µ P L · l ip )
Finally, this last term of the result above can be eliminated in favor of a Warsaw basis operator using field redefinitions (which at this order are equivalent to EOM identities). The field redefinitions are applied with the EOMSimplify routine:
In [8]:= LEFT0OnShell = LEFT0 //EOMSimplify; LEFT0OnShell -LSM //HcSimplify //NiceForm
Out[8]= ( 1 2 yE s yE r Ye sp 1 ME 2 H i H i H j (l r j · P R · e p ) - i 2 yE p yE r 1 ME 2 H i D µ H j (l r j · γ µ P L · l ip ) + H.c)
A final group identity, δ ik δ jl = 1 2 (δ ij δ kl + τ a ij τ a kl ), would be needed to recast the last term into elements of the Warsaw basis. At present, the automated reduction to the Warsaw basis is not implemented in Matchete and this and other identities need to be applied manually. In this example we were careful to apply EOMSimplify to the full EFT Lagrangian. Unlike the exact identities used by GreensSimplify, field redefinitions cannot be applied to individual terms. Eliminating operators with field redefinitions will typically shuffle all kinds of contributions between many other operators.
The dimension-six output at one-loop order is rather lengthy and is provided in the example notebook Examples/E VLL model.nb included in the public Matchete release along with the details of the comparison with Ref. [63].
Future Prospects
In this article, we have introduced the first version of Matchete and sketched out the workings of its routines. Already in this first version, it has great utility and versatility and can perform the matching of a wide range of UV models without any additional input for group theory or EFT bases. However, it is also clear that we can enhance the capabilities of Matchete even further. Our roadmap for future functionality includes addressing the following points:
• Currently, the matching is performed in strictly d dimensions, which prevents EFT simplifications to the four-dimensional basis. We intend to implement the methods of Ref. [64] for defining a physical projection on the operator space and matching the remnant evanescent operator to the physical space.
• After the implementation of routines for handling the evanescent operators, it will be possible to reduce EFT Lagrangians all the way to specific four-dimensional bases. The idea is to use this to obtain matching results as Wilson coefficients of known EFT bases, such as the Warsaw basis for the SMEFT or the LEFT basis of Ref. [7]. Hence, it will be possible to interface the result with phenomenology packages through export in the WCxf [81] format. The interface with other phenomenology codes and/or commonly used formats, such as UFO [82], would also be desirable.
• The restriction of heavy states to scalars and fermions is the primary limitation of Matchete v0.1.0. In weakly-coupled theories, heavy vectors must arise from spontaneous symmetry breaking. This results in a complicated interplay between vectors, ghosts, and Goldstone bosons, especially in the background field gauge. So as to avoid having to derive and input all interactions manually, we wish to provide (semi-)automated methods to determine the broken phase Lagrangian.
• With small changes to the matching procedure, it is possible to determine the EFT counterterms and, thereby, the RG functions. Implementing this functionality in
Matchete will allow for finding the RG functions for intermediate-scale EFTs and vastly simplify sequential matching scenarios.
In Figure 2 we show how the future functionalities fit into the Matchete workflow. The roadmap is our vision for the future of the Matchete package. It is of course subject to changes, as we determine what features are most important, or if the implementation of the functions becomes problematic. Automated tools like Matchete have the potential to fundamentally change the workflow of BSM physics. They allow the practitioner to focus less on mechanical tasks and instead concentrate on finding answers to open questions in physics. Even with its current limitations, the proof of concept for Matchete already provides a valuable tool for NP phenomenology, and it demonstrates that functional methods offer a natural formulation of the matching (and RG running) procedure.
A Defining the SM in Matchete
To illustrate the ease of defining general quantum field theories in Matchete, we showcase how to input the SM Lagrangian. When using Matchete for a BSM theory, it is not necessary to repeat the SM input, as its definition can be loaded running the LoadModel["SM"] routine, as shown in the examples in Sec. 3.2 and 3.3.
In a first step, we use the DefineGaugeGroup routine to define the SM gauge group SU(3) c × SU(2) L × U(1) Y by The Higgs is defined as massless, because the mass parameter of the Higgs doublet is tachyonic. We will therefore include it manually as an interaction below.
The last missing definitions are for the (non-gauge) coupling constants for which we use the DefineCoupling routine. The Yukawa matrices can be added with +i(e p ·γ µ P R ·D µ e p ) +i(l p i ·γ µ P L ·D µ l ip ) +i(q p ai ·γ µ P L ·D µ q aip ) +i(u p a ·γ µ P R ·D µ u ap ) + -Ye rp H i (l r i ·P R ·e p ) -Yd rp H i (q r ai ·P R ·d ap ) -Yu rp H i (q r aj ·P R ·u ap )ε ji + H.c.
where we used RelabelIndices to canonically relabel all indices in the Lagrangian, PlusHc to include the Hermitian conjugate of the Yukawa Lagrangian defined above, and HcSimplify to collect hermitian conjugated terms for the printing. We can also use the CheckLagrangian routine to perform a series of checks on the Lagrangian, ensuring for example that LSM is Hermitian, gauge-invariant, is free of gauge anomalies, and more:
In[11]:= CheckLagrangian[LSM]
Out [11]= True These checks are also internally applied by the Match routine, before performing the matching, which can only be done for Lagrangians passing all tests of CheckLagrangian.
The addition of further BSM fields, and couplings works analogously. For BSM theories with extended gauge symmetries, the BSM Lagrangian would have to be provided to Matchete in the broken phase, where only the residual SM gauge symmetry remains unbroken. The complexity of the resulting Lagrangian with ghosts, heavy vectors, and gauge fixing, and the precise relations among them needed to obtain meaningful results is why this is not presently supported.
Figure 1 :
1Schematic representation of the Matchete workflow.
In[ 8 ]
8:= LEFT = Match[LUV, LoopOrder -> 1, EFTOrder -> 6];
In[ 9 ]
9:= LEFTOffShell = GreensSimplify[LEFT];
rs Ye tp A 2 1 M 4 e s · P L · l ir l t i · P R · e p where the second argument specifies the field content of the operator(s) to be extracted, and the last argument gives the number of derivatives. The result shown above is not in the Warsaw basis, since the current version Matchete is not applying Fierz identities. Manually, using the identity (e s r )( t e p ) = − 1 2 ( t γ µ r )(e s γ µ e p ) gives the desired result for Q e in the Warsaw basis. Similarly, we extract the Q HW operator:In[8]:= SelectOperatorClass[LEFTOnShell,{Bar[H],H,W,W},H i H i W µνI 2 In general, results obtained with the SelectOperatorClass routine do not coincide with the matching conditions for the Warsaw basis. This is because of the Q H = (H † H) (H † H) operator being replaced in favor of the operator Q HD = (H † H)[(D µ H † )(D µ H)] in Matchete.
[3]:= DefineField[EE, Fermion, Charges -> {U1Y[-1]}, Mass -> {Heavy, ME}] and Yukawa coupling In[4]:= DefineCoupling[yE, Indices -> {Flavor}] The complete UV Lagrangian is then entered as In[5]:= LUV = LSM + FreeLag[EE] -PlusHc[yE[p] Bar[l[i, p]]**PR**EE[] H[i]]; where i and p are used for SU(2) L and flavor indices, respectively. As in previous examples, this Lagrangian can easily be matched to its EFT with the Match routine. For example, at tree level we have In[6]:= LEFT0 = Match[LUV, LoopOrder -> 0, EFTOrder -> 6];
Figure 2 :
2Roadmap for the future capabilities of Matchete. The workflow contained in the blue boxes are implemented in the proof-of-concept version, whereas the orange boxes are features expected in future releases. Here Standard format output refers to both EFT basis identification and interfacing with other EFT codes.
Table of Contents
ofIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector-like lepton BSM extension . . . . . . . . . . . . . . . . . . . . . 16 4 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A Defining the SM in Matchete . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
. .
2
2
Organization of the Package . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
Model setup and internal representations . . . . . . . . . . . . . . . .
4
2.2
Matching step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3
Using Matchete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1
Vector-like fermion toy-model . . . . . . . . . . . . . . . . . . . . . . . 12
3.2
Real singlet scalar BSM extension . . . . . . . . . . . . . . . . . . . . 14
3.3
Table 2 :
2Field redefinitions needed to remove redundant operators involving a given field type. SeeTable 1for the definitions of D (ψ).
CheckForUpdate[] command in a Mathematica notebook.Once Matchete is installed and loaded, the user can start implementing models and matching to their EFTs with the routines provided by the package. Below, we demonstrate the usage of the code with three illustrative examples.In[2]:= << MatcheteT
he user can check for updates and install them (when available) by simply running the
In[3]:= DefineGaugeGroup[U1e, U1, e, A]which initializes a gauge coupling e and the corresponding field-strength tensor, labeled A. Next, all matter fields are defined:In[4]:= DefineField[Ψ, Fermion, Charges -> {U1e[1]}, Mass -> {Heavy, M}]where we assign charges of +1 to both fermions under the U(1) gauge group, declare the field Ψ as heavy (for matching purposes) with mass label M, set ψ as massless, and set φ to be a real field with light mass, automatically generating the mass label mφ. Finally, we have to define the Yukawa coupling y After all symmetries, fields, and couplings are defined, the Lagrangian of the free theory can be automatically generated with the FreeLag routine. The interactions are manually added to obtain the UV LagrangianIn[6]:= LUV = FreeLag[] -PlusHc[y[] φ[] Bar[ψ[]]**PR**Ψ[]];DefineField[ψ, Fermion, Charges -> {U1e[1]}, Mass -> 0]
DefineField[φ, Scalar, Mass -> Light, SelfConjugate -> True]
In[5]:= DefineCoupling[y]
which by default is understood as a complex parameter that does not influence the EFT power
counting (i.e. EFTOrder -> 0).
Using these definitions, none of the couplings above carry a light mass dimension, i.e., we have µ = O(M ) and A = O(M ). The Lagranigian of the full NP model can then be specified withIn[2]:= DefineField[φ, Scalar, SelfConjugate -> True, Mass -> {Heavy, M}]
followed by the definition of all NP couplings:
In[3]:= DefineCoupling[A, SelfConjugate -> True]
DefineCoupling[κ, SelfConjugate -> True]
DefineCoupling[µ, SelfConjugate -> True]
DefineCoupling[λφ, SelfConjugate -> True]
In[4]:= LUV = LSM + FreeLag[φ] -
1
3!
µ[] φ[] 3 -
1
4!
λφ[] φ[] 4
-A[] Bar[H[i]]H[i]φ[] -
1
2
κ[] Bar[H[i]]H[i]φ[] 2 ;
Instead of showing the full result, we illustrate the use of CovariantLoop here. This Matchete routine provides the contribution from individual supertraces (or, equivalently, individual covariant loop topologies). For instance, to compute the contribution to the (H † H) 3 operator arising from the fermion hexagon graph (involving 3 vector-like and 3 SM lepton-doublet propagators), we simply run In[9]:= CovariantLoop[LUV, {Bar[EE], l, Bar[EE], l, Bar[EE], l}] //NiceForm yE p yE r yE s 1 ME 2 H i H j H j H i H j H k which coincides with the six NP Yukawa term in Eq. (6.15) of Ref. [63].Out[9]=
1
3
yE
p yE
r yE
s
In[6]:= DefineCoupling[Yu, Indices -> {Flavor, Flavor}] DefineCoupling[Yd, Indices -> {Flavor, Flavor}] DefineCoupling[Ye, Indices -> {Flavor, Flavor}] Similarly, we can define the parameters of the Higgs potential by In[7]:= DefineCoupling[µ, SelfConjugate -> True, EFTOrder -> 1] DefineCoupling[λ, SelfConjugate -> True, EFTOrder -> 0] where the option EFTOrder->1 specifies that the Higgs mass parameter µ should be counted as having light-mass dimension one in the EFT power counting. We can now write down the SM Lagrangian. Starting with the Yukawa interactions, we have In[8]:= YukawaL = Ye[p,r] Bar[l[i,p]]**e[r] H[i] + Yd[p,r] Bar[q[a,i,p]]**d[a,r] H[i] + Yu[p,r] Bar[q[a,i,p]]**u[a,r] CG[eps[SU2L], {i,j}] Bar[H[j]]; The scalar potential is written as In[9]:= HiggsPotential = -µ[] 2 Bar[H[i]]H[i] + λ[] 2 Bar[H[i]]H[i]Bar[H[j]]H[j]; Eventually, we can write the full SM Lagrangian In[10]:= LSM = FreeLag[q, u, d, l, e, H, G, W, B] -PlusHc[YukawaL] -HiggsPotential //RelabelIndices; LSM //HcSimplify //NiceForm +D µ H i D µ H i +µ 2 H i H i -1 2 λH i H j H i H j +i(d p a ·γ µ P R ·D µ d ap )Out[10]= -
1
4
B µν 2 -
1
4
G µνA 2 -
1
4
W µνI 2
We found Ref.[65] very useful as a primer and reminder for relevant Lie algebra methods.
Kinetic-mixing terms are ubiquitous in BSM models with new U(1) symmetries, which can mix with the hypercharge field. They are typically kept explicit in the Lagrangian until symmetry breaking.
We recommend placing the Matchete folder in the Applications folder of Mathematica's base directory. Then the location does not need be specified before loading the package.4 A preliminary implementation of this model in Matchete was also presented in[78].
Actually, agreement with this reference is only obtained forμ = M 2 , as we find that the results provided there for the log-terms contain contributions that cannot be generated by matching. The log-terms have been partially cross-checked against the Greens' basis results in Ref.[45], finding agreement for the non-logarithmic contributions as well.
The complete list of available models (including the one of this example) can be checked by GetModels[] in a Mathematica notebook or by looking into the Models folder of the public release.7 Matchete automatically applies the product rule for derivatives. Therefore, it is not possible to directly obtain Q H in the matching result. The issues related to this basis mismatch are discussed below.
We write EE for the field label because E is reserved in Mathematica for the Euler number.
AcknowledgementsWe thank José Santiago for his help with the cross-checks of the vector-like lepton example in Section 3.
. G Gs, Fundalphabet -> { , = DefineGaugeGroup[SU3c, SUa","b","c","d","e","f"}, AdjAlphabet -> {"A","B","C","D","E","F"}In[2]:= DefineGaugeGroup[SU3c, SU[3], gs, G, FundAlphabet -> {"a","b","c","d","e","f"}, AdjAlphabet -> {"A","B","C","D","E","F"}]
. W Definegaugegroup ; Gl, Fundalphabet -> {"i, SU2L, SU[2. j","k","l","m","n"}, AdjAlphabet -> {"I","J","K","L","M","N"}DefineGaugeGroup[SU2L, SU[2], gL, W, FundAlphabet -> {"i","j","k","l","m","n"}, AdjAlphabet -> {"I","J","K","L","M","N"}]
This automatically defines also the associated field-strength tensors, labeled G, W, and B, and the gauge couplings gs, gL, and gY. The optional arguments FundAlphabet and AdjAlphabet determine how fundamental and adjoint indices are displayed when using the NiceForm routine. Flavor indices and their printing style can be defined similarly. Definegaugegroup, U1Y, U1, gY, B] where the groups are labeled SU3c, SU2L, and U1Y, respectively. In[3]:= DefineFlavorIndex[Flavor,3,IndexAlphabet->{"p","r","s","t","u","v"}]DefineGaugeGroup[U1Y, U1, gY, B] where the groups are labeled SU3c, SU2L, and U1Y, respectively. This automatically defines also the associated field-strength tensors, labeled G, W, and B, and the gauge couplings gs, gL, and gY. The optional arguments FundAlphabet and AdjAlphabet determine how fundamental and adjoint indices are displayed when using the NiceForm routine. Flavor indices and their printing style can be defined similarly: In[3]:= DefineFlavorIndex[Flavor,3,IndexAlphabet->{"p","r","s","t","u","v"}]
. Definefield, d, Fermion, Indices -> {SU3c[fund], Flavor}, Charges -> {U1Y[-1/3]}, Chiral -> RightHanded, Mass -> 0DefineField[d, Fermion, Indices -> {SU3c[fund], Flavor}, Charges -> {U1Y[-1/3]}, Chiral -> RightHanded, Mass -> 0]
. Definefield, l, Fermion, Indices -> {SU2L[fund], Flavor}, Charges -> {U1Y[-1/2]}, Chiral -> LeftHanded, Mass -> 0DefineField[l, Fermion, Indices -> {SU2L[fund], Flavor}, Charges -> {U1Y[-1/2]}, Chiral -> LeftHanded, Mass -> 0]
; } Definefield, Chiral -> Righthanded, Mass -> 0Fermion, Indices -> {Flavor}, Charges -> {U1Y. DefineField[e, Fermion, Indices -> {Flavor}, Charges -> {U1Y[-1]}, Chiral -> RightHanded, Mass -> 0]
For example, the left-handed quark doublet is afterwards written in Matchete by typing q[a,i,p], where a, i, and p are the SU(3) c , SU(2) L , and flavor indices, respectively, given in the order used in DefineField. The Higgs doublet is defined similarly. In[5]:= DefineField[H, Scalar, Indices -> {SU2L[fund]}, Charges -> {U1Y[1/2]}, Mass -> 0For example, the left-handed quark doublet is afterwards written in Matchete by typing q[a,i,p], where a, i, and p are the SU(3) c , SU(2) L , and flavor indices, respectively, given in the order used in DefineField. The Higgs doublet is defined similarly: In[5]:= DefineField[H, Scalar, Indices -> {SU2L[fund]}, Charges -> {U1Y[1/2]}, Mass -> 0];
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Singlet night in Feynman-ville: one-loop matching of a real scalar. U Haisch, M Ruhdorfer, E Salvioni, E Venturini, A Weiler, 10.1007/JHEP04(2020)164JHEP. 041642003.05936U. Haisch, M. Ruhdorfer, E. Salvioni, E. Venturini and A. Weiler, Singlet night in Feynman-ville: one-loop matching of a real scalar, JHEP 04 (2020) 164, [2003.05936].
WCxf: an exchange format for Wilson coefficients beyond the Standard Model. J Aebischer, 10.1016/j.cpc.2018.05.0221712.05298Comput. Phys. Commun. 232J. Aebischer et al., WCxf: an exchange format for Wilson coefficients beyond the Standard Model, Comput. Phys. Commun. 232 (2018) 71-83, [1712.05298].
UFO -The Universal FeynRules Output. C Degrande, C Duhr, B Fuks, D Grellscheid, O Mattelaer, T Reiter, 10.1016/j.cpc.2012.01.022Comput. Phys. Commun. 1831108.2040C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, UFO -The Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201-1214, [1108.2040].
| {'fraction_non_alphanumeric': 0.07030701025922756, 'fraction_numerical': 0.053276436994544596, 'mean_word_length': 4.108138093062716, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 70, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 104, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Studying the impact of new-physics models on low-energy observables necessitates matching to effective field theories at the relevant mass thresholds. We introduce the first public version of Matchete, a computer tool for matching weakly-coupled models at one-loop order. It uses functional methods to directly compute all matching contributions in a manifestly gauge-covariant manner, while simplification methods eliminate redundant operators from the output. We sketch the workings of the program and provide examples of how to match simple Standard Model extensions. The package, documentation, and example notebooks are publicly available at https://', 'arxivid': '2212.04510', 'author': ['Javier Fuentes-Martín \nDepartamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de FuentenuevaE-18071GranadaSpain\n\nPRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics\nJohannes Gutenberg University\nD-55099MainzGermany\n', 'Matthias König \nPhysik Department T31\nTechnische Universität München\nJames-Franck-Str. 1D-85748GarchingGermany\n', 'Julie Pagès \nDepartment of Physics\nUniversity of California at San Diego\n9500 Gilman Drive, La Jolla92093-0319CAUSA\n', '‡ Anders ', 'Eller Thomsen §[email protected]¶[email protected] \nAlbert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics\nUniversity of Bern\nCH-3012BernSwitzerland\n', '§ ', 'Felix Wilsch \nPhysik-Institut\nUniversität Zürich\nCH-8057ZürichSwitzerland\n'], 'authoraffiliation': ['Departamento de Física Teórica y del Cosmos\nUniversidad de Granada\nCampus de FuentenuevaE-18071GranadaSpain', 'PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics\nJohannes Gutenberg University\nD-55099MainzGermany', 'Physik Department T31\nTechnische Universität München\nJames-Franck-Str. 1D-85748GarchingGermany', 'Department of Physics\nUniversity of California at San Diego\n9500 Gilman Drive, La Jolla92093-0319CAUSA', 'Albert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics\nUniversity of Bern\nCH-3012BernSwitzerland', 'Physik-Institut\nUniversität Zürich\nCH-8057ZürichSwitzerland'], 'corpusid': 254535576, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26600, 'n_tokens_neox': 22058, 'n_words': 12120, 'pdfsha': 'f5eb1639aae3cc443ef148214602c6d932af2a7d', 'pdfurls': ['https://export.arxiv.org/pdf/2212.04510v1.pdf'], 'title': ['A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories', 'A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories'], 'venue': []} |
arxiv |
Robertson-Schrödinger type formulation of Ozawa's noise-disturbance uncertainty principle
17 Oct 2013 (Dated: May 1, 2014)
Catarina Bastos
Alex E Bernardini
Orfeu Bertolami
Nuno Costa Dias
João Nuno Prata
Instituto de Plasmas e Fusão Nuclear
Departamento de Física e Astronomia
Faculdade de Ciências da Universidade do Porto
Departamento de Matemática
Instituto Superior Técnico Avenida Rovisco
Pais 1, Rua do Campo Alegre, 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal
Lusófona de Humanidades e Tecnologias Avenida Campo Grande
Universidade
376, 1749-024LisboaPortugal
Robertson-Schrödinger type formulation of Ozawa's noise-disturbance uncertainty principle
17 Oct 2013 (Dated: May 1, 2014)PACS numbers:
In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schrödinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that, for certain types of measurement interactions, it is covariant with respect to linear symplectic transformations of the noise and disturbance operators.
1. Introduction -Although, Heisenberg's uncertainty principle is one of the hallmarks of quantum mechanics, there has been some discussion about its formulation. Robertson's formulation of the uncertainty principle,
σ(A, ψ)σ(B, ψ) ≥ | ψ| [A, B] |ψ | 2 ,(1)
expresses an intrinsic uncertainty of the states in terms of the standard deviation of some pair of noncommuting observables A and B in a state ψ. This kind of formulation describes a limitation on the preparation of the state, but has no direct relevance for the accuracy of the measurement of an observable A with an apparatus and the disturbance caused by it on observable B. We shall refer to these formulations as kinematical uncertainty principles. Another kinematical inequality is the Robertson-Schrödinger uncertainty principle (RSUP). It can be stated in terms of the positivity of the matrix
Σ + i 2 J ≥ 0 ,(2)
where Σ is the covariance matrix of the state Σ = σ(X, ψ) 2 σ(X, P, ψ) σ(P, X, ψ) σ(P, ψ) 2 .
Here X is the particle position and P the momentum, σ(X, P, ψ) = σ(P, X, ψ) = ψ| {∆X, ∆P } |ψ are the covariance elements for position-momentum correlations, where {·, ·} is the anti-commutator and ∆X = X− < ψ|X|ψ >, etc. Also
J = 0 1 −1 0(4)
is the standard symplectic matrix. This formulation has several advantages over the one stated in Eq. (1). On the one hand, it is stronger than inequality (1), in the sense that in fact it implies Eq. (1). However, the converse is not true. On the other hand, it accounts for the position-momentum correlations, which are relevant in several contexts, e.g. states with strong positionmomentum correlations may lead to greater transparency of the Coulomb barrier during the interaction of charged particles. This is quite relevant, for instance, in the astrophysics of stars and in controlled nuclear fusion, where the action of the Coulomb barrier leads to a very low tunneling probability for low-energy particles [1]. For Gaussian states (which include coherent, squeezed and thermal states), the RSUP constitutes the necessary and sufficient condition of quantumness. Moreover, after a certain reflection transformation [2], it also establishes unequivocally the separable or entangled nature of Gaussian states. Experimentally, coherent and squeezed states play an important role in quantum optics [3], quantum computation of continuous variables [4] and investigations of the quantum-classical transition [5]. Finally, the RSUP is invariant under linear symplectic transformations, a property which is not shared by inequality (1). This is important in the context of semi-classical analysis [5] and in the search for directions of minimal uncertainty [6].
An experimental violation of the kinematical uncertainty principles, Eqs. (1) and (2) can only be attributed to either the failure of the Hilbert space formalism to correctly describe quantum systems -something which would have profound implications on the theoretical edifice of quantum mechanics -, or alternatively, to some modification of the position-momentum commutation relations. The latter possibility has been explored recently in Ref. [7]. To the best of our knowledge, no such experimental violation has ever been recorded.
In addition to the previous kinematical inequalities, there are other uncertainty principles (dynamical uncertainty principles) which try to account for the "unavoidable and uncontrollable disturbance" caused on observable B by a measurement of observable A. In his famous γ-ray thought experiment [8], Heisenberg argued that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less that /2. More generally, if ǫ(A, ψ) denotes the noise of the A measurement and η(B, ψ) the disturbance on B caused by that measurement, when the system is in state ψ, then the Heisenberg noise-disturbance relation reads
ǫ(A, ψ)η(B, ψ) ≥ | ψ| [A, B] |ψ | 2 .(5)
There have been various accounts of theoretical [9][10][11] and experimental [12,13] violations of inequality (5). This has prompted the search for an universally valid formulation of an uncertainty principle accounting for the noise and disturbance of the measurement interaction. Recently, M. Ozawa [14] considered a composite system of the object and the measuring device (the probe), initially prepared in a product state Ψ = ψ⊗ξ, where ψ and ξ describe the object and the probe, respectively. Working in the Heisenberg picture, he introduced the noise operator N (A) associated with observable A and the disturbance operator D(B). They are self-adjoint operators, defined by
N (A) = M out − A in , D(B) = B out − B in .(6)
Here For more details on the measurement interaction see Ref. [14]. The noise ǫ(A, ψ) and disturbance η(B; ψ) are defined to be the mean-square deviations of the noise and disturbance operators, respectively:
A in = A⊗ I, B in = B ⊗ I are observables A, B prior to the measurement interaction, B out = U † (B ⊗ I)U is the observable B immediatelyǫ(A, ψ) 2 = Ψ|∆N (A) 2 |Ψ , η(B, ψ) 2 = Ψ|∆D(B) 2 |Ψ .(7)
Since M and B are observables in different systems, they commute : [M out , B out ] = 0. Using this fact, Eq. (7), the triangle and the Cauchy-Schwartz inequalities, one can then prove Ozawa's uncertainty principle (OUP):
ǫ(A, ψ)η(B, ψ)+ǫ(A, ψ)σ(B, ψ)+σ(A, ψ)η(B, ψ)≥ | ψ|[A, B]|ψ 2 .
(8) Just as the kinematical uncertainty principle, Eq. (1), does not account for the position-momentum correlations, neither does the OUP, Eq. (8), account for the noise-disturbance correlations.
The purpose of this letter is then to derive a matrix formulation of the OUP, which encompasses the noisedisturbance correlations. It is related to the OUP in very much the same way as the RSUP relates to Eq. (1) as it is more general than the latter and admits nicer symmetry properties.
2. Matrix formulation of the Ozawa uncertainty principle -In the sequel, Latin indices i, j run in the set {1, · · · , n} and Greek indices α, β in the set {1, · · · , 2n}. For the sake of generality, we shall also consider a multidimensional system. Using Ozawa's notation, let A in i and B in j , i, j = 1, · · · , n denote some set of self-adjoint operators such that
A in i , A in j = B in i , B in j = 0 , A in i , B in j = iC ij ,(9)
for i, j = 1, · · · , n, and where {C ij } are some self-adjoint operators. If A and B are the particle's position and momentum, we simply have C ij = δ ij . We may write these collectively as Z in = A in 1 , · · · , A in n , B in 1 , · · · , B in n satisfying the commutation relations
Z in α , Z in β = iG αβ , α, β = 1, · · · , 2n .(10)
G = {G αβ } is the self-adjoint operator-valued skewsymmetric matrix
G = 0 C −C 0 ,(11)
with C = {C ij }. Again, if A and B are the position and momentum operators, then we simply have 2n × 2n standard symplectic matrix J. Let us define the noise and disturbance operators as
N = N (A) = (N 1 , · · · , N n ) ,(12)D = D(B) = (D 1 , · · · , D n ) .(13)
We can write these collectively as
K = (N 1 , · · · , N n , D 1 , · · · , D n ) .(14)
Then, we denote the output of the (commuting) probe observables M out and the output of B as
M out = M out 1 , · · · , M out n ,(15)B out = B out 1 , · · · , B out n .(16)
If we write Z out = (M out 1 , · · · , M out n , B out 1 , · · · , B out n ) then we have as before:
Z out α , Z out β = 0 , α, β = 1, · · · , 2n ,(17)
and Z out = Z in + K. Let {λ α } 1≤α≤2n denote an arbitrary set of complex numbers. Thus, we have:
2n α,β=1 λ α λ β ∆Z out α , ∆Z out β = 2n α,β=1 λ α λ β i G αβ + Z in α , K β + K α , Z in β + [∆K α , ∆K β ] = 0 .(18)
Now notice that writing K = α λ α ∆K α , we have 2n α,β=1
λ α λ β [∆K α , ∆K β ] = K † K − KK † ≤ K † K ,(19)
and so 2n α,β=1
λ α λ β [∆K α , ∆K β ] ≤ 2n α,β=1 λ α λ β ∆K α ∆K β = 2n α,β=1 λ α λ β {∆K α , ∆K β } + 1 2 [∆K α , ∆K β ] .(20) Thus, 2n α,β=1 λ α λ β [∆K α , ∆K β ] ≤ 2 2n α,β=1 λ α λ β {∆K α , ∆K β } .λ α λ β i G αβ + Z in α , K β + K α , Z in β +2 {∆K α , ∆K β } ≥ 0 .(22)
If we define the 2n × 2n real symmetric positive-definite matrix K αβ = {∆K α , ∆K β } , the 2n × 2n real skewsymmetric matrices G = G and
Γ αβ = 1 i Z in α , K β + K α , Z in β ,(23)
then we can rewrite Eq. (22) in the matrix form:
K + i 2 (Γ + G) ≥ 0,(24)
which is our matrix formulation of Ozawa's uncertainty principle. In Ozawa's terminology, if the measuring interaction is of independent intervention [14], i.e. if
Γ = 0,(25)
then we obtain
K + i 2 G ≥ 0,(26)
which is the matrix generalization of the Heisenberg noise-disturbance relation, Eq. (5), based on the γ-ray thought experiment. 3. On the connection with the Ozawa uncertainty principle -Here, we argue that our formulation of the uncertainty principle, Eq. (24), is in fact stronger than Ozawa's uncertainty principle. Indeed, let us consider for simplicity n = 1. We then have
K = {∆N, ∆N } {∆N, ∆D} {∆D, ∆N } {∆D, ∆D} ,(27)
while
Γ = 1 i 0 A in , D + N, B in D, A in + B in , N 0 .
(28) If Eq. (24) holds, then the matrix K + i 2 (Γ + G) must have non-negative determinant, and we obtain
{∆N, ∆N } {∆D, ∆D} ≥ {∆N, ∆D} 2 + 1 4 A in , D + N, B in + A in , B in 2 ≥ 1 4 A in , D + N, B in + A in , B in 2 .(29)
Taking the square root and writing ǫ(A) = {∆N, ∆N } 1/2 and η(B) = {∆D, ∆D} 1/2 , and using the inequality |a − b| ≥ | |a| − |b| |, we finally obtain
ǫ(A)η(B) ≥ 1 2 | A in , D + N, B in | − | A in , B in | .
(30) In particular
ǫ(A)η(B) ≥ 1 2 | A in , B in |− 1 2 A in , D + N, B in ,(31)
which is Ozawa's uncertainty principle before using the triangular identity and the Cauchy-Schwartz relation. The obvious question is now whether our uncertainty, Eq. (24), is equivalent to Ozawa's or whether it is in fact more restrictive. The latter is true and in the following we show it using a type of measuring interaction known as a backaction evading quadrature amplifier [14,15]. In this case, the system is described by a set of quadrature operators (X a , Y a ) and the probe by the operators
(X b , Y b ) with [X a , Y a ] = [X b , Y b ] = i 2 .(32)
Then we have the measuring interaction
X out a = X in a , X out b = X in b + GX in a , Y out a = Y in a − GY in b , Y out b = Y in b ,(33)
where G is the gain. The probe observable is then set to M = (1/G)X b , and thus
M out = X in a + 1 G X in b . (34) Moreover N (X a ) = 1 G X in b , D(X a ) = 0 , D(Y a ) = −GY in b .
(35)
Following our previous notation, we set A in = X in a , B in = Y in a . Then, Z in = X in a , Y in a and given that
N = N (X a ) = 1 G X in b , D = D(Y a ) = −GY in b ,(36)
one can write
K = 1 G X in b , −GY in b .(37)
We conclude that Γ = 0 and that the measuring interaction is of independent intervention for the pair (X a , Y a ). Also from (32):
G = 1 2 J.(38)
Now let us consider the state of the probe ξ to have a covariance matrix
Σ b = Σ b 11 Σ b 12 Σ b 12 Σ b 22 = 1 4 1 2 1 2 1 4 .(39)
Next, notice that from Eq. (37):
{N, N } = 1 G 2 X in b , X in b = 1 G 2 Σ b 11 = 1 4G 2 . (40)
In a similar fashion, we obtain
{N, D} = − X in b , Y in b = − 1 2 ,(41){D, D} = G 2 Y in b , Y in b = G 2 4 .(42)
Thus,
Σ = 1 4G 2 − 1 2 − 1 2 G 2 4 (43)
We have then from Eqs. (38) and (43)
K+ i 2 (Γ + G) = Σ+ i 4 J = 1 4G 2 − 1 2 + i 4 − 1 2 − i 4 G 2 4 .(44)
Since det(K+ i 4 J) = − 1 4 , we conclude that it is not a positive matrix and that our uncertainty principle Eq. (24) is violated. However, since ǫ(A) = {N, N } 1/2 = 1 2G and η(B) = {D, D} 1/2 = G 2 , we get that ǫ(A)η(B) = 1 4 , which exactly saturates the OUP. This proves that our inequality (24) is indeed stronger than OUP, Eq. (8).
4. Comments on the invariance properties of the matrix formulation -The matrix formulation of the noisedisturbance uncertainty principle Eq. (24) is universally applicable. Given the myriad of measurement interactions and apparatuses [16,17], it is virtually impossible to establish all transformations which leave Eq. (24) unchanged. There are nonetheless instances, where the inequality (24) is preserved under a certain type of transformation, while Eq. (8) is not.
Suppose that [K α , K β ] = iγJ αβ , where γ = 0 is some real constant and J is the standard symplectic matrix. Moreover, let the measurement interaction be of independent intervention (Γ = 0). Then Eq. (24) becomes K + iγ 2 J ≥ 0. Notice that this looks formally like the RSUP, Eq. (2). Suppose that the system undergoes a linear symplectic transformation K α → K ′ α = 1≤β≤2n S αβ K β , where S ∈ Sp(2n; R). Then the noisedisturbance covariance matrix transforms by similarity K → K ′ = SKS T . But given that S −1 J(S −1 ) T = J, we conclude that the matrix uncertainty principle remains unchanged: K ′ + iγ 2 J ≥ 0. But this may not happen for the Ozawa uncertainty relation. Indeed, let us consider again our previous example of the backaction evading quadrature amplifier. Remember that, for an interaction of independent intervention, the Ozawa inequality becomes simply the Heisenberg inequality
ǫη ≥ 1 4 .(45)
Suppose that we have now a noise-disturbance covariance matrix K = diag(ǫ 2 , η 2 ) and that the probe (and possibly the object) is subjected to a symplectic transformation such that:
K → K ′ = √ 2 2 1 1 −1 1 K.(46)
That is: the noise-disturbance vector K is rotated through an angle of π 4 . Such a transformation is easily implemented by a certain unitary transformation U (S) generated by an appropriate hermitian operator, quadratic in the variables X b , Y b of the probe.
Then the Ozawa uncertainty inequality is modified to
(ǫ ′ ) 2 + (η ′ ) 2 ≥ 1 + 4 {∆N ′ , ∆D ′ } 2 ,(47)
which is manifestly different from Eq. (45). We also remark that the noise-disturbance correlations naturally appear under such transformations. 5. Conclusions -In this work we presented a universal matrix formulation of the uncertainty principle which is more stringent than the noise-disturbance relation of Ozawa. Indeed, we have proved that our formulation implies Ozawa's, and showed that is possible to saturate the Ozawa uncertainty principle, while violating our universal form. Our inequality is also more general in the sense that, unlike Ozawa's relation, it is also accounts for the noise-disturbance correlations.
We recall that recent experimental work performed by Rozema et al. [12], using polarized entangled photons, and by Sulyok et al. [13], using neutron-spin measurements, proved the validity of Ozawa's relation using weak measurements. It would certainly be an interesting prospect to investigate the experimental validity of our relation with a similar experimental setup.
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arxiv |
BRExIt: On Opponent Modelling in Expert Iteration
Daniel Hernandez
University of York
UK
Hendrik Baier
Eindhoven University of Technology
The Netherlands
Michael Kaisers
Centrum Wiskunde & Informatica
The Netherlands
Sony Ai
BRExIt: On Opponent Modelling in Expert Iteration
Finding a best response policy is a central objective in game theory and multi-agent learning, with modern population-based training approaches employing reinforcement learning algorithms as bestresponse oracles to improve play against candidate opponents (typically previously learnt policies). We propose Best Response Expert Iteration (BRExIt), which accelerates learning in games by incorporating opponent models into the state-ofthe-art learning algorithm Expert Iteration (ExIt). BRExIt aims to (1) improve feature shaping in the apprentice, with a policy head predicting opponent policies as an auxiliary task, and (2) bias opponent moves in planning towards the given or learnt opponent model, to generate apprentice targets that better approximate a best response. In an empirical ablation on BRExIt's algorithmic variants against a set of fixed test agents, we provide statistical evidence that BRExIt learns better performing policies than ExIt.
Introduction
Reinforcement learning has been successfully applied in increasingly challenging settings, with multi-agent reinforcement learning being one of the frontiers that pose open problems [Hernandez-Leal et al., 2017;Albrecht and Stone, 2018;Nashed and Zilberstein, 2022]. Finding strong policies for multi-agent interactions (games) requires techniques to selectively explore the space of best response policies, and techniques to learn such best response policies from gameplay. We here contribute a method that speeds up the learning of approximate best responses in games.
State-of-the-art training schemes such as population based training with centralized control [Lanctot et al., 2017;Liu et al., 2021] employ a combination of outer-loop training schemes and inner-loop learning agents in a nested loop fashion [Hernandez et al., 2019]. In the outer loop, a combination of fixed policies is chosen via game theoretical analysis to act as static opponents. Within the inner loop, a reinforcement learning (RL) agent repeatedly plays against these static opponents, in order to find an approximate best response policy against them. In theory, any arbitrary RL algorithm could be used as such a policy improvement operator/best response oracle in the inner loop. The best response policy found is then added to the training scheme's population, and the outer loop continues, constructing a population of increasingly stronger policies over time. This technique has been successfully applied in highly complex environments such as Dota 2 [Berner et al., 2019] and Starcraft II [Vinyals et al., 2019], with PPO [Schulman et al., 2017] and IMPALA [Espeholt et al., 2018] respectively used as policy improvement operators.
The large number of training episodes required by modern deep RL (DRL) algorithms as policy improvement operators makes the inner loop of such training schemes a computational bottleneck. In this paper we accelerate approximating best responses by introducing Best Response Expert Iteration (BRExIt). We extend Expert Iteration (ExIt) [Anthony et al., 2017], famously used in AlphaGo [Silver et al., 2016], by introducing opponent models (OMs) in both (1) the apprentice (a deep neural network), with the aim of feature shaping and learning a surrogate model, and (2) the expert (Monte Carlo Tree Search), to bias its search towards approximate best responses to the OMs, yielding better targets for the apprentice. BRExIt thus better exploits OMs that are available in centralized training approaches, or also in Bayesian settings which assume a given set of opponents [Oliehoek and Amato, 2014].
For the case of given OMs that are computationally too demanding for search, but which can be used to generate training games, we also test a variant of BRExIt that uses learned surrogate OMs instead. In the game of Connect4, we find BRExIt learns significantly stronger policies than ExIt with the same computation time, alleviating the computational bottleneck in training towards a best response.
Related work
BRExIt stands at the confluence of two streams of literature: improving the sample efficiency of the ExIt framework (Section 2.1), and incorporating opponent models within deep reinforcement learning (Section 2.2).
Expert Iteration and improvements
Expert Iteration [Anthony et al., 2017] combines planning and learning during training, with the goal of finding a parameterized policy π θ (where θ ∈ R n ) that maximizes a re-ward signal. The trained policy can either be used within a planning algorithm or act as a standalone module during deployment without needing environment models.
ExIt's two main components are (1) the expert, traditionally an MCTS procedure [Browne et al., 2012], and (2) the apprentice, a parameterized policy π θ , usually a neural network. In short, the expert takes actions in an environment using MCTS, generating a dataset of good quality moves. The apprentice updates its parameters θ to better predict both the expert's actions and future rewards. The expert in turn uses the apprentice inside MCTS to bias its search. Updates to the apprentice yield higher quality expert searches, yielding new and better targets for the apprentice to learn. This iterative improvement constitutes the main ExIt training loop, depicted at the top of Figure 1.
Significant effort has been aimed at improving ExIt, e.g. exploring alternate value targets [Willemsen et al., 2021] or incorporating prioritized experience replay [Schaul et al., 2015], informed exploratory measures [Soemers et al., 2020] or domain specific auxiliary tasks for the apprentice [Wu, 2019]. BRExIt improves upon ExIt by deeply integrating opponent models within it.
Opponent modelling in DRL
Policy reconstruction methods predict agent policies from environment observations via OMs [Albrecht and Stone, 2018], which has been shown to be beneficial in collaborative [Carroll et al., 2019], competitive [Nashed andZilberstein, 2022] and mixed settings [Hong et al., 2018]. Deep Reinforcement Opponent Modelling (DRON) was one of the first works combining DRL with opponent modelling [He et al., 2016]. The authors used two networks, one that learns Q-values using Deep Q-Network (DQN) [Mnih et al., 2013], and another that learns an opponent's policy by observing hand-crafted opponent features. Their key innovation is to combine the output of both networks to compute a Q-function that is conditioned on (a latent encoding of) the approximated opponent's policy. This accounts for a given agent's Q-value dependency on the other agents' policies. Deep Policy Inference Q-Network (DPIQN) [Hong et al., 2018] brings two further innovations:
(1) merging both modules into a single neural network, and (2) baking the aforementioned Q function conditioning into the neural network architecture by reusing parameters from the OM module in the Q function.
OMs within MCTS have been shown to improve search [Timbers et al., 2022] when assuming access to ground truth opponent models, or partially correct models [Goodman and Lucas, 2020]. As auxiliary tasks for the apprentice, OMs have been used in ExIt to predict the follow-up move an opponent would play in sequential games [Wu, 2019]. BRExIt both learns opponent models as an auxiliary task, and uses them inside of MCTS.
Background
Section 3.1 introduces relevant RL and game theory constructs, followed by the approach to opponent modelling used in BRExIt in Section 3.2 and the inner workings of ExIt in Section 3.3.
Multiagent Reinforcement Learning
Let E represent a fully observable stochastic game with n agents, state space S, shared action space A and shared policy space Π. Policies are stochastic mappings from states to actions, with π i : S × A → [0, 1] denoting the ith agent's policy, and π = [π 1 , . . . , π n ] the joint policy vector, which can be regarded as a distribution over the joint action space. T : S × A × S → [0, 1] is the transition model (the environment dynamics), determining how an environment state s changes to a new state s given a joint action a. R i : S × A × S → R is agent i's reward function. G i t ∈ R is the return from time t for agent i, the accumulated reward obtained by agent i from time t until episode termination; γ ∈ (0, 1] is the environment's discount factor. When considering the viewpoint of a specific agent i, we decompose a joint action vector a = (a i , a −i ) and joint policy vector π = (π i , π −i ) into the individual action or policy for agent i, and the other agents denoted by −i.
The state-value function V π i : S → R denotes agent i's expected cumulative reward from state s onwards assuming all agents act as prescribed by π.
V π i (s) = a∈A π(a|s) s ∈S T (s |s, a i , a −i )[R i (s, a i , a −i , s ) + γV π i (s )]
Agent i's optimal policy depends on the other policies:
π * i (·|s) = arg max πi V (πi,π −i ) i (s)(1)
Assuming π −i to be stationary, agent i's optimal policy π * i is also called a best response π * i ∈ BR(π −i ), where BR(π −i ) denotes the set of all best responses against π −i . Our goal is to train a system to (1) predict and encode what is knowable about the opponents' policies π −i and (2) compute a best response to that.
Opponent modelling in DRL
We follow the opponent modelling approach popularized by DPIQN [Hong et al., 2018], which uses a neural network to both learn opponent models and an optimal Q-function. The latter is learnt by minimising the loss function L Q of DQN [Mnih et al., 2013]. Opponent modelling is an auxiliary task, trained by minimising the cross-entropy between one-hot encoded observed action for each agent j, a j , and their corresponding predicted opponent policies at state s, π j (·|s), defined as the policy inference loss L P I in Equation 2. These two losses are combined into L DP IQN with an adaptive weight to improve learning stability.
L P I = − 1 N N j=0
a j log(π j (·|s)) (2a)
L DP IQN = 1 √ L P I L Q + L P I (2b)
BRExIt makes use of both the policy inference loss for its OMs and the adaptive learning weight to regularize its critic loss. However, instead of learning a Q-function as a critic, BRExIt learns a state-value function V , as suggested by previous work [Hernandez-Leal et al., 2019].
Expert Iteration
We use an open-loop MCTS implementation [Silver et al., 2018] as the expert; tree nodes represent environment states s, and edges (s, a) represent taking action a at state s. Each edge stores a set of statistics:
{N (s, a), Q(s, a), P (s, a), i, A n } (3) N (s, a) is the number of visits to edge (s, a). Q(s, a) is the mean action-value for (s, a), aggregated from all simulations that have traversed (s, a). P (s, a) is the prior probability of following edge (s, a). ExIt uses the apprentice policy to compute priors, P (s, a) = π θ (a|s). One of our contributions is adding opponent-awareness to the computation of these priors. Finally, index i denotes the player to act in the node and A n indicates the available actions.
For every state s encountered by an ExIt agent during a training episode, the expert takes an action a computed by running MCTS from state s and selecting the action of to the root node's most visited edge (s, a). During search, the tree is traversed using the same selection strategy as AlphaZero [Silver et al., 2018]. Edges (s, a) are traversed following the most promising action according to the PUCT formula:
arg max a Q(s, a) + C P U CT P (s, a) a N (s, a ) 1 + N (s, a) ,(4)
where C P U CT is a tunable constant. The apprentice π θ is a distillation of previous MCTS searches, and provides P (s, a), thus biasing MCTS towards actions that were previously computed to be promising. Upon reaching a leaf node with state s , we backpropagate a value given by a learnt state value function V φ (s ) with parameters φ ∈ R n trained to regress against observed returns. After completing search from a root node representing state s, the policy π M CT S (·|s) can be extracted from the statistics stored on the root node's edges. This policy is stored as a training target for the apprentice's policy head to imitate:
π M CT S (a|s) = N (s, a) a N (s, a )(5)
As shown in the top right corner of Figure 1 ExIt builds a dataset containing a datapoint for each timestep t: {s t , π M CT S (·|s t ), G i t } (6) The top left corner of Figure 1 shows an actor-critic architecture, used as the apprentice, with a policy head π θ , and a value head V φ . A cross-entropy loss is used to train the actor towards imitating the expert's moves, and a mean-square error loss is used to update the critic's state value function towards observed returns G i t .
BRExIt: Opponent modelling in ExIt
We present Best Response Expert Iteration (BRExIt), an extension on ExIt that uses opponent modelling for two purposes. First, to enhance the apprentice's architecture with opponent modelling heads, acting as feature shaping mechanisms -see Subsection 4.1. Second, to allow the MCTS expert to approximate a best response against a set of opponents: π M CT S ∈ BR(π −i ) -see Subsection 4.3. We visually compare BRExIt and ExIt in Figure 1.
Learning opponent models in sequential games
Previous approaches to learn opponent models used observed actions as learning targets, coming from a game theoretical tradition where individual actions can be observed but not the policy that generated them [Brown, 1951]. However, the centralized population-based training schemes which motivate our research already require access during training to all policies in the environment, both for the training agent and the opponents' policies. We further exploit this assumption by separately testing two options for the learning targets in the policy inference loss from Equation 2: the one-hot encoded observed opponent actions, or the full distributions over actions computed by the opponents during play. Prior work has typically focused on fully observable simultaneous games, where a shared environment state is used by all agents to compute an action at every timestep. Thus, if agent i wanted to learn models of its opponents' policies, storing (a) each of i's observed shared states and (b) opponent actions, was sufficient to learn opponent models. We extend this to sequential games, where agents take turns acting based on individual states, in the following way. BRExIt augments the dataset collected by ExIt, specified in Equation 6, by adding (1) the state which each opponent agent j = i observed and (2) either only the observed opponent actions or the full ground truth action distributions given by the opponent policies in their corresponding turn. The data collection process for sequential environments is described in Algorithm 1. Formally, for the BRExIt agent acting at timestep t and its n o opponents acting at timesteps t + 1 to t + n o , BRExIt adds to its dataset either the observed action of every agent j, or their policy π j evaluated at the state of their turn.
{s t , π M CT S (·|s t ), {s t+j , π j (·|s t+j )} no j=1 , G i t }(7)
Apprentice Representation
BRExIt's three headed apprentice architecture extends ExIt's actor-critic representation with opponent models as per DPIQN's design [Hernandez-Leal et al., 2019], depicted on the bottom left of Figure 1 for a single opponent. This architecture reuses parameters from the OM as a feature shaping mechanism for both the actor and the critic. It takes as input an environment state s t ∈ S for a timestep t, which can correspond to the observed state of any agent. It features 3 outputs:
(1) the apprentice's actor policy π θ (s t ) (2) the state-value
critic V π θ ,π Ψ φ (s t ) (3) the opponent modelsπ ψj (s t ) ∈π Ψ ,
where Ψ contains the parameters for all opponent models and ψ j ⊂ Ψ the parameters for opponent model head j.
On certain sequential games the distribution of states encountered by each agent might differ, and so the actor-critic head and each of the OM heads could each be trained on different state distributions. In practice this means that the output of one of these heads might only be usable if the input state s t for a timestep t comes from the distribution it was trained on; however, the OM can in any case help via feature shaping. This does not apply to simultaneous games where all agents observe the same state. Figure 1: Illustration of ExIt (top half) and BRExIt (bottom half) for a 2-player game. BRExIt bases the decision on how to compute edge priors based on whose turn a node corresponds to. For opponent nodes, during training, these action priors can come from the true opponent's policy or from the apprentice's opponent model headsπ. BRExIt adds an opponent modelling loss by also gathering the states observed by the opponent st+1 and either the output of their policy π1(·|st+1), or a corresponding one-hot encoded action.
Opponent modelling inside MCTS
BRExIt follows Equation 8 to compute the action priors P (s, a) from Equation 4 for a node with player index j, notably using opponent models in nodes within the search tree that correspond to opponents' turns. This process is exemplified in the lower middle half of Figure 1. Such opponent models can be either the ground truth opponent policies (the real opponent policies π −i ) by exploiting centralized training scheme assumptions, or otherwise the apprentice's learnt opponent modelsπ ψ . This is a key difference from ExIt, which always uses the apprentice's π θ to compute P (s, a).
P (s, a) = π θ (a|s) j == BRExIt player index π j −i (a|s)
For ground truth modelŝ π ψ j (a|s) For learnt opponent models (8) By using either the ground truth or learnt opponent models, we initially bias the search towards a best response against the actual policies in the environment. However, note that initial P (s, a) values will be overridden by the aggregated simulation returns Q(s, a), as with infinite compute MCTS converges to best response against a perfectly rational player, whose actions may deviate from the underlying opponent's policy. Thus, BRExIt's search maintains the asymptotic behaviour of MCTS while simultaneously priming the construction of the tree towards areas which are likely to be explored by the policies in the environment.
In contrast to BRExIt, ExIt biases its expert search towards a best response against the apprentice's own policy, by using the apprentice π θ to compute P (s, a) at every node in the tree. MCTS here assumes that all agents follow the same policy as the apprentice, whereas in reality agents might follow any arbitrary policy. Not trying to exploit the opponents it is trained against, ExIt generates more conservative searches. Recent studies show that this conservativeness can be detrimental in terms of finding diverse sets of policies throughout training for population-based training schemes [Balduzzi et al., 2019;Liu et al., 2021]. Instead, they advocate for a more direct computation of best responses against known opponents as a means to discover a wider area of the policy space. This allows the higher level training scheme to better decide on which opponents to use as targets to guide future exploration.
We argue that BRExIt has this property built-in, by actively biasing its search towards a best response against ground truth opponent policies. Future work could investigate this claim.
We could have designed BRExIt to be even more exploitative towards opponent models, by masking actions sampled from these models as part of the environment dynamics. While this would yield actions very specifically targeted to respond to the modelled policies, it makes such best reponses very brittle, and can be problematic especially for imperfect OMs. In contrast, we propose using opponent models as priors in BRExIt, such that planning can still improve upon the opponent policies; this results in more robust learning targets.
Algorithms 1, 2 and 3 depict BRExIt's data collection, model update logic and overarching training loop respectively for a sequential environment. Coloured lines represent our contributions w.r.t ExIt.
Algorithm 1: BRExIt data collection
Input: (apprentice π θ , opp. modelsπ ψ −i , critic V φ ) Input: Opponent policies: π −i Input: Environment: E = (P, ρ 0 ) 1 Initialize dataset: D = [ ]; 2 Initialize time t ← 0; 3 Sample initial state s 0 ∼ ρ 0 ; 4 while s t is not terminal do 5 Search: a t , tree = M CT S(s t , π θ , π −i , V ψ ); 6 Act in the game s t+1 , r t ∼ P(s t , a t ); 7 Get from tree: π M CT S (s t , a) = Nroot(st,a) a Nroot(st,a ) ; 8 for j = 1, . . . , |π −i | do 9 Sample opp. action: a t+j ∼ π j −i (s t+j ); 10
Act in the game s t+j , ∼ P(s t+j , a t+j )
11 end 12 D ∪ {s t , π mcts (s t ), {s t+j , π j −i (s t+j )} |πo| j=1 , r i }; 13 t ← t + |π −i |; 14 end 15 return D;
Experiments & Discussion
We are trying to answer the following two questions: Primarily, is BRExIt more performant than ExIt at distilling a competitive policy against fixed opponents into its apprentice? Secondarily, are full distribution targets for learning opponent models preferable over one-hot action encodings?
The environment: We conducted our experiments in the fully observable, sequential two-player game of Connect4, which is computational amenable and possesses a high degree of skill transitivity [Czarnecki et al., 2020]. We decided on using a single environment in order to obtain statistically significant results through a larger number of runs over granular algorithmic ablations. We acknowledge the limitations of using a single test domain.
Test opponents:
We generated two test agents π weak , π strong by freezing copies of a PPO [Schulman et al., 2017] agent trained under δ = 0-Uniform self-Algorithm 2: BRExIt model update
Input: Three head network: N N = (π θ ,π ψ −i , V φ ) Input: Dataset: D 1 for t = 0, 1, 2, . . . do 2 Sample n datapoints from D: 3 (s t , r t , π M CT S (s t , ·), r i , {s t+j , π j −i (·|s t+1 )} |π ψ −i | j=1 ) 1,...,n ; 4 MSE value loss: L v = (v − V φ (s t )) 2 ; 5 CE policy loss L π = π M CT S (s t ) log (π θ (s t )); 6
CE policy inference loss:
L P I = 1 |π−i| |π −i | j=1 π j −i (s t+j ) log (π ψ j −i (s t+j )); 7 Policy inference weight: λ = 1 √ L P I ; 8 Weighted final loss L total = λ(L v + L π ) + L P I ; 9
Backpropagate ∇L total through θ, ψ, φ ; 10 end Algorithm 3: BRExIt training loop Input: Three head network: N N = (π θ ,π ψ −i , V φ ) Input: Opponent policies: π −i 1 for training iteration = 0, 1, 2, . . . Motivated by population-based training schemes, we also used an additional opponent policy π mixed , which randomly selects one of the test agents every episode.
Trained agents: We independently trained 7 types of agents for 48 wall-clock hours each, performing an additive construction from ExIt to BRExIt. ExIt is the original algorithm, ExIt-OMFS denotes ExIt using OMs only for feature shaping. BRExIt-OMS additionally uses learnt OMs during search and BRExIt uses the ground truth OMs during search. For the agents using OMs, we trained both a version using full action distributions as action targets and another with one-hot encoded action targets. Each algorithm was independently trained 10 times against the 3 test opponents, yielding a total of 280 training runs. Following statistical practices [Agarwal et al., 2021] we use Inter Quartile Metrics (IQM) for all results, discarding the worst and best performing 25% runs to obtain performance metrics less susceptible to outliers.
On BRExIt's performance vs. ExIt
To answer our first question, Figure 2 shows the evolution of the winrate of each of the ablation's apprentice policies throughout training. A datapoint was computed every policy update (i.e every 800 episodes) by evaluating the winrate of the apprentice policy against the opponent over 100 episodes. (Note that the difference in number of episodes between all ablations depends on the average episode length over the 48h of training time, which can vary as a function of both players involved.) Figure 3 analyzes these results and shows the probability of improvement (PoI) that one ablation has over another, defined as the probability that algorithm X would yield an apprentice policy which has a higher winrate against its training opponent that algorithm Y [Agarwal et al., 2021]. All agents are using full distribution OM targets here. Figure 2 shows that BRExIt style agents consistently achieve a higher winrate than ExIt agents. BRExIt regularly outperforms BRExIt-OMS (77% PoI), successfully exploiting the centralized assumption of having opponent policies available during search for extra performance. If opponent policies can only be sampled for training games but not during search, using learnt OMs during search is still beneficial, as we see that BRExIt-OMS consistently outperforms ExIt (90 % PoI). Surprisingly, ExIt-OMFS performs worse than ExIt by a significant margin (the latter has a 80% PoI against the former), providing empirical evidence that OMs with static opponents can be detrimental for ExIt if OMs are not exploited within MCTS. This goes against previous results [Wu, 2019], which explored OMs within ExIt merely as a feature shaping mechanism and claimed modest improvements when predicting the opponent's follow-up move. Differences may be attributed to the fact that we model the opponent policy on the current state, instead of the next state.
In summary, with BRExIt (using ground truth OMs) and BRExIt-OMS (using learnt OMs) featuring a > 97% and > 91% PoI respectively against vanilla ExIt, our empirical results warrant the use of our novel algorithmic variants instead of ExIt whenever opponent policies are available for training. Figure 3: Each row shows the probability (vertical marker) that the algorithm X (left) trains its apprentice's policy to reach a higher winrate than algorithm Y (right) after the allotted 48h. Colored bars indicate 95% bootstrap confidence intervals. Note that every training run for both BRExIt and BRExIt-OMS yielded higher performing policies than any run from ExIt-OMFS, and thus their comparisons have a 100% probability of improvement over ExIt-OMFS.
On full distribution VS one-hot targets
To answer our second question, we conducted Kolmogorov-Smirnov tests comparing full action distribution targets to one-hot encoded action targets for OMs. The samples we compared were the sets of winrates at the end of training, one datapoint per training run. Table 1 shows the results: There is no statistical difference between agents trained with algorithms using one-hot encoded OM targets when compared to using full action distributions. We obtain p 0.05 for each algorithmic combination, so we cannot reject the null hypotheses that both data samples come from the same distribution; the only exception being ExIt-OMFS, which shows a statistically significant decrease in performance when using one-hot encoded targets.
These results run contrary to the intuition that richer targets for opponent models will in turn improve the quality of the apprentice's policy. However, we observed that full distributional targets do yield OMs with better prediction capabilities, as indicated by lower loss of the OM during training. This hints at the possibility that OM's usefulness increases only up to a certain degree of accuracy, echoing recent findings [Goodman and Lucas, 2020]. Hence, while BRExIt does require access to ground truth policies during search, search in BRExIt-OMS achieves similar performance with opponent models trained on action observations, which is promising for transferring BRExIt-OMS into practical applications.
Conclusion
We investigated the use of opponent modelling within the ExIt framework, introducing the BRExIt algorithm. BRExIt augments ExIt by introducing opponent models both within the expert planning phase, biasing its search towards a best response against the opponent, and within the apprentice's model, to use opponent modelling as an auxiliary task. In Connect4, we demonstrate BRExIt's improved performance compared to ExIt when training policies against fixed agents. There are multiple avenues for future work. At the level of population based training schemes (such as self-play), future work can focus on measuring whether the quality of populations generated by different training schemes using BRExIt surpasses that of populations where ExIt is used as a policy improvement operator. In addition, search methods can struggle with complex games due to their high branching factor -simultaneous move games for example have combinatorial action spaces -which could be alleviated by using BRExIt's opponent models to narrow down the search space.
A Appendix
A.1 Training & benchmarking opponents
We used δ = 0-Uniform self-play [Hernandez et al., 2019], both for training our agents with BRExIt and its various ablations, and for training our test opponents with PPO. This self-play scheme makes a copy of the learning agent at the end of each episode, and adds it to its population -the set of available policies to be used as training opponents. At the beginning of each training episode, a policy is randomly sampled from the population as opponent for the learning agent. This self-play scheme is effective for relatively simple games like Connect4, because it ensures that the learning agent continues to face all policies discovered during training, preventing catastrophic forgetting and cyclic policy evolution, which leads to transitively better strategies.
Initially, we generated three internally monotonically stronger test opponents π test = [π 0 , π 1 , π 2 ], by freezing a copy of a PPO [Schulman et al., 2017] agent trained under δ = 0-Uniform self-play after 200k, 400k, and 600k episodes respectively. Table 2 shows the hyperparameters used to train these agents. No formal hyperparameter sweep was conducted, and the final values were chosen after a few manual trials. We experimented with frame stacking to add a temporal dimension to the state space, but it was ultimately discarded as it lead to weaker agents for our compute budget.
To evaluate the relative strength of the test opponents, we computed their corresponding winrate evaluation matrix to study pair-wise agent performances, as shown in Figure 4. We see that later agents can consistently beat earlier versions. This shows a transitive improvement within the population of agents that emerges from our single training run. However, the nature of PPO agents is reactive, as there is no planning involved during their training. This means that agents can be internally better because they learn how to exploit specific weaknesses of previous agents, which might not be present in unrelated agents, for example agents generated via planning based methods such as BRExIt. Because BRExIt uses planning to choose its actions during training, we ultimately want to give a ranking to our test agents against MCTS based methods. In order to label our test agents with the labels of weak and strong used in the main text, we use MCTS equivalent strength, which we define as the computational budget (number of MCTS iterations) required for an MCTS agent using random rollouts to reach a given winrate against a given agent policy π. Figure 5 shows a sweep of MCTS equivalent strength up to 80% winrate. Surprisingly, we see that the test agent with only 200K training episodes, even though it is the weakest when matched directly against other agents from the test population as seen in Figure 4, is the strongest against MCTS with an MCTS equivalent strength of 95. We hypothesize that in the relatively low computational budgets that we have used for the agents trained in Section 5 of the main text, the heavily stochastic behaviour of the 200K agent can thwart shallow plans made by low budget MCTS agents. The MCTS equivalent strength for both other agents is 79. Given the superiority of the agent trained for 600K episodes in Figure 4 when compared to the agent trained for 400K, and their equivalent strengths against MCTS, we ultimately decided to discard the 400K agent as a test opponents for our main experiments. It is from this strength analysis against MCTS that we label the 600K and 200K agents as weak and strong respectively in the main paper.
A.2 Neural network architectures
In order to level the playing field for all algorithms, we adjusted the actor-critic architecture, used by ExIt, and the opponent modelling enhanced actor-critic architecture, used by BRExIt and its ablations, to feature a similar number of parameters (approximately 27,000 trainable parameters).
The game of Connect4 is played on a 6 × 7 grid, and each grid position can either be empty, occupied by one of player 1's chips, or by one of player 2's chips. The input to PPO's and BRExIt's neural network models was a 3 × 6 × 7 tensor (a 3 channeled 6 × 7 grid): The first channel features 0s on all non-empty positions and 1s on empty positions; the second channel features 1s on positions taken by the current player and 0s everywhere else; the third channel features 1s on positions taken by the opponent and 0s everywhere else.
We used a similar network architecture to [Hernandez et al., 2019]. The input tensor was put through 5 convolutional layers, with residual connections skipping every other layer. Extra details are shown in Table 2. The output embedding found at the end of the last convolutional layer was fed into two parallel layers. One was a fully connected layer with a softmax activation function which represented the agent's policy (the actor). The other was a single neuron and no activation function representing a state-value function (the critic). Table 1 in Section 5 shows the p-values corresponding to the different two-sided Kolmogorov-Smirnov tests. This test measures if there is a significant statistical difference between 2 distributions f (x) and g(x). In our case, f, g correspond to an algorithmic ablation (i.e BRExIt, BRExIt-OMS and ExIt-OMFS), with g representing the same ablation as f but using one hot action encodings for OM targets instead of full action distributions.
A.3 Kolmogorov-Smirnov test
x denotes a test agent policy (π weak , π strong or π mixed ). Hence, f (x), g(x) denote the stochastic functions that determine the final winrate obtained by f or g respectively at the end of a training run against policy x. Each of the 6 training runs used for each algorithmc ablation (originally 10, but we discard the top and bottom 2 performing runs as per [Agarwal et al., 2021]) gives us one data point. Thus, our comparisons are between sets of 6 datapoints
f (x) = [f 1 (x), f 2 (x), f 3 (x), f 4 (x), f 5 (x)
, f 6 (x)] and similarly g(x) = [g 1 (x), g 2 (x), g 3 (x), g 4 (x), g 5 (x), g 6 (x)].
A.4 Hyperparameters and general performance improvements
Deep reinforcement learning algorithms require a large amount of simulated experiences to train policies, and more so for deep multi-agent reinforcement learning. This is further exacerbated when planning based algorithms are used as part of RL algorithms, as is the case with MCTS within BRExIt. A single step in the main game requires many simulated games inside of MCTS. On top of this, RL methods do not tend to be robust to hyperparameters, requiring a lot of manual tuning and stabilizing implementation details. Because of this, it is common place to add ad-hoc or environment specific methods to reduce the amount of compute required to train policies and to increase their robustness. Here we present some performance improvements shared across all algorithmic ablations, which are orthogonal to the novel contributions presented in this paper but were nonetheless required to train our agents within our computational budget. We argue that these improvement methods benefit all ablations equally, and thus do not affect our comparisons. We present them here for completeness and to aid reimplementations efforts. The set of hyperparameters used for all algorithmic ablations during the final training run is present in Table 3.
Reducing exploration: removing Dirichlet noise on MCTS root priors
The algorithm AlphaGo [Silver et al., 2018] introduced the notion of adding noise to the prior probabilities in the root node. By adding noise, we modify the priors over actions used in the selection phase, given by the apprentice's statevalue function. This is used as an exploration mechanism, by redistributing some probability weight given by the priors over all valid actions. The Dirichlet distribution is used to sample that noise: P (s, a) = 0.25 * Dirichlet(α) + 0.75 * P (s, a), where α = √ 2. During an MCTS search with many hundreds of iterations (as was the case with AlphaZero), the explorative effects of this Dirichlet noise on the root node's priors would eventually be washed out by state evaluations propagated upwards from further down the tree. In our case however, due to our relatively low MCTS budget of 50, we cannot afford to use Dirichlet noise, as this exploratory noise would randomize the output policy of the search π M CT S too much. Sample efficiency: Data augmentation via exploiting state space symmetries The board of Connect4, a 6 × 7 matrix, is symmetrical over the vertical axis. We can exploit this to reduce the complexity of the environment. Let's define a function over states σ(S) → S which swaps columns with indexes [1, 2, 3] with those with index [7, 6, 5]; and also over policies σ(Π) → Π by also swapping the probability weight over those column indices. Strategically speaking, the swapped situations are identical. AlphaZero was shown to be able to naturally learn this symmetry over enough training episodes [Silver et al., 2018]. We add this symmetry a priori into our algorithm to reduce computational costs. This is done by augmenting every data point at the time it is appended to the replay buffer by also adding a copy with the corresponding symmetric transformation, on both the state and the policy target (and the opponent policy target in case of BRExIt). The value target remains the same. Formally, for every data point we obtain a tuple:
{s t , π M CT S (·|s t ), G i t } → ({s t , π M CT S (·|s t ), G i t }, {σ(s t ), σ(π) M CT S (·|s t ), G i t })
Variance reduction on value targets: Averaging MCTS estimated returns with episodic returns In the original ExIt specification, as described in Equation 6, at every timestep of each training episode, the following information is stored in a replay buffer: {s t , π M CT S (·|s t ), G i t }, where G i t denotes player i's total reward observed at the end of the episode. What this means for our state value function V (s t ) is that we are regressing all observed states in an episode [s 0 , s 1 , . . .] to the same value G i t . This target features a large variance, as we are mapping potentially dozens of states to the same target value. We can reduce the variance, with the downside of introducing some bias, by re-using statedependent computation from each state-dependent MCTS call, meaning that we don't use the same value target for each state anymore. For every data point we substitute G i t by another variable z i t , for which we tried three different alternatives:
z i t = 1 2 * ( a∈A π M CT S (a|s) * Q(s, a) Definition of V π M CT S (s) +G i t )(9)
First, we averaged the original target with the root node value of MCTS. This includes the value of sub-optimal actions that were sampled during exploration. This in turn induced pessimism in V (s) estimations by our trained critics V φ (s), leading them to consistently estimate lower winrates than observed in practice.
z i t = 1 2 * (max a∈A * Q(s, a) Greedy w.r.t Q(s, ·) +G i t )(10)
Second, we tried using the value of the root node's highest valued child node. This also introduced a bias, although this time in a positive direction, because the policy that will actually be followed corresponds to MCTS's normalized child visitations (Equation 5) -which will put some probability weight on lower valued child nodes as well.
z i t = 1 2 * (Q(s, arg max a∈A N (s, a)) MCTS action selection +G i t )(11)
Ultimately, we decided to average the episode returns with the Q(s, a) value associated with the action actually selected and returned by MCTS. In our experiments, this was the most visited action at the root. Theoretically this still yields biased targets, as our actors π θ imitate an MCTS derived policy π M CT S while our critic V φ will regress against state-values that do not exactly match the actions taken by π M CT S . However, not only did this seem to work well in practice, but we saw little to no bias in our critic's estimations compared to the real episode returns using this bootstrapped estimate.
Extra exploitation: Temperature parameter It is beneficial to have a high degree of exploration on the initial states of an environment, specially during early stages of training. The policy targets derived from normalized child visits at the root node π M CT S feature a high degree of exploration, especially with our low computational budget, as MCTS would discard low quality moves given larger computational budgets. This is convenient early during training for the aforementioned exploration on initial states, but can slow down the learning of exploitative policies against fixed agents. Thus, ideally we would like to focus initial moves on exploration, and switch to exploitation later into an episode.
We achieve this with temperature parameter τ , which we use to exponentiate the visits to root node children before they are normalized:
We set τ = 1 for the first 10 moves, after which we reduce it to τ = 0.01, which effectively moves all probability weight to the most visited action. Therefore, the first 10 moves will feature increased exploration, and all other moves will focus on the most-visited action. We note that there exists research on this area, with a focus on removing exploration elements from MCTS policy targets with the hope of aiding interpretability [Soemers et al., 2019].
Gradient norm clipping
Many RL algorithms which involve computing gradients with respect to a loss function suffer from high variance in the estimation of these gradients. If the parameters of a model are updated by applying gradients of a large magnitude, these might move the model's parameters too far, even into poor performing sections of the parameter space. A proposed solution is to introduce gradient clipping [Pascanu et al., 2013]. Its most naive implementation involves clipping the value of each individual gradient to a hyperparameter threshold c. A more nuanced alternative is gradient norm clipping. It involves concatenating all gradients of all parameters into a single vector g. If the vector's nth norm is greater than a threshold value c, then the vector is normalized according to:
g = g if||g|| n ≤ c c g ||g||n otherwise(13)
For all experiments we used the L 2 norm. We tried thresholds of 5, 3 and 1. Even though we found that higher thresholds yielded a sharper increase in performance early on, a threshold of c = 1 ended up achieving better stability and final performance than the other values.
Parallel data generation: Parallelized MCTS with centralized apprentice
We simulate multiple games of Connect4 in parallel, With every MCTS procedure running on an individual CPU core. A single process hosts the apprentice's neural network, acting as a server, whose only function is to receive states from all MCTS processes. These are evaluated for (1) node priors P (s, a) and (2) state evaluations V (s). It busy polls all connections to MCTS processes for incoming states, batching all requests into a single neural network forward pass. This allows us to scale horizontally, theoretically linearly with the number of available CPUs. Because of the size of the network, Connect4's small state size and the average batch size, we saw no performance improvement hosting the apprentice in CPU vs GPU. Surprisingly, our main bottleneck came from Python's communication of PyTorch tensors among processes, stalling MCTS speed while requesting node evaluations V (s) and node prior generations P (s, a).
A.5 Description of computational infrastructure
The experiments presented in this publication were carried out in Snellius, the Dutch National supercomputer, a SLURM based computing cluster. Each of the 280 runs from Section 5 were individual SLURM jobs that run for 48h. They each had access to 1280GB of RAM and 64 CPUs with no GPUs.
do 2 Algo. 1 :
21D = DatasetCollection(N N, π −i ); 3 Algo. 2: N N = U pdateApprentice(N N, D); 4 end 5 return NN play [Hernandez et al., 2019] after 200k and 600k episodes.
Figure 2 :
2The evolution of winrates for each ablation during training vs fixed opponents for 48h wall-clock time. Lines represent the mean value the of final apprentice's winrate over all runs. Higher is better; shaded areas show 95% bootstrap confidence intervals.
Figure 4 :
4Winrates of πrow against πcolumn, where each entry was computed by playing 1000 head-to-head matches in the game of Connect4. The color of each agent (first player or second player) was chosen randomly at the beginning of each match to enforce symmetry. The agent IDs 0, 1 and 2 correspond to the 200K, 400K and 600K agents respectively.
π M CT S (a|s) = N (s, a) 1/τ a N (s, a ) 1/τ
Table 1 :
1Testing potential improvements of full distribution targets to one-hot encoded action targets. p-values are from two-sample Kolmogorov-Smirnov tests.Base Algorithm Opponent p-value Distr. targets
signif. better?
BRExIt
Weak
0.930
No
BRExIt-OMS
Weak
0.931
No
ExIt-OMFS
Weak
0.930
No
BRExIt
Strong
0.930
No
BRExIt-OMS
Strong
0.999
No
ExIt-OMFS
Strong
0.930
No
BRExIt
Mixed
0.142
No
BRExIt-OMS
Mixed
0.930
No
ExIt-OMFS
Mixed
0.025
Yes
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Table 2 :
2PPO hyperparameters used to generate test opponents. Empty entries in rightmost column indicate that a single value was used.Hyperparameter name
Value
Values explored
Horizon (T)
2048
512, 1024, 2048
Adam stepsize
10 −5
-
Num. epochs
10
5, 10
Minibatch size
256
128, 256, 512
Discount (γ)
0.99
-
GAE parameter (λ)
0.95
-
Entropy coefficient
0.01
-
Clipping parameter ( )
0.2
-
Gradient norm clipping
1
1, 5
Channels
[3, 10, 20, 20, 20, 1]
-
Kernel sizes
[3, 3, 3, 3, 3]
-
Paddings
[1, 1, 1, 1, 1]
-
Strides
[1, 1, 1, 1, 1]
-
Residual connections
[0,1], [1,2], [2,3], [3,4]
-
15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100
MCTS iteration budget
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Winrate
Trained episodes
200k
400k
600k
Figure 5: Evolution of winrate by MCTS against all candidate test
agents (excluding π mixed ). As the MCTS computational budget in-
creases on the horizontal axis, so does its playing strength. The black
horizontal dashed line at the 80% winrate mark is shown to compare
the required budget to reach such winrate against all test opponents.
Table 3 :
3BRExIt, BRExIt-OMS, ExIt and ExIt-OMFS hyperparameters. Hyperparameter tuning was done manually and in a limited way due to limited computational resources. Empty entries in the rightmost column indicate that a single value was used. The neural network architecture shared across all agents was adapted from the originalDPIQN [Hong et al., 2018].Hyperparameter name
Value
Values explored
MCTS budget
50
30, 50, 80
MCTS rollout budget
0
∞, 0
MCTS exploration factor
2.
1, 2, 5
Dirichlet noise
False
True, False
Batch size
512
128, 256, 512
Espisodes per generation (train every n episodes)
800
300, 500, 800, 1000
Epochs per iteration
5
1, 3, 5, 10
Reduce temperature after n moves
10
∞, 10
Channels
[12, 15, 20, 20, 20, 1]
-
Kernel sizes
[3, 3, 3, 3, 3]
-
Paddings
[1, 1, 1, 1, 1]
-
Strides
[1, 1, 1, 1, 1]
-
Residual connections
[[0,1], [1,2], [2,3], [3,4]]
-
Learning rate
1.5e-3
1e-3, 1.5e-3
Post feature extractor hidden units
[128, 128, 128]
-
Post feature extractor policy inference hidden units
[128, 64]
-
Post feature extractor actor critic hidden units
[128, 64]
-
Activation function
ReLu
-
Critic activation function
tanh
-
Gradient norm clipping
1
None, 1, 5
Iterative solution of games by fictitious play. Activity analysis of production and allocation. Agarwal, arXiv:1912.06680abs/1802.01561Advances in Neural Information Processing Systems. IEEE34Albrecht and StonearXiv preprintCEC 2020References [Agarwal et al., 2021] Rishabh Agarwal, Max Schwarzer, Pablo Samuel Castro, et al. Deep reinforcement learning at the edge of the statistical precipice. Advances in Neural Information Processing Systems, 34, 2021. [Albrecht and Stone, 2018] Stefano V Albrecht and Peter Stone. Autonomous agents modelling other agents: A comprehensive survey and open problems. Artificial In- telligence, 258:66-95, 2018. [Anthony et al., 2017] Thomas Anthony, Zheng Tian, and David Barber. Thinking fast and slow with deep learning and tree search. In NIPS, 2017. [Balduzzi et al., 2019] David Balduzzi, Marta Garnelo, Yoram Bachrach, et al. Open-ended learning in symmetric zero-sum games. In International Conference on Machine Learning, pages 434-443. PMLR, 2019. [Berner et al., 2019] Christopher Berner, Greg Brockman, Brooke Chan, et al. Dota 2 with large scale deep reinforce- ment learning. arXiv preprint arXiv:1912.06680, 2019. [Brown, 1951] George W Brown. Iterative solution of games by fictitious play. Activity analysis of production and allo- cation, 13(1):374-376, 1951. [Browne et al., 2012] Cameron B Browne, Edward Powley, Daniel Whitehouse, et al. A survey of Monte Carlo Tree Search methods. IEEE Transactions on Computational In- telligence and AI in Games, 4(1):1-43, 2012. [Carroll et al., 2019] Micah Carroll, Rohin Shah, Mark K Ho, et al. On the utility of learning about humans for human-ai coordination. In Advances in Neural Informa- tion Processing Systems, pages 5175-5186, 2019. [Czarnecki et al., 2020] Wojciech M Czarnecki, Gauthier Gidel, Brendan Tracey, et al. Real world games look like spinning tops. Advances in Neural Information Processing Systems, 33:17443-17454, 2020. [Espeholt et al., 2018] Lasse Espeholt, Hubert Soyer, Remi Munos, et al. IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures. CoRR, abs/1802.01561, 2018. [Goodman and Lucas, 2020] James Goodman and Simon Lucas. Does it matter how well I know what you're think- ing? opponent modelling in an RTS game. In IEEE Congress on Evolutionary Computation, CEC 2020, pages 1-8. IEEE, 2020.
Opponent modeling in deep reinforcement learning. International Conference on Machine Learning. et al., 2016] He He, Jordan Boyd-Graber, Kevin Kwok, and Hal Daumé III. Opponent modeling in deep reinforce- ment learning. In International Conference on Machine Learning, pages 1804-1813, 2016.
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arxiv |
Universal Power-law Decay in Hamiltonian Sys- tems? We thank Dima Shepelyansky for candid discussions
16 Oct 2002
M Weiss
Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik
Universität Göttingen
Ger-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany
L Hufnagel
Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik
Universität Göttingen
Ger-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany
R Ketzmerick
Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik
Universität Göttingen
Ger-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany
Universal Power-law Decay in Hamiltonian Sys- tems? We thank Dima Shepelyansky for candid discussions
16 Oct 2002arXiv:nlin/0210033v1 [nlin.CD] PACS numbers: 05.45.Mt [1] B. V. Chirikov and D. L. Shepelyansky, Phys. Rev. Lett. 82, 528 (1999). [2] M. Weiss, L. Hufnagel, and R. Ketzmerick, nlin.CD/0106021.
The understanding of the asymptotic decay of correlations and of the distribution of Poincaré recurrence times P (t) has been a major challenge in the field of Hamiltonian chaos for more than two decades. In a recent Letter, Chirikov and Shepelyansky [1] claimed the universal decay P (t) ∼ t −3 for Hamiltonian systems. Their reasoning is based on renormalization arguments and numerical findings for the sticking of chaotic trajectories near a critical golden torus in the standard map. We performed extensive numerics and find clear deviations from the predicted asymptotic exponent of the decay of P (t). We thereby demonstrate that even in the supposedly simple case, when a critical golden torus is present, the fundamental question of asymptotic statistics in Hamiltonian systems remains unsolved.As in Ref.[1] we study the standard mapat K = K c = 0.97163540631, where the golden torus is critical (Fig. 1, inset). We determine the Poincaré recurrence time distributions P (t) for trajectories starting below and above the critical golden torus by using the same numerical approach as in Ref.[1]. By considerably increasing the statistics we are able to extend the distribution by almost two orders of magnitude in recurrence times. We verify that our statistical data are not affected by the unavoidable finite numerical precision by comparing data for double (≈16 significant digits) and quadruple (≈32 digits) precision. The data for approaching the critical golden torus from above and below are presented inFig. 1. For times t < 10 8 our data agree with the results presented inFig. 2of Ref.[1]. For larger times, however, we find strong deviations from the predicted universal power law P (t) ∼ t −3 (dashed lines inFig. 1). The deviations might be explained in two ways: The onset of the claimed asymptotic decay might occur for larger times, which is in contradiction to the prefactors determined in Ref.[1]. On the other hand, the long-time trapping of chaotic trajectories might be dominated by islands of stability (non-principal resonances) that are neglected by the renormalization arguments. In fact, the latter possibility is supported by a detailed investigation [2]. If even in the supposedly simple case of a critical golden torus the decay P (t) ∼ t −3 is not observed, the claim for a universal existence of this decay cannot be maintained. It thus remains a fundamental challenge in the field of Hamiltonian chaos whether the asymptotic behavior of P (t) follows a universal power law and what the value of its exponent would be. 3.9⋅10 12 t −3 t q 0 2π p 0 π FIG. 1. The Poincaré recurrence time distribution P (t) for the standard map at K = Kc for trajectories approaching the critical golden torus from above (upper curve) and from below (lower curve, shifted by 10 −3 ). For large times we find clear deviations from the predictions of Ref. [1] (dashed lines and symbols). Inset: Phase space of the symmetrized standard map at K = Kc, with arrows pointing at the critical golden torus.
The understanding of the asymptotic decay of correlations and of the distribution of Poincaré recurrence times P (t) has been a major challenge in the field of Hamiltonian chaos for more than two decades. In a recent Letter, Chirikov and Shepelyansky [1] claimed the universal decay P (t) ∼ t −3 for Hamiltonian systems. Their reasoning is based on renormalization arguments and numerical findings for the sticking of chaotic trajectories near a critical golden torus in the standard map. We performed extensive numerics and find clear deviations from the predicted asymptotic exponent of the decay of P (t). We thereby demonstrate that even in the supposedly simple case, when a critical golden torus is present, the fundamental question of asymptotic statistics in Hamiltonian systems remains unsolved.
As in Ref.
[1] we study the standard map q n+1 = q n + p n mod 2π p n+1 = p n + K sin q n+1 , (1) at K = K c = 0.97163540631, where the golden torus is critical ( Fig. 1, inset). We determine the Poincaré recurrence time distributions P (t) for trajectories starting below and above the critical golden torus by using the same numerical approach as in Ref.
[1]. By considerably increasing the statistics we are able to extend the distribution by almost two orders of magnitude in recurrence times. We verify that our statistical data are not affected by the unavoidable finite numerical precision by comparing data for double (≈16 significant digits) and quadruple (≈32 digits) precision. The data for approaching the critical golden torus from above and below are presented in Fig. 1. For times t < 10 8 our data agree with the results presented in Fig. 2 of Ref.
[1]. For larger times, however, we find strong deviations from the predicted universal power law P (t) ∼ t −3 (dashed lines in Fig. 1). The deviations might be explained in two ways: The onset of the claimed asymptotic decay might occur for larger times, which is in contradiction to the prefactors determined in Ref.
[1]. On the other hand, the long-time trapping of chaotic trajectories might be dominated by islands of stability (non-principal resonances) that are neglected by the renormalization arguments. In fact, the latter possibility is supported by a detailed investigation [2]. If even in the supposedly simple case of a critical golden torus the decay P (t) ∼ t −3 is not observed, the claim for a universal existence of this decay cannot be maintained. It thus remains a fundamental challenge in the field of Hamiltonian chaos whether the asymptotic behavior of P (t) follows a universal power law and what the value of its exponent would be. We thank Dima Shepelyansky for candid discussions.
FIG. 1 .
1The Poincaré recurrence time distribution P (t) for the standard map at K = Kc for trajectories approaching the critical golden torus from above (upper curve) and from below (lower curve, shifted by 10 −3 ). For large times we find clear deviations from the predictions of Ref.[1] (dashed lines and symbols). Inset: Phase space of the symmetrized standard map at K = Kc, with arrows pointing at the critical golden torus.
M. Weiss 1,2 , L. Hufnagel 1 , and R. Ketzmerick 1 Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik der Universität Göttingen, Germany 2 EMBL, Meyerhofstr. 1, 69117 Heidelberg, Germany PACS numbers: 05.45.Mt [1] B. V. Chirikov and D. L. Shepelyansky, Phys. Rev. Lett. 82, 528 (1999). [2] M. Weiss, L. Hufnagel, and R. Ketzmerick, nlin.CD/0106021.1
| {'fraction_non_alphanumeric': 0.04388083735909823, 'fraction_numerical': 0.02509393451422437, 'mean_word_length': 4.537147102526003, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The understanding of the asymptotic decay of correlations and of the distribution of Poincaré recurrence times P (t) has been a major challenge in the field of Hamiltonian chaos for more than two decades. In a recent Letter, Chirikov and Shepelyansky [1] claimed the universal decay P (t) ∼ t −3 for Hamiltonian systems. Their reasoning is based on renormalization arguments and numerical findings for the sticking of chaotic trajectories near a critical golden torus in the standard map. We performed extensive numerics and find clear deviations from the predicted asymptotic exponent of the decay of P (t). We thereby demonstrate that even in the supposedly simple case, when a critical golden torus is present, the fundamental question of asymptotic statistics in Hamiltonian systems remains unsolved.As in Ref.[1] we study the standard mapat K = K c = 0.97163540631, where the golden torus is critical (Fig. 1, inset). We determine the Poincaré recurrence time distributions P (t) for trajectories starting below and above the critical golden torus by using the same numerical approach as in Ref.[1]. By considerably increasing the statistics we are able to extend the distribution by almost two orders of magnitude in recurrence times. We verify that our statistical data are not affected by the unavoidable finite numerical precision by comparing data for double (≈16 significant digits) and quadruple (≈32 digits) precision. The data for approaching the critical golden torus from above and below are presented inFig. 1. For times t < 10 8 our data agree with the results presented inFig. 2of Ref.[1]. For larger times, however, we find strong deviations from the predicted universal power law P (t) ∼ t −3 (dashed lines inFig. 1). The deviations might be explained in two ways: The onset of the claimed asymptotic decay might occur for larger times, which is in contradiction to the prefactors determined in Ref.[1]. On the other hand, the long-time trapping of chaotic trajectories might be dominated by islands of stability (non-principal resonances) that are neglected by the renormalization arguments. In fact, the latter possibility is supported by a detailed investigation [2]. If even in the supposedly simple case of a critical golden torus the decay P (t) ∼ t −3 is not observed, the claim for a universal existence of this decay cannot be maintained. It thus remains a fundamental challenge in the field of Hamiltonian chaos whether the asymptotic behavior of P (t) follows a universal power law and what the value of its exponent would be. 3.9⋅10 12 t −3 t q 0 2π p 0 π FIG. 1. The Poincaré recurrence time distribution P (t) for the standard map at K = Kc for trajectories approaching the critical golden torus from above (upper curve) and from below (lower curve, shifted by 10 −3 ). For large times we find clear deviations from the predictions of Ref. [1] (dashed lines and symbols). Inset: Phase space of the symmetrized standard map at K = Kc, with arrows pointing at the critical golden torus.', 'arxivid': 'nlin/0210033', 'author': ['M Weiss \nMax-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany\n', 'L Hufnagel \nMax-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany\n', 'R Ketzmerick \nMax-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany\n'], 'authoraffiliation': ['Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany', 'Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany', 'Max-Planck-Institut für Strömungsforschung and Institut für Nichtlineare Dynamik\nUniversität Göttingen\nGer-many 2 EMBL, Meyerhofstr. 169117HeidelbergGermany'], 'corpusid': 26883431, 'doi': '10.1103/physrevlett.89.239401', 'github_urls': [], 'n_tokens_mistral': 2064, 'n_tokens_neox': 1784, 'n_words': 1181, 'pdfsha': 'e1c1c32107195d7d1d1e81fe546d2bf4860eb729', 'pdfurls': ['https://export.arxiv.org/pdf/nlin/0210033v1.pdf'], 'title': ['Universal Power-law Decay in Hamiltonian Sys- tems? We thank Dima Shepelyansky for candid discussions', 'Universal Power-law Decay in Hamiltonian Sys- tems? We thank Dima Shepelyansky for candid discussions'], 'venue': []} |
arxiv |
Nonlinear spin dynamics of ferromagnetic ring in the vortex state and its application for spin-transfer nano-oscillator
25 May 2023
Vera Uzunova
Boris A Ivanov
Institute of Theoretical Physics
Faculty of Physics
Institute of Physics
University of Warsaw
ul. Pasteura 502-093WarszawaPoland
Institute of Magnetism
National Academy of Sciences of Ukraine
46 Nauky Ave03039KyivUkraine
and Institute for Molecules and Materials
National Academy of Sciences of Ukraine
03142KievUkraine
Radboud University Nijmegen
Heyendaalseweg 1356525 AJNijmegenThe Netherlands
Nonlinear spin dynamics of ferromagnetic ring in the vortex state and its application for spin-transfer nano-oscillator
25 May 2023(Dated: May 26, 2023)
We study a nonlinear spin dynamics of a ferromagnetic ring in a vortex state induced by the spin-polarized current. We also suggest to use the ferromagnetic ring as a free layer of a coreless vortex spin-transfer nano-oscillator. The calculated working frequency is about several GHz, that is much higher than the gyromode frequency of the disk-based vortex oscillator. The response of the vortex-state ring to the spin-polarized current has hysteretic behavior with the reasonable values of the thresholds current densities: ignition threshold is about 10 8 Acm −2 , and elimination current to maintain the oscillations has much lower values about 10 6 Acm −2 . The output signal can be extracted by the help of the inverse spin Hall effect or by the giant magnetoresistance. The output electromotive force averaged over all sample vanishes, and we suggest to use a ferromagnetic ring or disk in a vortex state as a GMR analyzer. For an inverse spin Hall analyser we advise to use two heavy metals with different signs of Spin-Hall angle. The ring-based STNO is supposed to increase the areas of practical application of the STNOs. * Electronic address: bor.a.
I. INTRODUCTION
Spin-polarized current flowing into a magnetic material creates a large torque acting on the magnetization due to the direct transfer of spin angular momentum, that opens up opportunities to manipulate mesoscopic magnetic elements [1]. Discovery of a spin transfer mechanism laid the foundation of a new generation of spintronic devices [2][3][4][5], including auto-oscillators [6][7][8][9], magnetic random access memory [10] and logic gates [11,12].
One of the key spintronic elements applicable in wireless communications technology are spin-transfer nanooscillators (STNOs), [13]. These devices are using the spin transfer torque to generate microwave signals by transforming energy from a dc electric current into high-frequency magnetic oscillations. Traditionally designed STNOs are fabricated as sandwich structures of magnetic and nonmagnetic nanolayers in forms of nanocontacts [14][15][16] or nanopillars, [17][18][19]. The main structural elements of the STNO are a free layer, i. e. a thin magnetic layer allowing rotation of the magnetization, a polarizer and an analyzer. Polarizer, the thicker magnetic layer with relatively fixed magnetization, gives a spin polarisation to direct current passing through the device. Above a certain critical current density its spin-transfer torque acting on a thinner free magnetic layer can locally overcome the intrinsic damping and excites a steady-state spin auto-oscillations on one or more spin-waves modes of the system. In the analyser these spin oscillations are converted to a microwave power by the magnetoresistive effect or by the Inverce Spin Hall effect.
Advantages of the STNOs are small size, broad range of working temperatures and easy integration into standard semiconductor technology. A characteristic feature of all STNOs is nonlinear dynamics of the free layer magnetization. Typical current densities for creating a steady-state spin dynamics are 10 7 − 10 8 Acm −2 in nanocontacts and 10 6 − 10 7 Acm −2 in nanopillars [20].
A promising alternative to the single-domain oscillators are vortex STNOs based on ferromagnetic nano-disks in a so-called vortex state. The vortex configuration has a closed magnetic flux and creates demagnetising fields only in a small region of the vortex core. Therefore magnetic vortex can realize the ground state of magnetic nanoparticles. The vortex STNOs are characterized by a uniquely narrow line width, require lower current densities and dissipate less energy comparing to single-domain oscillators [21,22]. They are useful for applications in arrays of oscillators, memory systems on vortex states, and others. The primary excitable mode in the vortex-state particle is the gyrotropic mode (gyromode) that corresponds to the slow precessional motion of the vortex around the disc center. The frequency of the gyromode is usually lower then 1 GHz. This slow dynamics of the vortex core is used in the standard design of the vortex-state STNO. Operating frequencies of the vortex oscillators are low compared to single-domain oscillators, where frequencies of 10 ÷ 15 GHz can be achieved.
The generated frequency can be doubled by using of a dual-free-layer STNO because of superposition of two opposite current-driven precessional motions of vortices in two free layers, [23]. One more way to increase the working frequency of the vortex-state STNOs is to use ferrimagnets near the point of compensation of the angular momentum. They have gyrotropic frequencies much higher than those of ferromagnets [24,25] that allows the operating frequency up to 20 GHz. Limitations of this design are related to the larger critical size of the vortex state ferrimagnetic disk with the thicknesses less than 10 nm, which is needed to spin-torque applications. Therefore, the problem how to rise the working frequency of the vortex-state STNO is quite challenging.
In this paper we investigate nonlinear spin dynamics of a ferromagnetic ring in a vortex state under an action of spin-polarized current and discuss the usage of the ferromagnetic ring as the base of a vortex-state STNO. Ferromagnetic nanorings are widely investigated due to their unique optical and electromagnetic properties, [26]; biomedical applications of vortex-state magnetic rings [27,28] are one of the current interests. The dynamics of magnetization in the vortex-state ring differs significantly from that for the vortex-state disk, [29]. The frequency of the lowest magnon mode for the ring is much higher then frequency of the gyromode of vortex-state magnetic disc, that opens up new opportunities to rise the working frequency of the STNO. The ring-based vortex oscillator is expected to have some significant advantages comparing to the case of oscillators based on vortex-state discs, providing high values of the working frequencies, and smaller size.
The adequate theoretical description of the non-linear spin dynamics for the vortex-state ring is much more complex then for the particles with uniform (or quasi-uniform) magnetization, which can be analysed within the macrospin approximation. In our work we used the collective variables approach, which is valid for highly-nonlinear regimes of oscillations for the wide variety of magnetic solitons, see e.g. [30]. We found the inertial behavior of the spin dynamics of the vortex-state ferromagnetic ring, formally common to one for antiferromagnetic STNO, with corresponding analogy of oscillators based on superconducting Josephson junction [31]. One of the important consequences is the lower value of the critical current, the elimination critical current, which is needed to support the generation once it has been started. Thus the ring-based STNO can have suitable geometric dimensions and threshold current densities and therefore is a good candidate for practical applications.
The article is organized as the following. In Section II basic equations describing magnetization dynamics in the system are given. The ground state of the ferromagnetic ring and spin dynamics above this ground state are considered in Sections III and IV, respectively. In Section V we obtain expressions for critical currents and working frequency. Finally, in Section VI we discus ways of extracting a useful ac electric output signal for the case of interest when the output electrical power averaged over the contact is zero.
II.
FERROMAGNETIC RING UNDER THE ACTION OF THE SPIN-POLARIZED CURRENT.
The problem under consideration is to investigate magnetization dynamics induced by the spin-polarized current in the a thin ferromagnetic ring. Spatial dimensions of the ring are: thickness L, inner and outer radii R i and R o respectively; ring is considered thin, L ≪ R i,o . Ground state of the ring is the vortex state, in which magnetization M aligns circularly around the ring axis. It is discussed in section III. In the Cartesian reference frame with the z-axis along the ring axis the components of the ground-state magnetization are M s (− sin θ 0 sin χ, sin θ 0 cos χ, cos θ 0 ), where χ is the azimuthal coordinate in the plane of the ring and θ 0 is the polar angle of the ground-state magnetization. The schematic picture of the vortex-state ring and its magnetization are shown in Fig. 1.
The spin-polarized current flows into the ring and its spin-polarization p is along z. The current flow is assumed to be uniform within the sample. Spin-polarized current creates a torque T, which tends to rotate the in-plane component of M in the plane of the ring.
In addition we are considering an option of applying a weak (with the value less than saturated) magnetic field H along z-axis: h = H/4πM s , where M s is the saturation magnetization of the ferromagnetic material of the ring. If the external magnetic field is zero, h = 0, magnetization M lies entirely in the plane of the ring and the polar angle θ 0 = π/2 , Fig. 1(a). The case of nonzero H is shown in Fig. 1(b). The small external field, h < 1, leads to appearance of a small out-of-plane component of magnetization equal to M s h. Then the polar angle θ 0 in the ground state is determined by the external field as cos θ 0 = h. We show below that small external field does not change the nature of the excitations but may give an additional way of the device control.
The dynamics of magnetization M is described by the Landau-Lifshitz equation Here W ≡ W [M] is the ferromagnet's energy written as a functional of the magnetization M, the vector −δW/δM has the sense of the effective field, γ = gµ B / is the gyromagnetic ratio, g is Lande factor, µ B is the Bohr magneton, is the Planck's constant, α is the dimensionless parameter of the Gilbert damping. The particular form of the energy functional W will be presented in the next section. The spin-transfer torque is given by
∂M ∂t = γ M × δW δM + α M s M × ∂M ∂t + T.(1T = σJM s [m × [m × p]],
where m = M/M s is the unit magnetization vector of the free layer, p is the spin-polarization of the current. The value of σ is given by the formula
σ = ǫgµ B 2eM s LS ,(2)
where e is the modulus of the electron charge, ǫ is dimensionless spin-polarization efficiency, and S is the area of the current-carrying region that in our case coincides with the area of the ring face.
Considering the spin dynamics of the ring it is convenient to parameterize the vector of normalized magnetization m by two angles: azimuthal angle ϕ and polar angle θ. Then the Cartesian components of m are (sin θ cos ϕ, sin θ sin ϕ, cos θ). The energy functional W can be also presented as a functional of these angles W = W [θ, ϕ]. Then the dynamic equations take the form
M s γ sin θ ∂ϕ ∂t = δW δθ + αM s γ ∂θ ∂t ,(3)− M s γ sin θ ∂θ ∂t = δW δϕ + αM s γ sin 2 θ ∂ϕ ∂t − σJM s sin 2 θ γ .(4)
To obtain the full time derivative of the energy in these terms we multiply Eq. (3) by ∂θ/∂t and Eq. (4) by ∂ϕ/∂t, sum them up and integrate over the ring volume. It gives
dW dt = − αM s L γ ∂θ ∂t 2 + sin 2 θ ∂ϕ ∂t 2 d 2 r + σJM s L γ sin 2 θ ∂ϕ ∂t d 2 r,(5)
where integration is performed over the area of the ring face. Since the ring is thin, the integration over z simply results in additional factor L.
Eq. (5) shows the change in energy due to non-conservative forces: the Gilbert damping and the spin pumping. The first term is the Gilbert dissipative function, leading to the friction force linear in velocity, as in Eqs. (3,4). Whereas the second term describes the income of energy due to the spin-transfer torque. As follows from Eq. (5), at a certain value of current J, these two terms can be compensated on average, giving dW/dt = 0. It means that a regime of a steady-state oscillations can exist in the system. Our goal is to investigate possible oscillations of the system and to find conditions necessary for this regime to appear. In the following two sections we consider in details the ground state of the ring and its excited states.
III. VORTEX STATE OF THE FERROMAGNETIC RING
Let us briefly discuss the ground state of the ring in the vortex state. The specific form of energy functional W [M] includes contributions from the exchange interaction energy and magnetostatic energy of the volume and surface magnetic charges,
W [M] = L A 2 (∇θ) 2 + sin 2 θ(∇ϕ) 2 + 2πM 2 s cos 2 θ − HM s cos θ d 2 r + W vol + W edge .(6)
For a thin magnetic particle, the problem is essentially two-dimensional and the vector r, over which the integration is performed, lies in the plane of the ring with an origin at its center. First term represents exchange energy with exchange constant A. The second term is the approximate contribution to the magnetostatic energy from upper and lower faces of the ring; this term can be treated as an effective easy-plane anisotropy, see e. g., [33]. This term is the largest of the magnetostatic terms in the case of a thin particle. The third term in the integrand is the Zeeman energy −(M · H) = −4πhM 2 s cos θ of the interaction with external magnetic field (if there is any) applied along the z-axis. Terms W vol and W edge denote the parts of the magnetostatic energy caused by volume (∇ · M) and edge (r · M)/r "magnetic charge densities", which are sources of demagnetizing field.
The ground state magnetization distribution in the ferromagnetic ring is the coreless magnetic vortex. Introducing planar coordinates (r, χ) in the plane of the ring and z-axis along the axis of the ring, we can write it as ϕ = χ + π/2 and θ = θ 0 (r), see Fig. 1. The function θ 0 (r) is a result of minimization of the energy functional (6). Here it is taken into account that volume and edge "magnetic charge densities" for both vortex-type configurations, in-plane and cone, do not contribute in the energy. So, the ground state distribution of θ 0 (r) is described by the solution of ordinary differential equation
l 2 0 d 2 θ 0 dr 2 + 1 r dθ 0 dr − 1 r 2 sin θ 0 cos θ 0 + sin θ 0 cos θ 0 − h sin θ 0 = 0,(7)
where l 2 0 = A/4πM 2 s is the exchange length, and h = H/4πM s is the normalized external magnetic field, see for details [35,36].
This equation describes vortex solutions for the out-of-plane projection of magnetization M ⊥ ; it is applicable to both disk-shaped and ring-shaped particles. For a zero external field, h = 0, the ground state corresponds to the in-plane vortex shown in Fig. 1(a), whereas for non-zero external field smaller then saturation value, h < 1, the cone vortex state shown in Fig. 1(b) is realized, [32]. Eq. (7) has a singularity at r → 0 that leads to appearance of vortex core (small region with out-of plane magnetization) for the ferromagnetic disk. However, for the ring the singular region of ferromagnet is just "cutted off" and the dynamics of excitations substantially changes. Size of the vortex core is about l 0 / √ 1 − h 2 and in what follows we assume that the field is not very close to saturation such that the core size do not exceed the inner radius of the ring. Therefore, for estimates in the main approximation in small parameters l 0 /R o and l 0 /R i asymptotic of Eq. (7) can be used: the out-of-plane component of magnetization far from the origin is M ⊥ = M s cos θ 0 = M s h(1 + l 2 0 /r 2 ), implying θ 0 (r) = π/2 in case h = 0. Note, that the magnetic field applied along z axis reduces the out-of-plane component of the magnetization and, as a consequence, suppresses the effect of the spin-polarized current.
IV. RADIALLY-SYMMETRIC NONLINEAR DYNAMICS
Excitation spectra of soft-magnetic nanoparticles are of interest both from a fundamental point of view and in context of their applications in spintronic devices. If such a particle is used as a working layer of STNO, the lower part of its spectrum determines the operating frequencies of the device. The linear dynamics of magnetic particles in the vortex state is well studied. Excitation modes in linear approximation are denoted by two integers: the number of radial nodes in the out-of-plane component of the dynamical magnetization, radial mode number n ≥ 1, and the azimuthal mode number m, which determines the angular dependence of this component. The spectrum of the vortex-state disk includes the gyrotropic mode (n = 1, m = 1) with low (sub-gigahertz) frequency, corresponding to the slow vortex core precession around the disc center, and a system of higher modes.
Contrary, in the spectrum of the vortex-state ring there is no low-frequency gyroscopic mode, since the vortex core is absent for the ring-shaped sample, [29]. At the same time, the higher mods for discs and rings are similar.
In the ring-shaped magnetic particle the mode with the lowest frequency is the breathing mode with the azimuthal number m = 0. It is characterized by the deviation of magnetization from the ground state of the form ϕ = χ + π/2 + ψ(r, t), θ = θ(r, t), with the small deviation ψ. The spin waves are described by the set of dynamic Landau-Lifshitz equations. There, Eqs. (3,4), the ferromagnetic energy W (θ, ϕ) contains, among other terms, the nonlocal contribution to the magnetostatic energy caused by the volume and edge magnetic charges, which greatly complicates the analytical description. However, in the linear approximation this problem can be solved for small values of parameter L/R o,i , see e.g., [33,34,37].
The challenge of our problem is that the spin dynamics excited by the spin-polarized current is essentially nonlinear and approaches suitable for small excitations do not work. For this reason, we use the collective variable approach, derived in works [38,39] to describe the nonlinear analogue of the breathing mode.
Within the collective variable approach applied to the case under cunsideration the Lagrangian of the ferromagnet can be written in the form
L = L γ M ⊥ dϕ dt d 2 r − W [M],(8)
where it is taken into account that magnetization M depends only on the in-plane coordinates. We assume that ϕ is coordinate-independent variable, and its time derivative dϕ/dt can be taken out of the integral, see for details [38,39].
With this assumption the breathing mode can be described as excitations with radial symmetry:
ϕ = χ + π/2 + ψ(t), θ = θ(r, t).
Here ψ(t) is the averaged value of ψ(r, t), the deviation from the ground state configuration. Then the Lagrangian describing the radially-symmetric dynamics of the vortex-state ring can be written through collective variables as
L = 1 γ µ dψ dt − W (µ, ψ),(9)
where ψ(t) has the meaning of generalized coordinate corresponding to the azimuthal rotation of the magnetization. The quantity µ/γ has the sense of generalized momentum conjugated to ψ(t); it is expressed through the total out-of-plane magnetic moment of the ring, µ, namely
µ γ = L γ M ⊥ d 2 r = LM s γ cos θd 2 r.(10)
In accordance with the collective coordinate approach, the Hamilton function W (µ, ψ) represents the ferromagnet's energy functional W [M] minimized over the distribution of magnetization M at fixed values of generalized variables µ and ψ. This dependencies of W (µ, ψ) need to be found in the main approximation, considering the ratios l 0 /R i,o to be small parameters.
The dependence of the energy W on the generalized momentum µ/γ can be understood by considering how the external magnetic field applied along the axis of the ring increases the out-of-plane component of the magnetization M ⊥ and transforms the in-plane magnetic vortex into the cone-state vortex. The minimum of the energy W (µ, ψ) at a given value of µ can be obtained by the method of indefinite Lagrange multipliers, see [39]. As a result, the extremals of the energy functional correspond to the solutions of the Eq. (7), where the role of magnetic field h is played by the Lagrangian multiplier h(µ). Thus, apart from the vortex core region we can assume that cos θ = h . The relation between µ and h is described by Eq. (10) and in the first order approximation has the form µ = M s LSh, where S = π(R 2 0 − R 2 i ) is the area of the ring face. To obtain the explicit form of the µ-dependence of the energy, we multiply Eq. (7) for the cone state by r 2 dθ/dr, integrate it over r, and substitute in the energy functional (6), see [40]. Finally,
W (µ) = 4πM s hµ + 4π 2 M 2 s l 2 0 L(1 − h 2 ) ln R o R i πΛ(h) − SL(2πM 2 s h 2 ),(11)
where the function Λ(h) ≈ 5.27 as h → 0 and linearly decreases to zero at h → 1, see [38].
The ψ-dependence of the ferromagnet energy W is determined by nonlocal effect of edge (r · M)/r = M s sin θ sin ψ and volume (∇ · M) = [(1/r)d(r sin θ)/dr] cos(ϕ − χ) magnetostatic charges, that can be estimated by asymptotic evaluation of the magnetostatic integral, [34,38]. In the leading (logarithmic) order we can write
W (ψ) = 2πM 2 s L 2 R o sin 2 θ 0 r=Ro ln ηR o L + R i sin 2 θ 0 r=Ri ln ηR i L sin 2 ψ.(12)
This expression includes the energy of the outer, r = R o , and inner, r = R i , edges of the ring. The contribution of the volume charge can be taken into account by a phenomenological coefficient η, it has values between 4 and 6, see. [33]. For the ring with R o,i ≫ L the value of sin θ 0 on the outer and inner edges can be replaced by their asymptotic value √ 1 − h 2 . Variables µ and ψ and can be considered as conjugated coordinate and momentum in accordance with Lagrangian (8,9) and the corresponding equations for them have the Hamiltonian form. Taking into account the explicit dependence of W on ψ, equations for collective variables µ and ψ can be written as
− 1 γ ∂µ ∂t = ∂W ∂ψ µ=const = (13) = 4π(1 − h 2 )M 2 s L 2 R o ln ηR o L + R i ln ηR i L sin ψ cos ψ, 1 γ ∂ψ ∂t = ∂W ∂µ ψ=const .(14)
To go further, note that the in-plane deviation of magnetization from the ground state weakly changes the ferromagnet energy, which makes possible nonlinear radial oscillations of the system. On the contrary, the deviation of the out-of-plane component M ⊥ strongly changes the vortex energy. This fact allows to separate in the main approximation the energy contributions from the azimuthal and out-of-plane deviations of magnetization. Since µ is a more rigid variable that does not contain any small parameter, it is sufficient to take into account µ in the linear approximation even for description of highly nonlinear dynamics with unlimited change of ψ. So, in the linear approximation on µ − µ 0 , the derivative ∂W/∂µ can be replaced by (µ − µ 0 )(d 2 W (µ)/dµ 2 )| µ=µ0 , where µ 0 the is equilibrium value of the out-of-plane component of the magnetic moment in the vortex core.
Then the dynamic equations take the form of the Hamilton equations for coordinate ψ and conjugated momentum p = (µ − µ 0 )/γ. Within this approximation, we can significantly simplify the Hamilton function of our problem
H = p 2 2m + W (ψ), 1 m = γ 2 d 2 W (µ) dµ 2 µ=µ0 ,(15)
where the effective mass m does not depend on ψ and p, and the potential energy W (ψ) does not depend on p. The form of the Hamiltonian is the same as for a massive particle subject to potential energy W (ψ). To the lowest order the effective mass can be evaluated using dW (µ)/dµ = h(µ) and the first order approximation relation between h and µ,
1 m ≈ 4γ 2 L(R 2 0 − R 2 i ) .(16)
The second Hamilton Eq. (14) takes the form µ = µ 0 + mγ(dψ/dt). Taking the time derivative of it and eliminating dµ/dt from Eq. (13) we come to the second-order differential equation for coordinate ψ(t) in the form
d 2 ψ dt 2 + ω 2 0 (1 − h 2 ) sin ψ cos ψ = 0,(17)
where
ω 0 = ω m R o ln η R o L + R i ln η R i L L π(R 2 0 − R 2 i ) ,(18)
and ω m = 4πγM s is a characteristic frequency of the ferromagnetic material (of the order of 30 GHz for permalloy). It can be seen that the frequency of ω 0 contains the small parameters L/R o,i , and is much lower than ω m . Note, that in the linear approximation, the frequency is √ 1 − h 2 ω 0 decreases not only with decrease of relative thickness of the ring, but also with the application of the external field h.
Eq. (17) has an integral of motion E = K + U that can be considered as a sum of kinetic and potential parts,
E = 1 2ω m dψ dt 2 + (1 − h 2 ) ω 2 0 2ω m sin 2 ψ.(19)
The first term can be treated as a kinetic energy and the second term as a periodic potential of the restoring force of a magnetostatic nature. Thus, within the above collective coordinate approach, the typical "inertial" features, common to ones that are known for usual mechanics of massive particles, are found here for in-plane magnetization dynamics in the vortex-state ferromagnetic ring. Before going further, it worth discussing these features. The inertial dynamics is widely accepted for antiferromagnets, whereas the spin dynamics of ferromagnets is commonly believed to be gyroscopic. In particular, the equation similar to Eq. (17) was obtained for antiferromagnetic STNO and was claimed as the clear manifestation of inertial character of spin dynamics [31]. For antiferromagnets, the inertial dynamics appears naturally within the sigma-model approach; it is based on the presence of the uniform exchange interaction between antiferromagnetic sublattices. It is valid up to extremely high frequencies of the order of the exchange frequency values, exceeding usual magnon frequencies for antiferromagnets. Inertial dynamics for some models of ferromagnets is known for a long time for some particular cases, see e.g., [41,42]; it was also employed for description of STNO based on easy-plane magnets in uniform state [43]. The models allowing such simplification are characterized by the presence of effective anisotropy with significantly different constants: hard for one angular variable and weak for the other one. In fact, it is the case of our problem. Then the inertial approach is valid up to frequency corresponding to "hard" anisotropy field. In our case, this upper limit frequency is ω m ; it is much lower than the exchange frequency, but much higher than the value of interest, ω 0 . Thus, the inertial approximation is valid for our problem.
The inertial approximation leads to the simple and transparent physical picture of the nonlinear motion of magnetization. Using the first integral (19), it is possible to write down solutions of Eq. (17) in quadratures and to present them explicitly through the elliptic integrals. However, it is useful to carry out a qualitative analysis using the phase plane, the phase diagram is shown in Fig. 2.
Stationary points of the function E define singular points of the phase plane, dψ/dt = 0 and sin 2ψ = 0, and the sign of cos 2ψ determines types of these points. There are two types of singular points: centers, ψ = 0, ±π and dψ/dt = 0, around which small oscillations can occur with the frequency √ 1 − h 2 ω 0 ; and saddle points, ψ = ±π/2 and dψ/dt = 0. Saddle points are characterized by the constant E = ω 2 0 /2ω m , and separatrices follow the equation
dψ dt = ±ω 0 1 − h 2 cos ψ.(20)
dψ dt ψ -π -π/2 π/2 π -1. This phase diagram shows that there are two regimes of oscillations: the regime with the finite oscillations of ψ around the equilibrium position and the regime with unlimited increase in ψ with time. These two regimes are separated by separatrices passing through the saddle points. As we will show below, these two regimes are essentially different when a ferromagnetic ring is used as an active element of the nanooscillator. In the finite regime the time average of dψ/dt equals to zero and the spin current cannot excite this type of oscillations. We are interested in the case when the initial vortex ground state of the ring loses its stability under the action of a spin-polarized current and the spin system goes into the regime of infinite circular oscillations. It takes place when the current overcome some critical values that we are about to estimate in the next section.
V. NON-CONSERVATIVE DYNAMICS AND CRITICAL CURRENTS
Taking into account changes in the energy of the system due to the spin-polarized current and damping we can write the equation for ψ in Newtonian form
d 2 ψ dt 2 + ω 2 0 (1 − h 2 ) sin ψ cos ψ + ω m (1 − h 2 ) α dψ dt − σJ = 0.(21)
This equation describes a wide variety of physically interesting systems. In particular, the same equation describes the planar dynamics of the Neel vector for biaxial antiferromagnet [31]. Note as well a simple mechanical analogy: after the substitution ψ = φ/2, equation (21) coincides with the equation of a physical pendulum, a massive suspended particle moving in the gravitational field with accounting for viscous friction force (the term with α here) and subject to some eddy force, which is perpendicular to the radius vector of the material point that mimicries the action of spin current, see e.g., [44,45]. Probably the most interesting analogy is that the above equation, written for the variable φ = 2ψ, also coincides with the equation for the phase φ of the superconducting order parameter of the point Josephson contact, see for details [46,47].
From the Eq. (21) it follows the law of the energy evolution
dE dt = M s LS γ (1 − h 2 ) σJ − α dψ dt dψ dt ,(22)
which can be rewritten in the form dẼ/dt = −α(dψ/dt) 2 , whereẼ = K +Ũ and
U = (1 − h 2 ) ω 2 0 2ω m sin 2 ψ − σJ(1 − h 2 )ψ.(23)
FunctionŨ includes the contribution of the spin-polarized current, which turns it into a "tilted washboard". So the eddy force in Eq. (21), that is proportional to σJ, can be formally presented as a part in the "potential"Ũ . Then its non-potential nature manifests itself in the fact, that functionŨ is changing after a full rotation of magnetization, i.e., after the angle ψ changes by 2π.
First note, that the system becomes absolutely unstable when local minima and maxima of potentialŨ (ψ) disappear, i.e., dŨ /dψ never equal to zero. Then the magnetization vector rotates through the full angle for any initial conditions. To achieve this regime, the current J should overcome a critical value that is natural to call the ignition threshold
J cr1 = ω 2 0 2ω m σ .(24)
If the current value is less than J cr1 , the behavior is more complicated, but its full qualitative analysis can be done through the phase plane method. To illustrate it, some characteristic phase trajectories obtained from the numerical solution of the Eq. (21) for various initial conditions and a fixed current value J are shown in Figs. 3,4. For non-conservative system, the maxima of the effective potential still are saddle points, whereas the centers for the weak enough dissipation transform to focuses, which, in principle, could be either stable or unstable (spiral sinks or spiral sources for the phase trajectories, respectively).
The steady-state motion and antidamping features are possible only for the infinite motion of ψ, i.e., for the full rotation of the magnetization. Thus, the minima of the effective potential corresponds to the stable focuses, and the phase trajectories approach them with spiralling as it shown on the figures 3 and 4 below. Obviously, for zero current it is the fate of the separatrix trajectories, which coming out from the saddle points, and it is clear that the same behavior will be present at small enough values of the current. This behavior is shown in Fig. 3. All trajectories, regardless of the initial conditions, are twisted around one of the focuses, and all oscillations in the system will decay in time.
For the oscillations to become steady-state the change of energy of the system over a period of rotation has to be zero. This behavior appears starting from some current value, such that the phase trajectories coming out from one saddle point come into the adjacent saddle point. This condition is held for all pairs of the neighboring saddle points, and the separatrix has the same form as for the system without dissipation and pumping, see Fig. 2 above.
The corresponding critical value of the current can be found from the condition of compensation of the damping and spin pumping on this separatrix trajectory. This brings us to a concept of the second critical current, the so-called elimination threshold, J cr2 < J cr1 .
Here and below we consider small damping regime, α ≪ 1, to obtain the analytical formulae for the characteristics of oscillations. This condition allows us to estimate the integrals over trajectories with use of non-perturbed equations
J cr2 = 1 − h 2 2αω 0 πσ .(25)
When the current value is above J cr2 , still some of the trajectories are finite, tending to the focuses on the phase plane. They correspond to the damped oscillations. But the trajectory coming out from one saddle point is going above the neighboring saddle point, and later on, these trajectories are going higher and higher, see Fig. 4. On the other hand, for the trajectories in the region of the phase plane with high enough "velocity" the dissipation is more efficient, and in the average they are going "downstairs". Such a behavior is typical to the system with limit circle: these two sets of trajectories can be fitted to each other if and only if we suppose that all of them are tending, from both above and below, to the some periodic trajectory, limit cycle, which corresponds to the steady-state oscillations. The center line of the limit cycle is horizontal showing that in this regime the action of spin current and damping are compensated at average. It is worth to stress here the difference of two regimes of oscillations. The ignition current is strong enough to remove the potential barriers and the system unavoidably begin to roll down the "washboard". The elimination current gives the system enough energy to inertially overcome the potential barriers of potential U like a pendulum uses its kinetic energy to maintain the oscillations.
Let us estimate the frequency of the steady-state oscillation, corresponding to the limit cycle on the phase diagram. It will play the role of the generation frequency ω of the STNO based on the ring-shaped vortex-state free layer. In the approximation α ≪ 1 the generation frequency is found by integrating of Eq. (19) over the period, that gives
ω = π 2 ω 0 √ 1 − h 2 kK(k) ,(26)
where K(k) is the elliptic integral of the first kind. To find the value of argument k, corresponding to the limit cycle, we use the condition that at the limit cycle the integral over the period of dE/dt is zero. The integrating of Eq. (22) gives the equation σJπk = 2α
√ 1 − h 2 ω 0 E(k), where E(k)
is the elliptic integral of the second kind. These two equations give the dependence of ω on the current in the implicit form, this dependence is shown in Fig. 5.
It turns out, that at current values 2-3 times higher than the threshold value J cr2 , the frequency grows almost linearly with the current, ω lin = σJ/α. For small current values, ω is more complicated function of the current; it vanishes at J → J cr2 . Despite the fact that the dependence ω(J) present in Fig. 5 is derived in the small-α approximation, ω lin coincides with the linear frequency expected at the extremely high current values. The reason is, All trajectories below the separatrix are finite. All trajectories above the separatrix are approaching the limit cycle. The limit cycle (red, color online) is shown between the two horizontal dashed lines. that at current values much larger than J cr1 the role of the potential U is negligible, and the limit cycle is presented by almost strait horizontal line, which corresponds to the constant solution of Eq. (21), namely dψ/dt = ω lin .
To evaluate practical benefits of the ring-based STNO we make estimates for the critical current values. We use the following parameters of the system: R 0 = 200 nm, R i = 150 nm, L = 5 nm, h = 0, η = 5, ǫ = 1. In typical magnetic materials which are used in spintronic devices (we use values for permalloy), the Gilbert damping is α ∼ 0.01, M s = 8 · 10 5 Am −1 , characteristic frequency ω m ≈ 30 GHz. For these parameters we obtain ω 0 = 11.4 GHz, J cr1 = 0.055A, J cr2 = 0.0017A. Dividing by the current carrying area S = 5.5 · 10 −10 cm 2 we obtain the critical current densities j cr1 = 10 8 Acm −2 . The value of j cr1 is high but reasonable [48], especially having in mind that it should be applied for a short time only to trigger the oscillations. After the ignition the applied current can be reduced to smaller densities above the elimination threshold j cr2 = 3.1 · 10 6 Acm −2 , which is sufficiently low. This makes the ring-based STNO promising for the practical implementation. Let's consider the configuration of this device in more detail.
VI. EXTRACTING AN ELECTRIC OUTPUT SIGNAL
In the previous sections we made theoretical description of the self-sustaining nonlinear dynamics of magnetization in the ferromagnetic ring. We suggest to use the ring as a free layer of the STNO. For its practical application as a microwave generator it is necessary to convert the energy of the oscillations into an alternating microwave electrical signal. There are two known phenomena allowing to extract electrical current from the magnetization oscillations: giant magnetoresistance (GMR) [49,50] and the inverse spin Hall effect (ISHE) [51,52]. In what follows we are considering both possibilities.
Geometries of the STNO for these two cases shown in Figs. 6,7. The basic design is a sandwiched structure with three main elements: polarizer, free layer, and analyser. The direct current J is injected in the device along its axis. Spin-polarization of electrons in the current, p, is provided by a thick ferromagnetic polarizer of magnetization M p , which assumed to be fixed. The vector p M p and directed perpendicular to the plane of the free layer. Polarizer and the free layer are separated by nonmagnetic spacer. As it is described above, the spin-polarized current flowing through the free layer leads to the nonlinear oscillations of its magnetization M. The Analyzer, in turn, converts the oscillations into a useful signal.
In contrast to mono-domain oscillators, in the vortex-state free layer the oscillations of the total magnetic moment are negligibly small. The deviation of the magnetization from the ground state during the oscillations lies in the plane of the free layer, and its value averaged over the layer area equals to zero. For this reason, the question of the signal extraction is nontrivial. We suggest a solution of this problem by using special non-uniform analyzers. First, we discuss the possibility to obtain an alternating electrical signal at the STNO output by the GMR effect. In the spin valve scheme based on GMR the analyzer consist of a ferromagnetic layer with the fixed magnetization M a and a thin nonmagnetic interlayer separating it from the free layer of magnetization M. The magnetization of analyser are usually fixed by usage of a hard ferromagnet of large thickness or by an additional antiferromagnetic layer. When the spin-polarized current passes through this system of layers, the scattering of electrons depends on the mutual orientation of the magnetizations of the layers. The GMR contribution to the resistance of the spin valve is locally (at each point of the layer area) proportional to the scalar product of these magnetizations (M · M a ).
In the case under consideration the ground state of the free layer is the vortex state, and projection of its magnetization on a fixed direction averaged over the layer area is zero. Moreover, the same is valid for the nonlinear oscillation of the breathing mode, see Fig. 1. Thus the use of a traditional homogeneous spin valve with M a = const is impractical.
Let consider an analyzer in a form of a thick magnetic ring (optionally a disk) with the saturation magnetization M as in a vortex state, which is stable enough even being in contact with free layer and under the action of the current. In order to stabilize the vortex ground state of the analyzer, it has to be made thick enough. Optionally, the analyzer in the vortex state can be made from a chiral magnet, where the vortex state is stabilized by the Dzyaloshinskii-Moriya interaction (DMI).
The magnetization of this vortex analyzer can be written as M a /M as = −e x sin(χ + ϕ 0 ) + e y cos(χ + ϕ 0 ), where e x and e y are orthogonal unit vectors in the plane of the layer, and e z is the unit vector along the device axis.
Here the constant ϕ 0 = 0 is for the "standard" vortices ( Fig. 1(a)) with closed magnetic flux, and ϕ 0 = ±π/2 for so-called radial vortices, stabilized by interfacial DMI. For the nonlinear regime of the oscillations the scalar product (M · M a ) = M s M sa sin θ 0 cos(ψ(t) + ϕ 0 ). So the GMR contribution to the electrical resistance ∆R(t) of this vortex spin valve scheme is proportional to sin θ 0 cos[ψ(t) − ϕ 0 ]. The factor sin θ 0 is determined by the external magnetic field, and the best efficiency of the vortex spin valve is achieved for zero external field, when θ 0 = π/2. We also consider an alternative possibility to extract a useful electric signal by the inverse spin Hall effect. Magnetization precession that appears in the ferromagnetic ring-shaped free layer can acts as a spin pump to an adjacent nonmagnetic metal. The direction of the spin current flow is perpendicular to the interface between the magnetic free layer and the nonmagnetic analyzer layer, which in our case coincides with z−axis. A spin current generated via the spin pumping mechanism, i.e. the direction of polarization of electrons in the current, j sp ∝ m × ∂m/∂t. It can be presented as a sum of in-plane and out-of-plane components, j sp = j in sp + j out sp . The component j in sp lies in the plane of the layers and j out sp is along e z . The spin pumping generated due to the nonlinear breathing mode is j sp ∝ 1 2 sin 2θ 0 [e x sin(χ + ψ) − e y cos(χ + ψ)] ∂ψ ∂t + e z sin 2 θ 0 ∂ψ ∂t ,
where ψ = ψ(t) determines the spin precession. If the external magnetic field is absent the ground state of the ferromagnetic ring is the in-plane vortex with θ 0 = π/2 and only nonzero component of the spin pumping is j out sp , which, as we show below, does not lead to the generation of an electrical signal. However, in the presence of the magnetic field, i.e., for the cone state vortex, the situation is absolutely different.
This spin current injected into nonmagnetic metal conductor attached to the free layer can be converted into a charge current j c via the inverse spin Hall effect
j c ∝ θ SH e z × j sp ,(28)
where θ SH is the spin Hall angle. It gives rise to an alternating electromotive force, the source of a useful microwave signal. In the considered geometry in-plane component of the spin current j in sp is responsible for the occurrence of the electric current, j c ∝ θ SH sin 2θ 0 [e x cos(χ + ψ) + e y sin(χ + ψ)] ∂ψ ∂t .
Due to of the factor sin 2θ 0 , the signal is nonzero only when external magnetic field is applied and the free layer is in the cone vortex state with θ 0 < π/2. The direction of the electric current j c lies in the plane of the layers perpendicular to j in sp . The maximum value of the sin 2θ 0 and hence the amplitude of the electrical signal is reached at θ 0 = π/4, that corresponds to the magnetic field H ≈ 2 √ 2πM s .
To extract the output signal by the inverse spin Hall effect the analyzer layer has to be made of the nonmagnetic metallic conductor. The signal that can be taken is proportional to the value of the current density (29) averaged over the interface between the free layer and the analyzer layer, i.e Ro Ri rdr 2π 0 dχ j c . For the homogeneous material of the analyzer this integral is equal to zero because of the alternating periodic functions sin(χ + ψ) and cos(χ + ψ). To solve this technical problem, it is advisable to fabricate different areas of the analyzer made of two heavy metals with different signs of spin Hall angle θ SH , e.g., platinum and tantalum, as has been realized in [53]. Then the electrical signals from different areas are summed up. For example, we can select areas according to the sign of sin χ, then averaged in-plane component of the spin current is along e x and the electrical current flows in the direction e y .
In summary, the proposed ring-based STNO is compatible with both known mechanisms of microwave signal extraction: the GMR effect and ISHE.
VII. CONCLUSIONS
We considered the nonlinear spin dynamics of the ferromagnetic ring in the vortex state under the action of the spin-polarized current and suggested the basic design of the ring-based vortex-state STNO.
Proposed nanooscillator is characterised by the working frequencies up to 10 GHz, that makes it advantageous over the disc-based vortex oscillators operating in the sub-GHz range. The high generation frequency is achieved due to the fact that there is no vortex core region in the ring, and so there is no mode of core precession. The first mode excited in the ferromagnetic ring by the spin-torque corresponds to the radially-symmetric in-plane oscillations of magnetization with small almost time-independent out-of-plane component. Due to the vortex state of the ring the in-plane total magnetic moment of the oscillations is zero.
We also found that the nonlinear spin oscillations are described by the simple and universal equation, Eq. (21). This equation appears for many physical problems; probably, the most important system is a point Josephson contact. This formal analogy allows usage of serious groundwork already developed for application of Josephson systems in electronics [46,47]. Despite the fact that the spin dynamics of considered device is described by the same equation as in cases of Josephson contact and antiferromagnetic oscillator, the ring-based oscillator has different operational properties. Indeed, the useful alternating electrical signal obtained by the ring-based STNO has different behavior with increasing of the spin-polarized current, and, as a result, different mechanisms of signal extraction are needed. In the antiferromagnetic oscillator the alternating part of the signal is related to inhomogeneity of precession of the Neel vector and decreases with the current. Due to the vortex state of the ring-based STNO the alternating signal can be obtained for the homogeneous precession of magnetization, and therefore there is no effect of signal decreasing. This fact simplifies the basic design of the analyzer for the ring-based STNO. Nevertheless, the signal extraction technology in the case has to be nontrivial, since the oscillation of magnetization averaged over the area of the ring is zero, which will result in a zero average electromotive force in output of standard schemes. We suggested to solve this problem by usage of special nonhomogeneous analyzers described in Sec. VI. We showed that the ring-based STNO can be used both in the spin valve circuit and in combination with the inverse spin Hall effect.
We expect the proposed device to have the following advantages. It can be made of convenient size: ring radii are of the order of hundreds of nanometers, and ring thickness is thin enough for spin transfer effects, namely 1 − 5nm. The system allows the easy adjustments of the main parameters, like working frequency, by application of a weak enough out-of-plane magnetic field. It is also worth noting the linear growth of the generation frequency with the current. The threshold current has usual values for the vortex STNOs: spin-transfer torque overcome the damping at current density about 10 6 Acm −2 , whereas the ignition current is of the order of 10 8 Acm −2 . In addition, it is possible to slightly reduce the elimination current by the magnetic field.
Aside from the continuous auto-oscillations discussed in the paper, note that the system also allows some impulsive regimes, which can be used, for example, for creation of artificial neurons for neuromorphic computing, see Refs. [43,55,56]. All of the above makes the proposed ring-based STNO interesting for practical applications. [1] K. Slonczewski, J. Appl. Phys. 45, 375 (1974).
FIG. 1 :
1Ground state and dynamics of magnetization M of the vortex-state ring under the action of spin-transfer torque: (a) standard in-plane vortex with the polar angle θ0 = π/2. (b) cone-vortex state, which realizes the ground state of the free layer when the external magnetic field H is applied along the z-axis, the angle θ0 = π/2. The spin-polarized current with polarization p flows into the free layer. Vector ω shows schematically the direction of rotation of magnetic moments in different points of the ring.
FIG. 2 :
2Phase plane representation of radially-symmetric oscillations without dissipation. Here we used the energy units of 2ωm/ω 2 0 (1 − h 2 ), and time units of ω0 √ 1 − h 2 . Black circles show the singular points of the center type and black rectangles show saddle points. Separatrix trajectories connect the saddle points. Closed phase trajectories below the separatrix correspond to the finite motion, whereas phase trajectories above the separatrix correspond to infinite dynamics with non-limited growing of ψ.
FIG. 3 :
3Phase plane representation of of the wave equation with the spin-polarized current J and damping for the case J < Jcr2. All trajectories are tend to the focuses. We also show some of the trajectories, passing as close as possible to the separatrix trajectories from above and below.found through Eq.(19), see the previous Section. All quantities are obtained in the linear approximation over the small parameter α.The value J cr2 is found by substituting the equation for the separatrix trajectory, Eq. (20) into Eq. (22) and integrating it over time. Finally,
FIG. 4 :
4Phase plane representation of the wave equation with the spin-polarized J current and damping for the case J > Jcr2.
FIG. 5 :
5The generation frequency ω via the value of the spin-polarized current J (red line, color online). Black dashed line shows the linear asimptotic ω lin . The frequency units are chosen asω = ω0 √ 1 − h 2 π/2.
FIG. 6 :
6Nanopillar geometry and the scheme of extracting the electric output signal by GMR. The analizer is a ferromagnetic ring (optionally a disk) in the vortex state. The gray arrow indicates the direction of the current flowing into the nanopiler (opposite to the direction of electron movement). Vector p shows the direction of spin polarization.
FIG. 7 :
7Devise geometry and the scheme of extracting the electric output signal by ISHE. The nonhomogeneous analyzer contains two parts made of heavy metals with different signs of spin Hall angle. The gray arrow indicates the direction of the current flowing into the nanopillar (opposite to the direction of electron' motion). Vector p shows the direction of spin polarization.
thank K. Byczuk for support and helpful discussions. The work of B.A. Ivanov was partly supported by National Scientific Foundation of Ukraine under Grant No. 2020.02/0261.
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| {'fraction_non_alphanumeric': 0.056870050483056725, 'fraction_numerical': 0.02699956452315285, 'mean_word_length': 3.905609621014321, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study a nonlinear spin dynamics of a ferromagnetic ring in a vortex state induced by the spin-polarized current. We also suggest to use the ferromagnetic ring as a free layer of a coreless vortex spin-transfer nano-oscillator. The calculated working frequency is about several GHz, that is much higher than the gyromode frequency of the disk-based vortex oscillator. The response of the vortex-state ring to the spin-polarized current has hysteretic behavior with the reasonable values of the thresholds current densities: ignition threshold is about 10 8 Acm −2 , and elimination current to maintain the oscillations has much lower values about 10 6 Acm −2 . The output signal can be extracted by the help of the inverse spin Hall effect or by the giant magnetoresistance. The output electromotive force averaged over all sample vanishes, and we suggest to use a ferromagnetic ring or disk in a vortex state as a GMR analyzer. For an inverse spin Hall analyser we advise to use two heavy metals with different signs of Spin-Hall angle. The ring-based STNO is supposed to increase the areas of practical application of the STNOs. * Electronic address: bor.a.', 'arxivid': '2305.16019', 'author': ['Vera Uzunova ', 'Boris A Ivanov ', '\nInstitute of Theoretical Physics\nFaculty of Physics\nInstitute of Physics\nUniversity of Warsaw\nul. Pasteura 502-093WarszawaPoland\n', '\nInstitute of Magnetism\nNational Academy of Sciences of Ukraine\n46 Nauky Ave03039KyivUkraine\n', '\nand Institute for Molecules and Materials\nNational Academy of Sciences of Ukraine\n03142KievUkraine\n', '\nRadboud University Nijmegen\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands\n'], 'authoraffiliation': ['Institute of Theoretical Physics\nFaculty of Physics\nInstitute of Physics\nUniversity of Warsaw\nul. Pasteura 502-093WarszawaPoland', 'Institute of Magnetism\nNational Academy of Sciences of Ukraine\n46 Nauky Ave03039KyivUkraine', 'and Institute for Molecules and Materials\nNational Academy of Sciences of Ukraine\n03142KievUkraine', 'Radboud University Nijmegen\nHeyendaalseweg 1356525 AJNijmegenThe Netherlands'], 'corpusid': 258887623, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19307, 'n_tokens_neox': 16647, 'n_words': 10676, 'pdfsha': '4fe27a3ac52aed5045926ad929987949b4486697', 'pdfurls': ['https://export.arxiv.org/pdf/2305.16019v1.pdf'], 'title': ['Nonlinear spin dynamics of ferromagnetic ring in the vortex state and its application for spin-transfer nano-oscillator', 'Nonlinear spin dynamics of ferromagnetic ring in the vortex state and its application for spin-transfer nano-oscillator'], 'venue': []} |
arxiv |
Characterizing Superradiant Phase of the Quantum Rabi Model
Yun-Tong Yang
School of Physical Science and Technology
Lanzhou University
730000LanzhouChina
Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000LanzhouChina
Hong-Gang Luo
School of Physical Science and Technology
Lanzhou University
730000LanzhouChina
Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000LanzhouChina
Beijing Computational Science Research Center
100084BeijingChina
Characterizing Superradiant Phase of the Quantum Rabi Model
Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalization to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus one is able to see how the phase transition happens and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition happens and further exhibits the statistics of Gaussian unitary ensemble as increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.
Introduction.-Continually increasing couplings between different degrees of freedom in hybrid quantum systems provides a huge opportunity to explore new physics and/or new phenomena emerging from the interplay between the constituents of the hybrid systems [1][2][3][4][5][6]. In particular, after the strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics has been firstly realized in 2004 [7], the ultrastrong coupling regime has been further achieved [8], where the interaction energy is comparable to mode frequency, as a consequence, light and matter can mix together more tightly. Furthermore, the deep-strong couplings [9] has also been reached in which the interaction is even larger than the mode frequency [10][11][12].
The achievement of strong hybridization not only leads to increased control of quantum systems [13][14][15][16][17][18][19][20] and to possible applications on, e.g., lasers, quantum sensing [21,22], and quantum information processing [23][24][25][26], but also provides chances to test many physical phenomena such as superradiance [27][28][29], predicted theoretically in strong coupling regime of the Dicke model [30,31], even of the Rabi model [32,33]. An early analysis of the ground state in the quantum Rabi model (QRM) showed that the ground state exhibits a squeezing in the deepstrong coupling regime [34,35], a precursor of distinguished physics of superradiance, which has been further explored and confirmed theoretically [36][37][38][39][40][41][42][43][44][45][46]. Very recently this phase transition has been observed in a single trapped ion [47] and stimulated experimentally in the platform of nuclear magnetic resonance [48].
Experimental observations of the superradiant phase transition further stimulate theoretical interest on the phase transition and the superradiant phase since the phases and phase transition are fundamental issues of modern condensed matter physics and related disciplines [49], in which the conventional paradigm is to identify the order parameter and broken symmetry associated under the framework of Landau phase transition theory [50]. Due to the solvability of the QRM [51,52], it is possible to solve accurately for the wavefunction of the QRM in full parameter regime ranged from weak to strong, ultra-strong, even deep-strong ones, though it is not an easy thing [53]. Thus we can see how the superradiant phase transition happens and how the photons populate in the superradiant phase based on wavefunction since it contains all information of the system. Here we provide a scheme consisting of two successive diagonalization, where the first one is made exactly in the two-level space and the second is done in the truncated Fock space in a controllable way in a sense that the convergence depends on the size of the truncated Fock basis, which in the present situation even a PC is enough. We confirm this convergence by comparison with those obtained by numerical exact diagonalization (ED).
With accurate wavefunctions at hand, we study the process of phase transition and see how the photons populate in Fock space of the superradiant phase by changing the coupling strength from weak to strong ones. In normal phase it is found that only ground state is populated and thus there are no photons in the system, in agreement with what one expects. Around the phase transition point, the high level states begin to be excited, and some photons begin to populate on them. Here it is helpful to borrow the distribution concept in the random matrix theory [54,55]. This population is found to follow the Poissonian-like distribution. Further increasing coupling strength, the population behaves like the statistics of Gaussian unitary ensemble (GUE). At the same time an effective potential with a double-well forms gradually around the center of the harmonic potential. Physical reason for this transition from the Poissonian statisticslike to the GUE one-like is due to the formed effective potential barrier, which blocks the tunneling between the low-lying levels bounded at two displaced minima. According to this picture, the photons populated on higher energy levels beyond the barrier should exhibit different behavior. Indeed, one checks the population of photons in excited states, and finds that the photons follow exactly the statistics-like of Gaussian orthogonal ensemble(GOE). The sprectra obtained here not only deepen understanding of the superradiant phase of the QRM in strong coupling regimes [56], but also have a profound implication on the nature of the superradiant phase of the QRM and its variants. An interesting issue on the integrability of related models [51,[57][58][59][60][61][62][63][64] deserves further investigation but is obviously beyond the scope of the present work.
Model and Method.-The Hamiltonian of the QRM consists of a single photon mode and a two-level atom and their coupling, denoting by
H = H 0 + H σ , where H 0 = ωa † a and H σ = ∆ 2 σ x + gσ z (a + a † ).
Here a † (a) is creation (destruction) operator of the single mode photon field and σ x , σ z are usually Pauli matrices denoting the two-level atom. For convenience, we rescale the Hamiltonian by the mode frequency ω, thus the twolevel interval ∆ and the coupling strength g used in the following are dimensionless. It is also useful to use dimensionless position-momentum operators related to the destruction (creation) operator by a = 1
√ 2 ξ + ∂ ∂ξ and a † = 1 √ 2 ξ − ∂ ∂ξ
to rewrite the Hamiltonian as
H 0 = 1 2 − ∂ 2 ∂ξ 2 + ξ 2 and H σ = 1 2 ∆σ x + 2 √ 2gσ z ξ with a matrix form H σ = 1 2 2 √ 2gξ ∆ ∆ −2 √ 2gξ .(1)
Thus Eq. (1) can be formally diagonalized and its eigenvales and eigenvectors read
± (ξ) = ± ∆ 2 1 + β 2 ξ 2 ,(2)φ ± (ξ) = 1 √ 2 ±(1 ± γ(ξ)) 1 2 , (1 ∓ γ(ξ)) 1 2 T ,(3)where β = 2 √ 2g ∆ and γ(ξ) = βξ √ 1+β 2 ξ 2 .
This finishes the first diagonization to solve the Schrödinger equation
H σ φ ± = ± φ ± ,(4)
which includes the position as a parameter. However, our aim is to solve the full Hamiltonian H satisfying with the full Schrödinger equation
HΨ E = (H 0 + H σ )Ψ E = EΨ E(5)
To proceed, it is useful to assume two complete basis |ξ, σ := |ξ |σ and |ξ, ± := |ξ |± , and one can write the total wavefunction Ψ E as Φ E (ξ, σ) = ξ, σ|Ψ E or ψ E ± (ξ) = ξ, ±|Ψ E . Thus, one uses the orthogonal basis 1 = ± dξ|ξ, ± ξ, ±|,
Φ E (ξ, σ) = ± dξ ξ, σ|ξ , ± ξ , ±|Ψ E = ± σ|± ξ, ±|Ψ E = ± φ ± (ξ)ψ E ± (ξ)(6)
To consider the Born-Oppenheimer approximation [65], one has
(H 0 + H σ )φ ± (ξ)ψ E ± (ξ) = Eφ ± (ξ)ψ E ± (ξ)(7)
Multiplying by φ * ± Eq. (7), one obtains
(H 0,± + ± )ψ E ± (ξ) = E ± ψ E ± (ξ),(8)
where H 0,± = φ * ± H 0 φ ± = H 0 which is easy to verify. Eq. (8) is the main result of the present work, which is the starting point of the following calculation.
In order to solve Eq. (8), one inserts the complete basis 1 = n |n n| of the standard harmonic oscillator into Eq. (8) to obtain
m n|(H 0 + ± )|m m|ψ E ± = E ±,n n|ψ E ± .(9)
In a truncated basis |n , n = 0, 1, · · · , N − 1, solving Eq. (9) is equivalent to diagonalize the following N × N matrix
0|H 0 + ± |0 · · · 0| ± |N − 1 . . . . . . . . . N − 1| ± |0 · · · N − 1|H 0 + ± |N − 1 (10)
This finishes the second diagonalization, which gives the spectra of the corresponding wavefunction. The convergence and its accuracy are verified in comparison with the numerical ED, as given in SM [66]. In the following we focus on the ground state and low-lying excited states given by the negative branch and the positive one has much high energy under the parameters we use below and will be explored in future.
Results and Discussion.- Figure 1 shows the ground state and the low-lying excited states energies (blue and red lines with odd and even parity, respentively) as functions of coupling strength scaled by g c = 1 + 1 + ∆ 2 16 roughly marked the superradiant phase transition point for the ground state [38]. To verify the accuracy of the present calculation, we also provide the results marked by symbols obtained by numerical ED for the same model parameter ∆ = 10. One sees that ranged from weak to strong coupling regimes, our results are in excellent agreement with the exact ones in a high precision. The phase transitions are found to happen smoothly from normal phase to superradiant phase. This is in contrast to some previous variational methods [38,[67][68][69][70][71][72] or approximations [73][74][75]. One can also refer to recent overviews on the related issues [76,77]. The superradiant phase transitions also happen in the low-lying excited states, but with larger coupling strengths, which are consistent with those reported in literature [78]. The inset shown in Fig. 1 gives the photons as functions of the coupling strength, which go up at the point of phase transition and further increase with increasing coupling strength. Impressive accuracy of our method can be further confirmed by wavefunctions presented in Fig. 2, in which the ground state wavefunction and those of the first three low-lying excited states are plotted for three coupling strengths g/g c = 0.5, 1.0, and 1.5, which correspond to the normal phase, roughly superradiant phase transition point, and the superradiant phase, respectively. Likewise, we also present the results obtained by numerical ED. The lines (red and blue ones denote the two components of the wavefunctions) denote our results and the symbols those of ED. Fig. 2(a1-a3) show the effective potential given exactly by V − (ξ) = 1 2 ξ 2 − ∆ 2 1 + β 2 ξ 2 , from which in weak-coupling regime, the quantity β is small, Τ g g c = 0.5 Τ g g c = 1.0 Τ g g c = 1.5 thus the potential is roughly the standard harmonic oscillator potential, and the local minimum locates at the point of ξ = 0. Increasing the coupling strength up to the critical point, the local minimum begins to become local maximum, and a tiny "Mexician Cap" forms, accompanying with a separation of the ground state wavefunction, as shown in Fig. 2(b2). A complete separation of the wave-packets is observed in Fig. 2 (b3), which represents that the system enters completely into the superradiant phase. Correspondingly, an effective double-well potential [79] develops well, as shown in Fig. 2(a3). This double-well potential can also be obtained by a Taylor expansion
(a3) (a2) (a1) (b3) (b1) (b2) (c3) (c1) (c2) (d3) (d2) (d1) (e3) (e2) (e1)V − (ξ) ≈ − ∆ 2 + 1 2 − β 2 ∆ 4 ξ 2 + β 4 ∆ 16 ξ 4 ,(11)
which corresponds to the standard form in the Landau phase transition theory except for a constant energy.
This double-well potential plays an important role in the photons population, as discussed later. Another important feature can be observed from the wavefunctions including the ground state and the excited states, namely, an obviously breaking of parity is absent up to the coupling strength g/g c = 1.5. We believe that this result is correct since it is also absent in numerical ED. Physical reason is still unclear and we would like to leave for future study. Next we move to the photon population calculated only from the wavefunctions. Fig. 3 shows the results for three typical regimes: the normal phase (g/g c = 0.5), the superradiant phase transition point (g/g c = 1.0), and the superradiant phase (g/g c = 1.5) and for four different ∆'s: 5 ( the first row, a1-a3), 10 (the second row, b1-b3), 20 (the third row, c1-c3), and 30 (the fourth row, d1-d3). The first column is in the normal phase, in which no photons are excited except for a tiny upward observed in Fig. 3(a1) with small ∆ = 5. The second column denotes the situation around the superradiant phase transition point, in which the high levels begin to be excited. An obvious difference of photon numbers in odd and even channels of Fock space is observed. The even parity channels are more easier to excite than those with odd parity. The reason is that the ground state of the system (single mode photon plus the two-level) is odd parity, therefore the Fock basis with even parity meet this requirement. The third column represents the situation of the superradiant phase, in which Fock basis with more higher energy levels are excited. This is physically reasonable since with increasing coupling strength the photon number also increases, as also observed in experiment [47]. It is noticed that in the superradiant phase the population of the lowlying energy levels are suppressed. As also mentioned above, this is due to the fact that an effective potential barrier forms with increasing the coupling strength, as shown in Fig. 2(a3). It blocks the tunneling of the states of low-lying energy levels around two separated local potentials.
g / g c = 0 . 5 E v e n O d d P ( n ) ( d 3 ) ( d 2 ) ( d 1 ) ( c 3 ) ( c 2 ) ( c 1 ) ( b 3 ) ( b 2 ) ( b 1 )
In order to further characterize the photon populations, it is helpful to borrow the distribution concept in random matrix theory [54,55], from which it is wellknown that there are three typical distributions
P P (s) = e −s ,(12)P GU E (s) = 32 π 2 s 2 e − 4s 2 π ,(13)P GOE (s) = π 2 se − πs 2 4 ,(14)
which correspond to the Poissonian statistics, the statistics of GUE and that of GOE [80]. In its original definition, the variable s denotes the energy intervals of adjacent levels [81], but here we replace it by the Fock basis. We fit the photon population by the above formulas and the details are presented in SM [66]. The results are plotted in Fig. 3 for the Poissonian-like statistics (dash lines) and the statistics of GUE-like (dashed-dot lines). The fitting is found well for all photon populations. Around the superradiant phase transition point, the both populations for odd and even components follow the Poissonian-like statistics. In the superradiant phase the photon populations become the statistics of GUE-like. The populations in low-lying energy levels are strongly suppressed due to the emerged potential barrier, as pointed out above. The same picture can also be applied to understand the photon population of the superradiant phase in the excited states of the system, as shown in Fig. 4, in which Fig. 4(a) is the same as that in Fig. 3(b3) and replotted here for comparison. The first excited state has a behavior of GUE-like, which indicates the first excited state is still influenced by the emerged potential barrier. More interesting is that the second and the third excited states behave like GOE. The inset in Fig. 4(d) shows the details of the tails in comparison to that of GUE. This transition of the populations is obviously due to the emerged double-well potential, by which the third and the fourth excited states should have a higher energy level than the induced potential barrier, therefore, its decay is more slower than those in the ground state and the first excited state.
Summary and Outlook.-We propose a scheme consisting of two successive diagonalization to obtain accurately wavefunctions of both ground state and excited states of the QRM in full parameter space ranged from weak to strong, ultra-strong, even deep-strong regimes in a controllable way. Based on the wavefunctions obtained, we characterize the process of superradiant phase transition and the nature of the superradiant phase. In particular, the photons population can be well characterized by the distributions borrowed from the random matrix theory, namely, Poissonian statistics, the statistics of GUE and that of GOE, dependent of the coupling strength and the excited states.
In the present work we focus on the wavefunctions of the ground state and the low-lying excited states and characterize their photon populations. Although we borrow the distribution concept in random matrix theory, we do not touch the issue of integrability of the QRM [51,[58][59][60][61][62]76] which involves the entire spectrum of the QRM. However, it keeps interest to explore the implication of the different photon populations and their transitions on the integrability of the QRM since the standard analysis of level statistics of the QRM is not sufficient to judge whether or not the QRM is integrable [60].
FIG. 1 .
1The energy levels of the ground state and the other 9 low-lying excited states as functions of the coupling strength scaled by gc = 1 + 1 + ∆ 2 16[38]. The lines (blue and red ones denote different parity) are our results and the symbols those obtained by numerical ED with the same parameter ∆ = 10. The inset presents the result of photons as functions of the coupling strength for the ground-state and the first three excited states.
FIG. 2 .
2The effective potentials (a1-a3) and the wavefunctions (blue and red lines) for the ground state (b1-b3) and the first (c1-c3), the second (d1-d3), and the third (e1-e3) excited states for three coupling strength g/gc = 0.5, 1.0, and 1.5. For comparison, we also present the results of numerical ED (symbols). The parameters used are the same as inFig. 1. From the wavefunctions obtained the parity of the states is not obviously broken.
FIG. 3 .
3The photon population P (n) in Fock space for the ground state in three coupling strengths g/gc = 0.5, 1.0, and 1.5 for different ∆'s: 5 (the first row, a1-a3), 10 (the second row, b1-b3), 20 (the third row, c1-c3), and 30 (the fourth row, d1-d3). The red dots and blue triangles denote the photon population in Fock basis with odd and even parity. The dash lines represent the fits of the Poissonian statistics-like and the dashed-dot lines represent the fits of the statistics of GUElike. The fitting parameters and details are given in SM[66].
FIG. 4 .
4The photon population of (a) the ground state, (b) the first, (c) the second, and (d) the third excited states in the superradiant phase. The parameters used read ∆ = 10 and g/gc = 1.5. The insets in (b) and (d) show the details of the tails of GUE and GOE, in which the GUE decays more rapidly than that of the GOE. In the second and third excited states the populations of lower low-lying energy levels are scattered and do not exhibit typical statistical features.
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Level clustering in the regular spectrum. M V Berry, M Tabor, J M Ziman, 10.1098/rspa.1977.0140Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 356375M. V. Berry, M. Tabor, and J. M. Ziman, Level clustering in the regular spectrum, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 356, 375 (1977).
| {'fraction_non_alphanumeric': 0.07923580123900706, 'fraction_numerical': 0.06574102153099684, 'mean_word_length': 4.014336917562724, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 23, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalization to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus one is able to see how the phase transition happens and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition happens and further exhibits the statistics of Gaussian unitary ensemble as increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.', 'arxivid': '2207.13285', 'author': ['Yun-Tong Yang \nSchool of Physical Science and Technology\nLanzhou University\n730000LanzhouChina\n\nLanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouChina\n', 'Hong-Gang Luo \nSchool of Physical Science and Technology\nLanzhou University\n730000LanzhouChina\n\nLanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouChina\n\nBeijing Computational Science Research Center\n100084BeijingChina\n'], 'authoraffiliation': ['School of Physical Science and Technology\nLanzhou University\n730000LanzhouChina', 'Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouChina', 'School of Physical Science and Technology\nLanzhou University\n730000LanzhouChina', 'Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouChina', 'Beijing Computational Science Research Center\n100084BeijingChina'], 'corpusid': 251104847, 'doi': '10.1088/0256-307x/40/2/020502', 'github_urls': [], 'n_tokens_mistral': 16541, 'n_tokens_neox': 13549, 'n_words': 7142, 'pdfsha': '4d62a7bb7cc52f569c53ca7529a992785891c26a', 'pdfurls': ['https://export.arxiv.org/pdf/2207.13285v1.pdf'], 'title': ['Characterizing Superradiant Phase of the Quantum Rabi Model', 'Characterizing Superradiant Phase of the Quantum Rabi Model'], 'venue': []} |
arxiv |
Cosmic Rays and Gamma Ray Bursts From Microblazars
Sep 1998
Arnon Dar
Max Planck Institut Für Physik
Department of Physics
Space Research Institute Technion
Werner Heisenberg Institut Fohringer Ring 680805MünchenGermany
Israel Institute of Technology
32000HaifaIsrael
Cosmic Rays and Gamma Ray Bursts From Microblazars
Sep 1998arXiv:astro-ph/9809163v1 13
Highly relativistic jets from merger and accretion induced collapse of compact stellar objects, which may produce the cosmological gamma ray bursts (GRBs), are also very eficient and powerful cosmic ray accelerators. The expected luminosity, energy spectrum and chemical composition of cosmic rays from Galactic GRBs, most of which do not point in our direction, can explain the observed properties of Galactic cosmic rays. PACS numbers: 98.70.Rz, 97.60.-s Typeset using REVT E X 1The origin of high energy cosmic rays (CR), which were first discovered by V. Hess in 1912, is still a complete mystery[1]. Their approximate broken power-law spectrum, dn/dE ∼ E −α with α ∼ 2.7 below the so called CR knee around 10 15.5 eV , α ∼ 3.0 above the knee and α ∼ 2.6 beyond the so called CR ankle around 10 19.5 eV , suggests different origins of CR with such energies. It is generally believed that CR with energy below the knee are Galactic in origin [2], those with energy above the knee can be either Galactic or extragalactic[3], and those beyond the ankle are extragalactic[4].
Galactic γ-ray, X-ray and radio emissions are [1], ǫ = E(dn/dE)dE ∼ 1 eV cm −3 and [6] M gas = ρdV ∼ρV ∼ 4.8 × 10 9 M ⊙ , respectively. Hence [7],
L M W [CR] ∼ M gas cEdn/dE X dE ∼ 1.5 × 10 41 erg s −1 .(1)
The only known Galactic sources which can supply this CR luminosity [8] are supernova explosions [9] and gamma ray bursts [10]. Approximately, E K ∼ 10 51 erg is released in supernova explosions (SNe) as nonrelativistic kinetic energy of ejecta at a rate R M W [SNe] ∼ 1/30 y −1 . If a fraction η ∼ 15% of this energy is converted into CR energy by collisionless shocks in the supernova remnants (SNR), then the total SNe luminosity in CR is,
L M W [CR] ≈ ηR M W [SNe] E K [SNe] ∼ 1.5 × 10 41 erg s −1 ,(2)
as required by eq. (1). The non thermal X-ray emission from SNR 1006 observed by ASCA and ROSAT [11], the GeV γ-ray emission from several nearby SNRs observed by EGRET [12], and the recent detection of SNR 1006 in TeV γ-rays by the CANGAROO telescope [13], were all used to argue that SNRs are the source of galactic CR. However, the TeV γ rays from SNRs can be explained by inverse Compton scattering of microwave background photons by multi-TeV electrons whose synchrotron emission explains their hard lineless X-ray radiation [11]. Furthermore, the mean lifetime of strong shocks in SNRs limits the acceleration of CR nuclei in SNRs to energies less than [14] ∼ Z×0.1PeV and cannot explain the origin of CR with much higher energies. In fact, most nearby SNRs in the Northern hemisphere have not been detected in TeV γ-rays [15]. Moreover, the Galactic distribution of SNRs differs significantly from that required to explain the observed Galactic emission of high energy (> 100MeV ) γ-rays by cosmic ray interactions in the Galactic ISM [16]. All these suggest that, perhaps, SNRs are not the main accelerators of Galacic CR.
Gamma ray bursts (GRBs) have already been proposed as CR sources [17,18,10]. Photon acceleration of CR in isotropic GRBs is probably limited to energies below TeV [17]. Shock acceleration in GRBs may accelerate particles to ultrahigh energies [18]. However, because of the assumed spherical symmetry, the total energy release in GRBs was severely underestimated (see below). Consequently, it was concluded that extragalactic GRBs may be the source of the ultrahigh energy CR in the Galaxy but they cannot supply the bulk of the Galactic CR [18]. In fact, spherical fireballs from merger and/or accretion induced collapse (AIC) of compact stellar objects [19] cannot explain the observed properties of GRBs and their afterglows [20]: They cannot explain their complex light curves and short time scale variability [21]. They cannot explain the lack of scaling and the diversity of their afterglows [22]. They cannot explain their delayed GeV γ ray emission [23]. In particular, isotropic emission implies enormous kinetic energy release in GRBs which cannot be supplied by merger/AIC of compact stellar objects [20]. Such release is necessary in order to sustain the long duration power-law fading of the observed afterglow of GRB 970228 [24]. It is also needed in order to produce the observed γ-ray fluence from GRBs 971214 and 980703 where the redshift z of the host galaxy has recently been measured [25],
E γ ∼ 4πd 2 L F γ 1 + z ∆Ω 4π ≥ 10 53 ∆Ω 4π erg,(3)
where ∆Ω is the solid angle that the emission is beamed into and d L is their luminosity distance (we assuming a Friedmann Universe with h ∼ 0.65, Ω m ≥ 0.2 and Ω m + Ω Λ ≤ 1).
However, if the relativistic ejecta is beamed into a narrow jet [26], most of the problems of the spherical fireball models can be avoided and the main observed properties of GRBs and their afterglows can be explained [20]. Moreover, in order to produce the same observed rate of GRBs, the number of jetted GRBs must be larger by a factor 4π/∆Ω than the number of GRBs with isotropic emission. Thus, for a fixed energy release per event, the total kinetic energy release in GRBs is larger by a factor 4π/∆Ω than that which was estimated [18] for spherical GRBs. Highly relativistic jets are also very eficient CR accelerators. Acceleration to CR energies can take place in the jets by diffusive shock acceleration or in front of the jets by the Fermi mechanism [27]. In this letter I show that jets from mergers/AIC of compact stellar objects may be the source of Galactic CR [10]. The predicted source luminosity, energy spectrum and chemical composition agree with those required by CR observations.
Highly relativistic jets seem to be emitted by all astrophysical systems where mass is accreted at a high rate from a disk onto a central black hole (BH). They are observed in galactic superluminal sources, such as the microquasars GRS 1915+105 [28] and GRO J1655-40 [29] where mass is accreted onto a stellar BH, and in many extragalactic blazars where mass is accreted onto a a supermassive BH. The emission of Doppler shifted Hydrogen
Lyα and Iron Kα lines from the relativistic jets of SS443 [30] suggest that the jets are made predominantly of normal hadronic plasma. Moreover simultaneous VLA radio observations and X-ray observations of the microquasar GRS 1915+105 indicate that the jet ejection episodes are correlated with sudden removal of accretion disk material into relativistic jets [31]. Highly relativistic jets may be the merger/AIC death throws of close binary systems containing compact stellar objects [10,20,26]. But, because the accretion rates and magnetic fields involved are enormously larger compared with normal quasars and microquasars, the bulk motion Lorentz factors of these jets perhaps are much higher, Γ ∼ 1000 as inferred from GRB observations. Such highly relativistic jets which point in (or precess into) our direction ('microblazars') can produce the cosmological GRBs and their afterglows [20]. Jetting the ejecta in merger/AIC of compact stellar objects can solve the energy crisis of GRBs [20,25] by reducing the total inferred energy release in GRBs by a factor ∆Ω/4π. If NS-NS and NS-BH mergers are the triggers of GRBs [32], then beaming angles ∆Ω/4π ∼ 10 −2 are required in order to match the observed GRB rate [21] and the currently best estimates of the NS-NS and NS-BH merger rates in the Universe [33]. Such angles are typical of superluminal jets from blazars and microquasars. The estimated rate of AIC of white dwarfs (WD) and NS in the observable Universe is ∼ 1 per second, i.e., larger than the estimated rate for NS-NS and NS-BH mergers [33] by about two orders of magnitude. Therefore, if GRBs are produced by jets from AIC of WD and NS then ∆Ω/4π ∼ 10 −4 . Note that either the 'firing' of many highly relativistic fragments into a small solid angle [20] or precessing jets [33] can produce
where the unknown GRB beaming angle has been canceled out. Note the agreement between eq. (1) and eq. (4). However, the estimated ratio between the rates of SNe and GRBs is approximated by its measured value in the local universe [37],
ρ L ∼ 1.8h × 10 8 L ⊙ Mpc −3 then R M W [GRB] ≈ R U N IV [GRB]L M W ρ L (1 + z) −1 (dV c /dz)dz .(5)
For a Friedmann Universe with Ω = 1 and Λ = 0, the volume integral yields
dV c 1 + z = 16π c H 0 3 ∞ 0 (1 + z − √ 1 + z) 2 (1 + z) 9/2 dz = 32π 30 c H 0 3 .(6)
Taking into consideration threshold and triggering effects, the estimated rate of cosmological GRBs which point in our direction from the BATSE observations [21] is ∼ 5 day −1 .
Consequently, for h ∼ 0.65 eq. The high collimation of relativistic jets over huge distances (up to tens of pc in microquasars and up to hundreds of kpc in AGN), the confinement of their highly relativistic particles, their emitted radiations and observed polarizations, all indicate that the jets are highly magnetized, probably with a strong helical magnetic field along their axis. Magnetic fields as strong as a few tens of mGauss in the jet rest frame have been inferred from microquasar observations [28], while hundreds of Gauss were inferred for GRB ejecta. The UV light and the X-rays from the jet (and accretion disk) ionize the ISM in front of the jet.
The swept up ISM/jet material can be accelerated by diffusive shock acceleration in the jet.
Alternatively, the jet magnetic field can act as a magnetic mirror and accelerate the ionized ISM particles to high relativistic energies through the usual Fermi mechanism [27]:
Let us denote by M the total ejected mass in an ejection episode, by Γ its initial bulk Lorentz factor and by n p m p the total mass of ionized ISM that is accelerated by the jet.
In the rest frame of ejecta with a bulk Lorentz factor γ, the charged ISM particles move towards the jet with energy γm p c 2 and are reflected back by the transverse magnetic field in the jet with the same energy. In the observer frame their energy is boosted to E = γ 2 m p c 2 .
Moreover, each time such a charged particle is deflected by an external magnetic field (of the ISM or a star) back into the jet, its energy is boosted again by a factor γ 2 . Thus, for n reflections the energy of the accelerated particle can reach E = γ 2n m p c 2 (neglecting radiation losses). Thus a GRB jet with Γ ∼ 10 3 can accelerate ISM protons to energies up to m p c 2 Γ 2 ∼ 10 15 eV in a single reflection, while two reflections can impart to them energies up to m p c 2 Γ 4 ∼ 10 21 eV ! For the sake of simplicity let us assume that the n multiple reflections take place simultaneously (in practice n ≤ 2), Let us also assume a pure hydrogenic composition (the generalization to an arbitrary composition is straight forward).
If the jet loses most of its energy by acceleration of the ISM and not by hadronic collisions or radiation (because of the small hadronic cross sections for binary collisions and radiation processes), conservation of energy and momentum reads,
d(Mc 2 γ) ≈ −dn p E; E = m p γ 2 n.(7)
Consequently, for an ISM with a uniform composition
dn p dE ≈ M m p 1 2nm p c 2 E m p c 2 −2+1/2n : E < Γ 2n mc 2 .(8)
Note that the power-law spectrum is independent of whether the ejecta is spherical, or conical or cylindrical. It is the same for ions and electrons as long as losses and escape are neglected.
It is also insensitive to variations in the ISM density along the radial direction. Under our assumed "ideal" conditions, the spectrum of accelerated particles approaches a dn/dE ∼ E −2 shape. Efficient acceleration continues until either the jet becomes non relativistic, or disperses, or the Larmor radius of the accelerated particles in the jet rest frame r L ∼ However, such simulations depend on too many unknown jet and ISM parameters. Instead, one may assume that the energy dependence of the escape probabilities of CR from their accelerators is similar to that for their escape from the Galaxy, i.e., τ ∼ (E/ZB) −0.6 is also valid for their escape from the CR accelerators. Then, Galactic CR are predicted to have a power-law spectrum with a power index α = 2 − 1/2n + 2 × 0.6 , i.e.,
dn dE ∼ C E E 0 −α with α = 2.70, E<E 0 2.95, E>E 0 ,(9)
where E 0 ∼ m p Γ 2 ∼ A P eV , with A being the mass number of the CR nuclei. These predictions agree well with the cosmic ray observations. Because the jet emits enormous fluxes of beamed radiation in all relevant wave lengths, the ISM in front of it must be completely ionized, Since the escape probability of accelerated nuclei decreases like Z −0.6 the abundances of CR nuclei are expected to be enhanced by approximately a factor ∼ Z 0.6 compared with the ISM abundances. Ionization potential effects are expected to be washed out in the CR composition at high energy. If the origin of the CR knee is the transition from 'single' to 'double' reflections then there should be no change in cosmic ray composition around the CR knee, as claimed by recent measurements [36].
In conclusion, Galactic GRBs may be the main source of Galactic CR.
the complex time structure of GRBs.The energy release in merger/AIC of compact stellar objects is bounded by E b ∼ M N S c 2 ≈ 2.5 × 10 54 erg, where M N S ∼ 1.4M ⊙ is the gravitational mass of a typical neutron star (NS). Then the typical kinetic energy release which is beamed into a solid angle ∆Ω may be of the order E K [GRB] ∼ 2.5 × 10 54 (∆Ω/4π) erg. Such kinetic energy release was inferred from the optical afterglows of GRBs[20]. It is also follows from the energy release in γ-rays from GRBs with measured redshifts[25] if the conversion efficiency of jet kinetic energy into γ-ray energy is a few percent. The rate of SNe and cosmological GRBs that point in our direction were estimated to be[34] R L * [SNe] ∼ 0.02 yr −1 and [35] R L * [GRB] ∼ 2 × 10 −6 yr −1 , respectively, per L * galaxy. If SNe and GRBs have similar histories (evolution functions) then the rate of GRBs in the Milky Way that point in our direction is R M W [GRB] ∼ 10 −4 (4π/∆Ω)R M W [SNe]. Thus, if most of the kinetic energy released in GRBs is converted into CR energy (see below) then the Galactic luminosity in CR due to Galactic GRBs is L M W [CR] ∼ 10 −4 R M W [SNe]E b ∼ 1.5 × 10 41 erg s −1 ,
sensitive to the choice of cosmological model, even if their histories were identical, This is because SNe and GRB observations employ different techniques, have different sensitivities and consequently sample different volumes of the Universe. Therefore, I have estimated L M W [CR] in other independent ways. For instance, I have assumed that the ratio between the Galactic rate of GRBs and the global rate of GRBs is equal to the ratio between the Galactic broad band luminosity [36], L M W ∼ 2.3 × 10 10 L ⊙ , and the luminosity of the whole Universe, L U N IV ∼ (1 + z) −1 ρ L (z)(dV c /dz)dz where ρ L (z) is the comoving luminosity density and the factor 1/(1 + z) is due to the cosmic time dilation. If one assumes that most of the contribution to the volume integral comes from redshifts where ρ L (z) is well
( 5 )
5yields L M W [CR] ∼ 10 41 erg s −1 , consistent with eq. (4). Moreover, if the 'standard candle' E K [GRB] is taken to be proportional to < E γ [GRB] > as obtained from eq. (3), then the estimated CR luminosity becomes insensitive to the choice of the specific cosmology.
3
× 10 15 (E[EeV ]/ΓZB[Gauss]) cm becomes comparable to the radius of the jet. Moreover, the assumption of instantaneous acceleration is unjustified when the Larmor radius of the accelerated particles ceases to be small compared with the typical deceleration distance of the jet. Our simple analytical estimates can be tested in detailed Monte Carlo simulations.
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| {'fraction_non_alphanumeric': 0.06912343776750557, 'fraction_numerical': 0.05709638760486218, 'mean_word_length': 3.7742133224356356, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Highly relativistic jets from merger and accretion induced collapse of compact stellar objects, which may produce the cosmological gamma ray bursts (GRBs), are also very eficient and powerful cosmic ray accelerators. The expected luminosity, energy spectrum and chemical composition of cosmic rays from Galactic GRBs, most of which do not point in our direction, can explain the observed properties of Galactic cosmic rays. PACS numbers: 98.70.Rz, 97.60.-s Typeset using REVT E X 1The origin of high energy cosmic rays (CR), which were first discovered by V. Hess in 1912, is still a complete mystery[1]. Their approximate broken power-law spectrum, dn/dE ∼ E −α with α ∼ 2.7 below the so called CR knee around 10 15.5 eV , α ∼ 3.0 above the knee and α ∼ 2.6 beyond the so called CR ankle around 10 19.5 eV , suggests different origins of CR with such energies. It is generally believed that CR with energy below the knee are Galactic in origin [2], those with energy above the knee can be either Galactic or extragalactic[3], and those beyond the ankle are extragalactic[4].', 'arxivid': 'astro-ph/9809163', 'author': ['Arnon Dar \nMax Planck Institut Für Physik\nDepartment of Physics\nSpace Research Institute Technion\nWerner Heisenberg Institut Fohringer Ring 680805MünchenGermany\n\nIsrael Institute of Technology\n32000HaifaIsrael\n'], 'authoraffiliation': ['Max Planck Institut Für Physik\nDepartment of Physics\nSpace Research Institute Technion\nWerner Heisenberg Institut Fohringer Ring 680805MünchenGermany', 'Israel Institute of Technology\n32000HaifaIsrael'], 'corpusid': 8943558, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8457, 'n_tokens_neox': 6806, 'n_words': 4101, 'pdfsha': '12aed7ea775fae9c77c851303373f373c8ceeecb', 'pdfurls': ['https://export.arxiv.org/pdf/astro-ph/9809163v1.pdf'], 'title': ['Cosmic Rays and Gamma Ray Bursts From Microblazars', 'Cosmic Rays and Gamma Ray Bursts From Microblazars'], 'venue': []} |
arxiv |
Finding Approximate Palindromes in Strings Quickly and Simply
1 Dec 2004 23 Nov. 2004
Lloyd Allison
School of Computer Science and Software Engineering
Monash University
3800ClaytonVictoriaAustralia
Finding Approximate Palindromes in Strings Quickly and Simply
1 Dec 2004 23 Nov. 2004TR 2004/162, CSSE, Monash U., .au --TR 2004/162, CSS (draft 19 Sept)
Described are two algorithms to find long approximate palindromes in a string, for example a DNA sequence. A simple algorithm requires O(n)space and almost always runs in O(k.n)-time where n is the length of the string and k is the number of "errors" allowed in the palindrome. Its worstcase time-complexity is O(n 2 ) but this does not occur with real biological sequences. A more complex algorithm guarantees O(k.n) worst-case time complexity.Code of the simple algorithm will be placed at
Introduction
An (exact) palindrome, p, is a string of symbols that reads the same forwards and backwards, i.e. either p = w.w ′ or p = w.c.w ′ where w is a string, c is a symbol and w ′ = reverse(w); for complementary palindromes in DNA (RNA) we have w ′ = reverse(complement(w)) where A and T (U) are complementary, as are C and G. The first case, p = w.w ′ , is called an even-palindrome and the second, p = w.c.w ′ , is called an odd-palindrome and either the "gap" between w and w ′ or the symbol c is called the centre of the palindrome. Finding palindromes within a long string leads to various classic computing problems, e.g. the longest palin-drome within a string can be found in linear-time by using a suffix-tree (Weiner 1973, McCreight 1976.
Palindromes can be interesting biologically (e.g. Tsunoda 1999, Rozen 2003 and reverse complementary palindromes are relevant to hair-pin loops in RNA folding. But, in biology, palindromes are often allowed to be approximate: k "errors" or "differences" are allowed between w and reverse(w ′ ), that is w and reverse(w ′ ) can have an edit-distance of k. Note that in general one approximate palindromic string, p, may correspond to multiple decompositions p = w.w ′ or p = w.c.w ′ (it is not necessary that |w| = |w ′ |), and a decomposition may correspond to multiple alignments of w and reverse(w ′ ); we prefer a cheapest decomposition and alignment and require costs≤k. Porto and Barbosa (2002) gave an (k 2 n)-time algorithm to find long approximate palindromes in a string. This paper gives a simple algorithm to find long approximate palindromes. It runs in O(n)space and, almost always, in O(k.n)-time; e.g. for k∼10, a million bases of real DNA can be processed in a few seconds on a p.c., most of that time being for I/O. A more complex algorithm guarantees O(k.n) running time.
[[ fig 1 near here ]] 1
It is convenient to describe the simple algorithm in terms of a distance matrix, related to those used in some alignment algorithms. The matrix is a variation on a triangular matrix (Figure 1). An odd exact palindrome is centered on one of the cells marked 'O', and an even exact palindrome on one of the cells marked 'E'. A marked cell is called an origin. Diagonals that run from an origin in a NE direction are important; note that an odd exact palindrome corresponds to an even-numbered diagonal and an even exact-palindrome to an odd-numbered diagonal. Also important are distances along diagonals ( Figure 2). It must be pointed out that the algorithm does not directly use a distance matrix; rather it operates on a different but equivalent matrix to be described.
[[ fig 2 near here ]] An approximate palindrome, p, together with an alignment of w and reverse(w ′ ) where p = w.w ′ or p = w.c.w ′ , implying a cost, is equivalent to a path ( Figure 3) which extends step by step N, E, and/or NE, some distance from an origin. A NE step represents a match or a mismatch. N and E steps represent indels. Each cell of the (notional) distance matrix holds the minimum cost of some optimal path from some origin, not necessarily on the same diagonal, to the cell. The position of any cell in the distance matrix specifies an approximate palindrome, p itself, without any associated alignment; the position fixes the start and the end of the string p. Obviously we want the minimum-cost for an approximate palindrome.
[ The simple algorithm's worst-case behaviour, O(n 2 )-time, is for strings such as A n , (AT ) n/2 , and similar. The cause is looping in order to check a run of matches to extend a path directly NE for zero cost; in practice the average run ends quickly on real DNA sequences. The complex algorithm is, in principle, formed by replacing the simple algorithm's loop by a constant-time step (following linear-time preprocessing) which uses a suffix-tree and a least-common-ancestor (LCA) algorithm such as that of Bender and Farach-Colton (2000).
Results
The simple algorithm was coded in Java and tested on a Linux p.c., AMD Athlon XP TM 2400+ processor, 512MB of memory. It confirmed O(k.n)-time 2 complexity in practice on real DNA, e.g. processing chromosome 3 (1.06Mb) of the malaria organism Plasmodium falciparum (Gardner et al 2002) as follows: k = 10 in 8.0s, k = 20 in 10.2s, k = 40 in 14.3s. Such DNA is approximately 80% AT-rich and is the kind of real DNA most likely to cause problems for this kind of algorithm if any will. The algorithm has not been observed to make more than 3.7(k + 1)n symbol comparisons on real DNA sequences.
References
O E O E O E O E O E O E O E O
Figure 1 :Figure 2 :Figure 3 :
123Odd Some Example Paths
[ fig 3 near here ]] The algorithm actually uses a different but equivalent matrix, reach[d][e], indexed by d which corresponds to a diagonal-number in the distance matrix, and by "error" count, e, where 0 ≤ e ≤ k. reach[d][e] holds the maximum distance along diagonal d of the distance matrix that can be reached by an approximate palindrome for a cost of at most e. The algorithm initially finds exact palindromes, e = 0, i.e. paths that move NE only, as long as this can be done for a cost of zero. It then iterates over the number of errors allowed, e = 1..k, and, within that, over diagonal-number, d, where it executes the general step: space is sufficient to find the approximate palindromes because reach[ ][e] only depends on reach[ ][e − 1]. If alignments (paths) are also required, either O(k.n)-space is required to keep all of reach[ ][ ] or, probably more sensibly assuming path lengths << n, paths can be recovered later by a separate process.reach[d][e] =
max(reach[d-1][e-1]+x,
reach[d ][e-1]+1,
reach[d+1][e-1]+x), where x=d & 1;
while endsMatch(d,reach[d][e]) do
reach[d][e]++ // extend for free
It is an instance of a greedy strategy (e.g. Ukko-
nen 1983). Other tests, not shown, check that the
ends of the string are not overrun. On termina-
tion, reach[d][k] holds the maximum NE-erly dis-
tance from an origin of an acceptable path ending
on diagonal d, thus giving long approximate palin-
dromes.
O(n)-
The LCA problem revisited. M A Bender, M Farach-Colton, Proc. of the. of the4Bender, M. A. and Farach-Colton, M. (2000) The LCA problem revisited. Proc. of the 4th
. Latin American Symp. on Theoretical Informatics. Latin American Symp. on Theoretical Infor- matics, pp.88-94.
Genome sequence of the human malaria parasite Plasmodium falciparum. M J Gardner, Nature. 419Gardner M. J., et al (2002) Genome sequence of the human malaria parasite Plasmodium fal- ciparum. Nature 419 pp.498-511.
A space-economic suffix tree construction algorithm. E M Mccreight, J. of the ACM. 232McCreight, E. M. (1976) A space-economic suffix tree construction algorithm. J. of the ACM 23(2) pp.262-272.
Finding approximate palindromes in strings. A H L Porto, V C Barbosa, Pattern Recognition. 35Porto, A. H. L. and Barbosa V. C. (2002) Finding approximate palindromes in strings. Pattern Recognition 35 pp.2581-2591.
Abundant gene conversion between arms of palindromes in human and ape Y-chromosomes. S Rozen, H Skaletsky, Nature. 423Rozen S., Skaletsky, H. et al (2003) Abundant gene conversion between arms of palindromes in human and ape Y-chromosomes. Nature 423 pp.873-876.
Time and memory efficient algorithm for extracting palindromic and repetitive subsequences in nucleic acid sequences. T Tsunoda, M Fukagawa, T Takagi, Pacific Symp. on Biocomputing. Tsunoda T., Fukagawa M. and Takagi T. (1999) Time and memory efficient algorithm for ex- tracting palindromic and repetitive subse- quences in nucleic acid sequences. Pacific Symp. on Biocomputing 4 pp.202-213.
On approximate string matching. E Ukkonen, Proc. Int. Conf. on Foundations of Computation Theory. Int. Conf. on Foundations of Computation TheoryBorgholm, SwedenUkkonen, E. (1983) On approximate string match- ing. Proc. Int. Conf. on Foundations of Computation Theory, Borgholm, Sweden, pp.487-495.
Linear pattern matching algorithms. P Weiner, 14th IEEE Symposium on switching and automata theory. Weiner, P. (1973) Linear pattern matching algo- rithms. 14th IEEE Symposium on switching and automata theory pp.1-11.
| {'fraction_non_alphanumeric': 0.06464848081614549, 'fraction_numerical': 0.0267243291195387, 'mean_word_length': 4.066853932584269, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Described are two algorithms to find long approximate palindromes in a string, for example a DNA sequence. A simple algorithm requires O(n)space and almost always runs in O(k.n)-time where n is the length of the string and k is the number of "errors" allowed in the palindrome. Its worstcase time-complexity is O(n 2 ) but this does not occur with real biological sequences. A more complex algorithm guarantees O(k.n) worst-case time complexity.Code of the simple algorithm will be placed at', 'arxivid': 'cs/0412004', 'author': ['Lloyd Allison \nSchool of Computer Science and Software Engineering\nMonash University\n3800ClaytonVictoriaAustralia\n'], 'authoraffiliation': ['School of Computer Science and Software Engineering\nMonash University\n3800ClaytonVictoriaAustralia'], 'corpusid': 6673745, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2790, 'n_tokens_neox': 2464, 'n_words': 1462, 'pdfsha': '2a76f9f190aaf54516197ea77267ee36f23a9262', 'pdfurls': ['https://arxiv.org/pdf/cs/0412004v1.pdf'], 'title': ['Finding Approximate Palindromes in Strings Quickly and Simply', 'Finding Approximate Palindromes in Strings Quickly and Simply'], 'venue': []} |
arxiv |
Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality
Lukas Janssen
Institut für Theoretische Physik
Technische Universität Dresden
01062DresdenGermany
Yin-Chen He
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality
We study the critical properties of the QED3-Gross-Neveu model with 2N flavors of twocomponent Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For N = 1, this theory has recently been suggested to be dual to the SU(2) noncompact CP 1 model that describes the deconfined phase transition between the Néel antiferromagnet and the valence bond solid on the square lattice. For N = 2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the QED3-Gross-Neveu model for all values of N . This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain QED3 (without scalar-field coupling) unstable at low values of N . The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter = 4 − D. We compute the scalar-field anomalous dimension η φ , the correlation-length exponent ν, as well as the scaling dimension of the flavor-symmetry-breaking bilinearψσ z ψ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.
I. INTRODUCTION
At zero temperature, strongly-correlated systems exhibit transitions between different phases of matter upon tuning non-temperature parameters, such as external pressure or chemical doping. Just as their classical counterparts, these quantum phase transitions are characterized by only a few universal properties that are governed by an associated continuum quantum field theory [1]. Most quantum phase transitions have a classical analog and can be characterized in terms of a local order parameter that allows to classify and distinguish different phases of matter-a property that is commonly referred to as Landau's symmetry breaking paradigm.
There exist, however, exotic phase transitions which are inherently quantum mechanical and for which the Landau theory is inapplicable. The most familiar example is the putative deconfined quantum critical point between two different symmetry-breaking phases of a spin-1/2 system on the square lattice-the Néel and valence bond solid (VBS) states [2,3]. The deconfined critical point is characterized by fractionalized bosonic spinons on the complex projective space CP 1 coupled to an emergent noncompact U(1) gauge field. These degrees of freedom emerge only directly at the critical point, but are "confined" in either phase. The appropriate theoretical description of the criticality is given by a strongly interacting gauge field theory-the noncompact CP 1 (NCCP 1 ) model. More recently, new types of such non-Landau transitions have been suggested, which are similarly governed by strongly interacting gauge theories [4][5][6]. This includes transitions between different long-range entangled phases, such as the Dirac and chiral spin liquid phases [7][8][9], between short-range entangled phases, e.g., symmetry-protected topological phases [10], and between phases with anticommuting fermion mass terms [11].
The strongly interacting gauge theories that describe the above deconfined critical points are also of wide fundamental interest with respect to various duality webs that were proposed recently. Via these dualities, several seemingly different theories can be mapped onto each other and themselves. The easy-plane version of the NCCP 1 model, for instance, has been argued to be self-dual [3,12], which can be understood as a consequence of the well-known bosonic particle-vortex duality [13][14][15]. Specifically, the spinon field content of the NCCP 1 model can be viewed as either two flavors of bosonic spinons or two flavors of bosonic vortices. Building on the Dirac theory of the half-filled Landau level [16], several works suggest a fermionic counterpart of the particle-vortex duality [17][18][19]. This has lead to a number of fascinating novel duality conjectures, including ones that relate purely bosonic systems to fermionic theories [20][21][22][23][24][25][26]. Early proposals of a duality between the easy-plane NCCP 1 model and quantum electrodynamics in 2+1 dimensions (QED 3 ) [27,28] have recently undergone various consistency checks, corroborating the intimate relationship between these seemingly different theories [22,29,30]. In a similar way, the SU(2) invariant NCCP 1 model has been argued to be self-dual as well as to be dual to QED 3 coupled to a critical real scalar field-a theory that was coined "QED 3 -Gross-Neveu" (QED 3 -GN) model [29]. An immediate consequence of this conjectured duality is the emergence of an enlarged SO(5) symmetry, which was numerically observed earlier [31,32].
While the infrared fate of QED 3 has extensively been discussed in the last three decades [33][34][35][36][37][38][39][40][41][42][43][44][45], the infrared structure of the QED 3 -GN model has, to the best of our knowledge, not been studied before. In this work, we demonstrate that the QED 3 -GN model exhibits a stable fixed point of the renormalization group (RG) for all fermion flavor numbers N . In particular, we demonstrate that the coupling to the critical scalar field prevents the mechanism of fixed-point annihilation that is responsible arXiv:1708.02256v2 [cond-mat.str-el] 9 Nov 2017 for the instability of plain QED 3 at low values of N [34][35][36][37]. The stable fixed point can be approached by tuning a single parameter, such as the scalar-field mass, and thus can be associated with a continuous quantum phase transition. The existence of this quantum critical point for two flavors of two-component fermions is a necessary condition for the NCCP 1 -QED 3 -GN duality to hold. We compute the critical exponents η φ (order-parameter anomalous dimension) and ν (correlation-length exponent) as well as the scaling dimension of the flavorsymmetry-breaking bilinearψσ z ψ, within an expansion around the upper critical space-time dimension of four. If the duality holds, the universal exponents at this quantum critical point in the physical space-time dimension of D = 2 + 1 can be uniquely mapped onto those of the SU(2) invariant NCCP 1 model, and we compare our leading-order estimates with numerical results for the bosonic systems [31,46,47]. Our work represents the first step towards a proper quantification of the critical behavior of the QED 3 -GN model. In the plain Gross-Neveu system (without the coupling to the gauge field), significant progress was made previously by employing high-order expansion [48,49], the functional renormalization group [50][51][52], the conformal bootstrap approach [53,54], and sign-free quantum Monte Carlo simulations [55][56][57][58][59]. Extending these advances to the QED 3 -GN case, and comparing with results for the NCCP 1 model, should allow to prove or disprove the duality conjecture in future studies.
The critical behavior of the QED 3 -GN model is of interest for yet another reason: This model for the case with four two-component fermion flavors has recently been suggested to describe the deconfined critical point between the chiral spin liquid and the U(1) Dirac spin liquid phases [6]. Both phases, and their transition, are potentially realizable in spin-1/2 systems on the kagome lattice [60][61][62][63]. Our finding of a stable fixed point corroborates this proposal, and the predictions for the critical behavior may facilitate a numerical test of it in the future.
The paper is organized as follows: In the following section, we define the QED 3 -GN theory and review the proposed dualities and the potential applicability to deconfined criticality. In Sec. III, we compute the RG flow in a fermionic language that allows to make contact with previous works on the plain QED 3 theory. The 4 − expansion of the QED 3 -GN theory is performed in Sec. IV. In Sec. V, we summarize our results and attempt some conclusions in light of the conjectured NCCP 1 -QED 3 -GN duality.
II. MODEL
We are interested in the QED 3 -GN theory, defined by the Lagrangian
L ψφ =ψ i [γ µ (∂ µ − ia µ )] ψ i + 1 2e 2 ( µνρ ∂ ν a ρ ) 2 + gφψ i ψ i + 1 2 φ(r − ∂ µ ∂ µ )φ + λφ 4 ,(1)
in D = 2 + 1 Euclidean space-time dimensions. The summation convention over repeated indices is assumed. We consider an even number 2N of two-component Dirac fermion flavors ψ i andψ i , i = 1, . . . , 2N . The parity symmetry is therefore explicitly preserved for any integer N and the flavor symmetry is U(2N ). The 2 × 2 Dirac matrices γ µ fulfill the Clifford algebra {γ µ , γ ν } = 2δ µν 1 2 , with µ, ν = 0, 1, 2. The fermions couple to the U(1) gauge field a µ with charge e 2 . The explicit calculations presented below are performed in a general R ξ gauge with undetermined gauge-fixing parameter ξ, by adding L gf = − 1 2ξ (∂ µ a µ ) 2 to the Lagrangian. This enables us to verify the gauge independence of our results. φ is a real scalar field that is odd under the time-reversal symmetry (TRS). It interacts with the fermions through the Yukawa coupling g, and with itself through the φ 4 coupling λ. r is a tuning parameter for the TRS breaking transition, indicated by the formation of a finite scalar-field expectation value, φ = 0. As mentioned in the introduction, this QED 3 -GN theory has interesting applications:
(1) By applying the boson-fermion duality [21,22,64], and building on earlier observations [28], the case N = 1 has recently been conjectured to be dual to the bosonic NCCP 1 theory [29],
L z = α=1,2 |(∂ µ − ib µ )z α | 2 + κ( µνρ ∂ ν b ρ ) 2 + λ 0 |z 1 | 2 + |z 2 | 2 2 + λ 1 |z 1 | 2 |z 2 | 2 .
(2)
Here, z = (z 1 , z 2 ) are complex bosonic fields and b µ is a U(1) gauge field. When λ 1 = 0, the theory has an explicit SU(2) symmetry. We will refer to this case as SU(2) NCCP 1 model. This theory is believed to describe the deconfined critical point between the Néel and VBS phases on the square lattice [2, 3]. For λ 1 = 0, the theory has an easy-plane anisotropy with a residual O(2) symmetry and is relevant for spin models with an XY symmetry. The postulated dualities between Eq. (1) and Eq.
(2) are as follows: (i) The plain QED 3 theory with the scalar field φ decoupled (formally corresponding to the limit of large tuning parameter r) is dual to the easy-plane NCCP 1 model with λ 1 = 0. (ii) The critical QED 3 -GN theory with r tuned such that φ becomes gapless is dual to the SU(2) NCCP 1 model with λ 1 = 0. While these proposed dualities have passed a number of consistency checks [29], we should emphasize that, at present, they lack any formal or numerical proof and should be considered as conjectural. The conjectures, however, do predict a number of nontrivial relations between the universal exponents that describe the critical behaviors of these theories, allowing in principle to verify or falsify the conjectures on a quantitative level. For the case (ii), the scalar field φ is identified with z † σ z z, which is an element of the Néel-VBS SO(5) order parameter. The scalar anomalous dimension η φ in the QED 3 -GN theory should therefore coincide with the anomalous dimensions η Néel and η VBS in the spin systems. Furthermore, the dual to the φ 2 operator corresponds to a rank-2 tensor representation of the SO(5) critical theory. The latter contains the operator z † z that tunes through the Néel-VBS transition, and therefore the correlation-length exponents ν QED3-GN and ν Néel-VBS in the two systems should also coincide. Another consequence of the proposed duality is that z † z can also be identified with the flavor-symmetry breaking fermion bilinearψσ z ψ. Therefore, the scaling dimension ofψσ z ψ should also coincide with the scaling dimension of φ 2 . This statement is particularly interesting, because it allows a nontrivial test of the duality conjecture fully within the QED 3 -GN theory-without the need to compare with a different system. A similar relation between η φ and the scaling dimensions of certain monopole operators also follows [29].
Obviously, a necessary condition for such a duality to hold is the existence of a stable interacting RG fixed point. In fact, for the case of plain QED 3 , the emergence of conformal invariance at low energy, and therewith the existence of a conformal fixed point, can be established when the number of fermions N is large [33,65]. At low values of N , however, a generic mechanism that may destabilize the conformal fixed point is the collision and subsequent annihilation with another, quantum critical, fixed point [34][35][36][37], very much like in the case of the Abelian Higgs model [32,66] as well as a number of further examples [67][68][69][70]. Such instability is driven by strong gauge fluctuations and is therefore ubiquitous in asymptotically safe gauge theories [68]. It leads to an essential singularity of physical observables at a critical flavor number N c , below which the conformal state becomes unstable. The actual value of N c in QED 3 , however, and with it the important question whether N c is above [33-35, 37, 38, 40-43] or below [39,44,45] the physically relevant values, as well as the nature of the low-N phase [36], has been a matter of intense debate within the last three decades. Similarly, on the bosonic side of the duality, the question whether the transition in the easy-plane NCCP 1 model is intrinsically first order, or if a lattice model that hosts a continuous transition between the XY antiferromagnetic and VBS states can be constructed, has been controversially discussed in the past [71][72][73]. Some very recent numerical studies suggest a continuous transition [30,74], with exponents that are potentially in agreement with the conformal phase of plain QED 3 [45].
By contrast, the infrared behavior of the QED 3 -GN model has, to the best of our knowledge, not been investigated before [75]. In the next section, we demon-strate that the coupling to the critical scalar field φ in fact stabilizes the theory, despite the presence of strong gauge fluctuations, and it leads to a stable fixed point that governs a continuous transition into a state with spontaneously broken TRS.
(2) The case N = 2 is relevant for the physics of spin liquid states on the kagome lattice. Despite tremendous efforts, the actual nature of the quantum ground state of the Heisenberg antiferromagnet on the kagome lattice to date has not been established beyond doubt. The most promising candidates are either a gapped Z 2 spin liquid [76][77][78] or a gapless U(1) Dirac spin liquid [63,79,80]. Longer-range spin interactions appear to stabilize yet another spin liquid phase, which is characterized by spontaneous breaking of time reversal symmetry-a chiral spin liquid with anyonic spinon statistics [60,61]. The transition into this state appears to be continuous [62], and if the ground state in the nearest-neighbor model is a Dirac spin liquid, the effective field theory that describes this transition would be the QED 3 -GN model with 2N = 4 flavors of two-component fermion flavors [6]. Determining the critical behavior of the QED 3 -GN model may therefore allow to prove or disprove this scenario if the critical behavior of the spin-liquid transition on the kagome lattice becomes possible to be quantified numerically.
III. QED3-GN QUANTUM CRITICAL POINT IN FERMIONIC RG
The presumed quantum critical point in the theory defined by Eq. (1) with r tuned to criticality demarcates the ordered phase in which the TRS is spontaneously broken, φ = 0, from the time-reversal-symmetric phase, φ = 0. The infrared behavior of the latter phase is governed by the conformal fixed point of plain QED 3 , which albeit in turn may be destabilized for low values of N by a collision with another fixed point [34][35][36][37]. In this section, we demonstrate that a critical point that can be identified with the TRS-breaking transition exists for all N . In particular, it survives when the conformal fixed point of plain QED 3 collides and annihilates with another fixed point when lowering N .
In order to make contact with the conformal phase of QED 3 , we approach the TRS-breaking transition from the symmetric side, φ = 0. On this side, we may neglect the quartic coupling λφ 4 for simplicity and integrate out φ. This way, we obtain a Gross-Neveu-type four-fermion interaction
u ψ i ψ i 2 , i = 1, . . . , 2N,(3)
with negative four-fermion coupling u < 0. In a RG picture, the transition towards the TRS-breaking state would in this formulation be indicated by an instability of the flow towards divergent u → −∞ at a finite RG scale. Once radiative corrections are taken into account, further terms that are not present in the initial action may be generated by the RG. However, symmetry strongly restricts the number of possible terms. On the level of four-fermion interactions, the only term that is compatible with the U(2N ) flavor symmetry is the Thirring interaction [81],
v ψ i γ µ ψ i 2 .(4)
A minimal low-energy effective theory is therefore given by a U(1) gauge theory with 2N flavors of twocomponent Dirac fermions, augmented with Gross-Neveu and Thirring four-fermion interactions:
L ψ =ψ i γ µ (∂ µ − ia µ )ψ i + u ψ i ψ i 2 + v ψ i γ µ ψ i 2 . (5)
Integrating over the momentum shell from Λ to Λ/b with b > 1, the RG flow of this theory reads, to the one-loop order
de 2 d ln b = (1 − η a )e 2 ,(6)du d ln b = −u + 16 3 e 2 u + 8e 2 v + 2e 4 − 4(2N − 1)u 2 + 8v 2 + 12uv, (7) dv d ln b = −v + 8 3 e 2 u + 4 3 (2N + 1)v 2 + 4uv,(8)
with the gauge-field anomalous dimension η a = 4 3 N e 2 . In order to arrive at the above beta functions, we have rescaled the couplings as e 2 /(2π 2 Λ) → e 2 , Λu/(2π 2 ) → u, and Λv/(2π 2 ) → v. The corresponding diagrams are depicted in Figs. 1 and 2. The above equations are consistent with previously published ones in the respective limit [36].
At large N , the fixed-point structure can be elucidated analytically. At zero charge e 2 = 0, there are two critical
They are believed to be of relevance in the context of interacting fermions on the honeycomb lattice [50,55,82,83], and have been extensively studied in the past [48,49,81,[84][85][86][87][88]. The charge e 2 , however, is RG relevant towards the infrared and flows to a finite fixed-point value e 2 * = 3 4N + O(1/N 2 ). In this "charged" plane, there are two quantum critical points when N is large. We find
QED 3 -GN : (e 2 * , u * , v * ) = 3 4N , − 1 8N , 0 + O(1/N 2 ),(11)
and therefore describes a transition into a state with u → −∞. This corresponds to the spontaneous breaking of TRS, and the fixed point should therefore be understood as the projection of the critical point in the full QED 3 -GN theory onto the four-fermion coupling space. The fixed point in Eq. (12) ("g-T") represents a gauged version of the Thirring fixed point. Moreover, we also rediscover [34][35][36][37][38] the fully infrared attractive fixed point that describes the conformal phase of QED 3 , which in the limit of large N is located at c-QED 3 : (e 2 * , u * , v * ) =
4N
, 0, 0 + O(1/N 2 ). (14) By evaluating the fixed-point equations at finite N numerically, we find that it is the g-T fixed point (and not the QED 3 -GN fixed point) that approaches c-QED 3 and eventually collides and annihilates with the latter at a critical flavor number N c . This is in agreement with the previous RG studies [34][35][36][37]. In our simple approximation, this fixed-point annihilation happens at N c ≈ 6, a number which should be expected to receive corrections when going beyond the present one-loop order. In any case, the point we would like to emphasize here is that the QED 3 -GN fixed point, in contrast to the c-QED 3 and g-T fixed points, survives across the transition at N c and continuous to exist for all values of N . For the case of N = 1, relevant to the duality conjecture, the RG flow in the coupling space spanned by e 2 , u, and v is illustrated in Fig. 3, showing the fixed points GN and T in the uncharged sector e 2 = 0 and the quantum critical
In the above equation, we have displayed only the leading-order value within the 1/N expansion, for which our one-loop flow equations are sufficient [89]. The com-putation of the 1/N correction requires the knowledge of the two-loop flow. This is left for future work. The scaling dimension of the TRS-breaking fermion bilinear ψψ can be determined by computing the flow of a small symmetry-breaking perturbation of the form ∆ψψ. At the one-loop order [36],
d∆ d ln b = 1 − 2(4N − 1)u + 6v + 8 3 e 2 ∆ + O(∆ 2 ).(16)
Therewith, we find
[ψψ] QED3-GN = 3 − [∆] = 1 + O(1/N )(17)
at the QED 3 -GN fixed point, corresponding to an anomalous dimension
η φ = 1 + O(1/N )(18)
of the TRS order parameter φ ∝ ψ ψ . Note that the scaling dimensions at the other critical fixed points, such as g-T and T, would be [ψψ] g-T = [ψψ] T = 2 + O(1/N ), and thus these fixed points are, pictorially speaking, "less unstable" towards the TRS-breaking perturbation. Along the same line, we can obtain the scaling dimension of the flavor-symmetry breaking bilinearψσ z ψ ≡
ψ i (σ z ⊗ 1 N ) ij ψ j . At the QED 3 -GN fixed point, it be- comes [ψσ z ψ] QED3-GN = 2 + O(1/N ).(19)
This corroborates our conclusion that the fixed point in Eq. (11) should be associated with the spontaneous breaking of TRS, and therewith represents the fourfermion version of the critical point in the original QED 3 -GN theory, Eq. (1). At the c-QED 3 fixed point, we find at large N
[ψψ] c-QED3 = 2 + O(1/N ) = [ψσ z ψ] c-QED3 ,(20)
consistent with known results [39]. We remark that the O(1/N ) corrections for the two operators are different [90].
IV. QED3-GN QUANTUM CRITICAL POINT IN 4 − EXPANSION
The above one-loop calculation in the four-fermion theory space spanned by u and v allows us to obtain a qualitative picture of the structure of the RG flow, and to make contact with the situation in plain QED 3 , when the order-parameter field φ is decoupled. However, in the physically interesting low-N limit, the fixed points are located at strong coupling in 2 + 1 dimensions, and the approximation ceases to be under perturbative control. One may therefore wonder whether it is possible to establish the existence and investigate the nature of the QED 3 -GN fixed point within a complementary approach. This is the subject of the present section. To this end, we turn back to our initial formulation of the theory in terms of L ψφ , Eq. (1). The Lagrangian can be generalized to arbitrary space-time dimension 2 < D < 4 by trading the 2N flavors of two-component spinors for N flavors of four-component spinors, and employing a 4 × 4 representation of the Dirac matrices. There are different possibilities on how to dimensionally continue the Dirac structure to noninteger D [38]. Here, we use the common prescription that fixes the form of the TRS-breaking fermion bilinearψψ in all 2 < D < 4, as commonly done in the plain Gross-Neveu-Yukawa models [49,83,91,92]. In general D, the couplings have engineering dimensions
[e 2 ] = 4 − D, [g] = 4 − D 2 , [λ] = 4 − D.(21)
Hence, all three couplings simultaneously become marginal when D 4. This observation suggests that the QED 3 -GN fixed point may be accessible perturbatively within an expansion near four space-time dimensions.
In this limit, we find the flow equations for the couplings e 2 , g, and λ to the one-loop order as
de 2 d ln b = ( − η a )e 2 ,(22)dg 2 d ln b = ( − η φ )g 2 + 6e 2 g 2 − 3g 4 ,(23)dλ d ln b = ( − 2η φ )λ − 36λ 2 + N g 4 ,(24)
with the anomalous dimensions
η a = 4 3 N e 2 , η φ = 2N g 2 ,(25)
where N is the number of four-component fermions and = 4 − D. Here, we have tuned the system to criticality with r ≡ 0, and have rescaled e 2 /(8π 2 ) → e 2 , g 2 /(8π 2 ) → g 2 , and λ/(8π 2 ) → λ. The corresponding diagrams are shown in Figs. 2, 4, and 5. Note that any dependence on the gauge-fixing parameter ξ in the beta functions has canceled out, as it should be. For e 2 = 0, the flow equations for g 2 and λ coincide with those for the ungauged Gross-Neveu-Yukawa model [50]. For g 2 = λ = 0, on the other hand, the flow equation for the charge agrees with the one for QED 4− [38].
In the full theory space spanned by e 2 , g, and λ, the above equations exhibit a unique infrared-stable fixed point at where f (N ) ≡ √ 4N 4 + 204N 3 + 1521N 2 + 2916N . The fixed-point structure in the plane spanned by λ and g 2 is illustrated for N = 1 in Fig. 6. For visualization purposes, there we have set the charge e 2 to its infrared fixed-point value e 2 * . The QED 3 -GN fixed point governs the continuous transition into the TRS-broken state with φ = 0, and should be understood as the Hubbard-Stratonovich-transformed version of the QED 3 -GN fixed point we have found in the fermionic language, Eq. (11). This is in full analogy to the equivalence of the critical points in the Gross-Neveu and Gross-Neveu-Yukawa theories [93].
QED 3 -GN : (e 2 * , g 2 * , λ * ) = 3 4N , 2N +9 2N (2N +3) , −2N 2 −15N +f (N ) 72N (2N +3) + O( 2 ),
In D = 2 + 1 dimensions, the QED 3 -GN fixed point is characterized by Lorentz invariance and U(2N ) flavor symmetry. In order to be relevant for real materials, these symmetries must be emergent in the lowenergy limit. Flavor-symmetry-breaking perturbations have previously been shown to be indeed RG irrelevant, at least near the ungauged version of the fixed point (GN) [94]. Here, we demonstrate that the Lorentz symmetry also emerges in the critical region. As long as the spatial spherical symmetry remains intact, the only potentially relevant symmetry-breaking perturbations are terms quadratic in the fields. Adding these perturbations is equivalent to allowing different fermion and boson velocities, v F and v B . Thus, we replace the kinetic terms in Eq. (1) by
γ µ D µ → γ 0 D 0 + v F γ · D, ∂ µ ∂ µ → ∂ 2 0 + v 2 B ∇ 2 ,(26)
where (D µ ) ≡ (D 0 , D) ≡ (∂ µ − ia µ ), µ = 0, . . . , D − 1, is the gauge-covariant derivative. v F and v B are mea-sured in units of the speed of light c ≡ 1. Lorentz invariance is emergent when both flow to unity in the infrared, v F,B → 1. The Lorentz-invariant subspace itself is invariant under the RG for symmetry reason. Allowing small symmetry-breaking perturbations out of this subspace as v F = 1 + δv F and v B = 1 + δv B with δv F,B 1, we find the flow equations
dδv F d ln b = − 8e 2 + g 2 3 δv F + g 2 3 δv B ,(27)dδv B d ln b = 2N g 2 δv F − 2N g 2 δv B .(28)
The corresponding stability matrix
∂(dδvF,B/d ln b) ∂δvF,B
has the eigenvalues
θ ± = −α ± α 2 − β 2 ,(29)
with α ≡ (N + 1 6 )g 2 + 4 3 e 2 > 0 and β 2 ≡ 16 3 N e 2 g 2 > 0. Consequently, we have Re θ ± < 0 everywhere and δv F and δv B are always irrelevant perturbations. The Lorentz symmetry is therefore emergent at low energy. This result is analogous to previous findings in related models with different order-parameter fields [95].
At the stable QED 3 -GN fixed point, the anomalous dimensions read, to the leading order in = 4 − D,
η a = ,(30)η φ = 2N + 9 2N + 3 + O( 2 ).(31)
We mention in passing that Eq. (30) is expected to not receive higher-order corrections due to the Ward identity associated with the U(1) gauge symmetry [36,96]. The correlation-length exponent is related to the scaling dimension of φ 2 via 1/ν = D − [φ 2 ]. We obtain
ν = 1 2 + 10N 2 + 39N + f (N ) 24N (2N + 3) + O( 2 ).(32)
From the viewpoint of the duality conjecture, it is interesting to also compute the scaling dimension of the flavor-symmetry breaking bilinearψσ z ψ at the QED 3 -GN fixed point. To the leading order, we find
[ψσ z ψ] = 3 − 2N + 6 2N + 3 + O( 2 ).(33)
Now, if we simply extrapolated Eqs. (30)-(33) towards large values of , the leading-order corrections become sizable, e.g., η φ 2.2 , ν 0.5+0.98 , and [ψσ z ψ] 3− 1.6 for N = 1. This obviously compromises the validity of the plain extrapolation. The qualitative behavior of the exponents at large can, however, be inferred from the behavior near the lower critical space-time dimension of two. From a calculation analogous to that leading to Eqs. (15)-(18), we find, to the lowest order,
1/ν = (D − 2) + O 1/N, (D − 2) 2 ,(34)η φ = 2 − (D − 2) + O 1/N, (D − 2) 2 ,(35)
This leading-order result coincides with the behavior of the plain Gross-Neveu model near the lower critical dimension [48,50], which can be attributed to the fact that the charge contribution to the flow of ∆ is subleading in 1/N , c.f. Eq. (16). In order to gain a reasonable estimate for the exponents in the physical situation in D = 3 and small N , we can thus search for a smooth interpolation between the boundary values near the upper and lower critical dimensions. We use a simple polynomial form as in Ref. 50, and therewith find, for D = 3,
η φ ≈ 4N + 9 2(2N + 3) , 1/ν ≈ 50N 2 + 51N − f (N ) 24N (2N + 3) ,(37)
and
[ψσ z ψ] ≈ 16N + 21 8N + 12 .
The interpolating polynomials together with the naive extrapolations are depicted for N = 1 as function of space-time dimension 2 < D < 4 in Fig. 7. In the large-N limit, Eqs. (37) and (38)
V. CONCLUSIONS
In this paper, we have studied the critical behavior of the QED 3 -GN model in three space-time dimensions. Just as in the corresponding plain Gross-Neveu universality class without a gauge field, there is a unique stable fixed point, which can be understood either as an ultraviolet fixed point of the four-fermion ("Gross-Neveu") theory, or as an infrared fixed point of the partially bosonized ("Gross-Neveu-Yukawa") theory [93].
We have employed the four-fermion language to clarify the correspondence of the QED 3 -GN fixed point with the previously-studied conformal fixed point of plain QED 3 [34][35][36][37][38]. Using this formulation, we have verified that the fixed-point annihilation mechanism that destabilizes the conformal phase of plain QED 3 at a critical flavor number N c , does not intrude upon the stability of the QED 3 -GN fixed point. In fact, the latter turned out to continue to exist across the transition at N c all the way down to N = 1, at least within the present one-loop approximation.
The equivalent partially bosonized QED 3 -GN theory, with a Yukawa interaction instead of the four-fermion term, can be dimensionally continued to noninteger space-time dimension D. We have used the fact that all three couplings present in the theory become simultaneously marginal when D 4 to set up an expansion around four space-time dimensions. This allows to establish the existence of the QED 3 -GN fixed point and to access the critical behavior in a controlled way. We have computed the critical exponents η φ and ν, the scaling dimension of the flavor-symmetry-breaking bilinear ψσ z ψ, as well as the gauge anomalous dimension η a to the leading order in = 4 − D. For the latter, we predict η a = 4−D for all N and 2 < D < 4 exactly, which follows as a consequence of the Ward identity associated with the U(1) gauge symmetry. For the other exponents, our best estimates for D = 3 are η φ ≈ 1.3(9) and 1/ν ≈ 0.3(4) in the case of N = 1. Here, we have taken the difference between the plain extrapolation and the polynomial interpolation, which makes use of additional information of the behavior of the exponents near the lower critical dimension, as a rough error estimate. The uncertainty becomes smaller for larger N , but for N = 1 it is significantly larger than the error of the corresponding leading-order estimates in the plain Gross-Neveu universality class [50]. It would therefore be desirable to extend our work to higher loop order, e.g., along the lines carried out recently for the ungauged Gross-Neveu-Yukawa [31,46,47]. The latter is presumably described by the SU(2) NCCP 1 model, for which we also quote the results of a field-theoretical approach [ (4) [z † z] ≈ -see abovemodel [49]. As a complementary approach, the QED 3 -GN fixed point should be accessible within the fourfermion formulation in an expansion around the lower critical space-time dimension of two. The analogous computation in the plain Gross-Neveu model has now been accomplished, in a technological tour de force, up to the four-loop order [48]. This necessitates to deal with the notorious evanescent operators, which render the theory nonunitary in dimensional regularization and are generically generated at high order in the expansion or when operators of high scaling dimension are analyzed [97].
The comparatively large uncertainty of our results notwithstanding, we consider our finding of a large orderparameter anomalous dimension of order unity or larger to be reliable. In fact, a large value of η φ appears to be characteristic to all known chiral universality classes that are driven by massless fermionic degrees of freedom [55][56][57][58][59]. Theoretically, this property can be traced back to the observation that in all critical fermion systems the order-parameter anomalous dimension has to approach unity in the limit of large flavor number. Furthermore, near the lower critical dimension, its boundary value is η φ = 2 + O(D − 2).
These findings are striking in the light of the recently conjectured duality of the N = 1 QED 3 -GN theory with the SU(2) NCCP 1 model [29], which in turn is believed to describe the deconfined critical point between the Néel and VBS phases of spin-1/2 systems on the square lattice [2, 3,31,32]. While the existence of a stable QED 3 -GN fixed point is a prerequisite for the duality scenario to hold, our leading-order results for its critical behavior is not entirely compatible with the critical (or pseudocritical) behavior measured in the spin systems. The largest discrepancy occurs in the case of the order-parameter anomalous dimensions, which in the spin systems have been determined as η Néel ≈ η VBS ≈ 0.25 . . . 0.35 [31,46,47]. This is about an order of magnitude larger than in the standard bosonic O(5) universality class [98], but still significantly smaller than our estimate of η φ ≈ 1. 3(9) in the QED 3 -GN theory. Direct simulations of the NCCP 1 model remain inconclusive as to whether the transition is continuous [12,99,100] or weakly first order [101]. In any case, as far as we are aware, at present no numerical data in the purely bosonic models appear to suggest an anomalous dimension of order unity or larger. Field-theoretical approaches to the critical behavior of the NCCP 1 model appear to be difficult, since the loop corrections are sizable [102,103]. Nevertheless, a functional RG approach finds values that are remarkably close to the most recent numerical results in the spin systems [104].
We have also computed the scaling dimension of the flavor-symmetry-breaking fermion bilinearψσ z ψ, which is identified with z † z in the bosonic NCCP 1 theory. The latter corresponds to the tuning parameter for the Néel-VBS transition. Therefore, the duality predicts 1/ν Néel-VBS = 3 − [ψσ z ψ]. Our calculation gives 3 − [ψσ z ψ] ≈ 1.2(5), while the numerical simulations of the Néel-VBS transition find 1/ν Néel-VBS ≈ 1.3 . . . 2.0 [31,46,47]. These values are not inconsistent with the duality prediction. The duality also predicts that the scaling dimension [ψσ z ψ] should coincide with [φ 2 ]. Our result for ν gives [φ 2 ] = 3 − 1/ν ≈ 2.7(4) which is only somewhat larger than [ψσ z ψ] ≈ 1.8 (5), but incompatible with the numerical ranges quoted for 3 − 1/ν Néel-VBS .
In conclusion, our estimate for [ψσ z ψ] seems to be not inconsistent with the duality proposal, but [φ] and [φ 2 ] show large discrepancies when comparing them with the corresponding measurements in the bosonic systems. This is summarized in Table I. We note, however, that if the transition in the spin systems is indeed continuous with an emergent SO(5) symmetry [32], then this unavoidably necessitates anomalous dimensions that are significantly above the ones currently observed [105]. We therefore believe that the possibility that higher-order computations in the QED 3 -GN model and forthcoming numerical calculations in the spin systems converge to common values in future works is as yet not excluded.
FIG. 1 .
1One-loop diagrams determining the flows of the four-fermion couplings u and v. Solid (wiggly) lines correspond to fermion (gauge) fields.
FIG. 2 .
2Diagrams that determine the gauge-field anomalous dimension ηa (left) and the fermion selfenergy (middle). In the flow equation for e 2 , the contributions from the fermion selfenergy and explicit vertex correction (right) cancel due to the Ward identity associated with the U(1) gauge symmetry.
have precisely one RG relevant direction in the (e 2 , u, v) space of couplings. The relevant direction of the former fixed point ("QED 3 -GN") is aligned along (e 2 , u, v) = (0, −1, 0) + O(1/N ),
FIG. 3 .
3RG flow for N = 1. The panels in (b)-(d) display the RG flow within different subspaces of the full theory space spanned by u, v, and e 2 , as schematically depicted in (a). (b) u-v plane for e 2 = 0. (c) u-v plane for e 2 = e 2 * = 3/4. (d) u-e 2 plane for v = 0. Besides the Gaussian fixed point (G), the only quantum critical point with just one relevant direction is the QED3-GN fixed point. It describes the TRS-breaking transition. There are furthermore two critical points in the uncharged sector e 2 = 0, which, however, receive a second relevant direction along the e 2 axis: the Gross-Neveu fixed point (GN) and the Thirring fixed point (T).
QED 3 -
3GN fixed point in the RG attractive plane e 2 = e 2 * . From the flow of the relevant direction [Eq. (13)], we obtain the correlation-length exponent at the QED 3 -GN fixed point as 1/ν = 1 + O(1/N ).
FIG. 4 .
4One-loop diagrams determining the flows of the Yukawa coupling g 2 and the φ 4 coupling λ, as well as the scalar-field anomalous dimension η φ . Dashed lines correspond to the scalar field.
FIG. 5 .FIG. 6 .
56Diagrams that cancel in the flow equation for the charge e 2 due to the Ward identity. In the flow equation for the Yukawa coupling g 2 , the contribution from the fermion selfenergy (right) cancels with the gauge-dependent part of the vertex correction (first diagram inFig. 4). RG flow for N = 1 in (λ, g 2 ) plane to leading order in = 4 − D. For visualization purposes, here we have put the charge to its infrared fixed-point value e 2 = e 2 * = 3 /4, and we tune the system to criticality with r ≡ 0. The infrared stable fixed point at g 2 * > 0 and λ * > 0 is the QED3-GN quantum critical point and governs the transition into the TRS-broken state with φ = 0. G and WF at g = 0 describe the Gaussian and Wilson-Fisher fixed points.
FIG. 7 .
7Critical exponents η φ (a) and 1/ν (b), and scaling dimension [ψσ z ψ] (c) as function of space-time dimension D in the critical QED3-GN theory for N = 1. Solid curves: polynomial interpolation between lowest-order 2 + expansion result and 4 − expansion result. Dashed curves near lower and upper critical dimensions, respectively: plain extrapolation of expansion for comparison. For 1/ν (b), the two dashed curves near D = 4 correspond to the inverse of Eq. (32) (upper curve) and the expansion of 1/ν itself (lower curve), cf. Ref. [50]. and [ψσ z ψ] = 1 + (D − 2) + O 1/N, (D − 2) 2 .
TABLE I .
IScaling dimensions of operators at the N = 1 QED3-GN fixed point in comparison with literature values for scaling dimensions of the corresponding dual operators at the Néel-VBS deconfined critical point
104].QED3-GN
SU(2) NCCP 1
[φ] ≈ (1 + 1.3(9))/2
[z † σ z z] ≈ (1 + 0.26(3))/2 [46]
≈ (1 + 0.35(3))/2 [47]
≈ (1 + 0.25(3))/2 [31]
≈ (1 + 0.22)/2
[104]
[ψσ z ψ] ≈ 3 − 1.2(5)
[z † z] ≈ 3 − 1.28(5)
[46]
≈ 3 − 1.47(9)
[47]
≈ 3 − 1.99(4)
[31]
≈ 3 − 1.79
[104]
[φ 2 ] ≈ 3 − 0.3
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| {'fraction_non_alphanumeric': 0.07099735062904464, 'fraction_numerical': 0.04716748946766464, 'mean_word_length': 3.9254135767256133, 'pattern_counts': {'":': 0, '<': 18, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study the critical properties of the QED3-Gross-Neveu model with 2N flavors of twocomponent Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For N = 1, this theory has recently been suggested to be dual to the SU(2) noncompact CP 1 model that describes the deconfined phase transition between the Néel antiferromagnet and the valence bond solid on the square lattice. For N = 2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the QED3-Gross-Neveu model for all values of N . This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain QED3 (without scalar-field coupling) unstable at low values of N . The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter = 4 − D. We compute the scalar-field anomalous dimension η φ , the correlation-length exponent ν, as well as the scaling dimension of the flavor-symmetry-breaking bilinearψσ z ψ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.', 'arxivid': '1708.02256', 'author': ['Lukas Janssen \nInstitut für Theoretische Physik\nTechnische Universität Dresden\n01062DresdenGermany\n', 'Yin-Chen He \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n'], 'authoraffiliation': ['Institut für Theoretische Physik\nTechnische Universität Dresden\n01062DresdenGermany', 'Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA'], 'corpusid': 54831306, 'doi': '10.1103/physrevb.96.205113', 'github_urls': [], 'n_tokens_mistral': 23644, 'n_tokens_neox': 19825, 'n_words': 11308, 'pdfsha': '6d91d5e3e370df199efae2d36d9e58439e9c3a1f', 'pdfurls': ['https://arxiv.org/pdf/1708.02256v2.pdf'], 'title': ['Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality', 'Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality'], 'venue': []} |
arxiv |
GALAXY CLASSIFICATION USING TRANSFER LEARNING AND ENSEMBLE OF CNNS WITH MULTIPLE COLOUR SPACES
Yevonnael Andrew
GALAXY CLASSIFICATION USING TRANSFER LEARNING AND ENSEMBLE OF CNNS WITH MULTIPLE COLOUR SPACES
and Machine Learning JUNE 2022 i
Big data is now a norm in astronomy. The growth of astronomical images makes it a very suitable domain for computer science research. It is common for astronomer uses the morphological of galaxies to classify galaxies into categories. This practice was first applied systematically by Hubble (1936). When the data is small in size, the classification process could be easily done by small teams or individuals. However, the exponential growth of data collected by modern telescopes made it impossible to rely on an expert to classify every single galaxy image.In December 2013, Winton Capital and Galaxy Zoo, together with the Kaggle team, created a Galaxy Challenge, where participants were asked to create a model to classify galaxies into categories. Since then, researchers worldwide have often used the Kaggle Galaxy Zoo dataset.This research will focus on investigating how colour space transformation could affect classification accuracy, and we will investigate whether the CNN architecture will affect this too. For this research, we will consider multiple colour spaces (RGB, XYZ, LAB, etc) and multiple CNN architectures (VGG, ResNet, DenseNet, Xception, etc). We will use a pre-trained model and weights. However, most of the pre-trained model was trained on a natural RGB image, so we will investigate the performance in predicting the transformed (non-RGB) nonnatural image (astronomical images).We test our hypothesis by first testing individual networks using RGB and transformed colour spaces. We also test multiple ensemble configurations of different networks and colour spaces.We did a minimal hyperparameter search to ensure we obtained an optimum result. Our experimental results show that using transformed colour spaces on individual networks produced a higher validation accuracy. Ensembles of network and colour spaces further increase the validation accuracy.Finally, this research aims to validate the usefulness of colour space transformation for astronomical images. This research will also become a benchmark that is useful for future research.ii
Galaxy Challenge, where participants were asked to create a model to classify galaxies into categories. Since then, researchers worldwide have often used the Kaggle Galaxy Zoo dataset.
This research will focus on investigating how colour space transformation could affect classification accuracy, and we will investigate whether the CNN architecture will affect this too. For this research, we will consider multiple colour spaces (RGB, XYZ, LAB, etc) and multiple CNN architectures (VGG, ResNet, DenseNet, Xception, etc). We will use a pre-trained model and weights. However, most of the pre-trained model was trained on a natural RGB image, so we will investigate the performance in predicting the transformed (non-RGB) nonnatural image (astronomical images).
We test our hypothesis by first testing individual networks using RGB and transformed colour spaces. We also test multiple ensemble configurations of different networks and colour spaces.
We did a minimal hyperparameter search to ensure we obtained an optimum result. Our experimental results show that using transformed colour spaces on individual networks produced a higher validation accuracy. Ensembles of network and colour spaces further increase the validation accuracy.
Finally, this research aims to validate the usefulness of colour space transformation for astronomical images. This research will also become a benchmark that is useful for future research.
ii Table 2: Literature review summary on feature extraction using a pre-trained model . 11 Table 3
LIST OF TABLES
Background
Big data is now a norm in astronomy due to a shared culture of cooperation and regulations, in which the collective data from those telescopes are made available online (Feigelson and Babu, 2012). SDSS, which began operating in 2000 (Gunn et al., 2006), now produces about 200 GB of data every night (Feigelson and Babu, 2012).
Fortunately, the growth of astronomical images happens simultaneously with the growth of computing power. Astronomy arguably is a perfect domain for computer science research because it pushes the boundaries of existing analysis (Kremer et al., 2017). This makes sense because as data increases, analysis becomes more difficult. It is common for astronomer uses the morphological of galaxies to classify galaxies into categories. This practice was first applied systematically by Hubble (1936). Small teams or individuals could quickly process the classification process when the data is small in size.
However, the exponential growth of data collected by modern telescopes made it impossible to rely on an expert to classify every single galaxy image.
To solve this, a crowdsourced project Galaxy Zoo project was launched (Lintott et al., 2008), which invites volunteers to assist in classification. The project was very successful because, in approximately 175 days, more than 100,000 volunteers were involved in assisting classification (Lintott et al., 2011).
Research Questions
After conducting literature reviews, several research questions would like to be addressed by this research.
• How does the colour transformation affect the classification accuracy in the astronomical images?
• Would the network learn different representations when we feed them with different colour spaces?
• How does the usage of pre-trained models and training from scratch affect classification accuracy?
• How does the usage of different base architecture for ensembling affect the classification accuracy?
• What are the best ensemble methods to combine multiple networks trained on different colour spaces in order to yield a higher accuracy?
Aim and Objectives
This research aims to understand the effect of colour transformation and ensembles of CNN trained on different colour transformations on the classification accuracy of astronomical images.
Based on the aim of this study, the research objectives are formulated as follows:
• To analyse the effect of colour transformation with respect to classification accuracy This research is intended to develop a novel methodology that can effectively combine networks trained on multiple colour spaces. We hypothesise that the combination of these networks will improve classification accuracy.
Significance of the Study
This research aims to contribute new pieces of knowledge to the deep learning and computer vision fields in the following ways:
• To validate the usefulness of colour space transformation to increase the classification accuracy of astronomical images
• To validate the usefulness of transfer learning for astronomical images • A benchmark for the comparison of different CNN architecture on galaxy classification
• A benchmark of multiple ensembling methods on galaxy classification
Scope of the Study
Taking into account the time and resource constraints, we will limit the scope of the study as follows:
• This work only focuses on astronomical data, i.e., Kaggle Galaxy Challenge data.
• Other methods, such as statistical feature extraction, will not be considered here.
LITERATURE REVIEW
Machine Learning in Astronomy
The earliest published work exploring the use of machine learning in astronomy was by Adorf and Meurs (1988), which used both supervised and unsupervised classification to classify the IRAS Point Source Catalog.
The earliest published work to automatically classify stars/galaxies using neural networks was by Odewahn et al (1992). The authors use both linear perceptron and multi-layer perceptron for the classification task. The classification is based on manually extracted features, e.g. diameter, ellipticity, average transmission, central transmission, gradients, etc.
The earliest review paper on neural network applications in astronomy was published by Miller (1993). The author identified several major areas of research: adaptive telescope optics, object classification, object matching, and detector event filtering. The state-of-the-art algorithm in this paper is the multi-layer perceptrons employing back-propagation learning (MLP) and selforganising maps (SOM).
The back-propagation algorithm is a learning procedure, in which the weights of connections are adjusted iteratively to minimise the difference between the true value and the predicted value (Rumelhart, Hinton and Williams, 1986). Years later, the back-propagation algorithm was applied to solve a real-world problem, recognising handwritten zip codes (LeCun et al.,
1989).
Neural networks and back-propagation were developed in the 1980s. The ability of the algorithms was also already demonstrated by LeCun in 1989. However, there was a fundamental problem in deep learning, which made researchers lose interest. By the late 1980s, it was known that traditional deep feedforward networks were hard to train by back-propagation (Schmidhuber, 2015). The reason is that deep neural networks suffer from a problem that is now famous as exploding and vanishing gradients (Hochreiter, 1991).
Due to the limitation of back-propagation of recurrent neural networks, i.e., exploding and vanishing gradients, an important concept called Long Short Term Memory (LSTM) was developed (Hochreiter and Schmidhuber, 1997). But, the LSTM breakthrough did little to fix the larger problem of neural networks and did not work very well; also, computers were not fast enough, algorithms were not smart enough, and people were not satisfied (Kurenkov, 2020 Forests are proven to be very effective and come with a sound mathematical theory (Kurenkov, 2020). This also contributes to the AI winter.
Convolutional Neural Networks
Convolutional neural networks, which are sometimes referred to as ConvNets or CNNs, are a specialised kind of neural network that is suitable for data that has a grid-like topology, like images, which are usually thought of as a two-dimensional grid of pixels (Goodfellow, Bengio and Courville, 2016).
AlexNet
In 2012, there was a competition called ImageNet Large Scale Visual Recognition Challenge,
which is now often referred to as ImageNet only. The winning solution achieved a top-5 error of 15.3%, which is substantially lower than the runner-up. This winning solution is now well known as AlexNet (Krizhevsky, Sutskever and Hinton, 2012).
AlexNet has eight layersfive convolutional and three fully-connected layers. To prevent overfitting, the network uses ReLUs non-linearity (Nair and Hinton, 2010) and dropout layer . While ReLU does not require input normalisation, applying normalisation after ReLU in certain layers successfully reduces error rates. This architecture also used overlapping pooling, which reduces the error rates compared with non-overlapping pooling.
To combat memory limitation, the network is trained across two GPUs, where each GPU has half of the neurons. The GPU communicates with each other on a certain layer, e.g., layer 3 take input from all kernel from two GPUs in layer 2; layer 4 only take input from the kernel of layer 3 within the same GPU.
AlexNet uses two data augmentation strategies. The first strategy is done by extracting random 224 x 224-pixel patches from 256 x 256 images, along with their horizontal reflections. This augmentation is done with little computation and is not stored in the disk. The second strategy is by altering RGB intensities, i.e., performing PCA on the RGB pixel values.
The model was trained using SGD (stochastic gradient descent) with batch size 128, the momentum of 0.9, and weigh decay 0.0005. The learning rate was initialised at 0.01, divided by 10 if the validation rate was not improving.
VGGNet
VGGNet is a class of neural networks that employs a very deep network for image recognition (Simonyan and Zisserman, 2014). This architecture won ImageNet Challenge 2014 competition -first place in the localisation track and second place in the classification track.
VGGNet uses a 3 x 3 receptive field, 1-pixel convolution stride, and spatial padding such that spatial resolution after convolution is preserved. Max-pooling window size is 2 x 2 pixels, with stride 2. The total pooling layer for each architecture is five. Thus, not every convolution is followed by pooling. The fully connected layers consist of two 4096 channels and one 1000 channels, with a soft-max layer as the final layer. All layers use ReLu for non-linearity. This paper also uses LRN normalisation (Krizhevsky, Sutskever and Hinton, 2012) in one of the architectures, but it does not improve the performance.
GoogLeNet
GoogLeNet, an Inception-based deep convolutional neural network with 22 layers, achieved a new state of the art in the ImageNet Challenge 2014 .
A set of techniques were adopted to obtain higher performance:
• Ensembling 7 models independently with the same architecture, same initialisation, and same learning rate policies. The difference between models is in the sampling methodologies and random order of input image.
• Data augmentation involves resizing, cropping, and mirroring, which leads to 144 crops for each image.
• The softmax is the average over multiple crops and over all the individual classifiers.
The authors tested that max-pooling over crops and averaging over the individual classifiers lead to inferior performance.
Xception
Xception (Chollet, 2017), "Extreme Inception," is a neural network architecture that uses a "depthwise separable convolution." Xception has a slightly better performance on the ImageNet dataset.
ResNet
ResNet (
DenseNet
Dense Convolutional Network (Huang et al., 2017) is a type of neural network architecture where each layer takes all preceding feature maps as input. This is the benefit of DenseNet, collective knowledge of preceding feature maps.
If L denotes the number of layers, then Densenet has
( +1) 2
connections. In DenseNet, each layer applies a non-linear transformation (•), which is defined as three consecutive operations: Batch Normalisation, ReLU, and 3 x 3 convolution.
The network is divided into multiple densely connected dense blocks. The layer between the dense blocks is referred to as transition layers, which consists of batch normalisation, 1 x 1 convolution layer, and 2 x 2 average pooling layer.
Pre-trained Model and Transfer Learning
The term "transfer learning" can be traced to its earliest work by Stevo andAnte (1976, 2020).
In machine learning and deep learning, the distributions of training and testing data are assumed to be the same. Thus, if there is a discrepancy between training and testing data, the model may not work well, and the model needs to be rebuilt from scratch (Pan and Yang, 2009). For example, a model trained to discriminate astronomical images captured by an old telescope may not work well to predict astronomical images from the newer telescope as they will have a different quality.
Fortunately, transfer learning allows us to use training and testing data that have different distributions and different tasks and domains (Pan and Yang, 2009). The use of transfer learning allows us not to train the model from scratch.
In computer vision, transfer learning is done by using a pre-trained model.
Transfer Learning in Astronomy
Transfer learning has been used in astronomy as well. There are two common types of transfer learning implementation in astronomy: transfer learning from one survey to another or transfer learning from ImageNet. The second one is interesting considering the pre-trained model usually was trained on the ImageNet dataset, which contains natural images like animals, cars, etc. (2019) use transfer learning from different surveys (NVSS and FIRST) for radio galaxy classification. This method is also known as deep feature extraction.
Pre-trained Model as a Feature Extractor
Sign language classification
Modified pre-trained Alexnet and VGG16 used as feature extractor, followed by SVM classifier.
Colour Space
Light in the visible region of the electromagnetic spectrum that falls upon the human retina is what we know as colour (Poynton, 1997). The human retina has three types of colour photoreceptor cells. Thus, three numerical components are needed to describe a colour.
The first defined quantitative links between electromagnetic spectrums and perceived colour by humans are by the Comission International de L'Éclairage (Smith and Guild, 1931;CIE, 1932), which created two colour spaces: CIE 1931 RGB colour space and CIE 1931 XYZ colour space.
CIE 1931 colour space model defines three parameters denoted as "X", "Y", and "Z", where Y is the luminance component, and additional X and Z components.
Transformation to XYZ
The
Transformation to YUV
YUV is created from RGB, where Y is created from weighted values of R, G, and B as a measure of luminance. U and V are computed as differences between the calculated Y and B and R.
Derived from BT.470-6 (1998), the formula to convert RGB to YUV is as follows (equasys
Transformation to YCbCr
YCbCr, sometimes written as YCBCR. From full-scale 8-bit RGB, YCbCr can be computed
Transformation to YDbDr
YDbDr, sometimes written YDBDR, is a colour space used in the SECAM TV system, which is used in France and some Eastern European countries. To convert RGB into YDbDr can be done
Color Space Transformation in CNN
This is a summary of the literature review for the use of colour space transformation in CNN.
We only include research that involves transforming original colour space into other colour spaces, e.g., RGB to LAB.
Deep Learning Ensemble
Ensemble learning combines several models to reduce generalisation error. If the models make independent errors, the ensemble will perform significantly better than its individual models (Goodfellow, Bengio and Courville, 2016).
Many machine learning competitions are won by the use of ensemble models or ensemble learning. Several reasons behind the success of ensemble learning are (Dietterich, 2000):
• Statistical reason. Ensemble reduces the risk arising from a single model and can help us find a good approximation to f.
• Computational reason. A single model may get stuck in local optima. An ensemble may start from different starting points to avoid getting stuck in the same position.
• Representational reason. In machine learning, sometimes, the function f cannot be represented by any hypothesis in a given space. By using an ensemble, the approximated function may expand beyond the given space.
Unweighted model averaging
Majority Voting
This approach is similar to unweighted averaging. Majority voting works by counting the number of all predicted labels from each individual learner. The final prediction depends on which label has the most votes. The majority voting is less sensitive to the excessively biased learners. (Ju, Bibaut and van der Laan, 2018)
RESEARCH METHODOLOGY
Introduction
This chapter presents the methodology that will be used in this research. Section 3.2.1 talks about the dataset that will be used in this research. The section also talks about converting the original dataset into a format suitable for classification. The original images are in the RGB format and need to be converted into multiple colour spaces, which are discussed in section 3.2.2. To get a better generalisation, data augmentation will be discussed in section 3.2.3. The modelling approach will be discussed in section 3.2.4, followed by an evaluation in section 3.2.5. The proposed method will be discussed in section 3.3.
Research Methodology
This section discusses the methodology used in this research based on literature reviews in chapter 2. The methodology described is end-to-end, from dataset creation to ensemble models.
Dataset
The dataset used in this research was downloaded from Galaxy Zoo -The Galaxy Challenge. Thus, to make it suitable for the classification task, we should convert them into classes. The conversion can be as simple as thresholding. For example, if we want to create a binary classification task, i.e., to predict whether the galaxy is smooth or features/disk, we can use a 50% vote as the threshold. This number should be determined carefully because we should incorporate statistics uncertainty. We will follow an appropriate threshold (Willett et al., 2013) to select clean samples for this research. The final data will be represented by five classes, i.e., "edge-on", "spiral", "completely round smooth", "cigar-shaped smooth", and "in-between smooth". The classes created from the given threshold are called clean samples because they are wellsampled.
Data Transformation
After obtaining the clean samples, the RGB images will be converted to multiple colour spaces using an open-source image-processing Python library, OpenCV and Scikit-image. The colour spaces available are HED, HSV, LAB, RGBCI E, XYZ, YCbCr, YDbDr, YIQ, YPbPr, and YUV. When we predict an image, a network that is trained on a specific colour space will only predict that specific colour space.
Data Augmentation
Data augmentation is a common practice in deep learning. We use data augmentation to make the network more robust to novel images, hence preventing overfitting and increasing the final accuracy (O'Gara and McGuinness, 2019). Following the paper results, we will apply data augmentation into our training process.
Modelling
There are a lot of readily available CNN architectures. We will test our transformed images We will consider two training scenarios:
• Train the entire model from scratch (using only the architecture) as a benchmark
• Using a pre-trained model as a base, freeze the convolutional layers and train only the fully-connected layers
According to the results that we obtained from transformed images trained on several CNN architectures, we will create an ensemble (Figure 1) to yield a higher classification accuracy.
The number of networks and what colour space we should use for each network are variables for this research.
Evaluation
We assume no class is more important than the other for galaxy classification. Thus, the model will be evaluated using accuracy. Figure 1 shows the general workflow used in this research. In this section, multiple ensembles will be tested. Two types of ensembles will be tested. The first one is to ensemble the prediction made by each network by using simple averaging or a meta-learner. The second one is to extract feature vectors by dropping the fully-connected layers and combine them with feature vectors from other networks to be fed into a machine learning classifier.
= = + + + +
Ensemble Method
Summary
All combinations in this research can be summarised as follows:
Introduction
This chapter describes the technical implementation of this research. Section 4.2. describes the dataset creation process, including label creation that is suitable for classification problems.
Dataset Creation
This section describes the dataset creation that will be used for this research.
Image Resizing
The dataset used in this research was downloaded from Kaggle. The original image was in JPG format with the size of 424 x 424 x 3, where 424 x 424 is the width and height of the image, and 3 is three layers of RGB.
Label Creation for Classification
The original dataset was meant for regression problems. Thus, to make it suitable for this research, we should create a label that can be used for classification. The label creation rule and threshold are described in chapter 3.
Colour Space Conversion
The next step of the dataset creation is to convert the image to multiple colour spaces.
Fortunately, TensorFlow has built-in colour space conversion for the following colour spaces: LAB, XYZ, HSV, YUV, and YDbDr. The built-in colour space conversion is done by using tfio.experimental.color.rgb_to_lab(x).
For the rest of colour spaces (HLS, LUV, YCrCb, YIQ, HED), we need to manually do the conversion using either OpenCV or scikit-image and save it to disk for future usage.
Building Network
Building a network in deep learning is not a simple task. There are a lot of options, parameters, hyperparameters, and configurations to choose from. This further could also be affected by the type of data we use (natural images, medical images, etc.) and the type of task we would like to accomplish (classification, regression, segmentation, etc.). Hence, in this section, we test several commonly used values for every parameter and choose the one that produces the best result for our networks.
Load Pre-trained Model Base Model and Weights
For this research, we used out-of-the-box pretrained model from TensorFlow. Architecture used in this research are:
• Xception • VGG16, VGG19 • ResNet50, ResNet101, ResNet152 • DenseNet121, DenseNet169, DenseNet201
When loading the model, we set include_top = "False" to use the base model only. To load pretrained weights, we set parameter weights = "imagenet". All images fed into networks will be resized to 128 x 128 x 3. We also set the base model to be non-trainable. The only trainable part in this research is the top layers.
Stack Classification Layers
For top layers, which are the trainable part of the network, we should strive for a balance between the number of parameters and performance. We test four configurations of top layers:
Configuring for Performance
To improve the training speed i.e., reduces step time, we implement prefetching and set the number of elements to prefetch using AUTOTUNE. We also implement caching, which will reuse the data cached for the next epoch. This configuration are chained and implemented using DATASET.cache().prefetch(buffer_size=AUTOTUNE).
Data Augmentation
Using data augmentation in the training pipeline is a common practice because it consistently improves final results. We benchmark two configurations in detail as follows:
• Configuration 1: Without data augmentation Figure 5 shows the validation accuracy of a network with data augmentation consistently higher than a network with no augmentation. Figure 6 also shows that the validation loss of networks without data augmentation is increasing, while the validation loss of networks with data augmentation seems still decreasing. Thus, we will use data augmentation (random horizontal flip and random rotation 0.2) for our network.
Learning Rate
Learning rate is a hyperparameter that determines how much to change model weights in response to the calculated error at each iteration. A learning rate that is too high will make the model become unstable, while a learning rate that is too low will be slower to reach the optimum performance. We will test three learning rates:
• Learning Rate 0.01
• Learning Rate 0.001 (the default of TensorFlow)
• Learning Rate 0.0005 differ much from the other learning rate. In other words, the learning rate of 0.01 is sub-optimal for our network. This leaves us two learning rates to choose from: 0.001 and 0.0005. It seems the performance is comparable between the two of them. We will try to test them again with bigger epochs in the next subsection.
Epochs
The number of epochs is influenced by and influences the learning rate and another hyperparameter. We will test two learning rates (0.001 and 0.0005) with a longer epoch. Figure 9 show that the learning rate of 0.0005 is slightly better than the learning rate of 0.001. Thus, we will use a learning rate of 0.0005 for our networks.
Batch Size
Batch size is a hyperparameter that determines the number of data that will be passed through the network before updating the model. We will check three batch sizes: Table 7 shows that as the batch size increases, the training time decreases. We will choose a batch size of 64 for our networks.
RESULTS AND DISCUSSIONS
Introduction
This chapter will discuss the final results of our research. We will begin to describe the results of a single network in section 5.2, followed by the results of the ensemble in section 5.3.
Evaluation of Single Network
Following the implementation described in chapter 4, we test each combination of colour spaces and network architectures. Results indicated with * means that the colour is not further preprocessed by the built-in preprocess_input layer in TensorFlow.
The technical details are repeated here for clarity: Table 8 shows the highest accuracy obtained from LUV and XYZ colour spaces. For ResNet and VGG family, LUV colour space returns the highest accuracy without using the preprocessing layer from the architecture. For DenseNet and Xception family, XYZ colour space returns the highest accuracy without using the pre-processing layer from the architecture. Table 9 shows the top 3 highest overall accuracy and its configuration. From the result presented in Table 8, the accuracy of converted images (non-RGB) is always higher than the original RGB images. The highest accuracy of RGB images is achieved by ResNet50 with the accuracy of 0.8937, while the highest overall accuracy is achieved by DenseNet201 with XYZ colour spaces with the accuracy of 0.9113. For each architecture, the highest accuracy is obtained from the image that is not pre-processed using the preprocess_input layer. This makes sense because preprocess_input layer is usually made for RGB images. Thus, it is not suitable for converted images (non-RGB). For future studies, we recommend that researchers design an optimum preprocess_input layer for each colour space.
Evaluation of Ensemble Networks
Chapter 5.2 shows that the top validation accuracy for single network achieved by DenseNet201 with XYZ colour spaces. Thus, because of time and resource constraint, for this research we will focus on DenseNet family and XYZ colour spaces. We will also include Xception because it just happens to have the highest accuracy achieved by XYZ. In this section, we will test multiple ensembles configurations using DenseNet, Xception, XYZ, and RGB. The technical detail for ensemble networks is the same as the single network. Figure 17) and decreases the validation loss ( Figure 18) by a marginal amount. This is not significant considering we achieve this by doubling the number of networks.
Using a smaller and fewer DenseNet, the accuracy increases becomes more significant. For example, using DenseNet169 and DenseNet201, shows that using XYZ* colour spaces increases the accuracy from 0.8812 to 0.9258, which means a rise of 4.46 percent. Looking at Table 10, there is something very interesting. Using fewer networks and parameters ({DN169 + DN201} XYZ*), we can achieve a similar or better results than using more networks and parameters ({Xception + DN121 + DN169 + DN201} RGB). This is achieved just by applying a colour space transformation (RGB to XYZ* in this case).
Summary
As described in this chapter, our findings can be summarised into three important points:
• Transforming image colour space into other colour spaces during pre-processing yields different results. Some colour spaces yield worse results, some colour spaces yield better results.
• By transforming colour space, we can use fewer networks and parameters to achieve results with similar or better results.
• Ensembles networks yield even better results. Thus, combining colour space transformation and ensembles produces superior results.
CONCLUSION AND RECOMMENDATIONS
Introduction
This chapter is the final chapter of this thesis. Section 6.2 will discuss the result of this research.
New contributions are described in Section 6.3. Future recommendations will be discussed in Section 6.4.
Discussion and Conclusion
The practice of colour space transformation in the pre-processing process has been used for some problems in the past. In this research, we conduct an experiment on whether colour space transformation will be helpful for astronomical problems.
The experimental result using individual network (DenseNet, ResNet, VGG, and Xception) and colour space transformation consistently showed a higher validation accuracy compared to a network that uses original RGB images. Ensembles network using multiple networks and colour spaces increases further the validation accuracy.
Contribution to Knowledge
Some contributions to knowledge from this research are:
• A pre-trained model that is trained on natural images can also be used for astronomical problems, which is considered to be a non-natural image.
• Colour space transformation can increase the classification accuracy of astronomical images (galaxy classification).
• Ensembles of multiple colour spaces and networks can further increase classification accuracy on astronomical images (galaxy classification).
Future Recommendations
Some configurations, tests, and experiments are not being done in this research due to the time and resource constraints (i.e., the author had limited access to the GPU computation power).
Future works can include experimenting with more configurations and combinations to see if any interesting pattern emerges.
There are some ideas that could be tried in future works:
• Finding a more optimal hyperparameter for each architecture. In this research, we only find an optimal hyperparameter for one network and apply the same values to all networks. A different network may need a different hyperparameter to achieve its best performance.
• Using different layers in the network architecture. For example, we use MaxPool2D for the pooling layer in this research. However, we also can test whether AveragePooling2D could achieve better results.
Figure 1 : 19 Figure 2 : 22 Figure 3 : 24 Figure 4 : 25 Figure 5 : 26 Figure 6 : 28 Figure 7 : 28 Figure 8 : 29 Figure 9 : 29 Figure 10 : 30 Figure 11 : 30 Figure 12 : 31 Figure 13 : 31 Figure 14 : 32 Figure 15 : 32 Figure 16 : 35 Figure 17 : 37 Figure 18 :
1192223244255266287288299291030113012311331143215321635173718: Literature review summary on colour space transformation in CNN . 16 Table 4: Text representation of classification flowchart (Willett et al., 2013) . 19Table 5: Classification threshold (Willett et al., 2013) . 20 Table 6: Combination of variables tested in this research . 23 Table 7: Batch Size and Training Time. 32 Table 8: Single Network ClassificationAccuracy . 33 Table 9: Top 3 Highest Overall Accuracy for Single Network . 34 Table 10: Validation Accuracy of Tested Ensemble Networks . 35 iii LIST OF FIGURES Classification flowchart for Galaxy Zoo (Willett et al., 2013) . Proposed Architecture . Sample images . Cropped Images . Sample Images from Each Class . Validation Accuracy of Multiple Top Layer Configurations. Validation Loss of Multiple Top Layer Configurations . Validation Accuracy of Data Augmentation Configuration . Validation Loss of Data Augmentation Configuration . Validation Accuracy of Multiple Learning Rate Configurations . Validation Loss of Multiple Learning Rate Configurations . Validation Accuracy of Learning Rates (100 Epochs) . Validation Loss of Learning Rates (100 Epochs) . Validation Accuracy of Multiple Batch Size Configurations . Validation Loss of Multiple Batch Size Configurations . Validation Accuracy of DenseNet201 with Multiple Colour Spaces . Validation Accuracy of Ensembles Networks with Multiple Colour Spaces ......... Validation Loss of Ensembles of Four Networks with Multiple Colour Spaces .... 37 iv LIST OF ABBREVIATIONS
•
To compare the effect of different CNN architectures used to train images with different colour space • To analyse whether using a pre-trained model suitable for astronomical images (considering most of the pre-trained model was trained on daily object images) • To identify relevant research related to colour space transformations on CNN • To identify relevant research on the usage of CNN in astronomical images • To compare between the ensembling methods to improve classification accuracy • Develop a new methodology so that they can combine the benefit of multiple colour spaces and ensembling
He et al., 2016). The network's convolutional layers mostly have filters of 3 x 3. The number of filters is designed to preserve the time complexity per layer. Downsampling is done by convolutional layers of stride 2. The network ends with a global average pooling layer. Then we will insert "shortcut connections." The input is processed by the following methods (Krizhevsky, Sutskever and Hinton, 2012; Simonyan and Zisserman, 2014). To reduce the internal covariate shift, Batch Normalization (Ioffe and Szegedy, 2015) is applied before activation and right after each convolution. The dropout layers are not needed because of the regularisation provided by BN. The model was trained using SGD (stochastic gradient descent) with batch size 256, momentum of 0.9, and weight decay of 0.0001. The learning rate was initialised at 0.01, which was divided by 10 when plateaus. The models trained for up to 60 x 10 4 iterations.
A pre-trained model can be used as a features extractor as an image representation (Sharif Razavian et al., 2014). The authors use the first fully-connected layer as a feature vector of size 4096, which is further trained on linear SVM for a classification task.
To get the final H value, H' is multiplied by 60 o , = ′ × 60 .
2.4. 3 2.4. 4 . 5
345Transformation to HSVThe H component of HSV is the same as the H component of HSL. The S and V components can be expressed as: Transformation to CIELAB LAB colour space, also known as CIE 1976 L*a*b*, and CIELAB colour space. The conversion from RGB to LAB cannot be done straightforward, but we need to convert it into XYZ, followed by the conversion into LAB using the following equations(Schanda, 2007): Transformation to CIELUV CIELUV is also known as the CIE 1976 L*, u*, v* colour space. The L* of CIELUIV is the same as that of the CIELAB. The other coordinates are defined as follows (Schanda, 2007): * = 13 * ( ′ − ′ ) and * = 13 * ( ′ − ′ ) Following the CIE 1960 UCS Diagram recommendation, the equations for u' and v' The value of ′ and ′ can be taken from the white reference's table (Poynton, 2012). For example, in CIE III C, ′ = 0.2009, ′ = 0.4609; in CIE III D 65 , ′ = 0.1978, ′ = 0.4683.
The RGB value should be in the interval of [0,1].
to YIQ YIQ has been used in NTSC TV systems for years. To convert RGB into YIQ can be done using the following equations (Broesch, 2008; Shi and Sun, 2019): 4.10 Transformation to HED HED (Haematoxylin-Eosin-DAB) is a special purpose colour space used in the medical field to analyse tissues (Ruifrok and Johnston, 2001), with the conversion equations as follow:
This approach is the most used approach in the literature. To get the final prediction, the outputs of learners are averaged. The averaging is performed either on the outputs of learners or by using the softmax function (Ju, Bibaut and van der Laan, 2018; Ganaie and Hu, 2021). This simple averaging of six ResNet models with different depths is used as the winning solution for ILSVRC 2015 classification tasks (He et al., 2016). VGGNet (Simonyan and Zisserman, 2014) also uses this ensemble method to get a lower test error. GoogleNet trained 7versions of the same model, with different sampling methodologies and random input order, to create ensemble prediction. However, this naïve averaging is not dataadaptive: it works well for networks that have comparable performance, and are sensitive to the excessively biased learners (Ju, Bibaut and van der Laan, 2018).
••
The dataset contains 61578 images in JPG and RGB format. This Galaxy Challenge vote fraction is a modified version of The Galaxy Zoo 2 project (Willett et al., 2013). The Galaxy Zoo 2 (GZ2) has a total of 11 tasks and 37 possible responses. The original challenge is a regression task, in which participants should predict the vote percentage, and the lowest RMSE determined the winner. The probability values in each set of responses are the likelihood of the galaxy falling in that category. The sum of all possible responses in one category is 1.0. For example, a galaxy had 75% of all users identify as smooth (Class 1.1), 15% as features/disk (Class 1.2), and 10% as a star/artefact. The total probability of Class 1 is 75% + 15% + 10% = 100%. Class Class 1.3 = 0.10
Figure 1 :
1Classification flowchart for Galaxy Zoo(Willett et al., 2013)
with several architectures, including AlexNet (Krizhevsky, Sutskever and Hinton, 2012), VGG (Simonyan and Zisserman, 2014), Inception (Szegedy et al., 2015), ResNet (He et al., 2016), DenseNet (Huang et al., 2017). DenseNet based ensembles are used in ColorNet (Gowda and Yuan, 2019).
Figure 2 :
2Proposed Architecture
Figure 3 :
3Sample imagesLooking atFigure 3, it is clear that the most important part of the image is in the centre. The outer part of the image is either not helpful for galaxy classification or contains unrelated information that may be messing up the model and prediction results. Cropping images also help to speed up the training process, even though it is not the primary motivation here.
Figure 4 :
4Cropped ImagesWe crop the outermost 100 pixels of the image. This left us with an image size of 224
Figure 5 :
5Sample Images from Each ClassThe final labels consist of:• completelysmooth: 8436 images • inbetweensmooth: 8069 images • cigarsmooth: 579 images • edgeon: 3903 images • spiral: 7806 images
••
Configuration Configuration C: Dense(512) + Dense(128) • Configuration D: Dense(512) + Dense(128) with Dropout(0.2)
Figure 6 :Figure 7 :
67Validation Accuracy of Multiple Top Layer Configurations Validation Loss of Multiple Top Layer ConfigurationsFigure 4 shows that the validation loss of layers with Dropout (configuration B and D) are more stable than layers without Dropout (configuration A and C). While the validation loss of configurations B and D are similar, Figure 3 shows that configuration D has the highest accuracy. Thus, we will choose configuration D -Dense(512) + Dense(128) with Dropout(0.2) for our top classification layers.
•Figure 8 :Figure 9 :
89Configuration Validation Accuracy of Data Augmentation Configuration Validation Loss of Data Augmentation Configuration
Figure 10 :Figure 11 :
1011Validation Accuracy of Multiple Learning Rate Configurations Validation Loss of Multiple Learning Rate ConfigurationsFigure 7 and Figure 8 show that the validation accuracy and validation loss of learning rate 0.01
Figure 12 :Figure 13 :
1213Validation Accuracy of Learning Rates (100 Epochs) Validation Loss of Learning Rates (100 Epochs) Figure 8 and
• 32 Figure 14 :Figure 15 :
321415Batch size of 16 • Batch size of 32 • Batch size of 64 Validation Accuracy of Multiple Batch Size Configurations Validation Loss of Multiple Batch Size Configurations Figure 11 and Figure 12 show that the performance of different batch size configurations is comparable. However, the training times are very different.
Figure 16 :
16Validation Accuracy of DenseNet201 with Multiple Colour SpacesFigure 16shows the multiple colour spaces trained on DenseNet201. In some cases, colour space conversion increases the classification accuracy. However, some colour spaces decrease the classification accuracy, which may be caused by information loss caused by the colour space conversion.
Figure 17 :Figure 18 :
1718Validation Accuracy of Ensembles of Four Networks with Multiple Colour Spaces Validation Loss of Ensembles of Four Networks with Multiple Colour SpacesFigure 17 shows the results of combining four networks (DenseNet121, DenseNet169, DenseNet201, and Xception) using three configurations: RGB only, XYZ* only, and a combination of XYZ and RGB (a total of eight networks). The figure shows that XYZ* colour spaces produce better results than the RGB images; it increases the accuracy from 0.9007 to 0.9297, which means a rise of 2.9 percent. Combining XYZ and RGB only increases the accuracy (
Table 1 :
1Literature review summary on transfer learning in astronomy . 11
Common astronomical images found in our daily life are the colourised RGB version. Most original datasets of astronomical images do not correspond to the wavelength range that is sensitive to the human eye (Rector et al., 2005). The way of astronomy image creation could affect how the image is interpreted. Hence, computer science researchers need to develop techniques that can address the nature of astronomical images, both in terms of their volumes and their uniqueness.
Shortly after the success of the ImageNet competition, ConvNet also began to be widely used in astronomical images. The availability of datasets collected from the Galaxy Zoo project makes it a perfect situation to boost the usage of ConvNets in astronomy. In December 2013, Winton Capital and Galaxy Zoo, together with the Kaggle team, created a Galaxy Challenge,where participants were asked to create a model to classify galaxies into categories. Since then, researchers worldwide have often used the Kaggle Galaxy Zoo dataset.CNN can also be combined with other networks for a particular task. For example, the combination of CNN and LSTM can be used to classify transient radio frequency inference(Czech, Mishra and Inggs, 2018). A digital image can be represented in different colour spaces (RGB, HSV, LAB, etc.). However, the most common colour space used in deep learning is RGB. The galaxy images from Kaggle Galaxy Challenge are also in RGB format. We are interested in how different colours can affect the model's performance. This research will consider all colour spaces and network architecture combinations on astronomical images. We will also compare the performance if we train the model from scratch and use the pre-trained model. Kaggle Galaxy Challenge asked participants to create a model to classify galaxies. Dieleman, Willett and Dambre (2015) developed a deep neural network model which exploits rotational and translational symmetry. Dieleman's winning solution in the Kaggle competition required simple ensembling by averaging over 17 network variants and 60 transformations. The network variants differ in their number of dense layers, filter size configuration, activation function on the dense layer, and the number of filters. Dieleman's winning solution incorporates the ensembling of networks. Ensembling is often used to increase the accuracy of the results. However, ensemble in deep learning is still not well studied. Most of the deep learning literature focuses on the design of the network, and most of them only applies naïve ensembling (Ju, Bibaut and van der Laan, 2018). The authors further test multiple ensembling methods on the CIFAR-10 dataset. Super Learner (Van der Laan, Polley and Hubbard, 2007) yield the highest accuracy over other ensembling methods. SuperLearner finds the best weights adaptively without human intervention, which means we can include all the weak learners in the library.The success of ConvNets in the ImageNet competition (2012) brought a revolution in computer
vision; since then, ConvNets was becoming a dominant approach for almost all image-related
tasks (LeCun, Bengio and Hinton, 2015). The architecture that won the ImageNet competition
(Krizhevsky, Sutskever and Hinton, 2012) is now known as AlexNet.
There is a lot of possible ConvNet application in astronomy. For classification alone, current
usage of CNN are galaxy classification (Dieleman, Willett and Dambre, 2015), solar radio
spectrum classification (Chen et al., 2017), sunspot group classification (Tang et al., 2021),
variable stars classification (Szklenár et al., 2020), pulsar candidate classification (Wang et al.,
2019).
1.2 Problem Statement and Related Research
Ensembling multiple networks has increased accuracy in numerous image-related tasks,
including astronomical images. However, none of them studied in-depth ensembling multiple
networks trained in different colour spaces.
ColorNet (Gowda and Yuan, 2019) shows that transforming RGB colour into different colour
spaces can significantly affect classification accuracy. Using a simple convolutional network
on the CIFAR-10 dataset on multiple colour spaces (RGB, HSV, LAB, etc) shows that LAB
colour spaces yield the highest accuracy. The experiment also shows that each class has
different accuracy for each colour space, which means there is no perfect correlation between
colour spaces. Finally, ColorNet proposed an ensemble of DenseNet based models with seven
colour spaces to obtain high classification accuracy.
In other fields, some works exist that tried to do colour space transformations before doing
further analysis. For example, in medical images, conversion to CIE Lab coluor space for
dysplastic nuclei segmentation achieves the best-averaged accuracy (dos Santos et al., 2020).
CIE Lab also yields the highest classification accuracy in histological image classification
(Velastegui and Pedersen, 2021). The authors also further show that despite CIE Lab yielding
the highest overall accuracy, some classes are more accurate when they are represented in other
colour spaces. FusionNet (Guo et al., 2020) used YIQ colour space for multi-modal medical
image fusion. In the field of steganography, StegColNet (Gowda and Yuan, 2021) shows that
the ensemble of the colour space model outperforms the recent state-of-the-art approach.
).LeNet, with a test error of 0.7%. It shows that the Optimal Margin Classifier, now known famous as Support Vector Machine, worked better or the same compared to the neural networks.A decision-tree-based classifier called Random Decision Forests(Ho, 1995) was developed, and the validity is demonstrated by experiments on recognising handwritten digits. RandomIn the 1990s, the enthusiasm and optimism on AI are at a low point. Thus, this period is often
referred to as AI Winter, when the funding and interest in AI research were reduced (AI
Newsletter, 2005). This situation was confirmed by work from LeCun et al, which compares
learning algorithms for recognising handwritten digits (LeCun et al., 1995). The paper
compares classification algorithms developed at Bell Laboratories and elsewhere, i.e., Linear
Classifier (Baseline), Nearest Neighbor Classifier (Baseline), Pairwise Linear Classifier, PCA
and Polynomial Classifier, RBF Network, Large Fully Connected Multi-Layer Neural Network,
LeNet 1, LeNet 4, LeNet 5, Boosted LeNet 4, Tangent Distance Classifier and Optimal Margin
Classifier. Using the Optimal Margin Classifier (Boser, Guyon and Vapnik, 1992), a test error
of 1.1% was reached. LeNet 4 has a test error of 1.1%, while the best algorithm is Boosted
Table 1 :
1Literature review summary on transfer learning in astronomy Transfer learning from different surveys (DES and HSC-SSP) (Farrens et al., 2022) Blended sources identification VGGTable 1 shows literature that uses transfer learning in the astronomy field. Except noted, all pretrained models are using the ImageNet dataset. For example, Tang, Scaife, and LeahyAuthors and Year
Objective/Purpose
Architecture/Methods Used
(George, Shen and
Huerta, 2018)
Glitch classification and
clustering of gravitational
waves
Inception, ResNet, VGG. The
trained CNN also used as a
feature extractor for clustering
(Ackermann et al.,
2018)
Galaxy merger detection
Xception
(Tang, Scaife and
Leahy, 2019)
Radio galaxy classification
Transfer learning from different
surveys (NVSS and FIRST)
(A. Khan et al., 2019) Galaxy classification
Xception. The model also used
as a feature extractor for
clustering
(Yang et al., 2020)
Lunar impact crater
identification and age
estimation
ResNet101 as a feature extractor
(Awang Iskandar et
al., 2020)
Planetary nebulae classification InceptionResNetV2,
DenseNet201, MobileNetV2
(Wei et al., 2020)
Star cluster classification
VGG19-BN, ResNet18
(Tanoglidis,
Ćiprijanović and
Drlica-Wagner, 2021)
Separating low surface
brightness galaxies from
artifacts
Table 2 :
2Literature review summary on feature extraction using a pre-trained modelAuthors and Year
Objective/Purpose
Architecture/Methods Used
(Chaib et al., 2017)
Remote sensing image
classification
CaffeNet + VGG-VD16. Features are
extracted from the first FC layer,
which then combined using either of
two options: addition or concatenation.
Table 3 :
3Literature review summary on colour space transformation in CNNAuthor
Colour Space
Task Type
Field/Dataset
(Rachmadi and
Purnama, 2015)
RGB, LAB, XYZ, HSV
Classification Vehicle images
(Kim and Ro, 2016) RGB, LAB, YCbCr, HSV, YIQ,
XYZ, RQCr, RIQ, YQCr
Face
Recognition
Multi-PIE
(Atha and
Jahanshahi, 2017)
RGB, YCbCr
Detection
Corrosion images
(Jafarbiglo, Danyali
and Helfroush,
2018)
LAB
Classification MITOS-
ATYPIA-14
(Gowda and Yuan,
2019)
RGB, HSV, LAB, YUV, YCbCr,
YpBPr, YIQ, XYZ, HED, LCH,
CMYK
Classification CIFAR-10,
CIFAR-100,
CVHN, ImageNet
(M. A. Khan et al.,
2019)
LAB
Classification Weizmann, KTH,
UIUC, Muhavi,
WVU
(Castro et al., 2019) RGB, HSV, LAB
Classification Fruit dataset
(Li et al., 2020)
RGB, HSV
Segmentation Medical
(Mohammadi
Lalabadi, Sadeghi
and Mireei, 2020)
RGB, HSV, LAB
Classification Fish dataset
(Gowda and Yuan,
2021)
RGB, HSV, LAB, YUV, YCbCr,
YpBPr, YIQ, XYZ, HED, LCH,
CMYK
Classification Bossbane,
BOWS2
Using LAB colour space for face recognition tasks results in higher accuracy than using RGB
(Kim and Ro, 2016), 79.77% and 71.50%, respectively. The authors also propose "collaborative
feature learning", a framework to aggregates features from multiple colour spaces, which
further improves accuracy to 90.81%.
However, colour space transformation does not always bring a better result. For example, a
publication on vehicle colour recognition (Rachmadi and Purnama, 2015) achieves slightly
higher accuracy using RGB colour space compared to XYZ, LAB, and HSV -0.9447, 0.9432,
0.9414, and 0.9372, respectively. The authors didn't address the stochastic factor of neural
network -whether the accuracy differences were just a random chance, nor evaluate the use of
different architecture on different colour spaces.
Table 4 :
4Text representation of classification flowchart(Willett et al., 2013) Task Question
Responses
Next
01
Is the galaxy simply smooth and
rounded, with no sign of disk?
smooth
features or disk
star or artefact
07
02
end
02
Could this be a disk viewed edge-
on?
yes
no
09
03
03
Is there a sign of a bar feature
through the centre of the galaxy?
yes
no
04
04
04
Is there any sign of a spiral arm
pattern?
yes
no
10
05
Table 5 :
5Classification threshold(Willett et al., 2013) Class Name
Tasks
Threshold
Completely round smooth
T01
T07
ℎ ≥ 0.469
≥ 0.50
In-between smooth
T01
T07
ℎ ≥ 0.469
−
≥ 0.50
Cigar-shaped smooth
T01
T07
ℎ ≥ 0.469
− ℎ
≥ 0.50
Edge-on
T01
/
≥ 0.430
Table 6 :
6Combination of variables tested in this researchVariable Tested
Variants
Architecture
ResNet, VGG, DenseNet, Xception
Colorspaces
RGB, HED, HSV, LAB, XYZ, YCbCr, YDbDr, YIQ,
YPbPr, YUV
Pre-trained and
transfer learning
Table 7 :
7Batch Size and Training TimeBatch Size Training Time16
2678 seconds
32
1691 seconds
64
1310 seconds
• Optimiser :
OptimiserAdam • Top layers: Dense(512) + Dense(128) with Dropout(0.2) • Data augmentation: random horizontal flip, random rotation of 0.2 • Learning rate: 0.0005 • Batch size: 64, epochs: 100 • Caching and prefetching enabled.
Table 8 :
8Single Network Classification AccuracyRN50
RN10
1
RN15
2
VGG1
6
VGG1
9
DN12
1
DN16
9
DN20
1
Xceptio
n
RGB
0.893
7
0.8836 0.8850 0.8451 0.8295 0.8407 0.8494 0.8651
0.8456
HLS*
0.813
7
0.8270 0.8187 0.7779 0.7607 0.6705 0.6448 0.6506
0.6598
HLS
0.827
7
0.8282 0.8282 0.7911 0.7864 0.7694 0.7930 0.7963
0.7576
HSV*
0.575
9
0.5478 0.5617 0.7701 0.7560 0.7053 0.7332 0.7338
0.7209
HSV
0.591
4
0.5653 0.5841 0.4297 0.4189 0.6030 0.6768 0.6923
0.5519
LAB*
0.840
4
0.8435 0.8378 0.8321 0.8051 0.8098 0.8003 0.8244
0.8057
LAB
0.863
7
0.8664 0.8498 0.8272 0.8154 0.8227 0.8288 0.8362
0.8532
LUV*
0.901
5
0.9034 0.8918 0.8687 0.8499 0.7893 0.8090 0.8301
0.7061
Table 9 :
9Top 3 Highest Overall Accuracy for Single NetworkConfiguration
Accuracy
DenseNet201 + XYZ*
0.9113
DenseNet169 + XYZ*
0.9074
ResNet101 + LUV*
0.9015
Table 10 :
10Validation Accuracy of Tested Ensemble NetworksTable 10shows the validation accuracy of multiple ensemble configurations. It is clear that combining two or more networks increases the accuracy. For example, the highest accuracy by a single model was achieved by DenseNet201 + XYZ* with an accuracy of 0.9113. Combining with an additional DenseNet201 network trained on RGB, the final accuracy increased to 0.9172. However, if we combine DenseNet201 + XYZ* with DenseNet169 + XYZ*, the final accuracy increased to 0.9258. Also, using RGB images only failed to surpass the XYZ* performances.Configuration
Accuracy
* = 500 [ ( ) − ( )]
* = 200 [ ( ) − ( )]
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| {'fraction_non_alphanumeric': 0.06315482100099282, 'fraction_numerical': 0.04591485136950301, 'mean_word_length': 4.299331517702401, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 9, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Big data is now a norm in astronomy. The growth of astronomical images makes it a very suitable domain for computer science research. It is common for astronomer uses the morphological of galaxies to classify galaxies into categories. This practice was first applied systematically by Hubble (1936). When the data is small in size, the classification process could be easily done by small teams or individuals. However, the exponential growth of data collected by modern telescopes made it impossible to rely on an expert to classify every single galaxy image.In December 2013, Winton Capital and Galaxy Zoo, together with the Kaggle team, created a Galaxy Challenge, where participants were asked to create a model to classify galaxies into categories. Since then, researchers worldwide have often used the Kaggle Galaxy Zoo dataset.This research will focus on investigating how colour space transformation could affect classification accuracy, and we will investigate whether the CNN architecture will affect this too. For this research, we will consider multiple colour spaces (RGB, XYZ, LAB, etc) and multiple CNN architectures (VGG, ResNet, DenseNet, Xception, etc). We will use a pre-trained model and weights. However, most of the pre-trained model was trained on a natural RGB image, so we will investigate the performance in predicting the transformed (non-RGB) nonnatural image (astronomical images).We test our hypothesis by first testing individual networks using RGB and transformed colour spaces. We also test multiple ensemble configurations of different networks and colour spaces.We did a minimal hyperparameter search to ensure we obtained an optimum result. Our experimental results show that using transformed colour spaces on individual networks produced a higher validation accuracy. Ensembles of network and colour spaces further increase the validation accuracy.Finally, this research aims to validate the usefulness of colour space transformation for astronomical images. This research will also become a benchmark that is useful for future research.ii', 'arxivid': '2305.00002', 'author': ['Yevonnael Andrew '], 'authoraffiliation': [], 'corpusid': 258426419, 'doi': '10.48550/arxiv.2305.00002', 'github_urls': [], 'n_tokens_mistral': 25372, 'n_tokens_neox': 20924, 'n_words': 12137, 'pdfsha': 'e0d3dc0652e0a2ef4cef31f9ea90a630d8cf8836', 'pdfurls': ['https://export.arxiv.org/pdf/2305.00002v1.pdf'], 'title': ['GALAXY CLASSIFICATION USING TRANSFER LEARNING AND ENSEMBLE OF CNNS WITH MULTIPLE COLOUR SPACES', 'GALAXY CLASSIFICATION USING TRANSFER LEARNING AND ENSEMBLE OF CNNS WITH MULTIPLE COLOUR SPACES'], 'venue': []} |
arxiv |
Quasi-Bayesian Nonparametric Density Estimation via Autoregressive Predictive Updates
Sahra Ghalebikesabi
University of Oxford
Chris Holmes
Novo Nordisk
Edwin Fong
Novo Nordisk
Brieuc Lehmann
University College London
Quasi-Bayesian Nonparametric Density Estimation via Autoregressive Predictive Updates
Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predictive density estimation without the need for posterior approximation. Although these estimators are computationally appealing, they struggle on non-smooth data distributions. This is due to the comparatively restrictive form of the likelihood models from which the proposed copula updates were derived. To address this shortcoming, we consider a Bayesian nonparametric model with an autoregressive likelihood decomposition and a Gaussian process prior. While the predictive update of such a model is typically intractable, we derive a quasi-Bayesian update that achieves state-ofthe-art results in small-data regimes.
INTRODUCTION
Modelling the joint distribution of multivariate random variables with density estimators is a central topic in modern unsupervised machine learning research [Durkan et al., 2019, Papamakarios et al., 2017. As well as providing insight into the statistical properties * equal contribution of the data, density estimates are used in a number of downstream applications, including image restoration [Zoran and Weiss, 2011], density-based clustering [Scaldelai et al., 2022], and simulation-based inference [Lueckmann et al., 2021]. In small-data regimes, Bayesian methods are a popular choice for a wide range of machine learning tasks, including density estimation, thanks to their attractive generalization capacities. For density estimation, the typical Bayesian approach is to target the Bayesian predictive density, p n (x) = f (x|θ)π n (θ)dθ, where π n denotes the posterior density of the model parameters θ after observing x 1 , . . . , x n , and f denotes the likelihood function.
De Finetti's representation theorem [De Finetti, 1937, Hewitt andSavage, 1955] states that an exchangeable joint density fully characterises a Bayesian model, which then implies a sequence of predictive densities. Further, Fong et al. [2021] recently showed that a sequence of predictive densities can be sufficient for full Bayesian posterior inference. This provides theoretical motivation for an iterative approach to Bayesian predictive density estimation by updating the predictive p i−1 (x) to p i (x) given observation x i for i = 1, . . . , n. The idea of recursive Bayesian updates goes back to at least Hill [1968], but was only recently made more widely applicable through the relaxation of the assumption of exchangeability in favour of conditionally identically distributed [Berti et al., 2004] sequences.
Here, we focus on a particular class of one-step-ahead predictive updates p i−1 (x) → p i (x) based on bivariate copulas, which were first introduced by Hahn et al. [2018] for univariate data, and extended by Fong et al. [2021] to the multivariate setting and to regression analyses. This class of updates is inspired by Bayesian models and thus retains many desirable Bayesian properties, such as coherence and regularization. However, we emphasize that the copula updates do not correspond exactly, nor approximately, to a traditional Bayesian likelihood-prior model, and we thus refer to them Figure 1: Density estimates of 600 observations from a chessboard distribution, reported with mean and standard deviation of test log likelihoods. For larger training sizes, see Supplement C.2. Our methods, AR-BP and ARnet-BP, outperform R-BP and AR neural networks.
as quasi-Bayesian [Fortini and Petrone, 2020]. The most related Bayesian density estimator proposed to date, henceforth referred to as the Recursive Bayesian Predictive (R d -BP), lacks flexibility to model highly complex data distributions (see Figure 1). This is because the existing copula updates rely on a Gaussian copula with a single scalar bandwidth parameter, corresponding to a Bayesian model with a likelihood that factorizes over dimensions. In contrast, popular neural network based approaches, such as masked autoregressive flows (MAFs) [Papamakarios et al., 2017], and rationalquadratic neural spline flows (RQ-NSFs) [Durkan et al., 2019] can struggle in small-data regimes (see Figure 1).
Contributions This motivates our main contribution, namely the formulation of a more flexible autoregressive (AR) copula update based on which we propose a new Dirichlet Process Mixture Model (DPMM) inspired density estimator. In particular:
• By considering a DPMM with an AR likelihood and a Gaussian process (GP) prior, we formulate a tractable copula update with a novel datadependent bandwidth based on the Euclidean metric in data space. Our method, Autoregressive Recursive Bayesian Predictives (AR-BP), outperforms traditional density estimators on tabular data with up to 63 features, and 10,000 samples.
• We observe in practice that the Euclidean metric used in AR-BP can be inadequate for highly non-smooth data distributions. For such cases, we propose using an AR neural network [Bengio and Bengio, 1999, Frey et al., 1998, Germain et al., 2015, Larochelle and Murray, 2011] that maps the observations into a latent space before bandwidth estimation. This introduces additional nonlinearity through the dependence of the bandwidth on the data, leading to a density estimator, ARnet-BP, that is more accurate on non-smooth densities.
BACKGROUND
We briefly recap predictive density estimation via bivariate copula updates, before describing a particular such update inspired by DPMMs.
UNIVARIATE PREDICTIVE DENSITY UPDATES
To compute predictive densities quickly, Hahn et al. [2018] propose an iterative approach. For x ∈ R, any sequence of Bayesian posterior predictive densities p i (x) with likelihood f and posterior π i , conditional on x 1:i , can be expressed as
p i (x) = f (x|θ)π i (θ)dθ = p i−1 (x)h i (x, x i ),(1)
for some bivariate function h i (x, x i ) [Hahn et al., 2018].
Rearranging for h i , we have
h i (x, x i )= p i (x) p i−1 (x) (a) = p i−1 (x|x i ) p i−1 (x) (b) = p i−1 (x, x i ) p i−1 (x)p i−1 (x i )(2)
where (a) holds by definition, and (b) p i−1 (x, x i ) = p i−1 (x|x i )p i−1 (x i ) = p i (x)p i−1 (x i ) holds by Bayes' law. Hahn et al. [2018] show that h i (x, x i ) is the transformation of a bivariate copula density. A bivariate copula is a bivariate cumulative distribution function (CDF) C : [0, 1] 2 → [0, 1] with uniform marginal distributions that is used to characterise the dependence between two random variables independent of their marginals: For each Bayesian model, there is thus a unique sequence of symmetric copula densities c i (u, v) = c i (v, u). This sequence has the property that c n (·, ·) → 1 converges to a constant function as n → ∞, ensuring that the predictive density converges asymptotically with sample size n.
In general, the above equation is intractable due to the posterior so it is not possible to compute the iterative update in (1) for fully Bayesian models. Alternatively, we will consider sequences of h i that match the Bayesian model for i = 1, but not for i > 1. As mentioned above, this copula update no longer corresponds to a Bayesian model, nor are the resulting predictive density estimates approximations to a Bayesian model. Nevertheless, if the copula updates are conditionally identically distributed, they still exhibit desirable Bayesian characteristics such as coherence and regularization, and are hence referred to as quasi-Bayes. Please refer to Berti et al. [2004] for details.
MULTIVARIATE PREDICTIVE DENSITY UPDATES
The above arguments cannot directly be extended to multivariate x ∈ R d since h i cannot necessarily be written as
c i {P i−1 (x), P i−1 (x i )} for d > 1. However,(2)
still holds, and recursive predictive updates with bivariate copulas as building blocks can be derived explicitly given a pre-defined likelihood model and a prior, which we now exhibit.
Hahn et al. [2018] andFong et al. [2021] propose to use DPMMs as a general-use nonparametric model. The DPMM [Escobar, 1988, Escobar andWest, 1995] can be written as
f (x|G) = Θ K(x|θ) dG(θ), with G ∼ DP(c, G 0 ) (3)
where θ ∈ Θ = R d are parameter vectors, the prior assigned to G is a Dirichlet process (DP) prior with base measure G 0 and concentration parameter c > 0 [Ferguson, 1973], and K(x|θ) is a user-specified kernel (not to be confused with the covariance function of a GP). In particular, Fong et al. [2021] consider the base measure G 0 = N (0, τ −1 I d ) for some precision parameter τ ∈ R >0 , and the factorized kernel
K(x|θ) = N (x|θ, I d ) where I d is the d-dimensional identity matrix. The like- lihood is then f (x|G) = d j=1 N x j | θ j , 1 dG(θ),(4)
where the dimensions of x are conditionally independent given θ. Following Hahn et al. [2018], we denote the dimension j of a vector y with y j . We note that the strong assumption of a factorised kernel form drastically impacts the performance of the regular DPMM and also influences the form and modelling capacity of the corresponding copula update.
This model inspires the following recursive predictive density update
p i (x) = h i (x, x i )p i−1 (x) for which the first d ∈ {1
, . . . , d} marginals take on the form
p i (x 1:d ) p i−1 (x 1:d ) =1−α i +α i d j=1 c u j i−1 (x j ), v j i−1 ; ρ 0 , (5) u j i−1 (x j ) :=P i−1 x j | x 1:j−1 , v j i−1 :=P i−1 x j i | x 1:j−1 i ,
where c(u, v; ρ 0 ) is the bivariate Gaussian copula density with correlation ρ 0 = 1/(1 + τ ), p 0 can be any chosen prior density, and α i = 2 − 1 i 1 i+1 (see Supplement A and Fong et al. [2021]). Note that the above update requires a specific ordering of the feature dimensions, and the Gaussian copula follows from the Gaussian distribution in the kernel and G 0 for the DPMM. Unlike the DPMM, there are now no underlying parameters (beyond ρ 0 ) in the copula update as we have integrated out θ, so we do not carry out clustering directly. While ρ 0 is a scalar here, Fong et al. [2021] also consider the setting with a distinct bandwidth parameter for each dimension. We refer to these recursive Bayesian predictives as R d -BP, or simply R-BP if the dimensions share a single bandwidth.
AR-BP: AUTOREGRESSIVE BAYESIAN PREDICTIVES
For smooth data distributions, the recursive update defined in (5) generates density estimates that are highly competitive against other popular density estimation procedures such as kernel density estimation (KDE) and DPMM [Fong et al., 2021]. Moreover, the iterative updates provide a fast estimation alternative to fitting the full DPMM through Markov chain Monte Carlo (MCMC). When considering more structured data, however, performance suffers due to the choices of the factorized kernel K(·|θ) = N (·|θ, I d ) and simple base measure G 0 = N (0, τ −1 I d ) in the DPMM. These choices induce a priori independence between the data dimensions, and are thus insufficiently flexible to capture more complex dependencies.
BAYESIAN MODEL FORMULATION
We therefore propose employing more general kernels and base measures in the DPMM and show that these inspire a more general tractable recursive predictive update. In particular, we allow the kernel to take on an autoregressive structure
K(x|θ) = d j=1 N x j | θ j x 1:j−1 , 1 ,(6)
where θ j : R j−1 → R is now an unknown mean function, and not scalar, for dimension x j , which we allow to depend on the previous j − 1 dimensions of x. Thus, specifying our DPMM requires a base measure supported on the function space in which (θ 1 , . . . , θ d ) is valued. We specify this base measure as a product of independent GP priors on the functional parameters
θ j ∼ GP(0, τ −1 k j ) for j = 1, ..., d(7)
where k j : R j−1 × R j−1 → R and k j can be any given covariance function that takes as input a pair of x 1:j−1 values. In practice, we use the same functional form of k for each j, so we will drop the superscript j. For later convenience, we have also written the scaling term τ −1 explicitly. We highlight that for j = 1, θ 1 ∼ N (0, τ −1 ). Under this choice, the mean of the normal kernels in the DPMM for each dimension j is thus a flexible function of the first j − 1 dimensions x 1:j−1 , on which we elicit independent GP priors. The conjugacy of the GP with the Gaussian DPMM kernel in (6) is crucial for deriving a tractable density update.
Remark. The proposed DPMM kernel in (6) is in fact more flexible than a general multivariate kernel, K(x | θ) = N (x | θ, Σ). This is because the multivariate kernel also implies an AR form like (6) but where the parameters θ j are restricted to be linear in x 1:j−1 ; see Wade et al. [2014] for details.
ITERATIVE PREDICTIVE DENSITY UPDATES
Computing the Bayesian posterior predictive density induced by the DPMM with kernel given by (6) and base measure given by (7) through posterior estimation is intractable and requires MCMC. However, as before, we can utilize the model to derive tractable iterative copula updates. In Supplement A.1, we derive the corresponding recursive predictive density update
p i (x) = h i (x, x i )p i−1 (x)
for the first d marginals and show that it takes on the form
p i (x 1:d ) p i−1 (x 1:d ) =1 − α i + α i · (8) d j=1 c u j i−1 (x j ), v j i−1 ; ρ j (x 1:j−1 , x 1:j−1 i ) , with u j i−1 (x j ), v j i−1 defined as in (5), α i = 2 − 1 i 1 i+1
, and the bandwidth given by
ρ j (x 1:j−1 , x 1:j−1 i ) = ρ 0 k x 1:j−1 , x 1:j−1 i ,(9)(a) Train: Estimate v j i = Pi−1(x 1:j i ) for each i Initialise u j 0 (xi) ← Φ(x j i ) For each preceding observation x k with k < i:
For each feature j:
Compute data-dependent bandwidth ρ j (x 1:j i , x 1:j k ) (9) Update conditional CDF u j i (x k ): = P j i (x k ) based on the similarity between u j i−1 (x k ) and v j i−1 (18) Set v j i ← u j i (xi) for all j (b) Test: Estimate predictive at test point pn(z) Initialise u j 0 (z) ← Φ(z j ) For each train observation xi:
For each feature j:
Compute data-dependent bandwidth ρ j (x 1:j i , z 1:j ) (9)
Update conditional CDF u j i (z): = P j i (z) based on the similarity between u j i−1 (z) and v j i−1 (18)
Update predictive density pi−1(z) → pi(z) (8) Figure 2: Simplified summary of AR-BP. We repeat the training update for each train datum x i to estimate v j i = P i−1 (x 1:j i ). These are needed at test time to update from p i−1 (z) → p i (z). All steps are averaged over different feature and sample permutations. The main step that induces autoregression in the observations is highlighted pink. Please see Supplement B.3 for detailed algorithms.
for ρ 0 = 1/(1 + τ ), and ρ 1 i = ρ 0 . Where appropriate, we henceforth drop the argument x for brevity. The conditional CDFs u j i−1 can also be computed through an iterative closed form expression similarly to (8) (Supplement B.3). Please see Figure 2 for a simplified overview of the density estimation pipeline.
Note that the estimation is identical to the update given in (5) induced by the factorized DPMM kernel, except for the main difference that the bandwidth ρ is no longer a constant, but is now data-dependent. More precisely, the bandwidth for dimension j is a transformation of the GP covariance function k on the first j − 1 dimensions. The additional flexibility afforded by the inclusion of k enables us to capture more complex dependency structures, as we do not enforce a-priori independence between the dimensions of the parameter θ. Similarly to the extension of R-BP to R d -BP, we can also define AR d -BP by introducing dimension dependence in ρ 0 . Finally, we highlight that extending R-BP to mixed data is possible as given in Appendix E.1.3 of Fong et al. [2021], which also extends naturally to AR-BP.
Remark. The data-dependent bandwidth also appears when starting from other Bayesian nonparametric models, such as dependent DPs and GPs (see Supplement A.2.2 for the derivation).
Our approach can be viewed as a Bayesian version of an online KDE procedure. To see this, note that a KDE trained on i − 1 observations -yielding the density estimate q i−1 (x) -can be updated after observing the i th observation
x i via q i (x) = (1 − α i )q i−1 (x) + α i d (x, x i ),
where α i = 1/i and d(·, ·) denotes the kernel of the KDE. Rather than adding a weighted kernel term directly, AR-BP instead adds an adaptive kernel that depends on a notion of distance between x and x i based on the predictive CDFs conditional on x 1:i−1 .
To better understand the importance of the datadependent bandwidth, we compare the conditional predictive mean of R-BP and AR-BP in the bivariate setting X × Y . Under the simplifying assumption of Gaussian predictive densities, we show in Supplement A.3 that the conditional mean of Y | X is given by
µ i (x) = µ i−1 (x) + α i (x, x i )ρ(x, x i )(y i − µ i−1 (x i )), α i (x, x i ) = α i c(P i−1 (x), P i−1 (x i ); ρ) 1 − α i + α i c(P i−1 (x), P i−1 (x i ); ρ) .
Note that ρ(x, x i ) = ρ 0 for R-BP. Intuitively, the updated mean is the previous mean plus a residual term at y i scaled by some notion of distance between x and x i . For R-BP, this distance between x and x i depends only on their predictive CDF values through α i (x, x i ). This can result in undesirable behaviour as shown in the upper plot in Figure 3(a), where the peak of α i (x, x i ), as a function of x, is not centred at x i . Counterintuitively, there is thus an x > x i where µ i (x) is updated more than at the actual observed x = x i . This follows from the lack of focus on conditional density estimates for R-BP, which is alleviated by AR-BP. In the AR case, ρ(x, x i ) takes into account the Euclidean distance between x and x i in the data space. We see in the lower plot in Figure 3(a) that the peak is closer to x i . Figure 3(b) further demonstrates this difference on another toy example -we see that R-BP struggles to fit a linear conditional mean function for n = 4, focussing density in data sparse regions, while AR-BP succeeds to assign significant density only to points on the data manifold.
Training the update parameters In order to compute the predictive density p n (x * ), we require the vector of conditional CDFs
[v j 1 , . . . , v j n−1 ] where v j i = P i (x j i+1 | x 1:j−1 i+1 )
. Given a bandwidth parameterization, obtaining this vector thus amounts to model-fitting, and each v j i requires i − 1 iterations (Supplement B.3), for i ∈ {1, . . . , n}. We note that the order of samples and dimensions influences the prediction performance in AR density estimators [Vinyals et al., 2015]. In practice, averaging over different permutations of these improves performance (Supplement B.3). Full implementation details can be found in Supplement B.
Computational complexity
The above procedure results in a computational complexity of O(M dn 2 ) at the training stage where M is the number of permutations. At test time, we have already obtained the necessary conditional prequential CDFs v j n in computing the prequential log-likelihood above. As a result, we have a computational complexity O(M dn) for each test observation. Note that the introduction of a data-dependent bandwidth does not increase the computational complexity at train or test time relative to R-BP and only adds a negligible factor to the computational time for the calculation of the bandwidth.
BANDWIDTH PARAMETERISATION
The choice of covariance function in (7) provides substantial modelling flexibility in our AR-BP framework. Moreover, the additional parameters associated with the covariance function allow us to tune the implied covariance structure according to the observed data. This formulation enables us to draw upon the rich literature on the choice of covariance functions for Gaussian processes [Williams and Rasmussen, 2006]. For simplicity we only consider the most popular such choice here, but study the more flexible rationalquadratic covariance in Supplement C.2. The radial basis function (RBF) covariance function is defined as k (x 1:j−1 ,
x 1:j−1 ) = exp[− j−1 κ=1 {(x κ − x κ )/ κ } 2 ], where ∈ R d−1
>0 is the length scale.
Neural parameterisation As we saw in the motivating example of the density estimation of a chessboard distribution in Figure 1, the RBF kernel can restrict the capacity of the predictive density update to capture intricate nonlinearities if the training data size is not sufficient. While the parameterization of the bandwidth in (9) was initially derived via the first predictive update for a DPMM, all we require is that the bandwidth function ρ j : R j−1 × R j−1 → R lies in (0,1). We would also like ρ j (x 1:j−1 , x 1:j−1 ) to take larger values when x 1:j−1 and x 1:j−1 are 'close' in some sense. Motivated by this observation, we now consider more expressive bandwidth functions that can lead to increased predictive performance. In particular, we formulate an AR neural network f w : R d → R d×d for d ∈ N with the property that the j th row of the output depends only on the first j − 1 dimensions of the input.
Let Z = f w (x) and denoting z j to be the j th row of the matrix Z, the covariance function is then computed as ρ j (x 1:j−1 ,
x 1:j−1 ) = ρ 0 exp(− j−1 κ=1 ||z κ − z κ || 2 2 )
. Numerous AR neural network models have been extensively used for density estimation [Dinh et al., 2014, Huang et al., 2018, Kingma et al., 2016. In our experiments, we use a relatively simple model with parameter
Figure 3: (a) Plots of α i (x, x i )ρ(x, x i ) for R-BP and AR-BP for ρ 0 ∈ {0.5, 0.7, 0.95} ( , , ) with new observation x i ( ).
Note that ρ(x, x i ) = ρ 0 for R-BP, and = 1 for AR-BP. (b) Density plots for R-BP and AR-BP trained on 4 sequential data points ( ). Both figures show that the update of R-BP, unlike AR-BP, is not centred around the new datum.
sharing inspired by NADE, an AR neural network designed for density estimation [Larochelle and Murray, 2011]. More advanced properties like the permutation invariance of MADE [Papamakarios et al., 2017] create an additional overhead that cannot be used in the copula formulation as the predictive update is not permutation-invariant. We refer to Bayesian predictive densities estimated using AR neural networks as ARnet Bayesian predictives (ARnet-BP).
Tuning the bandwidth function Recall that the bandwidths ρ i (·, ·) are parameterised by ρ 0 and the parameters of the chosen covariance functions or neural embedders. For AR-BP, these are the length scales of the RBF covariance function, while for ARnet-BP, these are the parameters w of the AR neural network. We fit these tunable parameters in a data-driven approach by maximising the prequential [Dawid, 1997] log-likelihood n i=1 log p i−1 (x i ) which is analogous to the Bayesian marginal likelihood -the tractable predictive density allows us to compute this exactly, and this approach is analogous to empirical Bayes. Specifically, we use gradient descent optimisation with Adam, sampling a different random permutation of the training data at each optimisation step (Supplement B.3).
RELATED WORK
Our work falls into the broad area of multivariate density estimation [Scott, 2015]. While AR networks have been previously used directly for the task of density estimation [Bengio and Bengio, 1999, Germain et al., 2015, Larochelle and Murray, 2011, we use them to elicit a data-dependent bandwidth in the predictive update to mitigate the smoothing effect observed in AR-BP. Neural network based approaches, however, often underperform in small-data regimes. Deep learning approaches that do target few-shot density estimation require complex meta-learning and pre-training [2021a,b, 2004] also focus on univariate predictive updates in the Bayesian nonparametric paradigm, specifically exploring the use of the conditionally identically distributed condition as a relaxation of the standard exchangeability assumption. Other studies have investigated quasi-Bayesian updates in the special case of the mixing distribution in nonparametric mixture models [Dixit and Martin, 2022, Fortini and Petrone, 2020, Martin, 2018, Tokdar et al., 2009, though these typically focus on univariate or low-dimensional spaces. See also Martin [2021] for a survey.
Finally, copulas are a well-studied tool for modelling the correlations in multivariate data (see e.g. Kauermann et al. [2013], Ling et al. [2020[ ], Nelsen [2007). Copula density estimation aims to construct density estimates whose univariate marginals are uniform [Gijbels and Mielniczuk, 1990], and often focus on modelling strong tail dependencies [Wiese et al., 2019]. In contrast, we employ bivariate copulas for generic multivariate density estimation as a tool to model the correlations between subsequent subjective predictive densities, rather than across the data dimensions directly.
EXPERIMENTS
We demonstrate the benefits of AR-BP, AR d -BP and ARnet-BP for density estimation and prediction tasks in an experimental study with five baseline approaches and 13 different data sets. The code and data used is
DENSITY ESTIMATION
We compared our models against KDEs [Parzen, 1962], DPMMs [Rasmussen, 1999] ]. These include low-dimensional data sets with up to 63 features, but at least 29,000 with up to 10 6 samples. In many circumstances, data sets of such a data size are not avialable. To investigate performance as a function of sample size, we trained the models on subsets of the full data set. We do not report results for the KDEs and the DPMM estimators here as these estimators performed significantly worse than the other approaches. Similarly, we do not report deep learning results for sample sizes smaller than 10 2 . See Supplement C.2 for complete results.
In the small-data regime, we observe that the R-BP methods significantly outperform the neural density estimators ( Figure 4). As the sample size increases, the gap in performance decreases until eventually the neural density estimators outcompete the R-BP methods.
The performance between the R-BP methods and our proposed AR extensions is largely similar, though we note that the AR-BP methods were generally more effective on the GAS dataset.
SUPERVISED LEARNING
R-BP methods, including AR-BP, can be used for prediction tasks such as regression and classification Fong et al. [2021]. In short, this is achieved by estimating the conditional predictive density p n (y|x) of the labels , and additionally report results on the MNIST data set, restricted to digits of class 0 and 1. We report the conditional test NLL − 1
n i log p n (y * i |x * i ) for a test set {(x * 1 , y * 1 ), . . . , (x * n , y * n )}.
We compared our models against a GP, a linear Bayesian model (Linear), and a one-hidden-layer multilayer perceptron (MLP) on several classification and regression tasks. To get a distribution over the predicted outcome in the regression case, we trained an ensemble over 10 MLPs. Our proposed methods were again highly competitive (Table 2). AR d -BP performed best on two regression tasks and one classification task. ARnet-BP was substantially better than the remaining methods on CONCR and also performed best on the PARKIN. On the other hand, the MLP model was best on MNIST.
DISCUSSION
Although Bayesian methods generally perform well in the small sample setting, the conventional Bayesian approach to density estimation, i.e. DPMM estimation via the posterior predictive, is computationally intensive. Here, we set out to propose a computationally efficient density estimator as an alternative to DPMM density estimation. We recommend its use for tabular data sets of up to 63 features, and 10,000 observations. Such data set sizes are ubiquitous in healthcare, finance, hyperparameter tuning, and survey data applications.
We expand upon the tractable recursive copula updates of Fong et al. [2021], Hahn et al. [2018] by incorporating regression methods, such as kernels and neural networks. This introduces a data-dependent bandwidth, thus increasing the flexibility of this class of models, with little computational overhead compared to R-BP. More generally, it would be of interest to integrate other machine learning methods with recursive copula updates. Furthermore, other Bayesian nonparametric models may inspire other recursive copula updatessee Appendix A.2 for an example based on GPs.
An appealing feature of AR-BP is that it requires no manual hyperparameter tuning. Further, on small data sets, AR-BP shows state-of-the-art generalization and is faster than competing deep learning models. It significantly increases the modelling capacity of the baseline R-BP via a data-dependent bandwidth. Additionally, ARnet-BP provides a useful illustration of how powerful neural network models can be incorporated into R-BP methods to improve density estimation. Future work can investigate alternative architectures for structured data. Our work adds to the rich body of density estimators and thus we do not anticipate any additional negative societal impact arising from our proposal.
This strong performance of AR-BP (and other copula methods) in the small data regime is likely due to its Bayesian-like regularization towards an initial density p 0 , as shown in the weighted sum in (8). Its weaker performance in the large data regime may be due to the importance of the sequence α i which governs how regularization decays, but further theoretical work is needed to understand AR-BP's asymptotic behaviour. A limitation of R-BP methods, including AR-BP, is the quadratic time dependence on the number of training observations. Subsampling techniques thus offer a particularly promising avenue to reduce the overall computational cost and warrant further investigation. Although the recursive updates depend on the sample and covariate ordering, it is possible to alleviate this dependence though by estimating the R-BP over multiple permutations in parallel, as we have done in the above experiments. Nevertheless, the algorithm is relatively fast: with a single GPU, we were able to train models with 100,000 observations in less than an hour.
The use of a GP prior greatly increases the flexibility of our framework. Moreover, it opens the door to future research to incorporate ideas from the vast GP literature to further boost performance in high-dimensional settings. Our use of the RBF kernel was illustrative; other kernels are discussed in Appendix C.2 where we find that the RBF kernel is . For example, we anticipate that the use of recent advances in convolutional kernels [Van der Wilk et al., 2017]
A DERIVATIONS
A.1 DERIVATION OF AR-BP
For illustration purposes, we first start by summarising the derivation of the update without autoregression, closely following Appendix E.1.2 in Fong et al. [2021].
A.1.1 No Autoregression (R-BP)
The multivariate DPMM with factorized kernel has the form
f G (x) = d j=1 N (x j | θ j , 1) dG(θ), G ∼ DP (a, G 0 ) , G 0 (θ) = d j=1 N (θ j | 0, τ −1 ). Given p i (x) = p i−1 (x)h i (x, x i ),p 1 (x) = p 0 (x)h 1 (x, x 1 )· From h i (x, x i ) = f (x|θ)f (x i |θ)π i−1 (θ)dθ f (x|θ)π i−1 (θ)dθ f (x n |θ)π i−1 (θ)dθ , it follows that h 1 (x, x 1 ) = E [f G (x) f G (x 1 )] p 0 (x) p 0 (x 1 )(10)
where the expectation is over G coming from the prior. Following the stick-breaking representation of the DP, Fong et al. [2021] write G as
G = ∞ k=1 w k δ θ * k where w k = v k j<k {1 − v j }, v k iidE ∞ j=1 ∞ k=1 w j w k K(x | θ * j ) K(x 1 | θ * k ) = 1 − E ∞ k=1 w 2 k E [K(x | θ * )] E [K(x 1 | θ * )] + E ∞ k=1 w 2 k E [K(x | θ * ) K(x 1 | θ * )]
where they have used the fact that ∞ k=1 w k = 1 almost surely. As p 0 (x) = E [K(x | θ * )], it follows that (10) can be expressed as
1 − α 1 + α 1 E [K(x | θ * ) K(x 1 | θ * )] p 0 (x) p 0 (x 1 ) ·
for some fixed α 1 . For R-BP, the kernel K factorises with independent priors on each dimension, and p 0 (
x) = d j=1 p 0 (x j ) = d j=1 N (x j | 0, 1 + τ −1 ), so E [K(x | θ * ) K(x 1 | θ * )] p 0 (x) p 0 (x 1 ) = d j=1 E K(x j | θ * j ) K(x j 1 | θ * j ) p 0 (x j ) p 0 (x j 1 ) ·(11)
Fong et al. [2021] then show that each univariate term corresponds to the bivariate Gaussian copula density,
c(u, v; ρ) = N 2 Φ −1 (u), Φ −1 (v) | 0, 1, ρ N {Φ −1 (u) | 0, 1} N {Φ −1 (v) | 0, 1} ,
where Φ is the normal CDF, and N 2 is the standard bivariate density with correlation parameter ρ = 1/(1 + τ ). They then suggest an alternative sequence h i which iteratively repeats h 1 , with the key feature that α i = (2− 1 i ) 1 i+1 . See Appendix E.1.1. in Fong et al. [2021] for a derivation of this sequence α i .
A.1.2 With Autoregression (AR-BP)
For the derivation of the AR-BP update, we can follow the arguments in the previous section until (11) where the factorised kernel assumption applies for the first time. For AR-BP, we instead have
E [K(x | θ * ) K(x 1 | θ * )] p 0 (x) p 0 (x 1 ) = d j=1 E K{x j | θ * j (x 1:j−1 )} K{x j 1 | θ * j (x 1:j−1 )} p 0 (x j ) p 0 (x j 1 ) ·(12)
The factorisation of the denominator follows from
p 0 (x) = E d j=1 K{x j | θ * j (x 1:j−1 )} = d j=1 E K{x j | θ * j (x 1:j−1 )}
as we have independent GP priors on each function θ * j . For notational convenience we write {y, x} in place of {x j , x 1:j−1 } in the following. With the autoregressive kernel assumption, there is the additional complexity
E [N {y | θ(x), 1} N {y 1 | θ(x 1 ), 1}]
where θ(·) ∼ GP{0, τ −1 k}. The marginal distribution of the GP is normal, so we have
[θ(x), θ(x 1 )] T ∼ N 2 (x, x 1 | 0, Σ x,x1 ) where Σ x,x1 = τ −1 τ −1 k(x, x 1 ) τ −1 k(x, x 1 ) τ −1 ·
Again from the conjugacy of the normal, we can show that
E [N {y | θ(x), 1}N {y 1 | θ(x 1 ), 1}] = N (y, y 1 | 0, K x,x1 )
where
K x,x1 = 1 + τ −1 τ −1 k(x, x 1 ) τ −1 k(x, x 1 ) 1 + τ −1 ·
Here p 0 (y) = E[N (y|θ(x))] is the same as above, since marginally θ(x) ∼ N (0, τ −1 ). Plugging in y = P −1 0 {Φ(z)} again gives us the Gaussian copula density with correlation parameter ρ 1 (x) = ρ 0 k(x, x 1 ) for ρ 0 = 1/(1 + τ ).
A.2 DERIVATION OF GAUSSIAN PROCESS POSTERIOR
In this section, we derive the copula sequence for the Gaussian Process, which is fully tractable. This section is mostly for insight, but it would however be interesting to investigate any potential avenues for methodological development.
A.2.1 First Update Step
We consider a univariate regression setting with {y, x}. For the GP, we have the model
f θ (y | x) = N (y | θ(x), σ 2 ), θ(·) ∼ GP(0, τ −1 k).
Like in the above, we can derive the function h 1 (x, x 1 ). Following a similar argument to the AR-BP derivation, the first step GP copula density is
N 2 y, y 1 | 0, K 2 + σ 2 I p 0 (y | x)p 0 (y 1 | x 1 ) where K i is the i × i Gram matrix, with kernel k(x, x ) = τ −1 exp −0.5(x − x ) 2 / .
Writing in terms of P 0 , we have
c {P 0 (y | x), P 0 (y 1 | x 1 ); ρ 1 (x)}
where c is again the Gaussian copula density, but we have the correlation parameter as
ρ 1 (x) = exp −0.5(x − x 1 ) 2 / 1 + τ σ 2 .
From this, we can derive the first step of the update scheme:
p 1 (y | x) = c{P 0 (y | x), P 0 (y 1 | x 1 ); ρ 1 (x)} p 0 (y | x)
where c (u, v; ρ) is again the Gaussian copula density, and p 0 (y | x) = N (y; 0, σ 2 + τ −1 ).
A.2.2 All Update Steps
We can even derive the copula update scheme for i > 1, as the Gaussian process posterior is tractable. After observing i − 1 observations, we have
π(θ x , θ xi | y 1:i−1 , x 1:i−1 ) = N (µ i−1 , Σ i−1 )
where each element of Σ i−1 has the entry
k i−1 (x, x ) = k(x, x ) − k(x, x 1:i−1 ) K i−1 + σ 2 I −1 k(x 1:i−1 , x )
where the subscript i − 1 indicates it is the posterior kernel and µ i−1 is the posterior mean vector of the GP at x and x i . Marginally, the GP copula after i − 1 data points is
N 2 y, y i ; µ i−1 , Σ i−1 + σ 2 I N y; µ y i−1 , k i−1 (x, x) + σ 2 N y i+1 ; µ yi−1 i−1 , k i−1 (x i , x i ) + σ 2
where µ y i−1 is the posterior mean of the GP at x and likewise for µ yi−1 i−1 . This is equivalent to the bivariate Gaussian copula density c (u, v; ρ
i (x)), where as before u = P i−1 (y | x) and v = P i−1 (y i+1 | x i+1 ). The correlation parameter is now ρ i (x) = k i−1 (x, x i ) {k i−1 (x, x) + σ 2 }{k i−1 (x i , x i ) + σ 2 }
In summary, we have the update
p i (y | x) = c{P i−1 (y | x), P i−1 (y i | x i ); ρ i (x)} p i−1 (y | x).
This gives the same predictives as fitting a full GP. While this update form does not offer any computational gains, it gives us insight into the GP update. The copula update corresponds to the regular normal update [Hahn et al., 2018] with a data-dependent bandwidth ρ i (x) which measures the distance between x and x i based on the posterior kernel. A potential interesting direction of research is to seek approximations of the expensive ρ i (x) to aid with the computation of the GP.
A.3 INTUITION FOR AR COPULA
As in the main paper, we consider bivariate data, (x, y). As shown in Fong et al. [2021], the update for the conditional density for R-BP takes the form
p i (y | x) = [1 − α i (x, x i ) + α i (x, x i ) c {P i−1 (y | x), P i−1 (y i | x i ); ρ}] p i−1 (y | x),(13)
where
α i (x, x i ) = α i c{P i−1 (x), P i−1 (x i ); ρ} 1 − α i + α i c{P i−1 (x), P i−1 (x i ); ρ} ·
To show the effect of the AR update, we make simplifying assumptions to derive the update for the conditional mean function, µ i (x) = y p i (y | x)dy. Let us assume that our predictive densities are normally distributed, that is
P i−1 (y | x) = N (y | µ i−1 (x), σ 2 y )
. This is an accurate approximation if the truth is normal and we have observed sufficient observations. Without loss of generalizability, we assume that σ 2 y = 1. This then gives the form P i−1 (y | x) = Φ(y − µ i−1 (x)), which will help us in the calculation of the bivariate Gaussian copula. If we multiply by y and integrate on both sides of (13), we get
µ i (x) = [1 − α i (x, x i )]µ i−1 (x) + α i (x, x i ) c (P i−1 (y | x), P i−1 (y i | x i ); ρ) y p i−1 (y | x) dy.
Plugging in P i−1 (y | x) = Φ{y − µ i−1 (x)} (and similarly for the density) to the above gives
c (P i−1 (y | x), P i−1 (y i | x i ); ρ) y dy = N (y, y i | [µ i−1 (x), µ i−1 (x i )], 1, ρ) N (y i | µ i−1 (x i ), 1) y dy·
The above is simply the expectation of a conditional normal distribution, giving us
c (P i−1 (y | x), P i−1 (y i | x i ); ρ) y dy = µ i−1 (x) + ρ(y i − µ i−1 (x i )).
Putting it all together, we thus have
µ i (x) = µ i−1 (x) + α i (x, x i )ρ(y i − µ i−1 (x i )).
In the autoregressive case, we have
µ i (x) = µ i−1 (x) + α i (x, x i )ρ(x, x i )(y i − µ i−1 (x i )),
where we use the notations ρ i (x) = ρ(x, x i ) interchangeably to highlight the dependence of ρ on the distance between x and x i . Further assuming P i−1 (x) = N (x | 0, 1) returns a tractable form for α i (x, x ), giving us Figure 3 in the main paper.
A.4 DERIVATION OF COPULA UPDATE FOR SUPERVISED LEARNING
We now derive the predictive density update for supervised learning tasks, closely following the derivations of Fong et al. [2021] for the conditional methods in Supplements E.2 and E.3. We assume fixed design points x 1:n ∈ R n×d and random response y 1:n ∈ R n .
A.4.1 Conditional Regression with Dependent Stick-Breaking
We follow Appendix E.2.2 in Fong et al. [2021], and derive the regression copula update inspired by the dependent DP. Consider the general covariate-dependent stick-breaking mixture model
f Gx (y) = N (y | θ, 1) dG x (θ), G x = ∞ l=1 w l (x) δ θ * l (x) .(14)
For the weights, we elicit the stick-breaking prior w l (
x) = v l (x) j<l {1 − v j (x)} where v l (x)
is a stochastic process on X taking values in [0, 1], and is independent across l. For the atoms, which are now dependent on x, we assume they are independently drawn from a Gaussian process,
θ * l (·) iid ∼ GP(0, τ −1 k),
where k is the covariance function. Once again, we want to compute
E f Gx (y) f Gx 1 (y 1 ) p 0 (y | x) p 0 (y 1 | x 1 ) ·
Following the stick-breaking argument as in Section A.1.1, we can write the numerator as
{1 − β 1 (x, x 1 )} E [K{y | θ * (x)}] E [K{y 1 | θ * (x 1 )}] + β 1 (x, x 1 )E [K{y | θ * (x)} K{y 1 | θ * (x 1 )}] where K{y | θ * (x)} = N {y | θ * (x), 1}, θ * (·) ∼ GP(0, τ −1 k), and β 1 (x, x 1 ) = ∞ k=1 E [w k (x)w k (x 1 )] .
As before, we have
E f Gx (y) f Gx 1 (y 1 ) p 0 (y | x) p 0 (y 1 | x 1 ) = c {P 0 (y | x), P 0 (y 1 | x 1 ); ρ 1 (x)}
where ρ 1 (x) = ρ 0 k(x, x 1 ) and ρ 0 = 1/(1 + τ ). We thus have the copula density as a mixture of the independent and Gaussian copula density. This then implies the copula update step of the form
p i (y | x) = [1 − β i (x, x i ) + β i (x, x i ) c {P i−1 (y | x), P i−1 (y i | x i ); ρ i (x)}] p i−1 (y | x),
where we write ρ i (x) = ρ d+1 i (x). As in Fong et al. [2021], we turn to the multivariate update for inspiration where we do not update P n (x) and instead keep it fixed at P 0 (x) = Φ(x) (for each dimension). This gives us
β i (x, x i ) = α i d j=1 c Φ x j , Φ x j i ; ρ j i (x 1:j−1 ) 1 − α i + α i d j=1 c Φ (x j ) , Φ x j i ; ρ j i (x 1:j−1 ) ·(15)
A.4.2 Classification with Beta-Bernoulli Copula Update
In the classification setting (Appendix E.3.1 in Fong et al.
[2021]), Fong et al.
[2021] assume a beta-Bernoulli mixture for y i ∈ {0, 1}. As the derivation is written w.r.t ρ, we simply replace ρ with our definition of ρ j i (x 1:j−1 ), giving the update
p i (y | x) = (1 − β i (x, x i ) + β i (x, x i ) b {q i−1 , r i−1 ; ρ i (x)}) p i−1 (y | x) where q i−1 = p i−1 (y | x), r i = p i−1 (y i | x i ), ρ i (x) as in Equation 9
, β i (x, x i ) similarly as in (15), and finally the copula-like function b given by
b{q i−1 , r i−1 ; ρ i (x)} = 1 − ρ i (x) + ρ i (x) q i−1 ∧ r i−1 q i−1 r i−1 if y = y i 1 − ρ i (x) + ρ i (x) q i−1 − {q i−1 ∧ (1 − r i−1 )} q i−1 r i−1 if y = y i ·
B METHODOLOGY
In this section, we provide more details on the methodology referred to in the main part of the paper.
B.1 GENERATIVE MODELLING
First, we consider three approaches to generative modelling
u ∼ U[0, 1], x * ∼ P −1 n (u).
As we cannot evaluate P −1 n (u) directly, we instead solve an optimisation problem
x * = argmin x |P n (x) − u|
Multivariate setting
The univariate procedure can be repeated iteratively in the multivariate setting given the conditional distribution
u 1 ∼ U[0, 1], x 1 = P −1 n (u 1 ) u 2 ∼ U[0, 1], x 2 = P −1 n (u 2 | x 1 ) . . . u d ∼ U[0, 1], x d = P −1 n (u d | x 1:d−1 )
B.1.2 Importance Sampling
In practice, inverse sampling is unstable and is highly dependent on the performance of the optimization. An alternative approach to data generation is importance sampling. This includes two steps 1. Sampling a set of particles z 1 , . . . , z B from the initial predictive p 0 .
2. Re-sampling z 1 , . . . , z B with replacement based on the weights w 1 = p n (z 1 )/p 0 (z 1 ), . . . , w B = p n (z B )/p 0 (z B ).
B.1.3 Sequential Monte Carlo
Importance sampling will perform poorly if p n and p 0 are far apart. Instead, we propose a SMC procedure. A similar SMC sampling scheme has been proposed for univariate imputation of censored survival data by Fong and Lehmann [2022]. Here, the goal is parameter inference, and thus only requires implicit sample observations by drawing the marginal CDF u j n from a uniform distribution. In our case, we generate new explicit data directly by sampling from the data space. Please see Algorithm 6 for a complete overview. As this sampling approach is similar to evaluating the density at test data points (Algorithm 5), we highlighted the differences in blue. In short, 1. We sample a set of particles z 1 , . . . , z B from the initial predictive p 0 , and set the particle weights to w 2. We update the predictive p i−1 → p i , and the particle weights w based on their weights.
[i] k = w [i−1] k · p i z [i−1] k /p i−1 z [i−1]
x 1
x 2
True Samples True Samples
x 1
Initial Samples
x 1 Final Samples (b) Importance Sampling x 1 x 2
True Samples
x 1
Initial Samples
x 1
Final Samples (c) Sequential Monte Carlo Figure 5: 100 samples generated from AR d -BP trained on 50 samples from a GMM with 4 components. All three sampling approaches manage to preserve the multi-modal data distribution.
Note that particle diversity can be improved by introducing move steps, for example using Markov kernels Chopin [2002], Gunawan et al. [2020].
In Figure 5, we see that inverse sampling struggles on a simple GMM example. On the other hand, importance sampling and SMC provide reasonable samples. Similar sampling schemes have been proposed for Restricted Boltzmann Machines [Larochelle andMurray, 2011, Salakhutdinov andMurray, 2008] where samples can only be drawn from the model approximately by Gibbs sampling.
B.2 SUPERVISED LEARNING
We briefly recap how joint density estimation can be extended to conditional supervised learning (regression and classification), as outlined by Fong et al. [2021]. Please see Supplement A.4 for the derivation. Given fixed design points x 1:n and random response y 1:n , the problem at hand is to infer a family of conditional densities {f x (y) : x ∈ R d }.
B.2.1 Regression
For the regression case, Fong et al.
[2021] posit a Bayesian model with the nonparametric likelihood being a covariate-dependent stick-breaking DPMM:
f Gx (y) = N (y | θ, 1) dG x (θ), G x = ∞ k=1 w k (x) δ θ * k ,(16)
where w k (x) follows an x-dependent stick-breaking process. Our contribution is to assume an autoregressive factorisation of the kernel and independent GP priors on θ * k . See Supplement A.4.1 for the derivation of the predictive density update that is now given by
p i (y | x) = [1 − β i (x, x i ) + β i (x, x i ) c {P i−1 (y | x), P i−1 (y i | x i ); ρ i (x)}] p i−1 (y | x),(17)
where ρ i (x) = ρ d+1 i (x) and β as in (15).
B.2.2 Classification
For y i ∈ {0, 1}, Fong et al.
[2021] assume a beta-Bernoulli mixture. As explained in Supplement A.4.2 and Fong et al. [2021], this gives the same update as in the regression setting with the difference that the copula c in (17) is replaced with
b{q i−1 , r i−1 ; ρ i (x)} = 1 − ρ i (x) + ρ i (x) q i−1 ∧ r i−1 q i−1 r i−1 if y = y i 1 − ρ i (x) + ρ i (x) q i−1 − {q i−1 ∧ (1 − r i−1 )} q i−1 r i−1 if y = y i , where ρ i (x) = ρ d+1 i (x), q i−1 = p i−1 (y | x), r i−1 = p i−1 (y i | x i )
and ρ y ∈ (0, 1).
B.3 IMPLEMENTATION DETAILS
Please see Algorithm 1 for the full estimation procedure, Algorithm 2 for the optimisation of the bandwidth parameters, Algorithm 4 for the fitting procedure of the predictive density updates, and eventually Algorithm 5 for the steps during test-time inference. All algorithms are written for one specific permutation of the dimensions, and are repeated for different permutations.
Note that at both training time and test time, we need to consider the updates on the scale of the CDFs, that is for the terms such as u j i (x j ), which appear in the update step (8). Given
u j i (x j ) = P i (x j |x 1:j−1 ) = x j −∞ p i (x 1:j−1 , x j )/p i (x 1:j−1 )dx j ,
and (8), the CDFs u j i (x j ) take on the tractable update
u j i = (1 − α i )u j i−1 + α i H u j i−1 , v j i−1 ; ρ j i k−1 r=1 c u r i−1 , v r i−1 ; ρ r i p i−1 x 1:k−1 p i (x 1:k−1 ) ,(18)
and set v j i−1 = u j i−1 (x i ) which holds by definition, where we dropped the argument x for simplicity from ρ j i and u j i , and H (u, v; ρ) denotes the conditional Gaussian copula distribution with correlation ρ, that is
H(u, v; ρ) = u 0 c(u , v; ρ)du = Φ Φ −1 (u) − ρΦ −1 (v) 1 − ρ 2 ·
The Gaussian copula density c (u, v; ρ) is given by
c(u, v; ρ) = N 2 Φ −1 (u), Φ −1 (v) | 0, 1, ρ N {Φ −1 (u) | 0, 1}N {Φ −1 (v) | 0, 1} ,
where Φ is the normal CDF, and N 2 is the standard bivariate density with correlation ρ ∈ (0, 1).
Ordering Note that the predictive density update depends on the ordering of both the training data and the dimensions. This permutation dependence is not an additional assumption on the data generative process, and the only implication is that the subset of ordered marginal distributions continue to satisfy (5) (main paper). In the absence of a natural ordering of the training samples or the dimensions, we take multiple random permutations, observing in practice that the resulting averaged density estimate performs better. More precisely, for a given permutation of the dimensions, we first tune the bandwidth parameters, and then calculate density estimates based on multiple random permutations of the training data. We then average over each of the resulting estimates to obtain a single density estimate for each dimension permutation, and subsequently take the average across these estimates to obtain the final density estimate. Importantly, our method is parallelizable over permutations and thus able to exploit modern multi-core computing architectures.
Algorithm 1 Full density estimation pipeline Input:
x 1:n : training observations;
_, {p (m) i−1 (x i )} i,m ← fit_conditional_predictive_cdf( R (s−1) , {x 1 , . . . , x nρ }, M , fit_density=True) 5: Compute L(x 1 , . . . , x nρ ) = − M m=1 nρ i=1 log p (m) i−1 (x i ) 6: R(Initialise u j 0 (z [0] k ) ← Φ(z [0] k ) 6: end for 7: Initialise w [0] k ← p 0 (z [0] k )/q(z [0] k ) 8:
end for 9:
C EXPERIMENTS
C.1 EXPERIMENTAL DETAILS
The UCI data sets [Asuncion and Newman, 2007] Initialisation We initialise the predictive densities with a standard normal, the bandwidth parameter with ρ 0 = 0.9, the length scales with l 2 = 1, ..., l d−1 = 1, and the neural network weights inside ARnet-BP by sampling from a truncated normal with variance proportional to the number of input nodes of the layer.
Data pre-processing For each dataset, we standardized each of the attributes by mean-centering and rescaling to have a sample standard deviation of one. Following Papamakarios et al.
[2017], we eliminated discrete-valued attributes. To avoid issues arising from collinearity, we also eliminated one attribute from each pair of attributes with a Pearson correlation coefficient greater than 0.98. On each data set, the hyperparameter search ran for more than 5 days. Please see Table 3 for the optimal parameters found. For the benchmark UCI data sets, we did not tune the hyperparameters for neither MAF nor RQ-NSF but instead used the standard parameters given by Durkan et al. [2019]. The kernel parameters of the GP are optimised during training, the α resp. λ intialization parameter of the linear model over the range from 1 to 2 resp. 0.01 to 0.1, and the hidden layer sizes of the MLP over the values {64, 128, 256}.
Hyperparameter tuning
Compute We run all BP and neural network experiments on a single Tesla V100 GPU, as provided in the internal cluster of our department. In total, these experiments required compute of approximately 4000 GPU hours. The remaining experiments were run on a single core of an Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz, using up a total of 100 hours.
C.2 ADDITIONAL EXPERIMENTAL RESULTS
Computational analysis For the computational study, we consider data sampled from a Gaussian mixture model (GMM). By default, we set the number of training samples to n = 500, the number of test samples to n = 500, the number of features to d = 2, the number of mixture components to K = 2, and the number of feature and samples permutations to 1. In Figure 6, we plot the compute in elapsed seconds w.r.t changes in these parameters.
Sensitivity analysis
For the sensitivity study, we consider the same simulated GMM data as in the computational study, and plot the results in Figure 7. As expected, we observe that the test NLL decreases in n, and in the number of permutations. It also decreases in the number of mixture components. One possible explanation for this is that, as noted by Hahn et al. [2018], R-BP can be interpreted as a mixture of n normal distributions. The NLL decreases in d, as the mixture components are easier to distinguish in higher dimensional covariate spaces. Figure 7 shows the test NLL of ARnet-BP and AR-BP for the above GMM example, as a function of the number of sample permutations, and number of feature permutations. We see that averaging over multiple permutations is crucial to the performance of AR-BP. In Table 4, we also show results on the small UCI datasets for:
Ablation study
• a different choice of covariance function, namely a rational quadratic covariance function, defined by Benchmark UCI data sets As we only presented a subset of the results on the benchmark data sets introduced by Papamakarios et al. [2017] in Section 5, we present more results for density estimation on the complete data set in Table 5. These results underscore that 1) MAF and RQ-NSF outperform any other baseline, the more data is available; 2) KDE underperforms in high-dimensional settings; 3) DPMM is not suitable for every data distribution. Note that evaluation of the R-BP variants take at least 4 days to run on any of the data sets with more than 800,000 observations which is why we omitted those results here. Image examples We provide preliminary results on two image datasets, digits and MNIST, in Table 6. Note that the AR-BP copula updates investigated here were not designed with computer vision tasks in mind. The rich parameterization allows the model to overfit to the data leading to a prequential negative log-likelihood of at least -684 at train time while the test NLL is considerably higher. ARnet-BP, on the other hand, helps to model the complex data structure more efficiently. We expect that further extensions based on, for instance, convolutional covariance functions [Van der Wilk et al., 2017] may prove fruitful. Figure 9: Illustration of the importance of an autoregressive kernel. We trained the models on 500 data points sampled according to a sine wave distribution (given in Figure 10). We visualise the predictive density after observing a different number, n, of observations, highlighting the last five points with . We observe that for highly non-linear relationships between x 1 and x 2 , the optimal bandwidth of R-BP is quite high (ρ = 0.93) which results in strong overfitting. Even when we choose ρ 0 = 0.93 for AR-BP and ARnet-BP, we observe that these models learn the true data distribution with fewer samples than R-BP does. Figure 8 shows density estimates for the introductory example of the checkerboard distribution in a large data regime. We observe that neural-network-based methods outperform the AR-BP alternatives. Nevertheless, AR-BP performs better than the baseline R-BP. An illustration of this behaviour on another toy example is also given in Figure 9. Figure 10 shows density estimates from AR-BP on a number of complex distributions.
Toy examples
pipelines[Gu et al., 2020, Reed et al., 2017.Our work directly extends the contributions of Hahn et al. [2018] and Fong et al. [2021] through an alternative specification of the nonparametric Bayesian model in the recursive predictive update scheme. R-BP has recently been used for nonparametric solvency risk prediction [Hong and Martin, 2019], and survival analysis [Fong and Lehmann, 2022]. Berti et al.
Figure 4 :
4Average NLL and standard errors over 10 runs for training sets of different size. Our models outperform neural methods for data sets up to 10,000 samples.provided in the Supplementary Material. See Supplement C for additional experimental details and results, including a sensitivity study, an ablation study, further illustrative examples, a preliminary investigation into image examples, and an empirical study of the computational complexity of the proposed methods.
k = 1, . . . , B
If the effective sample size (ESS) is smaller than half of the number of particles, we resample z 1 , . . . , z B and w[i] 1 , . . . , w[B] 1
x n+1:n+n : test observations; M : number of permutations over samples and features to average over; n ρ : number of train observations used for the optimisation of bandwidth parameters; Output: p n (x n+1 ), . . . p n (x n+n ): density of test points i ∈ {1, . . . , n}, j ∈ {1, . . . , d}, m ∈ {1, . . . , M } O(M n 2 d) 4: Evaluate density at test observations x n+1:n+n O(M nn d) 5: end procedure Algorithm 2 Estimate optimal bandwidth parameters Input: x 1:n : training observations; M : number of permutations over samples and features to average over; n ρ : number of train observations used for the optimisation of bandwidth parameters; maxiter: number of iterations; R (0) : initialisation of bandwidth parameters: -R (0) = {ρ and w initialised as implemented in Haiku by default) for ARnet-BP Output: R (maxiter) : optimal bandwidth parameters 1: procedure optimal_bandwidth_and_lengthscales 2: Subsample {x 1 , . . . , x nρ } from x 1:n 3:for s ← 1 to maxiter do 4:
We average over M = 10 permutations over samples and features. The bandwidth of the KDEs was found by five-fold cross validation over a grid of 80 log-scale-equidistant values from ρ = 0.1 to 100. For the DPMM, we considered versions with a diagonal (Diag) and full (Full) covariance matrix for each mixture component. We optimized over the weight concentration prior of the DPMM by five-fold cross validation with values ranging from 10 −40 to 1. The model was trained with variational inference using sklearn. The hyperparameters of MAFs and RQ-NSFs were found with a Bayesian optimisation search. For MAF and RQ-NSF, we applied a Bayesian optimisation search over the learning rate {3 · 10 −4 , 4 · 10 −4 , 5 · 10 −4 }, the batch size {512, 1024}, the flow steps {10, 20}, the hidden features {256, 512}, the number of bins {4, 8}, the number of transform blocks {1, 2} and the dropout probability {0, 0.1, 0.2}.
Figure 6 :
6Computational study: computational time measured in elapsed seconds for a simple GMM example. Note that R-BP has the same computational complexity and only saves an indiscernible constant time factor.
Figure 7 :
7Sensitivity analyis: Average test NLL over 5 runs reported with standard error for a simple GMM example over a range of simulation and parameter settings.
choice of initial distribution, namely a uniform distribution (unif).
Figure 8 :
8Scatter plot and density estimates of 60,000 observations sampled from a chessboard data distribution. Test log likelihoods are R-BP: 2.25 ±0.0 , R d -BP : 2.19 ±0.0 , AR-BP: 2.21 ±0.0 , AR d -BP: 2.10 ±0.0 , ARnet BP : 2.19 ±0.0 , MAF : 2.09 ±0.0 , RQ-NSF : 2.05 ±0.0 .
Figure 10 :
10Scatter plots of 60,000 samples from different data distributions in the first row, and corresponding autoregressive predictive density estimates in the second row.
Table 1 :
1Average NLL with standard error over five runs on data sets analysed byFong et al. [2021].WINE
BREAST
PARKIN
IONO
BOSTON
n/d
89/12
97/14
97/16
175/30
506/13
KDE
13.69±0.00
10.45±0.24
12.83±0.27
32.06±0.00
8.34±0.00
DPMM (Diag)
17.46±0.6
16.26±0.71
22.28±0.66
35.30±1.28
7.64±0.09
DPMM (Full)
32.88±0.82
26.67±1.32
39.95±1.56
86.18±10.22
9.45±0.43
MAF
39.60±1.41
10.13±0.40
11.76±0.45
140.09±4.03
56.01±27.74
RQ-NSF
38.34±0.63
26.41±0.57
31.26±0.31
54.49±0.65
−2.20±0.11
R-BP
13.57±0.04
7.45±0.02
9.15±0.04
21.15±0.04
4.56±0.04
R d -BP
13.32±0.01
6.12±0.05
7.52±0.05
19.82±0.08
−13.50±0.59
AR-BP
13.45±0.05
6.18±0.05
8.29±0.11
17.16±0.25
−0.45±0.77
AR d -BP
13.22 ±0.04
6.11 ±0.04
7.21 ±0.12
16.48±0.26
−14.75 ±0.89
ARnet-BP
14.41±0.11
6.87±0.23
8.29±0.17
15.32 ±0.35
−5.71±0.62
,MAFs [Papamakarios et al., 2017] andRQ-NSFs [Durkan et al., 2019]. The hyperparameters of the baselines were tuned with crossvalidation. Unless otherwise specified, we use respectively 10 permutations over samples and features to average the quasi-Bayesian estimates. We did not see substantial improvements with more permutations. We use the same few hyperparameters (initialisation of rho, l 1 , . . . , l d , number of permutations, neural network architecture, and learning rate) on all data sets as our method is robust to their choice. See Supplement C.1 for further information.Data sets analysed by Fong et al. [2021] SeeTable 1for the negative log-likelihood (NLL) estimated on five UCI data sets[Asuncion and Newman, 2007] of small size with up to 506 samples, as investigated by Fong et al.[2021]. Our proposed methods display highly competitive performance: AR d -BP achieved the best test NLL on four of the data sets, while ARnet-BP prevailed on ionosphere.Data sets analysed by Papamakarios et al. [2017]A number of UCI data sets have become the standard evaluation benchmark for deep AR models[Durkan et al., 2019, Huang et al., 2018, Papamakarios et al., 2017
Table 2 :
2Average NLL over five runs reported with standard error for supervised tasks y directly by assuming a dependent Dirichlet process likelihood. See Supplement B.2 for details. Again, we follow the experimental set-up of Fong et al.[2021]Regression
Classification
BOSTON
CONCR
DIAB
IONO
PARKIN
MNIST01
n/d
506/13
1,030/8
442/10
351/33
195/22
12,031/784
Linear
0.87±0.03
0.99±0.01
1.07±0.01
0.33±0.01
0.38±0.01
0.003±0.000
GP
0.42±0.08
0.36±0.02
1.06±0.02
0.30±0.02
0.42±0.02
0.035±0.000
MLP
1.42±1.01
2.01±0.98
3.32±4.05
0.26±0.05
0.31±0.02
0.003 ±0.000
R-BP
0.76±0.09
0.87±0.03
1.05±0.03
0.26±0.01
0.37±0.01
0.015±0.001
R d -BP
0.40±0.03
0.42±0.00
1.00±0.02
0.34±0.02
0.27±0.03
0.018±0.001
AR-BP
0.52±0.13
0.42±0.01
1.06±0.02
0.21±0.02
0.29±0.02
0.015±0.001
AR d -BP
0.37 ±0.10
0.39±0.01
0.99 ±0.02
0.20 ±0.02
0.28±0.03
0.017±0.001
ARnet-BP
0.45±0.11
−0.03 ±0.00
1.41±0.07
0.24±0.04
0.26 ±0.04
0.014±0.001
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Hahn et al.[2018] and Fong et al. [2021] derive the predictive density updates for R-BP by initally only considering the first step update h 1
we used are: wine, breast, parkinson (PARKIN), ionosphere (IONO), boston housing (BOSTON), concrete (CONCR), diabetes (DIAB), and digits.Code We downloaded the code for MAF and NSF from https://github.com/bayesiains/nsf, and the code for R-BP from https://github.com/edfong/MP/tree/main/pr_copula, and implemented EarlyStopping with patience 50, and 200 minimal, and 2000 maximal iterations. Note that we chose the autoregressive version of RQ-NSF over the coupling variant as the former seemed to generally outperform the latter in Durkan et al.[2019]. The neural network in ARnet-BP was implemented withHaiku [Hennigan et al., 2020]. The remaining methods are implemented in sklearn. For the DPMM with VI (mean-field approximation), we use both the diagonal and full covariance function, with default hyperparameters for the priors. The code used to generate these results is available as an additional supplementary directory.
Table 3 :
3Hyperparameters for MAF and RQ-NSFdata
batch size
learning
rate
flow steps
hidden
nodes
bins
transform
blocks
dropout
MAF
WINE
10000
0.0003
20
512
-
1
0.2
BREAST
10000
0.0004
20
512
-
1
0.2
PARKINSONS
10000
0.0004
20
512
-
1
0.2
IONOSPHERE
10000
0.0003
20
512
-
1
0.2
BOSTON
10000
0.0003
10
512
-
1
0.2
CONCRETE
1024
0.0003
10
512
-
1
0.2
DIABETES
10000
0.0004
20
512
-
1
0.2
CHECKERBOARD
10000
0.0003
20
512
-
1
0.2
RQ-NSF
WINE
10000
0.0004
20
512
8
1
0.2
BREAST
10000
0.0005
10
512
8
1
0.2
PARKINSONS
10000
0.0005
20
512
8
1
0.2
IONOSPHERE
10000
0.0003
10
512
8
1
0.2
BOSTON
10000
0.0003
10
512
8
1
0.2
CONCRETE
1024
0.0004
20
256
8
2
0.1
DIABETES
256
0.0004
10
512
8
2
0.2
CHECKERBOARD
1024
0.0004
10
512
8
2
0.1
10 2
10 4
n
1.2
1.4
1.6
1.8
test NLL
10 0
10 1
10 2
d
0.9
1.0
1.1
1.2
1.3
1.4
10 1
10 2
K
1.20
1.25
1.30
10 0
10 1
10 2
# sample permutations
1.20
1.25
1.30
10 0
10 1
10 2
# feature permutations
0.93
0.94
0.95
0.96
Model
AR-BP
ARnet-BP
Table 4 :
4Average NLL with standard error over five runs on five UCI data sets of small-to-moderate size
Table 5 :
5Average NLL with standard error over five runs on benchmark UCI data from Papamakarios et al.[2017] POWER
GAS
HEPMASS
MINIBOONE
BSDS300
n/d
1,659,917/6
852,174/8
315,123/21
29,556/43
1,000,000/ 63
Gaussian
7.73±0.00
3.59±0.00
27.93±0.00
37.20±0.00
56.45±0.00
KDE
29.39±0.00
−9.61 ±0.00
26.44±0.00
43.88±7.52
63.70±10.00
DPMM (Diag)
0.51±0.01
1.20±0.02
25.80±0.00
39.16±0.01
37.55±0.02
DPMM (Full)
0.33±0.00
−5.57±0.04
23.40±0.02
18.82±0.01
4.47±0.00
MAF
0.52±0.00
−2.21±0.54
21.10±0.04
12.81±0.08
2.76±0.17
RQ-NSF
0.00 ±0.01
−6.41±0.14
19.46 ±0.08
12.51 ±0.19
2.44 ±0.56
Table 6 :
6Image datasets: average test NLL over five runs displayed with standard errorDIGITS
MNIST
MAF
−8.76 ±0.10
−7.14±0.48
RQ-NSF
−6.17±0.13
−8.49±0.03
R-BP
−8.80±0.00
−9.04±0.07
R d -BP
−7.46±0.12
−7.73±0.07
AR-BP
−8.66±0.03
−7.31±42.54
AR d -BP
−7.46±0.18
−8.32±61.92
ARnet-BP
−7.72±0.28
−9.20 ±0.10
Compute ρ j i (z 1:j−1 ) ← ρ 0 k R z 1:j−1 , x 1:j−1 i where k R denotes the user-defined kernel if ρ =None 3:Compute the bivariate Gaussian copula densityCompute the conditional Gaussian copula CDFCompute u j i (z) = P j i (z|z 1:j−1 ) byreturn u i (z) 8: end procedure Algorithm 4 Estimate prequential CDFs at train observations Input:R: bandwidth parameters x 1:n : training observations; M : number of permutations over features to average over; compute_density (by default, False);Sample permutation π 1 ∈ Π(n), π 2 ∈ Π(d)4:Change the ordering of the training observations {xFor simplicity we will drop the superscript in the following 5:for j ← 1 to d do6:for k ← 1 to n do 7:u also depends on the permutation m, but since we do not reuse u after m is updated, we drop the index for simplicity 8: end for 9: end for 10:for k ← 1 to i do 13:for j ← 1 to d do 14:end for end for end forAverage density over permutations for i ← n + 1 to n + n doend for16:Compute density
UCI machine learning repository. Arthur Asuncion, David Newman, Arthur Asuncion and David Newman. UCI machine learning repository, 2007.
We observe that none of these ablations consistently outperforms AR d -BP. We observe that none of these ablations consistently outperforms AR d -BP.
| {'fraction_non_alphanumeric': 0.07257090527824599, 'fraction_numerical': 0.04898668825567043, 'mean_word_length': 3.6890285643163856, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 7, 'https://': 4, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predictive density estimation without the need for posterior approximation. Although these estimators are computationally appealing, they struggle on non-smooth data distributions. This is due to the comparatively restrictive form of the likelihood models from which the proposed copula updates were derived. To address this shortcoming, we consider a Bayesian nonparametric model with an autoregressive likelihood decomposition and a Gaussian process prior. While the predictive update of such a model is typically intractable, we derive a quasi-Bayesian update that achieves state-ofthe-art results in small-data regimes.', 'arxivid': '2206.06462', 'author': ['Sahra Ghalebikesabi \nUniversity of Oxford\n\n', 'Chris Holmes \nNovo Nordisk\n\n', 'Edwin Fong \nNovo Nordisk\n\n', 'Brieuc Lehmann \nUniversity College London\n\n'], 'authoraffiliation': ['University of Oxford\n', 'Novo Nordisk\n', 'Novo Nordisk\n', 'University College London\n'], 'corpusid': 257039098, 'doi': None, 'github_urls': ['http://github.com/deepmind/dm-haiku.', 'https://github.com/bayesiains/nsf,', 'https://github.com/edfong/MP/tree/main/pr_copula,'], 'n_tokens_mistral': 26432, 'n_tokens_neox': 22753, 'n_words': 12823, 'pdfsha': '88b36c675ec87711113ece8de556c3e31ef3ec08', 'pdfurls': ['https://export.arxiv.org/pdf/2206.06462v2.pdf'], 'title': ['Quasi-Bayesian Nonparametric Density Estimation via Autoregressive Predictive Updates', 'Quasi-Bayesian Nonparametric Density Estimation via Autoregressive Predictive Updates'], 'venue': []} |
arxiv |
Dipartimento di Fisica e Astronomia "G. Galilei"
Università di Padova
Via F. Marzolo 835131PadovaItaly
Gran Sasso Science Institute (GSSI)
I-67100L'AquilaItaly
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Mönchhofstr. 12-14D-69120HeidelbergGermany
Department of Physics and Astronomy
UCLA
90095Los AngelesCAUSA
Department of Physics and Astronomy, Mani L. Bhaumik Institute for Theoretical Physics
UCLA
90095Los AngelesCAUSA
Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory
Parks RoadOX1 3PUOxfordUK
Received: Accepted: Published:
Citation: Arca Sedda, M.; Naoz, S.; Kocsis, B. Merging compact objects in galactic nuclei. Universe 2022, 1, 0. https://doi.org/
Introduction
11
Most galactic nuclei (if not all) are expected to harbour in their centres either a dense 12 Nuclear Cluster (NC), a massive stellar conglomerate typically comprised of 10 4−9 stars [1-3], 13 a supermassive black hole (SMBH), with typical masses M SMBH 10 4−10 M [4, 5], or both 14 [6][7][8][9][10]. With densities up to several orders of magnitude larger than typical globular and 15 young massive clusters but similar half-mass radii, NCs constitute the densest stellar systems 16 in the Universe [1, 3,10]. The unique feature of hosting a massive NC or an SMBH -or possibly 17 both -makes galactic nuclei the ideal laboratories to study stellar dynamics at its extreme. 18 A particularly interesting aspect of dynamics in such extreme environments is the formation 19 of compact object binaries (COBs), comprised of stellar-mass black holes (BHs) or neutron 20 stars (NSs).
In fact, despite the observations of BHs in binary systems became recently possible by 53 observing the stellar companion's motion [60][61][62], and electromagnetic emission (EM) from 54 accreted material [63,64], even in the Galactic Centre [53], GW detections represent so far 55 the only known technique to provide uncontroversial proof of the existence of black hole 56 binaries (BBHs). Since the groundbreaking discovery of GWs emitted by a merging BBH [65], 57 the LIGO-Virgo-Kagra collaboration (LVC) performed three observing runs and assembled 58 a catalogue of almost 100 GW sources [65][66][67][68][69][70][71][72][73][74][75][76][77][78], making clear that the interpretation of these 59 observations requires a profound rethinking of our understandings of COBs formation and 60 evolution. So far, the so-called GWTC-3 catalogue of GW sources includes mergers from ∼ 60 61 BBHs, 2 double NSs, and 2 NS-BH binaries, the first merger with total mass in the IMBH mass 62 range above 100M , and the first merger involving one object with mass 2.6M , either the 63 lightest BH or the heaviest NS ever detected in a merging binary. With a number of detections 64 that doubles the number of detections performed via EM emission and proper motion, the 65 LVC BHs are becoming sufficiently numerous to enable placing constraints on the overall 66 properties of the underlying population of BHs in merging binaries [77]. Interpreting the 67 inferred properties of observed BBH mergers requires to understand the physics that regulates 68 the formation and evolution of single and binary COs and their stellar progenitors, and the 69 impact of different formation channels on their global properties. 70 Significant improvements in stellar evolution theories undoubtedly helped us to better 71 understand the processes that govern the evolution of single and binary massive stars and 72 how they can merge in galactic fields, despite the still many uncertainties about the CO mass 73 spectrum, natal spins, and kicks [for an extensive review on the topic, see 79]. In particular, 74 there are two aspects of stellar evolution particularly relevant to the formation and merger of 75 COBs, namely the so-called upper and lower mass-gaps.
Generally, stars with zero-age main sequence (ZAMS) masses in the 22 − 26M range 77 are expected to end their life in a supernova (SN) event. If the SN explosion happens on a 78 timescale ∼ 250 ms [rapid SN model,80] the remnant will have a mass falling in the 3 − 5M 79 range, whilst if the explosion timescale is order of seconds the star undergoes a failed SN and 80 directly collapse to a BH with a mass above the 3 − 5M range [delayed SN model,80]. This 81 opens a gap in the CO mass spectrum, called the lower mass-gap, whose existence is highly 82 uncertain observationally [e.g. 81,82] and intrisically relies on the uncertain physics of stellar 83 evolution [79]. Heavier stars that develop a He core with a mass ∼ 64 − 135M , instead, are 84 expected to undergo an explosive process, the pair-instability supernova (PISN), which rips the 85 star apart and leaves no remnant [83,84]. Stars with a lighter core (32 − 64M ) develop rapid 86 pulses that enhances mass-loss before the SN explosion -so-called pulsational pair instability 87 (PPISN) [84,85]. These two processes, PISN and PPISN, cause a dearth of BHs with masses in 88 the 40 − 150M range [85][86][87][88][89][90]. The extent of this "upper" mass-gap is rather uncertain, as it 89 depends on stellar rotation, nuclear reaction rates, and accretion physics [91][92][93][94][95]. 90 Broadly speaking, the formation channels of merging COBs are grouped into two main 91 channels: isolated, i.e. a stellar binary paired at birth which turns into a COB that eventually 92 merges without the intervention of other objects, and dynamical, i.e. a COB assembled dynami- 93 cally which eventually merges with the aid of multiple gravitational scatterings in star clusters. 94 Unfortunately, the localization accuracy of current GW detectors is too low to pinpoint the 95 location of the merger event, thus generally the impact of different formation channels onto 96 the overall merger population is assessed on a statistical basis. Several recent works tried to 97 untangle signatures of different formation scenarios in BBH merger populations by looking at 98 different parameters that can be retrieved from GW detections -like component mass, chirp 99 mass, effective spin parameter -but hugely depends on many uncertainties: the cosmic star 100 formation history or metallicity distribution, BH natal spins and mass spectrum, the physics of 101 single and binary stellar evolution, the properties of star clusters at birth, and the physics of 102 galactic nuclei [96][97][98][99][100][101][102]. 103 The mass of the merging objects represents one of the possible quantities that can be used 104 to discern between an isolated and dynamical origin. BBHs formed in isolation is expected to 105 have components with a mass below 40 − 60M , owing to PISN and PPISN mechanisms [e.g. 106 86,103], thus making hard to explain the existence of upper mass-gap objects -like the one 107 observed in GW190521 source [76] -via isolated stellar evolution only. Upper mass-gap objects 108 are easier to form in dynamically active environments, like star clusters and galactic nuclei, 109 where they can form either via stellar mergers [104][105][106][107][108][109][110], BH-star accretion events [105][106][107], 110 or repeated -so called hierarchical -mergers [98,99,105,[111][112][113][114][115][116]. These processes are affected 111 by uncertainties though: stellar evolution of post-merger stars is poorly known and requires 112 detailed hydrodynamical simulations [e.g. 109,110,117], the fraction of stellar matter actually 113 accreted onto a BH in a star-BH collision is rather unknown [118][119][120][121][122], and repeated mergers 114 are hampered by post-merger GW recoil kicks [123][124][125] that can eject the remnant BH from the 115 host cluster and avoid further mergers [98,98,[126][127][128][129]. Similarly, the development of mergers 116 with one component in the lower mass-gap, like GW190814 [74], seem to be unlikely in isolated 117 binary models [100], whilst they can more easily form dynamically [130][131][132][133][134][135][136]. 118 The level of spin alignment represents another quantity useful to discern isolated and 119 dynamical mergers. In fact, isolated mergers are expected to feature (approximately) aligned 120 spins [137][138][139], whilst in dynamical mergers the chaotic process forming the binary is expected 121 to distribute the spin-orbit angle isotropically [99,102,138,140]. 122 Hierarchical mergers, which are a natural byproduct of dynamics in high-density envi- 123 ronments, might have unique mass-spin features, thus a combined statistical analysis of such 124 quantities could help determining whether an observed GW source originated from remnants 125 of previous merger episodes. For example, assuming that the cosmic population of merging 126 BHs is characterised by a spin distribution peaked at relatively low values, as indicated by 127 GW observations [141], implies that a population of second generation mergers will be in- 128 evitably characterised by larger masses and higher spins, thus being clearly discernible from 129 the population of first generation mergers and possibly from isolated mergers too [99,142]. 130 Aside from masses and spins, there is a further binary parameter that could represent a 131 smoking gun of a dynamical origin, the binary orbital eccentricity. Placing constraints on the 132 eccentricity of observed mergers became possible only in recently [e.g. 143 -146], and led to 133 place constraints on the eccentricity of up to 4 LVC sources, for which possibly e > 0.1 [147][148][149][150]. 134 Interestingly, for eccentric sources the accuracy on the parameter measurement can significantly 135 increase, e.g.,the chirp mass(localization) accuracy of an eccentric 30 − 30M BBH can be 136 ∼ 10 1 (10 2 ) times higher than for the circular case [151]. Generally, LVC detectors at design 137 sensitivity could distinguish between circular and eccentric models provided that e 10 Hz > 0.04, 138 being the measurement error on the eccentricity around δe ∼ (10 −4 − 10 −3 )(D/100Mpc) 139 [151,152]. 140 In isolated COBs several processes (e.g.,tidal interactions or dynamical friction during 141 a common envelope phase) tend to circularise the orbit of the binary progenitor [e.g. 153-142 157], although some of the physical processes still partly unknown -like common envelope 143 -could produce mildly eccentric binaries [158]. Conversely, the eccentricity distribution of 144 dynamically assembled BBHs generally follows a thermal distribution, P(e)de ∼ 2e, which 145 implies a probability of 50% to form a binary with eccentricity e > 0.7. Theoretical models 146 predict that around 1 − 10% of mergers forming in globular clusters can have an eccentricity 147 e > 0.1 in the frequency band typical of LVC and ground-based GW detectors (i.e. 10 Hz) 148 [138,[159][160][161], and up to 30% could be eccentric in the LISA band (∼ 10 −3 Hz) [162][163][164]. 149 What about galactic nuclei? In such complex environments, the formation of COBs is 150 regulated by a variety of dynamical processes -some acting in concert, others acting in contrast 151 -which intrinsically affect the overall properties of those that eventually merge, and may leave 152 imprints that could differ from the general expectations of the dynamical channels. 153 This review aims at providing a broad overview of the processes that can aid or hamper 154 COB formation and mergers in galactic nuclei harbouring a central SMBH, either in its quiescent 155 or active phase, possibly surrounded by a NC, and to discuss the main properties and detection 156 prospects of GW sources formed in galactic nuclei. We organise the review according to the 157 main phases characterising the evolution of a galactic nucleus, following a zoom-in scheme, 158 from the possible formation of the central NC to the coalescence of COBs: 159
•
We start by briefly reviewing the current knowledge on the observational evidence of 160 single and binary COs in galactic nuclei (Section 2); 161 • We describe how the environment, stellar evolution, and dynamics can affect CO popula-162 tions (Section 3 and 4); 163 • Moving closer to the SMBH, we describe the main dynamical processes at play to form 164 COBs in galactic nuclei (Section 5) and discuss the impact of secular processes in quiescent 165 galactic nuclei (Section 6) and gaseous effects in AGNs (Section 7) on the formation of 166 merging COB; 167 • Finally, we discuss the main properties of merging COBs in galactic nuclei, focusing on 168 the prospects for current and future GW detections (Section 8).
169 Figure 1 sketches the main phases of COB formation in galactic nuclei, and provides a 170 schematic and ordered illustration of the themes touched on in this review. 171 172 Most if not all of high mass stars in the field reside in a binary or higher multiple configura- 173 tions [166][167][168][169]. Although challenging, observations of the inner pc of the Milky Way revealed 174 the presence of a handful spectroscopic and eclipsing binaries comprised of massive OB and 175 . Schematic illustration of the galactic nuclei zooniverse: a nuclear cluster (NC) forms via in-situ star formation and mass transport from infalling star clusters; in its inner parts a variety of dynamical interactions can trigger the formation of compact object binaries (COBs); in other cases, COBs promptly merge releasing gravitational waves (GWs); in some other cases, the COBs evolution is determined by the central supermassive black hole (SMBH) gravitational field, which can impinge periodic oscillations on the binary eccentricity and ultimately lead to their coalescence. The depicted interactions are actual N-body simulations carried out with the ARGdf code described in Arca-Sedda and Capuzzo-Dolcetta [165].
Observational evidence of binaries in galactic nuclei: the Milky Way test case
Wolf-Rayet (WR) stars [170][171][172][173][174]. These observations suggest that the fraction of spectroscopic 176 binaries attain values f b 0.34, whilst this quantity drops to f b ∼ 0.03 − 0.04 for eclipsing 177 binaries, similar to the binary fraction inferred in young clusters. More in general, the afore-178 mentioned observations suggest that the total fraction of massive binaries in the Galactic NC is 179 comparable to the field [e.g. 170,172]. The observation of X-ray sources [53,175], and diffuse 180 X-[e.g. [176][177][178] and γ-ray [179][180][181][182][183][184][185][186] emission at the Galactic Centre support the presence 181 of COBs in galactic nuclei, although it is unclear whether they are related to X-ray binaries, 182 millisecond pulsars, or cataclysmic variables [e.g. 53,56,59,[175][176][177][178]187,188]. 183 Complementary theoretical and modelling studies of these observations also suggested 184 that the binary fraction in the inner 1 pc at the centre of the Galaxy is high. For example, star 185 formation models suggest that in situ formation of stars and thus binaries is expected at the 186 centre of galaxies [e.g. 189,190]. Hyper velocity stars [e.g. [191][192][193][194][195] may imply the existence of 187 binaries that arrive on a nearly radial trajectory to the tidal breakup radius of the SMBH, known 188 as the Hills mechanism [196]. This mechanism may eject one star at a high velocity while the 189 other one may be captured on an eccentric orbit close to the SMBH, which was proposed to 190 explain the existence of the S-cluster [e.g. 58,[197][198][199][200][201][202]. 191 Other theoretical and observational analyses suggest that binaries can remain stable for 192 long-time even when interacting with neighboring stars [e.g. 173,203]. Thus, over the age of the 193 young stars of the nuclear star cluster, estimated as a few Mys [204], about 70% of the binaries 194 may retain their binary configuration [205]. Lastly, it was suggested that some of the peculiar 195 features of the stellar disc in the Galactic Centre [206,207], can be explained by the possible 196 observed high fraction of binaries [208]. 197 198 How can COBs form in galactic nuclei? This is one of the key questions that we try to 199 address in this review. This section is devoted to discuss how the presence of pristine stellar 200 binaries and the formation process of the galactic nucleus can impact the formation of COB 201 and COBs. 202 3.1. Binaries in galactic nuclei: primordial, dynamical, or hybrid? 203 Generally, it is possible to distinguish two main formation channels for stellar and CO 204 binaries: either the binary components were already paired at birth, in which case the binaries 205 are called primordial, or they found each other via multiple interactions with other stars and 206 COs forming dynamical binaries. In dense stellar environments, such as massive clusters and 207 NCs, a further possibility suggests that primordial binaries underwent interactions with other 208 members of the galactic nucleus and either suffered orbital modifications or exchanged one of 209 their components, thus they constitute a hybrid class, half-way between purely primordial and 210 dynamical binaries. 211 Either way, after their formation, the evolution of these binaries will inevitably be affected by dynamical encounters, which in galactic nuclei are typically more frequent and violent than in galactic fields. During each subsequent interaction, binaries and single objects suffer a change of their energy and angular momentum, up to a point where the system will have lost memory of its initial conditions. This process, called relaxation, occurs on a time-scale roughly given by [209,210]:
Environmental effects on binary formation in galactic nuclei
t relx = 4.2 Gyr 15 log Λ R h 4 3/2 M c 10 7 M ,(1)
where log Λ is the Couloumb logarithm, R h is the cluster half-mass radius, and M c its total 212 mass. 213 There is a plethora of dynamical processes that can concur with the formation and evolu-214 tion of binaries in a dense galactic nucleus. Some of them involve single or multiple dynamical 215 interactions with other stellar objects (see Section 5), like GW bremsstrahlung in single-single 216 encounters (cap), three-body encounters (3bb), binary-single (bs) and binary-binary (bb) en-217 counters, triple evolution (3ev). Others, involve secular effects impinged by the galactic nucleus 218 morphology, the gravitational field of the central SMBH, or the collective effect of all stars or-219 biting around the SMBH (see Section 6), including the eccentric-Kozai-Lidov (EKL) oscillations, 220 general relativistic effects arising from the motion around the SMBH (1pN), and scalar (rr,s) 221 and vector resonant relaxation (rr,v) processes, which torque the orbit of the i-th star owing to 222 the overall perturbation from all the N − i stars in the nucleus. 223 The concurring action of one process or another is regulated by their typical timescales, 224 which can vary over many orders of magnitude depending on the galactic nucleus' properties. 225 Figure 2 briefly summarizes how the timescales connected to these different mechanism vary at 226 varying COB separation and assuming a COB with mass M COB = 20M and an SMBH with 227 mass M SMBH = 10 6 M . To make the plot more readable we do not include the AGN typical 228 lifetime, which is expected to be a constant value in the range 1 − 100 Myr and the three-body 229 scattering time, which, for this specific example, is several orders of magnitude larger than all 230 other timescales. From the plot, we see that dynamical friction and some of the secular effects, 231 like EKL, clearly operates over timescales shorter than the typical timescale of stellar evolution 232 for CO progenitors (i.e. ∼ 1 − 10 Myr), suggesting that dynamics may play a role even before 233 massive stars turn into COs. 234 This dynamical process has crucial implications for the evolution of primordial binaries 235 in massive galactic nuclei like the one in the Milky Way. For example, it has been shown that 236 a population of stellar binaries, in a Milky Way-like NC, would be strongly affected by both 237 dynamical and secular mechanisms, undergoing several possible pathways that take place 238 prior to CO formation, like [see e.g. 59,205]: a) dissociation (75% of the population), b) merger 239 induced by eccentric-Kozai-Lidov oscillations (10%), c) shrinkage and circularization of the 240 orbit due to tidal synchronization (13%), or d) radial mergers and nearly head-on collisions 241 (2%).
242
Among binaries that shrunk via stellar evolution -i.e. the case c) above -only a fraction 243 of 0.5% decouple from the SMBH dynamics and evolve into a COB that possibly merge within 244 a Hubble time [59]. Note that assuming that the Galactic NC contains ∼ 2.5 × 10 7 stars and a 245 primordial binary fraction f b = 0.3 implies a number of primordial merging COBs of around 246 N COB ∼ 2, 500, although such an estimate certainly depends on the details of binary stellar 247 evolution. The population of merging COBs formed from binary stellar evolution is expected 248 to be comprised of BBH (∼ 75%) and NS-BH binaries (∼ 25%), whilst double NS are unlikely 249 to form [59]. These binaries may further interact with neighbouring single objects, possibly 250 leading to the formation of new binaries via component exchange or the binary abundance 251 may be reduced following ionization by strong encounters. 252 3.2. Nuclear cluster formation processes: in-situ versus dry-merger 253 In Section 3.1, we briefly described how primordial binary formation and dynamical and 254 secular processes can affect the formation of stellar and CO binaries. The dominance of one 255 process over another intrinsically depends on the environment. For example, the density and 256 velocity dispersion in the nucleus strongly affect the interaction rate of few-body interactions, 257 whilst matter density distribution determines how likely is for a star to orbit within a given 258 distance from the central SMBH, thus affecting the development of secular effects like EKL 259 indirectly. 260 In these regards, the presence of a NC in the galactic centre, which is clearly much denser 261 than the stellar distribution of the galactic field, can substantially affect the formation and 262 t relx t 1pN, in t EKL t ev P t rr, s t rr, v t df t bs P bin Figure 2. Timescales of several mechanisms affecting the formation of single and binary compact objects in a galactic nucleus similar to the Milky Way. Here we assume a binary mass M COB = 20M and a supermassive black hole mass M SMBH = 10 6 M . Different lines identify different processes: 2-body relaxation (t relx ), scalar (t rr,s ) and vector (t rr,v ) resonant relaxation, evaporation (t ev ), dynamical friction (t df ), binary-single scattering (t bs ), eccentric-Kozai-Lidov (t EKL ), general relativistic precession (t 1pN,in ), binary period (P bin ), orbital period of the binary about the supermassive black hole (P • ). For the sake of visibility, the plot does not include the AGN lifetime (nearly constant line 1 − 100 Myr), and the three-body and gravitational-wave capture (single-single) scattering time, which are much larger than all other timescales for the adopted example. evolution of both single and binary COs. As we will see in this and following sections, high 263 densities and large masses make NCs potential factories of COs and COBs and might provide 264 a significant contribute to the overall population of GW sources, but assessing the actual 265 properties of "nuclear" COB mergers demands to understand how the macro-physics that 266 regulates NC formation affects the micro-physics that governs COs and COBs in galactic nuclei. 267 NCs are pretty peculiar objects that exhibit a clear flattened morphology and rotation 268 [3,8,[211][212][213][214][215], a complex star formation history and multiple stellar populations [216,217], and, 269 in some cases, harbor a central SMBH [6][7][8]218]. 270 Although the actual fraction of NCs containing an SMBH is rather uncertain [3], this 271 sub-class of NCs represents, as we will discuss in the next section, the ideal place where 272 to study a huge variety of dynamical processes acting simultaneously. Further insights on 273 NCs harbouring an SMBH come from a special class of objects called ultracompact dwarf 274 galaxies (UCDs) [219]. Generally, a UCD is a compact stellar system characterised by a complex 275 star formation history, a central SMBH, and an associated stellar stream [220]. These objects 276 are thought to be the relics of galactic nuclei stripped in a galaxy merger event [221,222], and 277 their properties apparently overlap with those of massive globular clusters and NCs [223]. If 278 UCDs are what remain of NCs stripped away from their parent galaxies, the fraction of NCs 279 harbouring an SMBH could be as large as 75 − 90% [224]. Generally, the mass of NCs and 280 SMBHs seems to linearly scale with the host galaxy stellar mass [1, 225,226]. Moreover, the 281 SMBH-to-NCs mass ratio increases with the galaxy mass [3], and NCs become too faint to 282 be identified in galaxies with masses M g 10 10 M , i.e. when the SMBH-to-NCs mass ratio 283 becomes larger than 1 [7]. 284 These interesting peculiarities may be intrinsically related to the processes that regulate 285 NC formation, which are still partly unknown. 286 There are currently two most debated formation scenarios for NCs: in-situ and dry-merger. 287 In the in-situ scenario, gas funnelled toward the galactic centre, e.g.owing to the effect of the 288 galactic bar or inhomogeneities associated with a galaxy merger, cools and fragments, forming 289 stars in a dense NC [227][228][229][230][231][232][233][234][235][236]. The in-situ scenario may explain some of the features observed 290 in NCs, like rotation and flattening, the development of multiple episodes of star formations, 291 and the absence of NCs in galaxies heavier than 10 11 M , where the radiative feedback of the 292 central SMBH becomes sufficiently strong to quench star formation and hamper the NC growth 293 [e.g. 233]. 294 The basic idea behind the dry-merger formation scenario, instead, is that massive bodies 295 traveling through a sea of lighter particles, like a star cluster moving in the galaxy field, 296 experience a drag, called dynamical friction, that slowly forces it to spiral inward and ultimately 297 collide to form a NC [223,235,[237][238][239][240][241][242][243][244]. The typical timescale of this process can be relatively 298 short (∼ 0.1 − 1 Gyr) [242, 243,245]. As the clusters move inward, the increasing tidal force 299 exerted from the galactic field and possibly from the central SMBH strips their outer regions 300 while their core keeps spiralling in. The competing action between tidal forces and dynamical 301 friction ultimately determine whether spiralling clusters can efficiently build-up an NC. Both 302 observations [7,37], simulations [246], and semi-analytic models [235,242,245] seem to support 303 this picture, suggesting a typical host galaxy virial mass, M g 10 11 M , above(below) which 304 nuclei are dominated by a central SMBH(NC). Therefore, both the in-situ and dry-merger 305 scenario may be able to explain (some) observational features of NCs, making it difficult to 306 identify a single dominant process. In fact, the most recent works indicate that the in-situ 307 and the dry-merger scenario work in concert, with the former possibly leading NC formation 308 in galaxies heavier than 10 9 − 10 10 M and the latter being the dominant channel in smaller 309 galaxies [215,235,[246][247][248][249]. 310 Interestingly, it seems that the average age of NC stars increases at decreasing the host 311 galaxy mass [215,248,249], indicating that NCs in galaxies like the Milky Way likely formed 312 through either a recent star formation burst (in-situ) or the collisions of young massive clusters 313 born close to the galactic nucleus (dry-merger), whilst NCs in dwarf galaxies might be the relics 314 of ancient cluster collisions or star formation occurred during an early stage of the galaxy life. 315 Do NCs form first and SMBHs grow later or viceversa? This basic "chicken or egg" 316 question is still unanswered. From the observational point of view, while SMBHs candidates 317 have been observed up to redshift z 7 [250], it is practically impossible to distinguish a NC at 318 those cosmological distances, thus making impossible to date the oldest and farthest NCs. From 319 the theoretical point of view, some works propose that primordial NCs in high-redshift galaxies 320 can sustain the formation of SMBH seeds via BH merger and accretion events [e.g. 251-253], 321 whose subsequent growth could evaporate the NC. Whilst this possibility may explain the 322 dearth of NCs in the most massive galaxies, it is at odds with their observation in low-redshift 323 (z < 1) galaxies with masses M g < 10 11 M , thus suggesting that there may be different NCs 324 formation processes. Other theoretical works, instead, propose that the SMBH forms first and a 325 NC forms later either via fragmentation of gaseous clouds [233] or star cluster infall [238]. For 326 simplicity, in the following we assume that SMBHs are already fully grown at NCs formation. 327 3.2.1. The impact of nuclear cluster formation scenarios on the population of compact objects 328 in galactic nuclei 329 How can the NC formation scenario affect the formation of single and binary COs? 330 In the case of the in-situ scenario, stellar and CO binaries form either primordially during the star formation process, or dynamically via multiple encounters. The total number of CO progenitors is set by the NC mass and the adopted initial mass function (IMF) of stars, and could be possibly enriched by multiple star formation episodes over cosmic history. In this sense, both COs and their progenitors form in the same environment. Let us consider a NC formed purely through the in-situ mechanism, with a total mass of M NC = 10 7 M and an average stellar mass of m * = 1M . Assuming a typical initial mass function [254], and considering the fact that COs forms from the death of stars heavier than 18M , we expect that COs constitute a fraction f BH ∼ 10 −3 of the whole population. Therefore, the total number of BHs harbored in our in-situ NC should be simply given by
N BH,in = f BH M NC / m * = 10, 000 .(2)
The population of NSs, instead, are significantly affected by two processes. The first is 331 related to the fact that NSs at birth receive a natal kick that in 60 − 90% of the cases exceeds 100 332 km s −1 [see e.g. 255], thus NSs would be promptly ejected even if they formed in a dense NC. 333 Note that the problem of NS natal kicks is still actively debate. Observations of Galactic pulsars 334 suggest that the distribution of NS kicks is either Maxwellian, with a dispersion of σ = 265 km 335 s −1 [256], or bimodal [257][258][259]. From the theoretical standpoint, recent models focused on SN 336 explosion predicts that the kick amplitude depends on the mechanism that trigger the explosion 337 [e.g. 260,261], with the so-called electron-capture SN (ECSNe) possibly being the main source 338 of NS retention in star clusters [262]. Moreover, once BHs settle in the galactic centre owing 339 to mass-segregation, they will prevent the segregation of lighter stellar species, pushing them 340 onto wider orbits [e.g. 133,263]. Since the fraction of stars turning into a NS is expected to 341 be roughly f NS = 10 −3 , we would expect only N NC,in = f ret f NS M NC / m * = 6, 000 − 9, 000, 342 with f ret = 0.6 − 0.9 the NS retention fraction for an escape velocity of 100 km s −1 . Despite 343 being much lighter than BHs, also NSs may follow a cuspy surface density profile with r −1.5 , 344 as shown in [264][265][266]. 345 Let us consider now the dry-merger scenario. 346 In the case in which the galactic nucleus is entirely made up by spiralled star clusters, 347 the total amount of mass in BHs brought there will represent a fraction f BH of the cluster 348 mass. If the BHs were mass-segregated in the parent cluster, it is reasonable to assume that 349 M BH ∼ f BH M c . However, owing to the galactic field, only a fraction f i of the clusters mass 350 will reach the centre and build-up the NC, M NC = ∑ i f i M c . 351 Let us assume that a population of N c = 20 clusters with mass M c = 10 6 M fall in a galactic nucleus and lose a fraction f i = 0.5 of their initial mass [see e.g. 48]. The NC mass will thus be M NC = f i N c M c . In the process, the clusters will bring into the growing nucleus a fraction f BH ∼ 10 −3 of their total number of stars in the form of BHs, because BHs are likely segregated in the cluster core and they will not be affected by the cluster mass-loss process. Thus, the number of BHs lurking in the final nucleus will be equal to
N BH,dry = f BH N cl M cl / m * = 20, 000,(3)
around twice compared to the mass inferred for an in-situ NC, although the difference in the 352 numbers of BHs can be substantially affected by a number of unknown quantities, like the 353 star cluster mass function and galactocentric distribution, the SMBH mass, or the amount of 354 cluster mass actually brought into the galactic centre. In the case of the dry-merger scenario, the 355 timescale over which clusters collide to form an NC can crucially determine whether dynamics 356 and stellar evolution processes have already affected the population of COs in the infalling 357 clusters. This aspect is particularly important for the population of NSs that can be transported 358 into a galactic nucleus through this process. For clusters with escape velocity ∼ 40 km s −1 , 359 it has been shown that only 5-10% of NSs receives a kick sufficiently small to be retained 360 [255,262]. If the infall process proceeds slower than stellar evolution, NSs will form in their 361 parent cluster and most of them will likely be ejected well before reaching the galactic nucleus. 362 If we assume that only a fraction f ret ∼ 0.05 − 0.1, the number of NSs that can be accumulated 363 in a Milky Way-like nucleus is N NS,dry ∼ f ret f NS N cl M cl / m * ∼ 1, 000 − 2, 000, thus a factor 364 4-5 smaller than in the case of the in-situ NC formation scenario. Suppose the infall process, 365 instead, is faster than the stellar evolution process. In that case, stars will evolve into NSs after 366 settling into the galactic center, and their retention fraction will likely be similar to the one 367 inferred for the in-situ process, in which case N NS,dry ∼ 12, 000 − 18, 000.
368
Future observations capable of providing insights onto the population of BHs and NSs at 369 the Galactic Centre could thus provide crucial information about the NC formation history [267], 370 as different formation channels are expected to produce a substantially different population 371 of COs. Clearly, the arguments above serve as an order of magnitude estimate, and a more 372 detailed approach is needed to fully characterise the properties of COs in galactic nuclei and 373 the processes operating there. 374 Given a star cluster with mass M c , orbital radius r c and eccentricity e c , dynamical friction will drag it into the galactic centre over a timescale [243,268]
t df = 0.3Myr R g 1kpc 3/2 M g 10 11 M −1/2 M g M c 0.67 r c R g 1.76 f (e c , γ),(4)
where M g , R g ,and γ represent the galaxy total mass, length scale, and slope of the density profile. The term f (e c , γ) is a function of the infaller orbital eccentricity and the density slope [243]:
f (e c , γ) = (2 − γ) a 1 1 (2 − γ) a 2 + a 3 (1 − e c ) + e c ,(5)
where a 1 = 2.63 ± 0.17, a 2 = 2.26 ± 0.08, and a 3 = 0.9 ± 0.1. It is worth noting that this simple 375 expression for the dynamical friction timescale represents a relatively good approximation also 376 for COs orbiting inside a massive star cluster [263]. 377 As the cluster orbit around the galactic centre, and slowly sink, its internal dynamics will be regulated by several internal processes, the earliest of which will be mass-segregation [209,269], a mechanism driven by dynamical friction [270] by which the most massive stars rapidly segregate toward the cluster centre [269]. The mass-segregation time-scale can be expressed as [210]:
t seg = 0.42 Gyr 10m * m CO t relx 4.2 Gyr r CO R h 3/2 ,(6)
where m * ,CO is the average mass of stars(COs), and t relx is the half-mass relaxation time. 378 Mass-segregation gathers the most massive objects into the inner cluster regions, favouring 379 the development of strong interactions and ejection of the most massive components. This 380 mechanism is particularly effective once BHs have formed and settled in the centre of the 381 cluster. In fact, owing to their cross section, larger than for "normal star", the most massive 382 BHs tend to undergo the strongest interactions in cluster nuclei, pairing together and ejecting 383 each other from the parent cluster in what is called the BH burning process [e.g. 271,272]. 384 As a consequence, internal dynamics will have time to affect the CO population in star 385 clusters with sufficiently short segregation times, i.e. t seg < t df , before they reach the galactic 386 centre and collide to build-up the NC. If the segregation time is even shorter than the timescale 387 of stellar evolution for massive stars, though, interactions among the most massive stars 388 can trigger runaway stellar collisions and possibly the formation of a very massive star that 389 ultimately can collapse in an IMBH [e.g. [273][274][275]. [276], orbiting at a distance of r c = 393 50 − 100 pc in a galaxy with total mass M g = 10 11 M , scale radius R g = 1 kpc, and slope of 394 the density profile γ = 0.5. 395 The plot suggests that the population of massive stellar objects has already been "dynamically processed" in clusters with a mass M c < 10 5 M when they reach the galactic centre, whilst the population in the heavier cluster should be more representative of the cluster's initial population. Let us consider a population of N dec clusters each composed on average of N * = 10 6 members, a fraction f BH of which being in COs, and a fraction η of their COs in binaries. If a fraction δ of all COBs survives the cluster infall, the total number of delivered COBs via dry-merger mechanism will be [48] N dec = 2, 000δ
N dec 20
f BH 10 −3 N * 10 6 η 0.1 .(7)
Note that if the infalling clusters are "dynamically young", meaning that the cluster relaxation 396 time is much shorter than the infall time, their CO population will still be unaffected by 397 dynamical processes and a fraction δ = 0.7 − 0.88 of their BH population can be brought to the 398 galactic centre [48]. 399 Once COs are formed, or are brought, in the galactic centre, their subsequent evolution will 400 mostly driven by dynamics, making it hard to distinguish objects formed in-situ or delivered 401 by infalling clusters. Possible differences may arise in the number and orbital properties of 402 COBs and the mass spectrum of COs. For example, dynamical processes will have had time 403 to substantially affect the population of BHs and NSs in clusters falling into the NC over 404 timescales longer than clusters' dynamical times, likely reducing the number of COs and 405 the average BH mass -owing to the BH-burning process. Exploring how different can CO 406 populations in in-situ or dry-merger NCs is difficult, owing to the underlying uncertainties, 407 and in fact generally in the literature the initial properties of single and binary COs relies upon 408 agnostic guesses. Devising and developing self-consistent NCs models that implement both 409 the NC formation process and the detailed stellar dynamics and evolution is key to shed light 410 on the fingerprints of NC formation history on the population of single and binary COs in 411 galactic nuclei. DF -100 pc DF -50 pc SEG Figure 3. Dynamical friction timescale as a function of the cluster mass assuming that the cluster orbits at 50 pc (green) or 100 pc (red) in a galaxy with total mass M g = 10 11 M , scale radius R g = 1 kpc, and slope of the density profile γ = 0.5. For comparison, we show also the cluster mass-segregation time (grey). The shaded areas embrace the boundaries assuming COs with masses m CO = 5 − 50M . 413 After the NC assembly its stars and COs will inevitably undergo mass-segregation, which can be described in terms of the "individual" dynamical friction time of each CO or massive star from Equation 4:
Early black hole dynamics in nuclear clusters and galactic nuclei
t df = 1680yr R NC 2pc 3/2 M NC 2.5 × 10 7 M −1/2 M NC m BH 0.67 r R NC 1.76 f (e, γ),(8)
where now M NC , R NC , and γ represent the NC mass, length scale, and density slope, respec-414 tively, whilst m BH represents the CO mass and r its orbit. Note that the scaling values adopted 415 above correspond to the Galactic NC. Note that t df is valid also for galaxies without a central 416 NC -in which case one should consider the properties of the bulge in Equation 8 -and can 417 return a more accurate estimate of the mass-segregation time in the case of galactic nuclei, 418 which are often characterised by cuspy density profiles [2]. 419 An important element that needs to be taken into account to estimate how many COs 420 inhabit a galactic nucleus is related to the stellar evolution of COs progenitors, especially for 421 what concern stellar BHs. On the one hand, stellar progenitors are somewhat heavier than 422 their remnants, thus dynamical friction is more effective on them. On the other hand, COs at 423 formation can undergo a recoil, owing to supernova (SN) kick, which can delay their orbital 424 segregation. 425 The motion of a CO, or its progenitor, subjected to dynamical friction can be expressed as [277] r
CO (t) = r CO,0 1 − β t t df 1/β ,(9)
with β = 1.76.
426
From Equation 8, a star sufficiently massive ( 18 − 20M ) starting from 0.1 − 1 pc 427 from the SMBH in a MW-like nucleus will reach the centre in t df ∼ 1 − 80 Myr, thus it will 428 likely reach its last evolutionary stage while travelling through the galactic nucleus. If the 429 newborn CO receives a natal kick, as expected for both BHs and NSs [see e.g. 80,257,258,278], 430 the imparted momentum will suddenly displace the object from its orbit, or even eject it from 431 the galactic nucleus. If the kick amplitude is smaller than the host escape velocity, the CO will 432 eventually return toward the centre over a dynamical friction timescale. 433 In order to provide the reader with a simple view on how the interplay of single star 434 stellar evolution and dynamics can shape the evolution of COs in galactic nuclei we exploit 435 the B-POP population synthesis code [99], which combines stellar evolution models for single 436 and binary BHs obtained with the MOBSE tool [279] and semi-analytic recipes to describe the 437 motion and pairing of BHs via dynamics. Using B-POP , we consider a NC with mass M NC = 438 2.5 × 10 7 M and assume that a fraction ∼ 10 −3 of such mass consists of BH progenitors, 439 assuming that the underlying mass distribution follows a standard initial mass function [254]. 440 For each BH progenitor in the NC, we retrieve the final BH mass, the life-time, and the SN 441 kick. We divide the time in logarithmic bins and calculate the BH progenitor position via 442 Equation 9. As soon as the time exceeds the i-th progenitor life-time, we turn it into a BH 443 (assuming a metallicity Z = 0.0002) and assign a natal kick amplitude v kick , which is based on 444 the stellar evolution recipes implemented in the MOBSE population synthesis tool. Given the 445 kick, we assume that the newborn BH will reach a maximum distance r max in a travel time 446 t tr = r max /v kick , and then it comes back over a dynamical friction time. In order to simplify the 447 visualization of such complex system, we present in the sketch in the left panel of Figure 4 small, owing to the relatively small kick amplitude compared to the NC velocity dispersion, 452 suggesting that in a MW-like nucleus mass segregation is practically accomplished over a time 453 span of ∼ 10 7−8 yr. 454 The population of BHs accumulated in the galactic centre will be characterised by a 455 steeper density profile compared to normal stars and possibly by larger densities [268,280,281], 456 a feature that can crucially affect COB formation. Once BHs settled in, they can efficiently 457 evacuate lighter objects [e.g. 263], removing normal stars but also other COs like white dwarfs 458 and NSs, and hampering the possible formation of double NS or NS-BH binaries.
459
This shielding operated by BH dynamics can be alleviated by the BH burning process, 460 opening the possibility to NS-BH binary formation over a few relaxation times [130,136]. 461 In the next section we will describe how COs and especially BHs interact once they have 462 gathered into the innermost regions of the host galaxy centre. 463 464 5.1. Galactic threats: what binaries can survive around a supermassive black hole? 465 Regardless the formation scenario, in a dense environment around an SMBH, a binary 466 system frequently encounters other stars [e.g., 30,203,210,[282][283][284][285][286][287][288]. This process results in a 467 variation of the angular momentum and energy of interacting stars or COs with passing neigh-468 bours, displacing them from their position and forcing them to wander around the SMBH [e.g., 469 203]. When a passing star or a compact object approaches the binary with impact parameter on 470 the order of the binary's separation, it interacts more strongly with the closer binary member. 471 The outcome of this interaction depends on the ratio of the binary's gravitational binding 472 energy to the kinetic energy of the neighboring stars. We will discuss in detail the possible 473 outcomes of these "binary-single" interactions in Section 5.3.3. 474 Generally, the "resistance" of a binary against a strong interaction with another object can be quantified via the softness parameter [e.g., 282,289]
Dynamical formation of black hole and compact object binaries in galactic nuclei
s = E bind m * σ 2 < 1 Soft > 1 Hard(10)
where E bind = Gm 1 m 2 /(2a bin ) is the binary binding energy, m * is the average mass of the 475 objects (either stars or compact objects) in its vicinity and σ is the velocity dispersion of the 476 environment. If the softness is larger(smaller) than unity, the binary is labelled as hard(soft), 477 and a strong interaction will harden(soften) the binary further, a mechanism known as the 478 Heggie's law [282,286]. 479 In the case of hard -or immortal [237] -binaries, interactions become rarer and more 480 violent as they get harder and harder, until either the binary is kicked out from the environment 481 or, in the case of COBs, GW emission kicks in and drives them toward coalescence [127]. If 482 the hardening process involves massive stellar binaries and occurs over a timescale shorter 483 than the typical timescale of massive star evolution, i.e. before the formation of COs, it can 484 determine the onset of stellar collisions, which in turn can result in the formation of final BHs 485 with masses larger than expected, possibly falling in the so-called pair-instability mass gap [e.g. 486 51,99,111,128,290]. 487 Soft binaries, instead, are likely subjected to disruption, either via a single strong interaction, if the perturber passes at a distance comparable to the binary semimajor axis, or via a series of distant flybys [see 210]. In soft-binary flyby interactions, the passing object increases the energy of the binary system and widen the binary, eventually leading to its evaporation over a timescale given by the ratio between the energy gained/lost by the binary and the rate of change of the kinetic energy, or diffusion energy coefficient, which can be expressed as [205,210,277,291]:
t ev = √ 3σ 32 √ πGρ * ln Λa bin m bin m * = (11) = 1 Gyr M SMBH 4.3 × 10 6 M 1/2 r 0.5 pc −1/2 log Λ log(15) −1 × × ρ 2.8 × 10 6 M pc −3 a bin 1AU −1 m bin 30M m * 1M −1
and depends on the binary semi-major axis (a bin ), eccentricity (e bin ), as well as on several envi-488 ronmental properties, like the stellar density, ρ * , the velocity dispersion, σ = (GM SMBH /r) 1/2 , 489 and the average stellar mass, m * . 490 From the softness parameter defined above, we can determine a critical binary semimajor axis, a hard , that separates hard and soft binaries:
a hard = 2Gm bin σ 2 = 2m bin M SMBH r,(12)
where the latter equality represents a valid approximation at a distance r to an SMBH, where 491
σ 2 ∼ M SMBH /r. 492
Assuming a semi-major axis distribution flat in logarithmic values between 0.01 − 10 3
493
AU [13,127,292] implies that ∼ 30%(70%) of binaries are hard(soft) at the Galactic Centre 494 (r < 0.1 pc). Within r < 0.1 pc from SgrA*, the Galactic SMBH, binaries with a semimajor axis 495 a bin > 0.29 AU are soft, thus their existence will be endangered by the effect of the closeby 496 SMBH. 497 However, a typical equal-mass circular binary with mass m bin = 30M and semi-major 498 axis a bin = 0.1 AU orbiting at a distance r = 0.1 pc from SgrA*, evaporates in t ev = 24 Gyr, 499 thus, even though in a soft state, the binary could still survive in the dense environment of the 500 Galactic Centre, despite a single, strong encounter can disrupt it. Having typical binary masses in the range m bin = 3 − 5M , double NSs can, instead, 502 evaporate over a timescale shorter than a Hubble time, taking only t ev = 2 − 4 Gyr in the 503 aforementioned example. 504 Note that the fraction of binaries with a semi-major axis smaller than a hard in the case of a logarithmically flat distribution is given by
f (< a hard ) = log(a hard /a min ) log(a max /a min ) = 1 log(a max /a min ) log r a min m bin M SMBH ,(13)
which implies that the fraction of hard binaries diminishes closer to the SMBH. Assuming 505 an SMBH of mass M SMBH = (1 − 4.3 − 10) × 10 6 M and a min,max = 0.01 − 10 3 AU, the 506 percentage of hard binaries drops below 10% at a distance r 0 = (0.015 − 0.067 − 0.15) pc. 507 The depletion of hard binaries toward the galaxy centre has been also demonstrated via 508 self-consistent N−body models of the Milky Way nucleus [281].
509 Figure 5 shows how the hard binary fraction varies as a function of the distance to the 510 SMBH for the Milky Way centre, highlighting how only a tiny fraction of the most massive 511 binaries can still be hard within 1 − 10 mpc from the SMBH. At those distances, though, the 512 hard binary separation is a h ∼ 1 R (M SMBH /4.3 × 10 6 M )(r/1 mpc) −1 , thus it is highly 513 likely that a stellar binary would quickly merge. 514 Additionally, the evaporation process depends on the binary's eccentricity around the SMBH [e.g . 203]. In the case of soft binaries, each new interaction causes an increase of the semi-major axis with time, due to the single-fly by interactions. Additionally, in the case of soft stellar binary mass loss can further widen the orbit. Thus, the evaporation time of soft binaries changes with time. If we assume that the binary softens over time, we can find an upper limit on the evaporation time:
t ev,max = t ev S h(14)
where
S h = s h s 0(15)
represents the possible binary history, namely its semi-major axis evolution [289]. In particular, s 0 is the softness parameter calculated when the binary is observed and s h represents the hardest possible initial configuration. In other words,
s h = min[1, s(a bin = R 1 + R 2 )] ,(16)
where R 1,2 are the initial ZAMS stellar radii, assuming that the two stars were originally paired. 515 The hardest limit taken in Equation (16), is when the binary begins as a contact binary [289]. 516 The equations above for the evaporation timescale are derived under the assumption that the binary moves on a circular orbit about the SMBH. However, many of the stars in the Galactic Centre are in fact on an eccentric orbit [e.g. [37][38][39]44,206,207,[293][294][295]. A binary on an eccentric orbit may pass through a denser, inner regions of the Galactic Centre unlike a binary on a circular orbit. Therefore, extra term should be introduced (see for full derivation Ref. [203]), namely:
t ev,Ecc = t ev f (e SMBH ) (17) 517 f (e SMBH ) = (1 − e SMBH ) α+ 1 2 2 2 F 1 1 2 , − 1 2 − α; 1; 2e SMBH e SMBH − 1 + (1 + e SMBH ) α+ 1 2 2 2 F 1 1 2 , − 1 2 − α; 1; 2e SMBH e SMBH + 1 ,(18)
where 2 F 1 is the hypergeometric function. 518 Binary hardening and softening processes are highly sensitive to the underlining density 519 profile of the surrounding galaxy or NC [e.g. 48,203], with cusp-like density profiles leading to 520 more expedite hardening and softening processes. 521 In the next section, we will discuss in detail what dynamical processes intervene in the 522 formation of binaries in galactic nuclei. 523 5.2. Moving through a swarm: orbital evolution of compact binaries in galactic nuclei 524 As we have discussed in the previous section, binaries are more likely to orbit in an outer 525 layer of the galactic nucleus, where the effect of the central SMBH is less disruptive. 526 This has two crucial implications on the binary dynamics. On the one hand, the binary will 527 drift toward the galactic centre owing to dynamical friction. On the other hand, while migrating 528 the binary will sweep through regions at increasing density and velocity dispersion. This will 529 affect the boundary between hard-soft binaries, and can either boost the binary shrinkage or 530 cause its evaporation (or ionization). 531 The binary infall rate can be described via the dynamical friction timescale as:
dr dt = − r t df ,(19)
while the binary hardening rate due to binary-single interactions is given by [127,296] da
BBH dt = −H Gρ(r) σ(r) a 2 BBH .(20)
If the nucleus density is ρ ∼ r −γ we can express the hardening timescale as
t hard 1 a 2 BBH da BBH dt −1 ∝ r γ−1/2 M 1/2 SMBH ,(21)
thus the heavier the SMBH the smaller the hardening, and the steeper the density profile, for 532 γ > 1/2, the larger the hardening as the binary migrate inward. Note that values of γ < 1/2 533 produce an unphysical distribution of energies for a matter distribution around an SMBH [297], 534 and would imply that the hardening drops closer to the SMBH.
535
The fact that the hardening process proceeds at an increasing pace might have crucial 536 consequences on the late evolution of the binary. For example, also the hard-binary separation 537 changes getting closer to the SMBH, thus despite the increasing hardening a binary could still 538 become soft in some regions of the nucleus. Figure 6 shows the evolution of the semimajor axis 539 normalised to the hard binary separation (a BBH /a hard ) and distance to the SMBH (r/r 0 ) for a 540 BBH with mass M BBH = 30M and different values of a BBH . 541 It is quite evident that, depending on the binary properties and its initial position within 542 the nucleus, as the binary gets closer to the SMBH it can become softer and softer, and from 543 that point the evaporation time will represent a rough estimate of the binary lifetime. Note that 544 the a BBH /a hard increase is only due to the environment, which becomes denser and hotter. 545 Moreover, as the binary gets closer to the SMBH, the increasing tidal field can exert 546 important effects on the binary evolution, which we review in the next section. The accumulation of BHs toward the cluster centre will maximise the level of contraction 550 of the cluster central regions, a process known as core-collapse that ideally drives the central 551 density to grow up to infinity. The density increase onset the formation of binaries and multiple 552 systems that, strongly interacting with each other and with single object. Binaries thus act as 553 "heating sources" for the cluster nucleus, efficiently transferring energy to lighter single and 554 multiple stars and ejecting them from the inner cluster regions. As a consequence, the cluster 555 centre expands, causing a reduction of density and velocity dispersion that leads to a significant 556 reduction of the interaction rate, reversing the core-collapse process. The combination of 557 mass-segregation and core-collapse results in a different density distribution for COs, steep 558 and more concentrated, and stars, much shallower and sparse [see e.g. 268,281,298].
559
The energy exchange promoted by mass-segregation translates into a redistribution of 560 energy among the objects with different masses that can trigger the formation of bound binaries 561 via few-body interactions.
562
The earliest studies about the formation of binaries in dense stellar environments date 563 back to early 1960s, when the first Fokker-Planck, Monte Carlo, and N-body models were 564 developed [e.g. 11,282,[299][300][301][302][303][304][305][306][307][308]. 565 These pioneering experiments were developed to understand the complex evolution 566 of star clusters, and shed light on the fundamental dynamics regulating the formation and 567 disruption of binaries in star clusters and the feedback that binaries have on the whole cluster 568 evolution. As we will see in the next sections, such fundamental dynamics can be generally dissected 570 into several processes, namely single-single GW captures and three-body, binary-single, and 571 binary-binary scatterings. Aside three-body interactions, a binary can form also via GW bremsstrahlung during a single-single interaction, provided that the two objects come sufficiently close. The maximum impact parameter below which two BHs pair can be calculated by equalling the potential energy between the interacting objects and the energy lost to GWs during the closest passage, and can be expressed roughly as [265,309]
b bnd = 2.4R σ 100km s −1 −9/7 m 1 + m 2 10M η 0.25 ,(22)
where η = m 1 m 2 /(m 1 + m 2 ) 2 is the asymmetric mass-ratio. , where we assumed σ = (M SMBH /r) 1/2 and n = n 0 (r/r 0 ) γ . Figure 7 shows the GW capture 574 timescale for different values of the number density of BHs in the nucleus and the SMBH mass. 575 It appears evident how this process can operate quickly close to the most dense nuclei and 576 the most massive SMBHs. At formation, the pericentre of a GW-capture binary is typically 577 r p 7.4 × 10 −3 R (σ/100km s −1 ) −4/7 , thus these binaries are characterised by extremely short 578 merging times, order of minutes to hours, and their merger rate will directly follow the binary 579 formation rate, which is expected to be in the range 0.2 − 150 Gyr −1 [265].
580
As we will discuss further in Section 8, among all the channels for BBH formation in 581 galactic nuclei, the GW-capture enables the production of highly eccentric sources (e > 0.1) 582 in the typical frequency band where the sensitivity of ground-based interferometers like LVC 583 peaks, i.e. f > 1 − 10 Hz. By definition, a three-body interaction involves 3 unbound objects scattering onto each other. Statistically speaking, this type of interactions leads to the ejection of the least massive object and the formation of a bound pair. The formation rate for three-body binaries can be obtained by equalling the rate at which new binaries form and soft binaries are ionised via multiple or single interaction, and can be expressed as [11,311]
dn 3bb dt = α(Gm) 5 n 3 σ −9 ,(24)
where α is a normalization constant, m is the average mass of the interacting bodies, n is the 586 environment number density, and σ its velocity dispersion. The timescale associated to this 587 process, t 3bb ∼ m −5 n −2 σ 9 , immediately highlights how crucial is the environment, and its 588 evolution, in determining binary formation via three-body interactions.
589
The binary formation rate above is derived under the assumption that only hard binaries 590 are "immortal" [282]. However, in environments with a particularly large velocity dispersion, 591 thus a small hard-binary separation limit, soft binaries can harden via GW emission and cross 592 the soft-hard boundary, provided that the hardening time is shorter than the evaporation time. 593 This mechanism to harden soft binaries works for particularly dense nuclei where encounters 594 with a relative velocity close to the speed of light (v rel 0.3c) become possible [312].
595
At a distance r from an SMBH, the velocity dispersion is Keplerian, σ 2 ∝ M SMBH /r, 596 whilst the matter density is expected to follow a power-law [298,[313][314][315][316][317][318], n ∝ r γ , with 597 γ = −3/2 − m/4m max , and m(m max ) the average(maximum) stellar mass [317,319]. For a 598 monochromatic mass spectrum, the matter density around an SMBH follows the so called 599 Bahcall-Wolf cusp, with slope −7/4 [313]. In a multimass system, stellar BHs, which are the 600 heaviest objects in a stellar ensemble, are expected to distribute according to a Bahcall-Wolf 601 cusp or even a steeper one [265, 266,298,318,319], whilst lighter stars follow a shallower density 602 distribution [264,280,281,320]. 603 The energy transfer from the heaviest to the lightest object progressively leads to a decrease 604 of the heavy object velocity dispersion owing to the equipartition principle. Mass segregated 605 BHs are characterised by a velocity dispersion σ BH /σ * ∼ 1/ζ(m BH /m * ) −η , with ζ ≤ 1. In 606 the ideal case of "perfect equipartition", η = 0.5 [209], however simulations have shown that 607 equipartition is hard to reach in dense clusters, even in presence of a central massive object, 608 as shown for globular cluster models hosting a central IMBH, which suggest η = 0.08 − 0.15 609 [321].
Given the dependencies above, the three-body timescale can be re-written as Figure 8 shows the BH binary fraction (over the total population of BHs) as a function of 618 the distance to the Galactic Centre after a time t = 0.1 − 10 Gyr, assuming an average BH 619 mass m BH = 15M . The figure makes clear that a pure three-body formation process is 620 highly inefficient in the innermost Galactic regions, owing to the large velocity dispersion that 621 hampers three-body interactions. Nonetheless, it also highlights that, at distances 1 − 10 pc the 622 BBH fraction attains values f BBH ∼ 10 −3 − 10 −2 , which could imply the formation of a few tens 623 BBHs over a 10 Gyr time-span, but how many? Let us assume that the inner pc in the Milky 624 Way contains around N BH (< 1pc) = 20, 000 BHs [53] and that the BH density distribution at 625 the Galactic Centre is represented by a simple broken power-law [323], with a total number 626 of BHs N BH = 10 −3 N NC , a scale radius of a = 0.1 pc and density slope γ = 2 1 , it is possible 627 to show that at a distance 1 − 10 pc from the Galactic Centre we expect O(10 3 ) BHs, thus the 628 number of binaries formed in the Galactic NC in 10 Gyr solely via three-body scattering would 629 be N BBH ∼ 2 − 20. A way to increase the number of BBHs is via exchange in binary-single 630 scatterings involving a binary star and a single BH [e.g. 127,277].
t 3bb = (Gm BH ) −5 n −2 BH σ 9 BH =(f BBH = 1 2 1 − 1 √ 1 + t/t 3bb .(26)
631
For the NC in the Milky Way and CO average mass of m BH = 30M , the three-body 632 timescale is shorter than the dynamical friction timescale at distances r > 3.8 pc, thus suggesting 633 that binaries containing at least one BH could form in the outer NC regions. other mechanism to sustain their binary population, or need to harbor a substantial fraction of 639 "primordial" hard binaries.
640
It must be also noted that the steep dependence on the density and velocity dispersion 641 makes the three-body time estimates extremely susceptible to the choice of initial conditions, 642 as suggested by Figure 9, which shows how t 3bb varies at varying the SMBH and NC masses 643 and the distance to the galactic centre. This owes to the fact that t 3bb ∝ M −2 NC M 9/2 SMBH , thus 644 reducing the SMBH mass by 5 times causes a reduction of the three-body time by a factor 645 ∼ 1400. For the sake of comparison, note that for BHs with mass ∼ 10M in a typical globular 646 cluster, with density n ∼ 10 4 pc −3 and velocity dispersion σ = 5 − 10 km s −1 , the three-body 647 timescale reduces to t 3bb ∼ 0.004 − 2 Gyr. The presence of even a few binaries in the nucleus, either primordial or formed dynami-650 cally, will lead to a series of binary-single interactions with other members.
651
The cross-section of a strong encounter between a binary with mass m bin and semimajor axis a and a perturber m p from the binary can be expressed as The number of interactions per time unit that the binary will undergo while moving with velocity σ in an environment with stellar density n is dn nΣσdt. The associated time-scale of this process, t bs = (nΣσ) −1 , can be conveniently written as [13,127]: If the pertuber is a BH whilst the binary contains stars, the interaction will take place 653 proportionally to the fraction of binaries present in the cluster. In such a case, the binary will 654 likely acquire the BH if it is more massive than at least one component. In this case, Equation 655 28 remains valid even in the case of binary-binary interactions, provided that the perturber 656 mass is replaced with the typical binary mass. Binary-binary interactions represent a viable 657 way to produce CO triples. On the one hand, triples can undergo secular evolution, with the 658 least bound object -or outermost -possibly impinging EKL oscillations onto the most bound 659 pair, eventually driving it to coalescence [159,163,324]. On the other hand, triples formed out 660 of binary-binary interactions can undergo a chaotic unstable evolution that, in some cases, 661 can trigger the formation of highly eccentric merging binaries [159,163]. Figure 10 shows one 662 example of BBH-BBH strong scattering from numerical simulations described in [163]. The 663 plot highlights how, after the encounter between the two binaries, one of the BHs -the lightest 664 -is ejected away and an unstable triple forms. The triple eventually breaks-up and lead to 665 the formation of a tight binary that merges after 10 2 yr [see Figure 6 in 163]. The presence 666 Figure 11. Merging NS-BH binaries formed via binary-single encounters in dense clusters and corrected for ground-based GW detectors. Vertical lines correspond to the median value (dashed line) and 95% confidence level boundaries (stright lines) of the mass of GW190814 source components. Taken from Figure 4 in Arca Sedda, ApJL, 2021, 908, L38 ©AAS [136]. Reproduced with permission. of a SMBH in the galactic centre can significantly influence triple evolution, tidally limiting 667 them and enabling only the formation of triples whose maximum size remains within the 668 triple Roche lobe [325], and possibly boosting the COB merger rate owing to a more efficient 669 development of EKL mechanism [326]. 670 If, instead, the BH is already in a binary, it can interact with all the stars in the nucleus 671 and the actual binary-single scattering time becomes t bs,BBH ∼ f bin t bs , thus BHs in binaries can 672 undergo binary-single interactions at a rate up to 100 times larger than single BHs.
Σ = πa 2 (1 − e) 2 1 + 2G(m bin + m p ) 3σ 2 a(1 − e) .(27)t
673
If NSs or BHs are paired in a binary already, either because of previous dynamical processes 674 or owing to primordial stellar evolution, it has been shown that binary-single interactions 675 are particularly efficient at producing merging NS-BH binaries [130,132,133,136], especially in 676 dense galactic nuclei [130]. The typical properties of dynamical NS-BH mergers formed this way, 677 or at least of a sub-population of them, differ significantly from the observed properties of NS-678 BH binaries formed in isolation, permitting to clearly identify their origin in GW observation 679 [130,136, but see also Section 8]. Figure 11 shows the mass spectrum of merging NS-BH binaries 680 formed via binary-single interactions in dense clusters with velocity dispersion in the 10 − 100 681 km/s range, for two different values of the stellar metallicity. The distribution takes into 682 account observational biases, in particular the fact that LIGO-Virgo detectors can access a larger 683 volume for nearly equal mass binaries with large masses in comparison to low-mass or highly 684 asymmetric binaries [136,327].
685
The typical outcome of a binary-single encounter depends on the ratio between the energy transferred during the encounter, which is roughly given by ∆E bs ∼ (r p,e /a) 3/2 [282,328], where r p,e represents the minimum binary-perturber separation, and the kinetic energy calculated in the centre of mass of the triple, i.e.
[328]
∆E c = v 2 ∞ m p m bin 2(m p + m bin ) ,(29)
with v ∞ the velocity at infinity of the perturber in the binary reference frame. If the energy 686 transferred is small, i.e. ∆E bs < ∆E c , the binary has two possible fates, depending on its 687 binding energy E bin : 688 i if ∆E bs < E bin , the binary will harden or soften depending on the environment;
ii if ∆E bs > E bin , the binary will exchange one component, most likely the least massive, if 690 the perturber is heavier than the binary or its components, i.e. m p > (m 1 + m 2 ) or at least 691 m p > m 1,2 .
692
If the binary preserves its components, the pertuber will recede to infinity with velocity 693 v p < v ∞ , otherwise the former binary component with mass m exc will escape with velocity 694 v p < (m p /m exc )v ∞ .
695
In the case of a high-energy transfer, ∆E bs > ∆E c , the outcome of the scattering is a bit more complex, but can be determined statistically by comparing the interaction velocity v ∞ and the critical velocity value above which the perturber can promptly ionize the binary, defined as [329] v
2 c = Gm 1 m 2 (m p + m bin ) am p m bin .(30)
In such a way, the possible final states can be categorized as follows: ii if v ∞ < v c and ∆E bs < E bin and m p < m bin the system undergoes a resonant interaction 703 that generally leads to the ejection of the lighter component;
696 i if v ∞ < v704 iii if v ∞ > v c the binary undergoes: 705 - component exchange if ∆E bs < E bin , 706 - ionization if ∆E bs > E bin .
707
For a BH population with equal mass m BH and a binary population with semimajor axis 708 a BBH = 1 AU, the critical velocity is v c ∼ 50 km s −1 . The speed of the encounters around an 709 SMBH, instead, is σ SMBH = 65km s −1 M SMBH /10 6 M 1/2 (r/1pc) −1/2 .
710
This implies that the binary-single interactions inside the SMBH influence radius will be 711 characterised mostly by v ∞ ∼ σ SMBH > v c .
712
Depending on the masses involved and the semimajor axis of the binary, it can be shown 713 that the energy transferred in a binary-single encounter rapidly drops when σ/v c > k, where 714 k ∼ 0.1 − 1, whilst the critical energy transfer ∆E c increases with (σ/v c ) 2 . Thus it seems 715 reasonable to expect that in the closest vicinity of an SMBH binary-single encounters tend to 716 favour small energy transfer. Since the binary in these encounters is likely hard, thus E bin ∝ σ 2 , 717 binary-single interactions close to an SMBH might result statistically in the binary hardening, 718 i.e. case ∆E bs < ∆E c and ∆E bs < E bin . In other words, component exchange and binary 719 ionization might be significantly suppressed in galactic nuclei unless the binary semimajor 720 axis is close to the hard-soft separation value [see e.g. 277]. This simple line of thought find 721 support in recent scattering simulations tailored on binary-single scattering around a MW-like 722 SMBH [see e.g. 325]. After the interaction, the binary will recoil at a velocity v rec = kσ [127], 723 with k 0.3 for a monochromatic BH mass spectrum. stable binary have a tighter orbital configuration than the orbit of their mutual centre of mass 731 around the SMBH. In such a system, gravitational perturbations from the SMBH can induce 732 large eccentricities on the binary orbit, which can cause the binary members to merge. Below 733 we outline the relevant physical processes that characterise the COB-SMBH co-evolution. In the three-body approximation, dynamical stability requires that the system has either 736 circular, concentric, coplanar orbits or a hierarchical configuration, in which the inner binary, 737 with semimajor axis a 1 , is orbiting a third body (in this case the SMBH) on a much wider 738 orbit, the outer binary, with semimajor axis a 2 . In such hierarchical configurations it is possible 739 to apply the so-called secular approximation. In essence, this means that orbital period can 740 be averaged on, thus the interactions between two non-resonant orbits are equivalent to 741 treating the orbits as massive"wires" where the line-density is inversely proportional to orbital 742 velocity. Under this approximation, the gravitational potential is expanded in the ratio of 743 orbital separations -which, in this approximation, remains constant, since (a 1 /a 2 ) 1 [e.g. 744 14-16, 21,330].
745
The Hamiltonian of such a system can be written as:
H = − Gm 1 m 2 2a 1 − GM SMBH (m 1 + m 2 ) 2a 2 − R pert ,(31)
where the first two parts represent the Keplerian energies of the two orbits, and R pert is the pertrubation function [331]. The perturbation function can be written as a function of the orbital (Delaunay's) elements, the arguments of periastron, ω 1 and ω 2 , the longitudes of ascending nodes, Ω 1 and Ω 2 , and the mean anomalies µ 1 , and µ 2 for the inner and outer orbits, respectively. In particular:
R pert = G a 2 ∞ ∑ n=2 a 1 a 2 n µ n r 1 a 1 n r 2 a 2 n+1 P n (cos ψ) ,(32)
where r 1 (r 2 ) is the position radius of the inner (outer) orbit, P n are the Legendre polynomials, 746 and ψ is the 3D angle between the position vectors of the inner and outer orbits.
747
The nominal procedure in such a system is perform a canonical transformation to a set 748 of coordinates that eliminates the short-period angles of the inner and outer orbits. This 749 transformation is known as the Von Zeipel transformation, see for details appendix B in 750 Ref. [331].
751
The resulting (often called double-averaged) Hamiltonian is thus a multiple expansion of 752 a 1 /a 2 . The lowest order of approximation, (proportional to (a 1 /a 2 ) 2 ) is called the quadrupole 753 level, and corresponds to n = 2 in Eq. (32).
754
In early studies of high-inclination, hierarchical, secular perturbations [14,15], the outer 755 orbit (in this case, about the SMBH) was assumed to be circular, with one of the inner binary 756 members being a test (massless) particle. In this scenario, the component of the inner orbit's 757 angular momentum along the z-axis (set to be the total angular momentum) is conserved 758 -known as the Kozai's integral of motion-and the lowest order of the approximation, the 759 quadrupole approximation, is valid.
760
In this regime, the conservation of the vertical angular momentum and energy implies that the eccentricity and mutual inclination vary along the orbit. An illustrative case is represented by an initially circular binary inclined by an angle i 0 with respect to the binary-SMBH orbital plane, in which case the maximum eccentricity is given by [e.g. 19,188,332] e max = 1 − 5 3 cos 2 i 0 .
The eccentricity excitation is triggered above a minimum value of the initial inclination, given by cos i min = ± √ 3/5, thus implying that only orbits having 39.2 • < i min < 140.77 • can undergo eccentricity variations. The timescale for KL oscillations to develop is [e.g. 333] t KL = 16 30π
m 1 + m 2 + m 3 m 3 P 2 2 P 1 (1 − e 2 2 ) 3/2 ,(34)
761
About a decade ago, it has been shown that relaxing either one of these assumptions leads 762 to qualitatively different dynamical behaviors (already at the quadrupole level). Considering 763 systems beyond the test particle approximation, or a circular orbit, requires higher terms in 764 the Hamiltonian, called the octupole-level of approximation, proportional to (a 1 /a 2 ) 3 , i.e., 765 n = 3 in Eq. (32), [e.g. 16 -18,330]. In the past years, the octupole-level approximation was 766 proven to have an important role in the evolution of many triple configurations, for different 767 astrophysical settings [e.g. 19,20,188,205,291,331,332,[334][335][336][337][338][339][340][341][342][343][344].
768
In the octupole level of approximation, the inner orbit eccentricity can reach extremely high 769 values [17,331,338,345]. This process was coined as the eccentric Kozai-Lidov (EKL) mechanism 770 [21]. The hallmark of the EKL mechanism results in large eccentricity peaks, chaotic behaviour, 771 and also flips the inclinations of both the inner and outer orbits from prograde to retrograde 772 [19,332,334,[345][346][347][348]].
773
The quadrupole-level of approximation relates to the low-level resonance that causes 774 precession of the orbits and thus results in excitation of the orbital inclination and eccentric-775 ity [e.g. 331, [346][347][348][349]. The higher level approximation, results in resonances that often are 776 overlapping. These resonances result in extreme eccentricity values as well as flips and chaos 777 [e.g. 19,331,332,334,345,346,348]. These extreme eccentricities are valuable for the mergers of 778 compact objects. In Figure 12, left side, we show a representative example of the effect of 779 octupole on merging binary. In general, octupole effects shorten the timescale and thus have 780 significant effect on the merger rate [291], and see Figure 12 left panel for example. Aside two-body relaxation, which we discussed in the previous section, and the secular 783 effect of the SMBH field onto binary members, different physical and astrophysical processes 784 can contribute to the overall evolution of the system. These include short range forces between 785 the binaries as well as dynamical interaction with the neighboring objects in NCs. Below, we 786 briefly review these processes and their affects.
787
Tidal Dissipation: Considering the full evolution of binaries (i.e., starting from main-sequence 788 stars), requires the inclusion of tidal dissipation. As the system evolves, the inner orbit can 789 become highly eccentricity which could, on one hand drive the inner binary to merge, while on 790 the other hand, could allow tidal forces to shrink and circularize the orbit. The latter happens if 791 during the evolution the tidal precession timescale (or the general relativistic, GR, timescale) is 792 similar to that of the lowest EKL timescale, which corresponds to the quadrupole. Thus further 793 eccentricity excitations are suppressed [e.g. [349][350][351] and tides shrink the binary separation 794 forming a tight binary that is decoupled from the SMBH. Conversely, if the eccentricity is 795 excited on a shorter timescale than the typical tidal (or GR) precession timescale the orbit 796 becomes almost radial, and the stars can cross the Roche-limit. In that case, the tidal precession 797 does not have enough time to affect the evolution.
798
Most studies in this field use the equilibrium tides models [352,353]. The strength of the 799 equilibrium tide model is that it is self-consistent with the secular approach used throughout 800 this study. Further, assuming that the stars are polytropic, this model has only one dissipation 801 parameter for each member of the binary. Using this tidal description we are able to follow the 802 precession of the spin of each star in the binary due to the stars' oblateness and tidal torques 803 [e.g. 350]. The disadvantage of using equilibrium is that for sufficiently large eccentricity, they 804 tend to underestimate the efficiency of the tides compared to chaotic dynamical tides [e.g. 805 354,355]. 806 While the equilibrium tidal model seems to roughly consistent with the qualitative be-807 havior of polytropes with convective envelopes, tides for radiative stars are estimated to be 808 much weaker than for convective stars. Thus, within this framework a different tidal model for 809 (radiative) main-sequence and (convective) red-giant stars is invoked [e.g. 356,357]. Within the 810 context of EKL evolution only a few studies began including the less efficient radiative tides 811 [e.g. 59, 342,344,358,359].
812
Overall tides between the binary members largely influence the EKL evolution of the stars. 813 It tends to shrink and circularize the orbit [e.g. 205,350], thus hardening the binary to flyby 814 interactions [e.g. 203,360]. During the binary stellar evolution, if the eccentricity spikes take 815 place faster than the tides can suppress the high eccentricity values, the binary star may merge, 816 (typically of few ×10 M ), and is speculated to form a G2-like object [e.g. 59, 205,361,362]. 817 Surviving stars undergo a combination of EKL, tides, GR, and stellar evolution (see below). 818 Stellar evolution Stellar evolution plays an important role for the evolution of binaries and 819 especially massive binaries. Specifically, it was shown that mass loss can have a significant 820 effect on the dynamical evolution of binaries and triples [e.g. 188,205,337,[363][364][365][366]. For near 821 equal-mass binaries the mass loss associated with the AGB phase can re-trigger the EKL 822 mechanism, either by changing the mass ratio, or by expanding the semi major axis of the inner 823 orbit faster than that of the outer one [337,367]. 824 Often, in the literature, the post-main sequence evolution model is adopted from a com-825 bination of stellar evolution codes, such as BSE/SSE [154] and MESA [368], which are publicly 826 available. Furthermore, once the binaries cross each other Roche-limit the binary stellar evolu-827 tion is often followed using COSMIC binary stellar evolutionary code [369]. 828 The effects of supernova explosion and asymmetric and/or instantaneous mass loss (i.e., 829 pulse-like, instantaneous, relative to the secular timescale) can affect the orbital parameters of 830 massive binaries [e.g. [370][371][372][373] and triples [e.g. 374,375]. Sudden mass loss can cause a rapid 831 change of the mass ratio, but more importantly, it can change the eccentricity of the inner and 832 outer orbit due to a supernova kick.
833
In particular, there is a variety of dynamical outcomes resulting from SN kicks, in binaries 834 at the centre of galaxies. For example, it can result in hypervelocity star [e.g. 376] and binary 835 candidates [e.g. 374,377], as well as x-ray binaries [377]. Finally, GW events triggered by SN 836 kicks can result in binary merger events, and EMRIs [e.g. 374,376,377].
837
General Relativity precession, 1st pN In the 1st post Newtonian (pN) approximation the inner (and outer) binary exhibit an additional precession of the nodes. The typical timescale for this mechanism can be written as [18,378]
t 1pN = a 5/2 1 c 2 (1 − e 2 1 ) 3G 3/2 m 3/2 bin (35) =103 yr m bin 30M −3/2 a 1 0.1 AU 5/2 1 − e 2 1 .
If the precession timescale of the inner binary is shorter than the quadrupole timescale, 838 eccentricity excitations can be suppressed [e.g. 17,20,351,379]. If the GR timescale is comparable 839 to the octupole time scale, the system can be excited to larger eccentricities than the ones 840 We will see in Section 8 that these effects may leave an imprint detectable by future GW 849 detectors, like LISA.
850
During the EKL mechanism, the binary's spin axes undergo chaotic evolution, similarly to 851 non-compact object case [e.g. 334,336,350,383,384]. This process leads to a wide range of the 852 final spin-orbit misalignment (0 • − 180 • ), thus favouring the formation of merging binaries 853 with misaligned spins.
854
Unlike strong scatterings, which dominate mergers in massive clusters and produce an 855 isotropic distribution of spin orientations, the compact object spins in this EKL+1.5pN case are 856 strongly correlated with one another [e.g. 385]. Vector resonant relaxation, instead, tends to alter the orientation of the angular momentum, thus changing the binary's mutual inclination between the inner and the outer orbit over a timescale [e.g. 389,391,400] t rr,v = N −1/2 t rr,s , (37) where N is the number of stars within the outer orbit a 2 . While the vector resonant relaxation 872 rate depends on the underlying density distribution profile, it may be comparable to the EKL 873 timescale at distances of ≤ 0.5 pc, depending on the binary's separation [400]. Overall vector 874 resonant relaxation may work to enhance the merger rate by changing mutual inclination to a 875 more EKL-favorite regime of the parameter space [400]. Given the uncertainties, this regime 876 yields a BH merger rate similar to the one expected from the inner ≤ 0.1 pc region.
877
For even further distances from the SMBH, at the edge of the sphere of influence, deviation 878 from spherical potential of the nuclear star cluster can result in a long-term effect, similar to the 879 EKL [392]. This processes also yields a similar BH merger rate as the one expected from the 880 inner ≤ 0.1 pc merger rate. Below we discuss the overall expected merger rate and the relevant 881 uncertainties. 882
Initial Conditions and Unknowns
883
The description of the interplay among COs, COBs, and a central SMBH relies upon 884 several unknowns, which may significantly affect the possible development of both COB and 885 SMBH-CO mergers.
886
The number of binaries, their period and eccentricity distribution, the CO mass distri-887 bution, are poorly constrained by observations and largely influence the final outcomes of 888 dynamics. The density distribution of COs around the SMBH is another crucial, but poorly 889 constrained, quantity that affects all dynamical processes, from relaxation to binary hardening 890 [e.g. 48 Since binary's inner and outer period distribution are degenerate, the EKL efficiency is 892 less sensitive to the aforementioned uncertainties. While at first glance this may be counter 893 intuitive, it can be understood when considering the stability requirement. Long term stability 894 yields a hierarchical configuration and avoiding the breakup of the binary due to the SMBH 895 tidal forces (i.e., the Hills mechanism, [283]). Therefore, binaries closer to the SMBH may have 896 shorter inner period than those further away from the SMBH [e.g . 205].
897
Regarding the formation of COB mergers, surely one important parameter is represented 898 by the number of binaries that can actually form [see 291], which in turn depends on the star 899 formation rate in the galaxy centre and the rate at which infalling clusters can supply new COs 900 and COBs to the nucleus [277].
The Milky Way centre represents our benchmark, and in principle can be used to estimate 902 the star formation rate inside a NC. Within the SMBH sphere of influence, the Milky Way 903 contains about ∼ 10 7 M of stars and stellar remnants [e.g. 401,402], assuming a continuous 904 star formation rate, and the age of the galaxy of about 10 Gyr, this gives an approximate rate 905 of 10 3 M Myr −1 . Despite this rough estimate is consistent with recent observations and 906 theoretical advancements [e.g. 204,206,403], it remains highly uncertain and should be taken 907 with a grain of salt. The presence of an accretion disc feeding a central SMBH is a natural requirement to 910 explain the luminosity, variability, and spectra of observed AGN, although both the pathways 911 that can lead to the accretion disc formation and the physics regulating its evolution are still 912 not fully understood. In the classical framework depicted by Shakura and Sunyaev [404], and 913 extended by Novikov and Thorne [405] to include general relativistic effects, the disc settles 914 into a stationary configuration owing to inwards angular momentum transport due to an 915 effective viscosity in the disc.
916
Above a critical value of the disc surface density, the angular momentum and energy 917 transfer that a star experiences while passing through the disc becomes significant, causing 918 dissipation that can capture the star, forcing its orbit to settle into the disc plane and undergo 919 inward migration, causing a steepening of the radial density profile [406][407][408][409]. Two-body 920 relaxation with objects outside the disc competes with this dissipative process, leading to an 921 almost stationary state where two-body relaxation scatters stellar objects off the disc whilst dis-922 sipation captures them [410]. When the timescales of these competing process are comparable, 923 solar-mass objects are more easily accreted onto the central SMBH, possibly enhancing the rate 924 of tidal disruption events [27,411] and naturally leading to the formation of a nuclear stellar 925 disc [29]. Once stars and stellar compact objects are captured into the disc, the relative velocity 926 among disc-objects decreases, leading to a significant increase of the collision rate, σ c ∝ v −2 rel , 927 especially in the outer regions of the disc, possibly favouring the formation of an IMBH seed 928 [e.g. 26].
929
Depending on the disc properties, the swift growth of an IMBH-seed via gas accretion 930 can be so efficient to deplete gas from the AGN disc, a feature that is inconsistent with the 931 properties of observed AGNs. However, winds and jets launched by the accreting IMBH could 932 create a hole around the IMBH. This feedback mechanism could quench the IMBH growth in 933 MW-sized galaxies, and almost eradicate the disc in AGNs with an SMBH lighter than 10 5 M , 934 thus explaining the dearth of high-Eddington ratio AGNs in that SMBH mass range [412].
935
The stellar object motion is driven by different forces owed to accretion, dynamical friction 936 exerted by the gaseous medium, and stellar scatterings [33,413].
937
Whilst moving in the disc, the BH motion is subject to migration induced by the resonant 938 gravitational interaction with the gas disc [414]. If the gravitational torque of the moving body 939 is smaller than the gas viscous torque, the body undergoes the so-called Type I migration, 940 triggered by Lindblad and corotation resonances [25,[414][415][416]. The typical timescale of Type I 941 migration depends on the mass of the object and the central BH, the AGN suface density and 942 thickness, and the location of the captured object.
943
Sufficiently large gravitational torques, instead, enable the moving object to open a gap 944 in the disc. In such a case, the object migrates due to the torque exerted by the gas on the gap 945 boundary, leading to the Type II migration [415,[417][418][419]. The timescale of Type II migration 946 is generally larger than that of Type I migration, depending on the SMBH mass, the moving 947 object, and the viscosity of the gas. Non-axisymmetric streamers flowing across the gap may 948 significantly alter the migration rate [419,420].
949
Depending on the gas structure and properties, gas torques can also push the object away 950 from the disc centre, thus causing an outward migration [421,422]. Inward and outward torques 951 can cancel out in particular regions of the disc and form a so-called migration trap that halts 952 the migration. Migration traps are natural predictions of protoplanetary disc models [423] and 953 could be a possible mechanism to form the core of giant gaseous planets [424]. The location of 954 the trap inside the disc may be independent of the SMBH mass and the SMBH-to-BH mass ratio, 955 as these quantities only affect the amplitude of the torque [86]. However, details of migration 956 trap formation are still unclear and are mostly connected with the study of protoplanetary 957 disc formation: inefficient viscous mixing can cause torque saturation, damping migration for 958 sufficiently small objects [425] and leading to a mass dependency on the location of the traps 959 [426][427][428], the establishment of a dynamic torque can halt(boost) inward(outward) migration 960 [429,430], whilst a heating torque derived from accretion processes onto the moving body can 961 counteract inward migration [431][432][433]. Whilst some elements of protoplanetary disc dynamics 962 can be borrowed to explain compact object dynamics in AGNs, any parallelism must be 963 cautionary, taking into account that the latter are generally comprised of hotter, high-viscosity, 964 turbulent gas, where torque saturation is less likely [434,435].
965
Depending on the disc model adopted, these migration traps set at tens and a few hundred 966 times the SMBH gravitational radius, i.e. ∼ (0.4 − 3) mpc for an SMBH mass M SMBH = 10 8 M 967 [86].
968
More generally, even if such migration traps do not exist in AGN discs [434,435], a similar 969 accumulation of objects may take place at radii where the inward migration rate becomes 970 slower than the rate at which objects get captured in the disc. This may happen in regions 971 where the physical properties of the disc change (e.g.,source of opacity, self-gravity, etc.) or 972 where gaps are opened by the gravitational torques of objects in the disc [33]. Binary formation 973 and mergers may be especially abundant in these regions.
974
Once BHs and COs have settled in the disc, the formation of binaries can be triggered by 975 three-and few-body interactions as in galaxies harboring a quiescent SMBH, and by single-976 single triggered by gas dissipation [33]. Three-body scattering process can efficiently aid 977 binary formation in AGN discs, owing to the fact that gas damps stellar velocities via gaseous 978 dynamical friction and torques [33], thus providing low velocities and high densities. Further, 979 the presence of a gaseous medium offers a further mechanism: gas capture binary formation. 980 In such mechanism, two objects passing through their mutual Hill spheres can bind together if 981 the Hill sphere crossing time is longer than the timescale over which gas dynamical friction 982 damps the relative velocities of the two objects [436]. As discussed in Ref.
[33], the gas assisted 983 binary formation rate steeply increases toward the disc inner regions, and clearly overtakes 984 the formation rate associated with three-body scattering. Thus, gas capture binary formation 985 is expected to be the dominant process for binary formation in AGN discs. This also favours 986 the mass growth of IMBH-sized objects via gas accretion and repeated mergers with stars and 987 compact objects in the disc [26,33,34,434,437].
988
All these processes are nicely sketched in Figure 13, a schematic view on binary formation 989 activities in AGNs described in [33]. 990 Once binaries start forming in the disc, their long-term evolution is regulated by both 991 dynamical and gaseous processes. Having a relatively large cross section compared to single 992 objects, newborn hard binaries will further harden via binary-single scattering [11,210,282,328]. 993 Binary-single interactions with stars or compact objects in the spherical star cluster typically 994 kicks the binary from the gaseous disc to inclined orbits, while gas-dynamical friction drives the 995 binaries to settle back to the disc. Furthermore, binary-single interactions increase the binary 996 eccentricity in a disc configuration more than in isotropic scatterings, which accelerates the 997 subsequent GW-driven merger [35,438], and are expected to work even more efficiently close to 998 migration traps, where the binary-single interaction rate increases significantly [434,439]. Aside 999 binary-single interactions, which represent the dominant process in binary hardening [30,33], 1000 the gaseous medium in which the binary is embedded can trigger further binary hardening 1001 owing to dynamical friction [28] and type I/II torques exerted from a circumbinary disc formed 1002 around the moving binary [25,418,[439][440][441].
1003
Despite the many processes likely at play in an AGN, one seems clearly dominating in 1004 determining BBH formation, namely the gas-capture process, which accounts for 64 − 85% of 1005 all BBH mergers occurring around a MW-like SMBH [33]. Once formed, binaries merge very 1006 efficiently in AGN due to binary-single interactions, even if gas driven migration effects and 1007 accretion effects are completely neglected [33]. 1008 8. Black hole and neutron star mergers around supermassive black holes: implications for 1009 current and future gravitational-wave detections 1010 The various physical processes described in the previous sections leave an imprint on 1011 the population of merging COs. In the following, we discuss in more detail what properties 1012 characterise the population of merging BBHs and NS-BH binaries formed in galactic nuclei, 1013 differentiating between quiescent SMBHs and AGNs, and the consequences for current and 1014 future GW detectors.
1015
In these regards, it is worth recalling here that eccentric COBs emit broadband GWs with a characteristic peak frequency that can be expressed as [163,442] f peak = 4.6 Hz k 30
−3/2 m bin 30M −1 ζ(e bin ),(38)
where k = a bin /R Schw = c 2 a bin /(2Gm bin ) represents the ratio between the binary semimajor axis and the COB total Schwarzschild radius, in such a way that k = 3 corresponds to the innermost stable circular orbit (ISCO) for non-spinning BH. There are different expression for the function ζ(e bin ) [265,442] ζ(e bin ) = (1 + e bin ) 1.1954 /(1 − e 2 bin ) 3/2 Wen [442] (1 + e bin ) 1/2 /(1 − e bin ) 3/2 O'Leary et al.
[265](39)
From the Equation above we see that a typical hard BBH with mass m bin = 30M and 1016 a hard = 0.1 AU, i.e. k = 9 × 10 6 , will emit GWs with frequency peaked at f ∼ 0.35 mHz.
1017
Thus, merging BBHs represents potentially multiband GW sources [e.g. 443] that could be 1018 visible to both low-frequency GW detectors like LISA [444][445][446], TianQin [447], Taiji [448,449], 1019 mid-range frequencies like ALIA [450], DECIGO [451,452], and other decihertz observatories 1020 [453][454][455], and high-frequency detectors like LIGO-Virgo-Kagra [456,457], and future detectors 1021 like Einstein Telescope [458,459] or Cosmic Explorer [460,461].
1022
Next section will focus on the properties of COB mergers in galactic nuclei, attempting at 1023 highlighting how they could be used to identify the COB origin in different GW detectors. There are several parameters that could help unravelling BBH mergers with a dynamical 1026 origin, although the uncertainties in the theoretical models for both stellar evolution and 1027 dynamics make it harder to find a truly unique spot where dynamical mergers stand apart 1028 from the isolated ones.
1029
A merging COB can be characterised primarily through intrinsic quantities, like the jth component mass m j and adimensional spin χ j = cS j Gm 2 j (with S j the spin), and the binary eccentricity e and semimajor axis a. Nonetheless, there are further quantities that are particularly important from the perspective of GW detection. Among other, the so-called chirp mass, a parameter that, at the lowest order, can be linked to the frequency ( f ) and its variation (ḟ ) of the emitted GW via the relation M = (m 1 m 2 ) 3/5 /(m 1 + m 2 ) 1/5 = c 3 G
5 96 π −8/3 f −11/3ḟ 3/5 ,(40)
and thus can be directly measured by GW detectors [462][463][464].
1030
Similarly, also spins affect the evolution in phase and amplitude of the emitted GW, but it is hard to fully untangle the two spin vectors [465,466], even in space-borne detectors like LISA [e.g.,see 463,464]. Nonetheless, it is still possible to retrieve information encoded in the dominant spin effect by using a one-dimensional parametrization that, in the simplest form, can be written as a simple mass-weighted linear projection of the spins onto the binary orbital momentum vector L [467][468][469][470][471]
χ eff = (m 1 χ 1 + m 2 χ 2 ) ·L bin m 1 + m 2 = m 1 χ 1 L cos(θ 1 ) + m 2 χ 2 L cos(θ 2 ) m bin ,(41)
with θ j the angle between the binary angular momentum and the spin of the j-th binary 1031 component. As suggested by the BBH mergers detected during the three LVC observation runs 1032 [141], this parameter seems to cluster around low value [e.g. [472][473][474][475][476].
1033
Spins' information is encoded also in another measurable quantity, unfortunately poorly constrained in current detectors [68]: the precession spin parameter χ p [477], which measures the degree of in-plane spin and parametrizes the rate of relativistic precession of the binary orbital plane [141,478] χ p = max χ 1 sin θ 1 , 3 + 4q 4 + 3q qχ 2 sin θ 2 .
Aside from masses and spins, another parameter that could be measured from GW 1034 detectors is the binary eccentricity. This became possible only relatively recently [e.g. 143,144, 1035 479], because binary parameter estimation from the detected signal exploits Bayesian inference, 1036 which can be extremely computationally demanding. This led to the development of, on the one 1037 hand, numerical relativity simulations aimed at obtaining waveform templates [480], and, on 1038 the other hand, approximate methods, or approximants, that greatly reduced the computational 1039 cost of data analysis but generally they are generally constructed on the leading order of GW 1040 signal expansion [479, e.g.]. A recently developed technique, however, enabled the possibility 1041 to retrieve information on the binary eccentricity at a small computational cost [481,482], and 1042 permitted to place constrains on the eccentricity of LVC sources, suggesting that up to 4 of 1043 them might be eccentric sources [146,148]. In the frequency range of LVC detectors, i.e. f ∼ 10 1044 Hz, it has been shown that at design sensitivity it would be possible to discern an eccentric 1045 source provided that e 10Hz > 0.04, being the error on the eccentricity measurement around 1046 δe =∼ (10 −4 − 10 −3 )(D/100Mpc) [151,152].
1047
Merging COBs developing in different environments may exhibit peculiar values of the 1048 aforementioned quantities, thus their measurement is crucial to assess the origin of GW sources. 1049 For example, mergers with one or both components in the upper mass-gap, like GW190521 1050 [483], could represent the best candidates to identify signatures of a dynamical origin.
1051
Generally, the formation of mass-gap BH mergers in the isolated binary scenario is rather unlikely, whilst in dynamical environments hierarchical mergers represent a viable possibility to explain them, especially in the case of galactic nuclei. Close to an SMBH, the escape velocity is roughly given by
thus close to a MW-sized SMBH the escape velocity can favour the retention of first-generation 1052 merger products (1g), or second-generation BHs (2g), which are subjected to GW recoil kicks 1053 as large as 10 3 km s −1 . Once recoiled, the BH will eventually come back into the galactic 1054 centre over a mass-segregation time, form a new binary via binary capture [13] or three-body 1055 scattering [309], harden and merge. Depending on the nucleus properties, the whole process 1056 can take much longer than a Hubble time, owing to the steep dependence of the dynamical 1057 timescales (Eqs. 4, 25, 28) on the nucleus velocity dispersion and density. Nonetheless, the 1058 development of mergers involving 2g or 1g BHs, is more likely in galactic nuclei -O(10%) -1059 than in globular or young massive clusters [98,99,111,116,127,484,485].
1060
To discuss other potential characteristic traits of COBs mergers in galactic nuclei, let us 1061 divide the processes that can trigger a merger into three main categories: i) gravitational 1062 scatterings, ii) EKL-driven, iii) AGN-driven. [30,277,438,486]. BBH merger products formed via binary-single 1067 scatterings affect the overall BH mass spectrum, populating the upper mass-gap and extending 1068 beyond M BH 100M [114,116,277,446,484]. Binary-single scatterings in galactic nuclei can 1069 also favour the formation of NS-BH binary mergers, being the individual merger rate -i.e. the 1070 number of mergers per cluster -around 10(100) times larger compared to globular(young) 1071 clusters [e.g. 130, 136,310]. A substantial fraction of NS-BH mergers forming in galactic nuclei, 1072 and in dense star clusters in general, is expected to clearly differ from those formed in isolation, 1073 having larger chirp masses and primary components with a mass m 1 > 20M [130,132]. 1074 Binary-single scatterings and GW captures are efficient mechanisms to form eccentric binaries: 1075 50 − 90% BBH mergers formed this way have an eccentricity e 10Hz > 0.9 when sweeping 1076 through the typical LIGO band [151,152,265,487].
1077
The geometry of the scattering can be rather important in determining the merger char-1078 acteristics: if the scattering occurs in a plane, as it can happen in AGN discs, for example, 1079 the merger rate is boosted by a factor 10-100 owing to the fact that the merger efficiency 1080 in three-body scattering increases at decreasing the binary-single object mutual inclination 1081 [35,438]. The enhancement in the merger rate is also accompanied by an enhancement in the 1082 probability to form merging binaries with a high eccentricity in the LVC band [35]. Mergers 1083 induced by a "single" binary-single scattering, i.e. in which the merger occurs before the next 1084 interaction takes place, are more likely to emit GWs in the 1-10 mHz frequency range, where 1085 LISA and space-borne detectors are sensitive [35], whilst those formed chaotically during a 1086 3-body resonant interaction, or scramble, and via GW captures typically quickly merge in the 1087 1-10 Hz band, where LIGO and ground-based detectors are sensitive [35,265]. Given the chaotic 1088 nature of the gravitational scattering process, mergers formed this way are expected to feature 1089 an isotropic distribution of their spin orientation, which implies generally small χ eff values. 1090 For comparison, fully (anti)aligned spins implies (negative)positive values of χ eff . When the effect of the SMBH tidal field cannot be neglected, the EKL mechanism can aid 1093 the COB coalescence through eccentricity excitation. The merger efficiency for the EKL-driven 1094 channel weakly increases with the SMBH mass in the M SMBHs = (10 6 − 10 8 )M mass range 1095 and decreases for heavier SMBH mass values [101,277,291,399,486].
1096
Around 40% of EKL-assisted mergers have primary masses in the upper mass-gap, m 1 = 1097 (50 − 90)M , and around 30% have also a companion with mass m 2 < 20M , thus mass ratios 1098 q (0.1 − 0.6) [88,277]. Only a fraction ∼ 1 − 20% of EKL-driven mergers is expected to 1099 preserve an eccentricity e > 0.1 at > 10Hz [378,399], owing to the fact that the eccentricity 1100 increase driven by the EKL mechanism can drive to merger binaries that are initially relatively 1101 loose. Nonetheless, around 40% of these binaries may appear eccentric in the LISA band and 1102 up to 5% of them could preserve an e > 0.1 while sweeping through the deci-Hz band, thus 1103 representing potentially bright multiband sources [98]. Although LISA is unlikely to observe 1104 more than a few tens of BBH mergers [e.g. 358,488], it could be possible to observe eccentricity 1105 variation for BBHs in the Galactic Centre [489].
1106
The development of EKL resonances and the consequent increase of the binary eccentricity up to a maximum value e max reduces the merging timescale, possibly affecting the overall delay time of EKL-driven mergers. In such case, the merger time can be expressed as [378,490,491]:
t GW = t GW,0 (1 − e 2 max ) α ,(44)
with α = 1.5 − 2 − 2.5 − 3 in the eccentricity ranges e max = (0 − 0.6), (0.6 − 0.8), (0.8 − 0.95), 1107 and (0.95 − 1), respectively [378,490,491]. This implies that the larger the maximum eccentricity, 1108 the shorter the merging time. Let us consider an initially circular, equal-mass binary with 1109 m bin = 30M and semimajor axis a = 0.1 AU, its merging time is t GW,c = 4.8 Gyr [386]. 1110 Eccentricity increases owing to EKL up to e max = 0.5(0.9), thus the binary merges after 1111 t GW = 1.7(0.02) Gyr, a notable difference that highlights how crucial is the orbital distribution 1112 of compact objects around an SMBH.
If the SMBH is spinning, as indicated by several observations [e.g. 492,493], the coupling 1114 among the SMBH spin and the BBH-SMBH and BBH angular momenta can cause an efficient 1115 apsidal precession resonance, meaning that the outer and inner binaries have the same pericentre 1116 precession rate owing to both Newtonian and General relativistic effects [494]. This results 1117 in an angular momentum exchange between the BBH-SMBH and the BBH, which leads to a 1118 significant growth of the BBH eccentricity up to a maximum value that increases at increasing 1119 the BBH-SMBH initial eccentricity [494][495][496][497]. This mechanism can lead to an increase of the 1120 BBH even in cases in which the EKL resonance don't develop, e.g.,nearly coplanar systems, 1121 and could leave an imprint in the merging BBH waveform that might be detectable with LISA 1122 and help constraining the SMBH spin amplitude and its mass [495,498,499].
1123
A binary undergoing EKL around a Kerr SMBH can undergo de Sitter and Lense-Thirring 1124 precession mechanisms, which cause the chaotic re-orientation of the spins [e.g. 385], leading to 1125 merging binaries with misaligned spins and nearly zero χ eff [e.g., 96,98,99,385,490,491,500,501]. Merging BBHs in AGN discs are expected to exhibit several peculiarities, compared to 1128 those developing in quiescent nuclei via EKL oscillations or gravitational scatterings.
1129
Given the large escape velocities in AGNs, repeated mergers can be quite common in such 1130 environments, thus providing a further channel to produce mass-gap objects. If migration traps 1131 do exist, repeated merger might constitute up to 20 − 50% of all COB mergers [32,502], with 1132 more than 40% of all mergers involving one BH with mass ≥ 50M [32]. Even in the absence 1133 of migration traps, hierarchical mergers can constitute 20 − 45% of all mergers mediated by 1134 binary-single interactions and gaseous torques [33]. Nonetheless, the fraction of remnants in the 1135 high-end of the BH mass distribution us expected to remain generally low, around 3 − 11%, but 1136 can significantly vary depending on the adopted model [34]. The high occurrence of repeated 1137 mergers influences the mass ratio AGN mergers, whose distribution peaks at values around 1138 q = 0.2 − 0.7, depending on the models [31,502]. Similarly to quiescent nuclei, dynamical 1139 interactions can enable the formation of eccentric mergers also in AGNs via either binary-single 1140 interactions or GW captures. Given the flattened configuration of AGN discs, most of the 1141 mergers produced via binary-single scatterings are expected to produce eccentric binaries 1142 detectable by LISA [438]. Together with binary-single interactions, GW captures occurring 1143 within O(mpc) from the SMBH can be sufficiently energetic to trigger the formation of mergers 1144 preserving an eccentricity e > 0.3 in the LIGO-Virgo band with a probability of 20 − 70% 1145 [35,438]. 1146 Gaseous torques and accretion are expected to align both the BH spin vectors and the 1147 binary angular momentum, thus suggesting that BBHs in AGNs are characterised by either 1148 aligned or anti-aligned spins. BBHs moving on prograde orbits with respect to the AGN angular 1149 momentum are expected to rapidly accrete gas, spin up and align the spins over a timescale 1150 shorter than the AGN lifetime [503], whilst those on retrograde orbits are more likely to merge 1151 and leave behind a final BH with a spin antialigned with respect to the disc [25]. Given the 1152 alignment/antialignment of the orbital spins, AGN mergers are expected to have an effective 1153 spin distribution peaked at positive/negative values, likely around 0.4 [32]. The preponderance 1154 of hierarchical mergers would also favor the development of anti-/aligned mergers [32], thus 1155 possibly causing the formation of a sub-population of mergers characterised by large masses 1156 and χ eff < 0, a feature hardly reproducible through other channels. Nonetheless, even in 1157 AGNs is possible to generate low χ eff mergers, depending on the efficiency of radial migration 1158 and the binary-single scattering rate [504], although the overall distribution could preserve a 1159 small excess toward positive values dominated by mergers among high generation BHs [502]. 1160 Gaseous accretion can also favour the formation of COs in the lower mass-gap, thus providing 1161 suitable explanation for GW sources like GW190425 or GW190814 [34,134].
As shown in some numerical models, in AGNs mergers a combination of the effective 1163 spin parameter and the precession parameter, namely χ tot ≡ (χ 2 eff + χ 2 p ) 1/2 , increases with the 1164 merger chirp mass up to the maximum chirp mass allowed by the adopted, first-generation, 1165 mass spectrum, and then saturates toward a value χ typ ∼ 0.6 [505]. Despite such typical quan-1166 tity could represent an optimal parameter to identify AGN merger candidates, the precession 1167 spin parameter χ p is often unconstrained in observed data, making it hard to place reliable 1168 constraints on χ typ .
1169
One of the most intriguing predictions of the AGN channel is the possible production 1170 of EM signatures associated with the merger event. A well known example is the GW190521 1171 source, which has been proposed to be associated with an EM transient detected by the 1172 Zwicky Transient Facility [506], although it is rather hard to assess whether or not the GW 1173 and EM signals are truly associated [507,508], as GW190521 has a localization volume that 1174 likely contains ∼ 10 4 AGNs [509]. Nonetheless, future follow-up campaigns targeting the most 1175 massive BBH mergers could establish their association to an EM counterpart, provided that the 1176 fraction of BBHs producing an AGN flare is substantial (> 0.1) [509].
1177
Establishing whether a BBH merger has an AGN origin could be assessed statistically on 1178 the basis of spatial correlation, a technique that requires at least f AGN N det detections, where 1179 f AGN corresponds to the fraction of GWs coming from AGNs [510]. In principle, this implies 1180 that the missing evidence of a BBH-AGN connection in the light of 100 GW detections would 1181 suggest f AGN < 0.25, although the number of required detections to assess their AGN origin 1182 hugely varies with the AGN number density [510]. 1183
Expected Merger Rate and Prediction
1184
In the following, we retrieve from the literature merger rate estimates at redshift z = 0 1185 for both BBH and NS-BH mergers induced by dynamics around a quiescent SMBH or in an 1186 AGN. When the authors provide the number of events per galaxy and per year [e.g. 265] we 1187 multiply this quantity by the local density of Milky Way equivalent galaxies ρ MWEG = 0.0116 1188 Mpc −3 [511,512]. Figures 14 and 15 show the merger rate density interval for BBH and NS-BH 1189 mergers at redshift z ∼ 0 reported in the literature.
1190
Broadly speaking, the inferred merger rates for mergers around quiescent SMBHs and 1191 AGNs cover a wide range of values, up to 3-4 order of magnitudes, owing to the large uncer-1192 tainties affecting the models.
1193
For AGNs, there are several sources of uncertainties [for a detailed discussion, see 33,513]: 1194 the number density of galactic nuclei hosting an AGN, (10 −3 − 10 −2 ) Mpc −3 [e.g. 514], the BH 1195 number density in galactic nuclei (10 4−6 ) pc −3 [53,55,58], the AGN occupation fraction (0.01 − 1196 0.3) [515], the BH binary fraction (0.01 − 0.2) [55,287,434,516], the AGN lifetime (0.1 − 100) 1197 Myr [517][518][519].
1198
For quiescent SMBHs, instead, the uncertainties owe mostly to the number of BHs lurking 1199 in the galactic centre and the rate at which the BH population is replenished [277,291,399], the 1200 geometry of the NC surrounding the SMBH [291,392] and the SMBH mass [277,291,399], the 1201 properties of the BBH population [277,486].
1202
Adopting Milky Way like conditions and assuming that the fraction of galaxies hosting 1203 a NC is ∼ 0.5 yields a BBH merger rate of ∼ 1 − 20 Gpc −3 yr −1 [e.g. 59,358]. This rate is 1204 consistent with a simplified model that starts with objects at the BH stage [e.g. 291] and N-body 1205 calculations [277]. Vector resonant relaxation only slightly enhance the rate, for a Milky Way 1206 like conditions, but is much more effective for a smaller mass SMBH [e.g. 400]. On the other 1207 had, the EKL-derivative for a none spherical potential yields a rate of ∼ 1 − 15 Gpc −3 yr −1 1208 [392]. Thus, combining these three effects, the merger rate from within inner few to ten parsecs 1209 at the centre of a galaxy is expected to be 1 − 50 Gpc −3 yr −1 . The lower limit in this range is 1210 taken to be from the lower limit of one of the channels, and the upper limit is taken from the 1211 combination of the three chancels. Thus, the merger rate boundary limits for BBHs around quiescent SMBHs, R(0) (0.1 − 1216 10 2 )yr −1 Gpc −3 , is somewhat narrower with respect to AGNs -whose rate spans up to 7 orders 1217 of magnitudes in some models. Future numerical simulations that account simultaneously for 1218 the evolution of the galactic nucleus and its stellar population could help to further reduce the 1219 uncertainties.
1220
Another interesting class of GW sources is represented by NS-BH mergers, which may also 1221 produce EM emission promptly after the merger. However, the formation of NS-BH binaries 1222 is strongly suppressed in dense stellar environments, owing to the fact that BHs have larger 1223 cross sections than other COs, thus favouring the interaction among themselves. This implies 1224 that the formation of NS-BH binaries either takes place on long timescales or is a byproduct of 1225 primordial binaries. Either way, merging NS-BH binaries in quiescent galactic nuclei can form 1226 via binary-single scatterings (rate R ∼ 0.0001 − 0.01yr −1 Gpc −3 ) [130, 136], similarly to what 1227 happens in globular and young clusters [132,133], possibly contributing to the formation of 1228 low-mass ratio systems like GW190814 [136]. In AGN discs, instead, the production of NS-BH mergers proceeds, in principle, similarly 1234 to BBHs, but their amount relative to BBHs is rather uncertain. In general, double NS (DNS) 1235 and NS-BH mergers constitute a relatively small fraction among all mergers developing in 1236 an AGN, namely f DNS (0.5 − 3) × 10 −4 (DNS) and f NSBH = (0.8 − 6) × 10 −4 (NS-BH) [34], 1237 although according to some models these quantities can be as large as f DNS = 0.01( f NSBH = 0.3) 1238 for DNS(NS-BH) mergers [520]. These estimates corresponds to a merger rate density of 1239 f DNS = 400 f AGN yr −1 Gpc −3 (DNS) and f NS−BH = 10 − 300 f AGN yr −1 Gpc −3 (NS-BH) [520], 1240 although depending on the model assumptions these estimates can be up to 2-3 order of 1241 magnitude smaller [34].
1242
A list of values for the BBH and NS-BH merger rate density is reported in Tables 1, 2.
1243
Comparing the modelled merger rate densities listed in Tables 1 and 2 with that inferred 1244 from GW observations [141] makes evident that the galactic nuclei channel for BBH mergers 1245 can contribute quite significantly to the overall population but are unlikely to contribute much 1246 to the global population of NS-BH mergers.
1247
It is interesting to note that the estimated rates for BBH mergers galactic nuclei are 1248 comparable to those predicted for other channels [526, for a complete review on merger rates 1249 from different channels see]. Therefore, untangling the signatures of different formation 1250 channels in ground-based observations might be a quite hard task.
1251
Future detectors might help unravelling how different channels contribute to the overall 1252 population of BBH mergers. An equal-mass binary with total mass m bin emitting GWs at 1253 frequency f will merge in a timescale 1254 τ ∼ 5 yr f 10 mHz
−8/3 m bin 68M −8/3 ,(45)
thus combining LISA and ground-based observations has the potential to cover the full BBH 1255 frequency spectrum. Sources in the mass range between GW150914, the first GW sources ever 1256 detected [65], and GW190521 [76], emitting at 10 mHz could be observed in the last 5-1 yr prior 1257 to merger, enabling more precise measurement of localisation, eccentricity, spin amplitude 1258 and alignment, and binary mass [443,446]. More likely, ground based observations can be 1259 reverse engineered by seeking for the observed binaries in data previously acquired with LISA. 1260 Unfortunately, recent models suggest that LISA will observe only a few tens to a hundred BBHs, 1261 with only 10 potentially observable as multi-band sources, depending on LISA mission 1262 lifetime and the adopted SNR threshold [443,446,488,527,528].
1263
Deci-Hz observatories [450][451][452][453], filling the gap between space-borne and ground-based 1264 detectors, could permit us to pierce deeper in the cosmos and reconstruct BBH merger evo-1265 lutionary history. A DECIGO-like detector [451,452] has the potentiality to observe merging 1266 BBHs up to a redshift z ∼ 10 3 [453], well beyond the formation time of the first stars. More 1267 modest detector design, although still very ambitious, could still provide observation of stellar 1268 BBH mergers up to redshift z = 3 − 100 [453]. LISA has the potential capability to track signatures of the EKL mechanism in our own 1275 Galactic Centre [489,529,530] -and closeby galactic centres [531] -and probe the very existence 1276 of compact BBHs orbiting around an SMBH. As we described in the previous sections, a triple 1277 system composed of an inner binary whose centre of mass orbits around a distant perturber 1278 can be subjected to secular oscillations of the orbital inclination and eccentricity of the inner 1279 orbit.
1280
A compact binary with mass m bin , semimajor axis a, and eccentricity e emits GWs characterised by a broad frequency spectrum peaked at [265] f peak = 7.6 × 10 −6 Hz
(1 + e) 1/2 (1 − e) 3/2 a 0.1AU
−3/2 M 50M 1/2 ∼ 0.01 Hz.(46)
From the equation above it is apparent that a circular binary with mass m bin = (25 + 25)M 1281 and semimajor axis a = 0.1 AU -which implies a merging time of t GW = 1 Gyr [386] -would 1282 be invisible to LISA until its semimajor axis reduces by a factor O(10). However, the EKL 1283 mechanism at play can trigger eccentricity oscillations that can bring the GW peak frequency 1284 into the LISA band well before the binary shrinks, provided that the eccentricity peak reaches a 1285 sufficiently large value, e peak > 0.95 in the aforementioned example. Additionally, EKL will 1286 shorten the binary lifetime by a factor 10 4 [378,490].
1287
Such a binary moving about a SMBH like SgrA* will undergo a periodic shift of the 1288 frequency peak from 10 −5 Hz up to 0.1 Hz [489], thus fully covering the LISA sensitivity band. 1289 The period of such shift -∼ 1 month -can be much shorter than the LISA mission lifetime, 1290 thus making possible the detection of such clear signature of a BBH-SMBH triple [see also 1291 529,530]. EKL signatures could be observed also in closeby galactic nuclei, at a distance 1 − 10 1292 Mpc, depending on the binary and SMBH masses and the orbital properties [489,530,531].
1293
The same process might help spotting binaries containing an IMBH around an SMBH 1294 [531,532]. In general, a merging BBH emits GWs at a frequency that is redshifted by a factor (1 + 1305 z C ) −1 , with z C being the cosmological redshift. However, the signal emitted by the BBH while 1306 revolving around the SMBH suffers a time-dependent variation caused by its motion with 1307 respect to the observer.
1308
The variation in the distance between the emitter and the observer causes a Doppler shift 1309 in the arrival time of GW pulses, the gravitational time dilation causes relativistic Doppler shift 1310 and gravitational redshift, whilst the SMBH gravitational field causes a delay in the time arrival 1311 of the GW signal. The aforementioned three effects, called Roemer, Einstein, and Shapiro 1312 delay [533], causes a shift in the measured GW signal that might be measurable in both low-1313 [498,[534][535][536][537][538][539] and high-frequency detectors [165,540,541].
1316
The Doppler boost effect has potentially crucial implications for the interpretation of detected signals. In fact, aside the cosmological redshift, the signal emitted by a BBH revolving around an SMBH can suffer a Doppler shift,
(1 + z D ) ∼ (1 − v 2 /c 2 ) −1/2 (1 + v/c),
and a gravitational redshift [165,535,537,[540][541][542]. Defining the SMBH Schwarzschild radius (R S ) and the semimajor axis of the BBH orbit around the SMBH (r) enables us to calculate the Doppler shift at pericentre through the relation (v/c) 2 = (R S /2r)(1 + e)/(1 − e) and the gravitational redshift as
(1 + z G ) ∼ (1 − R S /r) −1/2 , under the simplistic assumption that the GW emitter and receiver are on the same side with 1317 respect to the SMBH [541]. Thus, the total shift induced by the SMBH can be written as 1318 (1 + z) = (1 + z C )(1 + z D )(1 + z G ) [165,541]. Since the measured BBH chirp mass scales with 1319 M obs ∝ f −11/5ḟ 3/5 and its luminosity distance D L,obs ∝ f 2/3 , measured and intrinsic quantities 1320 are linked by the total redshift as
1321 M obs = M(1 + z C )(1 + z D )(1 + z G )(47)D L,obs = D C (1 + z C )(1 + z D )(1 + z G ),(48)
where D C is the BBH comoving distance. Depending on the values of the additional corrections, 1322 the interpretation of observed sources might thus be biased toward larger values [541]. Note 1323 that this corrective factor depends only on the cosmological location of the merger, the SMBH 1324 mass, and the BBH-SMBH orbital semimajor axis and eccentricity. Figure 16 shows how the 1325 correction factor varies with the SMBH mass and the distance to the galactic centre assuming 1326 that the merger occurs at redshift z C = 0 and the BBH-SMBH orbital eccentricity e = 0 − 1327 0.5 − 0.9. Note that the spikes occurs when R S ∼ r or R S ∼ 2r(1 − e)/(1 + e). The picture 1328 makes evident that both the observed chirp mass and luminosity distance might overestimate 1329 the intrinsic properties of the merger by a factor up to 2 − 3 [541], and on average by around 1330 10 − 30% [165].
1331
It is worth noting that a BBH with mass m bin = 50M at a distance r = 1 mpc from an 1332 SMBH with mass M SMBH = 10 6 (10 8 )M revolves around the SMBH in a time P Figure 16. Total chirp mass correction factor as a function of the SMBH mass and for different values of the BBH-SMBH separation, assuming that theBBH undergoes merger at redshift z = 0 and moves around the SMBH on an orbit with eccentricity e = 0 − 0.5 − 0.7 − 0.9. orbital period could measure the Doppler modulation in the source signal [534][535][536]539] and 1335 related-effects [498,538,542,543]. 1336 While revolving around the SMBH, the source will appear to change its redshift by a 1337 factor δz ∼ a COM τ life /c [534,536]. The resulting progressive drift of the GW signal could be 1338 detected by LISA in principle, and since it is maximized in galactic nuclei, its detection could 1339 represent a strong indicator of the merger environment [534]. The effect of Doppler boost could 1340 be measurable up to a redshift z < 0.05, thus permitting to probe the BBH-SMBH interplay 1341 in the local Universe [535], but even in the case of no-detection the acceleration drift could 1342 be sufficient to bias the parameter estimation of future detected sources [536]. A sufficiently 1343 long monitoring of the source could also lead to detection of lensing signatures caused by the 1344 SMBH [538,542,544]. The probability to observe such an effect depends on the properties of 1345 BBHs forming around an SMBH and could be O(1 − 3%) with both LISA and TianGO [538,542], 1346 potentially enabling to measure the SMBH mass at a level of 0.01 − 1% [538,539]. Similarly to 1347 electromagnetic wavelets, GWs from BBHs around an SMBH could be affected by aberration, 1348 an effect that can cause a shift in the GW signal as significant as the "standard" Doppler shift 1349 and potentially detectable by a LISA-like mission, provided that the BBH is orbiting within a 1350 few 10 3 Schwarzschild radii [543]. The detection of a few hundred BBHs should thus help us 1351 starting to unveil how different channels contribute to the formation of merging BBHs [542]. 1352 Furthermore, the GW signal emitted by the merging COB can be scattered by the central 1353 SMBH. This can produce a secondary signal, a sort of "GW echo", which will have similar 1354 time-frequency evolution as the main signal but is delayed in time. For sources moving within 1355 10 − 10 4 Scwarzschild radii from the SMBH, around 10(90)% of the detectable echo arrives 1356 within ∼ 1(100)s(M SMBH /10 6 M ) after the primary signal [545].
1357
In the case of Kerr SMBHs, there are three further effects that might leave an imprint on 1358 the BBH emitted signal, namely i) the precession of the BBH-SMBH angular momentum around 1359 the SMBH spin owing to spin-orbit coupling associated with the 1.5 PN effect [382,496,499], ii) 1360 the precession of the BBH angular momentum around the BBH-SMBH angular momentum, a 1361 geodesic de Sitter-like precession induced by GR [546,547], iii) precession of the BBH angular 1362 momentum around the SMBH spin owing to the Lens-Thirring precession. The coupling of 1363 these effects can lead to a modulation in the GW signal of inspiralling BBHs, possibly being 1364 observable with space-borne detectors like LISA if the precession period is shorter than the 1365 observation time [498,499]. 1366 Finally, if the BBH is traveling in an AGN disc, the gaseous medium can leave an imprint 1367 on the binary waveform that translates into an overestimate of the binary chirp mass by a factor 1368 1 + τ gw /τ gas 3/5 and of the binary distance by a factor 1 + τ gw /τ gas , where τ gw and τ gas 1369
represents the GW and the hydrodynamical timescale, respectively [548].
Summary
1371
Galactic nuclei likely represents the most intricate environments in the Universe, where 1372 dynamics is regulated by a complex interplay among stellar interactions powered by the 1373 high-densities of NCs, secular effects driven by central SMBHs, and gaseous-driven effects in 1374 AGNs.
1375
Discoveries like the emission GWs emitted by merging COs, the high-energy emission 1376 from the Galactic Centre possibly triggered by COBs, the stellar motion in the immediate 1377 vicinity of SgrA*, the Galactic SMBH, recently revived the interest about how CO form, pair-up, 1378 and eventually merge in quiescent and active galactic nuclei harboring a NC, an SMBH.
1379
In this review we provided an overlook of the plethora of mechanisms that can contribute 1380 to the formation, and possibly coalescence, of COB in both galactic nuclei with a quiescent 1381 SMBH and AGNs. These mechanisms rely upon different theoretical frameworks devised to 1382 represent different aspects -e.g.,scatterings, EKL, gas torques -of the same environment.
We have highlighted the main imprints that one mechanism or another could leave in the 1384 population of COB it produces, and how they could be used to untangle the origin of some 1385 GW sources detectable with present-day or future GW observatories. 1386 The main peculiarities of the most recent theoretical frameworks can be summarised as 1387 follows:
1388 •
Dynamics plays a crucial role in determining COB formation in galactic nuclei. Three-1389 body scatterings, involving three initially unbound objects, are likely dominant in galaxies 1390 with a large NC-to-SMBH mass ratio, but become extremely inefficient close to the SMBH. 1391 Conversely, single-single interactions that form bound pairs via GW bremsstrahlung -or 1392 GW captures -are more efficient in the SMBH immediate vicinity and in the nuclei with 1393 the most massive SMBH. However, GW captures produce short-lived binaries that merge 1394 within days or hours from the formation and have a large chance to be highly eccentric 1395 when sweeping through high-frequency detectors.
1396
• Galaxies dominated by a quiescent SMBH can efficiently replenish their population of 1397 COBs -particularly BHs -via accretion of massive star clusters that undergo inward 1398 migration owing to dynamical friction.
1399
• A substantial population of primordial binaries can also play a crucial role in determining 1400 the properties of COBs in galactic nuclei, although most of them are likely destroyed by 1401 the SMBH tidal field.
1402
• Once binaries start forming in galactic nuclei, their further evolution is regulated mostly 1403 by binary-single interactions, which generally promote the formation of tighter and more 1404 massive binaries but, depending on the binary properties, can lead to their evaporation 1405 well before GW emission start dominating the binary evolution.
1406
• Owing to dynamical friction, or mass segregation, and dynamical interactions, COBs are 1407 expected to move through regions of the nucleus with different velocity dispersion and 1408 density. Variation of the environment structure can dramatically affect the COB fate: an 1409 initially hard binary moving inward can appear soft closer to the SMBH and eventually 1410 be disrupted by interaction with other stars and COs.
1411
• Around 20 − 70% of COBs formed in galactic nuclei are expected to suffer the effect of 1412 the SMBH gravitational field, which can cause periodic oscillations of their eccentricity 1413 called eccentric-Kozai-Lidov resonances. This mechanism can significantly shorten the 1414 COB lifetime, possibly affecting the delay time of merging COs. The development of EKL 1415 oscillations strongly depends on the binary properties (e.g.,general relativistic precession 1416 can suppress EKL), the distance to the SMBH, the eccentricity of the COB orbit about the 1417 SMBH.
1418
• In AGNs, the formation of COBs is favored by both gaseous torques and dynamical 1419 scatterings, whose efficiency is boosted by the nearly planar configuration. The possible 1420 existence of migration traps, where inward and outward torques cancel out, makes AGNs 1421 potential factories of multiple-generation COs mergers and IMBHs.
1422
• Mergers occurring in galactic nuclei features some peculiar traits: a significant fraction of 1423 mergers with one component in the upper mass-gap, a non-negligible fraction of multiple 1424 generation mergers that can affect the high-end of the BH mass distribution, fairly mis-1425 aligned spins, although in AGNs a noticeable fraction of high-generation mergers might 1426 have mildly aligned spins, and a quite significant probability to preserve an eccentricity 1427 e > 0.1 whilst sweeping through the frequency bands of both low-and high-frequency 1428 detectors.
1429
• The merger rate inferred for present-day GW detectors for BBH and NS-BH binary mergers 1430 in galactic nuclei is poorly constrained owing to the many model uncertainties. For BBH 1431 mergers, models for quiescent SMBHs and AGNs predicts similar estimates, which gener-1432 ally fall in the range R BBH = 10 −3 − 10 2 yr −1 Gpc −3 . For NS-BH mergers, instead, there 1433 are clear differences between the prediction for quiescent, R BBH = 10 −5 − 1yr −1 Gpc −3 , 1434 and active nuclei models, R BBH = 1 − 10 3 yr −1 Gpc −3 , partly owing to the relatively poor 1435 literature and the huge uncertainties.
1436
• The presence of an SMBH in the vicinity of a merging COB can leave some imprints on 1437 the emitted GW signal that could, in principle, be detected with future detectors. Among 1438 others, a shift in the peak frequency for mergers occurring in the Milky Way centre, a 1439 variation in the measured redshift induced by the rapid motion of the binary around the 1440 SMBH, and the development of a GW echo produced by the scattering of the emitted 1441 GWs onto the SMBH.
1442
Despite the current literature is very rich, and the amount of new work done in the field is 1443 constantly increasing, there are still a number of important elements that are missing in current 1444 models and deserve further development. Among others, we identify in the following some 1445 elements that may be key to place more stringent constraints on the model predictions:
1446 •
Initial conditions: probably the most important unknown that mostly affect all the models 1447 is the scarce knowledge of how stars form and pair in the extreme environment of a galactic 1448 nucleus. The initial binary fraction, the initial distribution of periods and masses, the 1449 metallicity spread in the galactic nucleus are all factors that crucially determine the COB 1450 properties: semimajor axis, eccentricity, component masses.
1451
• Interplay of mechanisms: as we have seen throughout the review, COB formation is 1452 likely regulated by many mechanisms likely operating simultaneously. However, most 1453 theoretical models focus on one specific aspect at the time. Fully self-consistent N-body 1454 simulations capable of taking into account stellar evolution of single and binary stars, the 1455 SMBH tidal field, and potentially the effect of an AGN disc exists, but their resolution is 1456 still to low and their computational cost too large to permit a one-to-one representation 1457 of a Milky Way-like nucleus. Simpler models relying on semi-analytic assumptions or 1458 few-body (scattering) simulations represent valid alternative, although they sometimes 1459 neglect potentially crucial elements, like the importance of flybys on the evolution of COBs 1460 undergoing EKL oscillations, the development of EKL resonances in binaries formed in 1461 AGNs, the role of star formation onto the actual population of COBs around an SMBH. 1462 • Observations: from an observational perspective, the observation of young massive 1463 binaries in galactic nuclei, a larger number of GW detections, a more precise localization of 1464 GW sources, and the future detection of inspiralling binaries in the Milky Way centre can 1465 definitely help us improving our knowledge of the processes regulating COB formation 1466 in galactic nuclei.
Figure 1
1Figure 1. Schematic illustration of the galactic nuclei zooniverse: a nuclear cluster (NC) forms via in-situ star formation and mass transport from infalling star clusters; in its inner parts a variety of dynamical interactions can trigger the formation of compact object binaries (COBs); in other cases, COBs promptly merge releasing gravitational waves (GWs); in some other cases, the COBs evolution is determined by the central supermassive black hole (SMBH) gravitational field, which can impinge periodic oscillations on the binary eccentricity and ultimately lead to their coalescence. The depicted interactions are actual N-body simulations carried out with the ARGdf code described in Arca-Sedda and Capuzzo-Dolcetta [165].
390 Figure 3
3903compares the dynamical friction and mass-segregation timescales for a population 391 of COs with mass m CO = 5 − 50M inhabiting star clusters with a mass in the range M c = 392 10 4−7 M and half-mass radius R h ∼ 1pc(M c /M ) 0.13
Figure 4 .
4a 448 cartoon showing the evolution of BH progenitor stars. More quantitatively, we show in the 449 right panel of the same figure the time evolution of the radii containing the 10%, 25%, 50%, 75% -450 referred to as lagrangian radii -of BH mass in this simple toy model. The SN-kick effect is rather 451 Left panel: schematic view of the orbit of a BH progenitor in a galactic nucleus. The occurrence of a SN event can impart a kick on the newborn BH and delay the segregation. Right panel: lagrangian radii enclosing 10%, 25%, 50%, and 75% of BH (or BH progenitor) mass as a function of time assuming a nucleus with mass M c = 2.5 × 10 7 M and an SMBH with mass M SMBH = 4.3 × 10 6 M .
Figure 5 .
5Fraction of hard binaries, normalised to the total population of binaries, as a function of the distance from an SMBH with mass M SMBH = 4.3 × 10 6 M , assuming a primordial semimajor axis distribution flat in logarithmic values and limited between 10 −2 − 10 3 AU.
547 5. 3 .
5473Multiple encounters make bound pairs: how dynamical processes aid binary formation in galactic 548 nuclei 549
Figure 6 .
6Time evolution of the BBH semimajor axis normalised to the local hard-binary separation (orange straight line) and the BBH position within the cluster normalised to the initial position (blue dashed line). We assume an SMBH with mass M SMBH = 4.3 × 10 6 M and a BBH with mass M BBH = 30M .
Figure 7 .
7Timescale for gravitational-wave capture assuming a black hole mass m BH = 15M as a function of the distance from the SMBH and for different values of the SMBH mass and BH number density.
used the fact that n = ρ 0 /m * (r/r 0 ) γ and σ 2 BH = (m * /m BH )GM SMBH /r. Note that 611 the Galactic NC is characterised by ρ 0 ∼ (2.8 ± 1.3) × 10 6 M pc −3 (r/r 0 ) −γ , with γ = 1.2(1.75) 612 inside(outside) the inner r 0 = 0.22 pc[39,322].613 Note that for NSs the three-body interaction time becomes incredibly long, owing to the 614 steep dependence on the CO mass. The t 3bb time becomes shorter than a Hubble time only if 615 we consider nuclei with considerably lighter SMBH, ∼ 10 5 M , and in the outermost (r > 5 pc) 616 regions of the NC.617Under the rather simplistic assumption that the NC properties do not vary significantly over time, Equation 24 can be integrated over time to obtain how the following, time-dependent, expression of the binary fraction [for a full derivation, see 277]:
634 From
634the equation above, for a Bahcall-Wolf cuspγ = −7/4 -the 3-body interaction 635 time increases toward the SMBH as t 3bb ∝ 1/r.
Figure 8 .
8that three-body scatterings are highly suppressed for objects lighter 637 than 30M , thus the formation of stellar binaries in such extreme environments must resort to 638 Fraction of binary black holes as a function of the distance to the Galactic Centre after an evolutionary time of 0.1 Gyr (dotted line), 1 Gyr (dashed line), and 10 Gyr (straight line).
Figure 9 .Figure 10 .
910Three-body timescale as a function of the NC (x-axis) and SMBH (y-axis) mass. We assume three equal mass BHs (m BH = 40M ) orbiting at r = 0.5 − 0.3 − 5 pc from the SMBH (from left to right, respectively). The star marks typical values in the Milky Way. The straight line marks the case M NC = M SMBH , whilst the dotted line separates the region with a t 3bb smaller (below the line) or larger (above the line) than the Hubble time. Binary-binary interaction involving four BHs in a dense star cluster. After the close encounter, one BH is ejected and the remaining three BHs form an unstable triple. Taken fromFigure 2in Arca Sedda et al, A&A, 2021, 650, A189, reproduced with permission ©ESO[163].
bin represents the binary fraction.
724 6 .
6Secular dynamical effects on binary evolution around a supermassive black hole 725 6.1. Secular perturbations on black hole binaries in galactic nuclei: the impact of a supermassive black 726 hole 727 While the effects of binary interactions have been shown to play a crucial role in the global 728 dynamical evolution of dense systems such as globular clusters, only recently the effect of 729 binaries in NCs have been investigated. Within the vicinity of an SMBH, the members of a 730
i = a 3 i /(Gm bin,i ) is the inner (i = 1) or outer (i = 2) binary orbital period.
Figure 12 .
12Left panel:A representative Example from Ref.[291], that depicts the evolution of BBH while solving the deafeningly equations up to the octupole level of approximation (red) and only up to the quadrupole level of approximation (blue). As shown the octupole-level evolution results in a merger after 1623 years, while the quadrupole-level approximation never merge.Right panel: volumetric BH merger rate as a function of the number of BBHs. This rate represents Monte Carlo results from Hoang et al, ApJ, 2018, 856(2), 140 ©AAS [291]. Reproduced with permission.
reached without the GR effects [e.g. 20,380]. Similar result takes place for the hexadecapole 841 level of approximation n = 4, in Eq. (32) [e.g. 381]. 842 Spin effects, 1.5pN: The compact objects' spins can cause a verity of precessions that may affect 843 the overall dynamics. For example, de-Sitter precession [e.g. 382] can cause the precession of 844 the compact object spin vector about the angular momentum of the binary. On the other hand 845 Lense-Thirring Precession can cause the precession of the angular momentum about the spin 846 of the compact object [e.g. 382], if the two vectors are initially misaligned. This effect translates 847 to eccentricity exaction as the angular momentum changes.
857 GW, 2 . 5pN :
85725pNThe higher pN terms affect the Gravitational Wave (GW) emission of compact 858 object binaries [e.g.386]. The large eccentricity, potentially produced via the octupole level of 859 approximation, can produce pulse-like GW emissions which shrinks the binary semi-major 860 axis, and later circularizes the orbit [e.g.387,388].861Resonant relaxation processes: Binaries embedded inside a dense stellar cluster are subjected 862 to a continuous influence from the gravitational field generated by all the other cluster members, 863 which can impinge secular effects on its orbital evolution [e.g.203,[389][390][391][392]. These effects 864 include different relaxation processes and precession of the orbit due to the extended and 865 possibly anisotropic gravitational potential.866 Within the sphere of influence of the SMBH the motion of stars and compact objects 867 is nearly Keplerian, therefore the encounters between stars are correlated. These correlated 868 gravitational encounters result in torques that can change both the direction and magnitude 869 of the angular momentum of the orbit of the binary about the SMBH, known as resonant 870 relaxation processes [e.g.389,391,[393][394][395][396][397][398].871Resonant relaxation causes the variation of magnitude and direction of the outer angular momentum (thus eccentricity) over a timescale [e.g.389,391,399]
908 7 .
7Dynamics of black hole binaries in active galactic nuclei 909
Figure 13 .
13Schematic illustration of the different processes that contribute to black hole binary formation in the disc of an active galactic nucleus. Taken from Tagawa et al, ApJ, 2020, 898(1), 25 ©AAS. [33]. Reproduced with permission.
1024 8. 1 .
10241Population properties: masses, mass-ratio, spins, and eccentricity 1025
are likely the main engine triggering COB mergers around 1065 SMBHs in the M SMBH = (10 6 − 10 8 )M mass range, being responsible of up to 20 − 70% 1066 of mergers in galactic nuclei
Figure 14 .
14Merger rate density of BH-BH mergers in quiescent and active galactic nuclei at redshift z = 0.
1212 Future
1212detectors, like LISA or DECIGO, could shed light on the population of merging 1213 binaries in our Galactic Centre. For example, recent models of the Milky Way centre suggest 1214 that LISA could identify up to 1 − 20(15 − 150) BBH (WD-WD binaries) [358].1215
1229
Mergers can be triggered by EKL oscillations induced onto binaries formed via GW cap-1230 tures (R ∼ (0.001 − 0.06)yr −1 Gpc −3 ) [310], dynamical interactions (R ∼ 0.06 − 0.1yr −1 Gpc −3 ) 1231[399], or binary stellar evolution -although likely with a rather low (∼ 0.6%) probability 1232 (R ∼ 0.2 − 2yr −1 Gpc −3 [59,358].
Figure 15 .
15Same as inFigure 14, but for NS-BH mergers.
1269 8. 3 .
12693Imprint of galactic nuclei on the gravitational-wave emission from merging compact objects1270 An SMBH in the vicinity of a merging BBH can leave several imprints, some of which 1271 potentially measurable with future space-borne and third generation detectors.
1295
Detection (or not) of even one such signatures would provide crucial insights on the 1296 formation probability of BBHs around an SMBH and the possible contribute of galactic nuclei 1297 BBHs to the overall population of BBH mergers.1298Several deci-Hz observatories, like DECIGO or the Big Bang Observatory, could enable 1299 an even more clear detection of such signatures[530]. Moreover, eventual synergies among 1300 different detectors operating at the same time could enable a simultaneous tracking of the EKL 1301 mechanism at play, helping to remove any possible degeneracy in the parameter space[530].
1314 For
1314example, LVC detectors could measure such shift in BBH binaries with masses m bin 1315 20M orbiting at r ∼ 1 AU from an SMBH with mass M SMBH ∼ 10 5 − 10 6 M [540].
2 3(0.3) 1333 yr, thus a detector with a mission lifetime (τ life ) longer than the BBH merging time and the 1334
For a MW-like nucleus, assuming a typical BH number density of 10 6 pc −3 and velocity dispersion σ ∼ 100 km s −1 , this condition implies around 0.01 − 0.1 binary captures per Gyr [e.g. 265,310]. The timescale for these interactions can be written as a function of the SMBHt cap =(nσπb 2
bnd ) −1
(23)
=353 × 10 12 yr
n
10 6 pc −3
−1
M SMBH
4.3 × 10 6 M
11/14
r 0
0.22 pc
−11/14 r
r 0
−11/14+γ
c the perturber cannot recede to infinity and the three bodies undergo a resonant 697 interaction that can culminate in the exchange of one binary component if:698
-
∆E bs > E bin ,
699
-
∆E bs < E bin and m p > m bin .
700
Either ways, the perturber or the exchanged component recedes to infinity and possibly 701
leaves the host system;
702
], dynamical friction [e.g. 268], collision rate [e.g.51,310] 891
BBH merger rates ProcessΓ [yr −1 Gpc −3 ] Ref.Table 1. BBH merger rate density for quiescent SMBHs and AGNs. Column 1: type of physical processes included -Kozai-Lidov (EKL), dynamical scatterings (single-single or binary-single, DYN), binary stellar evolution (SEV), gasesous capture via migration, and/or dynamical friction (GAS), migration (MIG), pairing in migration traps (TRP). Column2: merger rate density. Column 3: reference. When a volumetric rate density was not provided in the referred paper, we adopted when required a local galaxy number density of ρ glx = 0.0116 Mpc −3[511,512] and an average number of BBH binaries in a MW-like nucleus of N BBH = 200[277,291].Table 2. Same as in Table 1 but for NS-BH binaries. Note that the estimate from McKernan et al. [520] is obtained under the extreme assumption that all BBH mergers originate in AGNs.GWTC-3
-
17.9 − 44
The LIGO Scientific Collaboration et al. [141]
quiescent SMBH
EKL+SEV
4 − 24
Wang et al. [358]
EKL+DYN
3 − 8
Arca Sedda [277]
EKL+DYN
6 − 20
Arca Sedda [277]
EKL+SEV
10 − 20
Stephan et al. [59]
EKL
0.17 − 0.52
Fragione et al. [399]
EKL+DYN
1 − 10
Zhang et al. [486]
EKL
1 − 3
Hoang et al. [291]
DYN
10 2 − 10 4
Leigh et al. [30]
EKL
0.6 − 15
Petrovich and Antonini [392]
EKL
< 100
VanLandingham et al. [521]
EKL
0.1 − 48
Antonini and Perets [378]
DYN
0.21
O'Leary et al. [265]
AGN
GAS+TRP
27 − 37
Li [522]
DYN+GAS
6 − 19
Ford and McKernan [523]
MIG+TRP
0.66 − 120
Secunda et al. [524]
GAS
0.002 − 18
Gröbner et al. [525]
DYN+GAS
0.02 − 60
Tagawa et al. [33]
DYN+TRP
4
Yang et al. [31]
MIG
10 −3 − 10 4
McKernan et al. [513]
TRP
10 2 − 10 4
Leigh et al. [30]
DYN+GAS
3
Stone et al. [440]
NS-BH merger rates
Process
Γ [yr −1 Gpc −3 ]
Ref.
GWTC-3
-
7.8 − 140
The LIGO Scientific Collaboration et al. [141]
quiescent SMBH
DYN
3 × 10 −5 − 0.006 Arca Sedda [136]
EKL+SEV
0.025 − 0.3
Wang et al. [358]
EKL
0.06 − 1
Fragione et al. [399]
DYN
9 × 10 −5 − 0.015 Arca Sedda [130]
DYN
0.001 − 0.06
Hoang et al. [310]
AGN
MIG+TRP
10 − 300
McKernan et al. [520]
DYN+GAS
1.1 − 6.3
Yang et al. [134]
Funding: MAS acknowledges funding from the European Union's Horizon 2020 research and innovation 1468 programme under the Marie Skłodowska-Curie grant agreement No. 101025436 (project GRACE-BH, 1469 PI: Manuel Arca Sedda). SN acknowledges the partial support from NASA ATP 80NSSC20K0505, NSF 1470 through grant No. 2206428, and thanks Howard and Astrid Preston for their generous support. This work 1471 was supported by the Science and Technology Facilities Council Grant Number ST/W000903/1 (to BK). 14721467
This choice of parameters returns a number of BHs inside 1 pc similar to the value inferred from observations of the Galactic Centre
Acknowledgments:1473Conflicts of Interest: "The authors declare no conflict of interest."
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arxiv |
Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications
Roberto Garra [email protected]
Enzo Orsingher [email protected]
Federico Polito [email protected]
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, "Sapienza"
Dipartimento di Scienze Statistiche "Sapienza"
Università di Roma
Italy
Dipartimento di Matematica "G. Peano" Università degli Studi di Torino
Università di Roma
Italy, Italy
Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications
arXiv:1312.6019v2 [math.AP] 9 Apr 2014
In this paper we discuss some explicit results related to the fractional Klein-Gordon equation involving fractional powers of the D'Alembert operator. By means of a spacetime transformation, we reduce the fractional Klein-Gordon equation to a case of fractional hyper-Bessel equation. We find an explicit analytical solution by using the McBride theory of fractional powers of hyper-Bessel operators. These solutions are expressed in terms of multi-index Mittag-Leffler functions studied by Kiryakova and Luchko [8]. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.
I. INTRODUCTION
In this paper we study the following fractional Klein-Gordon equation
∂ 2 ∂t 2 − c 2 ∆ α u α (x, t) = −λ 2 u α (x, t),(1)
where α ∈ (0, 1], and x ∈ R N . Hence we consider a spacetime fractional order operator, that is a fractional power of the D'Alembert operator. In order to analyze the travelling wave-type solutions of (1) the main trick is based on the space-time transformation
w = c 2 t 2 − N k=1 x 2 k ,
which converts (1) into a case of the fractional hyper-Bessel equation
d 2 dw 2 + N w d dw α u α (w) = − λ 2 c 2α u α (w).(2)
In order to treat (2) we will use the theory developed by A.C. McBride in a series of papers on fractional power of hyper-Bessel-type operators. Here we recall one of his results by showing that the explicit representation of a general hyper-Bessel-type operator is given as a product of Erdélyi-Kober fractional integrals. This fact is also at the basis of the generalized fractional calculus developed by Kiryakova [6]. By means of this theory we find in an explicit form travelling wave solutions of the fractional Klein-Gordon equation. We consider both the one-dimensional and higher-dimensional cases. Similar results on fractional Klein-Gordon-type equations have been recently discussed in [3], where an application to the fractional telegraph-type processes has been investigated. A similar approach was adopted by Garra et al. [4] for the study of the fractional relaxation equation with time-varying coefficients. In view of these results, we also study the nonlinear fractional Klein-Gordon equation with power law nonlinearities. By recurring to the general theory, we are able to find in explicit form some particular solutions also in the nonlinear case. The main aim of this paper is to give new mathematical tools to solve linear and nonlinear space-time fractional equations that are strictly related to the propagation of linear dispersive waves. Moreover we show the way to treat fractional-Bessel equations that have wide applications in different fields of physics.
II. FRACTIONAL HYPER-BESSEL OPERATORS In this section we briefly recall some useful results on the fractional power of hyper-Bessel-type operators. We refer to the theory developed by McBride in a series of papers [13], [14], [12]. The hyper-Bessel operator considered in [12] is defined as
L = x a1 Dx a2 . . . x an Dx an+1 ,(3)
where n is a positive integer number, a 1 , . . . , a n+1 are complex numbers and D = d/dx. Hereafter we assume that the coefficients a j , j = 1, . . . , n + 1, are real numbers. The operator L was first introduced and studied, also with its fractional powers, by Dimovski [2] and served as a base for the generalized fractional calculus in Kiryakova [6]. In this book the whole chapter 3 is devoted to the hyper-Bessel operators, the solutions of differential equations involving it and to the development of its theory in terms of products of Erdélyi-Kober operators. Fractional power of second order version of
L Bn = x −n x d dx x d dx . . . x d dx n times .
are dealt with in section III and IV below. By using operational methods, the integer power of the operator L can be explicitly given in terms of a product of Erdélyi-Kober fractional derivatives (for further details see [12], pag. 527 and [6] pag.59-60). In what follows, we use the notations adopted in McBride works. Let us define the coefficients
a = n+1 k=1 a k , m = |a − n|, b k = 1 m n+1 i=k+1 a i + k − n , k = 1, . . . , n.
It is possible to prove the following result which was first formulated in [12] Lemma 2.1: Let r be a positive integer, a < n,
b k ∈ A p,µ,m :={η ∈ C : ℜ(mη + µ) + m = 1/p − ml, l = 0, 1, 2, . . . }, k = 1, . . . , n, where (p, µ) ∈ [1, +∞) × C.
Then
L r f = m nr x −mr n k=1 I b k ,−r m f,(4)
where, for α > 0 and ℜ(mη + µ) + m > 1/p
I η,α m f = x −mη−mα Γ(α) x 0 (x m − u m ) α−1 u mη f (u) d(u m ),
where the above notation is used for the Erdélyi-Kober fractional integrals; and for α ≤ 0
I η,α m f = (η + α + 1)I η,α+1 m f + 1 m I η,α+1 m x d dx f ,
which is pratically an Erdélyi-Kober derivative in the sense of Kiryakova [6]. The couple of parameters (p, µ) is related to the functional space to which f belongs [12]. The fractional generalization L α of the operator L is consequently defined as follows (see [12], pag. 527).
Definition 2.2: Let m = n − a > 0, α ∈ R, b k ∈ A p,µ,m , for k = 1, . . . , n.
Then,
L α f = m nα x −mα n k=1 I b k ,−α m f.(5)
Note that, for n = 1, a 1 = a 2 = 0 and α > 0, equation (5) coincides with the Riemann-Liouville fractional derivative of order α (see [17] Section 2.3). Moreover, we observe that the topic of fractional Bessel equations has been considered in recent papers with a different approach (see e.g. [16] and the references therein). An application to the description of corneal topography has been also suggested in [15]. A complete study of different approaches to fractional Bessel equations and their applications should be object of a further research.
The following lemma plays a relevant role for the next calculations. Lemma 2.3: Let be η + β m + 1 > 0, m ∈ N, α ∈ R, we have that
I η,α m x β = Γ η + β m + 1 Γ α + η + 1 + β m x β .(6)
III. FRACTIONAL KLEIN-GORDON EQUATION
A. The one-dimensional case
Let us consider the following fractional Klein-Gordon equation
∂ 2 ∂t 2 − c 2 ∂ 2 ∂x 2 α u α (x, t) = −λ 2 u α (x, t),(7)
x ∈ R, t ≥ 0, α ∈ (0, 1].
The classical Klein-Gordon equation (α = 1) emerges from the quantum relativistic energy equation. It is also used in the analysis of wave propagation in linear dispersive media (see, for example, [10]). The fractional Klein-Gordon equation was recently studied in the context of nonlocal quantum field theory, within the stochastic quantization approach (see [9] and the references therein). The fractional power of D'Alembert operator has been considered by [1] and [19], with different approaches. The transformation
z 1 = ct + x, z 2 = ct − x,
reduces (7) to the form
4c 2 ∂ ∂z 1 ∂ ∂z 2 α u α (z 1 , z 2 ) = −λ 2 u α (z 1 , z 2 ).(8)
The partial differential equation (8) involves in fact Riemann-Liouville fractional derivatives with respect to the variables z 1 and z 2 . The further transformation w = √ z 1 z 2
gives the fractional Bessel equation
d 2 dw 2 + 1 w d dw α u α (w) = − λ 2 c 2α u α (w).(9)
The Bessel operator (9) is a special case of L, when n = 2, a 1 = −1, a 2 = 1, a 3 = 0. By definition (5) and Lemma 2.1 we have that m = 2, b 1 = b 2 = 0 and thus
L B = d 2 dw 2 + 1 w d dw appearing in(L B ) α f (w) = 4 α w −2α I 0,−α 2 I 0,−α 2 f (w).(10)
We are now ready to state the following Theorem 3.1: Let α ∈ (0, 1], the fractional equation
(L B ) α u α (w) = − λ 2 c 2α u α (w),(11)
is satisfied by
u α (w) = w 2α−2 ∞ k=0 (−1) k λ 2 α c α 2k w 2αk [Γ(αk + α)] 2 .
(12) Proof: Let β > 0, we have that
(L B ) α w β = 4 α w −2α I 0,−α 2 I 0,−α 2 w β (13) = 4 α Γ β 2 + 1 Γ 1 − α + β 2 2 w β−2α .
By applying now the operator (L B ) α to the function (12) we obtain (since β = 2αk + 2α − 2)
(L B ) α w 2α−2 ∞ k=0 (−1) k λ 2 α c α w α 2k 1 [Γ(αk + α)] 2 = 4 α ∞ k=0 (−1) k λ 2 α c α 2k w 2αk−2 [Γ(αk)] 2 = − λ 2 c 2α u α (w),
as claimed. Remark 3.2: Let us note that our solution to equation (11) expressed in terms of the power series (12) can also be written by recurring to the multi-index Mittag-Leffler functions of Kiryakova and Luchko [8], that is defined as follows
E (n) (αi) n ,(µi) n (z) = ∞ k=0 z k Γ(α 1 k + µ 1 ) . . . Γ(α n k + µ n )
.
(14) Namely we have that (12) can be written as
u α (w) = w 2α−2 E (2) (α,α),(α,α) − λw α 2 α c α 2(15)
Returning to the original problem, the equation (7) admits the solution
uα(x, t) = (c 2 t 2 − x 2 ) α−1 ∞ k=0 (−1) k λ 2k (2c) 2αk c 2 t 2 − x 2 αk [Γ(αk + α)] 2 = (c 2 t 2 − x 2 ) α−1 E (2) (α,α),(α,α) − λ c 2 t 2 − x 2 α/2 2 α c α 2
which for α = 1, reduces to the Bessel function of the first kind
u 1 (x, t) = J 0 λ c c 2 t 2 − x 2 , |x| < ct.(16)
Remark 3.3:
We observe that within a similar approach, some particular solutions of the fractional wave equation with a source term of the type
∂ 2 ∂t 2 − c 2 ∆ α u α (x, t) = f (x, t),(17)
can be simply achieved. This kind of fractional generalization of the wave equation is new and can be of interest for the applications in the fractional approach to the electromagnetic theory (see e.g. [18] and [20]).
B. Relation with the linear damped wave equation
We recall that the linear damped wave equation for waves propagating on an elastically supported string, when the string motion is damped by air friction, has the form
∂ 2 ∂t 2 − ∂ 2 ∂x 2 + 2σ ∂ ∂t u = −u,(18)
where σ is the damping coefficient. It can be proved that by using the transformation
u(x, t) = e −σt v(x, t),(19)
equation (18) is trasformed into the linear Klein-Gordon equation
∂ 2 ∂t 2 − ∂ 2 ∂x 2 v(x, t) = (σ 2 − 1)v(x, t),(20)
when σ 2 < 1. For σ 2 > 1, we obtain the Helmholtz equation which is strictly related to the telegraph equation (see e.g. [3]). In our case we consider a space-time-fractional operator in the linear Klein-Gordon equation. From the point of view of the applications to the propagation of damped waves, our idea is to take into account damping effects in the classical way, i.e. with an exponential damping term such as in (19) and fractional effects in the wave propagation by directly generalizing the linear Klein-Gordon equation (20).
C. Higher-dimensional case
Higher dimensional fractional Klein-Gordon equations can be analyzed in a similar way. Let us consider the Ndimensional fractional Klein-Gordon equation, i.e.
∂ 2 ∂t 2 − c 2 ∆ α u α (x, t) = −λ 2 u α (x, t),(21)
α ∈ (0, 1], x ∈ R N .
By means of the transformation
w = c 2 t 2 − N k=1 x 2 k 1/2
, where x k is the k-th coordinate of the N -dimensional vector x, we transform (21) into
d 2 dw 2 + N w d dw α u α (w) = − λ 2 c 2α u α (w).(22)
The operator appearing in (22) can be considered again as a specific case of the operator (3) with a 1 = −N , a 2 = N , a 3 = 0, a = 0, n = m = 2, b 1 = N −1 2 and b 2 = 0. Hence, from (5) we have that
d 2 dw 2 + N w d dw α u α (w) = 4 α w −2α I 0,−α 2 I N −1 2 ,−α 2 u α (w).
By using arguments similar to those of the previous section, we can prove the following Theorem 3.4: A solution to the N -dimensional fractional Klein-Gordon equation (21), is given by
uα(x, t) = ∞ k=0 λ 2 α c α 2k (−1) k c 2 t 2 − N k=1 x 2 k αk+α−1 Γ(αk + α + N−1 2 )Γ(αk + α) = c 2 t 2 − N k=1 x 2 k α−1 (23) × E (2) (α,α),(α,α+ N −1 2 ) − λ c 2 t 2 − N k=1 x 2 k α/2 2 α c α 2 .
We observe that for α = 1, the solution of Theorem 3.2 reduces to the Bessel function
u 1 (x, t) = J N −1 2 λ c c 2 t 2 − N k=1 x 2 k c 2 t 2 − N k=1 x 2 k N −1 .
For N = 1 we retrieve result (16).
IV. THE NONLINEAR CASE
Here we consider the one-dimensional nonlinear fractional Klein-Gordon equation with power law nonlinearity,
∂ 2 ∂t 2 − c 2 ∂ 2 ∂x 2 α u α (x, t) = λu s α (x, t),(24)
x ∈ R, t ≥ 0, α ∈ (0, 1], λ ∈ R, s = 1.
The higher dimensional case can be handled in a similar way. In the recent literature, some attempts to find specific solutions to the nonlinear fractional Klein-Gordon equation were discussed (see e.g. [5]). However, these papers are based on the application of approximate methods such as the homotopy perturbation method and related to a different formulation of the fractional Klein-Gordon equation. In view of the previous discussion, we are going to find an explicit travelling wave solution of (24).
Theorem 4.1:
A travelling wave solution of (24) is given by
u α (x, t) = 4 α λ Γ 1 + α 1−s Γ 1 − α + α 1−s 2 1 s−1 (25) × c 2 t 2 − x 2 α/(1−s) .
Proof: We are going to study exact solutions in the travelling wave form u α ( √ c 2 t 2 − x 2 ). By means of the transformation w = c 2 t 2 − x 2 1/2 , we transform equation (24) in
d 2 dw 2 + 1 w d dw α u α (w) = λu n α (w).(26)
Assuming that the solution we are searching is in the form
u α (w) = kw β ,(27)
where β and k are real parameters that will be fixed in the next. Substituting (27) in (24) and using (13) we obtain
4 α Γ β 2 + 1 Γ 1 − α + β 2 2 kw β−2α = λk s w βs ,(28)
that is satisfied when
β = 2α 1−s k = 4 α λ Γ(1+ α 1−s ) Γ(1−α+ α 1−s ) 2 1 s−1 ,(29)
as claimed.
We observe that, for s < 1 we have bounded solutions for |x| ≤ ct, while for s > 1 the solutions are bounded for |x| < ct. We recall that similar specific solutions to the nonlinear Klein-Gordon equation (in the non fractional case) were investigated by [11]. In particular, for α = 1, we recover the solution (4.7a) in the two-dimensional case (space and time), that is Note that c = 1 in the original paper [11].
Similarly to Theorem 4.1, an exact solution of the nonhomogeneous equation
∂ 2 ∂t 2 − c 2 ∂ 2 ∂x 2 α u α (x, t) = λu s α (x, t) + γ c 2 t 2 − x 2 α/(1−s) ,
x ∈ R, t ≥ 0, α ∈ (0, 1], n = 1, γ, λ ∈ R, can be found. A case of particular physical interest is s = 3, where (24), for α = 1, is strictly related to scalar φ 4 theory. In the fractional case, we have the specific solution
u α (x, t) = 4 α λ Γ 1 − α 2 Γ 1 − 3 2 α 2 1 2 (31) × c 2 t 2 − x 2 −α/2 ,
which, for α = 1, becomes
u 1 (x, t) = [λ(c 2 t 2 − x 2 )] −1/2 ,(32)
that is the so-called meron solution in gauge theory.
AcknowledgementsF. Polito has been supported by project AMALFI (Università di Torino/Compagnia di San Paolo). We thank the anonymous reviewers for their accurate analysis of the first draft of the paper and for bringing some relevant papers to our attention.
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| {'fraction_non_alphanumeric': 0.08337002788786144, 'fraction_numerical': 0.04589265619648711, 'mean_word_length': 3.6539162112932604, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 38, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "In this paper we discuss some explicit results related to the fractional Klein-Gordon equation involving fractional powers of the D'Alembert operator. By means of a spacetime transformation, we reduce the fractional Klein-Gordon equation to a case of fractional hyper-Bessel equation. We find an explicit analytical solution by using the McBride theory of fractional powers of hyper-Bessel operators. These solutions are expressed in terms of multi-index Mittag-Leffler functions studied by Kiryakova and Luchko [8]. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.", 'arxivid': '1312.6019', 'author': ['Roberto Garra [email protected] ', 'Enzo Orsingher [email protected] ', 'Federico Polito [email protected] ', '\nDipartimento di Scienze di Base e Applicate per l\'Ingegneria, "Sapienza"\nDipartimento di Scienze Statistiche "Sapienza"\nUniversità di Roma\nItaly\n', '\nDipartimento di Matematica "G. Peano" Università degli Studi di Torino\nUniversità di Roma\nItaly, Italy\n'], 'authoraffiliation': ['Dipartimento di Scienze di Base e Applicate per l\'Ingegneria, "Sapienza"\nDipartimento di Scienze Statistiche "Sapienza"\nUniversità di Roma\nItaly', 'Dipartimento di Matematica "G. Peano" Università degli Studi di Torino\nUniversità di Roma\nItaly, Italy'], 'corpusid': 15442383, 'doi': '10.1109/icfda.2014.6967381', 'github_urls': [], 'n_tokens_mistral': 7738, 'n_tokens_neox': 6481, 'n_words': 3663, 'pdfsha': 'cea5e3dbb72534f7440bd2d20f3170d07562d1c3', 'pdfurls': ['https://arxiv.org/pdf/1312.6019v2.pdf'], 'title': ['Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications', 'Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications'], 'venue': []} |
arxiv |
Hierarchical Beam Training for Extremely Large-Scale MIMO: From Far-Field to Near-Field
24 May 2023
Student Member, IEEEYu Lu
Department of Electronic Engineering
Center for Information Science and Technology (BNRist)
Tsinghua University as well as Beijing National Research
100084BeijingChina
Student Member, IEEEZijian Zhang
Department of Electronic Engineering
Center for Information Science and Technology (BNRist)
Tsinghua University as well as Beijing National Research
100084BeijingChina
Fellow, IEEELinglong Dai [email protected].
Department of Electronic Engineering
Center for Information Science and Technology (BNRist)
Tsinghua University as well as Beijing National Research
100084BeijingChina
Hierarchical Beam Training for Extremely Large-Scale MIMO: From Far-Field to Near-Field
24 May 20231 2
Extremely large-scale MIMO (XL-MIMO) is a promising technique for future 6G communications.The sharp increase in the number of antennas causes electromagnetic propagation to change from farfield to near-field. Due to the near-field effect, the exhaustive near-field beam training at all angles and distances requires very high overhead. The improved fast near-field beam training scheme based on time-delay structure can reduce the overhead, but it suffers from very high hardware costs and energy consumption caused by time-delay circuits. In this paper, we propose a near-field two dimension (2D) hierarchical beam training scheme to reduce the overhead without the need for extra hardware circuits.Specifically, we first formulate the multi-resolution near-field codewords design problem covering different angle and distance coverages. Next, inspired by phase retrieval problems in digital holography imaging technology, we propose a Gerchberg-Saxton (GS)-based algorithm to acquire the theoretical codeword by considering the ideal fully digital architecture. Based on the theoretical codeword, an alternating optimization algorithm is then proposed to acquire the practical codeword by considering the hybrid digital-analog architecture. Finally, with the help of multi-resolution codebooks, we propose a near-field 2D hierarchical beam training scheme to significantly reduce the training overhead, which is verified by extensive simulation results.Index TermsExtremely large-scale MIMO (XL-MIMO), extremely large-scale antenna array (ELAA), beam training, codebook design.All authors are with the
I. INTRODUCTION
With the emergence of new applications such as digital twins, 6G is expected to achieve a 10-fold increase in spectrum efficiency than 5G [1], [2]. The extremely large-scale MIMO (XL-MIMO) is a promising technique for 6G to achieve ultra-high spectrum efficiency [3], [4]. In XL-MIMO systems, the base station (BS) usually deploys an extremely large-scale antenna array (ELAA), which consists of hundreds or even thousands of antennas. ELAA in the XL-MIMO system is expected to drastically improve spatial resolution to realize a high spatial multiplexing gain in 6G. In order to obtain spatial multiplexing gain, XL-MIMO should generate a directional beam with high array gain by beamforming. To support beamforming, beam training should be conducted to search the optimal beamforming vector, i.e., codeword, in the predefined codebook.
As the number of BS antennas in XL-MIMO systems is much larger than that of 5G systems, the high-dimensional XL-MIMO beam training overhead will be overwhelming.
A. Prior Works
There are two typical categories of beam training methods for MIMO, which are far-field beam training and near-field beam training respectively. For the first category, since the antenna number at BS is usually not very large in 3G-5G systems, the MIMO channel is modeled in the far-field region with the planar wave assumption, where the array response vector of the far-field channel is only related to the angle. In this case, the orthogonal Discrete Fourier Transform (DFT) codebook can be utilized in beam training to capture the physical angle information in the angle-domain of the channel paths [5], [6]. However, since the size of the DFT codebook is proportional to the number of antennas at BS, the DFT codebook suffers from very high training overhead when it comes to XL-MIMO systems. Thus, to reduce the beam training overhead, some hierarchical beam training schemes were proposed [7], [8]. The basic idea of the beam training is to search from the lowest-resolution codebook to the highest-resolution codebook layer by layer, where the angle range needed to be scanned reduces layer by layer gradually.
With the help of hierarchical beam training, the overhead becomes proportional to the logarithm of the antenna number at BS [8].
As the antenna number dramatically increases in 6G XL-MIMO systems, the near-field range expands by orders of magnitude, which can extend to several hundred meters [9]. Thus, the XL-MIMO channel should be modeled in the near-field region subjected to the spherical wave assumption. In this case, the existing far-field beam training schemes may not be valid for the near-field XL-MIMO channel. To cope with this problem, near-field beam training should be utilized to match the near-field XL-MIMO channel feature. For the second category, i.e., nearfield beam training, the array response vector of the near-field channel is not only related to the angle but also to distance. Thus, to capture the physical angle information as well as distance information of the channel paths, a polar-domain codebook [10] should be utilized instead of a DFT codebook. Accordingly, the size of the polar-domain codebook is the product of the antenna number at BS and the number of sampled distance grids. Since only one angle and one distance can be measured in each time slot, the exhaustive search method for near-field beam training has a very high overhead [11]. To address this problem, we have proposed a fast timedelay based near-field beam training for XL-MIMO with low overhead [12], where each antenna requires time-delay circuits to provide frequency-dependent phase shift. In specific, due to the near-field beam split effect in a wideband situation, near-field beams can be flexibly controlled by extra time-delay hardware circuits and then focus on different angles and distances at different frequencies in one time slot. However, the time-delay based beamforming structure will lead to not only high hardware costs but also very high energy consumption, especially for XL-MIMO systems with a large number of antennas.
B. Contributions
Thus, to design a general and low-overhead beam training scheme, we propose a nearfield two dimension (2D) hierarchical beam training scheme by designing the multi-resolution codebooks referring to the hierarchical beam training in the far-field scenario. Our contributions are summarized as follows.
1) We first formulate the problem of near-field codeword design. Specifically, compared with the far-field case, the ideal beam pattern of near-field codeword should not only cover a certain angle range but also a certain distance range. By considering ideal fully digital architecture, we provide the design problem of the near-field theoretical codeword. Then, based on the theoretical codeword, we formulate the problem of a practical codeword with assumptions of the hybrid digital-analog structure and quantized phase shifts in practice.
2) In order to design the near-field theoretical codeword, inspired by the Gerchberg-Saxton (GS) algorithm in phase retrieval problems for digital holography imaging, we propose a GS-based theoretical codeword design algorithm for a fully digital architecture. Different from the original GS algorithm, we modify the transformation methods from Fourier transform to polar-domain transform to match the near-field assumption. Additionally, the power constraint instead of amplitude measurements are considered in each iteration to control the power of the codeword.
3) Since fully digital architecture with high energy assumption is not available in a practical XL-MIMO system, we then design the practical codeword considering the hybrid digital-analog architecture. Based on the theoretical codeword, an alternating optimization algorithm is proposed to acquire the practical codeword, where the digital beamforming vector and the analog beamforming matrix are optimized iteratively. Specifically, in each iteration, the digital beamforming vector is obtained by a closed-form solution. Meanwhile, phases of the entries in the analog beamforming matrix are solved individually by a highefficient iterative search method. 4) Next, we generate multi-resolution codebooks based on the practical codewords obtained by the alternating optimization algorithm. With the aid of multi-resolution codebooks with different angle coverages and distance coverages, we propose a near-field two dimension (2D) hierarchical beam training scheme. Specifically, codewords are searched in multiresolution codebooks layer by layer, where angle and distance ranges are reduced gradually.
Moreover, we provide the analysis of the proposed beam training overhead, which is proportional to the sum of the logarithm of the antenna number and the sampled distance grid number. Simulation results show that the proposed beam training scheme can reach sub-optimal achievable rate performance with low overhead.
C. Organization and Notations
Organization: The rest of the paper is organized as follows. In Section II, we first introduce the signal model, the near-field channel model, and the formulation of the near-field codebook design problem. In Section III, we provide the design of the theoretical codeword by the proposed Gerchberg-Saxton algorithm considering fully digital architecture. In Section IV, we propose an alternating optimization scheme to design the practical codeword with hybrid digital-analog architecture. Then, the proposed near-field 2D hierarchical beam training scheme is described in Section V. Simulation results and conclusions are provided in Section VI and Section VII, respectively.
Notations: Lower-case and upper-case boldface letters a and A denote a vector and a matrix, respectively; a H and A H denote the conjugate transpose of vector a and matrix A, respectively; ∥a∥ 2 denotes the l 2 norm of vector a; ∥a∥ F denotes the Frobenius norm of vector a. 0 N ×M denotes N × M -dimensional null matrix. Finally, CN (µ, Σ) denotes the probability density function of complex multivariate Gaussian distribution with mean µ and variance Σ. U(−a, a) denotes the probability density function of uniform distribution on (−a, a).
II. SYSTEM MODEL
In this section, we will first introduce the signal model of the XL-MIMO system. Then, the existing near-field channel model will be briefly reviewed. Finally, we formulate the problem of codeword design in the near-field scenario.
A. Signal Model
We consider the scenario where the BS employs a N -element ELAA to communicate with a single-antenna user. Let h H ∈ C 1×N denote the channel from the BS to the user. Since the XL-MIMO channel h H is generally dominated by a few main paths, we only need to search the physical location of the main paths by beam training instead of acquiring the explicit channel information [13], [14]. Therefore, the main path is concerned in this paper, and the corresponding beam training method will be investigated to search for the optimal beamforming vector to align with the main path. Take downlink transmission as example, the received signal y can be represented by
y = h H vs + n,(1)
where v ∈ C N ×1 represents the beamforming vector at the BS, which is essentially a codeword chosen from the predefined codebook, s represents the symbol transmitted by the BS, and n ∼ CN (0 N , σ 2 I N ) represents the received noise with σ 2 representing the noise power. The beam training is to measure the power of y to find the best codeword from the codebook.
Next, we will briefly review the existing near-field XL-MIMO channel model for existing near-field beam training schemes.
B. Near-Field XL-MIMO Channel Model
When the distance between the BS and the UE is smaller than the Rayleigh distance [15], the near-field XL-MIMO channel should be modeled with the spherical wave assumption, which can be expressed by
h = √ N αb (θ, r) .(2)
where α is the complex path gain. b (θ, r) represents the near-field array response vector, which can be represented by [10] b(θ, r)
= 1 √ N [e −j 2π λ (r (1) −r) , · · · , e −j 2π λ (r (N ) −r) ] H ,(3)
where r represents the distance from the UE to the center of the antenna array, r (n) = r 2 + δ 2 n d 2 − 2rδ n dθ represents the distance from the UE to the nth BS antenna, and δ n = 2n−N −1 2 with n = 1, 2, · · · , N .
Before data transmission, beam training should be applied to estimate the physical angles and distances of near-field channel paths. The near-field response vector b(θ, r) implies that the optimal beam training codeword should focus on the spatial angle θ and BS-UE distance r. The existing near-field beam training scheme is conducting an exhaustive search in the polar-domain codebook [10], which can be represented as
A = [b(θ 1 , r 1 1 ), · · · , b(θ 1 , r S 1 1 ), · · · , b(θ N , r S N N )],(4)
where each column of polar-domain codebook A is a codeword aligned with the grid (θ n , r sn n ), with s n = 1, 2, · · · , S n , S n denotes the number of sampled distance grids at θ n . Therefore, the number of total sampled grids of the whole propagation environment is S = N n=1 S n . Apparently, in XL-MIMO systems, the codebook should not only sample angle but also distance, which leads to a large-size codebook and unfordable beam training overhead. Thus, to address this problem, we design the hierarchical near-field codebook with multi-resolution codebooks, and then propose the corresponding near-field 2D hierarchical beam training. To design the multi-resolution nearfield codebooks, we will first formulate the design problem of a near-field codeword with different angle coverage and distance coverage.
C. Formulation of Codebook Design Problem
Suppose the angle coverage and distance coverage of
codeword v are B v,θ ≜ [θ, θ + B θ ] and B v,r ≜ [r, r + B r ],
where B θ and B r are the angle sampled step and distance sampled step. The ideal beam pattern vector of the codeword v is denote as
g v = g v (θ 1 , r 1 1 ), · · · , g v (θ N , r 1 N ), · · · , g v (θ N , r S N N ) ,(5)
where g v (θ, r) = |g v (θ, r)| e jfv(θ,r) is the theoretical beamforming gain. The amplitude information |g v (θ, r)| of the ideal beam pattern can be further represented by
|g v (θ, r)| = √ C v , θ ∈ B v,θ , r ∈ B v,r 0, θ / ∈ B v,θ , r / ∈ B v,r .(6)
For the ideal beam pattern in (5), the amplitude information |g v (θ, r)| of ideal beam pattern vector in target angle coverage and distance coverage are fixed and flattened while other beamforming gains are zero. Meanwhile, the phase information f v (θ, r) of the ideal beam pattern vector can be designed flexibly. Compared to a far-field codeword, the near-field codeword should cover not only a certain angle range but also a certain distance range.
To evaluate the effectiveness of the codeword v, we reference G (v, θ, r) as the beamforming gain of v in the angle θ and the distance r. The G (v, θ, r) can be represented as
G(v, θ, r) = √ N b(θ, r) H v.(7)
Thus, according to the definition of polar-domain codebook A in (4), the beam pattern obtained by beamforming with codeword v can be presented as A H v.
The aim of designing a codeword is to make the beam pattern A H v obtained by beamforming with the codeword v as close as possible to the ideal beam pattern g v . Thus, the objective of the theoretical codeword v design can be express as
min v,f (θ,r) A H v − g v 2 2 . (P1)
In (P1), the ideal theoretical codeword v can only be realized by the fully digital architecture, where each antenna requires one dedicated radio frequency (RF) chain to realize fully digital signal processing. However, fully digital architecture in the XL-MIMO system results in unaffordable energy consumption. In fact, a hybrid digital-analog structure is usually preferred in XL-MIMO systems to improve energy efficiency [16]. In this structure, we need to design practical codewords considering the hardware constraints in terms of phase shifter resolution and the number of radio frequency (RF) chains N RF [17].
Specifically, based on the ideal theoretical codeword v, the design of the practical codeword v p ≜ F RF f BB can be represented as
min F RF ,f BB ∥v − F RF f BB ∥ 2 s.t. ∥F RF f BB ∥ 2 = 1, [F RF ] n,i = e jδ n,i , δ n,i ∈ Φ b n = 1, 2, . . . , N, i = 1, 2, . . . , N RF ,(P2)
where the F RF ∈ C N ×N RF and f BB ∈ C N RF ×1 are the analog beamforming matrix and the digital beamforming vector.
Φ b = π −1 + 1 2 b , π −1 + 3 2 b , . . . π 1 − 1 2 b
is the set of quantized phase shifters with b bits.
All the codewords in the codebook can be designed based on (P1) and (P2). Next, we introduce the design method of the theoretical codeword v in Section III and practical codeword v p Section IV.
III. PROPOSED GERCHBERG-SAXTON ALGORITHM BASED NEAR-FIELD THEORETICAL
CODEWORD DESIGN
In this section, we will first briefly review the Gerchberg-Saxton algorithm applied in the phase retrial problem in the hologram optical system, and the relationship between the phase retrieval problem and the codeword design problem is analyzed. Next, we propose a GS-based theoretical codeword design scheme. Finally, the convergence property of the GS algorithm in near-field codeword design is provided.
A. Preview of the Phase Retrieval Problem and Gerchberg-Saxton algorithm 1) Phase retrieval problem in digital holography imaging: In recent years, with the development of modern optics and computer science, digital holography imaging technology has changed the traditional imaging object-image relationship and structure by combining the frontend optical system design with the back-end signal processing. The back-end signal processing algorithm of the original data collected by the camera can break through the traditional imaging bottleneck.
In specific, in optical systems, the amplitude information is easy to measure, while the direct recording of the phase information is not allowed. The reason is that the electromagnetic field oscillates at a very high frequency that rare electronic measurement devices can follow [18].
Thus, in order to realize the imaging of the original object, one of the most important problems in digital holography imaging technology is conducting phase retrieval [19]. Fortunately, with the help of the measured amplitude information, some signal processing algorithms offer alternative methods for recovering the phase information of optical images without requiring sophisticated devices.
Reviewing the theoretical codeword design problem in (P1), it is obvious that the problem (P1) is similar to the phase retrieval in digital holography imaging, where the phase information (f v (θ, r) of the ideal beam pattern vector) should be obtained by measured amplitude information (|g v (θ, r)| of ideal beam pattern vector).
2) Gerchberg-Saxton algorithm: One of the most popular methods to solve the phase retrieval problem is Gerchberg-Saxton (GS)-based algorithm [20], [21] as shown in Fig. 1 (a), where two amplitude measurements are iteratively imposed in the object plane and diffraction pattern plane [22], [23]. It is worth noting that the diffraction pattern plane is also known as the Fourier Some modified versions of the GS algorithm have been proposed afterward [24] to match various imaging problems. Instead of utilizing the GS algorithm in the imaging problem, we improved the GS algorithm in the near-field codeword design problem. In specific, we replace one of updating processes with measured amplitude information by applying normalization to match the power constraint of the codeword.
B. Design of the Theoretical Codeword v
In order to solve the (P1), we draw the experience from the Gerchberg-Saxton (GS) algorithm, which is widely applied in phase retrieval problems in digital hologram imaging of optical systems. In the phase retrieval problem, the phase information needed to be obtained with the fixed amplitude information, which is the same as the phase information f v (θ, r) design of the ideal beam pattern in the problem (P1). Specifically, the proposed GS-based near-field codeword design procedure is shown in Algorithm 1.
For notation simplicity, in the description of the GS algorithm, we usev (s) , g (s) , g ′ (s) , and v ′ (s) to denote the designed codeword vector, the beam pattern vector realized by the designed codeword, the revised beam pattern vector with ideal beam pattern amplitude, and the codeword vector obtained by revised beam pattern vector in the s-th iteration of GS algorithm.
Before the GS algorithm starts, we should first obtain the initial beam pattern vector g (0) with randomly generated phase f (0) (θ, r) and amplitude information g v (θ, r) of ideal beam pattern vector g v . In this way, the g (0) can be represented as
g (0) = g v (θ 1 , r 1 1 ) f (0) (θ 1 , r 1 1 ),· · ·,|g v (θ N , r 1 N )|f (0) (θ N , r 1 N ), · · · , |g v (θ N , r S N N )|f (0) (θ N , r S N N ) .(8)
Algorithm 1: GS-based theoretical codeword design Inputs:
|g v |, C v , S max , A, B v,θ , B v,r .
Initialization: randomly generate f (0) (θ, r) and obtain the g (0) by (8).
1.v ′ (0) = AA H −1 Ag (0) 2. Obtainv (1) by normalizingv ′ (0)
3. for s = 1, 2, · · · , S max do 4.
calculate g (s) based onv (s) by (9) 5. calculate g ′ (s) based on g (s) and g v by (10) 6.
calculatev ′ (s) based on g ′ (s) by (11) 7.
if s < S max 8. calculatev (s+1) based onv ′ (s) by (12) 9.
end if 10. end for
11. v =v ′ (Smax) /||v ′ (Smax) || 2 Output: Theoretical codeword v.
In s-th iteration, with provided designedv (s) ,
g (s) = A Hv (s) .(9)
Then, in order to maintain the amplitude information of the ideal beam pattern vector g v to approach the ideal beam pattern, we assign the amplitude information |g v (θ, r)| of ideal beam pattern g v to g ′ (s) , and the phase information fv (s) (θ, r) of current beam pattern g (s) to g ′ (s) . In this case, the g ′ (s) can be presented as
g ′ (s) = g v (θ 1 , r 1 1 ) fv (s) (θ 1 , r 1 1 ), · · · , |g v (θ N , r 1 N )|fv (s) (θ N , r 1 N ), · · · , |g v (θ N , r S N N )|fv (s) (θ N , r S N N ) .(10)
Base on the (P1), given g ′ (s) , thev ′ (s) can be obtained by least square algorithm aŝ
v ′ (s) = AA H −1 Ag ′ (s) = A † g ′ (s) ,(11)
where the pseudo inverse of A H is denoted as A † . Finally, we normalize thev ′ (s) aŝ
v (s+1) =v ′ (s) /||v ′ (s) || 2 .(12)
After the iteration number reaches S max , we utilizev ′ (Smax) to obtain the designed theoretical codeword v.
C. Convergence Property of GS Algorithm in Near-Field Codeword Design
As mentioned before, the original GS algorithm assumes that the object and the diffraction pattern planes are connected through a Fourier Transform (FT). The convergence of the original GS algorithm with FT assumption is proved based on Parseval's theorem of FT [25], where the energy of wavefronts in the object and the diffraction pattern planes before and after FT and inverse FT are the same. However, the codeword vector plane and beam pattern vector plane in the proposed GS algorithm are connected with the polar-domain transformation, which does not satisfy Parseval's theorem. Thus, the convergence property of the proposed GS algorithm based on polar-domain transformation in near-field codeword design should be analyzed.
In this paper, the convergence of the proposed GS algorithm is supervised by the squared error in each iteration. Specifically, the squared error of the beam pattern plane in s-th iteration can be presented as
E (s) = ∥g (s) (θ, r) − g ′ (s) (θ, r)∥ 2 2 dθdr. = ∥A Hv (s) (u, w) − A Hv′ (s) (u, w)∥ 2 2 dudw(13)
It is worth noting that the codewords in the polar-domain codebook A H have been rearranged,
where the codewords aligned with the largest distance S n of each θ n are brought to the front columns of A H . Thus, the A H can be rewritten as
A H = [A 1 , A 2 ] H ,(14)
where
A 1 = [b(θ 1 , r S 1 1 ), b(θ 2 , r S 2 2 ), · · · , b(θ N , r S N N )], A 2 = [b(θ 1 , r 1 1 ),· · · ,b(θ 1 , r S 1 −1 1 ),· · · ,b(θ N , r 1 N ),· · · ,b(θ N , r S N −1 N )].
Since the S n in each column b(θ n , r Sn 1 ) of A 1 is larger than Rayleigh distance, b(θ n , r Sn 1 ) approximates to the far-field codeword aligned with the physical direction θ n . In this case, the A 1 is equal to a far-field w) is an FT process, which satisfies Parseval's theorem as
DFT codebook. Thus, A H 1 v (s) (u, w)−v ′ (s) (u,∥ v (s) (u, w)−v ′ (s) (u, w) ∥ 2 2 dudw = ∥A H 1 v (s) (u, w)−v ′ (s) (u, w) ∥ 2 2 dudw.(15)
Therefore, E (s) can be further expressed as
E (s) = ∥A H 1 v (s) (u, w)−v ′ (s) (u, w) ∥ 2 2 + ∥A H 2 v (s) (u, w)−v ′ (s) (u, w) ∥ 2 2 dudw. ≥ ∥ v (s) (u, w)−v ′ (s) (u, w) ∥ 2 2(16)
The squared error of the codeword vector plane of s + 1-th iteration for the GS algorithm can be expressed as
E 0 (s) = ∥v (s+1) (u, w) −v ′ (s) (u, w)∥ 2 2 dudw.(17)
Then, we provide Lemma 1 to show the change of squared error between adjacent iteration in codeword vector plane.
Lemma 1:
In the codeword vector plane of GS algorithm, the error betweenv (s) (u, w) and
v ′ (s) (u, w) not less than than the error betweenv
(s+1) (u, w) andv ′ (s) (u, w), i.e., ∥v (s) (u, w)−v ′ (s) (u, w)∥ 2 2 > ∥v (s+1) (u, w)−v ′ (s) (u, w)∥ 2 2 . proof: See Appendix A.
From the (16), (17), and Lemma 1, we can derive that
E 0 (s) ≤ ∥v (s) (u, w) −v ′ (s) (u, w)∥ 2 2 dudw ≤ E (s)(18)
On the other hand, E 0 (s) can be further expressed as
E 0 (s) = ∥v (s+1) (u, w) −v ′ (s) (u, w)∥ 2 2 dudw = ∥A † g (s+1) (θ, r) − A † g ′ (s) (θ, r)∥ 2 2 dθdr.(19)
Utilizing the uniqueness of pseudo inverses, we can easily know that A † = (A H ) −1 , 0 (S−N )×N .
In this case, since A H 1 g (s+1) (θ, r)−g ′ (s) (θ, r) is a inverse FT process, which also satisfies Parseval's theorem. Thus,
E 0 (s) = ∥g (s+1) (θ, r) − g ′ (s) (θ, r)∥ 2 2 dθdr.(20)
Similar to Lemma 1, we can obtain that
∥g (s+1) (θ, r)−g ′ (s) (θ, r)∥ 2 2 ≥ ∥g (s+1) (θ, r)−g ′ (s+1) (θ, r)∥ 2 2(21)
Thus, given (20) and (21) E 0 (s) ≥ ∥g (s+1) (θ, r) − g ′ (s+1) (θ, r)∥ 2 2 dθdr = E (s+1) .
Combining the equation (18) and (22), we can observe that
E (s+1) ≤ E 0 (s) ≤ E (s) ,(23)
which means that the squared error in each iteration decreases. Thus, the convergence property of the proposed GS algorithm is proven.
IV. PROPOSED ALTERNATING OPTIMIZATION BASED NEAR-FIELD PRACTICAL CODEWORD
DESIGN
It is well known that each antenna requires one dedicated radio-frequency (RF) chain to realize the fully digital architecture. In this way, an XL-MIMO system with a very large number of antennas leads to an equally large number of RF chains, which will result in unaffordable hardware costs and energy consumption. To solve this problem, hybrid digital-analog architecture is preferred in practice, where the fully digital beamforming matrix is decomposed into a highdimensional analog beamforming matrix and a low-dimensional digital beamforming vector.
Moreover, quantized phase shifts instead of continuous quantized phase shifts are accessible for realizing analog beamforming matrix. Thus, in this section, alternating optimization is proposed for practical codeword design considering the hybrid digital-analog architecture and quantized phase shifts.
Based on the theoretical codeword v obtained by Algorithm 1, we solve the practical codeword v p design problem (P2) by alternating optimizing the digital beamforming vector f BB and the analog beamforming matrix F RF considering the hardware constraints. Algorithm 2 provides the specific procedure to design the practical codeword.
For the given analog beamforming matrix F RF , the optimization problem of the digital beamforming vector f BB can be expressed as
min f BB ∥v − F RF f BB ∥ 2 , (P2.1)
which can be solved by least square aŝ
f BB = F H RF F RF −1 F H RF v(24)
Algorithm 2: Practical codeword design
Inputs: v, T max , P max , Φ b , N , N RF .
Initialization: randomly generate F 0 RF . 1. for t = 1, 2, · · · , T max do // Design the digital beamforming vector.
2. calculate the f t BB by (24) // Design the analog beamforming matrix.
3. for p = 1, 2, · · · , P max do
4.
for n = 1, 2, · · · , N do
5.
for i = 1, 2, · · · , N RF do
6.
Search δ n,i to satisfy (25
Output: f BB = f Tmax BB , F RF = F Tmax RF , v p = F Tmax RF f Tmax BB
Then, for the given analog beamforming vector f BB , the optimization problem of F RF can be expressed as min
F RF ∥v − F RF f BB ∥ 2 s.t. ∥F RF f BB ∥ 2 = 1, [F RF ] n,i = e jδ n,i , δ n,i ∈ Φ b , n = 1, 2, . . . , N, i = 1, 2, . . . , N RF , (P2.2)
The optimization of F RF problem (P2.2) can be converted to the minimization absolute value of each entry of the vector v − F RF f BB . Hence, the problem (P2.2) can be transformed into N sub-problems, which can be optimized one by one. The n-th sub-problem is rewritten as
min θ 1 ,θ 2 ,...,θ N RF [v] n − N RF i=1 [f BB ] i e jδ n,i s.t. δ n,i ∈ Φ b , i = 1, 2, . . . , N RF .(25)
To obtain the solution to (25), the exhaustive search is a obvious choice, where all the combination of δ n,1 , · · · , δ n,N RF are test to minimize the objective. However, the number of combination is 2 bN RF , which has prohibitively high computational complexity. For example, if b = 4, N RF = 32, the 2 bN RF ≈ 7.9 × 10 28 ! Thus, we need to investigate near-optimal search method to reduce complexity.
In this case, we propose a high efficient individual search method, where each δ n,i is determined separately in each iteration. The specific procedures are summarized in Algorithm 2. We firstly initialize the δ 0 n,1 , · · · , δ 0 n,N RF by choosing the entry from the Φ b and generate F 0 RF . In p-th iteration, we find best δ n,1 , · · · , δ n,N RF one by one. In step 6, for δ n,i , we search through the Φ b to find the optimal choice to satisfy the (25). This iterative process performs stop until the number of iterations reaches predetermined figure or δ p−1 n,i = δ p n,i . Then the n-th row of the designedF RF can be expressed as and B r , the corresponding codeword has a lower resolution, and the size of the corresponding codebook becomes smaller. As mentioned before, we can generate near-field multi-resolution codebooks with different angle coverage and distance coverage based on the Algorithm 1 and
Algorithm 2.
Then, these multi-resolution codebooks are applied to conduct near-field 2D hierarchical beam training. Compared with far-field scenario, the near-field 2D hierarchical beam training need to reduce the search range of angle and distance at the same time as shown in Fig. 2.
The specific near-field beam training procedure is summarized in Algorithm 3. First, as shown in Step2, for l-th codebook generation, we need to divide the angle coverage B l v k ,θ and distance coverage B l v k ,r based on angle samples step B l θ and distance samples step B l r for each codeword v k . Then, in Steps 3-4, the codewords design scheme based on Algorithm 1 and Algorithm 2 is applied to obtain the l-th codebook W l . Then, Steps 7-16 are operated to search the optimal codeword in multi-resolution codebooks layer by layer.
B. Comparison of the Beam Training Overhead
Beam training overhead refers to the number of time slots used for beam training. Generally, the beam training overhead is determined by the spatial resolutions of an antenna array on Algorithm 3: Near-field 2D hierarchical beam training Inputs: L, B 1 θ , B 2 θ , · · · , B L θ , B 1 r , B 2 r , · · · , B L r , y opt = 0, s opt = 0 // Generate L sub-codebooks 1. for l = 1, 2, · · · , L do 2. generate the collection of B l v l,k ,θ and B l v l,k ,r based on B l θ and B l r 3. generate |g v (θ, r)| for based on (6) 4. obtain the practical codewords in l-th sub-codebook W l based on Algorithm 1 and Algorithm 2.
end for
6. W = W 1 // Conduct beam training 7. for l = 1, 2, · · · , L do 8. for v l,k in W do 9. y l k = h H v l,k s + n 10.
if y l k > y opt then 11. k opt = k 12. end if 13. end for Far-field hierarchical scheme [26] L l U (l) 40
14. choose v l+1,k in W l+1 satisfied B l+1 v l+1,k ,θ ∈ B l v l,
Far-field exhaustive search scheme [27] U 512
Near-field exhaustive search scheme [10] US 8192
Time-delay based near-field scheme [12] S 16
Proposed near-field 2D hierarchical scheme After we conduct beamforming with the designed practical codeword, we can obtain Fig. 3 (b), which presents the beamforming gains of different locations in space with the designed practical codeword. From Fig. 3 (b) we can see that the target location has the largest beamforming gain and other locations have much lower beamforming gains. Moreover, for the codeword in the layer 2 codebook, the designed practical codeword can also approach the ideal beam pattern Fig. 3 (c) and (d). Since the codeword in the layer 1 codebook should cover a larger range than that of layer 2 codebook, we can observe that the beamforming gain of non-target position in Fig. 3 (b) is also larger than that in Fig. 3 (d). [26], far-field exhaustive search beam training scheme [27], the near-field exhaustive search beam training scheme [10], and time-delay based near-field beam training scheme [12]. We set the number of angle and distance grids as U = 512 and Fig. 4 (a), where the bandwidth is 100 MHz, we can observe that the proposed near-field 2D hierarchical beam training can achieve the best performance of all schemes with relatively lower overhead. For example, the proposed scheme outperforms the far-field angle-domain codebook with only half of the beam training overhead. The reason is that the existing far-field codebook can only capture the angle information of the channel path. Moreover, the time-delay based scheme has worse performance than the proposed scheme in this narrow-band condition. The principal reason is that the ability of timedelay circuits to control the beam split will decrease by reducing the bandwidth. Meanwhile, Fig. 4 (b) illustrates the wideband situation, where the bandwidth is 500 MHz. It can be observed that the time-delay based beam training scheme has better performance than the proposed scheme.
However, the proposed scheme has much lower hardware cost and is bandwidth-independent.
Thus, we believe that the proposed scheme provides a tradeoff between the performance and overhead in near-field XL-MIMO beam training in a more general and cost-saving way. Fig. 4. From Fig. 5 (a), i.e., narrow band condition, it is obvious that the proposed beam training scheme outperforms all existing far-field and near-field schemes. In specific, around 36.6% improvement in achievable rate is accomplished by the proposed method compared to the time-delay based near-field beam training in SNR = 2 dB. In addition, we can observe that Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training the proposed method can also achieve better performance as long as SNR is smaller than 4 dB in the wideband situation. The reason why the near-field beam training scheme is vulnerable to noise is that the time-delay based near-field beam training scheme has to utilize beams with different frequencies to search different locations. the time-delay based near-field beam training scheme can not accumulate the power from all frequencies to combat noise as the near-field exhaustive beam training approach. Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training Achievable Rate (bit/s/Hz) Far-field hierarchical beam training [26] Far-field exhaustive search beam training [27] Near-field exhaustive search beam training [10] Time delay based near-field beam training [12] Proposed near-field 2D hierarchical beam training
VII. CONCLUSIONS
In this paper, we proposed a low-overhead near-field 2D hierarchical beam training by designing the near-field multi-resolution codebooks. Specifically, we first formulate the problem of designing near-field codeword and generating multi-resolution codebooks. It is worth pointing out that the proposed Gerchberg-Saxton (GS) based near-field codeword design algorithm can be utilized in designing codewords to realize arbitrary beam patterns. Then, a low-overhead
( a )
aIllustration of GS algorithm in iterative phase retrieval problem. (b) Illustration of GS algorithm in codeword design Fig. 1. Comparisons of the original and improved GS algorithm.
plane since the complex-valued wavefronts in the object and the diffraction pattern planes are usually connected through a Fourier transform with each other.Specifically, the GS algorithm initializes in the object plane, where the initial complex-valued wavefronts are created by combining the measured amplitude information with the random phase information. The iteration process of the GS algorithm consists of four steps: i) The forward diffraction propagation of the wavefronts in the object plane provides complex-valued wavefronts in the diffraction pattern plane; ii) Update the complex-valued wavefronts in the diffraction pattern plane: the amplitude information is substituted with the measured amplitude information U ′ ; iii) The backward diffraction propagation provides the complex-valued wavefronts in the object plane; iv) Update the complex-valued wavefronts in the diffraction plane: The amplitude information in the object plane is substituted with the measured amplitude information. The result of the GS algorithm is the recovered complex-valued wavefronts in the diffraction pattern plane.
Fig. 2 .
2jδ n,1 , e jδ n,2 , . . . , e jδ n,N RF(26) After T max iteration, we can obtain the final practical codeword asv p = F Tmax RF f Tmax BB(27)V. PROPOSED NEAR-FIELD 2D HIERARCHICAL BEAM TRAININGIn this section, we first introduce the proposed near-field 2D hierarchical beam training scheme,where the angle and distance ranges are reduced gradually layer by layer in multi-resolution codebooks. Then, the analysis of the proposed beam training overhead is provided.A. Near-Field 2D Hierarchical Beam Training SchemeIn order to obtain the tradeoff between the near-field beam training overhead and the performance, one of the methods is to apply a hierarchical near-field codebook, which consists of multi-resolution codebooks. The sizes of codebooks are determined by the angle sample step and distance sample step of the codebook, i.e., B θ and B r in(6). Specifically, as the increase of B θ Near-field exhaustive search beam Comparison between the far-field exhaustive search, near-field exhaustive search and the near-field 2D hierarchical beam training.
Fig. 3 .
3k opt ,θ and B l+1 v l+1,k ,r ∈ B l v l,k opt ,r15. the chosen codewords v l+1,k compose the W16.end forOutput: The feedback optimal codeword index k opt from the user. the angle and distance, i.e., the number of sampled angle grids U and the number of sampled distance grids S. It is worth pointing out that U is usually set as the same as the number of antennas on the array. The training overhead of the exhaustive near-field beam training scheme is U S. Meanwhile, the training overhead of the time-delay based beam training is only related to the number of sampled distance grids S. For the proposed 2D hierarchical beam training method, the beam training overhead can be represented as O (log (U ) + log (S)). It is obvious that, the training overhead of the proposed 2D hierarchical beam training is much less than that of the exhaustive near-field beam training. Since the number of sampled angle grids U is usually Comparison of the beam patterns of different layers of the hierarchical codebook.large than the number of sampled distances S[12], the training overhead of the proposed 2D hierarchical beam training is larger than that of the time-delay based beam training. However, the performance of the time-delay based beam training heavily depends on the extra hardware overhead and wideband condition, which will be further verified by simulation results in Section VI.VI. SIMULATION RESULTSFor simulations, we assume that the number of BS antennas and RF chains are N = 512 and N RF = 100. The wavelength is set as λ = 0.005 meters, corresponding to the 60 GHz frequency. The quantified bits number of phase shifters is set as b = 5. The path gain α, angle θ and distance r are generated as following: α l ∼ CN (0, 1), θ l ∼ U (−1, 1), and r l ∼ U (20, 100) meters. The SNR is defined as 1/σ 2 .
Fig. 3
3shows the comparison of the ideal beam pattern and the normalized practical beam pattern obtained by conducting beamforming with the designed codeword. In these heat maps, the brighter the color, the greater the beamforming gain at this position. It is worth noting that, we utilize the rectangular coordinate system to present the beamforming gains of the locations in twodimension space to show the beam pattern more clearly, where the coordinates of the X-axis and Y-axis satisfy x = r cos(θ), and y = r cos(θ).Fig. 3 (a) presents an ideal beam pattern of the layer 1 codebook, where the beam should focus on the target location, i.e., x = [55, 75], y = [−5, 15].
S
= 16, respectively. The overhead of the far-field exhaustive search is set as the same as the number of sampled angle grids, i.e., 512. The overhead of the near-field exhaustive search beam training scheme is set as 512 × 16 = 8192. The overhead of time-delay based near-field beam training relates to the number of sampled distance grids, which is set as 16. For the farfield hierarchical beam training scheme, U (l) is the number of sampled angles in the l-th layer, where U (1) = 4, U (2) = 4, U (2) = 32. Thus, the overhead of far-field hierarchical beam training is L l U (l) = 4 + 4 + 32 = 40. For the proposed near-field 2D hierarchical beam training algorithm, we use a three-layer hierarchical codebook. The size of the layer 1 codebook can be calculated as 64 × 4 = 256, where the numbers of sampled angle and distance grids are set as 64 and 4. For the layer 2 and layer 3 codebooks, we only need to search 8 and 4 codewords. Thus the overhead of the proposed near-field 2D hierarchical beam training algorithm is 268, which is almost half of 512 and only 3.3 % of 8192.
Fig. 4
4presents the performance of achievable rate comparisons against the beam training overhead under different bandwidths. The training overhead increases from 0 to 1000. In the beam training process, we utilize the optimal beamforming vector with the largest achievable rate searched in the current time slots to serve the user. From
Fig. 5
5presents the performance of achievable rate comparisons against the SNR under different bandwidths, where SNR is from 0 dB to 5 dB. The simulation parameters are the same as those in
Fig. 4 .
4Achievable sum-rate performance comparison with respect to the beam training overhead under different bandwidths. (a) 100 MHz; (b) 500 MHz.
Fig. 5 .
5Achievable sum-rate performance comparison with respect to the SNR under different bandwidths. (a) 100 MHz; (b) 500 MHz.
Fig. 6
6presents the performance of achievable rate comparisons against the distance under different bandwidths, where the distance is from 25 m to 75 m at SNR = 5 dB. FromFig. 6(a), about 18.5% performance improvement compared to the time-delay based near-field beam training at distance = 55 m. Additionally, we can observe that the proposed method can also reach a 95.8% achievable rate of the time-delay based near-field beam training at distance = 55 m in the wideband situation.
Fig. 6 .
6Achievable sum-rate performance comparison with respect to the distance overhead under different bandwidths. (a) 100 MHz; (b) 500 MHz.
TABLE I
ICOMPARISONS OF BEAM TRAINING OVERHEADMethod
Overhead
Value
Table .
.I presents the comparison of beam training overhead for different methods. We com-
pare the proposed near-field 2D hierarchical beam training algorithm with the existing far-field
hierarchical beam training scheme
near-field 2D hierarchical beam training scheme is proposed to realize the tradeoff between the training overhead and performance. Significantly, the proposed scheme can achieve sub-optimal performance without restriction to the hardware cost and wideband condition.(16)can be further presented ascan be presented aswhere τ is the angle between thev (s+1) (u, w) andv ′ (s) (u, w). As shown in Step 8 of the Algorithm 1, where (12) presents the normalization of thev ′ (s) (u, w), thus, ∥v (s) (u, w)∥ 2 = ∥v (s+1) (u, w)∥ 2 = 1, and the phase information ofv (s+1) (u, w) andv ′ (s) (u, w) are the same, i.e., the angle between thev (s+1) (u, w) andv ′ (s) (u, w) τ is 0. In this case, (28)-(29) is written asSince ∥v ′ (s) (u, w)∥ 2 (1 − cos ϕ) is always greater than zero, we can obtain that ∥v (s) (u, w)−v ′ (s) (u, w)∥ 2 2 > ∥v (s+1) (u, w)−v ′ (s) (u, w)∥ 2 2 .
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| {'fraction_non_alphanumeric': 0.06567866127091765, 'fraction_numerical': 0.021911115972146268, 'mean_word_length': 4.066869862381084, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Extremely large-scale MIMO (XL-MIMO) is a promising technique for future 6G communications.The sharp increase in the number of antennas causes electromagnetic propagation to change from farfield to near-field. Due to the near-field effect, the exhaustive near-field beam training at all angles and distances requires very high overhead. The improved fast near-field beam training scheme based on time-delay structure can reduce the overhead, but it suffers from very high hardware costs and energy consumption caused by time-delay circuits. In this paper, we propose a near-field two dimension (2D) hierarchical beam training scheme to reduce the overhead without the need for extra hardware circuits.Specifically, we first formulate the multi-resolution near-field codewords design problem covering different angle and distance coverages. Next, inspired by phase retrieval problems in digital holography imaging technology, we propose a Gerchberg-Saxton (GS)-based algorithm to acquire the theoretical codeword by considering the ideal fully digital architecture. Based on the theoretical codeword, an alternating optimization algorithm is then proposed to acquire the practical codeword by considering the hybrid digital-analog architecture. Finally, with the help of multi-resolution codebooks, we propose a near-field 2D hierarchical beam training scheme to significantly reduce the training overhead, which is verified by extensive simulation results.Index TermsExtremely large-scale MIMO (XL-MIMO), extremely large-scale antenna array (ELAA), beam training, codebook design.All authors are with the', 'arxivid': '2212.14705', 'author': ['Student Member, IEEEYu Lu \nDepartment of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina\n', 'Student Member, IEEEZijian Zhang \nDepartment of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina\n', 'Fellow, IEEELinglong Dai [email protected]. \nDepartment of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina\n'], 'authoraffiliation': ['Department of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina', 'Department of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina', 'Department of Electronic Engineering\nCenter for Information Science and Technology (BNRist)\nTsinghua University as well as Beijing National Research\n100084BeijingChina'], 'corpusid': 255340541, 'doi': '10.48550/arxiv.2212.14705', 'github_urls': [], 'n_tokens_mistral': 16575, 'n_tokens_neox': 14596, 'n_words': 9179, 'pdfsha': 'e52ed4df1edeb56b95f12b0b984b81117d76a8f9', 'pdfurls': ['https://export.arxiv.org/pdf/2212.14705v2.pdf'], 'title': ['Hierarchical Beam Training for Extremely Large-Scale MIMO: From Far-Field to Near-Field', 'Hierarchical Beam Training for Extremely Large-Scale MIMO: From Far-Field to Near-Field'], 'venue': []} |
arxiv |
Mémoire de Mastère
2016-2017 5 Oct 2018
Mémoire de Mastère
2016-2017 5 Oct 2018Soutenu le 19/02/2018 devant le jury composé de : Mme Hajer Baazaoui Professeur à l'ISAMM Présidente M. Sami Zghal Maître-Assistant à la FSEGJ Rapporteur M. Sadok Ben Yahia Professeur à la FST Directeur de mémoire Au sein du laboratoire : LIPAH
Présenté en vue de l'obtention du diplôme de Mastère de recherche en Informatique Par Inès Osman Proposition d'une nouvelle méthode pour l'intégration sémantique des ontologies OWL en utilisant des alignements
Introduction générale
L'intégration des données est un vaste domaine qui permet d'unifier les données provenant des sources hétérogènes partageant des informations en commun, ou qui permet de les transférer d'une représentation à une autre, pour but de faire l'échange entre différents systèmes. Elle concerne des sources de données telles que les bases de données, les fichiers textes, et les ontologies, etc.
Pour ce faire, un sous-domaine a fait son apparition. Il s'agit de l'intégration des schémas tels que les schémas relationnels, orientés objet, XML (DTD, XML Schema), etc.). Rappelons que le schéma ou le modèle des données permet de décrire avec précision la structure d'un document, les conventions de structuration, de typage, et de nommage de ses données.
Par ailleurs, les ontologies ont été reconnues comme une composante essentielle pour la concrétisation de la vision du Web Sémantique. En définissant et décrivant les termes associés à des domaines particuliers, elles permettent d'annoter ou d'attacher les termes de multiples documents avec les mêmes termes propres à elles, ainsi elles arrivent à intégrer le contenu de différentes sources des données telles que les pages Web, les documents XML, les bases de données relationnelles, etc. L'utilisation de ces terminologies partagées permet un certain degré d'interopérabilité entre ces sources de données.
Cependant, cela ne résout pas complètement le problème d'intégration des données, car nous ne pouvons pas s'attendre à ce que tous les individus et toutes les organisations dans le Web sémantique s'accordent sur l'utilisation d'une terminologie ou d'une ontologie commune. Par conséquent, il est peu probable qu'une ontologie globale couvrant l'ensemble des systèmes distribués puisse être développée ; au contraire, un domaine donné pourrait avoir plusieurs ontologies concurrentes, chacune incomplète ou couvrant le domaine d'une certaine perspective. En effet, dans la pratique, les ontologies de différents systèmes sont développées indépendamment les unes des autres, par des communautés différentes, et pour des buts différents.
Suite à ce problème d'hétérogénéité, le domaine de l'intégration des ontologies, qui est aussi un sous-domaine de l'intégration des données, a fait son apparition. D'ailleurs, il ressemble énormément au domaine de l'intégration des schémas des bases de données, car les approches récentes de ces deux domaines se composent toutes les deux de deux étapes principales : l'étape de matching qui va réconcilier les différences en déterminant des correspondances (les similarités et les différences), puis l'étape de fusion (ou d'union) qui va exploiter le résultat du matching. L'intégration des ontologies de différents domaines vise à la construction d'une nouvelle ontologie pour un nouveau domaine plus large composé des domaines des ontologies intégrées. Elle est aussi appelée "composition" d'ontologies. L'intégration des ontologies de mêmes domaines vise à les unifier pour obtenir une ontologie plus complète qui couvre mieux ce même domaine. Elle est appelée "fusion" d'ontologies.
En général, les ontologies peuvent couvrir des domaines différents, ou bien des domaines identiques, proches (liés), complémentaires, ou interdisciplinaires dans lesquels les termes se chevauchent et les niveaux de détail (de leur conceptualisation) diffèrent. Ainsi, si les connaissances et les données doivent être partagées (e.g. dans le Web, ou par des entreprises en collaboration), il faudrait au moins établir des correspondances sémantiques ou des liens entre les ontologies qui les décrivent (matching des ontologies).
La tâche d'intégration des ontologies (composée d'une étape de matching, puis d'une étape d'union) est particulièrement importante dans les systèmes d'intégration puisqu'elle autorise la prise en compte conjointe des ressources décrites par des ontologies différentes. Ce thème de recherche a donné lieu à de très nombreux travaux.
Dans un contexte plus large, les ontologies produites sur le Web peuvent être graduellement d'une très grande taille. Ainsi, le processus d'intégration des ontologies nécessitera l'utilisation de mécanismes de prise en charge pour le passage à l'échelle des techniques d'intégration.
Pour conclure, l'idée de base est de réconcilier et d'intégrer des ontologies ou des fragments d'ontologies pour en fédérer d'autres qui encapsulent les données des ontologies initiales. Ainsi, l'objectif de ce mastère est de proposer une nouvelle méthode d'intégration des ontologies qui cherche à mettre en place une solution capable de raisonner sur des ontologies déjà existantes pour en produire une nouvelle en utilisant des techniques d'intégration.
Motivations et contributions :
Les techniques actuelles d'intégration des ontologies sont encore non efficaces en termes de temps d'exécution, semi-automatiques (qui reposent beaucoup sur l'intervention humaine), non extensibles (non scalables), générant une ontologie de mauvaise qualité (ayant énormément de contradictions sémantiques / logiques), et non complètes (avec perte d'informations précieuses, car ils n'arrivent pas à préserver toutes les connaissances des ontologies sources surtout les disjonctions).
Dans ce mémoire, nous proposons de développer une méthode automatique d'intégration des ontologies (OIA2R) ayant pour but d'intégrer deux ou plusieurs ontologies (de toute taille) en utilisant les mappings (ou les alignements) entre elles, pour former à la fin une nouvelle ontologie qui conserve toutes les informations des ontologies sources et des mappings, tout en les personnalisant (refactoring). Autrement dit, il s'agit d'une ontologie de pont qui englobe les ontologies d'entrée et les "bridging" axiomes qui les réconcilient. Notre algorithme produit une ontologie de sortie de bonne qualité en des temps d'exécution compétitifs. Nous proposons aussi une nouvelle classification des terminologies ambiguës utilisées dans le domaine de l'intégration des ontologies, car il y a une très grande confusion dans les appellations de l'état de l'art.
Introduction
Dans ce chapitre, nous rappelons les grandes étapes de l'évolution du Web, citons la définition du terme Web sémantique et Ontologie, présentons le langage OWL, rappelons les types d'hétérogénéité entre les ontologies, et présentons tous les domaines de l'ingénierie des ontologies et leurs applications dans le monde réel. Enfin, nous terminons ce chapitre par une conclusion qui introduit les causes du recours aux domaines de l'intégration et la fusion des ontologies dans lesquels nous allons entrer en détail dans le chapitre 2.
Notion du Web sémantique
Introduction au Web sémantique
Le Web actuel est un ensemble de documents (données et pages) dédiés aux humains, stockés et manipulés d'une façon purement syntaxique. Voici les deux principaux problèmes du Web actuel.
D'une part, il y a énormément de sources de données, du fait que n'importe qui peut facilement publier un contenu (sachant qu'il n'a pas la moindre idée sur la probabilité que ce contenu soit trouvé par autrui) ; il n'a qu'à l'annoter, ainsi les moteurs de recherche à base de mot-clé auront la tâche de l'indexer pour pouvoir l'afficher aux utilisateurs lorsqu'ils font une recherche. Par conséquent, l'information sur Internet est tellement énorme que l'utilisateur a du mal à la retrouver.
D'une autre part, les résultats de recherche sont imprécis, très sensibles au vocabulaire, et assez longs à trouver. En effet, les moteurs de recherche ne sont capables de répondre qu'à deux questions principales :
-Quelles sont les pages contenant ce terme ? et ; -Quelles sont les pages les plus populaires à ce sujet ?
Le Web est essentiellement syntaxique, et l'Homme est le seul à pouvoir interpréter son contenu (des documents et des ressources) inaccessible et non interprétable par la machine ; lui seul doté 1.1. NOTION DU WEB SÉMANTIQUE de la capacité de comprendre ce qu'il a trouvé et décider en quoi cela se rapporte à ce qu'il veut vraiment chercher. Finalement, nous ne pourrons pas se passer de l'intervention humaine pour naviguer, chercher, faire le tri des documents manuellement, interpréter, et combiner les résultats.
Pour conclure, le Web actuel ne peut pas être manipulé de façon intelligente par les programmes informatiques car il y a un vrai manque de sémantique.
Voici un exemple qui illustre ces problèmes : Supposons que nous voulons rechercher un fabricant de portes et de fenêtres pour construire une maison, nous tapons les mots "gates" et "windows" dans Google, nous aurons des résultats non satisfaisants concernent en grande partie Bill Gates et Microsoft Windows. Idéalement, les résultats devraient contenir les deux sens équitablement, ou selon le contexte de l'utilisateur.
Objectifs du Web sémantique
L'intérêt croissant porté à la recherche d'information sur le Web a donné lieu à l'initiative du Web sémantique. De nos jours, le souci du Web n'est plus vraiment l'augmentation continuelle de sa taille d'informations, mais plutôt l'amélioration de la recherche dans cette énorme masse d'informations, et la réalisation de systèmes permettant de filtrer et délivrer les informations de façon "intelligente".
Le but ultime du Web de troisième génération est de permettre aux utilisateurs d'exploiter tout le potentiel du Web en s'aidant par les machines qui pourront accomplir les tâches encore réalisées par l'Homme comme la recherche ou l'association d'informations, et ainsi atteindre un Web intelligent qui regroupera l'information de manière utile et qui apportera à l'utilisateur ce qui cherche vraiment.
Définition du Web sémantique
En 1993, Tim Berners-Lee a fournit une solution au problème du partage de connaissances entre les applications Web à l'aide d'un mécanisme à base d'ontologies qui structure les données d'une manière compréhensible par la machine. En 2001, il a envisagé un WWW accessible pour les machines et les humains, de telle sorte qu'ils soient mis dans une position égale.
Le Web sémantique (nommé aussi Web intelligent ou Web des données) est un ensemble de connaissances, où toutes les machines peuvent lier sémantiquement les données du Web, ainsi comprendre leurs significations, y accéder plus intelligemment, pour améliorer le dialogue entre les applications et l'interaction avec l'utilisateur en lui offrant une meilleure qualité des tâches de recherche (d'association des informations, et d'apprentissage, etc.).
Il peut être vu aussi comme une couche supplémentaire de connaissances (au-dessus du Web actuel) ou une extension du Web actuel.
De la même manière que le Web actuel, le Web sémantique est construit principalement autour des identifiants (URIs) et du protocole HTTP, mais il est par contre basé sur le langage RDF et non plus sur le HTML, pour but de séparer l'information qui décrit le sens et le contexte 1.2. ONTOLOGIE des données, de l'information qui décrit la présentation des données. Il est basé principalement sur les bases de connaissances et non seulement sur les bases de données. La recherche aussi va s'affecter et devenir une recherche par concept, non plus par mot clé.
Dans le Web sémantique, toutes les données du Web, textuelles ou multimédia, doivent être annotées sémantiquement par des métadonnées pertinentes, car les machines (les agents logiciels) ne pourront comprendre les données et prendre des décisions qu'à travers une explication plus spécifique du contenu, et cela en utilisant un mark-up sémantique nommé "méta-données". L'annotation de ces ressources d'information repose sur l'accès à des représentations de connaissances (des ontologies) partagées sur le Web.
Pour résumer, le Web sémantique donnera naissance à un nouvel aspect intelligent basé sur la recherche, le raisonnement, et la prise de décision automatique, faisant ainsi croître la productivité et les capacités des moteurs de recherche.
Ontologie
Ontologies et Web
Le domaine des ontologies est né d'une volonté de pallier les limites du Web (déjà évoquées). Les ontologies font partie intégrante des normes du W3C pour le Web sémantique, car elles sont indispensables pour représenter la sémantique des documents (les connaissances) qui coexistent dans le Web, en structurant et en définissant la signification des termes actuellement collectées et normalisées. En effet, les ressources du Web telles que les pages Web, les bases de données, ou les documents XML, etc. sont annotées par (attachées à) la signification des termes (concepts) de sorte que nous aurons besoin du même concept de la même ontologie pour représenter la même chose dans l'indexation de ces différentes ressources. C'est ici que se manifeste le rôle et l'utilité des ontologies.
Elles sont utilisées pour publier des bases de connaissances réutilisables et faciliter l'interopérabilité entre plusieurs systèmes hétérogènes et bases de données. Ainsi, nous pouvons considérer les ontologies comme une représentation pivot qui a pour but d'intégrer les sources de données hétérogènes.
Elles sont utilisées dans beaucoup de filières telles que : la gestion des connaissances, l'intelligence artificielle, ou le Web sémantique. Et elles aident à réaliser de nombreuses applications comme la recherche d'informations, la réponse aux requêtes, la recherche documentaire, et la synthèse de texte, etc.
Nous pouvons conclure que l'ontologie est un outil essentiel permettant l'exploitation automatique (le traitement machine) des connaissances, et la concrétisation des principes de réutilisabilité et du partage de l'information entre différentes sources de données, et cela grâce au vocabulaire commun fourni pour un domaine de connaissances réel ou imaginaire. Caldarola and Rinaldi (2016) constatent que les ontologies disponibles dans la littérature sont en train de devenir de plus en plus volumineuses en termes de nombre d'entités, à un tel point qu'elles peuvent être considérées comme de la Big Data.
Revue des définitions d'une ontologie
L'ontologie est un terme qui est apparu dans la Métaphysique avec Aristote qui considérait que l'ontologie est une "Science qui étudie l'être en tant qu'être et les attributs qui lui appartiennent essentiellement". Dans ce contexte, élaborer une ontologie, revient à faire l'étude philosophique de la nature de l'être et de l'existence, i.e. l'étude des propriétés générales de ce qui existe, en définissant l'ensemble des connaissances sur le monde.
Pendant la dernière décennie, les informaticiens ont repris le terme "Ontologie" qui est devenu très utilisé dans le domaine de l'informatique. C'est au début des années 90 qu'il est apparu pour la première fois dans le cadre des recherches sur les Systèmes à Base de Connaissances (SBC).
Une des premières définitions a été donnée par Neches et al. (1991) : "Une ontologie définit les termes et les relations de base comportant le vocabulaire d'un domaine, aussi bien que les règles pour combiner ces termes et ces relations afin de définir des extensions du vocabulaire". Studer et al. (1998) ont conclu qu'"une ontologie est une spécification formelle et explicite d'une conceptualisation partagée d'un domaine de connaissances".
Le terme "conceptualisation" ou conceptualiser un domaine veut dire faire une abstraction décrivant un phénomène quelconque du monde réel de ce domaine ; faire les choix quant à la manière de décrire ce domaine particulier (par des entités).
Une "spécification" est une conceptualisation représentée dans une forme concrète. Une spécification de la conceptualisation est par conséquent une définition formelle des termes qui décrivent un domaine, des relations entre eux, et des axiomes qui les contraignent (Nous en parlerons en détail juste après).
Le terme "formelle" signifie qu'une ontologie doit être interprétable et lisible par la machine.
Le terme "explicite" veut dire que les entités et les axiomes doivent être explicitement définis.
Le terme "partagée" indique qu'une ontologie doit annoter multiples sources de données, être consensuelle et accessible par tous les utilisateurs d'une communauté particulière. Gruber et al. (2009) définissent une ontologie comme suit : "Dans le contexte des sciences de l'informatique et de l'information, une ontologie définit un jeu de primitives représentatives avec lequel un domaine de connaissance ou un univers de discours peut être modélisé".
Le terme "jeu de primitives" est la traduction la plus fidèle possible du monde réel à représenter.
Constituants d'une ontologie
Une ontologie est une collection structurée de termes, de relations entre les termes, et d'un ensemble de règles d'inférence sur ces termes. Elle est nommée avec un IRI. Et puisqu'elle est un document Web, elle est ainsi référencée par un URI (IRI physique) qui doit pointer sur (coïncider avec) la localisation de l'URL choisi pour la publier.
ONTOLOGIE
Dans la syntaxe abstraite, une ontologie OWL est une séquence d'axiomes (de règles ou de contraintes) logiques et non logiques (y compris les faits), et éventuellement de références à d'autres ontologies (des importations) qui sont considérées incluses dans l'ontologie.
La particularité des ontologies réside dans l'existence d'une sémantique (de théorie) de logique mathématique. En effet, les relations entre les entités peuvent être formellement modélisées par la logique de description formelle de premier ordre.
L'ontologie est formalisée par des entités pouvant avoir chacune un IRI qui est une référence d'URI. Il existe cinq types d'entités : les concepts (ou classes), les propriétés (relations, attributs, slots, rôles, ou actions), les individus (objets, instances, ou extensions des classes), les types de données, et les valeurs de données. La déclaration de ces entités dans l'ontologie est faite par des axiomes non logiques :
• Les concepts sémantiques de l'ontologie correspondent aux abstractions d'une partie de la réalité (du domaine). Ce sont les concepts auxquels nous nous référons, choisis en fonction des objectifs que nous nous donnons et de l'application envisagée pour l'ontologie. Ils sont les entités principales d'une ontologie. Ils peuvent représenter des concepts abstraits (une notion, une intention, une idée, une croyance, un sentiment, etc.), ou bien des concepts spécifiques (un objet matériel, un ensemble ou un groupe d'individus de caractéristiques similaires, etc.). § Les concepts sont organisés hiérarchiquement à travers la relation conceptuelle "Sous classes de" ou "is a" d'héritage ou de spécialisation, utilisée pour construire une taxonomie / hiérarchie de concepts ; Cette relation peut aussi signifier une relation d'agrégation ou de composition "Partie de" ou "has a". D'autres relations prédéfinies telle que l'équivalence et la disjonction peuvent également lier les concepts pour véhiculer plus de sémantique.
• Les propriétés permettent de définir des liens pour les individus présents dans le domaine. Les propriétés sont des relations non prédéfinies et non taxonomiques utilisées pour exprimer la sémantique qui relie deux concepts, et c'est justement l'apport des ontologies qui peuvent définir d'autres relations spécifiques non prédéfinies.
• Le premier type de propriétés, nommé propriété d'objet, est défini tel que le premier argument de la relation corresponde au domaine (un concept pour lequel est définie la propriété) et que le deuxième argument corresponde au co-domaine / à l'image (un concept relié au domaine par la propriété). Ainsi, il définit une relation entre deux individus.
• Le deuxième type de propriétés, nommé propriété de type de données, est utilisé pour exprimer les attributs des concepts. Les attributs sont des relations dans lesquelles le domaine est un concept, et le co-domaine / l'image est un type de donnée (un littéral) tel que "String", "Integer", "Double", "Date", etc. Ainsi, il définit une relation entre un individu d'une classe et une valeur de donnée. § Ces deux types de propriétés peuvent être organisés hiérarchiquement, et liés par des relations conceptuelles prédéfinies telles que l'équivalence, la disjonction, et beaucoup d'autres. Ces propriétés sont instanciées à l'aide de la relation d'affectation qui leur associe une valeur de domaine (un individu) et une valeur de co-domaine (un individu ou une valeur de donnée).
• Le troisième type de propriétés, nommé propriété d'annotation, ne se conforme pas à la définition des propriétés décrite ci-dessus. Le rôle de cette propriété est d'annoter les entités ou les ontologies. Son domaine peut être une entité (classe, propriété, ou individu) ou une ontologie, et son co-domaine peut être une entité, un littéral généralement de type "String", ou une ontologie. § Les propriétés d'annotation peuvent également être organisées hiérarchiquement.
• Les individus constituent la définition extensionnelle / l'extension des concepts, et ainsi l'extension (les données) de l'ontologie. Les IRIs des individus sont utilisés pour faire référence aux ressources. Ce sont des objets particuliers instanciés par les concepts à l'aide de la relation d'instanciation prédéfinie "Instance de" ou "is kind of" ou "type". Ils peuplent les classes et véhiculent les connaissances à propos du domaine. (Il existe aussi des individus anonymes qui sont des individus non utilisés en dehors de l'ontologie. Ils sont identifiés par un ID local plutôt qu'un IRI global). § Les instances peuvent être liées par des relations conceptuelles d'identité et de différence.
• Les types de données sont des parties particulières du domaine qui spécifient des valeurs. Les ontologies référencent des types de données intégrés de XML Schema (des littéraux) au moyen d'une référence URI à ce type de données.
• Les valeurs de données sont, contrairement aux autres entités, des valeurs simples qui n'ont pas d'IRI. Ce sont les valeurs des types de données.
• Les axiomes logiques constituent des assertions liées aux entités. Au lieu de compter sur les labels et les termes des entités (qui sont destinés aux humains) pour transmettre la sémantique, le concepteur d'ontologies doit contraindre l'interprétation possible des entités à travers une utilisation judicieuse d'axiomes logiques pour rende leurs sens beaucoup plus précis. § Ils sont aussi utilisés pour vérifier la consistance de l'ontologie, car ils permettent à un "raisonneur" d'inférer des connaissances additionnelles qui ne sont pas déclarées directement. Plus les axiomes exprimés dans les ontologies sont complexes, plus ils transportent des connaissances implicites qui peuvent être inférées par le raisonneur.
• Les faits sont des axiomes qui énoncent des informations sur les individus, telles que les classes auxquelles les individus appartiennent, et les propriétés et les valeurs des propriétés de ces individus.
Définitions formelles d'une ontologie
Selon Kalfoglou and Schorlemmer (2003), une approche algébrique plus formelle identifie une ontologie comme étant une paire <S, A>, où S est la signature des entités de l'ontologie (modélisée par une structure mathématique comme un treillis ou un ensemble non structuré) et A est l'ensemble des axiomes ontologiques qui spécifient l'interprétation voulue de la signature dans un domaine donné. Udrea et al. (2007), les ontologies modélisent la structure des données (i.e., les ensembles de classes et de propriétés), la sémantique des données (sous la forme d'axiomes tels que les relations d'héritage ou les contraintes sur les propriétés), et les instances des données (les individus). Ainsi, les entités d'une ontologie se composent d'une partie "structure", et d'une partie "donnée".
ONTOLOGIE
Selon Cheatham and Pesquita (2017), les informations des classes, des propriétés, et des axiomes qui restreignent leur interprétation, sont appelées la "structure", le "schéma", ou "Tbox" (comme Terminologie) de l'ontologie, et les informations des instances et leurs axiomes sont appelées "données", "données d'instances" ou "A-box" (comme Assertions) et contiennent des assertions sur des instances utilisant des données du T-box. D'après Zhang et al. (2017), une ontologie est un modèle en arbre, à cause du principe de l'hyponymie (la subsomption -is-a -) qui fait que chaque entité (classe ou propriété) soit héritée d'une seule super-entité directe, formant ainsi une structure de graphe acyclique enracinée Raunich and Rahm (2012). Mais dans le cas d'un héritage multiple, l'ontologie devient un modèle en réseau qui peut contenir des cycles et dans lequel plusieurs chemins peuvent mener à une entité.
Langage d'ontologie OWL
Il existe une grande variété de langages pour exprimer les ontologies. Quelques exemples de langages incluent RDF, RDFS, OWL, KIF, F-Logic, UML, SQL DDL, ou XML Schema, etc.
Le défi du Web sémantique est de fournir un langage qui exprime à la fois des règles, des structures, et des données sur lesquelles il va raisonner (à l'aide de ces règles). Par la suite, les règles de n'importe quel système de représentation de connaissances pourront être exportées dans le Web sémantique. Un individu peut ne pas avoir de classe(s) qui l'instancie(nt) ; dans ce cas, il sera implicitement une instance de la classe « owl:Thing ».
OWL (Web
Il y a une variété de syntaxes (formats) pour persister, partager, et éditer des ontologies OWL, telles que Functional OWL, RDF/XML, Turtle, OWL/XML, Manchester OWL, OBO, KRSS, etc. La spécification OWL décrit ce qui constitue une ontologie d'un point de vue structurel de haut niveau, qui est ensuite mappée en diverses syntaxes concrètes. RDF/XML est la syntaxe d'échange officiellement recommandée par W3C, que tout outil OWL doit pouvoir prendre en charge.
La diversité du monde réel est une source de richesse et d'hétérogénéité. En effet, dans les systèmes ouverts et distribués, tels que le Web sémantique, l'hétérogénéité ne peut pas être évitée. Plusieurs ontologies de mêmes domaines ou de domaines proches peuvent exister, à cause du développement déconnecté qui se focalise sur des applications particulières de différents buts et intérêts. Par ailleurs, les concepteurs ont des habitudes et des pré-requis différents, et modélisent les connaissances avec des niveaux de détails différents et des outils différents. Tout cela va influencer de différentes manières leurs décisions de conception. Par conséquent, la conception des ontologies ne peut jamais être un processus déterministe ; même deux ontologies de même domaine ne vont pas être identiques. Toutes ces raisons mènent à diverses formes d'hétérogénéité.
Prenons l'exemple du domaine biomédical. Il y a neuf ontologies qui décrivent une maladie neurologique, allant des ontologies très spécifiques couvrant une seule maladie (e.g. l'épilepsie, l'Alzheimer) à des ontologies couvrant toutes sortes de maladies telles que la "Disease Ontology". Il en résulte plusieurs ontologies qui décrivent les mêmes concepts sous des modèles légèrement différents. Klein (2001) distingue deux niveaux d'hétérogénéité qui peuvent exister entre les ontologies :
Hétérogénéité des langages
Ce sont les différences au niveau du langage, du méta-modèle, ou des primitives du langage utilisées pour spécifier une ontologie. Ce sont des différences entre les mécanismes (à partir desquels les entités vont être définies). Nous pouvons classifier ces différences en quatre catégories de difficulté croissante :
Syntaxe
Les différents langages d'ontologie utilisent souvent des syntaxes différentes, e.g., pour définir la classe de chaises dans RDF Schema (RDFS), nous utilisons <rdfs:Class ID="Chair">. Dans LOOM, l'expression (defconcept Chair) est utilisée pour définir la même classe. L'exemple typique d'incompatibilité de "syntaxe seulement" est quand un langage d'ontologie a plusieurs représentations syntaxiques, comme les différentes syntaxes de OWL.
Représentation logique
La différence de représentation des notions logiques, e.g. dans certains langages, il est possible d'indiquer explicitement que deux classes sont disjointes (A disjoint B), alors que dans d'autres langages, il est nécessaire d'utiliser la négation dans des instructions de sous-classes (A subclass-of (NOT B), (B subclass-of (NOT A)) pour indiquer la disjonction.
Sémantique des primitifs
Une différence plus subtile au niveau du méta modèle est la sémantique des constructions du langage. Malgré le fait que parfois le même nom est utilisé comme un constructeur dans deux langages, la sémantique peut différer, e.g. il existe plusieurs interprétations de A equalTo B.
Même lorsque deux langages d'ontologie semblent utiliser la même syntaxe, la sémantique des constructeurs peut différer.
Expressivité des langages C'est la différence fondamentale qui a le plus d'impact. Cette différence implique que certains langages sont capables d'exprimer des choses qui ne sont pas exprimables dans d'autres langages, e.g. certains langages ont des constructions pour exprimer la négation, d'autres non ; également pour le support des listes, des ensembles, et des valeurs par défaut, etc.
Hétérogénéité des modèles du domaine
Ce sont des différences dans la façon dont le domaine est modélisé. Elles sont décrites par Visser et al. (1998) :
Différence de conceptualisation (de sémantique) C'est une différence dans la façon dont un domaine est interprété (conceptualisé / modélisé), ce qui entraîne différents concepts, différentes relations entre les concepts, ou différentes instances des concepts. Elle est classée en deux catégories :
Portée Quand il s'agit de deux classes qui semblent représenter le même concept mais qui n'ont pas exactement les mêmes instances (extensions) bien que l'ensemble de leurs instances se croise (se chevauche), e.g. les concepts "Student" et "TaxPayer".
Couverture et granularité du modèle C'est une différence dans la partie du domaine couverte par les deux ontologies (e.g., les ontologies des employés universitaires et des étudiants), ou une différence dans le niveau de détail avec lequel le modèle est modélisé / couvert (e.g., une ontologie peut avoir un concept "Person" alors qu'une autre peut distinguer "YoungPerson", "MiddleAgedPerson" et "OldPerson").
Tenons l'exemple d'une ontologie sur les voitures : -Une ontologie peut modéliser des voitures mais pas des camions.
-Une autre pourrait représenter les camions mais les classer seulement dans quelques catégories. -Alors qu'une troisième pourrait faire des distractions très fines entre les types de camions en se basant sur leur structure physique générale, poids, but, etc.
Différence d'explications
C'est une différence dans la façon dont la conceptualisation est spécifiée. Elle se base sur la manière d'exprimer les entités. Elle est classée en trois catégories :
Style de modélisation Une différence dans le style de modélisation qui résulte des choix explicites du modélisateur :
Paradigme De différents paradigmes peuvent être utilisés pour représenter certains concepts tels que le concept du temps (e.g., représentation basée sur les "intervalles" vs représentation basée sur les "points"), l'action, les plans, la causalité, les attitudes propositionnelles, etc.
Description des concepts ou convention de modélisation Les différences dans la description des concepts ou les conventions de modélisation peuvent se manifester par l'utilisation de différentes structures pour représenter des informations identiques ou similaires, e.g. une distinction entre deux classes peut être modélisée en utilisant un attribut qualificatif (une propriété), ou en introduisant une autre (sous-) classe. Un autre choix dans les descriptions des concepts est la manière dont la hiérarchie is-a "<" est construite, en effet, les entités peuvent être augmentées ou réduites dans la hiérarchie, e.g. la classe "thèse" ou "dissertation" peut être modélisée comme dissertation < livre < publication scientifique < publication, ou comme dissertation < livre scientifique < livre < publication, ou même comme sous-classe de "livre" et de "publication scientifique".
Terminologie Une différence terminologique peut voir deux cas :
Termes synonymes Lorsque deux concepts sont équivalents, mais représentés en utilisant des noms différents. Un exemple trivial est l'utilisation du terme "Car" dans une ontologie et du terme "Automobile" dans une autre. Un type particulier de ce problème est le cas où le langage naturel avec lequel les ontologies sont décrites diffère.
Termes homonymes Lorsque le même nom est utilisé pour des concepts différents, e.g. dans le domaine musical, le terme "Conductor" (le chef d'orchestre) a une signification différente que celle dans le domaine de l'ingénierie électrique (le conducteur électrique).
Différence de codage Une différence de codage se produit lorsque les valeurs dans les différentes ontologies sont codées dans différents formats, e.g. une date peut être représentée par "jj/mm/aaaa" ou "mm-jj-aa", la distance peut être décrite en "mile" ou en "kilomètre", le poids peut être décrit en "gramme" ou en "pound", le prix peut être décrit en différentes monnaies, etc.
Ingénierie ontologique
L'ingénierie des ontologies est un contexte dans lequel les utilisateurs sont confrontés à des ontologies hétérogènes. Et plus généralement, c'est la tâche de concevoir, mettre en oeuvre, et maintenir des applications basées sur les ontologies Euzenat and Shvaiko (2013). Elle doit traiter plusieurs ontologies distribuées et évolutives.
Dans le but d'atténuer l'hétérogénéité croissante et la complexité des ontologies modernes, plusieurs domaines de recherche connexes ont vu le jour au cours des dernières années, tels que le "matching", le "mapping", l'"alignement", l'"intégration", la "fusion", le "versionning", et l'"évolution" des ontologies qui sont les domaines les plus répandus. Caldarola and Rinaldi (2016) Nous n'allons expliciter que les domaines liés à notre thème.
Médiation
La médiation des ontologies est un vaste domaine de recherche qui vise à déterminer et réconcilier les différences entre les ontologies afin de permettre leur réutilisation dans différentes applications hétérogènes dans le Web sémantique. De Bruijn et al. (2006) distinguent deux types principaux de médiation ontologique : le mapping et la fusion. Leung et al. (2014) distinguent trois types de médiation : le matching, la fusion, et l'intégration (qui sont basées sur le matching).
Réconciliation
La réconciliation des ontologies est un processus qui harmonise le contenu de deux ou de plusieurs ontologies. Il exige typiquement de faire un matching entre deux ontologies, et des changements dans un des deux côtés, ou dans les deux côtés Euzenat and Shvaiko (2013). Dans ce cas, il ne s'agit pas d'une fusion ou d'une intégration d'ontologies, mais plutôt d'une coévolution. Sachant que la réconciliation des ontologies peut être effectuée pour le but de fusionner ou d'intégrer deux ontologies.
Matching (Appariement)
Le matching des ontologies peut être une solution au problème de l'hétérogénéité sémantique des systèmes car il permet que la connaissance et les données exprimées dans les ontologies correspondues soient interopérables. C'est le processus de découverte des relations sémantiques ou des correspondances entre des entités provenant de deux différentes ontologies (ou de plusieurs ontologies dans le cas du matching multiple). Ces entités sont généralement des entités nommées (des classes, des propriétés, ou des individus), mais elles peuvent aussi être des entités anonymes i.e. des expressions plus complexes (des formules).
Le matching des ontologies peut concerner des ontologies entières (i.e. tout type d'entités : T-box et A-box), ou bien uniquement la partie "schéma" (la structure) des ontologies (i.e. T-box : seulement les classes et les propriétés) Cheatham and Pesquita (2017). La correspondance est la relation sémantique détenue ou supposée être détenue entre deux entités des différentes ontologies. La relation entre les deux entités n'est pas limitée à la relation d'équivalence, elle peut être plus sophistiquée, e.g. la subsomption, la disjonction, l'instanciation, et même des relations floues. Certains auteurs, utilisent le terme mapping, au lieu de correspondance.
Le résultat du matching, l'"alignement" (éventuellement le "mapping"), exprime, avec de différents degrés de précision, les relations sémantiques entre les ontologies mises en correspondance. Plusieurs auteurs utilisent le terme "alignement" (qui est le résultat du matching), au lieu de "matching", et utilisent le terme "mapping" au lieu d'"alignement" (que nous expliquerons juste après) Euzenat and Shvaiko (2013). Le type le plus simple de relations à trouver est l'équivalence ou la disjonction (l'exclusion) un à un (1-à-1) entre deux entités appartenant chacune à une ontologie. Le niveau de complexité suivant est la relation de subsomption (d'inclusion) 1-à-1. Pour trouver des relations 1-à-1, une recherche exhaustive doit comparer chaque entité de la première ontologie avec chaque entité de la deuxième ontologie, ce qui peut être réalisable pour de petites ontologies, mais infaisable pour des ontologies contenant des millions d'entités. C'est pour cela que les systèmes de matching peuvent employer une étape de filtrage ou de hachage pour déterminer les entités qui valent la peine d'être comparées Cheatham and Pesquita (2017).
Les relations un-à-plusieurs (1-à-m) sont encore plus difficiles à trouver. Tenons comme exemple une relation d'équivalence entre une classe de la première ontologie et l'union de trois classes de la deuxième ontologie. Ce type de relation cause un problème de complexité. Pour trouver des relations 1-à-m, une approche exhaustive aurait besoin de comparer chaque entité de la première ontologie avec toutes les combinaisons possibles des m entités de la deuxième ontologie, ce qui n'est pas possible Cheatham and Pesquita (2017).
Trouver des relations plusieurs-à-plusieurs (n-à-m) arbitraires est la tâche d'alignement la plus complexe. Une relation arbitraire signifie tout type de relation, non seulement l'équivalence, la disjonction, et la subsomption Cheatham and Pesquita (2017). § Les systèmes de matching actuels traitent l'identification des relations 1-à-1. Ils sont devenus très compétents dans la découverte des relations d'équivalence 1-à-1 entre les classes et les instances, mais moins performants dans la découverte des relations entre les propriétés. Leur compétence et leur exactitude est due principalement aux mesures de similarité syntaxiques (de chaînes de caractères). § Les travaux qui traitent un matching multiple sont très spécifiques pour le moment, et seul un petit nombre d'algorithmes le considère.
Voici les travaux qui ont été menés au sein de notre laboratoire LIPAH concernant le matching des ontologies : Zghal et al. (2007aZghal et al. ( ,b,c,d, 2011Zghal (2010); Kachroudi et al. (2011Kachroudi et al. ( , 2012Kachroudi et al. ( , 2013bKachroudi et al. ( ,c,a, 2014Kachroudi et al. ( , 2015Kachroudi et al. ( , 2016Kachroudi et al. ( , 2017b; Djeddi et al. (2015); El Abdi et al. (2015).
Méthodes de matching
Pour évaluer la similarité des entités, les systèmes de matching utilisent différentes approches. Ils peuvent utiliser zéro ou plusieurs approches de mesure de similarité, soit en combinant 1.5. INGÉNIERIE ONTOLOGIQUE leurs valeurs pour former une seule mesure, soit en les appliquant en série pour filtrer les correspondances et ne mesurer que les correspondances candidates Cheatham and Pesquita (2017).
La similarité reflète à quel point deux entités ont des choses en commun, c'est une mesure du degré qu'une entité puisse être utilisée à la place d'une autre. En général, la mesure ou le degré de confiance nous renseigne à quel point la correspondance est correcte et fiable. Plus elle est élevée, plus la relation qui la détient est solide. Généralement, c'est un nombre réel appartenant à un ensemble ordonné qui varie dans l'intervalle [0 1], mais il existe des systèmes qui utilisent simplement les booléens "vrai" et "faux" où le plus grand élément (1) est interprété en tant que "vrai", et le plus faible élément (0) est interprété en tant que "faux" Euzenat and Shvaiko (2013). Un seuil peut être mis pour ne pas afficher les correspondances de mesure de similarité inférieure à ce seuil.
Parmi les méthodes utilisées par les approches de matching des ontologies, nous citons Abels et al. (2005) :
Méthode basée sur les chaînes Elle compare deux entités en se basant sur les chaînes de caractères associées à elles. Les chaînes de caractères sont généralement les labels de l'entité, mais ils peuvent aussi inclure les commentaires et d'autres annotations de l'entité. Plus les chaînes sont similaires, plus elles sont susceptibles de désigner les mêmes concepts. § Cette approche souffre lorsque les concepts sémantiquement identiques sont modélisés avec des noms différents, i.e. lorsqu'il s'agit de synonymes Fahad et al. (2010).
Méthode linguistique Telle que la suppression de mots inutiles (stop-words), la tokenisation, la stemmatisation du texte, la considération des préfixes ou des suffixes, etc. pour gérer les noms des entités, e.g. cette méthode détecte que les classes "house" et "houses" sont identiques.
Méthode sémantique Elle tente d'utiliser les sens des labels de l'entité, plutôt que leurs orthographes. Des ressources linguistiques externes comme les lexiques, les thésaurus, les dictionnaires, les encyclopédies, et les moteurs de recherche du Web sont souvent utilisées afin d'identifier les synonymes, les hyperonymes (is-a), ou les hyponymes (is-a). Il est courant d'utiliser la base de données lexicale WordNet, l'ontologie de référence (UMLS), ou les règles d'articulation (les mappings), pour identifier les relations entre les entités Cheatham and Pesquita (2017). § L'inconvénient de cette méthode c'est qu'elle est spécifique au domaine particulier de la ressource externe utilisée, et ne produit des résultats efficaces que lorsqu'elle est utilisée pour des ontologies dans ce même domaine. Elle manquerait de précision si elle aurait été appliquée à des ontologies de domaine plus général ou totalement différent Fahad et al. (2010).
Méthode taxonomique / structurelle Elle ne considère que la relation de spécialisation (héritage). Son intuition est que la spécialisation (is-a) relie des termes qui sont déjà similaires (étant interprétés comme un sous-ensemble ou un sur-ensemble de l'autre), par conséquent, leurs voisins peuvent aussi être en quelque sorte similaires. Elle examine le voisinage de deux entités pour déterminer leur similarité Euzenat and Shvaiko (2013).
INGÉNIERIE ONTOLOGIQUE
Méthode basée sur les attributs / propriétés Elle examine les attributs de deux concepts pour déterminer leur similarité Fahad et al. (2010). § Son inconvénient est qu'elle produit des correspondances inexactes lorsque de différents concepts ont les mêmes attributs, e.g. le concept "Person" et "Company" sont supposés être les mêmes sur la base des labels de leurs attributs qui sont identiques, tels que les attributs "name", "adress", et "phone", etc.
Méthode extensionnelle Elle se base sur l'intuition qui dit : si deux classes ont les mêmes instances, alors ce sont des classes similaires. § L'inconvénient majeur de cette méthode se manifeste lorsque des concepts sémantiquement distincts ayant des instances en commun sont considérés comme identiques Fahad et al. (2010).
Méthode basée sur les graphes Cette méthode interprète la représentation graphique de la structure de deux ontologies et regarde les chemins, les enfants et les feuilles pour identifier leurs structures similaires en recherchant leurs parties identiques. Elle se base sur l'intuition qui dit : si deux noeuds de deux ontologies sont similaires, alors leurs voisins doivent aussi être plus ou moins similaires Euzenat and Shvaiko (2013), e.g. deux entités qui ont la même superclasse et qui partagent quelques instances en commun, sont considérées plus similaires que deux entités n'ayant pas ces choses en commun ; deux classes de deux ontologies sont similaires ou identiques si elles ont les mêmes attributs et les mêmes classes voisines.
Nous pouvons trouver une première classification qui groupe ces méthodes de matching en des approches syntaxiques, structurelles, et sémantiques ; et une autre classification qui les groupent en des approches élémentaires (qui calculent les correspondances en analysant les entités isolément, en ignorant leurs relations avec les autres entités), et des approches structurelles (qui calculent les correspondances en analysant l'apparition des entités ensemble dans une structure).
En pratique, il n'existe aucun système de matching automatisé qui peut générer des alignements complètement corrects. En effet, les alignements manqueront toujours quelques correspondances correctes, contiendront quelques correspondances incorrectes, ou bien les deux en même temps Cheatham and Pesquita (2017).
Concernant notre sujet
Ces quinze dernières années, la grande majorité des recherches sur l'intégration des ontologies s'est concentrée surtout sur l'étape de matching des ontologies et a négligé la partie de fusion des ontologies qui vient après Raunich and Rahm (2014). En effet, la résolution de l'hétérogénéité par les stratégies de matching des ontologies est considérée comme une phase interne nécessaire et très importante pour l'intégration (ou la fusion) des ontologies en une nouvelle ontologie les regroupant. L'amélioration du processus de matching va améliorer considérablement les résultats de l'intégration (et de la fusion) des ontologies Umer and Mundy (2012).
Les systèmes de matching peuvent faire à la fin une vérification d'incohérence et une réparation à l'alignement (ou le mapping) produit, en supprimant les correspondances incorrectes ou incohérentes i.e. celles qui sont correctes mais qui causent une incohérence logique dans l'ontologie produite suite à l'intégration ou la fusion des ontologies sources à l'aide de cet alignement-là (Nous détaillerons ce volet dans le chapitre 3).
Résolution de coréférence
Les systèmes de matching des ontologies se concentrent généralement sur la recherche des relations entre les entités de schéma / T-box (les classes et les propriétés), alors que les systèmes de résolution de coréférence se concentrent sur l'identification des mêmes individus qui sont référencés par différents URIs Cheatham and Pesquita (2017).
Les relations cherchées par les algorithmes de résolution de coréférence sont uniquement des identités 1-à-1, car deux individus ne peuvent être qu'identiques ou distincts, tandis que 1.5. INGÉNIERIE ONTOLOGIQUE les matchings (de schéma / T-box) impliquent (aussi) des classes et des propriétés, et ainsi peuvent avoir toute relation traditionnelle qui existe entre deux ensembles comme la subsomption, l'exclusion (la disjonction), etc.
Le nombre d'instances (A-box) d'un data set (dans les linked data du Web) est souvent beaucoup plus grand que le nombre de ses entités de schéma (T-box), ainsi ce n'est pas faisable de comparer chaque individu d'un data set avec chaque individu d'un autre data set pour déterminer s'ils sont identiques ou pas. Par conséquent, une méthode de filtrage est utilisée pour décider si deux individus sont suffisamment proches pour valoir la peine d'être comparés ; s'ils le sont, un algorithme de comparaison va se produire en mesurant la similarité entre les individus, ou bien entre les individus et les noms des propriétés auxquelles elles sont liées. La mesure de similarité la plus utilisée est la similarité syntaxique (de chaînes de caractères). Enfin, le système doit prendre le résultat de la comparaison de deux individus et décider s'ils sont identiques ou pas en spécifiant souvent un seuil (une valeur empirique malheureusement) Cheatham and Pesquita (2017).
Le matching (des schémas) a un plus grand historique de recherche que celui de la résolution de coréférence qui vise l'intégration des linked data Cheatham and Pesquita (2017). Etant donné deux ou plusieurs ontologies (dans le cas d'un matching multiple), l'alignement est un ensemble de correspondances (relations) sémantiques entre des paires d'entités appartenant à différentes ontologies. Rappelons que l'alignement est la sortie du processus de matching des ontologies Euzenat and Shvaiko (2013).
Alignement
Plusieurs auteurs, utilisent le terme "mapping" au lieu d'"alignement". Dans le reste de ce mémoire, nous utiliserons le mot "alignement" dans ce sens.
Puisque la relation est une relation binaire valable dans les deux sens et pouvant être décomposée en une paire de fonctions totales, Kalfoglou and Schorlemmer (2003) supposent que l'alignement des ontologies peut être décrit au moyen d'une paire de mappings (chacun contenant des correspondances dans un seul sens). Ils introduisent la notion de l'ontologie intermédiaire commune O 0 (ou l'articulation) qui peut être créée à travers cet alignement. Euzenat and Shvaiko (2013). L'alignement peut avoir des correspondances ayant la même entité source, i.e. une entité source peut avoir plus qu'une relation avec des entités cibles.
Les alignements peuvent être utilisés dans des tâches variées, telles que la réponse aux requêtes, la liaison des données, la navigation dans le Web sémantique, la transformation des ontologies, l'intégration et la fusion des ontologies, et le raisonnement sur les ontologies.
Concernant notre sujet
Il est possible d'utiliser des relations à partir d'un langage ontologique pour exprimer un alignement. Tenons l'exemple du langage OWL qui peut être considéré comme un langage d'expression de correspondances entre les ontologies. En effet, dans OWL, les primitifs "equi-valentClass", "equivalentProperty" et "sameAs" ont été introduits initialement pour lier les éléments des ontologies de même domaine ; d'ailleurs, dès qu'une ontologie OWL implique des entités provenant d'autres ontologies, elle exprime implicitement des alignements. Par conséquent, il est possible d'utiliser ces constructeurs pour relier les entités de deux ontologies mises en correspondance ou pour créer une ontologie OWL intermédiaire. (2012), le "mapping" des ontologies est une approche pour l'intégration des ontologies où l'ontologie intégrée O contient les règles de correspondance entre les entités des ontologies A et B. Klein (2001) considère également le mapping comme une intégration virtuelle. (C'est la même notion d'ontologie intermédiaire O 0 ou d'articulation rencontrée dans la partie "alignement").
Mapping
Définition 1
Selon Umer and Mundy
Selon Ziemba et al. (2015), le mapping permet d'obtenir un résultat similaire à l'ontologie de pont (c'est l'ontologie que nous allons réaliser dans ce mémoire). Cependant, dans l'ontologie de pont, par opposition au mapping, les ontologies sources et les connexions entre elles sont stockées ensemble, or que dans le mapping, les connexions sont à part. Le mapping entre les ontologies forme des "ponts sémantiques" De Bruijn et al. (2006).
Définition 2 (consensuelle)
Le mapping est la version orientée d'un alignement où une entité d'une première ontologie est correspondue à une entité d'une deuxième ontologie, et pas l'inverse. Il assigne chaque entité d'une ontologie à au plus une (exactement une ou aucune) entité de l'autre ontologie. Il se conforme à la définition mathématique d'une fonction totale (une relation unidirectionnelle (injective)), et non pas à la définition d'une relation générale bidirectionnelle (bijective). Selon Euzenat and Shvaiko (2013), cette définition mathématique exige que l'entité mise en correspondance soit égale à son image, i.e. que la relation soit une relation sémantique d'équivalence ou d'identité.
Selon Flouris et al. (2006), un mapping peut être perçu comme une collection de règles (ou d'axiomes) toutes orientées dans la même direction, de telle sorte que les entités de l'ontologie source et cible apparaissent au maximum une fois. Ils ajoutent que les deux ontologies mappées doivent partager le même domaine de discours (ou des domaines proches). Ceci est implicite sinon nous n'aurons pas de mapping. Dans le reste de ce mémoire, nous utiliserons le terme "mapping" dans ce sens.
D 'après De Bruijn et al. (2006), le mapping, comme l'alignement, est stocké séparément des deux ontologies, ainsi il n'est pas incorporé dans les définitions de ces ontologies. Processus d'intégration des ontologies Il s'agit du mapping entre une ontologie globale et des ontologies locales dans le processus de l'intégration des ontologies : Il décrit les relations (les correspondances) entre l'ontologie globale (cible) et les ontologies locales (sources) la composant. Il peut être aussi utilisé pour exprimer une entité de l'ontologie globale dans une vue ou une requête sur les autres ontologies (approche global-centric), ou l'inverse (approche local-centric).
Processus de fusion ou d'alignement Il s'agit du mapping entre des ontologies sources dans le processus de fusion ou d'alignement (définition 1) des ontologies. Il identifie les similitudes (synonymies) entre les différentes ontologies pour pouvoir les fusionner ou les aligner.
Processus de transformation des ontologies Il s'agit du mapping entre deux ou plusieurs ontologies sources dans le processus de transformation des ontologies : Il peut être utilisé pour transformer les entités sources en des entités cibles en se basant sur leurs correspondances, i.e. leurs relations d'équivalence sémantique dans le mapping. Il fournit une interopérabilité entre les différentes ontologies qui ne peuvent pas être intégrées ou fusionnées à cause d'une inconsistance mutuelle de leurs informations. § Les utilisations du mapping dans une ontologie intermédiaire, une ontologie de pont, une transformation des ontologies, une interconnexion des données, ou une requête, s'avèrent très utiles pour les environnements dynamiques, ouverts et distribués, et évitent également la complexité et les coûts de l'intégration ou de la fusion des ontologies sources. En effet, le mapping forme une sorte de couche ou d'interface commune entre les ontologies.
Mophisme
Selon Flouris et al. (2006), le morphisme des ontologie est une collection de correspondances sous forme de fonctions qui relient non seulement les signatures (les vocabulaires, les entités) de deux ontologies, mais aussi leurs axiomes (les syntaxes, les formalismes, les constructeurs des langages).
Selon Euzenat and Shvaiko (2013), le terme "morphisme" est utilisé pour représenter un mapping entre différents types de modèles*. Il contient des relations binaires sur deux ensembles d'identificateurs d'objets (OIDs) et il peut être inversé et composé.
* Les modèles, tels que les schémas relationnels ou les schémas XML, sont représentés implicitement (intérieurement) sous la forme de graphes étiquetés et orientés, dans lesquels les noeuds désignent les éléments du modèle (les relations et les attributs). Chacun de ces éléments est identifié par un identifiant d'objet (OID) Euzenat and Shvaiko (2013).
Transformation
La transformation des ontologies est le processus de changement / de traduction des entités (vocabulaire, signature) d'une ontologie par les entités d'une autre ontologie. Elle est utile quand nous voulons exprimer une ontologie par rapport à une autre. En général, les deux ontologies initiales sont inchangées et une troisième ontologie (le résultat de la transformation de la première ontologie par rapport à la deuxième) est créée. Les conséquences de la première ontologie sont aussi les conséquences du résultat de la transformation Euzenat and Shvaiko (2013).
Ce terme est très confondu à la notion de traduction (que nous allons expliquer juste après). La transformation des ontologies n'est pas bien supportée par les outils. Elle peut être particulièrement utile dans la connexion d'une ontologie à une autre ontologie (réconciliation des ontologies), ou dans la connexion d'une ontologie locale à une ontologie globale dans le cadre de l'intégration ou la fusion des ontologies. Elle est utilisée aussi pour importer des données sous une autre ontologie sans importer l'ontologie elle-même.
Traduction
La traduction des ontologies est le processus qui transforme la représentation formelle de l'ontologie d'un langage (d'un formalisme de représentation) à un autre, tout en préservant la sémantique, e.g. de "Ontolingua" à "Prolog". Elle change la forme syntaxique des axiomes, mais pas le vocabulaire (pas la signature) de l'ontologie Klein (2001); Kalfoglou and Schorlemmer (2003); Flouris et al. (2006); Euzenat and Shvaiko (2013).
Interconnexion des données
L'interconnexion des données est le processus qui consiste à établir des liens explicites, principalement des déclarations de la relation d'identité "owl :sameAs" entre les instances de deux ensembles de données RDF différents dans le Web de données (Linked Data). Il est possible de traiter les alignements en tant que spécifications de liaison, ainsi l'interconnexion des données pourrait être exprimée par l'opérateur Interlink(d, d , A) = L dans lequel un alignement A résultant de chaque couple d'ontologies (O et O ) sous lesquels deux ensembles de données (d et d ) sont exprimés, est utilisé pour les lier, et générer un ensemble de liens L entre les ressources (les URIs des instances) de ces deux data sets Euzenat and Shvaiko (2013). Bien qu'il y a une quantité très énorme de liens de type "owl:sameAs" entre les instances des data sets du LOD, il n'existe que quelques liens rares de type "owl:equivalentClass" ou "owl:equivalentPropery" entre leurs classes et leurs propriétés Zhao and Ichise (2014).
Dans le Web des données, le "matching" et la "résolution de coréférence" sont utiles dans l'aide à la génération de ces liens qui fournissent le contexte nécessaire pour rendre les données plus utiles Cheatham and Pesquita (2017). Ils sont effectués hors ligne et sans contraintes de temps de telle sorte que les correspondances résultantes soient correctes, mais pouvant être non exhaustives (non complètes).
Dans ce contexte, citons le travail de Hamdi et al. (2015), membre de notre laboratoire LIPAH, qui a exploité l'ontologie du Web de données FOAF pour les réseaux sociaux.
Intégration / Fusion
Nous allons l'expliquer en détail dans le chapitre 2
Raisonnement
Le raisonnement consiste en l'utilisation des alignements comme des règles pour raisonner sur les ontologies mises en correspondance. Les "bridging" axiomes utilisés dans l'intégration (l'ontologie de pont) sont des règles. Cet ensemble de règles est vu comme une ontologie O qui doit être écrite dans un langage ontologique supportant les règles ou les expressions des axiomes de pont. C'est la même notion de l'ontologie intermédiaire ou l'articulation des ontologies (définition 1 d'un mapping). Le raisonnement peut être décrit par la fonction T ransf ormAsRules(A) = O où A est un alignement entre deux ontologies O et O Euzenat and Shvaiko (2013).
Toute transformation des alignements sous une forme adaptée au raisonnement, telle que SWRL ou OWL, peut être utilisée par les moteurs d'inférence (les raisonneurs) de ces langages, tels que Pellet ou HermiT.
Enrichissement
L'enrichissement est le processus qui cherche de nouvelles entités (généralement à partir de ressources textuelles externes) et les place correctement au sein de l'ontologie à enrichir.
Voici quelques travaux d'enrichissement d'ontologies réalisés au sein de notre laboratoire LIPAH : Kamoun et al. (2010); Hamdi et al. (2012); Ben Yahia (2012c,a,b, 2014).
Conclusion
A présent, les entreprises ont migré vers l'adoption des stratégies de mondialisation et d'internationalisation. En effet, traditionnellement, les entreprises partageaient seulement les biens physiques en collaboration, mais maintenant elles ont aussi besoin de partager et intégrer leurs connaissances. C'est pourquoi la notion de l'interopérabilité s'impose car elle permet aux systèmes informatiques hétérogènes de communiquer, interpréter et traiter l'information échangée.
Pour ce faire, les ontologies se présentent comme le meilleur outil pour communiquer et partager des connaissances en fournissant une compréhension commune d'un domaine donnée. Malheureusement, les concepteurs des ontologies eux-mêmes appliquent des visions différentes du même domaine lors du développement des ontologies, et ceci engendre le problème de l'hétérogénéité sémantique qui est l'un des principaux obstacles de l'interopérabilité sémantique ; Il se produit lors de l'utilisation des ontologies de même domaine, i.e. quand des ontologies hétérogènes réutilisent les mêmes connaissances. L'intégration sémantique est indispensable pour remédier à ce problème. Elle se base sur la sémantique des systèmes inter-opérants pour comparer leurs différents concepts et déduire leurs correspondances, et éventuellement les associer et créer des bases de connaissances intégrées. Par conséquent, l'intégration sémantique mènera inévitablement à un matching inter ontologique qui est une étape essentielle dans l'intégration des ontologies.
Introduction
A présent, il existe une très grande confusion dans l'utilisation des termes "intégration" et "fusion" dans la littérature. En effet, il arrive que nous trouvons des travaux sur la fusion que les auteurs nomment "intégration", et des travaux sur l'intégration que les auteurs nomment "fusion" ; il y a des cas où les auteurs utilisent les deux termes comme synonymes et choisissent l'un d'entre eux comme titre de l'article (ils choisissent généralement le terme "intégration" car il parait plus général, ainsi vrai dans les deux cas). Par ailleurs, plusieurs auteurs utilisent le terme "intégration" dans le titre de leurs travaux, sans pour autant faire une intégration ; ils se contentent d'un matching ; et même s'ils font une intégration, ils ne l'explicitent et ne l'évaluent pas, ils se concentrent seulement sur l'évaluation du matching.
Toutes les définitions et les approches qui vont suivre vont être ordonnées chronologiquement dans chaque section.
Dans ce chapitre, nous citons les différentes définitions et approches du terme fusion des ontologies dans la littérature. Puis, nous citons les différents types d'intégration des ontologies, les définitions du terme intégration des ontologies dans la littérature, et ses principales approches existantes. Par la suite, nous évoquons les avantages de l'intégration et la fusion, leurs différences et leurs points communs, et nous expliquons les causes des erreurs qui peuvent se produire suite à ces deux processus. Ensuite, nous consacrons une petite section pour parler de ce qui nous intéresse parmi toutes ces définitions et approches évoquées. Enfin, nous clôturons ce chapitre par une conclusion qui résume le tout.
Fusion des ontologies
Puisque de nombreuses ontologies se réfèrent au même domaine et aux mêmes objets, il existe un besoin croissant de les fusionner et les organiser. En effet, le but ultime de la fusion est de représenter une meilleure perspective des connaissances d'un domaine.
En général, la fusion des ontologies est utilisée dans le domaine de l'intégration des données, mais elle peut être aussi perçue comme une technique utilisée dans le domaine de l'enrichissement des ontologies (de domaine) qui consiste à insérer dans l'ontologie des connaissances connexes en moins de temps et de coût.
La façon dont le processus de fusion est effectué est encore très peu claire. En effet, il n'y a pas de consensus sur la méthodologie à suivre pour fusionner les ontologies. La seule phase commune est la phase initiale qui prend en entrée un ensemble d'ontologies (deux ou plus). Certains commencent directement par toutes les ontologies à fusionner (méthode non incrémentale), d'autres commencent par un groupe initial sélectionné d'ontologies (généralement par une seule ontologie) qui est élargi ensuite de manière incrémentielle par les autres ontologies (méthode incrémentale) Pinto and Martins (2004).
Définitions de la fusion
Fusion de Noy D'après Noy and Musen (2000), dans la fusion, une ontologie unique qui est une version fusionnée des ontologies d'entrée est créée. Souvent, les ontologies sources couvrent des domaines similaires ou liés. C'est une définition très vague (et qui peut convenir aussi au terme "intégration des ontologies").
Fusion de Pinto (consensuelle)
Selon Pinto (1999), dans la fusion, nous avons, d'une part, un ensemble d'ontologies (au moins deux) qui vont être fusionnées (O 1 , O 2 , . . . , O N ) et, d'autre part, l'ontologie résultante du processus de fusion (O). Ainsi, cette méthode est non incrémentale. Le sujet des ontologies sources et de l'ontologie résultante est le même (S), bien que certaines ontologies sources soient plus générales que d'autres (leur niveau de généralité peut ne pas être le même). Le but est de remplacer les ontologies existantes, portant sur un sujet particulier, par une ontologie plus riche et plus large qui couvre mieux ce même sujet en fusionnant leurs connaissances (les terminologies, les définitions, et les axiomes des ontologies sources).
Selon eux, dans la fusion (l'unification qui est le troisième cas d'intégration de Sowa (1997)), les ontologies sources sont unifiées en une seule. Dans certains cas, les connaissances des ontologies sources sont homogénéisées et modifiées par l'influence d'une ontologie source sur une autre (à l'aide des opérations d'abstraction, de généralisation, de transformation (mapping)). Dans d'autres cas, les connaissances provenant d'une ontologie source particulière sont dispersées et mêlées avec les connaissances des autres sources. Malik et al. (2010) donnent une autre définition proche qui considère que la fusion est le fait de former des ontologies mieux modélisées à partir d'ontologies mal définies ou plus petites (i.e. qui ne couvrent pas tout le domaine). (2014), la fusion peut être symétrique ou asymétrique par rapport aux ontologies d'entrée. Ils exigent, dans ces deux types d'union, que la propriété de "préservation de l'égalité" soit assurée, ce qui signifie que les entités correspondues (comme prescrit dans le mapping entre les deux ontologies d'entrée) doivent être fusionnées dans la même entité afin qu'elles ne soient représentées qu'une seule fois dans l'ontologie résultante. Selon eux, en fusionnant les entités équivalentes ainsi, ils réduisent le chevauchement sémantique (même si que l'héritage multiple est aussi une source de conflits), ainsi le résultat de la fusion sera plus compact et moins redondant qu'une simple union directe des ontologies d'entrée (i.e. une ontologie de pont avec des "bridging" axiomes).
Approche symétrique L'approche symétrique est la plus courante et vise à fusionner les ontologies d'entrée avec la même priorité (en préservant toutes les entités de toutes les ontologies). C'est une approche "Full Merge" qui prend l'union des ontologies d'entrée et qui combine leurs entités équivalentes (en une seule entité). Mais elle engendre une quantité importante de conflits sémantiques due à l'organisation hétérogène des mêmes concepts dans les ontologies d'entrée et à l'introduction de l'héritage multiple dans les entités fusionnées, ce qui génère des chemins redondants (plusieurs chemins conduisant à une même entité), réduisant ainsi la compréhensibilité de l'ontologie résultante.
Approche asymétrique L'approche asymétrique, prend l'une des ontologies d'entrée comme cible, dans laquelle les autres ontologies sources seront fusionnées (d'une façon incrémentale) pour l'étendre, donnant la préférence uniquement à l'ontologie cible dont toutes les entités à elle seule doivent être préservées. Les entités des ontologies sources ne doivent pas obligatoirement faire partie de l'ontologie résultante (cible). Ici, nous n'aurons plus affaire à l'héritage multiple, ainsi nous n'aurons pas (ou presque pas) de conflits sémantiques dans l'ontologie résultante qui aura bien une structure d'arbre (où un seul chemin conduit à une entité). § Mais d'après nous, cette approche asymétrique est un enrichissement de l'ontologie cible, plutôt qu'une fusion des ontologies sources.
Fusion comme étant synonyme à l'intégration
Raunich and Rahm (2012)
Fusion comme étant une intégration
Chatterjee et al. (2017) considèrent la création d'une ontologie à l'aide de la fusion comme étant un processus incrémental où des ontologies de petites tailles, de différents domaines, et de développement indépendant, devraient être fusionnées en une seule ontologie pour former un domaine (interdisciplinaire) plus vaste. (C'est la définition de la composition / l'intégration).
Ils donnent l'exemple du domaine de l'agriculture qui peut se composer de plusieurs sous domaines tels que les pesticides et les engrais, la récolte, la terre (le sol), les prévisions météo, l'infrastructure d'irrigation, la gestion de la sécheresse, la gestion du bétail, l'infrastructure de marketing, le suivi des régimes et des programmes, etc.
Fusion comme étant une ontologie de pont
Selon De Bruijn et al. (2006), dans la seconde approche de fusion (l'ontologie de pont), les ontologies originales ne sont pas remplacées, elles sont conservées après l'opération de fusion, c'est plutôt une "vue", appelée "Bridge Ontology", qui est créée. Elle importe les ontologies originales et spécifie des correspondances entre elles pour relier les entités de ces ontologies par des axiomes de pont. Ces "Bridging" axiomes sont des règles de transformation utilisées pour connecter la partie de chevauchement des ontologies sources. D'après Euzenat and Shvaiko (2013), la fusion des ontologies est la création d'une nouvelle ontologie O qui lie les différentes entités de deux ontologies O et O (qui se chevauchent) par des axiomes de pont ou des axiomes d'articulation, comme prescrit dans l'alignement entre O et O . Ils expriment la fusion par l'opérateur suivant : f usion(O, O , A) = O . Selon eux, les ontologies sources sont inchangées et l'ontologie résultante est supposée contenir les connaissances des ontologies initiales de sorte que les conséquences de chaque ontologie source soient les conséquences de la fusion.
Dans la fusion de Abbas and Berio (2013), une nouvelle ontologie peut être créée à partir d'ontologies sources, en établissant des correspondances entre les ontologies sources (un matching), puis en les combinant avec ces correspondances trouvées. Ils ne spécifient pas également les domaines des ontologies sources. § C'est l'approche que nous allons suivre. En général, leurs algorithmes consistent en une itération de trois étapes principales : 1. Trouver un endroit où il y a un chevauchement dans les deux ontologies (trouver des entités candidates identiques ou apparentées) ; 2. Relier ces entités (qui sont sémantiquement proches) via des relations d'équivalence ou de subsomption ; ou les fusionner (après avoir transformé et uni les ontologies). 3. Vérifier la consistance, la cohérence et la non-redondance de la nouvelle structure de l'ontologie résultante, et les résoudre (trouver des solutions possibles à ces conflits). Si deux ou plusieurs entités (concepts ou relations) des ontologies sources sont équivalentes à une certaine entité cible, elles seront automatiquement fusionnées pour former une seule entité dans l'ontologie cible ; Si une entité source est subsumée par une entité cible, elle sera importée dans l'ontologie cible avec le consensus des experts du domaine ; La même approche sera appliquée si une entité source subsume une autre entité cible ; Si une entité source est disjointe à toutes les entités cibles, elle peut être non pertinente et ainsi rejetée, ou peut être considérée comme une nouvelle entité qui enrichit éventuellement l'ontologie cible.
Mais ce processus nécessite beaucoup de travail manuel. Bien qu'ils déclarent que leur outil assure que chaque aspect des ontologies sources soit présent dans l'ontologie de sortie, ils n'expliquent pas la manière avec laquelle ils ont fait les mises à jour de tous les axiomes sources qui appellent les entités nouvellement modifiées suite à la fusion (en effet, deux ou plusieurs entités similaires formeront une nouvelle entité).
Ils n'ont pas évoqué non plus le traitement des conflits sémantiques (les incohérences) générés certainement suite à la fusion. ¶ La sortie de ces deux étapes est un modèle en réseau où toutes les paires de concepts équivalents sont fusionnées générant ainsi un héritage multiple. Selon eux, il ne s'agit plus d'une ontologie (qui doit être un modèle en arbre), mais plutôt d'un réseau. C'est pourquoi ils ont ajouté deux autres étapes pour transformer le modèle initial de fusion en une structure d'arbre 3. La décomposition du modèle (de réseau) en plusieurs blocs dont les concepts fusionnés sont les frontières.
4. La reconstitution de ce modèle, de sorte que les concepts contenus dans les blocs (à part les concepts fusionnés) soient réorganisés pour former un seul chemin acyclique entre les deux concepts fusionnés. Cette réorganisation va être réalisée à l'aide d'un matching de subsomption / d'inclusion (is-a) entre les concepts de chaque bloc.
Dans la figure 2.5, ils donnent un exemple de correspondances entre deux fragments d'ontologies à fusionner. Dans l'image 2.6, ils donnent un modèle qui illustre le résultat des deux premières étapes, où les concepts fusionnés ont plus qu'un super-concept direct (héritage multiple), ce qui forme une structure de réseau. Dans la figure 2.7, ils donnent le modèle qui illustre le résultat des deux dernières étapes, où la sortie finale est une ontologie ayant les concepts réorganisés sous forme d'arbre. Il s'agit bien de très petites ontologies fusionnées en un temps relativement long.
FUSION DES ONTOLOGIES
INTÉGRATION DES ONTOLOGIES
Intégration des ontologies
En général, les techniques d'intégration des ontologies sont utilisées dans le développement des ontologies ou dans le domaine de l'intégration des données.
L'intégration des ontologies joue un rôle important dans le développement des ontologies en réutilisant des ontologies publiques existantes pour construire une ontologie en cours de développement ; ce qui réduit le coût de l'ingénierie des ontologies et favorise la réutilisation des modules d'ontologies standards. Tenons l'exemple de la construction d'une ontologie de catalogage des bibliothèques qui peut nécessiter l'assemblage d'ontologies dans les domaines des personnes, des livres, des sujets, des coordonnées géographiques, des numéros d'identification des livres, etc.
Types d'intégration
Voici les trois types d'intégration d'ontologies selon Keet (2004) :
Intégration sémantique
Elle se focalise sur le sens voulu des entités, e.g. découvrir si le concept C1 dans l'ontologie I est synonyme (ou hyponyme ou hyperonyme) au concept C2 dans l'ontologie II. C'est le type auquel nous nous intéressons.
Intégration structurelle
Quand la sémantique est (convenue d'être) identique mais l'organisation des entités (la catégorisation, le schéma) ne l'est pas et doit ainsi être alignée et intégrée. Il faut noter que la distinction entre la sémantique et la structure n'est pas aussi claire que cela puisse paraître, car la structure transporte une interprétation sémantique de la conceptualisation.
Intégration syntaxique
Elle se concentre sur la réalisation d'un formalisme uniforme à partir des formalismes avec lesquels les ontologies sources sont exprimées, tels que la description logique, KIF, OWL, F-logic etc. Dans la méthodologie, ce type d'intégration vient logiquement en troisième position (après l'intégration sémantique et structurelle), car c'est inutile de faire correspondre les formalismes si le sens de ce qui est intégré n'est pas compatible. Cependant, ces traductions, telles que la représentation syntaxique d'un concept dans deux langages formels, peuvent être recherchées indépendamment du domaine de l'intégration (le domaine de traduction des ontologies).
Définitions de l'intégration
Intégration comme étant une fusion Comme le montre la figure 2.8, pour Mena et al. (1996), l'intégration relie les entités des ontologies à intégrer, en traversant les hyponymes, les hyperonymes, et les synonymes entre eux. C'est en effet une fusion des ontologies. Selon la quantité de changement nécessaire pour dériver C de A et B, Sowa (1997) distingue trois niveaux d'intégration qui ressemblent un peu à la classification de Heflin and Hendler (2000) :
Alignement C'est la définition 1 de l'alignement (expliquée dans le chapitre 1). Il s'agit du plus bas niveau d'intégration qui ne nécessite aucun changement dans A et B. Il supporte l'interopérabilité la plus limitée (le mapping de Heflin and Hendler (2000)).
Compatibilité partielle Elle nécessite plus de changements dans A et B, et permet une interopérabilité moyenne. Toute inférence exprimée dans une ontologie en utilisant seulement les entités alignées, peut être traduite en une inférence équivalente dans l'autre ontologie (les révisions de mappings et l'intersection des ontologies de Heflin and Hendler (2000)).
Unification (Compatibilité totale) Elle nécessite des changements ou des réorganisations majeures dans A et B, pour entraîner l'interopérabilité la plus complète (le plus haut niveau d'intégration). En effet, tout ce qui peut être fait avec une ontologie peut être fait d'une manière exactement la même avec l'autre. Autrement dit, toute inférence exprimée dans une ontologie, peut être traduite en une inférence équivalente dans l'autre (C'est la fusion de Pinto and Martins (2004)).
Intégration de Heflin et Hendler
Selon Heflin and Hendler (2000), l'intégration des ontologies implique généralement l'identification des correspondances entre deux ontologies, la détermination des différences dans les définitions des entités, et la création d'une nouvelle ontologie qui résout ces différences. Selon eux, la simple création d'une nouvelle ontologie intégrée ne résout pas le problème d'intégration de l'information sur le Web. En effet, puisque d'autres ontologies et pages Web dépendent des ontologies intégrées, tous les objets dépendants devraient être révisés pour refléter la nouvelle ontologie. Vu que cette tâche est impossible, ils ont suggéré trois façons d'incorporer les résultats de l'intégration dans le Web comme le montre la figure 2.9 :
Mapping des ontologies C'est la définition 1 du mapping (expliquée dans le chapitre 1 ) les règles qui mettent en correspondance les entités de O 2 par rapport à O 1 (A ne pas confondre avec la notion de révision de mapping qui veut dire le débogage ou la réparation de mapping ! ! !). § Ils pensent que l'inconvénient de ces deux premières approches est que les concepts partagés entre deux domaines pourraient être également utilisés dans plusieurs autres domaines connexes, ainsi chaque nouveau domaine aurait besoin d'un ensemble de règles pour le mapper à tous les autres domaines en chevauchement. Et cela peut devenir très lourd. . § Ils considèrent cette troisième approche comme l'approche la plus naturelle d'intégration des ontologies car elle a l'avantage que l'équivalence des termes soit déterminée dans la phase de pré-traitement plutôt que lors de l'exécution de la requête.
INTÉGRATION DES ONTOLOGIES
Intégration de Pinto (consensuelle)
Dans l'intégration ou la composition des ontologies, Pinto (1999) considère que nous avons, d'une part, une (ou plusieurs) ontologies (O 1 , O 2 , ..., O N ) qui vont être intégrées dans l'ontologie cible, et d'une autre part, l'ontologie cible en cours de construction (O) qui sera issue du processus de l'intégration. Ainsi, cette méthode est incrémentale. Les domaines des ontologies sources (à intégrer dans l'ontologie cible) sont généralement différents entre eux, et différents du domaine de l'ontologie cible, mais il peut y avoir une relation entre eux (D 1 , D 2 , ..., D k ).
Il s'agit de deux ou plusieurs ontologies sources de sujets différents (ou de sujets liés) qui vont tout simplement être assemblées, composées, agrégées, ou combinées pour former une ontologie résultante, peut-être après que les ontologies sources aient subi quelques changements, comme l'extension, la spécialisation, la transformation, ou l'adaptation.
L'intégration vise à créer une ontologie d'un nouveau domaine plus large composé de tous les domaines des ontologies d'entrée. C'est un processus de réutilisation qui vise à construire des ontologies à partir d'autres ontologies existantes.
Les ontologies à intégrer doivent répondre à certaines exigences avant de leur appliquer le processus d'intégration, e.g., le domaine, l'abstraction, le type, la généralité, la modularité, l'évaluation, etc.
L'ontologie résultante doit avoir toutes les propriétés d'une bonne ontologie : consistante, cohérente, complète, ayant un niveau adéquat de détail, et décrivant seulement le vocabulaire nécessaire pour le domaine, etc. Il ne devrait pas avoir une ontologie existante similaire à la résultante, sinon il faudrait tout simplement réutiliser l'ontologie existante.
Avant leur inclusion dans l'ontologie résultante, les entités de l'ontologie à intégrer peuvent être :
• incluses (utilisées telles quelles) ;
• spécialisées (conduisant à une ontologie plus spécifique dans le même domaine) ;
• augmentées (étendues) par de nouvelles entités manquantes (soit par des entités plus générales, soit par des entités de même niveau) ; • adaptées (modifiées) pour les corriger ou les améliorer en changeant :
-leurs terminologies (pour se conformer aux règles de normalisation des noms, ou introduire une terminologie standard ou plus usuelle), -leurs documentations (pour les mettre à jour ou améliorer leur clarté), -leurs définitions (pour les mieux représenter dans le domaine concerné) ; • retirées (à cause de leur non pertinence). Ces adaptations transforment l'ontologie source choisie en l'ontologie voulue.
Ils précisent que des problèmes tels que la cohérence, la consistance, et le niveau de détail de l'ontologie résultante doivent être traités (i.e. elle ne doit pas avoir des parties de niveau de détail exagéré et d'autres de niveau adéquat).
La figure 2.10 illustre leur définition. Pinto (1999) interprétée par Keet (2004) Intégration comme étant synonyme à la fusion Klein (2001) considère l'intégration et la fusion comme égales. Il les définit par la création d'une nouvelle ontologie à partir de deux ou plusieurs ontologies existantes qui se chevauchent. C'est une définition très vague qui ressemble à la définition de la fusion des ontologies de Noy and Musen (2000). (2012) Autres définitions D'après Kokla (2006), une intégration d'ontologies génère une nouvelle ontologie intégrée sans modifier les originales. (2012), l'intégration des ontologies est un processus de construction d'une nouvelle ontologie en utilisant des ontologies disponibles. Elle peut être divisée en trois différents scénarios : -Le mapping (définition 1 de l'alignement dans le chapitre 1) ; -L'intégration (la réutilisation) ; -La fusion.
Raunich and Rahm
Selon Umer and Mundy
Selon Wróblewska et al. (2012), il existe différents types d'intégration des ontologies : -L'alignement, le mapping (définition 1 de l'alignement dans le chapitre 1) -La transformation -La fusion -L'intégration, etc.
Principaux outils d'intégration et leurs limites
Les approches récentes d'intégration des ontologies suivent un schéma modulaire qui décompose ce problème en sous-problèmes indépendants tels que le matching et puis la composition. De cette façon, elles profitent des grands progrès déjà réalisés dans le domaine du matching automatique des ontologies qui identifie les entités à intégrer dans le deuxième sous-problème (la composition).
La plupart des outils actuels d'intégration des ontologies sont semi-automatiques, car les experts de domaine et les ingénieurs de connaissances sont souvent nécessaires pour aider à la prise de décisions.
L'outil ILIADS
L'algorithme ILIADS de Udrea et al. (2007) prend en entrée deux ontologies consistantes O1 et O2, et retourne en sortie un alignement A entre O1 et O2, de telle sorte que l'intégration future de O1 et O2 à travers A soit consistante et cohérente. Les auteurs combinent un algorithme de matching de similarité (lexicale, structurelle, extensionnelle, et de clusters) avec un algorithme d'inférence logique qui raisonne sur les conséquences des relations (des correspondances) de l'alignement. Les relations de l'alignement sont exprimées comme des axiomes OWL (owl:equivalentClass, owl:equivalentProperty, owl:sameAs) ajoutés à l'ontologie résultante (qui compose O1 et O2).
Ils ont utilisé le raisonneur Pellet pour vérifier, à la fin de chaque expérience, si l'ontologie résultante est consistante ou pas. ILIADS a produit des inconsistances dans moins de 0,5% de leurs essais.
Mais ils n'ont pas évoqué les incohérences (e.g. le nombre de classes insatisfiables), car une ontologie peut être consistante tout en ayant une multitude de classes insatisfiables. Leur approche qui consiste en un processus interactif semi-automatique composé de quatre étapes principales (le calcul des correspondances, le calcul des nouvelles inférences, la détection des erreurs, et la réparation des erreurs identifiées à l'aide de l'utilisateur) a été appliquée pour réaliser l'outil "ContentMap", un plugin téléchargeable destiné à être utilisé dans Protégé 4.
ContentMap permet à l'utilisateur de choisir un ou plusieurs outils d'alignement à sélectionner (comme OLA, AROMA, CIDER, etc.) en leur attribuant différents poids. Il lui permet aussi de filtrer automatiquement les correspondances en choisissant un seuil de confiance minimal.
Le calcul des nouvelles inférences a été fait à l'aide de la notion de "différence déductive" qui compare les axiomes de l'ontologie résultante U (la composition de O1, O2, et M (Mapping)) avec les axiomes de chacune des ontologies initiales (O1, O2, et M ) pour détecter ceux qui existent dans l'ontologie de sortie mais qui n'existent pas dans les ontologies initiales, i.e. le système calcule la différence logique entre les inférences avant et après l'application des correspondances. Pour aider l'utilisateur à comprendre les conséquences sémantiques de U , ils lui montrent les justifications et les explications derrière la manifestation de ces nouveaux axiomes.
Les inférences imprévues de l'ontologie de sortie U (i.e. les inférences qui se trouvent dans l'ontologie résultante U mais qui ne se trouvent pas dans les ontologies individuelles O1, O2, et M ) seront présentées à l'utilisateur qui aidera à réparer U en choisissant la ou les source(s) d'axiomes à modifier (O1, O2, et/ou M ) et en supprimant les axiomes non désirés (qui sont, selon lui, générateurs d'incohérences) :
• Il peut choisir de supprimer uniquement des axiomes provenant de l'ontologie M et laisser les axiomes des ontologies initiales O1 et O2 intacts, puisque M est généralement considérée la plus grande source d'erreurs.
• Mais si deux entités correspondues sont originairement contradictoires dans O1 et O2, et leur relation dans M est correcte, l'utilisateur sera en face d'un dilemme ; soit supprimer l'axiome de la correspondance correcte, soit supprimer les axiomes originaux dans l'une ou les deux ontologies O1 et O2.
Par suite, le système exécute un algorithme de réparation qui essaie de supprimer le minimum d'axiomes tout en préservant les inférences jugées valides par l'utilisateur. (b) Sélectionner la première ontologie source à intégrer ; ouvrir l'ontologie cible "Intégrée" dans l'éditeur Protégé et importer l'ontologie source sélectionnée ;
(c) Utiliser l'option "Merge ontologies" de Protégé pour intégrer l'ontologie source dans la destination ("Intégrée"). De cette façon, l'ontologie source sera incluse dans le même fichier de l'ontologie cible ;
(d) Puis, en utilisant les outils de refactoring, changer l'IRI de tous les éléments de l'ontologie source en "Intégrée" qui est l'IRI de l'ontologie cible. 3. La vérification de la consistance et la cohérence de l'ontologie cible (en utilisant un raisonneur dans Protégé) et la vérification de l'absence de redondances, puis leur résolution en éliminant les relations "is-a" redondantes entre les entités (causées par les relations d'équivalence).
Les étapes 2 et 3 sont itératives (elles se répètent pour chaque ontologie source à intégrer).
Cette approche est simple et claire, et bien que nous n'ayons pas la moindre idée sur la qualité du matching réalisé, son inconvénient majeur est qu'elle est toute manuelle (elle est difficile à appliquer pour les grandes ontologies).
Dans les expérimentations, ils ont créé une ontologie cible vide dans laquelle ils ont intégré l'ontologie source eQual. Puis ils ont intégré l'ontologie source Ahn dans l'ontologie cible (contenant déjà eQual) en reliant les paires d'entités mises en correspondance par des relations d'équivalence. De la même manière, ils ont intégré trois autres ontologies sources (SiteQual, Website Evaluation Questionnaire, et Web Portal Site Quality) dans l'ontologie cible. Toutes ces ontologies sources appartiennent au domaine de l'évaluation de la qualité des sites Web. Dans chacune de ces itérations, le raisonneur a détecté des incohérences et des relations de subsomption redondantes dans l'ontologie résultante qu'il fallait corriger. Il s'agit d'une fusion, non pas d'une intégration. Dans ce travail, il n'y a pas une partie d'évaluation à l'aide d'une référence ou des résultats d'un travail concurrent pour se comparer avec.
L'outil FITON pour l'intégration des data sets
Zhao and Ichise (2014) ont proposé un système semi-automatique, nommé FITON, qui prend en entrée des data sets (deux ou plusieurs ontologies) du LOD (Linked Open Data) et qui retourne en sortie une ontologie intégrée (et enrichie).
Dans ce cas, la problématique c'est qu'il existe très peu (ou pas) de liens d'équivalence préétablis entres les différentes classes et propriétés des data sets (contrairement aux liens "sameAs" disponibles avec abondance entre leurs instances -des centaines de millions -) ; ce qui rend difficile l'extraction directe des classes et des propriétés équivalentes pour faire l'intégration de ces data sets.
Pour ce faire, ils ont intégré tout d'abord les instances identiques (informations déjà fournies sous forme de propriétés "owl:sameAs" dans les data sets) et ont extrait les classes et les propriétés qui décrivent ces instances pour former un graphe nommé graphe "sameAs" à partir duquel ils vont découvrir les similarités entre les différentes classes et propriétés contenues dans le graphe en combinant des méthodes de matching (terminologiques et sémantiques) pour parvenir enfin à intégrer tous les types d'entités des data sets.
Puisqu'il n'existe pas de benchmark pour les ensembles de données du LOD, un expert a créé manuellement un alignement de référence entre les deux data sets DBpedia et Geonames. DBpedia, la version "linked data" de Wikipédia de domaine transversal, contient (au moment Au final, leur ontologie de sortie contenait seulement 135 classes et 453 propriétés. Nous concluons qu'ils ont fait une sorte d'intersection entre les deux ontologies plutôt qu'une intégration, car la conservation des informations initiales des ontologies d'entrée n'est pas respectée.
Puisqu'ils comptent sur l'analyse des instances inter-liées (sameAs) pour découvrir les alignements entre les data sets, leur outil ne pourra pas fonctionner s'il n'existe pas (ou s'il n'existe que peu) de liens "sameAs" entre les instances. Ainsi, l'inconvénient de FITON est qu'il ne peut fonctionner efficacement que lorsqu'au moins 4% des instances d'un data set soient liées (identiques) aux instances de l'autre data set.
Différences entre la fusion et l'intégration
L'intégration et la fusion des ontologies sont tous les deux des processus de construction d'une nouvelle ontologie en se basant sur les informations de deux ou plusieurs ontologies sources. Cependant, dans la fusion, il y a beaucoup de connaissances en chevauchement entre les entités des ontologies sources, alors que dans l'intégration (la composition), il y a peu ou pas de chevauchement Pinto (1999).
Par conséquent, la différence principale entre ces deux processus est que, après le processus d'intégration (composition), nous pouvons identifier dans l'ontologie résultante les régions issues des ontologies sources, car les connaissances ont été laissées plus ou moins inchangées. En effet, l'ontologie résultante est composée de modules (sous-ontologies). Tandis qu'après le processus de fusion, il est généralement difficile d'identifier dans l'ontologie résultante les régions issues des ontologies sources car les connaissances ont été mêlées ou unifiées et homogénéisées ainsi modifiées.
Avantages de ces deux processus
Dans le contexte de l'ingénierie des ontologies, la réutilisation des modèles de connaissances existants est recommandée comme étant un facteur clé pour le développement d'ontologies rentables et de haute qualité. Ziemba et al. (2015) ont conclu que la construction d'une ontologie à l'aide de l'intégration ou de la fusion des ontologies sources, est beaucoup moins complexe que son processus de construction à partir de zéro.
L'intégration des ontologies réduit le coût et le temps requis pour la conceptualisation des domaines à partir de zéro, et améliore la qualité des ontologies nouvellement développées pour une application particulière en réutilisant des composants déjà validés.
La fusion des ontologies évite la confusion qui peut être générée à partir de plusieurs représentations du même domaine et renforce l'orchestration et l'harmonisation des connaissances.
CONSÉQUENCES SÉMANTIQUES DE CES DEUX PROCESSUS
Ces deux processus sont aussi particulièrement utilisés lorsqu'il est nécessaire d'effectuer un raisonnement impliquant plusieurs ontologies. Jiménez-Ruiz et al. (2009) affirment que quand nous raisonnons sur les ontologies à intégrer et leurs correspondances, sur l'ontologie produite suite à l'intégration (contenant les axiomes des correspondances), ou sur l'ontologie produite suite à la fusion, il est souvent nécessaire de détecter des conflits ; ainsi des erreurs vont probablement se manifester. Ces erreurs sont dues à deux causes principales :
Conséquences sémantiques de ces deux processus
• Les correspondances (générées généralement par un outil de matching automatique) peuvent comporter quelques fautes ou être erronées (incorrectes).
• Même si les correspondances trouvées étaient toutes correctes, les ontologies à intégrer peuvent contenir des descriptions contradictoires des entités correspondues, et ceci est à cause de la représentation variable des ontologies sources. En effet, selon Cheatham and Pesquita (2017), les correspondances qui forment l'alignement ne sont pas indépendantes les unes des autres :
• Il y a des cas dans lesquels seulement une parmi plusieurs correspondances peut être vraie (parmi les correspondances ayant la même entité source). • Dans d'autres cas, plusieurs correspondances, réunies ensemble, peuvent conduire à une inférence involontaire et indésirable ou une classe insatisfiable.
• Ou la combinaison de ces deux causes.
Ces erreurs sont des conséquences logiques imprévues (e.g. des classes insatisfiables, de nouvelles subsomptions (suite aux axiomes d'équivalence), etc.) difficiles à détecter, à comprendre, et à réparer. Par conséquent, l'ontologie globale créée, associée aux règles de correspondances, est très prédisposée aux erreurs et nécessite une supervision d'un expert de domaine ou une supervision automatique supportée par les applications.
Récemment, la recherche a été conduite au debugage et à la révision des alignements, ainsi que le debugage et la réparation des insatisfiabilités dans les ontologies OWL. Cependant, la suppression de certaines insatisfiabilités peut entraîner une perte d'informations précieuses provenant des ontologies sources. D'ailleurs, c'est le plus grand inconvénient de l'intégration ou de la fusion des ontologies dans les travaux actuels.
Exemples
Les mêmes données peuvent être décrites par des ontologies de différentes perspectives. Cependant, même en se concentrant sur une même perspective, la multitude d'ontologies actuellement utilisées pour les décrire, empêche leur intégration transparente. En effet, l'intégration de deux ontologies de différents modèles peut causer des incohérences logiques.
Supposons que A est une classe satisfiable dans O1, et B et C sont des classes disjointes dans O2, et supposons que deux correspondances (d'un alignement entre O1 et O2) disent que A est une sous classe de B, et que A une est sous classe de C. Si ces correspondances, exprimées sous forme d'axiomes, sont ajoutées à la composition des deux ontologies O1 et O2, ils vont créer un problème car une classe ne peut pas être une sous classe de deux classes mères disjointes. Par conséquent, A sera insatisfiable et l'ontologie résultante sera incohérente. Et si jamais A avait une instance, l'ontologie résultante serait inconsistante Abbas and Berio (2013).
Voici un autre exemple : Fahad et al. (2010), les outils de fusion ou d'intégration existants ne préservent pas la connaissance disjointe des ontologies sources (pour but de produire à la fin une ontologie cohérente) et sont en général semi-automatiques nécessitant beaucoup d'intervention humaine pour la validation des correspondances suggérées et également la résolution des conflits générés durant ou après la construction de l'ontologie résultante. Par conséquent, le résultat est incomplet (il viole la description des disjonctions) et très dépendant de l'observation et de l'intelligence de l'utilisateur.
Conséquences dans les outils existants
D'après
La raison de l'inexactitude de ces outils c'est qu'ils n'exploitent pas la sémantique cachée (et précisément les disjonctions) des ontologies sources pendant la phase de matching, et détectent ainsi des correspondances non fiables qui créent des situations erronées dans l'ontologie de sortie.
CONSÉQUENCES SÉMANTIQUES DE CES DEUX PROCESSUS
Remarque
Les alignements peuvent être réalisés pour aider l'interrogation distribuée ou bien le raisonnement logique Cheatham and Pesquita (2017) :
• Pour l'interrogation, le rappel (i.e. les résultats pertinents) des correspondances est un aspect important. Cela signifie que, pour les applications centrées sur les requêtes, ce n'est pas grave si les correspondances provoquent d'inconsistance logique, l'essentiel c'est que les relations soient correctes. Ce cas s'applique dans le contexte des linked data, et l'intégration des ontologies tout en les maintenant séparées.
• Pour le raisonnement logique, le rappel n'est pas suffisant car les correspondances peuvent être correctes, mais générant de conflits. En effet, les applications qui ont l'intention d'employer un raisonneur sur les données intégrées ne peuvent pas utiliser un alignement qui génère une inconsistance logique. Ce cas s'applique dans le contexte de l'intégration des ontologies tout en les regroupant en une seule ontologie (la fusion et l'ontologie de pont entre autres).
Discussion et synthèse
Dans notre mémoire, nous allons nous intéresser aux ontologies de pont, qui, suivant les définitions ci-dessus, peuvent entrer dans le cadre de la fusion et de l'intégration, les deux à la fois, mais vu que le terme "intégration" peut être également un terme générique qui inclut la fusion, nous avons choisi d'utiliser le terme intégration, d'où le titre de notre mémoire. L'ontologie de pont peut être considérée comme le plus faible niveau de "fusion" des ontologies car dans nos expérimentations, nous avons uni des ontologies ayant un domaine identique ou proche ; cependant, elle peut être considérée aussi comme une "intégration", car il s'agit bien d'une composition des ontologies sources de telle manière que les éléments de chacune des ontologies sources soient facilement reconnus dans l'ontologie résultante.
Ce type d'intégration est utilisé par exemple dans le cas où des entreprises en coopération veulent unir leurs connaissances sans tout de même changer leurs ontologies de base, ainsi changer toutes les données qui s'y conforment. En d'autres termes, elles veulent coopérer tout en restant indépendantes. Dans l'ontologie résultante, les noms et les descriptions des entités issues des ontologies sources restent comme ils le sont originairement, sans changer tout un système qui en est dépendant.
Conclusion
L'intégration des ontologies est encore plus difficile avec les ontologies de domaines identiques, similaires, complémentaires, et surtout interdisciplinaires. Les difficultés apparaissent aussi avec les ontologies de différents niveaux formels (les ontologies légères et lourdes). Or, les ontologies qui ont beaucoup de points communs dans la structuration et l'organisation de leurs entités ont plus de chance de ne pas avoir des conflits et des difficultés d'intégration.
Il y a de multiples possibilités pour intégrer les ontologies. Les approches peuvent être distinguées par trois facteurs principaux : le niveau d'intégration (d'interopérabilité sémantique), le(s) domaine(s) des ontologies d'entrée, et la méthode d'intégration (incrémentale ou non).
Introduction
Dans ce chapitre, nous décrivons les étapes de notre méthode et celle de la référence, citons les conditions favorables pour avoir les meilleurs résultats, et proposons une nouvelle terminologie à utiliser au lieu des notions floues et mal appropriées de la littérature.
Description de la nouvelle méthode
En général, le processus d'intégration des ontologies passe par deux étapes majeures : le matching des ontologies d'entrée, puis la composition / l'union / l'agrégation de ces ontologies avec les mappings (ou les alignements) générés suite à l'étape de matching. La figure 3.1 illustre ce processus.
Mais dans notre travail, nous avons utilisé des alignements de référence comme entrée, sans avoir fait l'étape de matching par nous-mêmes, ainsi nous allégeons notre charge de travail et nous nous concentrons sur l'intégration concrète. Les temps d'exécution de nos expérimentations ne comprennent pas le temps de matching. En effet, le vrai temps d'exécution sera la somme de notre temps global et celui du matching (supposé fait). Ainsi, le temps d'exécution réel de notre méthode dépendra du temps d'exécution de l'algorithme du matching utilisé, et la qualité de l'ontologie résultante dépendra de la qualité des alignements d'entrée utilisés. La
Approche générale de OIA2R
Notre implémentation et également celle de la référence sont divisées en deux parties majeures :
1. La première consiste à composer (assembler) les ontologies d'entrée. Avec ce code seul, nous obtiendrons une ontologie composée des sous-ontologies sources, ainsi contenant les axiomes de description des entités des ontologies sources, sans "bridging" axiomes entre elles. § Pour l'instant, il s'agit d'une intégration (composition) simple sans interopérabilité sémantique.
2. La deuxième consiste à ajouter (aux axiomes créés dans la première étape) des axiomes de pont qui sont en fait des axiomes d'équivalence entre les différentes entités. Ce sont des axiomes qui traduisent fidèlement les correspondances provenant des alignements entre les ontologies sources. § Ces deux étapes font une intégration qui produit une ontologie dite "ontologie de pont" qui permet l'interopérabilité sémantique.
Ontologie de sortie = axiomes des entités sources + "bridging" axiomes 3.
Démarche détaillée de OIA2R
Introduction au refactoring
Un IRI d'une ontologie (Internationalised Resource Identifier) identifie l'ontologie d'une façon unique. Il est considéré comme son nom. Il peut être un IRI physique, i.e. un fichier .owl local, ou bien un URI à publier dans le Web, i.e. une adresse Web contenant ce fichier .owl. L'IRI d'une entité (classe, propriété, ou instance) est composé d'un "IRI de préfixe" (un préfixe) suivi du "nom" court de l'entité (un suffixe). En général, la partie "préfixe" de l'entité est exactement l'IRI de l'ontologie actuelle (ou d'une autre ontologie existante), mais elle peut aussi contenir en plus un "ID" (identifiant), juste après l'IRI de l'ontologie :
Dans une ontologie, nous ne pouvons pas avoir deux IRIs identiques, i.e. si deux entités (deux objets créés dans OWL API) ont le même IRI, alors après leur création, il s'agira de la même entité. Ainsi, dans une ontologie, une entité (un IRI complet d'une entité) doit être unique. Mais quand nous allons intégrer des ontologies de même domaine en une seule ontologie, il y aura un grand risque de rencontrer des entités, même si originaires d'ontologies différentes (donc d'IRIs de préfixe différents), mais ayant exactement le même "nom local" (suffixe). Ceci nous causera un problème, car nous voulons, conformément aux standards, que les entités de notre future ontologie aient comme "IRI de préfixe" l'IRI de cette dernière. Dans ce cas, pendant la création de ces entités ayant le même "nom", seulement une sera créée, et elle aura dans sa description toutes les informations de ces entités. Elle sera bien évidemment sémantiquement erronée.
Voici un exemple pour concrétiser ce que nous étions en train de dire. Les entités de nom "Paper", "Person", et "Conference" existent dans au moins trois ontologies de la base "Conference" : cmt (Ont1), conference (Ont2), et confOf (Ont3) :
• Voici les IRIs complets originaux des entités ayant le nom "Paper" :
- Pour pallier cette redondance et ne pas mettre les informations (les définitions) de toutes ces entités dans l'ontologie de sortie dans une seule entité ayant ce nom, nous avons choisi d'ajouter un ID aux IRIs de préfixe des entités de notre ontologie, pour pouvoir les différencier.
Nous allons attribuer à chaque ontologie un numéro ; celle qui sera parsée la première aura le numéro 1, la suivante aura le numéro 2, et ainsi de suite. L'ID sera le numéro de l'ontologie originale d'où venait l'entité en question. Par conséquent, nous pourrons garder intacte la partie "nom" des entités redondantes, sans être obligés de la modifier. C'est seulement la partie "IRI de préfixe" qui va changer. Nous avons défini l'ID sur quatre caractères, ainsi les quatre derniers caractères du préfixe de chaque entité seront réservés à l'ID.
• Voici comment vont paraître les IRIs complets des entités de nom "Paper" dans notre ontologie de sortie : -http ://intégration/001#Paper -http ://intégration/002#Paper -http ://intégration/003#Paper
• Voici comment vont paraître les IRIs complets des entités de nom "Person" dans notre ontologie de sortie : -http ://intégration/001#Person -http ://intégration/002#Person -http ://intégration/003#Person
• Voici comment vont paraître les IRIs complets des entités de nom "Conference" dans notre ontologie de sortie : -http ://intégration/001#Conference -http ://intégration/002#Conference -http ://intégration/003#Conference
DÉMARCHE DÉTAILLÉE DE OIA2R
De cette manière, le "nom" redondant d'une entité quelconque sera préservé, aura un IRI unique dans la nouvelle ontologie, et toutes ses informations liées (sa définition dans son ontologie originale) seront conservées correctement.
Première étape
En détail
Pendant le parsing des classes des ontologies d'entrée, nous extrayons pour chaque classe sa définition qui consiste en ses annotations utilisées (ses labels, ses commentaires, ses propriétés d'annotation), ses superclasses, ses classes équivalentes, et disjointes (informations avec lesquelles nous créons les classes de notre future ontologie). Et nous remplissons au fur et à mesure le HMap des classes qui contiendra comme "clé" l'URI original de la classe, et comme "valeur" le numéro de l'ontologie dont il est issu.
Pendant le parsing des propriétés d'objet/data des ontologies d'entrée, nous extrayons pour chaque propriété sa définition qui consiste en ses annotations utilisées (ses labels, ses commentaires, et ses propriétés d'annotation), ses super-propriétés, ses domaines, ses images, ses propriétés inverses (pour les propriétés d'objet seulement), équivalentes, disjointes, et son type (informations avec lesquelles nous créons les propriétés de notre future ontologie). Et nous remplissons au fur et à mesure les deux HMaps (des propriétés d'objet, et des propriétés data) qui contiendront comme "clé" l'URI original de la propriété, et comme "valeur" le numéro de l'ontologie dont il est issu.
Pendant le parsing des instances des ontologies d'entrée, nous extrayons pour chaque instance sa définition qui consiste en ses annotations utilisées (ses labels, ses commentaires, et ses propriétés d'annotation), ses classes qui l'instancient, ses instances identiques et différentes, les propriétés d'objet/data qu'elle appelle et leurs valeurs (informations avec lesquelles nous créons les instances de notre future ontologie). Et nous remplissons au fur et à mesure le HMap des instances qui contiendra comme "clé" l'URI original de l'instance, et comme "valeur" le numéro de l'ontologie dont il est issu.
Pendant le parsing des propriétés d'annotation des ontologies d'entrée, nous extrayons pour chaque propriété sa définition qui consiste en ses labels, ses commentaires, ses superpropriétés, ses domaines, et ses images (informations avec lesquelles nous créons les propriétés d'annotation de notre future ontologie).
Pendant le parsing des individus anonymes des ontologies d'entrée, nous extrayons pour chaque individu (qui est sous forme d'un ID local unique) sa définition qui consiste en ses labels et ses commentaires, etc. (informations avec lesquelles nous créons les individus anonymes de notre future ontologie). § Ainsi, lors du parsing des ontologies originales, nous avons créé des entités propres à notre future ontologie, et en même temps, nous avons rempli les quatre HMaps correspondants à chaque type d'entité -classes, propriétés objet, propriété de données, et instances-(où l'URI original de l'entité forme la "clé", et le numéro de son ontologie forme la "valeur"), et tout cela pour remédier au problème de redondance des entités expliqué dans la section précédente.
Deuxième étape
Nous allons représenter les correspondances (entre les différentes entités des ontologies sources) par des axiomes OWL, car cette représentation est sémantiquement correcte et permet de réutiliser l'infrastructure et le vocabulaire du langage OWL.
En général
Nous parcourons les correspondances (les paires d'entités) de chaque alignement d'entrée (i.e. les cellules qui ont une mesure supérieure ou égale à un seuil que l'utilisateur a fixé), et au fur et à mesure, nous ajoutons à la nouvelle ontologie des axiomes de pont (des liens d'équivalence) traduisant ces correspondances entre les entités déjà créées dans notre ontologie.
En détail
Dans OWL API, nous ne pouvons pas lier les entités directement par des axiomes. En effet, il existe quatre types de méthodes de création d'axiomes, chacun destiné à un type d'entité (-classes, propriétés d'objet, propriétés de données, et instances-). Ceci s'applique entre autres pour les axiomes d'équivalence que nous allons utiliser dans notre cas. Les voici : Figure 3.4 -Création des axiomes d'équivalence dans OWL API Sachant que dans les alignements, nous ne pouvons pas savoir le type des entités mises en correspondance, et que les entités sont citées par leurs URIs originaux complets (comme elles ont été définies dans leur ontologie originale), nous avons eu recours aux quatre HMaps remplis déjà dans l'étape précédente, pour savoir le type de chaque entité, et son ID (le numéro de l'ontologie dont elle est issue), afin de pouvoir créer des "bridging" axiomes en liant les couples d'entités (citées dans les cellules des alignements) par des axiomes d'équivalence, en changeant leurs URIs de préfixe originaux par l'URI de notre ontologie + l'ID correspondant.
A la fin, dans notre ontologie de sortie, les supposés "bridging" axiomes créés ne vont plus être considérés comme ceci. En effet, selon nous, ils vont plutôt être perçus comme des axiomes d'équivalence normaux et originaux liant des entités d'une nouvelle ontologie indépendante (la nôtre), comme si celle-ci n'était pas le résultat d'une intégration ; car les entités la composant ont toutes un "URI de préfixe" propre à elle (les URIs des entités de notre nouvelle ontologie ne sont pas des URIs des ontologies sources déjà publiées). Autrement dit, nous avons fait un refactoring (une personnalisation) des URIs des entités sources que nous avons réutilisées pour former notre ontologie.
Démarche détaillée de la référence
Comme il n'y a pas de benchmark pour les approches d'intégration des ontologies, nous avons créé une autre version d'intégration avec laquelle nous pourrons nous comparer. Nous considérerons l'ontologie résultante comme notre ontologie de référence, car il s'agit d'une ontologie de pont sans perte de connaissances (i.e., complète). Nous appellerons cette version comme l'intégration de "référence" ou la "pseudo-référence". En principe, la démarche de référence consiste en la composition (l'union / l'intégration) automatique des ontologies sources et l'ontologie qui correspond aux alignements entre elles. Supposons que nous avons deux ontologies O 1 et O 2 à intégrer, et un alignement A entre elles. L'ontologie résultante serait O 3 = O 1 + O 2 + O A , après avoir converti A en une ontologie. En effet, puisque le format d'alignement est exprimé en RDF, ainsi il est librement extensible, l'"Alignment API" permet de convertir les correspondances (les cellules) d'un alignement en des axiomes OWL d'équivalence, de subsomption, et de disjonction (à l'aide de la méthode OWLAxiomsRendererVisitor(ObjectAlignment)) pour générer une ontologie comprenant à la fois les entités alignées et les axiomes OWL de pont. Malheureusement, cette tâche n'a pas pu être effectuée correctement, et nous n'avons pas pu transformer directement l'alignement en une ontologie. Pour ce faire, nous avons appliqué l'idée de notre approche (déjà expliquée).
Première étape
OWLOntologyMerger() est une méthode prédéfinie dans OWL API qui unit toutes les ontologies chargées (loaded) dans le OWLOntologyManager. Nous n'avons qu'à spécifier à la méthode createMergedOntology() le nouvel URI de l'ontologie que nous allons créer :
OWLOntologyMerger integration = new OWLOntologyMerger(manager) ; OWLOntology newOnto = integration.createMergedOntology(manager, newIRI) ; Il faut noter que les termes "Merger" et "MergedOntology" utilisés par OWL API sont faux et accentuent encore plus la mécompréhension du terme "fusion" dans la communauté. En effet, c'est une composition (union, intégration, association), ce n'est pas une fusion. D'ailleurs, Protégé fait exactement la même erreur avec le terme "Merge ontologies" dans le menu "refactor".
Nous aurons avec seulement cette étape une ontologie intégrée (composée, agrégé) qui ne manque aucune information des ontologies d'entrée, et qui maintient les URIs d'origine des entités sources en les mettant dans l'ontologie de sortie telles qu'elles sont originairement dans leurs ontologies d'entrée. Nous parcourons les correspondances de chaque alignement d'entrée (i.e. les cellules qui ont une mesure supérieure ou égale au seuil que nous avons fixé), et au fur et à mesure, nous ajoutons à la nouvelle ontologie des liens d'équivalence (des "bridging" axiomes) qui traduisent fidèlement ces correspondances entre les entités.
Deuxième étape
3.4. COMPARAISON ENTRE OIA2R ET LA RÉFÉRENCE § Sachant que dans OWL API, nous ne pouvons pas lier les entités directement par des axiomes, mais plutôt à travers quatre types de méthodes de création d'axiomes pour chacun des types d'entité (-classes, propriétés objet, propriétés de données, et instances-), et sachant que dans les alignements, nous ne pouvons pas savoir le type des entités correspondues, nous avons pu, à partir des quatre HSets déjà créés et remplis, savoir le type de chaque entité des cellules parcourues et créer des "bridging" axiomes (lier les entités citées dans les cellules des alignements par des axiomes d'équivalence en gardant leurs URIs originaux (tels qu'ils sont cités dans les alignements)).
Comparaison entre OIA2R et la référence
L'intégration de référence n'a absolument aucune perte d'informations des ontologies originales, pas le moindre axiome perdu, par contre la nôtre ne parvient pas à tout parser, nous perdons les entités anonymes et les restrictions, car toutes les entités que nous parsons sont nommées.
La seule différence entre ce travail et la référence, c'est que toutes les entités de notre ontologie résultante ont un URI de préfixe propre à nous, contrairement à l'intégration de référence qui garde les URIs originaux des entités et qui ne fait aucun refactoring. Ainsi notre ontologie est tout à fait originale et ne pointe pas sur des entités appartenant à des ontologies externes déjà existantes.
Conditions favorables pour de meilleurs résultats
3.5.1 Mapping (1-à-1) au lieu d'alignement (1-à-N) Comme expliqué dans le chapitre 1, le matching retourne un alignement ou un mapping. Dans l'alignement, une entité d'une première ontologie peut être mise en correspondance avec une ou plusieurs entités d'une deuxième ontologie (ces correspondances n'ont pas la même pertinence). Il s'agit de correspondances "1-à-N". Tandis que dans le mapping, une entité d'une première ontologie peut être mise en correspondance avec zéro ou une seule entité d'une deuxième ontologie. Il s'agit de correspondances "1-à-1". D'après les expérimentations (que nous détaillerons dans le chapitre 4), nous avons remarqué que, dans l'ontologie résultant d'une intégration qui utilise des alignements, le nombre de classes insatisfiables est beaucoup plus important que celui de l'ontologie résultant d'une intégration qui utilise des mappings.
Ci-dessous un exemple qui montre comment se forme l'insatisfiabilité d'une classe dans une ontologie de pont. La figure 3.6 montre les justifications que nous a affichées le debugueur de OWL API (à l'aide du raisonneur HermiT) pour une des classes insatisfiables (002#Tissue_Dissection) générées dans une de nos expérimentations.
La figure 3.6 est la représentation graphique de axiomes de justification générés par le debugueur, où les classes de couleur rouge sont les classes insatisfiables. figure 3.7 montre les deux correspondances qui ont causées toutes ces incohérences. La relation " ?" veut dire une relation d'équivalence "=" correcte mais qui génère des insatisfiabilités dans l'ontologie intégrée.
CONDITIONS FAVORABLES POUR DE MEILLEURS RÉSULTATS
Sachant que la relation d'équivalence (des "bridging" axiomes) est en réalité égale à deux relations de subsomption dans les deux sens, le schéma devient comme le montre la figure 3.8. Le pire avec les alignements, c'est que les entités cibles (ou sources) correspondues avec la même entité source (ou cible) sont généralement toutes proches (i.e., voisines) ainsi susceptibles d'avoir entre elles des relations de disjonction qui sont la première source des incohérences.
CONDITIONS FAVORABLES POUR DE MEILLEURS RÉSULTATS
La figure 3.9 montre ce que devient si nous ne gardons qu'une seule correspondance pour l'entité source "003#Clinical_finding". Dans notre approche, nous avons la possibilité de filtrer les correspondances ayant la même entité source ou la même entité cible en gardant uniquement la correspondance ayant la plus grande confiance (c.f., figure 3.10).
En premier lieu, nous créons deux HMaps : la première contiendra comme clé l'IRI de l'entité source de chaque cellule d'un alignement, et comme valeur l'IRI de l'entité cible. La deuxième contiendra comme clé l'IRI de l'entité source de chaque cellule d'un alignement, et comme valeur la mesure de confiance de la relation. Pendant le parsing des alignements d'entrée, nous remplissons ces deux HMaps au fur et à mesure, de telle sorte que si nous trouvons une cellule dont l'entité source est déjà existante comme clé dans les HMaps et dont la mesure de confiance est supérieure à celle rencontrée avant, nous mettons à jour les valeurs correspondantes à cette clé dans les deux HMaps ; sinon nous ne faisons rien (i.e. si la cellule dont l'entité source est déjà existante comme clé dans les HMaps et dont la mesure de confiance est inférieure ou égale à celle rencontrée avant, elle ne sera pas stockée, car nous avons celle qui est plus pertinente déjà enregistrée dans les HMaps).
En deuxième lieu, nous allons refaire la même chose, mais à l'envers. Nous créons deux HMaps "inverses" : la première contiendra comme clé l'IRI de l'entité cible de chaque entrée de la première HMap issue de l'étape précédente, et comme valeur l'IRI de l'entité source. La deuxième contiendra comme clé l'IRI de l'entité cible de chaque entrée de la première HMap issue de l'étape précédente, et comme valeur la mesure de confiance de la relation. Pendant le parcours des entrées de la première HMap déjà remplie dans l'étape 1, nous remplissons ces deux HMaps "inverses" au fur et à mesure, de telle sorte que si nous trouvons une entrée dont l'entité cible est déjà existante comme clé dans les HMaps "inverses" et dont la mesure de confiance est supérieure à celle rencontrée avant, nous mettons à jour les valeurs correspondantes à cette clé dans les deux HMaps "inverses".
Au lieu d'utiliser les alignements originaux (i.e., d'entrée), nous utiliserons le premier HMap "inverse" qui contiendra toutes les correspondances filtrées des alignements d'entrée. Ce HMap exprime les correspondances supposées former des mappings.
Réparation des alignements
La réparation des alignements ou des mappings (en supprimant les correspondances qui sont susceptibles d'engendrer des insatisfibilités lorsqu'elles seront associées aux ontologies d'entrée) aide à diminuer les incohérences dans l'ontologie résultante. D'après Cheatham and Pesquita (2017), actuellement, peu de systèmes de matching d'ontologies ne supportent la gestion de l'incohérence logique, et encore moins pour les grandes ontologies. L'approche la plus basique consiste à filtrer les correspondances qui violent une série de règles sémantiques (comme le fait YAM++ (2012)). Des approches plus sophistiquées reposent sur des procédures automatisées capables d'identifier les correspondances impliquées dans l'incohérence logique et de sélectionner celles à supprimer pour atteindre la cohérence, comme AML (2015) et LogMap (2011). Abbas and Berio (2013) • D'autres auteurs trouvent que les incohérences peuvent être causées soit par les alignements, soit par les ontologies (ContentMap de Jiménez-Ruiz et al. (2009)). Ils ont déduit que lorsque les correspondances sont celles qui sont prévues (sont correctes) et lorsque l'ontologie résultante contient quand même des incohérences logiques, alors ces incohérences doivent être forcément dues aux ontologies sources qui sont incompatibles à cause des différences de leurs conceptualisations. Ils proposent alors une solution pour réparer les ontologies (supprimer les axiomes qui causent des contradictions dans l'ontologie de sortie) à l'aide d'un ingénieur du domaine.
Dans notre approche, nous avons exploité les outils LogMap et ALCOMO qui prennent en entrée deux ontologies sources et un alignement entre eux, et génèrent un alignement réparé (après avoir fait un calcul des inférences entre eux pour décider quelles correspondances supprimer). Nous utilisons l'un de ces deux outils pour tous nos alignements d'entrée afin de minimiser au maximum les incohérences dans notre future ontologie résultante.
LogMap 1 est un système de matching d'ontologies basé sur la logique créé par Jiménez-Ruiz and Grau (2011). Il effectue une réparation des alignements (i.e., une transformation d'un alignement incohérent en un alignement cohérent) en exécutant un raisonnement (parfois incomplet). Il supprime ou modifie les correspondances qui causent l'apparition des classes insatisfiables.
Créé par Meilicke (2011), ALCOMO 2 est un système de debugage des alignements qui permet de transformer un alignement incohérent en un alignement cohérent en lui supprimant certaines correspondances. Il est complet car il détecte toute forme d'insatisfiabilité entre les ontologies causée par les alignements.
Nouvelle définition de la notion d'intégration
Dans la littérature, il n'y pas un accord général sur les définitions de l'intégration et la fusion des ontologies. Flouris et al. (2006), et Euzenat and Shvaiko (2013) ont essayé de faire une clarification et une désambiguïsation de toutes les terminologies de l'ingénierie des ontologies. Et Pinto (1999) a fait la même chose pour les termes "intégration" et "fusion". Mais malgré leurs efforts, les termes "intégration" et "fusion" sont toujours mal définis, mal compris, et mal placés.
Comme le dit Pinto (1999), l'intégration concerne des ontologies de différents ou de proches domaines pour former une ontologie de domaine plus large englobant tous les domaines sources ; et la fusion concerne des ontologies de même domaine pour former une ontologie décrivant mieux ce domaine-là. La confusion réelle réside dans le sens naturel de ces termes. En effet, dans la littérature, la plupart des auteurs parlent de fusion lorsqu'ils vont fondre les entités équivalentes pour les remplacer par une seule, ou lorsqu'ils vont changer et mêler les hiérarchies des ontologies sources en répartissant leurs entités autrement dans l'ontologie cible ; et parlent d'intégration ou de composition lorsqu'ils vont regrouper et assembler directement les ontologies, telles qu'elles sont, dans une autre ontologie, sans fusionner leurs entités équivalentes et sans toucher leurs hiérarchie originale. Les processus de fusion ou d'intégration peuvent s'appliquer concrètement à toute ontologie (de domaine = ou =). C'est seulement le but à atteindre qui différencie vraiment les deux définitions consensuelles. Autrement dit, le problème réside dans le fait que nous pourrons faire une fusion (au sens propre du mot) pour deux ontologies de domaines différents (car elles peuvent contenir quand même des chevauchements entre elles), et une composition / intégration (au sens propre du mot) pour des ontologies de même domaine. Ce qui est contradictoire aux définitions soi-disant standardisées.
Dans le cas d'une ontologie de pont créée à partir d'ontologies de même domaine, si nous nous conformons à ces définitions, nous sommes en train de faire une fusion des ontologies, car il s'agit bien d'ontologies de même domaine, mais réellement, nous ne fusionnons pas les entités, ne mélangeons pas leurs emplacements, et ne modifions pas leurs structures, nous faisons une ontologie de pont où les entités originales et leurs hiérarchies sont intactes. Pas de fusion dans un processus de fusion. En contre parti, notre travail respecte les règles de la définition de l'intégration des ontologies qui dicte que les parties provenant des ontologies sources soient identifiables facilement dans l'ontologie de sortie et qu'il s'agit juste d'une inclusion, d'une agrégation, ou d'un assemblage des ontologies sources pour former un tout. Avions-nous fait une intégration ou une fusion ?
Par conséquent, nous proposons que le terme "intégration des ontologies" soit le terme général de toutes les définitions rencontrées dans le chapitre 2, par analogie avec le terme "intégration des données", et nous proposons qu'il soit appliqué sur des ontologies de mêmes ou de différents sujets, ainsi pour de différents objectifs. Bien évidemment, les ontologies de même sujet seront les plus dures à intégrer ; les ontologies de très différents domaines n'auront pas (beaucoup) de correspondances entre elles, donc seront toujours plus faciles à intégrer. Nous distinguons cinq niveaux d'intégration selon le niveau d'interopérabilité sémantique. Nous expliquons chaque type avec le plus simple cas qui intègre seulement deux ontologies (c.f., figure 3.11) :
Alignement Défini par Noy and Musen (2000) (définition 1), il implique deux ontologies séparées et un alignement (deux mapping dans les deux sens, une articulation, ou une ontologie intermédiaire comme le nomme Kalfoglou and Schorlemmer (2003)) à travers lequel une ontologie peut interroger l'autre et vice-versa.
Interconnexion des données Nommée aussi "révisons de mapping" par Heflin and Hendler (2000), terme qui peut être confondu avec la notion de debugage et de réparation des mappings, elle consiste en l'ajout de correspondances d'un alignement ou d'un mapping dans les deux ontologies. En d'autres termes, c'est l'ajout des relations sémantiques entre les entités de l'ontologie en question et les entités de l'autre ontologie comme prescrit dans l'alignement ou le mapping entre elles. Réconciliation / Coévolution Définie par Euzenat and Shvaiko (2013), elle peut être une transformation des entités de l'une des ontologies par les entités de l'autre comme le prescrit l'alignement entre les deux ontologies, ou bien une transformation des entités des deux ontologies, nommée "intersection d'ontologies" par Heflin and Hendler (2000), après la standardisation des termes correspondus.
Ontologie de pont Introduite par De Bruijn et al. (2006), elle met les deux ontologies et les correspondances de leur alignement dans une même ontologie qui les englobe sans rien modifier.
Fusion / Unification Appelée par Pinto (1999) "Fusion", et appelée "unification" ou "compatibilité totale" par Sowa (1997), elle génère une ontologie en sortie et peut se faire de différentes manières ; la plus simple est d'unir les ontologies sources et de fusionner leurs entités équivalentes (comme décrites dans l'alignement) ; et la plus difficile est d'exploiter, en plus des correspondances d'équivalence et de disjonction, des correspondances de subsomption qui changeront énormément la hiérarchie originale des ontologies.
Selon Calvanese et al. (2001), dans le cas de plusieurs ontologies sources, l'intégration des ontologies peut impliquer soit une approche "global-centric", où les entités de l'ontologie globale sont mises en correspondance avec les entités des ontologies locales, soit une approche "localcentric", où les entités des ontologies locales sont mises en correspondance avec les entités de l'ontologie globale. Mais nous ajoutons une autre approche que nous nous permettons d'appeler "non-centric" où les entités des paires d'ontologies sont mises en correspondance. Ainsi, il n'existe pas d'ontologies globale et locales, elles ont toutes la même priorité.
Concernant l'ontologie de pont et l'ontologie de fusion, le processus d'intégration de plusieurs ontologies peut être fait soit d'une manière incrémentale, où il y a une ontologie cible prioritaire ou une ontologie vide (nommée ontologie globale) dans laquelle les autres ontologies sources (locales) seront intégrées l'une après l'autre (c.f., figure 3.12 and 3.13) ; soit d'une manière non incrémentale, où toutes les ontologies sources ont la même priorité et seront intégrées les unes avec les autres en même temps pour former ensemble l'ontologie cible.
Remarque
Rappelons que les termes "Merger" et "MergedOntology" utilisés par des méthodes dans OWL API, et "Merge ontologies" dans le menu "refactor" de Protégé sont faux, car ils font juste une composition/union/agrégation d'ontologies (c.f., figure 3.17); ils mettent les ontologies sources, telles qu'elles sont, dans une nouvelle ontologie qui les englobe, sans créer des liens de pont en elles et sans les fusionner. Nous n'avons pas ajouté ce simple processus dans les types d'intégration, car il ne permet aucune interopérabilité sémantique entre les agrégats (les sous-ontologies sources) de l'ontologie résultante. Conclusion L'ontologie de pont que nous allons implémenter est une intégration d'interopérabilité moyenne. Elle peut être exploitée réellement quand deux entreprises veulent coopérer, collaborer et intégrer leurs ontologies utilisées sans modifier les noms des entités et leurs descriptions, pour ne pas être obligés de modifier tout un système (ou une application) alimenté avec ces ontologies. Les liens de pont ajoutés dans l'ontologie de pont vont permettre l'interopérabilité sémantique entre elles.
En général, le type d'intégration choisi dépend des circonstances à faire face et des buts des applications. Tenons l'exemple des linked data dans le Web distribué et réparti, l'intégration dans ce contexte consiste à ajouter à chaque ontologie des liens d'équivalence et d'identité (sameAs) qui pointent vers des entités d'autres ontologies. C'est l'un des niveaux les plus bas d'interopérabilité, mais qui convient le mieux à cette situation.
Introduction
Dans ce chapitre, nous allons présenter notre environnement de travail, les bases de test utilisées dans les expérimentations, et les critères d'évaluation de la qualité de l'ontologie produite suite à l'intégration. Par la suite, nous allons étaler et évaluer les résultats de nos expérimentations faites sur des ontologies de différentes tailles. Ce volet d'expérimentations va nous permettre de valider notre méthode d'intégration qui s'est avérée efficace, et de prouver sa capacité de produire une ontologie de bonne qualité.
Environnement de réalisation
L'environnement de développement de notre méthode est constitué des outils suivants :
• Java : un langage de programmation orienté objet.
• Eclipse : un environnement de développement intégré (IDE) Java, libre, gratuit et multiplateforme.
• OWL API 1 (Version 4.1.4) : une interface de programmation pour le développement, la manipulation, et la sérialisation des ontologies OWL.
• HermiT 2 (Version 1.3.8) : un moteur d'inférence OWL 2 DL (Description Logic) créé à l'université d'Oxford et publié sous la licence LGPL. Il est supporté par OWL API et Protégé (qui est un outil de création et de gestion d'ontologies). Il permet de réaliser les services de raisonnement suivants : l'inférence, la classification, la satisfiabilité, et la consistance. Mais il ne fournit pas un diagnostic ou une solution pour les deux derniers problèmes.
• Alignment API 3 (Version 4.9) : une interface de programmation développée en java permettant d'exprimer, d'accéder et de manipuler des alignements ontologiques sous le format d'alignement (qui est le format le plus utilisé pour représenter les alignements). Les
CRITÈRES D'ÉVALUATION
fondateurs de cette API ont conçu le format d'alignement pour exprimer les alignements disponibles de manière uniforme et pouvoir les partager sur le Web. Ce format est écrit en langage RDF, ainsi il est librement extensible. Dans sa représentation, chaque correspondance entre deux ontologies (nommée "cellule") contient l'URI de l'entité source, l'URI de l'entité cible, la relation qui existe entre ces deux entités (égalité, subsumption, exclusion, ou instanciation etc.), et la force de cette relation (une valeur décimale comprise entre 0 et 1, inclusivement).
Les tests ont été effectués sur un PC doté d'un système d'exploitation Windows 10, d'une mémoire centrale de 4 Go, et d'une horloge possédant une fréquence de 2 GHz.
Critères d'évaluation
Selon Flouris et al. (2006), le problème de l'évaluation des techniques d'intégration ou de fusion des ontologies est encore ouvert. Une comparaison générale et objective est difficile, car nous ne savons pas comment l'évaluation de ces outils pourrait être mesurée.
En effet, déterminer la qualité d'un résultat d'intégration ou de fusion nécessiterait de le comparer avec un résultat parfait ou presque parfait. Mais ce résultat est impossible à obtenir manuellement pour les grandes ontologies, et même inexistant car il pourrait y avoir plus qu'un résultat idéal. Il n'y a pas de benchmark qui pourrait être utilisé pour évaluer la qualité de l'approche proposée, e.g., en utilisant des mesures de qualité telles que la précision, le rappel ou la F-Mesure Raunich and Rahm (2012).
Un benchmark de la fusion ou de l'intégration des ontologies devrait être en mesure d'évaluer équitablement la qualité des différents outils. Pour ce faire, Raunich and Rahm (2012) ont défini des métriques de qualité telles que :
• La qualité de l'ontologie de sortie : Elle se reflète par la quantité de ses chevauchements sémantiques qui peuvent être palliés en évitant l'introduction de chemins supplémentaires (relations redondantes). La qualité de l'ontologie dépend fortement de la qualité des ontologies d'entrée ; Idéalement, les ontologies d'entrée sont correctes et ne représentent pas (ou très peu) de conflits et d'incohérences ; idéalement, les alignements d'entrée sont aussi corrects bien que ce n'est pas évident d'en obtenir pour les grandes ontologies.
• La couverture (Préservation de l'information) : C'est une exigence clé, afin que toutes les informations représentées dans les ontologies d'entrée soient conservées dans l'ontologie résultante : * Pour le "Full Merge" où chaque paire d'entités équivalentes devient une entité fusionnée, la taille de l'ontologie résultante doit être égale à la somme du nombre d'entités des deux ontologies d'entrée, moins le nombre d'entités fusionnées, ou plutôt moins le nombre de correspondances d'équivalence (=) dans l'alignement d'entrée. § En dépit que ce soit considéré comme une perte d'information, le fait de ne pas couvrir toutes les entités d'entrée peut être un choix volontaire pour éviter les conflits qui sont dus à l'héritage multiple dans l'ontologie de sortie (ce qui est appelé "fusion asymétrique" par Raunich and Rahm (2012)). * Pour l'ontologie de pont (notre cas), la taille de l'ontologie résultante doit être égale à la somme du nombre d'entités des deux ontologies d'entrée.
• L'efficacité : C'est le temps d'exécution de l'algorithme d'intégration ou de fusion.
• L'effort manuel : C'est l'intervention de l'utilisateur ou de l'expert nécessaire pour le bon déroulement du processus. La base de test "Conference" est composée essentiellement de sept petites ontologies (cmt, conference, confOf, edas, ekaw, iasted, et sigkdd) décrivant le contexte de l'organisation des conférences. OAEI fournit un alignement de référence entre chaque paire de ces sept ontologies, pour avoir en tout, 21 alignements de référence.
Présentation des ontologies utilisées
La base de test "Anatomy" est composée de deux ontologies de taille moyenne : "mouse" qui décrit l'anatomie de la souris adulte, et "human" (une partie de NCIT) qui décrit l'anatomie humaine. OAEI fournit un alignement de référence entre elles.
La base de test "Large Biomedical Ontologies" est composée de trois ontologies volumineuses et sémantiquement riches : FMA (Foundational Model of Anatomy), SNOMED CT (Clinical Terms), et NCI (National Cancer Institute Thesaurus) contenant des dizaines de milliers de classes. La base Large Bio se subdivise en trois catégories de taille croissante (qu'on nommera 1, 2, et 3) dont la troisième est la complète. Pour FMA, il y a FMA1 (5%), FMA2 (13%), et FMA3 (100%). Pour NCI, il y a NCI1 (10%), NCI2 (36%), and NCI3 (100%). Et pour SNOMED, il y a SNOMED1 (5%), SNOMED2 (17%), et SNOMED3 (40%). OAEI fournit trois alignements de référence pour la troisième catégorie, i.e. entre chaque paire des trois ontologies complètes. Les relations "?" contenues dans les correspondances des alignements de référence veulent dire des relations d'équivalence "=" correctes mais qui génèrent des insatisfiabilités dans l'ontologie intégrée. Ces correspondances qui causent l'incohérence de l'alignement sont détectées par le système de débugage ALCOMO et/ou les systèmes de réparation de Logmap et/ou AML.
Nous n'avons pas testé notre Framework sur d'autres ontologies de plus grandes tailles, car il n'existe pas d'alignements disponibles publiés sur Internet pour de telles ontologies. Voici ce que donne l'exécution de tout le code (avec ses deux parties) qui réalise une intégration (i.e., une ontologie de pont avec -les "bridging" axiomes-) :
PRÉSENTATION DES ONTOLOGIES UTILISÉES
Pour la référence : les axiomes sont exactement identiques aux originaux, mais des axiomes d'équivalence (de pont) s'y ajoutent en plus (Figure 4.3). Pour OIA2R : les axiomes sont décrits exactement comme les originaux, mais des axiomes d'équivalence (de pont) s'y ajoutent, tout en personnalisant les IRIs de toutes les entités mentionnées (Figure 4.4). 4.5 Notions à clarifier 4.5.1 Insatisfiabilité C'est un terme dédié aux entités. Une classe insatisfiable est une classe ayant une description fausse (contradictoire), ce qui signifie qu'il n'est pas possible pour une instance de répondre à toutes les exigences requises pour être membre de cette classe. Elle ne peut et ne doit jamais avoir d'instances (tout à fait comme la classe "owl:Nothing"), car il n'existera aucune instance qui pourra la satisfaire Sattler et al. (2013).
Inconsistance
C'est un terme dédié aux ontologies. Une ontologie est consistante s'il lui existe une interprétation satisfaisante, e.g. une ontologie à partir de laquelle nous pouvons déduire que l'individu x est différent de l'individu y et qu'il est en même temps identique à lui, ne peut pas avoir une interprétation satisfaisante. L'inconsistance peut se manifester lorsqu'il y a au moins une violation des restrictions d'une classe, une instanciation d'une classe insatisfiable, une instanciation de deux classes disjointes, ou une contradiction sémantique entre les individus etc. Bail (2013).
Dans une ontologie inconsistante, toutes les classes sont insatisfiables, i.e. aucune de ses classes ne peut avoir d'individu. En effet, elle n'a pas de modèle. Elle est considérée comme une ontologie sévèrement endommagée contenant une grave erreur qui doit être réparée car aucune connaissance utile ne peut en être inférée. Elle ne peut pas être publiée et utilisée dans les applications.
Raisonneur
Un raisonneur est un composant clé dans le domaine des ontologies. Puisque la connaissance dans une ontologie OWL peut ne pas être explicite, la classification et l'interrogation d'une ontologie (qui sont les deux tâches basiques d'un raisonneur) doivent être faites par un raisonneur, pour pouvoir déduire les connaissances implicites et obtenir des résultats d'interrogation corrects. Les raisonneurs existants détectent l'inconsistance et l'incohérence, mais ne leur fournissent pas un diagnostic et une solution Sattler et al. (2013).
Voici les travaux qui ont été menés en collaboration avec notre laboratoire LIPAH concernant l'évaluation des performances des raisonneurs existants : Alaya et al. (2015dAlaya et al. ( ,a,c,b, 2016Alaya et al. ( , 2017.
Classification
Selon Sattler et al. (2013), un raisonneur détermine toutes les inférences de la forme "A subClassOf B" d'une ontologie donnée, i.e. il détermine sa hiérarchie, en appliquant les tests suivants :
-Si A = "owl: Table 4.11 -Qualité de l'ontologie résultant de l'intégration des ontologies de Anatomy
Anatomy
Nombre de classes insatisfiables Al originaux (1-à-N) Al filtrés (1-à-1) OIA2R
Réf OIA2R Réf Intégration 2-à-2 0 0 0 0 4.6. RÉSULTATS ET ÉVALUATION (366 467 + 20 881) § Ces axiomes d'équivalence vont aussi affecter la hiérarchie (la classification) des classes et des propriétés de l'ontologie résultante. En effet, le raisonneur pourra ne pas pouvoir s'arrêter (dans son calcul) et ainsi ne pas avoir un nombre précis de niveaux dans la hiérarchie.
Ci-dessous un exemple qui montre comment se forme l'insatisfiabilité d'une classe dans une ontologie de pont. La figure 4.5 montre les justifications que nous a affichées le debugueur de OWL API à l'aide du raisonneur HermiT pour une des classes insatisfiables (Cytoplasmic_Matrix) de l'ontologie résultant d'une intégration des ontologies FMA 1 (Ont1) et NCI 1 (Ont2). Nous remarquons qu'après l'ajout des "bridging" axiomes d'équivalence, la classe "002#Cy-toplasmic_Matrix" (provenant de l'ontologie NCI 1 (Ont2)) devient par inférence sous-classe des deux classes "001#Portion_of_body_structure" et "001#Anatomical_structure" qui sont disjointes (information extraite de l'ontologie FMA 1 (Ont1)). Ceci est contradictoire, car une classe ne peut pas être sous-classe de deux classes disjointes. La même chose s'applique pour l'autre classe coloriée en rouge.
La référence ne manque aucun axiome des ontologies d'entrée, alors que la nôtre a des pertes d'informations telles que les entités anonymes et les restrictions. C'est pour cette raison que le raisonneur infère plus d'informations et ainsi détecte plus de classes insatisfiables dans l'ontologie de référence. D'ailleurs, pour calculer toutes ses classes insatisfiables, le raisonnement HermiT prend beaucoup plus de temps pour terminer son calcul.
L'ontologie d'une intégration N-à-N produit beaucoup plus de classes insatisfiables que l'ontologie d'une intégration 2-à-2 ou 1-à-N à cause des relations redondantes. Plus il y a de relations redondantes (dans ce cas, des relations d'équivalence redondantes entre les entités sources), plus il y a d'incohérence. Les correspondances redondantes peuvent être déduites automatiquement par un raisonneur, ainsi elles sont inutiles, et surtout source de conflits sémantiques.
Il est important également de noter que la réparation de l'alignement est dédiée à l'intégration de deux ontologies seulement. Autrement dit, si nous intégrons deux ontologies en utilisant un alignement réparé entre elles, nous obtiendrons une ontologie consistante et cohérente sans aucune classe insatisfiable. Cependant, si nous intégrons plusieurs ontologies à l'aide de plusieurs alignements réparés (i.e., entre les paires d'ontologies), nous obtiendrons une ontologie comportant plusieurs classes insatisfiables. En effet, les systèmes actuels de réparation des alignement prennent en entrée deux ontologies (à intégrer ultérieurement) et un alignement entre elles. Ils ne sont pas en mesure de prendre en compte plusieurs ontologies et alignements entre elles pour but de les intégrer simultanément.
Atouts de OIA2R
Notre framework est automatique et générique. Il prend en entrée toute ontologie et tout alignement avec lesquels il produira une nouvelle ontologie qui les englobe tous. Ce processus est rapide même pour les plus grandes ontologies et les plus grands alignements. L'ontologie résultante est assez complète et cohérente.
Nous donnons la possibilité de convertir les alignements sources en des mappings, et également de les réparer à l'aide d'outils externes, afin de minimiser les erreurs dans l'ontologie de sortie.
Notre approche permet un refactoring (une personnalisation) de toutes les entités des ontologies et des alignements sources pour former une ontologie propre à nous (dont les entités ne pointent pas sur les URIs externes des ontologies sources déjà publiées). En effet, l'utilisateur n'a qu'à entrer l'URI qu'il désire pour la future ontologie, et par la suite toutes les entités l'auront comme URI de préfixe.
Temps d'exécution
Ce sont les temps d'exécution moyens d'une intégration complète N-à-N avec des alignements transformés en mappings, mais qui sont déjà réparés à l'avance. Nous voulons dire par "loading", le temps de chargement des ontologies dans le manager de OWL API, car pour les grandes ontologies, leur loading prend une bonne part du temps d'exécution (comme le montrera le tableau 4.15), et ce temps-là ne fait pas partie du temps effectif de notre intégration. Le temps "avec loading" est le temps d'exécution de tout le programme (du début jusqu'à la fin), et le temps "sans loading" est le temps d'exécution exact de notre framework. Rappelons que notre framework prend en entrée des alignements pré-établis, ainsi, les temps de d'exécution fournis ne comprennent pas le temps du matching. Les temps d'exécution CPU ne dépassent pas 0,7 min pour les plus grandes ontologies. Le temps global de l'intégration de référence est plus long, car la référence prend plus de temps pour sauvegarder tous les axiomes d'entrée (à la fin du programme) et créer l'ontologie de pont complète. Cependant, le temps global de l'intégration de OIA2R prend moins de temps pour sauvegarder les axiomes d'entrée, car OIA2R manque les axiomes complexes. Le temps effectif de l'intégration de référence prend moins de temps, car la référence ne perd pas du temps à parser, refactoriser les axiomes des ontologies d'entrée et créer de nouveaux axiomes refactorisés comme le fait OIA2R, elle assemble directement les axiomes des ontologies d'entrée et ajoute les axiomes de pont.
Pour découvrir le niveau d'intégration qui génère plus d'incohérences dans son ontologie résultante (soit l'ontologie de pont, soit la fusion), nous avons réalisé un autre travail qui fait une fusion de deux ontologies (uniquement deux), i.e. il fusionne les paires d'entités équivalentes en une seule entité. (2012)) que la fusion complète des ontologies génère toujours moins de conflits qu'une ontologie de pont (appelée "fusion directe" selon eux).
Ci-dessous un exemple qui montre comment se forme l'insatisfiabilité d'une classe dans une ontologie résultante d'un processus de fusion complète. Dans la figure 4.8, nous schématisons les justifications que nous a affichées le debugueur de OWL API (à l'aide du raisonneur HermiT) pour une des classes insatisfiables générées suite à la fusion de FMA 1 (Ont1) et SNOMED 1 (Ont2). Nous avons choisi le ID "/000" pour les entités issues de la fusion de deux entités équivalentes. Et nous avons choisi de leur donner comme noms, une concaténation des noms des paires d'entités fusionnées (juste pour pouvoir voir clairement dans l'ontologie résultante ce qui a été fusionné). Dans l'état de l'art, les auteurs choisissent généralement l'un des deux noms des entités fusionnées (peut-être après avoir défini une ontologie prioritaire), ou bien créent un code unique tout en ajoutant les deux noms originaux comme labels.
Dans la figure 4.9, nous modélisons la représentation graphique des axiomes de justification ci-dessus, où la classe en rouge est la classe insatisfiable. Nous remarquons qu'après la fusion des classes "001#Extracellular_space" provenant de FMA 1 (Ont1) et "002#Intercellular_space" provenant de SNOMED 1 (Ont2), la classe "000#Extracellular_space=Intercellular_space" devient par inférence sous-classe des deux classes "001#Immaterial_anatomical_entity" et "001#Material_anatomical_entity" qui sont disjointes (information extraite de l'ontologie FMA 1 (001)). Ceci est contradictoire, car une classe ne peut pas être sous-classe de deux classes disjointes.
Conclusion
Enfin, nous déduisons que si les ontologies modélisent des vues différentes et incompatibles du même domaine, il est impossible de les intégrer aveuglément et d'assurer à la fois la complétude et la cohérence dans l'ontologie résultante. Dans un contexte d'intégration d'ontologies, assurer la cohérence et la consistance est une priorité car l'ontologie résultante doit être logiquement correcte pour être réellement utile. Dans ce cas, nous ne pourrons jamais réaliser une interopérabilité sémantique complète entre les ontologies d'entrée car nous serons dans l'obligation d'abandonner des correspondances sémantiques. Et au cas où nous souhaiterions atteindre la complétude, les incompatibilités des ontologies sont insolvables automatiquement et l'intervention d'un expert devient nécessaire, ce qui est impossible pour les grandes ontologies. Dans l'ensemble, l'intégration des ontologies ayant des vues incompatibles reste toujours un problème ouvert.
Dans ce chapitre, nous avons présenté les expérimentations sur notre nouvelle approche d'intégration des ontologies OIA2R. L'analyse des résultats a montré les performances de notre approche et la validité de l'ontologie résultante. En effet, OIA2R a produit des résultats encourageants dans des temps minimes. De même, les expérimentations ont montré une possibilité d'amélioration de ces résultats pour avoir une qualité optimale de l'ontologie de sortie.
Conclusion générale
Les services Web et les moteurs de recherche peuvent améliorer leurs performances dans l'échange des informations et la précision des résultats de recherche en exploitant la représentation sémantiquement enrichie des informations qu'ils partagent à travers les ontologies. Actuellement, la recherche et le développement dans le domaine du Web sémantique (qui est un Web distribué et ouvert) ont atteint un stade où un grand nombre d'applications et de services tels que le commerce électronique, le renseignement gouvernemental, la médecine, la fabrication, etc, sont alimentés par des ontologies de toute taille, développées par différentes personnes, groupes de recherche, ou organisations, et contenant beaucoup de chevauchements (similarités) sémantiques entre elles. En effet, les différentes sources de données modélisent leurs ontologies de différentes manières selon leurs propres besoins, exigences, et buts, et n'utilisent pas nécessairement des ontologies déjà existantes. Par conséquent, il devient difficile de récupérer les informations provenant de différentes sources dans le Web.
Pour une représentation efficace et homogène des domaines de connaissances, il serait alors nécessaire d'intégrer (ou de fusionner) toutes les ontologies pour former de nouvelles ontologies plus complètes et mieux modélisées qui les remplacera.
Dans le premier chapitre, nous avons passé en revue les principales définitions des notions essentielles pour cerner le champ d'étude. Une étude bibliographique sur le Web Sémantique, l'ontologie, et l'ingénierie ontologique a été menée.
Nous avons établi au deuxième chapitre un bilan des définitions et des méthodes existantes qui s'appliquent dans le cadre de l'intégration des ontologies. Notamment, parmi ces méthodes ou définitions, il y a celles qui sont restées uniquement théoriques.
Dans le troisième chapitre, nous avons mis au point une nouvelle méthode qui permet d'intégrer deux ou plusieurs ontologies à l'aide des alignements entre elles, pour générer une nouvelle ontologie qui les englobe.
Dans le quatrième chapitre, nous avons décrit notre environnement de travail et les critères d'évaluation des outils d'intégration, nous avons appliqué et évalué notre approche dans la pratique, et nous avons prouvé qu'elle est générique, efficace et scalable.
Figure 1 . 1 -
11Fragment d'une ontologieCheatham and Pesquita (2017)
Formellement
, le processus de matching peut être vu comme une fonction f qui, à partir d'une paire d'ontologies O et O à mettre en correspondance, un ensemble de paramètres p, et un ensemble de ressources externes r, retourne en sortie un alignement A (éventuellement un mapping) entre ces deux ontologies : A = f (O, O , p, r).
Figure 1 . 2 -
12Matching De Bruijn et al. (2006)
Figure 1 . 3 -
13Processus général du matchingCheatham and Pesquita (2017)
Figure 1 . 4
14Noy and Musen (2000) pensent que dans l'alignement, les ontologies sources(généralement de domaines complémentaires) doivent être toujours séparées et consistantes les unes avec les autres, tout en ayant des liens entre elles. Les auteurs Zhu et al. (2009) le pensent aussi et définissent l'alignement par le processus qui combine deux ontologies et qui établit ensuite une collection de relations binaires (correspondances) entre elles.(a)Noy and Musen (2000) (b)Zhu et al. (2009)
Formellement
, étant donné deux ontologies O et O (ayant les langages L et L ) et un ensemble de relations d'un alignement A, une correspondance est un triplé (e, e , r), tel que e ∈ O, e ∈ O , et r ∈ Θ. La correspondance (e, e , r) déclare que la relation bidirectionnelle r relie les entités e et e ; mais elle est souvent accompagnée aussi par un identifiant et une confiance, ainsi, elle sera représentée généralement par un tuple (id, e, e , r, n) où id est son identifiant unique, et n est sa mesure de confiance
Figure 1 . 5 -
15Alignement Abels et al. (2005)
Figure 1
1Choi et al. (2006), le mapping des ontologies est utilisé principalement dans trois situations :
Formellement, la transformation des ontologies à l'aide d'un alignement (mapping) A entre deux ontologies O et O , génère une ontologie O qui transforme les entités de O par celles de O suivant les correspondances dans A. Elle peut être exprimée par l'opérateur suivant : T ransf orm(O, A) = O . Les opérations de transformation sont orientées, i.e. la transformation a une source et une cible identifiées, ainsi, à partir d'un alignement, il est possible de générer deux opérations (dans les deux sens) selon la source et la cible.
Figure 2 . 1 -
21Fusion dePinto (1999) interprétée parKeet (2004) Figure 2.2 -Fusion deAbels et al. (2005) Fusion comme étant une ontologie intermédiaire SelonKalfoglou and Schorlemmer (2003), la "forte" notion de fusion peut être détendue en prenant l'articulation (l'alignement des deux ontologies O et O ) avec laquelle une ontologie O pourrait être définie.Fusion de De Bruijn et al.D'après De Bruijn et al. (2006), la fusion des ontologies est la création d'une nouvelle ontologie qui unie deux ou plusieurs ontologies en se basant sur les correspondances entre elles. Selon eux, la nouvelle ontologie doit capturer toutes les connaissances des ontologies sources et refléter toutes les correspondances entre elles pour pouvoir les remplacer. Nous notons qu'ils n'évoquent pas les domaines des ontologies sources (différents ou similaires). Ils distinguent deux approches distinctes dans la fusion des ontologies :• Fusion complète (Full Merge) : Chaque paire d'entités équivalentes est fusionnée en une seule entité. • Ontologie de pont : Nous allons l'expliquer tout de suite.
Figure 2 . 3 -
23Fusion complète De Bruijn et al. (2006) Fusion dans la symétrie et l'asymétrie Selon Raunich and Rahm
et Zhang et al. (2017) utilisent les termes "fusion" et "intégration" comme des synonymes.
Figure 2 . 4 -
24Ontologie de pont De Bruijn et al. (2006) 2.1.2 Principaux outils de fusion et leurs limites Outils célèbres Les approches les plus connues de fusion des ontologies telles que PROMPT* Noy and Musen (2000), Chimaera McGuinness et al. (2000), et FCA-Merge Stumme and Maedche (2001) sont des Full Merge, semi-automatiques (nécessitant l'intervention d'experts et introduisant un effort manuel important, surtout pour les grandes ontologies) qui ne se basent pas sur les mappings, i.e. elles n'appliquent pas une séparation entre le matching et la fusion Raunich and Rahm (2012).
L
'outil de Chatterjee et al. Dans leur expérimentation, Chatterjee et al. (2017) ont choisi de créer une nouvelle ontologie dans le domaine de l'agriculture, en fusionnant des ontologies de sous-domaines (de la récolte, les engrais, la terre (le sol), et la météo). § Ce travail est en réalité une intégration (non pas une fusion), car les ontologies d'entrée appartiennent à différents domaines, et l'ontologie résultante est de domaine (interdisciplinaire) plus large qui englobe ces sous-domaines (c'est une composition de sous-domaines). Ils font le parsing des fichiers .owl des ontologies d'entrée et extraient leur ensemble d'entités (en utilisant la bibliothèque "Owlready" en Python), puis ils font le matching de chaque couple d'ontologies (en combinant plusieurs techniques de matching (i.e. de niveau élémentaire et structurel). A l'aide des alignements générés par le matching, ils appliquent une fusion complète des entités similaires et les mettent dans l'ontologie résultante O M , puis ils copient les entités restantes (non fusionnées) des ontologies sources dans O M , et génèrent un fichier .owl correspondant à O M .
et al. (2017), le processus de fusion des ontologies avec la méthode OIM-SM prend deux ontologies et retourne une nouvelle ontologie (sous forme d'arbre, i.e. sans héritage multiple). Il se compose des étapes suivantes : 1. Le matching d'équivalence sémantique entre les concepts des deux ontologies. 2. La fusion de toutes les paires de concepts équivalents, pour produire un nouveau concept à la place de chaque paire. Concernant les instances et les propriétés de chaque couple de concepts (A et B), ils ont proposé d'appliquer deux règles de fusion pour former le nouveau concept C : * L'ensemble des instances de C est l'union de l'ensemble des instances de A et de B. * L'ensemble des propriétés de C est l'intersection de l'ensemble des propriétés de A et de B.
Figure 2 . 5 -
25Correspondances entre deux ontologies Zhang et al. (2017)Figure 2.6 -Fusion initiale des fragments d'ontologies Zhang et al. (2017) Dans les expérimentations, ils ont fusionné l'ontologie BCO (Biological Collections Ontology) qui contient 127 concepts, avec l'ontologie ACO (Animal in Context Ontology) qui contient 510 concepts, dans 9 minutes ; et ils ont utilisé comme référence une fusion retournée artificiellement.
Figure 2 . 7 -
27Résultat final de la fusionde Zhang et al. (2017)
Figure 2 . 8 -
28Intégration des ontologiesMena et al. (1996) Intégrationde Sowa Selon Sowa (1997), l'intégration est "le processus de recherche de points communs entre deux ontologies A et B et de dérivation d'une nouvelle ontologie C facilitant l'interopérabilité entre les systèmes informatiques basés sur les ontologies A et B. La nouvelle ontologie C peut remplacer A ou B, ou peut être utilisée comme intermédiaire entre un système basé sur A et un autre basé sur B". Il n'a pas spécifié les domaines des ontologies à intégrer.
Intersection des ontologies où une nouvelle ontologie O N fusionne et standardise les termes des entités en correspondance de O 1 et O 2 , tout en les renommant dans O 1 et O 2 (qui sont des nouvelles versions de O 1 et O 2 ) par les termes fusionnés et standardisés. (C'est la transformation des termes des entités correspondues en des termes communs)
Figure 2 . 9 -
29Intégration deHeflin and Hendler (2000)
Figure 2 .
210 -Intégration de
et Zhang et al. (2017) utilisent les termes "fusion" et "intégration" comme des synonymes.Intégration comme étant une ontologie de pontSelonUdrea et al. (2007), l'intégration des ontologies est l'ajout des axiomes de l'alignement A (entre O1 et O2) à l'union de O1 et O2 produisant à la fin une ontologie consistante et cohérente. Les correspondances de A sont utilisées pour créer des liens logiques (des axiomes) qui représentent la sémantique des relations entre les différentes entités (l'équivalence, la subsumption, la disjonction etc.).Selon Euzenat and Shvaiko(2013), l'intégration des ontologies est l'inclusion dans une ontologie O d'une autre ontologie O et des assertions exprimant des liens entre ces deux ontologies (des axiomes de pont). L'ontologie résultante O est censée contenir la connaissance des deux ontologies initiales (O et O ). Il n'y a pas vraiment de différence entre leurs définitions de fusion et d'intégration, à part le fait que, selon eux, contrairement à la fusion qui ne modifie pas les ontologies d'entrée, dans l'intégration, l'ontologie source O est inchangée tandis que l'ontologie initiale O est modifiée (plutôt augmentée par O ). En d'autres termes, l'intégration, selon eux, se fait d'une manière incrémentale, alors que la fusion se fait d'une manière non incrémentale. § C'est l'approche que nous allons implémenter.
(Nous expliquerons plus les notions d'inconsistance et d'incohérence dans le chapitre 4).
L
'outil ContentMap (le plus proche de notre travail) Pour Jiménez-Ruiz et al. (2009), un ensemble de correspondances (d'un Mapping) est représenté par une ontologie M , où les correspondances sont des axiomes de la forme subClassOf (e, e ), equivalentClass(e, e ), et disjointW ith(e, e ) pour la relation de subsomption, d'équivalence, et de disjonction respectivement ; et les identifiants et les valeurs de confiance des correspondances sont des annotations d'axiomes qui n'ont aucun effet sur les inférences.
Dans leurs expérimentations, ils ont utilisé quatre petites ontologies qui décrivent toutes le domaine des références bibliographiques, mais qui sont développées séparément. Leur taille varie de 130 entités (58 classes, 46 propriétés d'objet, et 26 propriétés de données) à 49 entités (18 classes, 12 propriétés d'objet, et 19 propriétés de données) et de 235 axiomes à 96 axiomes. O-INR est l'ontologie avec laquelle les trois autres ontologies (O-MIT, O-UMBC, et O-AIFB) ont été intégrées chacune à part. Il existe trois alignements de référence : un alignement pour O-MIT et O-INR contenant 119 correspondances, un alignement pour O-UMBC et O-INR contenant 83 correspondances, et un alignement pour O-AIFB et O-INR contenant 98 correspondances. Ils ont intégré O-MIT, O-UMBC, et O-AIFB séparément avec O-INR en utilisant leurs alignements de référence correspondants, puis ils ont évalué les conséquences sémantiques de leurs ontologies résultantes. Dans tous les cas, ils ont trouvé un nombre signifiant d'inférences imprévues. Par exemple, lors de l'intégration de O-AIFB et O-INR, ContentMap a détecté 34 concepts insatisfiables (originaires des deux ontologies) pour lesquels il y a eu un nombre énorme de justifications (généralement complexes) qui ont rendu la réparation manuelle extrêmement difficile. Egalement, lors de l'intégration de O-MIT et O-INR, ContentMap a détecté des nouvelles subsomptions qui ont été identifiées et réparées automatiquement. D'un point de vue temps, le goulot d'étranglement est le calcul de toutes les justifications des inférences imprévues. Une fois les justifications calculées, le temps de réparation est, selon eux, relativement court (ils ne l'ont pas précisé). Ils ont remarqué aussi que l'utilisation des correspondances générées automatiquement a abouti à un plus grand nombre d'inférences imprévues, e.g. quand ils ont intégré O-AIFB et O-INR en utilisant l'outil de matching CIDER avec un seuil de confiance égale à 0.1, ils ont trouvé 55 concepts insatisfiables et 34 subsomptions imprévues qui sont des erreurs causées principalement par des correspondances incorrectes.Pour conclure, les auteurs ont fait une fusion (non pas une intégration), précisément une ontologie de pont, semi-automatique à de petites ontologies de même domaine, et malgré la réparation des alignements d'entrée, ils trouvent toujours énormément de classes insatisfiables dans l'ontologie de sortie. Ils n'évoquent pas le temps d'exécution de ces expérimentations. L'outil de Ziemba et al.L'algorithme d'intégration des ontologiesde Ziemba et al. (2015) utilise essentiellement les outils de refactoring et d'intégration de l'éditeur "Protégé" qui facilitent énormément leur processus d'intégration. Il est divisé en trois parties :1. L'intégration de la première ontologie :(a) Créer une nouvelle ontologie vide dans l'éditeur Protégé (Ils ont choisi de lui donner l'IRI (l'identificateur) : "Intégrée") ;
2 .
2La sélection, l'importation et le refactoring d'une nouvelle ontologie source dans l'ontologie cible, et faire l'alignement entre elles, en utilisant les dictionnaires, les thésaurus, et les outils de "Protégé", pour l'introduire ensuite sous forme de relations d'équivalence et de subsomption entre les entités des ontologies source et cible.
Figure 2 .
211 -Incohérence de la fusion de O1 etO2 Fahad et al. (2010) L'exemple de la figure 2.12 illustre une incohérence logique causée par deux correspondances entre l'ontologie "National Cancer Institute Thesaurus" (NCIT) et l'ontologie "Foundational Model of Anatomy" (FMA). Cela se produit car, lors de l'intégration, la classe Fibrillar_Actin devient (suite à l'équivalence) une sous-classe de Anatomic_Structure_System_or_Substance et de Gene_Product, qui sont deux classes disjointes. § Résoudre ces incohérences est loin d'être facile.
Figure 2 .
212 -Incohérence de l'ontologie de pont Cheatham and Pesquita (2017) Udrea et al. (2007) exigent l'exploitation de la sémantique des ontologies pendant la génération des correspondances entre elles (pendant le matching), pour parvenir à créer une ontologie consistante et cohérente suite au processus d'intégration (ou de fusion). Et Fahad et al. (2010) exigent de prêter une attention particulière aux conflits sémantiques générés à cause des relations disjointes dans les ontologies sources.
appelé notre algorithme d'intégration "OIA2R" (Ontology Integration : Alignment Reuse and Refactoring). Ce que notre algorithme génère est une ontologie de pont, i.e., l'union des ontologies d'entrée et des alignements entre eux. Puisque nous allons convertir les correspondances contenues dans les alignements d'entrée en des axiomes OWL, ces alignements sont considérés comme des ontologies OWL intermédiaires (constituées d'entités ayant des relations d'équivalence). Ainsi, implicitement, c'est une union des ontologies sources et des ontologies intermédiaires (l'articulation). Dans le cas de deux ontologies O 1 et O 2 , ayant un alignement A qui peut être vu comme une ontologie O A , le résultat sera une nouvelle ontologie O 3 de sorte que O 3 = O 1 + O 2 + A, ou plutôt O 3 = O 1 + O 2 + O A . Le schéma de la figure 3.3 illustre nos dires.
Figure 3 . 1 -
31Processus général de l'intégration des ontologies
Figure 3 . 2 -
32Processus général de notre méthode d'intégration des ontologies (OIA2R)
Figure 3 . 3 -
33Ontologie de pont (Ont3)
le parsing des classes (des ontologies d'entrée) et de leurs définitions (descriptions), et au fur et à mesure, nous créons les axiomes correspondants à ces classes et à leurs définitions dans notre nouvelle ontologie. Nous faisons le parsing des propriétés d'objet (des ontologies d'entrée) et de leurs définitions, et au fur et à mesure, nous créons les axiomes correspondants à ces propriétés d'objet et à leurs définitions dans notre nouvelle ontologie. Nous faisons le parsing des propriétés de données (des ontologies d'entrée) et de leurs définitions, et au fur et à mesure, nous créons les axiomes correspondants à ces propriétés de données et à leurs définitions dans notre nouvelle ontologie. Et, nous faisons le parsing des individus anonymes (des ontologies d'entrée) et de leurs définitions, et au fur et à mesure, nous créons les axiomes correspondants à ces individus et à leurs définitions dans notre nouvelle ontologie.
Nous parsons les classes des ontologies d'entrée, et nous remplissons au fur et à mesure le HSet des classes par tous les URIs originaux des classes. Nous parsons les propriétés d'objet des ontologies d'entrée, et nous remplissons au fur et à mesure le HSet des propriétés d'objet par tous les URIs originaux des propriétés d'objet. Nous parsons les propriétés de données des ontologies d'entrée, et nous remplissons au fur et à mesure le HSet des propriétés de données par tous les URIs originaux des propriétés de données. Nous parsons les individus des ontologies d'entrée, et nous remplissons au fur et à mesure le HSet des individus par tous les URIs originaux des individus.
Figure 3 . 5 -
35Debugage d'une classe insatisfiable dans une ontologie de pont
Figure 3
3Figure 3.6 -Formation des classes insatisfiables
Figure 3 . 7 -
37Deux correspondances ayant la même entité source
Figure 3 . 8 -
38Cause de l'incohérence d'une ontologie de pont nant de l'ontologie Ont3) avait été correspondue avec une seule classe de l'autre ontologie (002), précisément la classe avec laquelle elle a la plus grande mesure de similarité, nous aurons évité toutes ces incohérences.
Figure 3 . 9 -
39Résolution des insatisfiabilités
Figure 3 . 10 -
310Transformation d'un alignement (1-à-N) en un mapping(1-à-1)
Figure 3 .
311 -Nouvelle définition de l'intégration sémantique
Figure 3 .
312 -Intégration incrémentale(cas 1)
Figure 3 .
313 -Intégration incrémentale (cas 2)Nous proposons trois types d'intégration non incrémentale :• Intégration 2-à-2 : Les ontologies sont intégrées uniquement à l'aide des alignements entre les paires d'ontologies consécutives (c.f.,figure 3.14),• Intégration 1-à-N : Les ontologies sont intégrées à l'aide des alignements entre une ontologie choisie (préférée ou prioritaire) et les autres ontologies (c.f., figure 3.15), • Intégration N-à-N : Les ontologies sont intégrées à l'aide des alignements entre toute paire d'ontologies possible (c.f., figure 3.16). L'intégration N-à-N est la plus complète, par contre les deux autres types d'intégration (1-à-N et 2-à-2) ne le sont pas (i.e., n'assurent pas une interopérabilité complète). Tenons l'exemple de cinq ontologies appartenant au même domaine (O 1 , O 2 , O 3 , O 4 , and O 5 ) pour faire une intégration 2-à-2. Supposons que toutes les ontologies contiennent une classe A, à part l'ontologie O 4 . Si O 4 était dans la quatrième position, alors la classe A de l'ontologie O 5 ne va pas avoir une
Figure 3 Figure 3 .
3315 -Intégration non incrémentale(1-to-N)
Figure 3 .
316 -Intégration non incrémentale (N-to-N) correspondance d'équivalence avec une classe de l'ontologie O 4 . Ce qui fait qu'elle ne sera pas intégrée avec les autres classes A des autres ontologies (ayant la position 1, 2 et 3). L'intégration 1-à-N par contre résout ce problème. Tenons le même exemple en choisissant l'ontologie O 1 comme l'ontologie préférée, avec qui les quatre autres ontologies (O 2 , O 3 , O 4 , and O 5 ) seront alignées. Dans ce cas, la classe A de l'ontologie O 1 va être correspondue à toutes les classes A existantes, y compris celle de l'ontologie O 5 . Malheureusement, cette méthode ne va pas garantir une interopérabilité complète. Par exemple, si une classe B existe seulement dans l'ontologie O 3 et O 4 , ces deux classes ne vont pas être intégrées (parce que ce type d'intégration n'utilise pas l'alignement entre O 3 and O 4 ). Ces inconvénients sont tous palliés par l'intégration N-à-N. Mais dans le cas d'une ontologie de pont, cette dernière va générer de multiple redondances et cycles. Par exemple, parmi les cinq ontologies d'entrée, les ontologies O 1 , O 2 , et O 3 ont une entité A en commun. Dans une ontologie de pont, nous aurons des liens d'équivalence entre O1:A et O2:A, O2:A et O3:A, O1:A et O3:A. L'équivalence entre O1:A et O3:A peut être déduite des deux autres équivalences. Il s'agit donc d'un lien redondant. Et sachant qu'une relation d'équivalence est formellement composée de deux relations de subsumption (chacune dans un sens), il y aura deux cycles de subsomption entre ces trois entités (un cycle dans chaque direction).
Figure 3 .
317 -Agrégation des ontologies
Figure 4 . 2 -
42Agrégation de OIA2R
Figure 4 . 3 -
43Intégration de "référence"
Figure 4 . 4 -
44Intégration de OIA2R
Figure 4 . 5 -
45Debugage d'une classe insatisfiable dans une ontologie de pont Dans la figure 4.6, nous modélisons la représentation graphique des axiomes de justification ci-dessus, où les classes en rouge sont les classes insatisfiables.
Figure 4 . 6 -
46Formation des classes insatisfiablesSachant que la relation d'équivalence (des "bridging" axiomes) est en fait égale à deux relations de subsomption dans les deux sens, le schéma devient comme le montre la figure 4.7.
Figure 4 . 7 -
47Cause de l'incohérence d'une ontologie de pont
Figure 4 . 8 -
48Debugage d'une classe insatisfiable après une fusion
Figure 4 . 9 -
49Formation d'une classe insatisfiable après une fusion complète
Table des matières
desTable des figures ivListe des acronymes
ALCOMO
Applying Logical Constraints on Matching Ontologies
AML
AgreementMakerLight
API
Application Programming Interface
AROMA
Association Rule Ontology Matchinh Approach
CIDER
Context and Interface baseD ontology alignER
ContentMap
LogiC-based ONtology inTEgratioN Tool using MAPpings
DTD
Document Type Definition
FITON
Framework for InTegrating ONtologies
F-logic
Frame logic
FMA
Foundational Model of Anatomy
HTML
HyperText Markup Language
ILIADS
Integrated Learning In Alignment of Data and Schema
IRI
Internationalized Resource Identifier
KIF
Knowledge Interchange Format
LGPL
Lesser General Public Licence
LOD
Linked Open Data
LogMap
Logic-based Methods for Ontology Mapping
NCI
National Cancer Institute Thesaurus
OAEI
Ontology Alignment Evaluation Initiative
OIM-SM
Ontology Integration Method based on Semantic Mapping
OLA
OWL Lite Alignment
OWL
Ontology Web Language
RDF
Resource Description Framework
RDFS
RDF-Schema
SWRL
Semantic Web Rule Language
SNOMED CT
SNOMED Clinical Terms
URI
Uniform Resource Identifier
URL
Uniform Resource Locator
WWW
World Wide Web
W3C
WWW Consortium
XML
Extensible Markup Language
YAM++
Yet Another Matcher
Ontology Language) est un langage de représentation de connaissances qui permet d'écrire (de construire) des ontologies Web, et tout comme RDF, il est un langage profitant de l'universalité syntaxique de XML. OWL devient une recommandation du W3C en 2004, et OWL 2 le devient en 2009. A part sa capacité de définir et de décrire des classes, des propriétés, et des individus de classes, OWL permet aussi de définir des relations entre les classes (union, intersection, disjonction, équivalence, subsomption etc.), des contraintes de cardinalité pour les valeurs des propriétés (nombre minimum, maximum, ou exact), des relations spéciales pour les propriétés (transitive, symétrique, fonctionnelle, inverse, réflexive, etc.), et des restrictions sur le domaine et le co-domaine des propriétés, etc. Par conséquent, OWL possède une logique très développée qui permet le raisonnement sémantique sur ces règles. Comparé aux langages RDF et RDFS, OWL offre aux machines une plus grande capacité d'interprétation du contenu Web, grâce à son vocabulaire riche et sa sémantique formelle.Les classes définies par l'utilisateur sont toutes des enfants de la superclasse « owl:Thing » (qui représente l'ensemble de tous les individus) et des parents de la sous-classe « owl:Nothing » (qui représente l'ensemble vide).Les propriétés d'objet et de type de données définies par l'utilisateur sont toutes respectivement des enfants des super propriétés « owl:TopObjectProperty » et « owl:TopDataProperty », et des parents des sous-classes « owl:BottomObjectProperty » et « owl:BottomDataProperty ». Les propriétés d'annotation telles que « owl:versionInfo », « rdfs:label », « rdfs:comment », « rdfs:seeAlso », « owl:priorVersion » etc. sont des constructeurs intégrés dans OWL.
2.1. FUSION DES ONTOLOGIES * Le plugin de PROMPT est dépassé et non fonctionnel maintenant (il n'est plus disponible pour le téléchargement). L'outil de Caldarola et Rinaldi Le Framework de Caldarola and Rinaldi (2016) contient quatre blocs principaux : 1. Le bloc de récupération des ontologies 2. Le bloc de normalisation des ontologies 3. Le bloc de matching des ontologies : Ce bloc est responsable de l'obtention d'un ensemble d'alignements (A) qui sont des correspondances entre les entités des ontologies d'entrée et de l'ontologie cible. Le matcher implique trois types d'opérations de matching (à base de chaînes, sémantique, et linguistique) qui vont également aider à faire une analyse automatique qui évalue et découvre les ontologies d'entrée les plus similaires à l'ontologie cible (i.e. les ontologies pertinentes) parmi lesquels les ingénieurs de connaissances vont sélectionner les ontologies locales à réutiliser en les intégrant dans l'ontologie cible. 4. Le bloc de fusion des ontologies : Il est responsable de l'intégration des ontologies d'entrée sélectionnées dans une ontologie OWL globale qui sera plus riche. Selon les mesures des correspondances contenues dans l'ensemble des alignements (A), les opérations suivantes seront effectuées sur les entités appariées des ontologies :
). Il s'agit d'une nouvelle ontologie O M , positionnée entre les deux ontologies O 1 et O 2 , qui contient les règles / les axiomes (de transformation) pour mettre en correspondance des entités entre O 1 et O 2 . Révisions de Mapping où O 1 , une nouvelle version de O 1 , contient (à part les entités et les axiomes de O 1 ) les règles qui mettent en correspondance les entités de O 1 par rapport à O 2 , et O 2 , une nouvelle version de O 2 , contient (à part les entités et les axiomes de O 2
remarquent deux perceptions de réparation et de debugage dans les travaux de fusion ou d'intégration des ontologies :• Quelques auteurs considèrent que les ontologies (à intégrer ou fusionner) sont correctes et
toujours plus fiables que les alignements, et s'il y a d'incohérence ou d'inconsistance, c'est
3.6. NOUVELLE DÉFINITION DE LA NOTION D'INTÉGRATION
forcément à cause des alignements (LogMap et ALCOMO). Ils cherchent alors à trouver
l'ensemble minimal de conflits entraînant l'incohérence de l'alignement, et suppriment les
correspondances qui causent les insatisfiabilités dans les classes de l'ontologie résultante
pour minimiser leur impact.
L'expérimentation est réalisée à l'aide des bases de test Conference, Anatomy, et Large Biomedical disponibles dans le cadre de la compagne OAEI (Ontology Alignment Evaluation Initiative). Menée depuis 2004 par un groupe de chercheurs sur le matching des ontologies, l'initiative de l'évaluation des alignements d'ontologies (OAEI) est une plate-forme internationale standard d'évaluation des outils de matching. Elle vise à améliorer les différents matchers d'ontologies en évaluant et en comparant leurs forces et leurs faiblesses à l'aide d'une suite d'alignements de référence qu'elle fournit. Les résultats des campagnes d'évaluation de chaque année, ainsi que l'ensemble des bases et des alignements de référence peuvent être téléchargés sur le site Web de OAEI.
Table 4 .
41 -Caractéristiques des ontologies de la base ConferenceConference Classes Niv Prop Obj Niv Prop Data Niv Instances Axiomscmt
29
4
49
1
10
1
0
226
conference
59
7
46
2
18
1
0
285
confOf
38
3
13
1
23
1
0
196
edas
103
4
30
1
20
1
114
739
ekaw
73
6
33
2
0
0
0
233
iasted
140
6
38
1
3
1
4
358
sigkdd
49
4
17
1
11
1
0
116
Total/Max
491
7
226
2
85
1
118
2 153
Table 4 .
42 -Caractéristiques des ontologies de la base AnatomyAnatomy
Classes Niv Prop Obj Niv Prop Data Niv Instances Axioms
human
3 304
13
2
1
0
0
0 11 545
mouse
2 743
7
3
1
0
0
0
4 838
Total/Max
6 047
13
5
1
0
0
0 16 383
Table 4 .
43 -Caractéristiques des ontologies de la base LargeBioLargeBio
Classes Niv Prop Obj Niv Prop Data Niv Instances Axioms
FMA
78 988
21
0
0
54
1
0
79 218
NCI
66 724
17
123
6
67
1
0
96 046
SNOMED
122 464
34
55
3
0
0
0 191 203
Total/Max 268 176
34
178
6
121
1
0 366 467
Table 4 .
44 -Les alignements de référence de la base ConferenceTable 4.5 -Les alignements de référence de la base AnatomyTable 4.6 -Les alignemens de référence de la base LargeBioAlignment
Cellules
cmt-conference
15 (14)
cmt-confOf
16
cmt-edas
13
cmt-ekaw
11
cmt-iasted
4
cmt-sigkdd
12
conference-confOf
15
conference-edas
17
conference-ekaw
25
conference-iasted
14
conference-sigkdd
15
confOf-edas
19
confOf-ekaw
20 (19)
confOf-iasted
9
confOf-sigkdd
7
edas-ekaw
23
edas-iasted
19
edas-sigkdd
15
ekaw-iasted
10
ekaw-sigkdd
11
iasted-sigkdd
15
Total
305 (303)
() : filtré
Alignment
Nombre de cellules
Original (1-to-N) Filtré (1-to-1)
human-mouse
1 516
1 491
Alignment
Original (1-to-N)
Filtré (1-to-1)
=
? Taille
=
? Taille
FMA-NCI
2 686
338
3 024
2 337
170
2 507
FMA-SNOMED
6 026 2 982
9 008
5 186 2 568
7 754
SNOMED-NCI
17 210 1 634 18 844 13 358
740 14 098
Total
25 922 4 954 30 876 20 881 3 478 24 359
Thing", et B = "owl:Nothing", alors il s'agit d'un test de consistance. -Si A = "une classe", et B = "owl:Nothing", il s'agit d'un test de satisfiabilité. -Si A et B sont toutes les deux des classes, il s'agit d'un test de subsomption.Table 4.7 -Caractéristiques de l'ontologie résultant d'une intégration ou d'une agrégation Sortie = Ontologie résultante Classes Propriétés d'objet Propriétés data Individus4.6 Résultats et évaluation
4.6.1 Résultats
Entrées =
ontologies (+ alignements)
« Conference »
491
226
85
118
« Anatomy »
6 047
5
0
0
« Large Bio »
268 176
178
121
0
Table 4 .
48 -Qualité de l'ontologie résultant d'une agrégationTable 4.9 -Qualité de l'ontologie résultant de l'intégration des ontologies de ConferenceOntologies
d'entrée
Ontologie de sortie
Classes
insatisfiables
Axioms
logiques
Consistance
OIA2R
Réf OIA2R
Réf OIA2R Réf
Conference
0
0
1 860
2 153
Anatomy
0
0
6 635 16 383
LargeBio
0
0 244 942 366 467
Conference
Nombre des classes insatisfiables
Al originaux (1-à-N) Al filtrés (1-à-1)
OIA2R
Réf OIA2R
Réf
Intégration
2-à-2
0
5
0
5
Intégration
1-à-N
0
0
0
0
Intégration
N-à-N
54
54
Table 4 .
410 -Préservation des axiomes après l'intégration des ontologies de ConferenceConference
Nombre des axiomes logiques
Al originaux
Al filtrés
OIA2R
Réf Attendus OIA2R
Réf Attendus
Intégration
2-à-2
1 957 2 250
2 250
(2 153 +
97)
1 956 2 249
2 249
(2 153 +
96)
Intégration
1-à-N
1 931 2 224
2 224
(2 153 +
71)
1 930 2 223
2 223
(2 153 +
70)
Intégration
N-à-N
2 165 2 458
2 458
(2 153 +
305)
2 163 2 456
2 456
(2 153 +
303)
Table 4 .
412 -Préservation des axiomes après l'intégration des ontologies de AnatomyAnatomy
Nombre des axiomes logiques
Al originaux
Al filtrés
OIA2R
Réf Attendus OIA2R
Réf Attendus
Intégration
2-à-2
8 151 17 899
17 899
(16 383
+ 1 516)
8 126 17 874
17 874
(16 383
+ 1 491)
Table 4 .
413 -Qualité de l'ontologie résultant de l'intégration des ontologies de LargeBioLargeBio
Nombre de classes insatisfiables
Al originaux
Al filtrés
OIA2R
Réf OIA2R
Réf
Intég
2-à-2
Al
originaux
120 743 190 486
67 342 141 941
Al
réparés
11 978
-11 078
-
Intég
1-à-N
Al
originaux
58 608 118 579
27 773 65 043
Al
réparés
56
-
48
96
Intég
N-à-N
Al
originaux
136 301 206 232
80 320 157 121
Al
réparés
14 655
-12 919
-
Table 4 .
414 -Préservation des axiomes après l'intégration des ontologies de LargeBioLargeBio
Nombre des axiomes logiques
Al originaux
Al filtrés
OIA2R
Réf Attendus OIA2R
Réf Attendus
Intég
2-à-2
Al
originaux
266 810 388 335
388 335
(366 467
+ 21 868)
261 547 383 072
383 072
(366 467
+ 16 605)
Al
réparés
264 838 386 363
386 363
(366 467
+ 19 896)
260 637 382 162
382 162
(366 467
+ 15 695)
Intég
1-à-N
Al
originaux
256 974 378 499
378 499
(366 467
+ 12 032)
255 203 376 728
376 728
(366 467
+ 10 261)
Al
réparés
253 654 375 179
375 179
(366 467
+ 8 712)
252 465 373 990
373 990
(366 467
+ 7 523)
Intég
N-à-N
Al
originaux
275 818 397 343
397 343
(366 467
+ 30 876)
269 301 390 826
390 826
(366 467
+ 24 359)
Al
réparés
270 864 392 389
392 389
(366 467
+ 25 922)
265 823 387 348
387 348
Table 4 .
415 -Temps d'exécution d'une intégration N-à-NTps d'exécutionCPU (s)Notre intégration
La référence
+loading -loading +loading -loading
Conference
1,531
0,406
1,375
0,171
Anatomy
3,093
0,703
3,562
0,453
Large Bio
36,859
8,406
41,375
4,890
Table 4 .
416 -Qualité de l'ontologie intégrée (LargeBio OAEI Task 1) FMA1-NCI 1Table 4.17 -Qualité de l'ontologie intégrée (LargeBio OAEI Task 3)Table 4.18 -Qualité de l'ontologie intégrée (LargeBio OAEI Task 5)Table 4.19 -Qualité de l'ontologie intégrée (LargeBio OAEI Task 2)Nombre de classes insatisfiables
Al original
Al filtré
OIA2R Full merge OIA2R Full Merge
Al original
1 727
826
410
173
Al réparé
0
0
0
0
FMA2-SNOMED1
Nombre de classes insatisfiables
Al original
Al filtré
OIA2R Full merge OIA2R Full Merge
Al original
13 508
7 212 10 048
4 379
Al réparé
0
0
0
0
NCI 2-SNOMED2
Nombre de classes insatisfiables
Al original
Al filtré
OIA2R Full merge OIA2R Full Merge
Al original
34 639
19 132 25 637
12 990
Al réparé
0
0
0
0
FMA3-NCI 3
Nombre de classes insatisfiables
Al original
Al filtré
OIA2R Full merge OIA2R Full Merge
Al original
7 175
6 272
1 158
995
Al réparé
0
0
0
0
Table 4 .
420 -Qualité de l'ontologie intégrée (LargeBio OAEI Task 6) Nous remarquons (comme l'avaient ditRaunich and Rahm NCI 3-SNOMED3
Nombre de classes insatisfiables
Al original
Al filtré
OIA2R Full merge OIA2R Full Merge
Al original
92 149
76 280 49 825
42 331
Al réparé
0
0
0
0
§
.5. INGÉNIERIE ONTOLOGIQUE
.1. FUSION DES ONTOLOGIES
Dans la littérature, le problème de l'intégration des ontologies a été largement étudié au cours des dernières années, mais il reste toujours un défi si nous voulons réaliser une intégration de manière automatique, efficace, sur de grandes ontologies, en préservant toutes les données originales, et sans produire d'erreurs (conflits sémantiques / logiques).
Nous faisons le parsing des propriétés d'annotation (des ontologies d'entrée) et de leurs définitions, et au fur et à mesure, nous créons les axiomes correspondants à ces propriétés d'annotation et à leurs définitions dans notre nouvelle ontologie. Nous faisons le parsing des individus / instances (des ontologies d'entrée) et de leurs définitions, et au fur et à mesure, nous créons les axiomes correspondants à ces individus et à leurs définitions dans notre nouvelle ontologie.
Nous remarquons qu'après l'ajout des "bridging" axiomes d'équivalence, la classe "002#Tissue_Dissection" devient par inférence une sous-classe des deux classes "002#Fin-dings_and_Disorders_Kind" et "002#NCI_Kind" qui sont disjointes (information extraite de l'ontologie originale (Ont2)). Ceci est contradictoire, car une classe ne peut pas être une sous-classe de deux classes disjointes. Aucune instance ne peut la satisfaire. La même chose s'applique pour les autres classes coloriées en rouge. Si la classe "003#Clinical_finding" (prove-
. https://github.com/ernestojimenezruiz/logmap-matcher 2. http://web.informatik.uni-mannheim.de/alcomo/
Nous allons passer tout de suite à la concrétisation de notre approche décrite dans ce chapitre.
RemerciementsJe remercie Monsieur Sadok Ben Yahia, Professeur à la Faculté des Sciences de Tunis et directeur du Laboratoire d'Informatique en Programmation, Algorithmique et Heuristique (LIPAH), pour la confiance qu'il m'a accordée en acceptant de diriger mes travaux de mastère. Je remercie sa disponibilité continue et je voudrais lui éprouver toute mon admiration.RésuméCe travail est accompli dans le cadre d'un projet de mémoire de mastère de recherche. Le but est d'intégrer deux ou plusieurs ontologies (de mêmes ou de différents domaines) dans une nouvelle ontologie OWL consistante et cohérente pour assurer leur interopérabilité sémantique. Pour ce faire, nous avons choisi de créer une ontologie de pont qui inclut toutes les ontologies sources et leurs axiomes de pont dans une nouvelle ontologie. Par la suite, nous avons introduit un critère qui aide à obtenir une ontologie de meilleure qualité (ayant le minimum de conflits sémantiques / logiques). Nous avons proposé également une nouvelle terminologie qui clarifie les notions floues et mal placées utilisées dans les travaux de l'état de l'art. Enfin, nous avons testé et évalué notre outil OIA2R à l'aide des ontologies et des alignements de référence de OAEI. Il s'est avéré qu'il est générique, efficace, scalable, et assez performant.Mots clés: Ontologie, Intégration des ontologies, Fusion des ontologies, Matching, Alignement, Mapping, Consistance, Cohérence, Insatisfiabilité, OWL, Réparation des alignements, debugage des alignements.AbstractThis work is done as part of a research master's thesis project. The goal is to integrate two or more ontologies (of the same or different domains) in a new consistent and coherent OWL ontology to insure semantic interoperability between them. To do this, we have chosen to create a bridge ontology that includes all source ontologies and their bridging axioms in a new ontology. Subsequently, we introduced a new criterion for obtaining an ontology of better quality (having the minimum of semantic / logical conflicts). We have also proposed a new terminology that clarifies the unclear and misplaced notions used in state-of-the-art works. Finally, we tested and evaluated our OIA2R tool using OAEI ontologies and reference alignments. It turned out
Pour but de mettre en relief les axiomes d'équivalence ajoutés à l'union des ontologies sources et bien les montrer dans nos captures, nous avons choisi de faire une intégration N-à-N. Nous avons choisi d'imprimer le résultat de l'intégration des plus grandes ontologies pour prouver que notre Framework monte à l'échelle facilement. comme le montreront les temps d'exécutionPour but de mettre en relief les axiomes d'équivalence ajoutés à l'union des ontologies sources et bien les montrer dans nos captures, nous avons choisi de faire une intégration N-à-N. Nous avons choisi d'imprimer le résultat de l'intégration des plus grandes ontologies pour prouver que notre Framework monte à l'échelle facilement (comme le montreront les temps d'exécution).
La première ontologie insérée en entrée est. FMA. 3wholeLa première ontologie insérée en entrée est "FMA 3" (whole);
. La , NCI. 3wholeLa deuxième ontologie est "NCI 3" (whole);
. La , SNOMED 3" (wholeLa troisième ontologie est "SNOMED 3" (whole).
Voici ce que donne l'exécution de la première partie du code qui réalise une agrégation / composition simple des ontologies d'entrée (i.e., sans les "bridging" axiomes) : Pour la référence : les axiomes sont exactement identiques aux originaux. Figure 4.1Voici ce que donne l'exécution de la première partie du code qui réalise une agrégation / composition simple des ontologies d'entrée (i.e., sans les "bridging" axiomes) : Pour la référence : les axiomes sont exactement identiques aux originaux (Figure 4.1).
Pour OIA2R : les axiomes sont décrits exactement comme les originaux sauf que nous personnalisons les IRIs de toutes les entités mentionnées. Figure 4.2Pour OIA2R : les axiomes sont décrits exactement comme les originaux sauf que nous personnalisons les IRIs de toutes les entités mentionnées (Figure 4.2).
En effet, si nous ne gardons pas d'axiomes de disjonction, nous n'obtiendrons aucune classe insatisfaisable, et toutes nos ontologies de sortie seraient cohérentes et consistantes, mais incomplètes (i.e., manquant des informations de disjonction précieuses). Notez que toutes les classes insatisfiables sont causées par la préservation des connaissances de disjonction des alignements d'entrée. Nous concluons que, lorsque toutes les correspondances sont des correspondances d'équivalence, la seule cause de conflits est les relations. DisjointWith" issues des ontologies d'entréeNotez que toutes les classes insatisfiables sont causées par la préservation des connaissances de disjonction des alignements d'entrée. En effet, si nous ne gardons pas d'axiomes de disjonction, nous n'obtiendrons aucune classe insatisfaisable, et toutes nos ontologies de sortie seraient cohérentes et consistantes, mais incomplètes (i.e., manquant des informations de disjonction précieuses). Nous concluons que, lorsque toutes les correspondances sont des correspondances d'équivalence, la seule cause de conflits est les relations "DisjointWith" issues des ontologies d'entrée.
Nous remarquons aussi que dans l'ontologie résultant d'une intégration N-à-N, le nombre de classes insatisfiables est beaucoup plus important que celui de l'ontologie résultant d'une intégration 2-à-2 ou 1-à-N. L'ontologie de référence contient toujours plus de classes insatisfiables que notre ontologie. Par conséquent, nous serions dans le risque d'avoir une inconsistance possible si jamais il y avait un individu instancié par une des classes insatisfiables, ou si jamais des individus avaient un conflit entre eux suite à des axiomes de pont de type "sameAs". Dans notre cas, les ontologies sources utilisées ne contiennent pas d'instances, à part "edas" et "iasted" dont toutes les instances sont instanciées par des classes qui n'ont aucune correspondance avec d'autres classes, donc sont hors de danger. Nous constatons aussi qu'une intégration d'ontologies de différents domaines. Constatations Dans tous les cas d'intégration, notre ontologie finale est complète dans le sens où elle conserve toutes les entités et la hiérarchie des ontologies d'entrée, et toutes les correspondances des alignements d'entrée. Dans l'agrégation (sans "bridging" axiomes), notre ontologie n'aura aucun ajout de classe insatisfiable, et le nombre de niveaux de sa hiérarchie sera toujours égal au nombre maximal des niveaux de hiérarchie des ontologies d'entrée. Dans l'ontologie de pont, nous constatons que suite à l'ajout des "bridging" axiomes, le raisonneur HermiT génère beaucoup trop de classes insatisfiables. comme dans "Anatomy") génère toujours moins de conflits qu'une intégration d'ontologies de même domaine. comme dans "Conference" et "Large Bio")Constatations Dans tous les cas d'intégration, notre ontologie finale est complète dans le sens où elle conserve toutes les entités et la hiérarchie des ontologies d'entrée, et toutes les correspondances des alignements d'entrée. Elle ne parvient pas pourtant à préserver tous les axiomes des ontologies d'entrée, contrairement à l'ontologie de "référence". Dans l'agrégation (sans "bridging" axiomes), notre ontologie n'aura aucun ajout de classe insatisfiable, et le nombre de niveaux de sa hiérarchie sera toujours égal au nombre maximal des niveaux de hiérarchie des ontologies d'entrée. Dans l'ontologie de pont, nous constatons que suite à l'ajout des "bridging" axiomes, le raisonneur HermiT génère beaucoup trop de classes insatisfiables. Nous remarquons aussi que dans l'ontologie résultant d'une intégration N-à-N, le nombre de classes insatisfiables est beaucoup plus important que celui de l'ontologie résultant d'une intégration 2-à-2 ou 1-à-N. L'ontologie de référence contient toujours plus de classes insatisfiables que notre ontologie. Par conséquent, nous serions dans le risque d'avoir une inconsistance possible si jamais il y avait un individu instancié par une des classes insatisfiables, ou si jamais des individus avaient un conflit entre eux suite à des axiomes de pont de type "sameAs". Dans notre cas, les ontologies sources utilisées ne contiennent pas d'instances, à part "edas" et "iasted" dont toutes les instances sont instanciées par des classes qui n'ont aucune correspondance avec d'autres classes, donc sont hors de danger. Nous constatons aussi qu'une intégration d'ontologies de différents domaines (comme dans "Anatomy") génère toujours moins de conflits qu'une intégration d'ontologies de même domaine (comme dans "Conference" et "Large Bio").
nous ne parvenons pas toujours à avoir un nombre de niveaux fixe (un niveau maximal) dans la hiérarchie des classes de notre ontologie, car le raisonneur qui découvre les niveaux (et les classes de chaque niveau) ne termine pas son raisonnement. On dirait que, suite à l'ajout des "bridging" axiomes, le raisonneur rencontre une boucle infinie. un cercle vicieux) dans son raisonnementA part cela, nous ne parvenons pas toujours à avoir un nombre de niveaux fixe (un niveau maximal) dans la hiérarchie des classes de notre ontologie, car le raisonneur qui découvre les niveaux (et les classes de chaque niveau) ne termine pas son raisonnement. On dirait que, suite à l'ajout des "bridging" axiomes, le raisonneur rencontre une boucle infinie (un cercle vicieux) dans son raisonnement.
Interprétations Ces insatisfiabilités sont dues au fait que l'axiome d'équivalence entre deux entités est formellement équivaut à deux axiomes de subsomption réciproques : equivalentClass (C1 C2) = subClassOf (C1 C2) + subClassOf. C2 C1Interprétations Ces insatisfiabilités sont dues au fait que l'axiome d'équivalence entre deux entités est formellement équivaut à deux axiomes de subsomption réciproques : equivalentClass (C1 C2) = subClassOf (C1 C2) + subClassOf (C2 C1)
Perspectives : Dans nos prochains travaux, nous allons nous projeter sur la fusion des ontologies qui constitue le plus haut niveau d'interopérabilité sémantique entre les ontologies, et cela pour but de minimiser au maximum les erreurs de l'ontologie résultante de ce processus. En effet, la fusion génère toujours moins d'insatisfiablités que l. § Et l'ajout de ces subsomptions implicites aux équivalences infère de nouvelles connaissances qui peuvent être contradictoires. ontologie de pont que nous avons réalisée dans ce mémoire§ Et l'ajout de ces subsomptions implicites aux équivalences infère de nouvelles connaissances qui peuvent être contradictoires. Perspectives : Dans nos prochains travaux, nous allons nous projeter sur la fusion des ontologies qui constitue le plus haut niveau d'interopérabilité sémantique entre les ontologies, et cela pour but de minimiser au maximum les erreurs de l'ontologie résultante de ce processus. En effet, la fusion génère toujours moins d'insatisfiablités que l'ontologie de pont que nous avons réalisée dans ce mémoire.
Nous exploiterons aussi d'autres relations sémantiques à part la relation d'équivalence dans les alignements, telles que la subsomption et la disjonction, pour que l'interopérabilité soit maximale et que toute hétérogénéité soit traitée. Nous exploiterons aussi d'autres relations sémantiques à part la relation d'équivalence dans les alignements, telles que la subsomption et la disjonction, pour que l'interopérabilité soit maximale et que toute hétérogénéité soit traitée.
nous comptons exploiter le domaine de la fouille de données dans nos prochains travaux de fusion ou d'intégration des ontologies. En effet, les contributions de notre laboratoire LIPAH ont atteint un niveau avancé dans ce domaine, ce qui va énormément nous aider. Fca-Merge Suivant L'exemple De, Maedche Stumme, Citons en quelques travaux intéressants. Bouzouita et al.Suivant l'exemple de FCA-Merge Stumme and Maedche (2001), nous comptons exploiter le domaine de la fouille de données dans nos prochains travaux de fusion ou d'intégration des ontologies. En effet, les contributions de notre laboratoire LIPAH ont atteint un niveau avancé dans ce domaine, ce qui va énormément nous aider. Citons en quelques travaux intéressants : Bouzouita et al. (2006);
. Gasmi, Gasmi et al. (2007);
. Cellier, Cellier et al. (2008);
. Ben Othman, Yahia, Othman and Ben Yahia (2008);
. Hamrouni, Hamrouni et al. (2008);
. Ayouni, Ayouni et al. (2010, 2011);
. Brahmi, Brahmi et al. (2010, 2011);
. Hamdi, Hamdi et al. (2013);
. Bouker, Bouker et al. (2014)
Il faut également noter qu'un autre axe de recherche très intéressant et plus difficile se présente. C'est le domaine de réparation, de debugage, ou de révision des ontologies et des alignements. Ce domaine aidera énormément à avoir des ontologies de bonne qualité suite à l. intégration des ontologiesIl faut également noter qu'un autre axe de recherche très intéressant et plus difficile se présente. C'est le domaine de réparation, de debugage, ou de révision des ontologies et des alignements. Ce domaine aidera énormément à avoir des ontologies de bonne qualité suite à l'intégration des ontologies.
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arxiv |
Bound on Lyapunov exponent in Einstein-Maxwell-Dilaton-Axion black holes
3 Dec 2022
Chengye Yu
School of Science
Xihua University
610039ChengduChina
Deyou Chen
School of Science
Xihua University
610039ChengduChina
Chuanhong Gao 3e-mail:[email protected]
School of Science
Xihua University
610039ChengduChina
Bound on Lyapunov exponent in Einstein-Maxwell-Dilaton-Axion black holes
3 Dec 2022Lyapunov exponentangular momentumEinstein-Maxwell-Dilaton-Axion black holes
In this paper, we investigate the influence of the angular momentum of a charged particle around non-extremal and extremal Einstein-Maxwell-Dilaton-Axion black holes on the Lyapunov exponent. The angular momentum's ranges and spatial regions where the bound of the exponent is violated are found for certain values of the rotation parameter and dilatonic constant of the black holes. This violation always exists when the rotation parameter is large enough and the rotation directions of the particle is opposite to those of the black holes. The spatial regions outside the extermal black hole for the violation is relatively large. In the near-horizon regions of the extremal black holes, the violation depends on the rotation directions of the black holes and particle, and does not depend on the value of the angular momentum.
Introduction
Chaos is an important physical phenomenon in nonlinear dynamic systems. In classical chaos, trajectories of dynamics sensitively depend on initial conditions. A small change of the conditions leads to that the trajectories deviate exponentially from their initial locations. Its sensitivity is characterized by a Lyapunov exponent. However, it is not easy to study the chaos in quantum systems. Recently, people have proposed that the chaos can be efficiently diagnosed from a holographic approach by using out-of-time-order correlators (OTOCs) [1][2][3][4][5].
In holographic systems at finite temperature, the OTOC is
< V (0)W (t, x)V (0)W (t, x) > < V (0)V (0) > W (t, x)W (t, x) > = 1 − ε ∆ V ∆ W exp λ L (t − t − | x| v B ) ,(1.1)
where ε ∆ V ∆ W is a multiplicative factor and contains information about the operators V and W , λ L is a Lyapunov exponent, t is a scrambling time, and v B is a butterfly velocity.
In the seminal work [6], Maldacena, Shenker and Stanford conjectured that there is a universal bound on the exponent of chaos in thermal quantum systems with a large number of degrees of freedom,
λ ≤ 2πk B T ,(1.2)
where T is the temperature of the system. This conjecture relies on two assumptions. The first one is that certain time-ordered correlation functions approximately factorize. Another assumption is that there is a large hierarchy between the dissipation time and scrambling time. This work provides a cornerstone for the development of the AdS/CFT correspondence, and promotes the research on black holes. After the conjecture was put forward, it attracted great attention and was confirmed by a lot of work . It was found that the bound is saturated in the Sachdev-Ye-Kitaev (SYK) model, which provides an important basis for the theory dual to gravity [41]. In the study of particle motions near black holes, when electromagnetic or scalar forces on a particle are large enough, the particle can be very close to the event horizons without falling into them. Based on this, Hashimoto and Tanahashi found that the value of the exponent is independent on the external forces and the particle mass, and obeys an inequality λ ≤ κ, where κ is the surface gravity of the black hole [42]. From the relation between the temperature and surface gravity of the black hole, this inequality is identical to Eq. (1.2). However, as the authors said in [6], there are cases where the bound on the exponent (chaos bound) is not applicable [43][44][45][46][47][48][49][50][51][52][53]. For example, the string corrections to the bound and the cases that does not meet the assumptions. In the classical limit of the SYK model, the linear relation of the exponent dependent on temperature was found [45]. Its slope is different in parameters from that obtained in the quantum case. In the research of the chaos in anti-de Sitter (AdS) spacetimes, the exponent for the motion of classical closed strings was modified as λ = 2πT n for winding strings in the bulk, where n is the winding number of the string [46]. Sub-leading terms in near-horizon expansions have an important influence on the exponent's value. When a charged particle is in equilibrium outside an event horizon of a black hole by a Lorentz force, one can adjust the charge mass ratio of the particle to let the particle close to the event horizon. In the near-horizon regions, the bound is violated by the Einstein-Maxwell-Dilaton, Einstein-Born-Infeld and Einstein-Gauss-Bonnet Maxwell black holes, and satisfied by Reissner-Nordström (RN) and RN-AdS black holes [47]. In [47], the influence of the angular momentum of the particle was neglected. In fact, the angular momentum plays an important role in the exponent. When this influence is considered, Kan and Gwak studied the bound via the effective potential of the particle [51]. The violation for the bound was found for the specific values of the black hole's parameter. The exponent can also be obtained by the matrix method. Using this method, Lei and Ge found that the bound in the near-horizon regions of the RN and RN-AdS black holes is violated when the angular momentum of the particle and the charge of the black holes are large enough [53].
In this paper, we investigate the influence of the angular momentum of a charged particle around extremal and non-extremal Einstein-Maxwell-Dilaton-Axion (EMDA) black holes on the Lyapunov exponent, and find the angular momentum's ranges and spatial regions where the bound is violated. The exponent is derived by the effective potential of the particle and affected by the angular momentum. We first investigate the exponent at a certain distance from the event horizons by numerical calculations. Then the exponent in the near-horizon regions is discussed. In the investigation, the same and opposite rotation directions of the particle and black holes have different influences, which are considered.
The rest is organized as follows. In the next section, we review the EMDA black holes and derive the Lyapunov exponent by the effective potential. An auxiliary field is introduced and a static gauge is used. In Section 3, we investigate the influence of the angular momentum of the particle around the non-extremal and extremal EMDA black holes on the exponent, and find the spatial regions where the bound is violated. The last section is devoted to our conclusion and discussion.
EMDA black holes and Lyapunov exponent
The EMDA black hole is a solution of field equations arising in the low energy heterotic string field theory and describes a rotating charged spacetime. From the action,
S = d 4 x √ −g R − 2g µν ∂ µ φ∂ ν φ − 1 2 e 4φ g µν ∂ µ κ 0 ∂ ν κ 0 − e −2φ F µν F µν − κ 0 F µνF µν , (2.1)
where R is a scalar Riemann curvature, F µν is an electromagnetic tensor field and its dual isF µν = − 1 2 √ −g µναβ F αβ . φ is a dilaton field, and κ 0 is an Axion scalar field dual to the three-index anti-symmetric tensor field H = −exp(4φ) * dκ 0 /4. From the action, the solutions of rotating black holes were obtained [54][55][56]. In [55], Garcia, Galtsov and Kechkin got the EMDA black hole solution, which is given by
ds 2 = − ∆ Σ dt − a sin 2 θdφ 2 + sin 2 θ Σ adt − (Σ + a 2 sin 2 θ)dφ 2 + Σ ∆ dr 2 + Σdθ 2 , (2.2)
with an electromagnetic potential
A µ dX µ = Qr Σ dt − aQr sin 2 θ Σ dφ,(2.3)
where ∆ = r 2 − 2M 0 r + a 2 = (r − r + )(r − r − ), Σ = r 2 + 2br + a 2 cos 2 θ,
M 0 = M − b = M − Q 2 2M . (2.4)
r + and r − are the event and inner horizons, respectively. a is a rotation parameter of the black hole, M is the ADM mass and Q is the charge. b is a dilatonic constant and relates to the ADM mass and charge by b = Q 2 2M . When a = 0, the metric describes a charged, non-rotating dilatonic black hole. When b = 0, the metric is reduced to the Kerr metric. The event (inner) horizons r + (r − ) and surface gravity are
r ± = M 0 ± M 2 0 − a 2 , κ = r + − r − 2r + (r + + r − + 2b) ,(2.5)
respectively. When the inner and event horizons coincide with each other, the black hole is extremal and the surface gravity is zero. We consider a charged particle with mass m and charge q moving around the EMDA black hole. Then the action of the particle is [51]
S = ds 1 2e(X(s)) g µν (X(s))Ẋ µ (s)Ẋ ν (s) − e(X(s)) 2 m 2 − qA µ (X(s))Ẋ µ (s) . (2.6)
In the above equation, e is an auxiliary field and s is adopted to parameter the geodesic of the particle. Without loss of generality, we use a static gauge and let s be equal to the time t. We focus our attention on the motion of the particle in the equatorial plane of the black hole, where θ = π 2 . From the metric (2.2) and action (2.6), the Lagrangian is
L = 1 2e (r 2 + 2br)ṙ 2 ∆ + 2a[∆ − (r 2 + 2br + a 2 )]φ r 2 + 2br + [(r 2 + 2br + a 2 ) 2 − ∆a 2 ]φ 2 r 2 + 2br − ∆ − a 2 r 2 + 2br − e 2 m 2 − qQ r + 2b + aqQ r + 2bφ , (2.7)
whereφ appears, and the corresponding angular momentum is
L = ∂L ∂φ = a[∆ − (r 2 + 2br + a 2 )] e(r 2 + 2br) + [(r 2 + 2br + a 2 ) 2 − ∆a 2 ]φ e(r 2 + 2br) + aqQ r + 2b . (2.8)
The equation of motion of the auxiliary field satisfies −e 2 m 2 =Ẋ µẊ µ . The auxiliary field is solved and takes the form e = (r + 2b)
√ r ∆ 2 −[(r 2 +2br+a 2 ) 2 −∆a 2 ]ṙ 2 α .
Then the effective Lagrangian of the particle is
L ef f = L − Lφ = − h −ṙ 2 f − Φ, (2.9) where h = rα χ 2 , f = χ∆ 2 rα , χ = (r 2 + 2br + a 2 ) 2 − ∆a 2 , Φ = a[aqQ − L(r + 2b)][∆ − (r 2 + 2br + a 2 )] (r + 2b)χ + qQ r + 2b , α = ∆[m 2 (r + 2b)[(r 2 + 2br + a 2 ) 2 − ∆a 2 ] + r[aqQ − L(r + 2b)] 2 ].
(2.10)
When the particle moves slowly around a local maximum of a potential and its velocity obeyṡ r 1. Eq. (2.9) is expanded and rewritten as follows
L ef f =ṙ 2 2 √ hf − V ef f (r) + O(ṙ 4 ), (2.11) where V ef f (r) = √ h + Φ, (2.12)
is an effective potential, O(ṙ 4 ) contains higher order terms ofṙ and is neglected. The local maximum of the potential is obtained at a location r 0 and determined by V ef f (r) = 0, where " " represents a derivative in term of r and
V ef f (r) = h 2 √ h + Φ . (2.13)
At this location, we introduce a small perturbation (s) to let r(s) = r 0 + (s). Then the effective Lagrangian is
L ef f = 1 2 √ hf (˙ 2 + λ 2 2 ).
(2.14)
In the above derivation, the constant and higher-order terms were neglected. λ is defined as a Lyapunov exponent and given by
λ 2 = − √ hf V ef f (r) r=r 0 , (2.15)
The stability of the system is determined by the exponent. When λ 2 > 0, the system is unstable and a chaos appears. λ 2 < 0 corresponds to the stable system, and λ 2 = 0 reflects that the system is marginal. It is not difficult to get χ| r=r 0 > 0 and √ hf r=r 0 > 0. Then the sign is determined by the value of V ef f (r). Due to the appearance of the angular momentum of the particle, we need to consider the influence of the angular momentum on the exponent when the bound of the exponent is discussed.
Bound on Lyapunov exponent and its violation in EMDA black holes
In this section, we use Eq. (2.15) to investigate the influence of the angular momentum of the particle around the non-extremal and extremal EMDA black holes on the exponent, and find the angular momentum's ranges and spatial regions where the bound is violated.
Lyapunov exponent in non-extremal EMDA black holes
We first investigate the influence of the angular momentum of the particle around a nonextremal EMDA black hole on the exponent. In [47], the authors found that when the charge mass ratio of the particle is large, the particle is in equilibrium near the horizons. In this subsection, we order M = 1, m = 1, q = 15. When b = 1 3 , we use Eq. (2.5) and get the location of the horizon and the value of the surface gravity. There are r + = 1.26091 and κ 2 = 0.0563241 when a = 2 7 . When a = 1 3 , we obtain r + = 1.24402 and κ 2 = 0.0538476. a = 2 5 yields r + = 1.20000 and κ 2 = 0.0493827. a = 1 2 yields r + = 1.10763 and κ 2 = 0.0396232. It is found from Eq. (2.13) that different values of the angular momentum and rotation parameter lead to different locations corresponding to the local maximum of the effective potential. Their relations are listed in Table 1. The positive sign in front of the angular momentum indicates that the particle and black hole rotate in the same directions, and the negative sign indicates that they rotate in the opposite directions. Using Eqs. (2.15), we get the values of the exponent by numerical calculations in Figure 1. In the figure, the bound is violated in certain ranges of the angular momentum which corresponds to specific spatial regions. The ranges of the angular momentum and spatial regions for the violation increase with the increase of the rotation parameter's value. For the fixed a and b, the angular momentum's ranges where the bound is violated when the black hole and particle rotate in the opposite directions is larger than those when they rotate in the same directions. It is more likely to cause the violation when they rotate in the opposite directions. Due to different values of the rotation parameter, the values of the angular momentum corresponding to the maximum values of the exponent are different. There is λ 2 − κ 2 > 0 when the angular momentum is zero, which means the black hole can violate the bound without depending on the angular momentum of the particle. The values of λ 2 − κ 2 tend to constants when the angular momentum is large enough. When the rotation parameter is large enough and their rotation directions are opposite, the spatial region is relatively large. When a = 0, the metric (2.2) describes a charged, non-rotating dilatonic black hole. To investigate the bound, we first derive the positions of equilibrium orbits for different values of b and L, and then list them in Table 2. The horizon is located at r + = 1.66667 when b = 1 6 , at r + = 1.33333 when b = 1 3 , and at r + = 0.66667 when b = 2 3 . When the angular momentum increases, the location of r 0 is gradually far from the horizon. The influence of the angular momentum on the exponent is plotted in Figure 2. In the figure, the violation occurs only for the certain values of the dilatonic constant and angular momentum. For example, the bound is violated in the range 1.22 < L < 19.16 (the spatial region is 1.02042099r + < r 0 < 1.45644077r + ) when b = 2 3 . When the dilatonic constant is less than a certain value, there is no violation no matter how the angular momentum increases. When the value of the dilatonic constant is greater than a certain value, one can take a specific value of the angular momentum to violate the bound.
When b = 0, the metric (2.2) is reduced to the Kerr metric. The values of the exponent of the chaos for a neutral particle around the Kerr black hole is plotted in Figure 3 directions of the particle and black hole are opposite, the bound is violated by increasing the angular momentum. The violation occurs in the range L < −17.40(2.47909864r + < r 0 < 2.52593108r + ) when a = 5 6 . Although the range of the angular momentum where the bound is violated is large, the corresponding spatial region is not large. There is no violation for the bound when the rotation parameter is less than a certain value, or when the particle and black hole rotate in the same direction.
Lyapunov exponent in extremal EMDA black holes
For an extremal EMDA black hole, the inner and event horizons coincide with each other, and the surface gravity is zero. From Eq. we get the effective potential at different positions in Figure 4. In this figure, the potential has maximum values for different a and b, which leads to V ef f (r) < 0. To investigate the influence of the angular momentum of the particle around this extremal black hole on the exponent, we draw Figure 5. In the figure, the angular momentum's range and spatial region decrease with the increase of the rotation parameter when the bound is violated. The bound is always violated when the particle and black hole rotate in the opposite directions. The values of the exponent approach positive constants when the angular momentum is very large. The spatial regions where the bound is violated for the extremal EMDA black hole are significantly larger than those for the non-extremal EMDA black hole. One reason is the disappearance of the surface gravity for the extremal black hole. Figure 5: The influence of the angular momentum of the particle around the extremal EMDA black hole on the Lyapunov exponent. The bound on the Lyapunov exponent is violated in the range L < 2.54(1.00304200r+ < r0 < 4.44949278r+) when a = 2 3 , violated in the range L < 2.52(1.00756435r+ < r0 < 4.30940108r+) when a = 3 4 , violated in the range L < 2.49(1.00119340r+ < r0 < 4.23606798r+) when a = 4 5 , and violated in the range L < 2.38(1.00577200r+ < r0 < 4.19089368r+) when a = 5 6 .
Kan and Gwak have studied the Lyapunov exponent of the chaos of the particle around the extremal Kerr black hole in [51], where the rotation parameter took several specific values. Here we simply discuss the violation of the bound by taking into account several different values of the rotation parameter. Now Q = 0 in Eq. (2.12) and the effective potential is plotted in Figure 6. In this figure, there are maximum values in the effective potential for different values of the rotation parameter, which indicates the violation of the bound. This result is consistent with that gotten by them.
Lyapunov exponent in near-horizon regions of EMDA black holes
In this subsection, we investigate the exponent of the chaos of a particle in the near-horizon regions of the non-extremal and extremal EMDA black holes. We first focus our attention on the non-extremal black hole, and consider that the location of an equilibrium orbit is very close to the horizon. Let
r 0 = r + + ,(3.1)
where 0 < r + . Inserting the above relation into Eq. (2.13) yields V ef f (r + + ) = r 2 + − r + r − 2 m 2 (r + + 2b)(r 2 + + 2br + + a 2 ) 2 + r + [aqQ − L(r + + 2b)] 2 (r 2 + + 2br + + a 2 ) 2 − 1 2 + 1 (r 2 + + 2br + + a 2 ) 2 a 2 (r + − r − ) r 2 + + 2br + + a 2 − 2(r + + b) (aL + qQr + ) + qQ r 2
+ + 2br + + a 2 − aL(r + − r − ) + O( 1 2 ). (3.2)
Without loss of generality, we let
q L = r + + 2b aQ . (3.3)
Using Eqs. (3.2) and (3.3) and ordering V ef f (r + + ) = 0 yield
L = ma 2 (r + + 2b)(r + − r − )r −1 + − 1 2 + O( 1 2 ),(3.4)
where O( 1 2 ) contains the higher order terms of and is neglected. In the above relation, when the angular momentum is large enough, is very small and r 0 is very close to the horizon. It indicates that the assumption Eq. (3.1) makes sense. Thus, the exponent in the near-horizon region is gotten as follow,
λ 2 − κ 2 = 3(r + − r − ) 8(r 2 + + 2br + + a 2 ) 2 + O( 2 ). (3.5)
It is clearly that λ 2 > κ 2 , which shows that the bound is violated in the near-horizon region of the non-extremal EMDA black hole. When a = 0, we use Eq. (2.13) and get
V ef f (r + + ) = √ r + − r − m 2 (r 2 + + 2br + ) + L 2 2(r 2 + + 2br + ) − 1 2 − qQ (r + + 2b) 2 + O( 1 2 ). (3.6)
This implies that when the angular momentum and charge of the particle take certain values, one can get a small value of . Now the exponent is
λ 2 − κ 2 = 8(r + + b)[m 2 (r 2 + + 2br + ) + L 2 ] − r + (r + + 3)(r + + 2b) 2r + (r + + 2b) 3 [m 2 (r 2 + + 2br + ) + L 2 ] + O( 2 ). (3.7)
It is not difficult to prove that the first term at the right hand of the equal sign is always larger than zero in the large-L limit, which leads to the violation of the bound in the near-horizon region of the charged dilatonic black hole. When the EMDA black hole is extremal, ∆ = 2 is obtained from Eqs. (2.4) and (3.1). We insert this relation into Eq. (2.13) and get
V ef f (r + + ) = √ Hr + − 2aL(r + + b) 4r 2 + (r + + b) 2 − 1 8r 3 + (r + + b) 3 2r 2 + (r + + b) √
Hr + (4m 2 br + (r + + b) 2 + (aqQ − L(r + + 2b))(aqQ − 2bL)) − 2ar + (aqQ − L(r + + 2b))]
+ O( 2 ). (3.8)
It is obviously that there is a solution in the above equation when is very small. From Eq.
(2.15), we get λ 2 = 2 √ Hr + µ + (r 2 + + 2br + + a 2 )(ρ + ) r + (r 2 + + 2br + + a 2 ) 3 H H = m 2 (r + + 2b)(r 2 + + 2br + + a 2 ) 2 + r + [aqQ − L(r + + 2b)] 2 , µ = aL(6r 2 + + 12br + + 4b 2 + a 2 ) + qQ[r 2 + (3r + + 4b) − 2a 2 (r + + b)], ρ = m 2 (r 2 + + 2br + + a 2 )[(r 2 + + 2br + + a 2 )(3r 2 + + 8br + + 4b 2 + a 2 )r + − (2r + + 1)(r + + 2b)],
= 2r 2 + [aqQ − (r + + 2b)L][(r + + 2b − 1)L − aqQ].
(3.10)
We use numerical calculations to evaluate whether the coefficient of 3 in Eq. (3.9) is larger than zero. In Figure 7, there is λ 2 < 0 when a > 0 and L < 0, which shows that the bound is satisfied in the near-horizon region when the particle and black hole rotate in the opposite directions. When a > 0 and L > 0, the values of the exponent are always larger than zero. This implies that there is always a violation for the bound when the particle and black hole rotate in the same directions, and the angular momentum only affects the value of the exponent.
Conclusion and discussion
In this paper, we investigated the influence of the angular momentum of the charged particle around the non-extremal and extremal EMDA black holes on the Lyapunov exponent. The angular momentum's range and spatial region where the bound of the exponent is violated were found. For the non-extremal black hole, the bound is violated when the dilatonic constant is fixed at 1 3 and a = 2 7 , 1 3 , 2 5 or 1 2 . The spatial regions for the violation increase with the increase of the value of the rotation parameter a. For the extremal black hole, the violation was also found when a = 2 3 , 3 4 , 4 5 and 5 6 . The angular momentum's range and spatial region decrease with the increase of the rotation parameter when the bound is violated. The bound is always violated when the particle and black hole rotate in the opposite directions. It is more likely to cause the violation when the particle and black holes rotate in the opposite directions. The spatial regions where the bound is violated for the extremal black hole are relatively larger than those for the non-extremal black hole. In the near-horizon regions, there always exists the violation for the non-extremal black hole when the angular momentum is very large. The violation occurs when the particle and the extremal black hole rotate in the same directions.
The violation for the bound in the Kerr-Newman and Kerr-Newman AdS black holes was studied in [51,52]. For the non-extremal Kerr-Newman black holes, the authors found that the bound is violated when the particle and black holes rotate in the opposite directions. The bound is also violated in the near-horizon region when the angular momentum of the particle is very large. For the extremal Kerr-Newman black holes, there are violations when the particle and black holes rotate in the opposite directions, or the rotation parameter and black holes' charge take different signs. In [52], they found that the negative cosmological constant reduces the chaotic behavior of the particle. In our work, the violation occurs within certain ranges of the angular momentum when the particle and non-extremal (or extremal) black holes rotate in the opposite directions. In the near-horizon regions, the violation occurs when the particle and extremal black hole rotate in the same directions, and doesn't occur when they rotate in the opposite direction.
In this paper, although we got the violation for the bound, this violation may be not contrary to the conjecture in [6]. Because they conjectured the upper bound of the exponent in the general thermal quantum systems with a large number of degrees of freedom. While we investigated the bound by using the motion of a single particle outside the horizon. As elaborated in [31], this result does not necessarily show that the bound conjectured in [6] is violated.
Figure 1 :
1The influence of the angular momentum of the particle around the non-extremal EMDA black hole on the Lyapunov exponent, where b = 1 3 . The bound on the Lyapunov exponent is violated in the range −27.48 < L < 0.67 (the corresponding spatial region is 1.01494866r+ < r0 < 1.60347830r+) when a = 2 7 , violated in the range −35.76 < L < 1.16(1.01322326r+ < r0 < 1.73241588r+) when a = 1 3 , violated in the range −59.63 < L < 1.84(1.01115833r+ < r0 < 1.96475153r+) when a =2 5 , and violated in the range L < 2.90(1.00838728r+ < r0 < 2.52098625r+) when a = 1 2 .
. In the figure, when the rotation parameter and angular momentum are large enough, and the rotation
Figure 2 :
2The influence of the angular momentum of the particle around the charged dilatonic black hole on the Lyapunov exponent. The bound on the Lyapunov exponent is violated in the range 1.22 < L < 19.16(1.02042099r+ < r0 < 1.45644077r+) when b = 2 3 . There is no violation when b = 1 3 and b = 1 6 .
Figure 3 :
3(2.5), we get r + = M 0 and a = ±M 0 . Here we also let M = 1, m = 1 and q = 15. Since the value of √ hf r=r 0 in Eq. (2.15) is always positive, the sign of the value of the Lyapunov exponent depends on that of V ef f (r). We evaluate the violation of the bound by the positive and negative values of V ef f (r). The appearance of the maximum of the effective potential implies that V ef f (r) is less than zero. Using Eq. (2.12), The influence of the angular momentum of the particle around the non-extremal Kerr black hole on the Lyapunov exponent. The bound on the Lyapunov exponent is violated in the range L < −17.40(2.47909864r+ < r0 < 2.52593108r+) when a = 5 6 .
Figure 4 :
4The effective potentials at different positions outside the extreme EMDA black hole are plotted, where L = ±7. The cases that the particle and black hole rotate in the same directions is plotted in the left figure, and that they rotate in opposite directions is plotted in the right figure.
Figure 6 :
6The effective potential of the particle at different positions outside the extreme Kerr black hole, where L = ±7. L and a have the same signs in the left picture and different signs in the right picture.
Figure 7 :
7The influence of the angular momentum of the particle in the near-horizon region of the extremal EMDA black hole on the Lyapunov exponent.
Table 1 :
12.19796 2.02433 1.71543 1.18062 1.15620 1.21716 1.25265 Locations of equilibrium orbits of the particle around the non-extreme EMDA black hole are gotten when b = 1 3 .L
-30
-20
-10
0
10
20
30
r 0
a = 2
7
2.06440 1.92212 1.67266 1.29859 1.38420 1.49575 1.56067
a = 1
3
2.09597 1.94687 1.68472 1.28002 1.34305 1.44363 1.50221
a = 2
5
2.13829 1.97951 1.69912 1.24754 1.27777 1.36285 1.41242
a = 1
2
Table 2 :
2Locations of equilibrium orbits of the particle around the charged dilatonic black hole are gotten for different values of the dilatonic constant.
+ O( 4 ), (3.9)where
AcknowledgmentsThis work is supported by the NSFC (Grant No. 12105031) and Tianfu talent project.
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arxiv |
Sum Rules and Asymptotic Behaviors for Optical Conductivity of Nonequilibrium Many-Electron Systems
22 Aug 2011
Akira Shimizu
Tatsuro Yuge
IIAIR
Tohoku University
Aoba-ku980-8578SendaiMiyagiJapan
Department of Basic Science
University of Tokyo
Meguro-ku153-8902Komaba, TokyoJapan
Sum Rules and Asymptotic Behaviors for Optical Conductivity of Nonequilibrium Many-Electron Systems
22 Aug 2011Typeset with jpsj3.cls <ver.1.1> Lettersum rulenonlinear nonequilibriumoptical propertiespump-probe
For many-electron systems, we consider a nonequilibrium state (NES) that is driven by a pump field(s), which is either an optical field or a longitudinal electric field. For the differential optical conductivity describing the differential response of the NES to a probe optical field, we derive exact sum rules and asymptotic behaviors, which open wide possibilities for experiments. In deriving these results, we have also derived universal properties of general differential response functions of time-dependent NESs of general systems.
Introduction -The optical conductivity tensor σ eq αβ (ω) describes the response of an equilibrium state to a probe optical field. It gives much information on electronic properties of condensed matter. [1][2][3][4][5][6][7][8][9][10][11][12] In particular, it has been shown that the integrals of Re σ eq αβ (ω) and ωIm σ eq αβ (ω) over the frequency ω are directly related to basic properties of the system such as the singleparticle distribution and band dispersion. [1][2][3][4][5][6][7][8][9][10][11][12] Such relations, called sum rules, are therefore useful for exploring electron systems, [1][2][3][4][5][6] and have been successfully utilized for analyzing a large variety of electron systems. [6][7][8][9][10][11] However, since an equilibrium state (of each system) is uniquely determined by a small number of parameters (such as temperature), the number of controllable parameters that affect the sum (integral) values is very small. This fact has severely limited the usage of sum rules.
This limitation can be removed by considering the optical conductivity of a nonequilibrium state (NES). A NES can be created and driven by a pump field A, which is assumed to be an optical field and/or a longitudinal electric field (generated by, say, a battery). The response of the NES to a probe optical field is characterized by the differential optical conductivity tensor σ A αβ [defined by eqs. (1)-(3)]. [11][12][13][14][15] Unlike equilibrium states and σ eq αβ , the NES and σ A αβ depend strongly on the magnitude and functional form of A(t). Therefore, by tuning A(t) as a new controllable parameter, one will be able to make the sum rules for σ A αβ much more informative than those for σ eq αβ . However, the problem was that, until now, the sum rules for σ A αβ were unknown. Note that two different configurations are possible in experiments on σ A αβ : (i) A(t) is turned off before a(t) is applied and (ii) A(t) is present when a(t) is applied. We here call both configurations pump-probe experiments. In configuration (i), the NES (created by A(t) beforehand) might sometimes be approximated as a quasi-equilibrium state (QES), and the sum rules of σ eq αβ are often substituted for those of σ A αβ . 11) However, in general, the tran-sient NES is not well approximated as a QES, and this substitution has not been justified. In configuration (ii), such substitution is obviously wrong because the NES driven by A(t) is far from quasi-equilibrium because, for example, strong mixing phenomena such as frequency mixing take place. Therefore, until now, reliable sum rules for σ A αβ were unknown in either configuration. In this paper, we derive sum rules for σ A αβ (ω) [eqs. (23) and (24)], and its asymptotic behaviors [eqs. (25) and (26)], for a general class of models for many-electron systems. They hold rigorously in both configurations (i) and (ii), even when many-body interactions are strong.
Differential optical conductivity of NESs -Suppose that an optical field, described by a vector potential A(t) (in the Coulomb gauge), and/or a longitudinal electric field, described by a scalar potential φ(r, t), is applied to an electron system. Since A and φ induce optical excitation and electrical conduction, respectively, the system becomes a NES, whose density operator is denoted bŷ ρ A (t). We therefore call A ≡ (A, φ) the pump field. It can be strong such that perturbation expansion in powers of A breaks down. [13][14][15] Furthermore, we do not assume any specific functional form (such as periodicity) for the time dependence of A.
One can study properties of a NES created by A by measuring the response to another optical field a(t), which we call a probe field. It brings the system into another NES,ρ A+a (t). We are interested in the change, induced by a(t), in the current density j,
∆j(t) ≡ ĵ A+a t − ĵ A t ,(1)
where
· A+a t ≡ Tr[ρ A+a (t) · ] and · A t ≡ Tr[ρ A (t) · ]
. When a(t) is weak, ∆j(t) is well described in terms of the differential optical conductivity tensor σ A αβ as
∆j α (t) = β t −∞ σ A αβ (t − t ′ ; t)f β (t ′ ) dt ′ + o(f ).(2)
Here, f (t) = −ȧ(t) is the probe electric field, and α, β = x, y, z. Since the NES varies as a function of time, so does σ A αβ . That is, σ A αβ depends not only on the time delay τ ≡ t − t ′ but also on t. Furthermore, as eq. (15) shows, σ A αβ is generally a nonlinear functional of A [while it is independent of a]. Throughout this paper, the superscript A, such as those in σ A αβ and · A t , denotes such a functional dependence. Equations (1) and (2) and the causality,
σ A αβ (τ ; t) = 0 for τ < 0,(3)
define the differential optical conductivity tensor of the NES driven by A. It contains much more information than that of equilibrium states, σ eq αβ (ω), as we will discuss later.
Experimentally, a(t) is usually taken as monochromatic, and thus f (t) = f e −iωt + c.c. Then, eq. (2) reads
∆j α (t) = β σ A αβ (ω; t)f β e −iωt + c.c. + o(f ), (4) where σ A αβ (ω; t) ≡ ∞ −∞ σ A αβ (τ ; t)e iωτ dτ is the Fourier transform (FT) with respect to the time delay τ . 16) One can measure σ A αβ (ω; t) directly by such experiments us- ing eq. (27). Since σ A αβ (τ ; t) is real, Re σ A αβ (ω; t)
and Im σ A αβ (ω; t) are even and odd functions of ω, respectively. We study sum rules for them. For example, we consider
W A αβ (t) ≡ ∞ −∞ Re σ A αβ (ω; t)dω,(5)
which is called the optical spectral weight. This quantity is of central interest in many theories and experiments. [2][3][4][5][6][7][8][9][10][11][12] Model and definitions -We consider a many-electron system in the presence of electron-electron and electronphonon interactions as well as random potentials. The electrons move on a regular lattice, whose dimensionality and symmetries are arbitrary.
We assume that the system is described, in the energy scale of interest, by the general Hamiltonian;
H 0 =Ĥ e +Ĥ ei +Ĥ ee +Ĥ ep +Ĥ p .(6)
Here,Ĥ e is the kinetic-energy term of electrons;Ĥ e ≡ k,σ ε(k)n kσ , where ε(k) denotes the energy dispersion of the band of interest, andn kσ ≡ĉ † kσĉ kσ . Here, c kσ ≡ l e ik·lĉ lσ / √ N , whereĉ lσ annihilates an electron on site l with spin σ, and N is the number of unit cells. H ei ≡ l,σ u lnlσ is a random potential (with a random on-site energy u l andn lσ ≡ĉ † lσĉ lσ ), which may be produced, for example, by impurities. Furthermore,Ĥ ee is the sum of electron-electron interactions. We assume that H ee is a function ofn lσ 's.Ĥ ep is the electron-phonon interaction, andĤ p denotes the Hamiltonian of free phonons. This general model includes many models such as the Hubbard model (for whichĤ ee = U ln l↑nl↓ , H ei =Ĥ ep =Ĥ p = 0). Our results hold irrespective of the details and magnitudes ofĤ ee ,Ĥ ei andĤ ep .
For later use, we define the velocity vector and inverse mass tensor as
v α (k) ≡ 1 ∂ ∂k α ε(k), m −1 αβ (k) ≡ 1 2 ∂ 2 ∂k α ∂k β ε(k).
To consider interactions with A and a, we assume that the spatial variations of A and a can be neglected. This approximation is good in most experimental configurations. The directions of A, ∇φ and a are arbitrary. Under these conditions, we may incorporate the interactions with A and a by the Peierls substitution, and the interaction with φ by the Coulomb interaction with the charge of electrons. Then, the Hamiltonian in the presence of A, φ and a is given bŷ
H A+a = k,σ ε(k − (e/ )A(t) − (e/ )a(t))n kσ +Ĥ ei +e l σn lσ − n bg l φ(l, t) +Ĥ ee +Ĥ ep +Ĥ p .(7)
Here, e is the electron charge, and −en bg l is a background charge on site l. By differentiatingĤ A+a with A + a, we obtain the current density aŝ
j α = e V k,σ v α (k − (e/ )A(t) − (e/ )a(t))n kσ (8) =ĵ v α +ĵ m α + o(a),(9)
wherê
j v α ≡ e V k,σ v α (k − (e/ )A(t))n kσ ,(10)j m α ≡ − e 2 V k,σ,β m −1 αβ (k − (e/ )A(t))n kσ a β (t).(11)
When A = 0,ĵ m α represents the diamagnetic current induced by a. [2][3][4][5][6] When A = 0, the diamagnetic current is induced by both A and a, and thus is included in botĥ j v α andĵ m α . Sinceĵ m α is O(a), ∆j(t) defined by eq. (1) is given by
∆j(t) = ∆j v (t) + j m (t) + o(a).(12)
Here
, ∆j v α (t) ≡ ĵ v α A+a t − ĵ v α A t and j m α (t) ≡ − β d A αβ (t)a β (t), where d A αβ (t) ≡ e 2 V k,σ m −1 αβ (k − (e/ )A(t)) n kσ A t .(13)
For a simple cubic lattice, for example, αβ d A αβ (t) is proportional to the expectation value of the kinetic energy.
While j m (t) responds to a(t) instantaneously, ∆j v (t) responds with a finite delay as
∆j v α (t) = β t −∞ Φ A αβ (t − t ′ ; t)a β (t ′ ) dt ′ + o(a).(14)
Here, Φ A αβ (τ ; t) is the response function describing the differential response of ∆j v (t) to a(t). We denote its FT with respect to the time delay τ by Ξ A αβ (ω; t). Since f (t) = −ȧ(t), eqs. (2) and (12)-(14) yield the differential optical conductivity tensor as
σ A αβ (ω; t) = −i ω + i0 Ξ A αβ (ω; t) − d A αβ (t) .(15)
Both Ξ A αβ and d A αβ are nonlinear functionals of A, and so is σ A αβ . Universal properties of response functions of timedependent NESs -To derive sum rules for σ A αβ , we note that Ξ A αβ in eq. (15) should satisfy all the universal properties that were found in ref. 13 for general response functions of general systems. Since ref. 13 assumed steady NESs driven by a static pump field, we here generalize its theory to time-dependent NESs, which are realized, for example, by the application of a time-dependent pump field. For this general discussion, we omit vector and tensor indices.
We denote the pump and probe fields by A(t) and a(t), respectively. In nonequilibrium statistical mechanics (e.g., in the Kubo formula 1) and in refs. [13][14][15], it is usually assumed (implicitly) that an observable of interest is independent of a(t). However, we here consider the general case where an observable of interest, denoted bŷ Q a(t) , is a function of a(t), because this is the case forĵ α given by eq. (8). Then, by expandingQ a(t) in powers of a(t), we obtainQ
a(t) =Q +Q 1 a(t) + o(a),(16)
whereQ andQ 1 are operators independent of a(t). We have obtained such an expansion in eq. (9), whereQ =
j v α andQ 1 a(t) =ĵ m α . The response to a(t), ∆Q a(t) ≡ Q a(t) A+a t − Q a(t) A t , is therefore given by ∆Q a(t) = ∆Q(t) + Q 1 A t a(t) + o(a),(17)
where ∆Q(t) ≡ Q A+a t − Q A t . Since the response function of the second term on the right-hand side is simply given by Q 1 A t , let us consider the non-trivial term ∆Q(t). Unlike Q 1 A t , ∆Q(t) depends onρ A+a (t) (the NES in the presence of both A and a). We therefore have to use the theory of ref. 13 to evaluate ∆Q(t).
When a(t) is sufficiently weak, ∆Q(t) responds to a(t) linearly as
∆Q(t) = t −∞ Φ A (t − t ′ ; t)a(t ′ ) dt ′ + o(a).(18)
This and the causality condition, Φ A (τ ; t) = 0 for τ < 0, define the differential response function Φ A (τ ; t) of the NES. Its FT with respect to the time delay τ is denoted by Ξ A (ω; t). It is straightforward to generalize the theory of ref. 13 to the case where A and the NES are timedependent. We then obtain the following results.
The dispersion relations, such as
Re Ξ A (ω; t) = ∞ −∞ P ω ′ − ω Im Ξ A (ω ′ ; t) dω ′ π ,(19)
are satisfied. Furthermore, the sum rules
∞ −∞ Re Ξ A (ω; t) dω π = Ĉ A t ,(20)∞ −∞ ω Im Ξ A (ω; t) − Ĉ A t dω π = D A t (21) hold. Here,Ĉ ≡ [R,Q]/i andD ≡ −[Q, [R,Ĥ A + H ′ ]]/ 2
, whereR denotes the operator that couples to a(t) via the interaction term −Ra(t),Ĥ A is the Hamil-tonian of the target system in the presence of A [such as eq. (7) with a = 0], andĤ ′ is the interaction between the target system and other systems such as heat reservoirs and electric leads. 13) In general, these operators (such asQ andR) are additive operators or their densities. 13,15) Equation (21) also gives the asymptotic behavior for large ω as
ω Im Ξ A (ω; t) → Ĉ A t .(22)
In deriving these results following ref. 13, we have used the von Neumann equation for the density operator of a huge system, which includes not only the target system of interest but also environments and a source of the pump field, as well as all interactions among them. [Although such a huge system is analyzed, we have successfully derived, as in ref. 13, the relations among quantities of only the target system.] Therefore, these results are rigorous and apply to all physical systems, as long as the linear relation given by eq. (18) holds. 13,14) Main results -Let us apply the above results to σ A αβ of the system described by eq. (7). By expandingĤ A+a in powers of a(t), we find thatR = Vĵ v β for a β (t For ω Im σ A αβ , on the other hand, eqs. (15) and (20) yield the following sum rule:
∞ −∞ ω Im σ A αβ (ω; t) − d A αβ (t) dω = 0.(24)
This and eq. (22), respectively, give the asymptotic behaviors for large ω as
ω Im σ A αβ (ω; t) → d A αβ (t),(25)ω 2 Re σ A αβ (ω; t) → 0.(26)
Equations (23)-(26) are our main results. They are rigorous (to the same degree as the Kubo formula is) within the general model defined by eq. (7), even when A(t), φ(t),Ĥ ee ,Ĥ ep andĤ ei are strong. For example, our results hold for any possible phases of the system that is described by eq. (7). That is, our results are completely valid as long as the target system is well described by the Hamiltonian of eq. (7). Conversely, if experimental results disagree with our results, it means that the system is not described by eq. (7) (because, say, transition to another band takes place). Such rigor seems important for the application of the sum rules and asymptotic behaviors.
Note that the effects ofĤ ee ,Ĥ ep ,Ĥ ei and φ on the sum and asymptotic values appear only through the distribution function n kσ A t . In contrast, the effects of A on
). For Ξ A αβ , which is the FT of Φ A αβ of eq. (14),Q =ĵ v α . The sum rules for σ A αβ are obtained from the properties of Ξ A αβ through eq. (15). For the optical spectral weight [defined by eq. (5)], eq. (19) for ω = 0 yields 17) Note that this result relies only on eqs. (15) and (19). That is, this sum rule is derived only from the causality [eq. (3)] and the specific form of the current [eq. (9)]: No other relations are necessary for deriving this sum rule.W A
αβ (t) = πd A
αβ (t).
(23)
A. Shimizu and T. Yuge the sum and asymptotic values appear not only through n kσ A t but also through m −1 αβ (k−(e/ )A). In either case, the decoherence of electrons affects the sum and asymptotic values only through the broadening of n kσ A t . Possible applications -For σ eq αβ , the sum rule for W eq αβ ≡ ∞ −∞ Re σ eq αβ (ω)dω reads W eq αβ /π = d eq αβ ≡ (e 2 /V ) k,σ m −1 αβ (k) n kσ eq . 1-6) For each system, d eq αβ depends only on the temperature T and doping density n d . In pump-probe experiments, in contrast, W A αβ /π = d A αβ (t) can be studied as a function of T, n d and A. This opens wide possibilities for studying many-electron systems. For example, suppose that an ordered phase is realized as an equilibrium state. By measuring σ eq αβ , one obtains the value of d eq αβ for the ordered phase. Then, a static A = (0, φ) is applied to induce a DC electric current while keeping T equal to that for A = 0 (by, for example, using a good heat sink). By measuring σ (0,φ) αβ by applying a(t), one now obtains, from eq. (23) or (25), the value of dfor a non-ordered phase, because the order would be destroyed by the electric current if |∇φ| was larger than a certain value. One thus obtains the values of d αβ with and without the order at the same T and n d . Alternatively, suppose that no order is present in an equilibrium state. Then, a coherent optical field A = (A(t), 0) is applied. This would induce an electron-hole (eh) correlation. Hence, by measuring σ A αβ , one obtains the value of d A αβ for the state with the eh correlation.Method of measuring σ A αβ (ω; t) -σ A αβ (ω; t) can be measured, for example, by the following process.Step 1: Prepare the system in some initial state at an initial time t = 0. Apply a pump field A(t) only, and measure the current density j(t) continuously for a sufficiently long time. Then, turn off A(t), and at another initial time prepare the system in the same initial state as that at t = 0. Redefine the origin of time (t = 0) as this new initial time. Apply the same pump field A(t) again, and measure the current density j(t) continuously. By repeating these procedures sufficiently many times, one obtains many independent records of j(t). The average of these records gives ĵ A t .Step 2: Perform the same sequence of experiments using the pump and probe fields instead of the pump field. Here, the pump field A(t) is taken to be the same as that of Step 1. One then obtains ĵ A+a t . From this and the result of Step 1, one obtains ∆j(t) = ĵ A+a t − ĵ A t . If one takes the probe field as a monochromatic one, f (t) = f e −iωt + c.c., and if one takes f parallel to the β-axis (i.e., f α = f δ αβ ), then eq. (4) yields ∆j α (t) = σ A αβ (ω; t)f e −iωt + c.c. + o(f ).Step 3: Perform the same sequence of experiments using the same pump field and another (phase shifted) probe field f ′ (t) = f ′ e −iωt + c.c., where f ′ = if . One then obtains ĵ A+a ′ t . From this and the result of Step 1, one obtains ∆jFrom these experimental results, one can evaluateConcluding remarks -Our results hold in both configurations (i) and (ii), which were discussed in the introduction. In configuration (i), eq. (23) readsComparing this with the corresponding result for σ eq αβ , 1-6) W eq αβ = (πe 2 /V ) k,σ m −1 αβ (k) n kσ eq , we find that the result for W A αβ (t) is obtained simply by replacing the equilibrium electron distribution n kσ eq with the nonequilibrium one n kσ A t . Hence, the analysis of the pumpprobe experiments in ref. 11, which substituted the sum rule of W eq αβ for that of W A αβ (t), is now justified. In configuration (ii), on the other hand, eq. (23) readsSince the pump field enters the inverse mass tensor, the simple replacement of n kσ eq with n kσ A t in the sum rule of W eq αβ does not yield the correct result. Finally, we point out that the present results can be generalized. Suppose that the current density takes a general form;Here,Ĵ A α andD A αβ are arbitrary vector and tensor operators, respectively, which may be functions of A. [Equation (9) takes this form.] Then the sum rules eqs. (23) and (24) are respectively generalized asFurthermore, generalizations to the case where the probe field is a longitudinal AC electric field and to higher-order responses [following ref. 15]are straightforward. We thank T. Oka and N. Tsuji for directing our attention to this problem and for helpful discussions. This work was supported by KAKENHI Nos. 22540407 and 23104707, and by a Grant-in-Aid for the GCOE Program "Weaving Science Web beyond Particle-Matter Hierarchy".
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When A(t) is monochromatic, one might be interested in σ A αβ (ω; ν), which is the FT of σ A αβ (ω; t) with respect to t. When A(t) is monochromatic, one might be interested in σ A αβ (ω; ν), which is the FT of σ A αβ (ω; t) with respect to t.
If A(t) is sufficiently weak, σ A αβ (ω; ν) has a peak at ν = Ω, where Ω is the frequency of A(t). As A(t) becomes stronger, σ A αβ (ω; ν) has more peaks as a function of ν, until. it has a broad spectrum in addition to multiple peaksIf A(t) is sufficiently weak, σ A αβ (ω; ν) has a peak at ν = Ω, where Ω is the frequency of A(t). As A(t) becomes stronger, σ A αβ (ω; ν) has more peaks as a function of ν, until, for a strong A(t), it has a broad spectrum in addition to multiple peaks.
A similar result has recently been obtained independently by. N. Tsuji et al.private communicationA similar result has recently been obtained independently by N. Tsuji et al., private communication.
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arxiv |
High-order density-matrix perturbation theory
24 Jul 2003
Michele Lazzeri
Laboratoire de Minéralogie Cristallographie de Paris
ParisFrance
Francesco Mauri
Laboratoire de Minéralogie Cristallographie de Paris
ParisFrance
High-order density-matrix perturbation theory
24 Jul 2003(Dated: March 22, 2022)
We present a simple formalism for the calculation of the derivatives of the electronic density matrix ρ at any order, within density functional theory. Our approach, contrary to previous ones, is not based on the perturbative expansion of the Kohn-Sham wavefunctions. It has the following advantages: (i) it allows a simple derivation for the expression for the high order derivatives of ρ; (ii) in extended insulators, the treatment of uniform-electric-field perturbations and of the polarization derivatives is straightforward. PACS numbers: 71.15.-m,71.15.Mb
I. INTRODUCTION
Linear response methods, 1,2 within the density functional theory approach (DFT), 3 have been successfully applied to compute a wide range of properties in real materials such as phonon dispersions, dielectric constants, effective charges, 1,2 and NMR spectra. 4 Beyond linear response, perturbation theory applied to the Kohn-Sham (KS) orbitals allows the calculation of the derivatives of the energy at any order. 5 This kind of approach has two disadvantages: (i) although the final result is gauge invariant, i.e. invariant with respect to an arbitrary unitary rotation in the space of the occupied KS-orbitals, 5 the formulation of the theory depends on the chosen gauge. This becomes apparent in the application of the KS-orbitals orthonormality constraints at high order. (ii) In the case of periodic systems, the treatment of a perturbation due to a uniform electric field is not trivial, because the position operator, necessary to describe such a perturbation, is ill-defined in periodic boundary conditions. Much effort has been devoted throughout the years to overcome this last problem. Early treatments of the electric field perturbation for the calculation of the second and third order susceptibilities are particularly complex. 6 A simpler formalism for the calculation of the second order susceptibility was obtained in Ref. 7, taking advantage of the 2n + 1 theorem and of a Wannier representation of the orbitals. Only very recently Nuñes and Gonze 8 were able to give an expression for the derivatives of the DFT energy, with respect to uniform electric fields, at any order, by introducing in the Hamiltonian an additional term depending on the polarization Berry phase. 9 We remark that, although a perturbation due to a macroscopic uniform electric field is ill-defined on an individual Bloch states, such a perturbation is well defined on individual Wannier states, 7,10 which can be obtained by a different choice of gauge. This consideration suggest that the two problems, mentioned in the previous paragraph, are related, and that both problems might possibly disappear using a perturbative approach which is not based on the perturbative series of the single KS orbitals, but is solely based on the properties of the electronic density matrix ρ, which is a gauge independent operator.
In a recent paper 11 we gave an expression for the second order derivative of ρ which allowed the efficient computation of Raman spectra. 11 In the present paper we derive a general expression for the n-th order derivative of ρ, using the two relations ρ 2 = ρ, and [ρ, H] = 0, being [, ] a commutator and H the KS Hamiltonian.
To fix our notation we define the electronic density matrix as
ρ = v |ψ v ψ v |,
where, throughout the paper, v or v ′ is an index running on the occupied valence states, and |ψ v are normalized KS-eigenstates, i.e. H|ψ v = ǫ v |ψ v . Given a perturbation associated with a small parameter λ, for a generic quantity F , we consider the perturbation series:
F (λ) = F (0) + λF (1) + λ 2 F (2) + λ 3 F (3) + . . . .(1)
The generalization to the case of different perturbations λ 1 , . . . λ n is straightforward. ρ and H stand for ρ(λ) and H(λ). We call P V , and P C , respectively, the projectors on valence and conduction band states, i.e. P V = ρ (0) , P C = 1 − ρ (0) . Given an Hermitean operator A we define
A CC = P C AP C , A V V = P V AP V , A CV = P C AP V , and A V C = (A CV ) † .
The work is organized as follows. In Sec. II, we use the relation ρ 2 = ρ to express ρ (n) as a function of the operators ρ CV that can be easily computed using standard linear response techniques. In Sec. IV, we show that, within our formalism, the perturbations due to a uniform electric field are well defined in extended insulators. In Sec. V, we derive a simple expression for the derivatives of the polarization.
II. ρ (n) AS A FUNCTION OF {ρ (i) CV }, WITH i ≤ n
We decompose ρ in ρ CC + ρ V V + ρ CV + ρ V C , and we consider these four terms separately. The idempotency condition, ρ 2 = ρ, implies that P C ρP C = P C ρρP C = P C ρ(P C + P V )ρP C , or
ρ CC − ρ CC ρ CC = ρ CV ρ V C .
When all the eigenvalues of ρ CC are lower than 1/2, i.e. for λ sufficiently small, this relation between the two operators ρ CC and ρ CV ρ V C can be inverted to obtain:
ρ CC = 1 − √ 1 − 4ρ CV ρ V C 2 ,(2)
where the right-hand side denotes the operator obtained substituting ρ CV ρ V C in the Taylor series
1 − √ 1 − 4x 2 = x + x 2 + 2x 3 + 5x 4 + . . . .(3)
In a similar way, defining
∆ρ V V = ρ V V − ρ (0) , ∆ρ V V ∆ρ V V + ∆ρ V V = ρ V V ρ V V − ρ V V = −ρ V C ρ CV .
When all the eigenvalues of ∆ρ V V are larger than −1/2, i.e. for λ sufficiently small, ∆ρ V V can be expressed as a function of ρ V C ρ CV :
∆ρ V V = ρ V V − ρ (0) = − 1 − √ 1 − 4ρ V C ρ CV 2 .(4)
Finally, ρ (n) can be expressed as a function of the {ρ
(i) CV }, with i ≤ n, using the relation ρ (n) = ρ (n) CV + ρ (n) V C + ρ (n) CC + ρ (n) V V ,(5)
and taking the n-th order variation of Eq. (2) and Eq. (4) through Eq. (3). As examples, observing that ρ
(0) CV = 0, is easy to show that ρ (1) = ρ (1) CV + ρ (1) V C (6) ρ (2) = ρ (2) CV + ρ (2) V C + [ρ (1) CV , ρ (1) V C ](7)ρ (3) = ρ (3) CV + ρ (3) V C + [ρ (1) CV , ρ (2) V C ] + [ρ (2) CV , ρ (1) V C ](8)ρ (4) = ρ (4) CV + ρ (4) V C + [ρ (1) CV , ρ (3) V C ] + [ρ (2) CV , ρ(2)V C ] + +[ρ (3) CV , ρ (1) V C ] + [ρ (1) CV ρ (1) V C ρ (1) CV , ρ (1) V C ].(9)
Note that each ρ
(i)
CV is a gauge independent operator. In Eqs.
(6)-(9), ρ (n) is expressed as ρ (n) CV + ρ (n)
V C plus a commutator, for n ≤ 4. This property is used in Sec. V to compute the derivatives of the polarization. It holds at any order n. Indeed, as we show in the appendix:
ρ (n) = ρ (n) CV + ρ (n) V C + n−1 i=1 [ρ (i) CV , O (n−i) V C ] = = ρ (n) CV + ρ (n) V C + n−1 i=1 [O (n−i) CV , ρ (i) V C ],(10)
where n ≥ 2, and
O V C = ρ V C 1 − √ 1 − 4ρ CV ρ V C 2ρ CV ρ V C = = 1 − √ 1 − 4ρ V C ρ CV 2ρ V C ρ CV ρ CV .(11)III. COMPUTATION OF ρ (n) CV
In order to compute ρ (n)
CV we introduce the wavefunction |η
(n) v = P C ρ (n) |ψ (0) v , |ψ (0) v
being an unperturbed KS eigenvector. We have:
ρ (n) CV = v P C ρ (n) |ψ (0) v ψ (0) v | = v |η (n) v ψ (0) v |.(12)
Equating to zero the n-th order term of the perturbation series of [H, ρ] = 0, we find:
n i=0 [H (i) , ρ (n−i) ] = 0.
Multiplying this relation on the left by P C and applying to |ψ
(0) v
to the right, we derive:
H (0) − ǫ (0) v |η (n) v = − n i=1 P C [H (i) , ρ (n−i) ]|ψ (0) v .(13)
Solving the linear system of Eq. (13) one can obtain |η (n) v and, thus, ρ (n) . Since the right-hand side of Eq. (13) depends on H (n) , that in turn depends on ρ (n) , the system is to be solved self-consistently, e.g. by using an iterative procedure. Eq. (13) needs to be solved only for a finite number of |η (n) v functions, running the index v on the sole valence states. The linear system of Eq. (13) is analogous to the one that is to be solved in the standard density functional perturbation theory (DFPT), 1,2 thus Eqs. (10,12,13) give an efficient algorithm that can be easily implemented in available DFPT codes (as the PWSCF 12 or the ABINIT 13 code), to compute the derivatives of ρ at any order.
Alternatively, Eq. (13) can be written as
|η (n) v =G v n i=1 [H (i) , ρ (n−i) ] |ψ (0) v ,(14)
whereG
v = c |ψ (0) c ψ (0) c |/(ǫ (0) v − ǫ (0)
c ) is the unperturbed Green function operator projected on the conduction band, and the sum c is restricted to the empty conduction-band states. From Eq. (14) one can recognize that |η wavefunctions, for the three lowest order:
(1) v = P C |ψρ (1) = v |η (1) v ψ (0) v | + |ψ (0) v η (1) v | (15) ρ (2) = v |η (2) v ψ (0) v | + |ψ (0) v η (2) v | + |η (1) v η (1) v | + − v,v ′ |ψ (0) v η (1) v |η (1) v ′ ψ (0) v ′ | (16) ρ (3) = v |η (3) v ψ (0) v | + v |η (2) v η (1) v |+ − v,v ′ |ψ (0) v η (2) v |η (1) v ′ ψ (0) v ′ | + (. . . ) † .(17)
We already used Eq. (16) in Ref. 11, to compute the Raman tensor.
IV. TREATMENT OF THE ELECTRIC FIELDS
Thanks to the commutators in Eq. (13), all the quantities needed to compute ρ (n) are well defined in an extended insulator, even if the perturbation λ is the component E α of a uniform electric field, i.e., if H (1) = −er α + ∂V Hxc /∂E α , 14 being r α the α th Cartesian component of the position operator r, e the electron charge, and V Hxc the Hatree and exchange-correlation potential. In particular, in an insulator, the commutator [r, ρ (n−1) ], which appears in Eq. (13), is a well-defined bounded operator, since the variation of the density matrix is localized ( r ′′ |ρ (n−1) |r ′ goes to zero exponentially for |r ′′ − r ′ | → ∞).
To prove the localized nature of ρ (n−1) in a periodic system, we notice that ρ (n−1) can be written (see Eq. (16)) as a sum of operators of the type
D = kv |α kv β kv |,
where |α kv and |β kv are Bloch wavefunction, i.e. |α kv = e ik·r |α kv / √ N and |β kv = e ik·r |β kv / √ N , being N the number of unit cells, |α kv and |β kv wavefunctions periodic in the lattice, normalized on the unit cell. In an insulator, the operators
D k = v |α kv β kv |
are analytic in k and periodic in the reciprocal space. Cloizeaux has shown in Ref. 16 that an operator having the properties of D is exponentially localized.
The representation of ρ (n−1) in terms of D is also useful to obtain a practical expression for the calculation of the [r, ρ (n−1) ] commutator. In the limit of a converged kpoints grid,
1 N k ∂ ∂k α D k = Ω c d 3 k (2π) 3 ∂ ∂k α D k = 0,
since the integral over its period of the derivative of a periodic analytic function is zero. Ω c is the unit-cell volume. From this it can be easily demonstrated that
[r α , D] = i N kv e ik·r ∂|α kv β kv | ∂k α e −ik·r .(18)
The terms required in Eq. (13), when the perturbation is a uniform electric field, can thus be computed using kv / √ N , and the bra-ket products on the right-hand side are performed on the unit cell. In practical implementation, the derivative with respect to k α in the right-hand side of Eq. (19) can be computed numerically by finite-differentiation, using an expression independent from the arbitrary wavefunction-phase, as in Refs. 8,15.
ψ (0) kc | [r α , D] |ψ (0) kv = i v ′ ψ (0) kc | ∂|α kv ′ β kv ′ | ∂k α |ψ (0) kv ,(19)
V. DERIVATIVES OF THE POLARIZATION
Finally, with the present formalism, the computation of the n-th order variation of the polarization density P (n) , becomes natural. The components of P (n) can be written as:
P (n) α = − 2e N Ω c T r{r α ρ (n) },(20)
where the factor two accounts for the spin degeneracy and T r{A} is the trace of the operator A. We substitute ρ (n) from Eq. (10)
P (n) α = − 2e N Ω c T r [r α , ρ (0) ]ρ (n) V C − [r α , ρ (0) ]ρ (n) CV + 1 2 n−1 i=1 [r α , ρ (i) CV ]O (n−i) V C − [r α , ρ (i) V C ]O (n−i) CV = = 2e N Ω c kvv ′ Im 2 η (n) kv | ∂|ψ (0) kv ′ ψ (0) kv ′ | ∂k α |ψ (0) kv + n−1 i=1 χ (n−i) kv | ∂|η (i) kv ′ ψ (0) kv ′ | ∂k α |ψ (0) kv ,(21)
where n ≥ 2, Im(z) is the imaginary part of the complex number z, and we have written the operators O
(i) V C as O (i) V C = k |ψ (0) kv χ (i) kv |, being |χ (i) kv = O (i) CV |ψ (0)
kv .
VI. CONCLUSIONS
Concluding, we presented a formalism for the calculation of the derivatives of the electronic density matrix at any order, within the density functional theory approach. Beside being simple, this formalism allows the treatment of extended systems in the presence of an external uniform electric fields in a natural way, without introducing in the Hamiltonian an additional term depending on the polarization Berry-phase.
APPENDIX
The operators defined in Eq. (11) are well defined for λ sufficiently small, since the series
1 − √ 1 − 4x 2x = 1 + x + 2x 2 + 5x 3 + . . .CC = n−1 i=1 ρ (i) CV O (n−i) V C = n−1 i=1 O (n−i) CV ρ (i) V C ρ (n) V V = − n−1 i=1 O (n−i) V C ρ (i) CV = − n−1 i=1 ρ (i) V C O (n−i) CV .
Eq. (10) of the text easily follows.
Writing O (n)
V C as a function of ρ (i) CV , at the lowest orders we have:
O (1) V C = ρ (1) V C O (2) V C = ρ (2) V C O (3) V C = ρ (3) V C + ρ (1) V C ρ (1) CV ρ (1) V C O (4) V C = ρ (4) V C + * i,j,k ρ (i) V C ρ (j) CV ρ (k) V C δ i+j+k,4 + O (5) V C = ρ (5) V C + * i,j,k ρ (i) V C ρ (j) CV ρ (k) V C δ i+j+k,5 + 2ρ (1) V C ρ (1) CV ρ (1) V C ρ (1) CV ρ (1) V C ,
where * is a sum on positive integers. These equations allows to compute ρ (n) , with n ≤ 6.
CV , having i ≤ n. In Sec. III, we use the relation [H, ρ] = 0 to obtain an expression for ρ (n)
projection on the conduction band of the KS orbital v in the paralleltransport gauge of Ref. 5. However, at higher orders, there is not such a simple relation between the |η (i) v functions, defined in the present paper, and the variations of the KS-orbitals. Finally, as examples, we write ρ (i) as a function of the |η (i) v
are the periodic part of the Bloch wavefunctions, i.e. |ψ (0) kv = e ik·r |ψ (0)
has the same convergence properties of Eq. (3). From Eq. (11) of the text,ρ CC = ρ CV O V C = O CV ρ V C and ρ V V − ρ (0) = −O V C ρ CV = −ρ V C O CV .For n ≥ 2 the n-th order variation of ρ CC and ρ
into Eq. (20). Using the trace property T r{[A, B]C} = T r{A[B, C]}, is easy to arrive at:
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The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors. The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors (URL http://www.abinit.org).
If one is interested in a perturbation with respect to a macroscopic uniform electric field, one can set to zero the uniform component of the Hartree potential in ∂V Hxc /∂Eα and in its higher-order derivatives. If one is interested in a perturbation with respect to a macroscopic uniform electric field, one can set to zero the uniform component of the Hartree potential in ∂V Hxc /∂Eα and in its higher-order derivatives. 1
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| {'fraction_non_alphanumeric': 0.10236679491663751, 'fraction_numerical': 0.04103999067272939, 'mean_word_length': 3.147727272727273, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We present a simple formalism for the calculation of the derivatives of the electronic density matrix ρ at any order, within density functional theory. Our approach, contrary to previous ones, is not based on the perturbative expansion of the Kohn-Sham wavefunctions. It has the following advantages: (i) it allows a simple derivation for the expression for the high order derivatives of ρ; (ii) in extended insulators, the treatment of uniform-electric-field perturbations and of the polarization derivatives is straightforward. PACS numbers: 71.15.-m,71.15.Mb', 'arxivid': 'cond-mat/0307603', 'author': ['Michele Lazzeri \nLaboratoire de Minéralogie Cristallographie de Paris\nParisFrance\n', 'Francesco Mauri \nLaboratoire de Minéralogie Cristallographie de Paris\nParisFrance\n'], 'authoraffiliation': ['Laboratoire de Minéralogie Cristallographie de Paris\nParisFrance', 'Laboratoire de Minéralogie Cristallographie de Paris\nParisFrance'], 'corpusid': 118994561, 'doi': '10.1103/physrevb.68.161101', 'github_urls': [], 'n_tokens_mistral': 6656, 'n_tokens_neox': 5978, 'n_words': 3404, 'pdfsha': '92cc43a7b5d2512810070f0731b483fd286bb2ab', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0307603v1.pdf'], 'title': ['High-order density-matrix perturbation theory', 'High-order density-matrix perturbation theory'], 'venue': []} |
arxiv |
Occlusion-Robust FAU Recognition by Mining Latent Space of Masked Autoencoders
Minyang Jiang
Department of Electrical and Computer Engineering
University of British Columbia
VancouverBCCanada
Yongwei Wang
Joint NTU-WeBank Research Center on Fintech
Nanyang Technological University
Singapore
Martin J Mckeown
Department of Medicine
University of British Columbia
VancouverBCCanada
Z Jane Wang
Department of Electrical and Computer Engineering
University of British Columbia
VancouverBCCanada
Occlusion-Robust FAU Recognition by Mining Latent Space of Masked Autoencoders
Occlusion-robust FAU recognitionmasked autoencodersknowledge distillation
Facial action units (FAUs) are critical for fine-grained facial expression analysis. Although FAU detection has been actively studied using ideally high quality images, it was not thoroughly studied under heavily occluded conditions. In this paper, we propose the first occlusion-robust FAU recognition method to maintain FAU detection performance under heavy occlusions. Our novel approach takes advantage of rich information from the latent space of masked autoencoder (MAE) and transforms it into FAU features. Bypassing the occlusion reconstruction step, our model efficiently extracts FAU features of occluded faces by mining the latent space of a pretrained masked autoencoder. Both node and edge-level knowledge distillation are also employed to guide our model to find a mapping between latent space vectors and FAU features. Facial occlusion conditions, including random small patches and large blocks, are thoroughly studied. Experimental results on BP4D and DISFA datasets show that our method can achieve state-of-the-art performances under the studied facial occlusion, significantly outperforming existing baseline methods. In particular, even under heavy occlusion, the proposed method can achieve comparable performance as state-of-the-art methods under normal conditions. did part of the work at UBC, and he has moved to NTU.
Introduction
The Facial Action Coding System (FACS) [1] is a comprehensive system that breaks down facial expressions into individual components of muscle movement, which are called Action Units (AUs). It is widely adopted to describe fine-grained facial behaviors. Automatic action unit detection enables efficient facial analysis and can be used in a wide range of applications including security, clinic, entertainment, and education [2].
With the recent advancement of deep neural networks (DNNs) and high-quality image datasets, the performance of computer vision tasks has been improved tremendously including facial action unit (FAU) detection. Some pioneering studies (e.g., [3,4,5,6,7]) take advantage of DNNs to extract local and global facial appearance features, and they have remarkably improved the accuracy of FAU detection over traditional approaches using hand-crafted features [8,9]. More recently, works [10,11] further improve the detection performance by combining the appearance features with domain knowledge of dependencies between AUs. To further capture these AU dependencies automatically, Luo et al. [12] propose ME-GraphAU, a node and edge feature learning approach that can achieve state-of-the-art performance in FAU recognition. Despite the promising performance of these methods, they all rely on high-quality images and videos gathered from well-controlled lab environments with full facial region and action units exposed. As a result, directly applying these methods often suffers from significant performance degradations in the presence of occlusion, particularly for heavily occluded face images. Indeed, even for ME-GraphAU [12], the state-of-the-art approach, our preliminary studies show that its F1-score drops sharply from 65.5% to 30.7% by randomly masking 50% of facial regions.
In many real-world scenes, the captured face images can be partially or even heavily occluded. Thus, the inference on occluded facial images has been a long-standing problem in face recognition-related tasks, e.g., occlusion-roust face identification and recognition [13,14,15,16]. However, to our best knowledge, there have been no explorations yet specifically developed for the occluded FAU recognition task. Facial occlusion is often characterized as an intractable problem, thus it presents particular challenges to existing occlusion-unaware methods [16]. First, a large portion of data that contains discriminative appearance features may be missing, which leads to severe performance degradation. Earlier region-based AU prediction models that only infer from local features will fail because of missing information on occluded regions [5]. Some other typical models considering dependencies among AUs by graph models will also encounter the problem of missing node features [10,17,12]. Second, we often do not have prior information about the occluded regions (e.g., position, shape), further increasing the difficulty of producing accurate FAU predictions.
Recent studies in compressed sensing theory reveal an intriguing phenomenon that, image signals contain much redundant information such that missing image regions may be recovered with high probability under proper sampling conditions [18,19,20]. Moreover, different facial AUs often mutually influence each other [12], thus the activation status of one missing AU may be inferred from neighboring AUs. Based on these observations, as the first attempt, we aim to exploit and reconstruct missing AUs from occluded facial images prior to FAU recognition. To show the feasibility of the concept, we explore the adoption of masked autoencoder (MAE) structure [21] for image reconstruction, which achieves state-of-the-art performance in the self-supervised learning regime. In particular, MAE has been demonstrated to well recover an image even with 75% randomly masked missing regions. Despite the effectiveness of MAE in reconstructing natural images, there are two key issues that may hinder the direct employment of MAE for occluded FAU recognition. First, the decoding process of MAE requires the location information of occluded regions as a priori; while usually we don't have such information or need extra efforts to obtain it in a real test scene. Moreover, the decoding network in MAE also causes a large computational overhead.
To address these challenges above, we propose a simple yet effective and efficient framework based on off-the-shelf masked autoencoders. In our preliminary study, we leverage a pretrained MAE to predict occluded missing facial regions and observe surprisingly good overall reconstruction quality. This key observation indicates that the bottleneck layer of a pretrained MAE is capable of capturing essential knowledge of relations between different action units, thus well recovering missing facial action units. We are then motivated to mine discriminative feature information from the latent space of the MAE. Meanwhile, we can bypass the redundant decoding process to be much more efficient. To make the learning process more effective, we propose to perform node and edge knowledge distillation simultaneously to further aid the model find the mapping between the latent space vector of MAE and features needed for FAU recognition. The superior performance is validated through experiments on benchmark datasets under different facial occlusion conditions. The contributions of this paper can be summarized as follows:
1. As the first attempt, we specifically explore the facial action unit recognition task and investigate its feasibility under heavily occluded conditions. 2. We propose a novel and effective reconstruction-based FAU recognition approach by mining the latent space of off-theshelf masked autoencoders. 3. We further improve the efficiency of the occlusioninsensitive model by transferring the latent space feature of the masked autoencoder to FAU features. 4. We perform experiments on two benchmark datasets and demonstrate that our method can achieve comparable performances with occlusion-free images even for 50% heavily occluded facial images.
Related Work
In this part, we will briefly review existing works that are closely related to our proposed approach, including deep learning-based FAU detection models, masked autoencoders, and knowledge distillation techniques.
Facial Action Unit Detection
Early works treat action unit detection as a patch-learning problem where detected landmarks define the region of interest. In JPML [3], joint patch and multi-label learning are introduced where a discriminative subset of patches are used to identify target AUs. The authors further improved their method in DRML [22] by combining deep region and multi-label learning into a unified deep network using a specifically designed region layer to replace conventional convolutional layers. The region layer can capture the local appearance change of different facial regions. In EAC-Net [6], authors design a fixed attention map based on facial landmarks to enhance the AU feature learning in regions of interest (ROI). JAA-Net [11] jointly estimates the location of landmarks and the presence of action units. In this work, the adaptive attention map for each action unit is computed separately using estimated landmarks, yielding precise local features. Work [23] uses the ROI attention module to predict attention maps directly using the supervision from landmarks. SEV-Net [24] combines the embeddings of semantic description of AUs with visual features to generate a cross-modality attention map, assisting the model to learn discriminative features from meaningful regions.
Besides learning better local features, the focus of FAU detection gradually shifts towards AU relationship modeling. DSIN [7] uses a recurrent neural network to perform structure inference on fused local and global features. The authors propose an iterative structure inference process to simulate the fully connected graph which captures the relationship between AUs. AU-GCN [10] proposes a graph convolutional network-based framework for modeling AU relationships. Individual AU features are fed into a GCN as nodes, and a fixed connection matrix is constructed based on statistical results on each training set. In the work of UGN [17], a probabilistic mask is used on graph edges to simultaneously capture dependencies and underlying uncertain information among AUs. The uncertainties are also used to select hard samples to improve training efficiency. Unlike the previous GCN-based methods where the adjacency matrix only represents the connectivity between nodes, ME-GraphAU [12] employs edge feature learning where a pair of multi-dimensional edge features are learned between each pair of AUs. The combination of node and edge features captures both the activation status of each AU and the association between them. This method extracts reliable task-specific relationship cues for AU recognition and achieved state-of-the-art results on two widely used AU datasets. Although learning rich node and edge features, ME-GraphAU still relies on the visibility of the full facial region where heavy occlusion causes significant degeneration in performance. This work improves ME-GraphAU in the presence of heavy occlusions.
Masked Image Modeling
Performing computer vision tasks on masked images is called masked image modeling (MIM). Models can learn meaningful representations by reconstructing masked images, and it is promising to apply MIM for self-supervised pre-training. BEiT [25] is one of the first works to use masked image modeling tasks to pretrain vision transformers where the goal is to recover original visual tokens from a masked image. SimMIM [26] proposes a simple framework to demonstrate that masked image modeling provides the model superior representation-learning performance by experiments on large-size masked patches, simple pixel-wise regression, and lightweight prediction heads. MAE [21] adopts an asymmetric encoder-decoder architecture to produce informative latent representation by training image reconstruction tasks on 75% masked images. MAE can learn models with high capacities that generalize well, not only producing high-quality image reconstruction but also improving the performance of downstream learning tasks. Here we consider heavy occlusion as a masked image modeling task where representational features learned from non-occluded parts should benefit from reconstructing the occlusion portion and provide meaningful information to the FAU detection task.
Knowledge Distillation
In a real-world deployment, it is desirable that an FAU recognition model is lightweight, to be resource-efficient. However, a smaller FAU model is often associated with performance degradation. Therefore, we aim to leverage knowledge distillation (KD) to create a lightweight FAU model while preserving high accuracy.
KD is a popular model compression method where smaller models learn from models with higher knowledge capacity [27,28]. KD is commonly used in multi-class classification tasks. Different KD methods have been proposed, such as logitbased KD [29], feature-based KD [30], self-supervision signals guided KD, etc [31,32]. KD is also commonly used in multilabel classification tasks to simplify large models size or improve performances through distilled knowledge. [33] builds an efficient multi-label image classification model by distilling knowledge from a weakly supervised detection model. CPSD [34] boosts the performance of multi-label image classification through self-distillation. [35] proposes uncertainty distillation to address the problem of hard samples in multi-label image classification. In this work, we intend to perform KD for both nodes and edge features, where we formulate them as multi-label KD and multi-class KD, respectively.
Methodology
In this section, we present the framework of the proposed method as shown in Fig. 1. The overall framework consists of three modules: the MAE reconstruction module, the FAU detection module, and the knowledge distillation module. We elaborate on each component individually as follows.
MAE Reconstruction Module
The MAE reconstruction module is rooted in the idea of the masked autoencoder (MAE), a state-of-the-art masked imaging modeling method where the original image can be reconstructed by observing only partial signals. More specifically, the encoder MAE firstly maps observed patches into a latent space representation. Then, empty latent space vectors representing masked patches are added at corresponding positions in the latent representation and then projected back to the image space through a decoder. Finally, the pixel values of masked patches are reconstructed in the encoding and decoding process. In this module, there are mainly four components: masking, patch encoding, latent space representation decoding, and image reconstruction.
The masking process is to simulate random occlusions on facial images. MAE is a vision transform (ViT) [36] based method that operates on image tokens, where images are divided into non-overlapping patches. To better simulate real-world random occlusions, we consider two types of patch-based masking occlusions: random small-patch masking and large-block masking. The former type of masking is similar to conventional MAE settings where a random subset of small patches is selected. This type of masking strategy is to simulate random and small-patch occlusions on faces (e.g., hair, sunglasses, fingers). The latter masking strategy is used to simulate large-block occlusion regions (e.g., covered by facial masks or palms). In this case, a random large-block region consisting of many patches is chosen to be masked. Such large block occlusion further increases the difficulty of both image reconstructions and FAU detection since only signals from distant patches are available.
The encoder component of the reconstruction module only takes in unmasked patches. The encoder follows standard ViT [36] operations to obtain a latent space representation: applying a linear projection to construct patch embedding, incorporating positional embedding to provide the position information of each image patch, and passing through a series of transformer blocks. Taking advantage of self-generated masks with known positions, the encoder by design can only operate on visible patches saving a large amount of computation and memory.
A decoder component is used to map the latent space representation back to the image space including the reconstruction of previously masked patches. Different from the large encoder that operates only on unmasked patches, the smaller decoder takes both encoded visible patches and learned mask tokens as input. A shared vector representing mask token is filled at each position where the missing patch needs to be predicted. After filling tokens to the set, additional positional embedding is added on all tokens to provide necessary location information, especially for mask tokens. A smaller amount of transformer blocks is used in the decoder to save computation.
As for the image reconstruction step, an MAE directly reconstructs the pixel values of masked patches. MAE uses the pixel-wise L 2 metric to measure the quality of reconstruction on masked image patches, and the loss can be written as,
L recons = 1 N M i∈M x i − x i 2(1)
where M represents the masked pixels set, N M is the number of masked pixels,x and x corresponds to predicted pixel values and original pixel values, respectively. However, the reconstructed image suffers from significant block artifacts due to unconstrained visible patch reconstruction. To reduce the noise caused by such artifacts on downstream FAU tasks, we again use the positional information of masks to combine the visible patches from the original image with reconstructed masked patches and form a better-quality facial image for the next FAU detection module.
FAU Detection Module
This module is a self-contained facial action unit detection where the inputs are regular RGB images and the outputs are the activation probability of each facial action unit. To guarantee a good FAU detection performance, this module is based on the state-of-the-art FAU detection method [12]. The detection module can be further split into AU feature generation and graph learning components.
To generate AU features, the face feature map F ∈ R H×W×C (H, W, and C correspond to the height, width, and channels of the feature map) is extracted by standard computer vision backbones. N different fully connected layers are used on N AUs respectively to selectively extract features that are specific to each AU from the full-face feature map. Global average pooling is used on each AU-specific feature map to generate N feature vectors v i ∈ R C as AU-specific representations.
To better model the relationships between AUs, we adopt a graph neural network approach incorporating both node and edge feature learning. This is a two-stage learning method. Node feature learning is conducted in the first stage where the learning target is to produce node features containing both the AU activation status and associations with each other on each facial display. In this stage, each AU representation is used as graph node features and the similarity between these node features determines the connectivity among nodes. More specifically,
s i, j = Sim(v i , v j ) and a i, j = 1, if s i, j ∈ Top K (s i ),
where a i, j represents the connection between node i and j in adjacency graph A, and K represents the out-degree of each node to their closest neighbors. After the construction of nodes and edges, one GCN layer is used to update the AUs' activation status by fusing information from most related AUs. The updated AU representations can be written as,
V new = σ V + BN(A · g 1 (V) + g 2 (V)) (2) where V {v i ∈ R C } N i=1 , σ(·)
is a non-linear activation function, BN(·) represents the batch normalization function, and g 1 , g 2 denote linear layers with weight and bias. To provide a probabilistic prediction of the activation status of each action unit, a similarity calculation strategy is used here, where cosine similarity is computed between a trainable vector t i ∈ R C and an updated representation vector v new i ∈ R C . The vector t i is trained to be a representation of the active status of the i−th AU. The probability of the i−th AU being activated can be written as,
p i = S c σ(t i ), σ(v new i )(3)
where S c (·) denotes a function that computes the cosine similarity between two vectors. To supervise the learning of node features, a multi-label classification loss is adopted. However, there are two significant label imbalance issues due to the nature of FAU dataset collection process: First, the negative label dominates on each AU; Second, the occurrence frequency of each AU is dramatically different. To address these issues, we adopted the weighted asymmetric loss proposed in ME-GraphAU and added additional degrees of freedom to compensate for the noisiness caused by image reconstruction. We first applied asymmetric probability shifting [37] on the estimated probability p m i ,
p m i = max(p i − m, 0)(4)
where m denotes a margin to discard low-probability negative samples. This strategy helps reject mislabeled negative samples generated in the image reconstruction process. The AU loss now can be written as,
L AU = − 1 N N i=1 w i y i log(p m i ) + (1 − y i )(p m i ) γ log(1 − p m i ) (5) where w i = N(1/r i ) N j=1
(1/r j ) is pre-generated by occurrence rate r i of the i-th AU in the training dataset, y i is the ground truth binary label, and γ is the hyperparameter only applies to negative sample to adjust the contribution from easy negative samples. In this stage, only AU nodes with similar feature representations are connected, which forces the model to extract AU features containing both activation and association information.
The second stage builds on top of the AU features learned in the first stage. In addition to associations encoded in node features, this stage aims for learning edge features that describe fine-grained relationships between AUs through additional supervision. Edge features contain much richer information than binary connectivity in the adjacency matrix. Far away nodes in terms of similarity can still have critical relationship information contributing to the detecting activation of AUs. To acquire meaningful edge features, the model conducts two cross-attention operations,
Cross-Attention(A, B) = softmax AW q (BW k ) T √ d k BW v(6)
where W q , W k , W v are learned weights that apply a linear transformation on the query, key, and value in the attention mechanism, and d k is a scaling factor that is equal to the number of channels in BW k term. Firstly, a cross-attention operation is conducted between each AU-specific feature map and the full-face feature map, acquiring the AU activation status in terms of global face feature representation,
F f ace i = Cross-Attention 1 (F AU i , F f ace )(7)
Then, between each pair of AUs, another cross-attention operation is used to extract features that are related to both AUs,
F rel i, j = Cross-Attention 2 (F f ace i , F f ace j ) (8)
With global average pooling on the above features map describing the relationship between pair of AUs, the edge feature vectors E {e i, j ∈ R C } N i, j=1 is obtained. Thus, we can form a graph G 0 = (V 0 , E 0 ) containing both node and edge features. Multiple layers of GatedGCN [38] are used on the graph to allow information propagation between AUs leading to more accurate AU activation status and richer edge features. The activation probability is generated using the similarity calculation strategy as in the node feature learning in the first stage. To further guide edge feature learning, one additional classification head is added to the final edge features. The classification head classifies 1 of the 4 possible activation status combinations of two AUs that the edge connects to. Categorical cross-entropy loss is used for edge classification, and it can be written as,
L E = 1 |E| N i=1 N j=1 CCE y e i, j , softmax(z i, j )(9)
where CCE(·) is the categorical cross-entropy function, y e i, j ∈ R 4 is a one-hot vector indicating 1 of the 4 co-occurrence patterns of the edge between the i-th and j-th nodes, z i, j ∈ R 4 denote the logits output from the edge classification head. In the second stage, the model focuses on node and edge feature learning with an MAE reconstruction module. The loss in this stage can be written as,
L stage2 = L AU + λL E(10)
where λ is a hyperparameter that adjusts the importance of edge classification results.
Student Module
Using the above two modules, we now have a complete occlusion-robust FAU detection pipeline that can be trained end to end by learning from regular images with generated masks. By using reconstructed images from the MAE reconstruction module as input, the FAU detection module can estimate the activation status of the heavily occluded face. The purpose of reconstructing the masked facial image in RGB space is to provide supervision on creating high-quality FAU-aware face reconstruction, and reconstructed images allow maximum flexibility in terms of FAU detection model selection. However, because of the process of facial image reconstruction and FAU feature extraction, the complexity of the model greatly increases. Besides, during the testing phase, the reconstruction process requires explicit knowledge of occluded positions which is often unknown in practice. Furthermore, the square artifacts in the reconstructed facial images can potentially cause error propagation in the downstream FAU detection tasks. In the next module, we want to maintain the effectiveness of this occlusion-robust FAU detection pipeline (the teacher model) while addressing the problems by introducing feature alignment and knowledge distillation.
Since the latent space representation in MAE is capable of reconstructing masked patches of arbitrary images, it should contain generic information extracted from visible patches. Meanwhile, in the previous pipeline, FAU-related features are then extracted from the reconstructed image. This indicates that the FAU-specific features can be derived directly from the generic MAE latent space features without reconstructing the missing patches. In standard MAE training, masks are generated during the forward propagation process and the position information of masks is used in several places including selecting visible patches before the encoder, adding masked tokens in the corresponding place in the latent space representation, and combining visible patches with reconstructed ones to form better quality images. In the proposed student network, to better simulate random occlusions in a realistic scenario, we intentionally avoid the use of position information of occlusion. In the student network, tokens of all patches are fed into the encoder with additional positional embeddings indicating the location of patches. This produces the latent space representation F mae with the same dimensionality as standard MAE latent space representation after adding mask tokens. To mitigate the gap between MAE latent space representation and the FAU face feature map, we add a simple feature alignment component between these two feature spaces. The proposed feature alignment component uses a downsampling layer and multiple fully connected layers to selectively project MAE latent space representation into features that are significant for FAU detection.
With features projected into the same space as features generated in the FAU detection module, the AU-specific feature generation and graph learning components from the original pipeline can be seamlessly adopted. Nevertheless, there is a large gap in terms of knowledge capacity between the model with reconstruction and the student model directly projecting features from the latent space of MAE. To efficiently transfer the rich knowledge of the well-trained large model to this compact model, we apply knowledge distillation losses on both FAU detection and edge classification targets.
FAU detection is modeled as a multi-label classification problem where the ground truth label indicates the occurrence of certain AU. The output of the teacher model contains probabilistic estimations of the occurrence which provides meaningful likelihood information that binary ground truth labels do not have. To incorporate probability information learned from the teacher model, we minimize the per AU Kullback-Leibler (KL) divergence between the output from the teacher and student model. The AU distillation loss can be written as,
L AU kd = 1 N N i=1 D KL (p s i , p t i ) + D KL (1 − p s i , 1 − p t i )(11)
where D KL (·) denotes the KL divergence function and p s i , p t i are activation probability outputs for the i-th AU from the student and teacher models, respectively.
The edge features are also critical in the detection algorithm, so a 4-class classification problem is set up to guide the model to learn representative association features between AUs. To efficiently transfer knowledge in terms of edge features, we adopt a typical knowledge distillation on the logit layer output of the edge classification head [29], the edge distillation loss can be written as,
L edge kd = T 2 D KL softmax(z s /T ), softmax(z t /T )(12)
where T is the temperature hyperparameter that adjusts the smoothness of probability, and z s , z t are logit output from the edge classification head of student and teacher models, respectively. The overall knowledge distillation loss can be written as,
L kd = L AU kd + βL edge kd(13)
where β are hyperparameters to adjust the relative weight between AU distillation loss and edge distillation loss. Finally, we have our overall training loss for the student network by combining with the AU detection loss, edge classification loss, and knowledge distillation loss,
Loss = L AU + λL E + αL kd(14)
Experiments
In this section, we will empirically demonstrate the effectiveness of the proposed FAU recognition method in the presence of different occlusion conditions. We first describe our experimental setup (e.g., datasets, metrics) and then present comparison results with state-of-the-art methods on two benchmark datasets. Experimental results show that the proposed method significantly outperforms state-of-the-art methods in the presence of heavy occlusions.
Datasets
Our occlusion-robust model is evaluated on two widely-used datasets for AU detection: BP4D [39] and DISFA [40]. Descriptions and sample images for AUs contained in these datasets are shown in Table 1. For both datasets, we evaluated our proposed method using three-fold cross-validation and reported the mean performance values over the folds. For a fair comparison, we adopt the folds split following prior works [11,12].
The BP4D dataset [39] contains images from 41 young adults (18 male and 23 female) of various ethnicity. Each subject is asked to perform 8 tasks corresponding to different target emotions. There are 328 videos collected, including around 140,000 frames with binary occurrence AU labels (present or absent) on 12 AUs (1,2,4,6,7,10,12,14,15,17,23,24). The original resolution of the frames is 1392 × 1040 and each contains exactly one front-facing face in the middle of the frame. To avoid the situation that testing data and training data share images from the same person, training and testing partitions in each fold only contain images from different people.
The DISFA [40] dataset contains video recordings from 27 subjects (15 males and 12 females) when watching video clips. The dataset contains around 130,000 valid face color images with a resolution of 1024 × 768, each of which is with intensity labels on 8 AUs (1,2,4,6,9,12,25,26). Following prior work, AUs with an intensity equal to or greater than 2 are considered present while others are treated as absent. The same training and testing split strategy as in BP4D is applied to this dataset. This dataset has a more significant AU occurrence imbalance problem, where certain AU could have 5 times more occurrence when compared to the ones with lower occurrence.
Due to a lack of real-world annotated FAU occlusion datasets, we simulate the occlusions by masks following ideas from existing face recognition work [15]. Specifically, we generate two types of masks: small block patches with a size of 16 × 16 to simulate randomly placed gadgets (e.g., sunglasses, stains), and a relatively large block to simulate large objects e.g., facial masks or palms. In the first type of mask (Figure 2 (a)), considering that FAU describes very subtle facial muscle movement, we limited the overall masking ratio to around 50% to avoid covering all FAU-related regions. In the second type (Figure 2 (b)), the block region is set to be 30%. It is worth mentioning that our proposed method can be applied to many real-world occlusions by simply adding a pre-processing procedure, i.e., detecting occlusion regions and converting them to binary masks, though obtaining AU annotations for such real-world occluded images remains a challenge currently.
Evaluation Metric
We follow the previous AU detection studies [11,23,12] and use the F1-score as the metric to evaluate the performance of our approach. F1-score is the harmonic mean of the precision P and the recall R, i.e., F 1 = 2 P·R P+R , and it is considered a better metric in the case of imbalanced classes (e.g., here negative classes dominate in AU detection). The F1-score for each individual AU is reported, and the average score of all AUs is also computed for comparison purposes.
Implementation Details
For each face image, we performed face alignment through a series of similarity transformations based on the provided face landmarks from both BP4D and DISFA datasets. The transformation is shape-preserving which has no impact on the activation status of AUs. The face alignment gives 256 × 256 colored images, from which we perform data argumentation including cropping, horizontal flipping, and color jittering, and obtain 224 × 224 images as training inputs. During training, we use AdamW [41] with β 1 = 0.9, β 2 = 0.999 and the weight decay of 5e −4 . For training in the first stage, we choose to use K = 4 nearest neighbors for both BP4D and DISFA datasets. In both the second stage teacher training and knowledge distillation student training, we give the edge classification loss weight λ = 0.01.
In knowledge distillation training, we adopt T = 2 in L edge kd and select the weight α = 1, β = 0.1 for overall loss and L kd respectively. For teacher model training, we train up to 30 epochs in stage 1 with an initial learning rate of 1e −4 and 7.5e −5 on BP4D and DISFA, respectively. We also train 20 epochs for the second stage with an initial learning rate of 1e −6 and 1e −5 on BP4D and DISFA datasets, respectively. For knowledge distillation on the student model, 10 epochs of training are used with an initial learning rate of 1e −5 on both datasets. All the phases of training are done on a single RTX 3090 GPU with a batch size of 48. The initialization of MAE models used in both teacher and student networks are trained on ImageNet [42].
To fine-tune a pretrained MAE on facial datasets, we set the learning rate on MAE model parameters to 1/100 of the learning rate of rest parameters. The backbone used in the FAU detection module is also pretrained on ImageNet.
Experimental Results
In this section, we compare our results with several state-ofthe-art methods on both datasets under a few different settings. Table 2 reports the occurrence detection results of 12 AUs on BP4D dataset in terms of F1-score. The top section of the table contains results under the occlusion-free conditions from different recent baseline methods (i.e. JAA-Net [11], AU-GCN [10], SEV-Net [24] and ME-GraphAU [12]), which are reported to have better performance than representative earlier methods including JPML [3], DRML [22], EAC-Net [6] and DSIN [7] etc. The bottom section contains experimental results under various occlusion conditions. As we can see from the table, even with 30% to 50% occlusions, both the teacher and student models we proposed can achieve the same level of performance as other models under occlusion-free conditions. As mentioned in the previous section, the state-of-the-art models trained on regular high-quality images degenerated significantly when we introduce different occlusion conditions. E.g., for ME-GraphAU, the F1-score drops from 65.5% to 30.73% and 50.81% on 50% and 30% occlusion respectively. Reconstructing occluded images using the ImageNet pre-trained MAE does help in the AU detection performance by filling in the missing information, especially for the high percentage sparse occlusion case where the F1-score is increased from 30.73% to 54.3% for ME-GraphAU. However, this accuracy is still over 10% away from its original performance on occlusion-free images. By contrast, our proposed models are forced to learn rich AU features and reliable AU relationships from only visible areas. And the performance of AU detection under 50% random occlusion is boosted to 63.38% and 62.56% in our reconstruction-based teacher model and the efficient student model respectively. In the case of 30% occlusion, our proposed models again significantly improve the performance by 10% over the ME-GraphAU model, achieving non-occlusion level performance with an F1-score of 62.09% and 61.24% when using the teacher and student models respectively.
In Table 3, we show experimental results on the DISFA dataset using the same 30% block and 50% random occlusion configurations. From the table, our proposed models again improve the performance of occurrence detection on 8 AUs under the occluded conditions by a large margin. Our proposed models can achieve 60.62% and 61.25% under the 50% random occlusion condition, which is even better than most other methods under the occlusion-free condition. For the 50% random occlusion condition, the student model using latent space features achieves a better result than the reconstruction-based teacher model in our experiments. Unlike the BP4D dataset, the data variance of DISFA is much smaller. The reconstruction module could suffer from overfitting the training data by reconstructing similar images again and again because of the small number of unique images from the DISFA dataset. The overfitting phenomenon is further exaggerated on the block masking experiments on DISFA dataset where we found some occluded regions are reconstructed with blocks from other faces from the training dataset. Such an overfitting problem could cause huge noise in supervision and thus limit the ability for efficient learning. As seen from the 30% block masking experimental results, though better than the stateof-art ME-GraphAU, the performance gains of our proposed models are limited.
By comparing the results from the 30% block occlusion condition and the 50% random occlusion condition, we can see that the occlusion condition has a very pronounced effect on the relationship and feature learning. Although the single large block type of occlusion has a smaller coverage, it brings more challenges in AU detection. Normally, under the non-occlusion condition, the activation of each AU is mostly determined by local features with the aid of the relationship between AUs. Under the large block occlusion, all nearby regions could be occluded and the models are forced to use only features from the far-away region and inter-relationship information between occluded and visible regions to do the inference. We can see that, for the BP4D dataset, the state-of-the-art model ME-GraphAU has only a 2% performance gain from reconstruction under the large block occlusion setting, while a 24% gain under the random occlusion setting. Our models also have lower performance gains under the large block occlusion than under the random occlusion, because limited local information can be extracted for reconstruction as well as for AU detection.
Based on the above experiments, we observe that our proposed models can significantly improve the AU detection performance under various heavy occlusions, achieving comparable performances with other state-of-the-art models under occlusion-free conditions. Fig. 3: The trend of F1-score over an increasing percentage of random occlusion (%) on BP4D dataset when using ME-GraphAU [12] and our student model.
Model
Computational Complexity No. of Parameters ME-GraphAU [12] 21.
Robustness Assessment
In this section, we aim to show the robustness of our proposed model against different levels of occlusions. In this study, we choose 1 of the 3 folds from the BP4D dataset. Our model is trained with 50% small patch random occlusions. We reported in Fig. 3 the F1-score comparisons between our model and the SOTA model ME-GraphAU on the testing dataset under various levels of occlusions. As we can see, the performance of our proposed model in terms of F1-score is relatively stable under occlusion rations from 30% to 70%, confirming that our model is robust against different occlusions. In particular, even on face images under 70% occlusion conditions, our model can still achieve an F1-score as high as 55.8%, significantly outperforming the SOTA model ME-GraphAU.
Computation Efficiency Comparison
In addition to performance comparisons, this section compares the computation efficiency between our proposed model and the state-of-the-art FAU recognition model. In Table 4, we show the computational complexity and the number of parameters of ME-GraphAU, our teacher and student models. We can note that, although our reconstruction-based teacher model generally achieves the best performances under occlusion settings on BP4D and DISFA datasets, it also requires a high computational cost (50% more than that of ME-GraphAU) and a larger number of model parameters (100% more than that of ME-GraphAU). While our proposed student model, maintaining high performance under heavy occlusion conditions, has slightly less computational complexity as well as model size when compared with ME-GraphAU.
Conclusion
This work proposed a novel framework for facial action unit recognition under heavy occlusion conditions. Our reconstruction-based model, taking advantage of masked image modeling, is robust against heavy occlusions by learning the rich FAU-related features only from the visible parts of the facial image. The proposed models incorporate graph edge feature learning to further mitigate the influence of occlusion by shifting the focus from local feature learning to AU relationship learning. Further, we improve the efficiency of our model by transferring the latent space features of the masked autoencoder to FAU features by performing both edge-level and node-level knowledge distillation. The results on two commonly used datasets demonstrate that the proposed models under the 50% random occlusion can achieve comparable results with the state-of-the-art method under occlusion-free conditions. Our proposed occlusion-robust facial action unit recognition methods are modular by design and can be easily extended to other similar problems to enhance the robustness under heavy occlusions.
FAU
Fig. 1 :
1Overview of the proposed framework for occlusion-robust FAU recognition. The blue path on top is the reconstruction-based teacher model (Section 3.1). The green path on the bottom is the student network learning through knowledge distillation (Section 3.3). The orange part in the middle represents the GCN structure shared by both the teacher and student models, conducting AU feature and edge relationship learning (Section 3.2).
Fig. 2 :
2Visualization of two typical facial occlusion conditions for FAU detection: (a) 50% small-patch occlusion; (b) 30% large-block occlusion.
Table 2 :
2Comparison of F1-scores (%) for 12 AUs on BP4D dataset. The top section of the table are results from different baseline models on occlusion-free images. The middle and bottom sections contain results from the SOTA method ME-GraphAU[12] and our method on images with different types of occlusions. The best results of each section are highlighted with underline, bold font and double underline, respectively.Method
AU
Avg.
1
2
4
6
9
12
25
26
JAA-Net[11]
43.7
46.2
56.0
41.4
44.7
69.6
88.3
58.4
56.0
AU-GCN[10]
32.3
19.5
55.7
57.0
61.4
62.7
90.9
60.0
55.0
UGN-B[17]
43.3
48.1
63.4
49.5
48.2
72.9
90.8
59.0
60.0
SEV-Net[24]
55.3
53.1
61.5
53.6
38.2
71.6
95.7
41.5
58.8
ME-GraphAU[12]
54.6
47.1
72.9
54.0
55.7
76.7
91.1
53.0
63.1
ME-GraphAU:
50% random mask
25.29 22.91 47.72 27.83 25.44 50.63 69.53 31.30 37.58
ME-GraphAU:
50% random mask
reconstructed
40.82 34.37 61.64 31.85 44.19 73.09 89.84 61.95 54.72
Ours (teacher):
50% random mask
60.58 48.64 60.48 46.60 44.08 73.43 91.14 59.96 60.62
Ours (student):
50% random mask
53.83 54.31 67.02 49.15 41.22 73.19 91.06 60.19 61.25
ME-GraphAU:
30% block mask
35.66 23.47 54.87 26.42 28.13 57.04 67.84 42.66 42.01
ME-GraphAU:
30% block mask
reconstructed
40.03 28.92 55.91 26.03 35.00 66.01 74.48 49.31 47.00
Ours (teacher):
30% random mask
48.43 42.04 61.69 48.13 46.35 69.98 83.41 48.98 56.13
Ours (student):
30% random mask
50.66 43.20 59.64 47.62 38.94 68.52 81.53 44.88 54.37
Table 3 :
3Comparison of F1-scores (in%) for 8 AUs on DISFA dataset. The top section of the table are results from different baseline models on occlusion-free images. The middle and bottom sections contain results from the SOTA method ME-GraphAU[12] and our method on images with different types of occlusions. The best results of each section are highlighted with underline, bold font and double underline, respectively.
Table 4 :
4Comparisons of computational efficiency between different model backbones. Best performances have been highlighted in bold font.
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| {'fraction_non_alphanumeric': 0.0434572903101336, 'fraction_numerical': 0.027701409990514316, 'mean_word_length': 4.689197841397603, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Facial action units (FAUs) are critical for fine-grained facial expression analysis. Although FAU detection has been actively studied using ideally high quality images, it was not thoroughly studied under heavily occluded conditions. In this paper, we propose the first occlusion-robust FAU recognition method to maintain FAU detection performance under heavy occlusions. Our novel approach takes advantage of rich information from the latent space of masked autoencoder (MAE) and transforms it into FAU features. Bypassing the occlusion reconstruction step, our model efficiently extracts FAU features of occluded faces by mining the latent space of a pretrained masked autoencoder. Both node and edge-level knowledge distillation are also employed to guide our model to find a mapping between latent space vectors and FAU features. Facial occlusion conditions, including random small patches and large blocks, are thoroughly studied. Experimental results on BP4D and DISFA datasets show that our method can achieve state-of-the-art performances under the studied facial occlusion, significantly outperforming existing baseline methods. In particular, even under heavy occlusion, the proposed method can achieve comparable performance as state-of-the-art methods under normal conditions. did part of the work at UBC, and he has moved to NTU.', 'arxivid': '2212.04029', 'author': ['Minyang Jiang \nDepartment of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada\n', 'Yongwei Wang \nJoint NTU-WeBank Research Center on Fintech\nNanyang Technological University\nSingapore\n', 'Martin J Mckeown \nDepartment of Medicine\nUniversity of British Columbia\nVancouverBCCanada\n', 'Z Jane Wang \nDepartment of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada', 'Joint NTU-WeBank Research Center on Fintech\nNanyang Technological University\nSingapore', 'Department of Medicine\nUniversity of British Columbia\nVancouverBCCanada', 'Department of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada'], 'corpusid': 254408994, 'doi': '10.2139/ssrn.4342053', 'github_urls': [], 'n_tokens_mistral': 16644, 'n_tokens_neox': 14417, 'n_words': 9398, 'pdfsha': '63d933b505466050130b28fab152aebcbf7417a9', 'pdfurls': ['https://export.arxiv.org/pdf/2212.04029v1.pdf'], 'title': ['Occlusion-Robust FAU Recognition by Mining Latent Space of Masked Autoencoders', 'Occlusion-Robust FAU Recognition by Mining Latent Space of Masked Autoencoders'], 'venue': []} |
arxiv |
Lorentz breaking terms from Einstein gravity
12 Nov 2016
Takayuki Hirayama
College of Engineering
Chubu University
1200, 487-8501Matsumoto, KasugaiAichiJapan
Lorentz breaking terms from Einstein gravity
12 Nov 2016
We construct an action which is invariant under the foliation preserving diffeomorphism from the Einstein Hilbert action. Starting from the Einstein Hilbert action, we introduce the gauge invariance under the anisotropic rescaling by using the Stückelberg method. We then introduce the gauge field corresponds to the anisotropic rescaling, and the Stückelberg field turns out to be the Nambu-Goldstone boson. The Nambu-Goldstone boson, however, is not completely eaten by the massive gauge field and the Nambu-Goldstone boson can be integrated out from the action. Then the resultant action is a Horava-Lifshitz type action which contains R 3 term.
Introduction
It have been known that the quantum theory of general relativity is non renormalizable [1] and there are many attempts to construct a renormalizable quantum gravity. Many different approaches seem suggesting that at high energy the dimension of space time approach to two in some sense. One proposal by Horava [2] is that if Lorentz symmetry is abandoned and the dispersion relation between the energy ω and momentum k is given ω 2 = k 6 at high energy, the spectral dimension becomes two in four dimensions [3] suggesting that the Horava-Lifshitz gravity theory becomes renormalizable.
Although the diffeomorphism invariance is broken, the Horava-Lifshitz gravity, instead, requires the invariance under the foliation preserving diffeomorphism at high energy, and the theory is characterized by the invariance under the anisotropic space time rescaling with the dynamical critical exponent z,
t → b z t, x i → bx i ,(1.1)
where b is a scaling parameter, and t and x i are time and spacial directions respectively. The action consists of second order time derivative kinetic terms and various higher order (up to 2z order) spacial derivative potential terms. Since the theory does not have the full differmorphism invariance, a scalar mode, in addition to spin two modes, generically appears in the metric [4]. Therefore there are technical difficulties to successfully recover Einstein gravity at low energy. Although the higher order spacial derivative terms are allowed by the symmetry, we can ask how these terms apear. In this paper, we try to discuss the origin of those terms. We will derive some of those terms by integrating out the Nambu-Goldstone mode associated with the breaking of the anisotropic rescaling. To do this, at first, we modify the Einstein Hilbert action such that the action has the foliation preserving difeomorphism and anisotropic rescaling invariance, but has only second order spacial derivatives. After that we show that the Nambu-Goldstone mode can be integrated out in an unitary gauge and the resultant action is a Horava-Lifshitz type action which contains higher order spacial derivative terms.
The Stückelberg method
When an action has a global symmetry, for example U(1) global symmetry, we can gauge the global symmetry by introducing the corresponding gauge fields A µ and replacing the derivatives with the covariant derivatives. The transformation law for the gauge fields is, A µ → A µ − ∂ µ λ for U(1) with the transformation parameter λ. We can also introduce the kinetic term for the gauge field, −(1/4)F µν F µν .
In the same way, we construct a gravity action which is invariant under the anisotropic rescaling in this section. We start with the four dimensional Einstein gravity,
S 0 = d 4 x √ −g R (4) − 2Λ , (2.1)
where the signature of the metric is (−, +, +, +). If the cosmological constant is positive (negative), the dS (AdS) space, R µν = Λg µν , is a solution of Einstein equations. This action is invariant under the diffeomorphism transformation, x ′µ = x µ − ξ µ , (µ = 0, 1, 2, 3),
δ ξ g µν = (∂ µ ξ ρ )g ρν + (∂ ν ξ ρ )g µρ + ξ ρ ∂ ρ g µν = ∇ µ ξ ν + ∇ ν ξ µ . (2.2)
It takes two steps to obtain the action which is invariant under the anisotropic rescaling with the dynamical critical exponent z and the parameter b,
t → b z t, x i → bx i ,(2.3)
where t = x 0 , and i = 1, 2, 3 are the indices for the three spacial directions.
First we introduce the Stückelberg [5] field X in the following way,
g µν = g µν X 2 . (2.4)
We substitute this redefinition in the action, and we obtain 5) which is nothing but the Einstein action with a conformally coupled scalar field X. We then use the ADM decomposition, 8) where N i = γ ij N j . The action becomes
S 0 = d 4 x − g X 2 R (4) + 6( ∂X) 2 − 2X 4 Λ ,(2.g 00 = −( N 2 − N i N i ), g 0i = N i , g ij = γ ij , (2.6) g µν = 1 N 2 −1 N i N j N 2 γ ij − N i N j ,(2.7)ds 2 = − N 2 dt 2 + γ ij (dx i + N i dt)(dx j + N j dt),(2.S 0 = d 4 x √ γ N X 2 ( K ij K ij − K 2 ) + X 2 R (3) − 6X 2 N 2 (∂ t ln X − N i ∂ i ln X) 2 + 6X 2 γ ij (∂ i ln X)(∂ j ln X) − 2X 4 Λ − 2X 2 ∂ t ( √ γ K) + 2 √ γX 2 ∇ i ( KN j ) − 2X 2 √ γ∆ N, (2.9) K ij = 1 2 N (−∂ t γ ij + ∇ i N j + ∇ j N i ), K = K ij γ ij ,(2.10)
where the indices are contracted with γ ij . R (3) and ∇ i , (∆ = ∇ i ∇ i ), are the Ricci scalar and the covariant derivatives constructed from γ ij . The diffeomorphism transformations are then,
δ ξ γ ij = δ (3) γ ij + N i ∂ j ξ t + N j ∂ i ξ t + ξ t ∂ t γ ij , (2.11) δ ξ N i = γ ij ∂ t ξ j + δ (3) N i + N i ∂ t ξ t + ξ t ∂ t N i − ( N 2 − N j N j )∂ i ξ t , (2.12) δ ξ N = δ (3) N + ξ t ∂ t N + N∂ t ξ t − N N i ∂ i ξ t , (2.13) δ ξ X = δ (3) X + ξ t ∂ t X, (2.14) δ (3) γ ij = (∂ i ξ k )γ kj + (∂ j ξ k )γ ik + ξ k ∂ k γ ij = ∇ i ξ j + ∇ j ξ i , (2.15) δ (3) N i = N j ∂ i ξ j + ξ j ∂ j N i , δ (3) N = ξ i ∂ i N, δ (3) X = ξ i ∂ i X. (2.16)
The second step is the redefinition of N,
N = NX z−1 . (2.17)
After this replacement, we have
S 0 = d 4 x √ γN X −z+3 (K ij K ij − K 2 ) + X z+1 R (3) − 6X −z+3 N 2 (∂ t ln X − N i ∂ i ln X) 2 + 6X z+1 γ ij (∂ i ln X)(∂ j ln X) − 2X z+3 Λ + 4 √ γX −z+3 K(∂ t − N i ∂ i ) ln X − 4 √ γNX z+1 2(∂ i ln X)(∂ i ln X) + ∆ ln X , (2.18) K ij = 1 2N (−∂ t γ ij + ∇ i N j + ∇ j N i ). (2.19)
Here we used the integration by parts in the last line in (2.18). So far we have only redefined the fields and thus we have not changed the theory at all. And this action, of course, is invariant under the diffeomorphism transformations which are now,
δ ξ γ ij = δ (3) γ ij + N i ∂ j ξ t + N j ∂ i ξ t + ξ t ∂ t γ ij , (2.20) δ ξ N i = γ ij ∂ t ξ j + δ (3) N i + N i ∂ t ξ t + ξ t ∂ t N i − (X 2z−2 N 2 − N j N j )∂ i ξ t , (2.21) δ ξ N = δ (3) N + ξ t ∂ t N + N∂ t ξ t − NN i ∂ i ξ t , (2.22) δ ξ X = δ (3) X + ξ t ∂ t X. (2.23)
In addition to the diffeomorphism invariance, this action is invariant under the anisotropic rescaling,
N → NY z , N i → N i Y 2 , γ ij → γ ij Y 2 , X → Y −1 X. (2.24)
This anisotropic rescaling is equivalent with the anisotropic space time rescaling in the following way, This is achieved from the combination of the general coordinate transformation, t ′ = b z t and x ′i = bx i , and the anisotropic rescaling with Y = b. Since N i and X transform non trivially, this is not exactly same as the anisotropic rescaling of space and time. We come to this point later.
The infinitesimal transformation of the anisotropic rescaling (2.24), Y = 1 + λ, is
δ λ N = zλN, δ λ N i = 2λN i , δ λ γ ij = 2λγ ij , δ λ X = −λX. (2.26)
Thus, ∂ µ ln X transforms as a corresponding gauge field does,
∂ µ ln X → ∂ µ ln X − ∂ µ λ. (2.27)
Therefore a heuristic way of constructing a gauge theory is introducing the gauge field A µ by replacing ∂ µ ln X with A µ . After the replacement, we finally have,
S 1 = d 4 x √ γN X −z+3 K − X z+1 V z+1 − X z+3 V z+3 , (2.28) K = K ij K ij − K 2 − 6 N 2 (A t − N i A i ) 2 + 4 K N (A t − N i A i ), (2.29) K ij = 1 2N (−∂ t γ ij + ∇ i N j + ∇ j N i ), (2.30) V z+1 = −R (3) + 2γ ij A i A j + 4∇ i A i , V z+3 = 2Λ. (2.31)
The transformations under the differmorphism and anisotropic rescaling for A µ are given
δ ξ A t = A i ∂ t ξ i + δ (3) A t + A t ∂ t ξ t + ξ t ∂ t A t , (2.32) δ ξ A i = δ (3) A i + A t ∂ i ξ t + ξ t ∂ t A i , (2.33) δ (3) A t = ξ i ∂ i A t , δ (3) A i = A j ∂ i ξ j + ξ j ∂ j A i , (2.34) δ λ A µ = −∂ µ λ,(2.35)
where δ ξ A t and δ ξ A i are easily computed from δ ξ A µ = A ν ∂ µ ξ ν + ξ ν ∂ ν A µ . We can think the action (2.28) is an anisotropic rescaling gauge theory in an unitary gauge where the Nambu-Goldstone boson X is eaten by the massive gauge fields A µ . However, the Nambu-Goldstone boson is not completely eaten by the gauge fields. We notice that this replacement changes the theory. For example, some of the symmetries are lost, and dS 4 (AdS 4 ) space for the positive (negative) cosmological constant is no longer a solution. Instead, as we will see in the next section that Minkowski space is a solution independent of the value of the cosmological constant. We can check explicitly which of the symmetries remain, and find that the anisotropic rescaling symmetry remains, but the diffeomrophism invariance is broken down to the foliation preserving diffeomorphism invariance. The foliation preserving diffeomorphism is generated by ξ i (t, x) and ξ t (t). Under the transformation of the anisotropic rescaling, the X −z+3 , X z+3 and X z+1 parts are independently invariant. The transformation laws for the three dimensional part of the diffeomorphism generated by ξ i (t, x) are same as those of three dimensional diffeomorphism, ξ i (t, x) = ξ i (x), except that δ ξ N i has the additional term γ ij ∂ t ξ i and δ ξ A t has the additional term A i ∂ t ξ i . Because of these, the combination ∂ t γ ij −∇ i N j −∇ j N i and A t − N i A i are invariant, and thus the action is invariant. Under the diffeomorphism transformation generated by ξ t (t, x), one can check explicitly that the action is not invariant and the deviation of the action δS is proportional to A µ − ∂ µ ln X. However since the action does not explicitly depend on time, it is easily checked the action is still invariant under the diffeomorphism transformation by ξ t (t, x) = ξ t (t). Therefore the action (2.28) has the foliation preserving diffeomorphism and anisotropic rescaling invariance. We give two comments. We can add the kinetic term for the gauge field, − g g µρ g ντ F µν F ρτ , and this term introduces additional X z−1 and X −z+1 terms. We can add − gX 2 g µν Y µ (A ν − ∂ ν ln X) in the action where Y µ fields are Lagrange multipliers in order to keep the resultant action equivalent with the Einstein Hilbert action.
Higher order spacial derivative terms
In the resultant action (2.28), X field appears only in front of K, V z+1 and V z+3 and thus can be integrated out. We do this for the case z = 3 in which we are most interested.
For z = 3, the action becomes simpler,
S 1 = d 4 x √ γN K − X 4 V 4 − X 6 V 6 , (3.1) K = K ij K ij − K 2 − 6 N 2 (A t − N i A i ) 2 + 4 K N (A t − N i A i ), (3.2) V 4 = −R (3) + 2γ ij A i A j + 4∇ i A i , V 6 = 2Λ. (3.3)
The equation of motion for X is
4X 3 V 4 + 6X 5 V 6 = 0. (3.4)
This has a non trivial solution,
X = − 2V 4 3V 6 . (3.5)
We notice that the value of action is real, even when X is pure imaginary. Therefore this non trivial solution is allowed no matter what the sign of V 4 and that of V 6 are. We substitute this non trivial solution into the action, and we obtain
S 1 = d 4 x √ γN K − 4(V 4 ) 3 27(V 6 ) 2 . (3.6)
We can easily see that (R (3) ) 3 term from (V 4 ) 3 has higher order spacial derivative terms. However these are interaction terms and are not the higher order spacial derivative bi-linear terms for the spin two graviton around Minkowski space. One can check from the action (3.1) or (3.6) that Minkowski space, i.e. γ ij = δ ij , N = constant and A µ = 0, is always a solution.
Here we quote the Horava-Lifshitz gravity action with z = 3 in [6],
S Horava = d 4 x √ γN K ij K ij − λK 2 − V , (3.7) K ij = 1 2N (−∂ t γ ij + ∇ i N j + ∇ j N i ), (3.8) V = 2Λ − ηR + µ 1 R 2 + µ 2 R ij R ij +ν 1 R 3 + ν 2 RR ij R ij + ν 3 R i j R j k R k i + ν 4 ∇ i R∇ i R + ν 5 ∇ i R jk ∇ i R jk ,(3.9)
where R and R ij are the Ricci scalar and Ricci tensor constructed from γ ij . The dispersion relations of spin two and zero modes are,
ω 2 2 = ηk 2 + µ 2 k 4 + ν 5 k 6 ,(3.10
)
ω 2 0 = 1 − λ 1 − 3λ − ηk 2 + (8µ 1 + 3µ 2 )k 4 + (8ν 4 + 3ν 5 )k 6 ,(3.11)
where ω 2 and ω 0 are the energy of spin two and zero modes and k is the momentum. Thus, what we have successfully obtained are limited terms of Horava gravity theory. We can also integrate out A t which appears in K. The equation of motion for A t tells
A t − N i A i = NK 3 ,(3.12)
and K becomes
K = K ij K ij − 1 3 K 2 . (3.13)
Since the coefficient in front ot K 2 is −1/3, the kinetic term for spin zero part of the graviton does not appear. In fact, we can solve the equations of motion for A i from the action (3.1) and A i = ∂ i ln N + 4∂ i ln X. We substitute this solution and have spacial derivative terms for X. However we still do not have kinetic term for X and then can integrate out X field. It will be interesting to study the resultant action.
Discussion
We have derived some of higher order spacial derivative terms as a result of spontaneous breaking of anisotropic rescaling symmetry. The breaking is occurred once we take a background, for example Minkowski space. We do not obtain ∇ i R jk ∇ i R jk term which give k 6 contribution to the dispersion relation of the spin two graviton. This is because ∇ i R jk ∇ i R jk can not be rewritten a polynomial of scalar field X. One may think what terms will be introduced if we introduce matter fields in the action. Once we introduce a scalar field φ, there are the additional terms γ ij (∂ i φ)(∂ j φ) to V 4 , and m 2 φ 2 to V 6 , respectively. Then we have (∂ i φ) 6 interacting term from (V 4 ) 3 /(V 6 ) 2 . Since the theory (2.28) is no longer same as the original theory, the dS (AdS) space is no longer a solution. We have to resolve the equations of motion to study the background and perturbation theory of the action (3.6). Although Minkowski space is a solution, the solution space is more limited compared with the one in the original theory, since the gauge fields A µ also do not have kinetic terms and are not dynamical. Therefore we come to think about the kinetic term for A µ . The kinetic term which does not break the symmetries of the theory is,
d 4 x − g − 1 4 g µν g ρσ F µρ F νσ = d 4 x √ γN X −z+1 1 2N 2 γ ij F ti F tj − 1 N 2 N j γ ik F ti F jk + X z−1 − 1 4 γ ij γ kl F ik F jl , (4.1) where F µν = ∂ ν A ν − ∂ ν A µ as usual.
Then the kinetic term will induce more complex potential terms and it is interesting to study the solutions of equations of motion. We can keep the theory (2.28) equivalent with Einstein gravity by introducing the following term,
d 4 x − gX 2 g µν Y µ (A ν − ∂ ν ln X) = d 4 x √ γN − X −z+3 N 2 (Y t − N i Y i )(A t − N j A j − ∂ t ln X + N j ∂ j ln X) + X z+1 γ ij Y i (A j − ∂ j ln X) ,(4.2)
where Y µ are Lagrange multipliers, and the action is invariant under the full diffeomorphism by giving appropriate transformations for Y µ . It is also interesting to study the effects of this term. We could not induce the terms which contribute to k 6 in the dispersion relation. If we are allowed to introduce spin two graviton mass terms [7] (for review [8]) in the action and fine tune the cosmological constant to be zero, (V 4 ) 3 /(V 6 ) 2 term is schematically written as,
(V 4 ) 3 /(V 6 ) 2 ∼ (∂ i h T T jk ) 6 /(m 2 (h T T jk ) 2 ) 2 ∼ k 6 (h T T jk ) 2 ,(4.3)
where h T T jk denotes the spin two graviton mode. Therefore we have higher order spacial derivative bi-linear terms.
Under the space time rescaling in (2.25), N i and X transform. If we further redefine N i = N ′ i X z−1 , N ′ i does not transform under the space time anisotropic rescaling. Then 5) and all the N i are replaced with N ′ i X z−1 in the action (2.28). Therefore only X field remains transforming. This X field naturally appears in the theory which admits the Lifshitz geometry [9]. The metric of the Lifshitz geometry is,
K ij = 1 2N − ∂ t γ ij + X z−1 ∇ i N ′ j + ∇ j N ′ i + (z − 1)((∂ i ln X)N ′ j + (∂ j ln X)N ′ i ) , (4.4) → 1 2N − ∂ t γ ij + X z−1 ∇ i N ′ j + ∇ j N ′ i + (z − 1)(A i N ′ j + A j N ′ i ,(4.ds 2 = − dt 2 r 2z + γ ij dx i dx j r 2 + dr 2 r 2 . (4.6)
This metric is invariant under the anisotropic space time rescaling, t → b z t and x i → bx i provided with r → br. We then replace r in the metric (4.6) by the radion field R and R acts as X −1 . This is clear if we compactify r direction and see the form of the effective action [10].
The radion field has an ordinary kinetic term, and has a stable non trivial minimum when the sign of V 4 is negative and that of V 6 is positive for z = 3. This is realized if γ ij in (4.6) describes dS 4 space for instance. In this case, when we consider the perturbation of graviton or matter fields around the background, we have higher order spacial derivatives as higher order perturbations from R −4 V 4 and R −6 V 6 terms once we take into account of the back reaction to R field. In other words, once we firstly integrate out R field in the functional integral, the higher spacial derivative terms are induced. We expect there are many directions to be explored by using the method in this paper.
t → b z t, x i → bx i , N i → b −z+1 N i , X → b −1 X.(2.25)
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arxiv |
Getting inflationary models using the deformation method
15 Aug 2014
J J Rodrigues
Universidade Estadual da Paraíba
58233-000Araruna, ParaíbaBrasil
M A M Souza
Instituto Federal de Educação
Ciência e Tecnologia do Piauí64215-000Parnaíba, PiauíBrasil
Getting inflationary models using the deformation method
15 Aug 2014(Dated: August 18, 2014)
We show as the dynamics for the inflaton, under slow-roll regime, can be treated in a other dynamics, following the deformation procedure. In a direct way we present a relationship between two slow-roll inflationary potentials, and we apply this framework to show how to construct an eternal inflation from chaotic inflation, or even, a natural inflation from hilltop inflation, easily. PACS numbers: 98.80.-k, 98.80.Cq
I. INTRODUCTION
The early universe is full of challenging problems that remain to fill the tables of many high energy theoretical physics. Several of these problems can be addressed through the inflation theory [1], where a scalar field, called inflaton, can be invoked to drive the evolution of the early universe. Particularly challenging is the choice of a dynamics to the inflaton, provided that a large number of parameters should be adjusted to ensure the success of the model and then bring the universe to graceful exit [2]. Following this line, theoretical developments to simplify and put the inflation models within the observational constraints are welcome.
In this paper we introduce a framework to implement a correlation between two dynamics for the inflaton under the slow-roll regime [3] (see also references therein), where the inflation parameters present a direct relationship with a potential that drives the solution. Especially, we show that the choice of dynamics to inflaton can be treated in another correspondence model, and this is based in the deformation procedure [4].
The work is organized as follows: Sec. II presents the generalities about the deformation procedure in field theory. The cosmological background is described in Sec. III, where we established that the inflaton following a slow-roll dynamics. In the Sec. IV we apply the deformation procedure to show as a new inflation solution can be studied, from a known inflation solution; being the Sec. V dedicated to show just a few examples, that illustrate this framework. The comments are presented in the Sec. VI.
Throughout this work we use units in which c = 4πG = H 0 = 1, where c is the speed of light in vacuum, G is the gravitational constant, H is the Hubble parameter and the subscript '0' refers to the initial time.
II. DEFORMATION PROCEDURE IN FIELD THEORY
A result of advances in research in high energy physics, an innumerable class of models described by a scalar field has been proposed. However the difficulty lies in the fact that many of these models do not provide an analytical description of the system studied, which hinders the complete understanding of these systems. It is therefore necessary to find a method that can generate potentials with the analytical solutions and of physical interest. A rather effective method is the socalled deformed procedure, proposed by Bazeia et al [4], consisting of generating new solutions from a potential, or a solution, of a known model, with the aid of a properly deformation function.
The advantage of the method is the fact that the description of the characteristics of the new model can be done analytically, without having to resort to computational methods, or numerical analysis. The relationship between the potential of the original model V (χ), and the potential of the obtained model, V (φ), is given for [4]
V (φ) = V (χ → f (φ)) f 2 φ (φ)(1)
where f (φ) is the deformation function. In this case, if χ(x) is a static solution of the starting model, then we get that
φ(x) = f −1 (χ(x))(2)
where φ(x) is a solution of the new deformed model. We can still see that for the case of topological solutions a deformed defect φ(x) connects the corresponding minimums of the solutions χ(x) of the original model, given forυ i = f −1 (υ i ), i = 1, 2, 3, ..., n.
III. COSMOLOGICAL BACKGROUND
We shall consider a model in which the dynamics of the early universe is described by the Einstein-Hilbert action, where a scalar field χ is minimally coupled to gravity, i.e.
S = d 4 x √ −g − 1 4 R + L(χ, X) (3)
R is a scalar curvature, being L(χ, X) the lagrangians scalar field with X = χ ,µ χ ,µ /2 and a comma represents a partial derivative. In the following it will be assumed that χ plays the role of the inflaton field. Consider a flat Friedmann-Robertson-Walker background with line element
ds 2 = dt 2 − a 2 (t) dx 2 + dy 2 + dz 2 ,(4)
where t is the physical time and x, y and z are comoving spacial coordinates. We assume that the energy-momentum tensor for the inflaton field can be written in a perfect fluid form
T µν = (ρ + p)u µ u ν − pg µν ,(5)
by means of the following identifications
u µ = χ ,µ √ 2X , ρ = 2XL ,X − L , p = L(X, χ) . (6)
In Eq. (5), u µ is the 4-velocity field describing the motion of the fluid (for timelike χ ,µ ), while ρ and p are its proper energy density and pressure, respectively. Solving the action to the metric above the Einstein equations reduce to
H 2 = 2 3 ρ (7) andä a = − 1 3 (ρ + 3p)(8)
where H =ȧ/a and dot represents a derivative with respect to physical time.
If we consider the standard dynamics, described by lagrangian density
L = 1 2 χ ,µ χ ,µ − V (χ)(9)
the continuity equation can be written as
χ + 3Hχ + V χ = 0(10)
the index χ presents a derivative related to the field. The pressure and energy density are given by
ρ = 1 2χ 2 + V, p = 1 2χ 2 − V(11)
In this way, we can rewrite the solutions of Einstein equations. We obtain
H 2 = 1 3χ 2 + 2 3 V (12) andḢ = −χ 2(13)
Inflationary solutions for which the energy density of the universe is dominated by the potential term V (χ) enable us choose a known approach as slow-roll approximation, where the inflaton does not vary too rapidly and we can neglect the kinetic term in the Friedmann equation and the acceleration term in the equation of motion of the scalar field, which leads us naturally to first-order equations, such as
H 2 ≈ 2 3 V (χ) (14) 3Hχ + V χ ≈ 0(15)
These equations show that the choice of the potential allows us to apply limits to the inflationary parameters. The number N of e-fold, written as N = ln(a end /a), where a end is the scalar factor in the end of inflation, can be obtained, it is given that
a = a 0 exp te ti 2 3 V (χ) 1/2 dt(16)
i.e. N = t end t H dt, being the inflaton evolution determined for (16). In a similar way, to establish the flatness condition, we need that the slow-roll parameters, defined as [5]
ǫ = 1 4 V χ V 2 , η = 1 2 V χχ V(17)
are such that |ǫ| << 1, as well as, |η| << 1, and all parameters are sensible the choice of the potentials. The deformation procedure creates a class of the analytical potentials and this can be an easy way to analyze these parameters, or even, all parameters that are functions of the potential. We are going to see now as this procedure can be applied to slow-roll inflation scenario.
IV. DEFORMING SLOW-ROLL INFLATIONARY MODELS
Initially we consider that the inflaton has its dynamics described by the lagrangian density
L = 1 2 χ ,µ χ ,µ − V (χ)(18)
where V (χ) presents the potential field. The continuity equation in this dynamics takes the form
ρ χ + 3Hχ = 0(19)
Since we know that
H 2 = 2 3 ρ(20)
and squared (19) we obtain the useful relation
6ρχ 2 = ρ 2 χ(21)
Now we consider another dynamics for the inflaton evolution described by the lagrangian density
L = 1 2 φ ,µ φ ,µ −Ṽ (φ)(22)
Similarly to the previous model results
6ρφ 2 =ρ 2 φ(23)
The key point of this description is to redefine the dynamics field via the relation
χ = f (φ) (24)
where f (φ) is a so called deformation function. As a direct consequence of this definition we can writė
φ =χ f φ(25)
in which f φ = df /dφ. Using (21), and (23) we come tõ
ρ 2 φ ρ = 1 f 2 φ ρ 2 χ ρ χ=f (φ)(26)
this presents a generic correspondence between two energy densities describing two dynamics scalar field. This is a natural way to deform the energy density for two fluids in the same background provided that these fluids have a conserved energy-momentum tensor, or better, provided that they satisfy a continuity equation. The slow-roll condition applied to the equation of motion of the scalar field allows us to rewrite it as
3Hχ = −V χ (27) now 6Vχ 2 = V 2 χ(28)
as well as
6Ṽφ 2 =Ṽ 2 φ(29)
and we haveṼ
2 φ V = 1 f 2 φ V 2 χ V χ=f (φ)(30)
this presents a generic correspondence between two potentials describing the inflaton dynamics, under slowroll approximation.
The solutions in the new model are obtained with φ = f −1 (χ), this is the inverse deformation function calculated with the solutions of original model. An important implication of this framework is based under the possibility of obtaining an analytical description for new inflation solutions and allows to analyze the parameters for these solutions, provided we have known the parameters for the original inflationary model, minimizing or even annulling the numerical techniques, making the search for more complicated vacuum configurations more accessible.
The limit in that slow-roll condition ceases to be valiḋ χ 2 /2 = V (χ) depends on the potential chosen and we have
t end − t ini = χ end χini dχ √ 2V (31)
which leads to t end −t ini =t end −t ini , since that (30) is a valid relation between the potential of the original model and the potential of the deformed model. In this sense the deformation procedure does not show the correlation between two slow-roll sectors, but between two potentials that have a slow-roll regime by construction, which can be seen by analyzing the deformation of the slow-roll parameters.
V. APLICATIONS
To illustrate this framework initially we deal with a model based in the chaotic inflation [6]. In this model the dynamics field is driven by the quadratic potential V (χ) = V 0 χ 2 . The deformation procedure can lead us directly to an eternal inflation model [7], described by potentialṼ (φ) =Ṽ 0 φ p , where we choose p > 2, assuming the deformation function
f (φ) = χ = −4 V 0 V 0 φ − 1 2 (p−4) p(p − 4) ,(32)
applied to potential chaotic inflation and using (30). Once the potentials are known we can obtain the slowroll parameters. To the original potential we have ǫ = η = χ −2 . It applying the deformation procedure, these parameters are obtained in the other frame and now we writeǫ
= p 2 4φ 2 ,η = p(p − 1) 2φ 2(33)
The end of inflation occurs now with an additional choice of parameter p for the deformed model, while only conditions for fields are necessary for the original model. The e-fold number can be estimated in this framework as
N = 1 2 (χ 2 end − χ 2 ini )(34)
and to the deformed model
N = 1 p (φ 2 end − φ 2 ini ),(35)
We note that the case p = 4 must be analyzed separately.
To this choice, we writeṼ (φ) = λφ 4 and following the previous results, the deformation function is
f (φ) = χ = V 0 λ ln φ 2(36)
Now the slow-roll parameters arẽ
ǫ = 4 φ 2 ,η = 6 φ 2(37)
To this case the e-fold number is given by sameÑ for p = 4. We now consider as the initial model the hilltop inflation [8], being
V (χ) = V 0 − λ p χ p/2 2(38)
This potential can be deformed directly in the natural inflation potential [9], given byṼ (φ) =Ṽ 0 cos 2 (rφ), when we choose the following deformation function
f (φ) = χ = 4 Ṽ 0 r 2 (p − 4) λ arctanh(cos(rφ)) 2/(p−4)(39)
being the integration constant such that f (φ = π/(2r)) = 0.
Here, the slow-roll parameters for the original potential are ǫ = 1 4
χ p−2 χ p/2 p − V 0 λ 2 (40) η = p − 1 2p χ p−2 − 1 4 V 0 λ (p − 2)χ (p−4)/2 χ p/2 p − V 0 λ 2(41)
In the other way, for the deformation potential we havẽ ǫ = r 2 tan 2 (rφ),η = r 2 (tan 2 (rφ) − 1)
The e-fold number can be computed as
N = χ 2 1 p + 4 V 0 λ χ −p/2 p − 4 χ end χini(43)
and for the deformed model we havẽ For p = 4 we come to
N = 1 r 2 ln sin(rφ ini ) sin(rφ end )(44)f (φ) = χ = sin(rφ) 1 + cos(rφ) ±λ 2 √Ṽ 0r 2(45)
and the integration constant is such that f (φ = π/(2r)) = 1. The slow-roll parameters ǫ and η are the same as before above, with p = 4 in their respective expressions. The e-fold number is now
N = χ 2 4 − 2 V 0 λ ln χ χ end χini(46)
In both cases we can see that the parameters in the slow-roll inflation regime of the new model are constructed, through the deformation procedure, considering known results of the original model. We take special deformation functions only to illustrate the procedure, exploiting well-established results in the literature. However, if we take the choice of arbitrary deformation functions, we can generate new potentials, expanding the analytical range of solutions in the slowroll inflation regime.
VI. SUMMARY
In this paper we have shown as the deformation procedure works on cosmological background, most especially in the slow-roll regime of the inflationary phase, since for this situation the first-order differential equations favor the application of the method. This does not impose a limit to the procedure, because we can reduce the order of the higher order differential equations, which is a direct way to improve the method for further analysis in the cosmological background, provided that a scalar field drives the evolution of the universe, that will be the focus of our interest on the current study in future papers.
FIG. 1 :
1Plots of the chaotic inflation potential (solid line) and of the ethernal inflation potential to p = 3 (dotted line), p = 4 (dashed line), and p = 5 (dotted-dashed line). FIG. 2: Plots of the natural inflation potential (solid line) and of the hilltop inflation potential to p = 4 (dotted line), p = 8 (dashed line), and p = 12 (dotted-dashed line). The parameter λ was properly chosen so that the zeros of the natural inflation potential were coincident with the zeros of the hilltop inflation potential.
The authors would like to thank CNPq Brasil and CAPES for partial support.
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| {'fraction_non_alphanumeric': 0.07349827259556448, 'fraction_numerical': 0.04909171960325421, 'mean_word_length': 3.645871084649236, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We show as the dynamics for the inflaton, under slow-roll regime, can be treated in a other dynamics, following the deformation procedure. In a direct way we present a relationship between two slow-roll inflationary potentials, and we apply this framework to show how to construct an eternal inflation from chaotic inflation, or even, a natural inflation from hilltop inflation, easily. PACS numbers: 98.80.-k, 98.80.Cq', 'arxivid': '1408.3449', 'author': ['J J Rodrigues \nUniversidade Estadual da Paraíba\n58233-000Araruna, ParaíbaBrasil\n', 'M A M Souza \nInstituto Federal de Educação\nCiência e Tecnologia do Piauí64215-000Parnaíba, PiauíBrasil\n'], 'authoraffiliation': ['Universidade Estadual da Paraíba\n58233-000Araruna, ParaíbaBrasil', 'Instituto Federal de Educação\nCiência e Tecnologia do Piauí64215-000Parnaíba, PiauíBrasil'], 'corpusid': 119269599, 'doi': '10.1088/0031-8949/90/4/045301', 'github_urls': [], 'n_tokens_mistral': 6294, 'n_tokens_neox': 5307, 'n_words': 3197, 'pdfsha': 'b348b39f18943435ecfbd3830529d6bbd75fe679', 'pdfurls': ['https://arxiv.org/pdf/1408.3449v2.pdf'], 'title': ['Getting inflationary models using the deformation method', 'Getting inflationary models using the deformation method'], 'venue': []} |
arxiv |
FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting
XXX-XXXCopyright XXX-XXX2020
Maowei Jiang [email protected]
Pengyu Zeng [email protected]
Kai Wang [email protected]
Huan Liu [email protected]
Wenbo Chen [email protected]
Haoran Liu [email protected]
Maowei Jiang
Pengyu Zeng
Kai Wang
Huan Liu
Wenbo Chen
Haoran Liu
Shenyang Institute of Automation
Chinese Academy of Sciences Shenyang
China
Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang
University of Chinese Academy of Sciences
BeijingChina, China
Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang
Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang
Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang
University of Chinese Academy of Sciences
BeijingChina, China, China, China
Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang
University of Chinese Academy of Sciences
BeijingChina, China
University of Chinese Academy of Sciences
BeijingChina
FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting
FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting. PVLDB
XXX-XXX1412020PVLDB Reference Format: †These authors contributed equally to this work. *Corresponding author. PVLDB Artifact Availability: The source code, data, and/or other artifacts have been made available at https://github.com/Zero-coder/FECAM.
Time series forecasting is a long-standing challenge due to the real-world information is in various scenario (e.g., energy, weather, traffic,economics, earthquake warning). However some mainstream forecasting model forecasting result is derailed dramatically from ground truth. we believe it's the reason that models' lacking ability of capturing frequency information which richly contains in real world datasets. At present, the mainstream frequency information extraction methods are Fourier transform(FT) based. However,use of FT is problematic due to Gibbs phenomenon. If the values on both sides of sequences differ significantly, oscillatory approximations are observed around both sides and high frequency noise will be introduced. Therefore We propose a novel frequency enhanced channel attention that adaptively modelling frequency interdependencies between channels based on Discrete Cosine Transform which would intrinsically avoid high frequency noise caused by problematic periodity during Fourier Transform, which is defined as Gibbs Phenomenon. We show that this network generalize extremely effectively across six real-world datasets and achieve stateof-the-art performance, we further demonstrate that frequency enhanced channel attention mechanism module can be flexibly applied to different networks. This module can improve the prediction ability of existing mainstream networks, which reduces 35.99% MSE on LSTM, 10.01% on Reformer, 8.71% on Informer, 8.29% on Autoformer, 8.06% on Transformer, etc., at a slight computational cost,with just a few line of code. Our codes and data are available at https://github.com/Zero-coder/FECAM.
ABSTRACT Time series forecasting is a long-standing challenge due to the real-world information is in various scenario (e.g., energy, weather, traffic,economics, earthquake warning). However some mainstream forecasting model forecasting result is derailed dramatically from ground truth. we believe it's the reason that models' lacking ability of capturing frequency information which richly contains in real world datasets. At present, the mainstream frequency information extraction methods are Fourier transform(FT) based. However,use of FT is problematic due to Gibbs phenomenon. If the values on both sides of sequences differ significantly, oscillatory approximations are observed around both sides and high frequency noise will be introduced. Therefore We propose a novel frequency enhanced channel attention that adaptively modelling frequency interdependencies between channels based on Discrete Cosine Transform which would intrinsically avoid high frequency noise caused by problematic periodity during Fourier Transform, which is defined as Gibbs Phenomenon. We show that this network generalize extremely effectively across six real-world datasets and achieve stateof-the-art performance, we further demonstrate that frequency enhanced channel attention mechanism module can be flexibly applied to different networks. This module can improve the prediction ability of existing mainstream networks, which reduces 35.99% MSE on LSTM, 10.01% on Reformer, 8.71% on Informer, 8.29% on Autoformer, 8.06% on Transformer, etc., at a slight computational cost,with just a few line of code. Our codes and data are available at https://github.com/Zero-coder/FECAM.
INTRODUCTION
Time series forecasting (TSF) enables decision-making with the estimated future evolution of metrics or events, thereby playing a crucial role in various scientific and engineering fields such as weather forecasting [11,13,30], estimation of future illness cases [10,16,31], energy consumption management [32,36,37,44], traffic flow [24,39,45,46],and financial investment [2,3,7,41], to name a few.
With the growing data availability and computing power in recent years, it is shown that deep learning-based TSF methods can achieve much better prediction performance than traditional approaches [21].
In recent year, Transformers [35] have achieved progressive breakthrough on extensive areas [5,6,8,23]. Especially in time series forecasting, credited to their stacked structure and the capability of attention mechanisms, Transformers [18,35,47] can naturally capture the temporal dependencies among time points, thereby fitting the series forecasting task perfectly. Despite the promising results of TSF methods, we found that the prediction of those methods,like transformers and LSTM is way derailed from the distribution of the ground truth of datasets, such as baseline drift in Fig.1(a) and temporal drift in Fig.1(c), we believe it's the reason that models' lacking ability of capturing frequency information which richly contains in real world datasets (Fig.2), Therefore, the thing is that there still have room for improvement for these TSF mainstream methods to exploiting the natural property of time series data what we call frequency during modeling. Some efforts has been done for getting frequency representation and reconstructing temporal signal based on Fourier Transform and it's inverse transform. However, Fourier Transform (FT) would introduce high-frequency components for its problematic periodity, causing error value for boundary information which call Gibbs phenomenon and new round of computation consumption for inverse operation for avoiding complex operation in networks. Unlike FT/IFT based methods, Our method is based on Discrete Cosine Transform which would intrinsically eradicate Gibbs Phenomenon mentioned above and save unnecessary consumption of inverse transform, and for better exploiting utility of relationship between different time-series variate, we propose Frequency Enhanced Channel Attention Mechanism as a general framework, which empowers Transformer-based method and other mainstream models like LSTM, with better predictive ability for real-world time series.Consequently, effectively utilizing frequency information of time series enable us to perform forecasting with reasonable accuracy. Our method achieves state-of-the-art performance on six real-world benchmarks as a model. Furthermore,as a module FE-CAM can generalize to various Networks for further improvement, with just few line codes.
To this end, we propose a general feature extraction method for sequence modeling and forecasting, named frequency enhanced channel attention mechanism, which intrinsically eradicate Gibbs Phenomenon caused by Fourier Transform for the first time in time series forecasting.Our method achieves state-of-the-art performance on six real-world datasets, and can be generalized to other model architectures with just few line codes. The contributions of this paper are summarized as follows:
• We theoretically prove that our method can mitigate Gibbs phenomenon which would introduce high frequency noise during Fourier Transform,and we demonstrate that GAP is the lowest frequency component of DCT. • Based on above proof, We build the channel attention in frequency domain and propose our method with frequency enhanced channel mechanism for time-series forecasting. For generalization, we generalize frequency-enhanced channel attention into module that can be easily and flexibly adapted into other mainstream time series forecasting models to get better performance on six real-world datasets. • Extensive experiments on various TSF datasets show that FECAM as a general method consistently boosts four mainstream Transformers and non-transformer based methods like LSTM by a considerable margin and achieves state-ofthe-art performance on six real-world datasets.
RELATED WORK AND PRELIMINARY 2.1 Deep Learning Models for Times series forecasting
In recent years, deep learning models with meticulously designed architectures have achieved excellent progress in TSF tasks. RNNbased models [25,29,33,38,43] are proposed for application in an auto-regressive manner for sequence modeling, but the recurrent structure can suffer from problem of modeling long-term dependency. Shortly afterwards, Transformer [35] emerges and shows great power in sequence modeling and gains great achievements in various downstream tasks. To solve the quadratic computation consumption on sequence length, subsequent works aim to decrease Self-Attention's complexity. Particularly in long-term time series forecasting, Informer [47] extends Self-Attention with KLdivergence criterion to select dominant queries. Reformer [18] introduces local-sensitive hashing (LSH) mechanism to approximate attention by allocated similar queries. Not just improvement of reduction complexity, the following models further develop delicate building blocks for time series forecasting. Autoformer [40] coalesce the decomposition blocks into a canonical structure and designs Auto-Correlation to capture series-wise connections. Pyraformer [22] designs pyramid attention module (PAM) to capture temporal dependencies with different hierarchies. Transformer-based models have taken the place of RNN-based models in almost all sequence modeling tasks, thanks to the effectiveness and efficiency of the self-attention mechanisms. Various Transformer-based TSF methods are proposed in the literature. These works typically focus on the challenging long-term time series forecasting problem, taking advantage of their remarkable long sequence modeling capabilities. Although the transformers can capture long-range dependency in the time domain, it does not explicitly model the pattern occurrences in the frequency domain that plays an important role in tracking and predicting data points over various time cycles.
Different from previous works focusing on architectural design based on transformers, we analyze the series forecasting task from the natural view of frequency, which is the essential property of time series.It is also notable that as a general block, our proposed frequency-enhanced channel block can be easily applied to various models with a few operation. In the following subsection, we highlight our insights and motivate our work.
Frequency Representation for time series forecasting
Frequency is an indispensable information of time series, and real world datasets often contain rich frequency information as shown in Fig.2, which allows better utilization of the capabilities of deep learning models. To utilize frequency information,Auto-former [40] use FFT in efficient computing of auto-correlation function, FNO [14] is used as an inner block of networks to perform representation learning in low-frequency domain, DCTnet [42] use Discrete Cosine Transform to compress information for keeping more original picture information in CV task. Most of these work was based on Fourier Transform which is helpful for extracting frequency features. However, most of the FT-based methods use Fourier Transform to get the frequency information and use Inverse Fourier Transform to reconstruct temporal information for avoiding complex-number training, which introduces new amount of computation, which is avoidable if using DCT for time-frequency transformation, what's more the implicit periodicity of DFT gives rise to boundary discontinuity that result in significant high-frequency content which is known as Gibbson Phenomeneon. After quantization, Gibbs Phenomeneon causes the boundary points to take on erraneous values. SENET [15] only use GAP which is the lowest component of DFT and DCT for channel representation,meaning discarding other frequency-component information.
Problem of Gibbs Phenomenon
The Gibbs phenomenon [9,26,34] involves both the fact that Fourier sums overshoot at a jump discontinuity, and that this overshoot does not die out as more sinusoidal terms are added. And this would cause high frequencies noise which is supposed to be avoidable for time series forecasting. We demonstrate the phenomenon for square wave (In Fig.3) with the additive synthesis of a square wave with an increasing number of harmonics. The Gibbs phenomenon is visible especially when the number of harmonics is large. We give the mathematic description of Gibbs Phenomenon below. Formal mathematical description of the phenomenon: Let : R → R be a piecewise continuously differentiable function which is periodic with some period > 0. Suppose that at some point 0 , the left limit ( − 0 ) and right limit ( + 0 ) of the function differ by a non-zero jump of :
( + 0 ) − ( − 0 ) = ≠ 0.(1)
For each positive integer , let ( ) be the th partial Fourier series
( ) : = ∑︁ − ≤ ≤ ( ) 2 = 1 2 0 + ∑︁ =1 cos 2 + sin 2 ,(2)
where the Fourier coefficients ( ), , are given by the usual formulae
( ) := 1 ∫ 0 ( ) −2 / := 2 ∫ 0 ( ) cos 2 := 2 ∫ 0 ( ) sin 2 .(3)
Then we have:
lim →∞ 0 + 2 = ( + 0 ) + · (0.089489872236 . . . ) (4) and lim →∞ 0 − 2 = ( − 0 ) − · (0.089489872236 . . . ) (5) but lim →∞ ( 0 ) = ( − 0 ) + ( + 0 ) 2 .(6)
More generally, if is any sequence of real numbers which converges to 0 as → ∞, and if the jump of is positive then lim sup
→∞ ( ) ≤ ( + 0 ) + · (0.089489872236 . . . )(7)
and lim inf
→∞ ( ) ≥ ( − 0 ) − · (0.089489872236 . . . )(8)
If instead the jump of is negative, one needs to interchange limit superior with limit inferior, and also interchange the ≤ and ≥ signs, in the above two inequalities.
FECAM: FREQUENCY ENHANCED CHANNEL ATTENTION MECHANISM
Frequency is a natural auxiliary means to analyze time series. It is important and intuitive to introduce frequency information into time series models. However, most time series models tend to ignore the impact of frequency information on time series tasks, resulting in failure to learn the inherent characteristics of time series information. Most methods of extracting frequency information are based on FT and IFT, However,methods based on FT and IFT tends to introduce high-frequency noise due to problematic periodity which is known as Gibbs Phenomenon, Frequency Enhanced Frequency Channel Attention Mechanism can intrinsically avoid problem mentioned above and automatically acquire the importance of each channel through learning, it also suppresses features that are not useful for the current task.
We expect the learning of channel interdependencies features to be enhanced by explicitly modelling in frequency domain.
Channel Attention and DCT
We first elaborate on the definitions of discrete cosine transform and channel attention mechanism.
3.1.1 Revisiting Channel attention. The channel attention mechanism has been successfully introduced to CNNs. Squeeze-andexcitation (SE) block [15] models the interdependencies between the channels of feature maps with global information and recalibrate the feature maps to improve representation ability. It consists of squeeze and excitation two steps which are depicted in Fig .4. For time-series signals ∈ × , is the number of channels, is the length of the temporal sequence, this type of tensor could be anywhere in the time-series model.
For temporal signals, the squeeze step applies GAP on temporal dimension to generate channel wise descriptor. Officially, a statistic ∈ is generated by shrinking through its temporal dimension , such that the -th item of is calculated by:
= ( ) = 1 ∑︁ =1 ( )(9)
Where , represent the channel, and temporal dimension respectively. The scalar is the -th element of , Then the excitation step aims to modelling channel-wise dependencies by using two fully-connected layers 1 and 2 with a bottleneck architecture and non-linearity:
= ( 2 ( 1 ))(10)
where ∈ is the learned attention vector which dot multiplies to the original feature map to re-scale each channel, and refer to ReLU and sigmoid activation function respectively.
Frequency representation for time series.
Sometimes, frequency information contains more information that can be found, but it is difficult to mine in the time domain. For example, when a signal is disturbed by noise, its waveform will become messy, Or we can't distinguish it from noise in time domain. But it can be clearly distinguished from the frequency domain. Instead of well-known Fourier Transform,Our method introduce frequency information by Discrete Cosine Transform which can intrinsically avoid Gphenomenon and inverse transform operation.
Discrete Cosine Transform (DCT) Typically, the basis function of one-dimensional (1D) DCT is:
= ( ( + 1 2 ))(11)
Then the 1D DCT can be written as:
1 = −1 =0 1(12)
s.t. ∈ {0, 1, · · · , − 1},In which 1 ∈ is the 1D DCT frequency spectrum, 1 ∈ is the input, is the length of 1 . Correspondingly, the inverse 1D DCT can be written as:
1 = −1 =0 1(13)
s.t. ∈ {0, 1, · · · , − 1},In which 1 ∈ , Please note that in Eqs. 2 and 3, some constant normalization factors are removed for simplicity, which will not affect the results in this work.
Frequency Enhanced Channel Attention Mechanism
In this section, we first theoretically discuss the problem of existing channel attention mechanisms. Based on the theoretical analysis, we then elaborate on the network design of the proposed method. Although GAP is a widely used operation in many attention mechanism as a standard squeezing method, we argue that simply use average-pooling on temporal dimension cause inadequate information extraction from time series which would even leads to information loss. Since GAP is the lowest frequency component of DCT and DFT, we mitigate this problem by introducing more frequency information. Rather than DFT, We use DCT to evade the Gibbs phenomenon mentioned many times before.
0 = −1 =0 1 0 ( + 1 2 ) = −1 =0 1 = ( 1 )(14)
In Eq.14, 1 0 represents the lowest frequency component of 1D DCT, and it is proportional to GAP. In this way, Theorem 1 is proved.
According to Theorem 1, without any surprise, we can sure that using GAP for feature extraction in channel attention means only the lowest frequency in obtained. All other frequency components are ignored, which supposed to be included in presenting channels. Discrete Cosine Transform is actually the DFT whose input signal is a real even function (proved in Derivation of DCT in Appendix A). Since Discrete Cosine Transform is using symmetric expansion for it's periodic extension (In Fig.6). Therefore, followed with eq.1, we have:
+ 0 − − 0 = = 0 = ( 0 ) ⇔ + 0 = − 0(15)
Equation 15 means that there's no jump discontinuity which is necessary condition of Gibbs Phenomenon.
Then follow with eq.4 and eq.5,then we have:
lim →∞ 0 + 2 = + 0 + 0 · (0.0894 . . .)(16)lim →∞ 0 − 2 = − 0 − 0 · (0.0894 . . .)(17)
Follow with eq.7 and eq.8,
⇒ lim →∞ S ( 0 ) ≤ + 0(18)
lim
→∞ ( 0 ) = + 0 + − 0 2 = ( 0 )(20)
As we can see, the limit of the formula converges at this point, no oscillation observed, thus fundamentally eliminating the Gibbs effect. Because IFT and FT is consistent in mathematical nature, it is also true for the inverse DFT. It's worthy to note that in the DFT case the periodic extension introduces discontinuities, which not happen for the DCT due to its property of symmetry extension, it is the elimination of this artificial discontinuity which contains a lot of high frequencies make the DCT is much more energy efficient than Discrete Fourier Transform. To this extent, Theorem 2 is proved. And we have done experiments to validate our Theorem 2 in section 4.4.
According to Theorem 2, we can found that in the DFT case the extension introduces discontinuities and this does not happen for DCT, due to the symmetry of its periodic extension, then our method eliminate this artificial discontinuity which contains a lot of high frequencies.
For capturing more time series information from feature map, we try to introduce DCT for getting more frequency components instead of only the GAP for lowest frequency [15]. Since DCT weight are constant, it can be pre-calculated only once and saved in advance, what's more, results are real number, which means no training time for inverse transform and number of network parameters. Therefore,we propose frequency enhanced channel attention mechanism (FECAM) which can not only be used as a model for forecasting with just adding a projection layer but also can be seamlessly added to the existing time series forecasting models for improving their prediction performance. The overall structure of FECAM is shown in Fig.5.
First, FECAM splits the input feature maps along the channel dimension into sub-groups as [ 0 , 1 , · · · , −1 ], in which = 1× , ∈ {0, 1, · · · , − 1}, = , Subsequently,for sub-group will be processed by a corresponding DCT frequency component ranging from low frequency to high frequency, Every single channel will processed by the same frequency component, In this way we have:
= ( ) = = −1 =0 ( :, )(21)
s.t. ∈ {0, 1, · · · , − 1}, ∈ {0, 1, · · · , − 1}, in which are the frequency component 1D indices corresponding to ,and ∈ is the dimensional vector after the discrete cosine transformation. The whole frequency channel vector can be obtained by stack operation.
= ( ) = ([ 0 , 1 , · · · , −1 ])(22)
In which ∈ × is the attention vector for ∈ × . Once we obtain , the attention weight can be learned through neural structure as SE-block. The whole frequency enhanced channel attention mechanism framework can be written as:
− = ( 2 ( 1 ( )))(23)
By doing so, each channel features interact with every frequency components to acquire important temporal information comprehensively from frequency domain,which would encourages networks to enhance the diversity of extracted features. In the subsequent experiment section 4.3, we visualize the frequency channel attention tensor Fig.10, demonstrating that FECAM learned the importance of different channels in the frequency domain and the importance of different frequency component pairs in each channel.
EXPERIMENTS
We conduct extensive experiments to evaluate the performance of frequency enhanced channel mechanism network on six realworld time series forecasting benchmarks and further validate the generality of the proposed method on various mainstream Transformer variants and non-transformer based models.As a module embedding to other Networks,we have also done experiment of parameters increment and performance promotion and the visualization of frequency channel attention tensor to prove proposed method's effectiveness and efficiency.
Datasets: Here are the descriptions of the datasets: Electricity 1 : records the hourly electricity consumption of 321 clients from 2012 to 2014.
ETT 2 : contains the time series of oil temperature and power load collected by electricity transformers from July 2016 to July 2018. ETTm1 /ETTm2 are recorded every 15 minutes, and ETTh1/ETTh2 are recorded every hour.
Exchange 3 : collects the panel data of daily exchange rates from 8 countries from 1990 to 2016. Table 1 summarizes overall statistics of the datasets. We follow the standard protocol that divides each dataset into the training, validation, and testing subsets according to the chronological order. The split ratio is 3:1:1 for the ETT dataset and 7:2:2 for others.
Baselines: We evaluate the single full-connected layer equipped by the Frequency Enhanced Channel Attention mechanism in both multivariate and uni-variate settings to demonstrate its effectiveness. For multivariate forecasting, we include six state-of-the-art deep forecasting models: Autoformer [40], Pyraformer [22], Informer [47], LogTrans [20], Reformer [18] and LSTNet [19]. For univariateforecasting, we include seven competitive baselines:N-HiTS [4],N-BEATS [27],Autoformer [40], Pyraformer [22], Informer [47], Reformer [18] and ARIMA [1]. In addition, we adopt the proposed framework on both the canonical and efficient variants of Transformers and classical RNNs:Transformer [35],Informer [47],Reformer [18] and Autoformer [40] and LSTM [12] to validate the generality of our framework.
Implementation details: All the experiments are implemented in PyTorch [28] and conducted for three runs on a single NVIDIA GeForce RTX 3090 24GB GPU. Each model is trained by ADAM [17] using L2 loss with the initial learning rate of 10e-4 and batch size of 32. Each Transformer-based model contains two encoder layers and one decoder layer. We report the test MSE/MAE under different prediction lengths as the performance metric. A lower MSE/MAE indicates better performance of time series forecasting.
Main Results
Forecasting results: As for multivariate forecasting results, Our proposed method with a projection layer for forecasting achieves state-of-the-art performance in all benchmarks and prediction lengths (Table 2). Notably, Frequency Enhanced Channel Attention Mechanism outperforms other deep models impressively characterized by much less model parameters. Compared with Autoformer, the proposed FECAM yields an overall 21.52% relative MSE reduction and relative 10.78% MAE reduction. With the prediction length of 24 and 48,FECAM achieve an 39.67% MSE reduction(3.483→2.101) and 24.9%(3.103→2.330) respectively on ILI, with the the prediction length of 96,192,336,720, FECAM achieve 36.40% relative MSE on Exchange compared to previous state-of-the-art results, which indicates that the potential of deep model is still constrained on ability of modelling in frequency domain. We also list the univariate results of two typical datasets with different frequency distribution (as shown in Fig.2). FECAM still realizes remarkable forecasting performance. Module generality: We apply our proposed method to four mainstream Transformers (as shown in Fig.7(a)) and a mainstream recurrent neural network LSTM (as shown in Fig.7(b)) and report the performance promotion of each model ( Table 4), and thereby their computational complexities can be preserved. It validates that Frequency Enhanced Channel Attention Mechanism is an effective and lightweight tool that can be widely applied to Transformer-based models and RNNs with a few line code, and enhances their ability of modelling in frequency domain to achieve state-of-the-art performance. By analyzing the results of Table 5, we can obviously find that the module gains of the FECAM are large in datasets Exchange, ETTm2, and weather, but small in the dataset traffic. By observing the frequency spectrum of each dataset (as shown in the Fig.2), we can safely say that datasets like Exchange, ETTm2, and Weather have a lot of energy information at low frequencies, while there is little energy information in the frequency spectrum of the Traffic dataset, ant this might be the reason why the gains of FECAM module is not so profitable on the traffic dataset.
Model Analysis
Qualitative results: As shown in Fig.8, we plot the prediction results of vanilla Transformer,Transformer with our FECAM block,and Our FECAM method (FECAM with a projection layer) on Exchange dataset,and plot the prediction results of vanilla LSTM, LSTM with FECAM block,and our FECAM method on ETTm2 dataset. When the input length is 96 steps and the output horizon is 336 steps, Transformer and LSTM both fail to capture the scale and bias of the future data on Exchange and ETTm2 respectively (as shown in Fig.8(a,d)) Moreover, transformer can hardly predict a proper trend on aperiodic data such as Exchange-Rate. With Our FECAM module, both Transformer and LSTM have a better predictability compared to their vanilla version (as shown in Fig.8(b,e)). We can see FECAM with just a projection layer can have a remarkable prediction result than other models with FECAM module, it could be the reason that parameters of FECAM is much less small than Transformer and LSTM which means Transformer and LSTM are more likely to get overfitting than FECAM with just a projection layer. These phenomena further indicate the inadequacy of existing mainstream models modelling in frequency for the TSF task. Our proposed method is beneficial for an accurate prediction of the detailed series variation, which is vital in real-world time series forecasting. Figure 10: Visualization of frequency enhanced channel attention and output tensor of encoder layer of transformer.xaxis represents channels,y-axis represents frequency from low to high,performing on datasets weather and exchange.
The interpretability of FECAM
Energy Compaction
For a signal with 16 sampling points, DCT and DFT are respectively used for reconstruction. During reconstruction,DCT and DFT use = {5, 10, 15} number of components starting from low frequency. The effect is as shown in Fig.9.
This experiment intuitively verified that for a signal with more energy concentrated in the low frequency, DCT can better reconstruct the signal with using less components.So Discrete Cosine Transform is more efficient in Energy compaction than Discrete Fourier Transform.
CONCLUSION
This paper addresses time series forecasting from the perspective of modelling in frequency domain. Unlike previous studies that most frequency extraction method are FT-based which could bring high-frequency noise to the results due to the problematic periodity, which is known as Gibbs Phenomenon. We propose Frequency enhanced channel mechanism based on Discrete Cosine Transform could intrinsically avoid G-phenomenon, and we theoretically prove the feasibility of the method. By modeling in the frequency domain, FECAM can assign channel weights to different channels, and learn the importance of different frequencies of each channel, so as to learn the frequency domain representation of time series. In the experimental stage, we visualize the frequency domain information extracted by FECAM, which verify our conjecture and proved its validity. Most importantly, for its generalization, we design this method into a module for accessibility, which can flexibly and easily use in other mainstream model like transformer-based methods and RNNs methods, etc., with just a few lines to add. Our work achieve state-of-the-art on six real-world benchmarks. This impressive generality and performance of proposed frequency enhanced channel attention mechanism can be interesting of future research for time series forecasting.
Figure 1 :
1The discrepancy between ground truth and forecasting output on real-world dataset ETTm2,(a) is from the vanilla LSTM,(b)is from vanilla Informer(c)is from vanilla transformer (d)is from vanilla autoformer.
Figure 2 :
2six real world datasets visualization in Frequency domain, we can see most energy is contained in low frequency range.
Figure 3 :
3Gibbs Phenomenon with increasing harmonics component.
Figure 4 :
4SENET channel attention(Squeeze and excitation Network), Fsq(.) represent usg1d-global average pooling to extract global information a full connected layer for from channel and redistribute weight for each channel.
Figure 5 :
5Structure of Frequency Enhanced Channel Attention Mechanism. split every sequence of multivariate time series in each channel and Fdct(.) stands for Discrete Cosine Transform,stack&realign each channel together.
Theorem 1 .
11d-GAP is a lowest component of 1D DCT, and its result is proportional to the lowest frequency component of 1d-DCT.
Theorem 2 .
2Discrete Cosine Transform can intrinsically avoid Gibbs Phenomenon caused by periodic problem of Discrete Fourier Transform and Inverse Discrete Fourier Transform, and have a more efficient energy compaction than Fourier Transform.
Figure 6 :
6Extension of Discrete Fourier Transform and Discrete Cosine Transform.
4 : collects the ratio of influenza-like illness patients versus the total patients in one week,which is reported weekly by Centers for Disease Control and Prevention of the United States from2002 and 2021.
Traffic 5 :
5contains hourly road occupancy rates measured by 862 sensors on San Francisco Bay area freeways from January 2015 to December 2016. Weather 6 : includes meteorological time series with 21 weather indicators collected every 10 minutes from the Weather Station of the Max Planck Biogeochemistry Institute in 2020.
Figure 7 :
7FECAM as a module embedded into other Networks, figure (a) represent module is put behind the encoder of Transformers and figure (b) represent module is put between LSTM output layer and projection layer.
Figure 8 :
8Visualization of ETTm2 and Exchange predictions given by different models.
Figure 9 :
9Signal reconstruction contrast between DCT and DFT with different number of frequency components.
Fig. 10 (
10a) and (b) visualize the tensor of channel attention in FECAM and encoder layer of Transformer on the weather dataset, and Fig.10(c) and (d) visualize the tensor of channel attention in FECAM and encoder layer of Transformer on the exchange dataset. we can see FECAM can extract the importance of different channels and the significance of different frequencies with obvious patterns compared with output tensor of encoder layer in transformer.
Table 1 :
1Statistics of datasets.Dataset
Variable Number
Sampling Frequency
Total Observations
Exchange
8
1 Day
7,588
ILI
7
1 Week
966
ETTm2
7
15 Minutes
69,680
Electricity
321
1 Hour
26,304
Traffic
862
1 Hour
17,544
Weather
21
10 Minutes
52,695
Table 2 :
2Forecasting results comparison under different prediction lengths ∈ {96, 192, 336, 720}.The input sequence length is set to 36 for ILI and 96 for the others.Models
Ours
Autoformer
Pyraformer
Informer
LogTrans
Reformer
LSTNet
Metric
MSE
MAE
MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE
Exchange
96
0.085 0.208 0.197 0.323 0.852 0.780 0.847 0.752 0.968 0.812 1.065 0.829 1.551 1.058
192 0.210 0.338 0.300 0.369 0.993 0.858 1.204 0.895
1.04
0.851 1.610 1.020 1.477 1.028
336 0.344 0.445 0.509 0.524 1.240 0.958 1.672 1.036 1.659 1.081 2.226 1.192 1.507 1.031
720 0.921 0.717 1.447 0.941 1.711 1.093 2.478
1.31
1.941 1.127 1.802 1.131 2.285 1.243
ILI
24
2.101 0.939 3.483 1.287 5.800 1.693 5.764 1.677 4.475 1.444 4.400 1.382 6.026 1.770
36
2.330 0.951 3.103 1.148 6.043 1.733 4.755 1.467 4.799 1.467 4.783 1.448 5.340 1.668
48
2.557 1.061 2.669 1.085 6.213 1.763 4.763 1.469 4.800 1.468 4.832 1.465 6.080 1.787
60
2.531 1.093 2.770 1.125 6.531 1.814 5.264 1.564 5.278 1.560 4.882 1.483 5.548 1.720
Table 3 :
3Univariate results with different prediction lengths ∈ {96, 192, 336, 720} on datasets ETTm2 and Exchange. The input sequence length is set to 96.Models
Ours
N-HiTs
N-BEATS
Autoformer Pyraformer
Informer
Reformer
ARIMA
Metric
MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE
Table 5 )
5. Our method consistently improves the forecasting ability of different models. Overall, it achieves averaged 35.99% promotion on LSTM, 10.01% on Reformer, 8.71% on Informer,8.29% on Autoformer and 8.06% on Transformer, making each of them surpass previous state-of-the-art. Compared to vanilla models, only a few parameters are increased by applying our method (See
Table 4 :
4Parameters increment and performance promotion of FECAMModels
LSTM
Reformer
Informer
Autoformer
Transformer
Vanilla
13.2K
5.79MB
11.33MB
10.54MB
10.54MB
Vanilla+Ours
13.5K
5.85MB
11.39MB
10.69MB
10.60MB
Parameters increment
2.27%
1.03%
0.53%
0.57%
0.57%
Performance promotion
35.99%
10.01%
8.71%
8.29%
8.06%
Table 5 :
5Performance promotion by applying our proposed method method to Transformers and RNNs.We report the aver-
aged MSE/MAE of all prediction length (stated in Table 2) and the relative MSE reduction ratios(Promotion) by our method.
Complete results can be found in Appendix B
Dataset
Exchange
ILI
ETTm2
Electricity
Traffic
Weather
Model
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
LSTM
2.104
1.221
6.537
1.828
2.394
1.177
0.559
0.549
1.010
0.541
0.443
0.453
+Ours
1.294
0.946
4.305
1.442
1.338
0.896
0.381
0.437
0.755
0.430
0.277
0.333
promotion
38.49%
34.14%
44.11%
31.84%
25.24%
37.47%
Transformer
1.556
0.969
4.774
0.445
1.344
0.814
0.272
0.367
0.667
0.363
0.681
0.576
+Ours
1.271
0.874
4.471
1.394
1.254
0.806
0.256
0.364
0.662
0.359
0.615
0.537
promotion
18.31%
6.77%
6.69%
5.88%
0.75%
9.69%
Informer
1.550
0.998
5.136
1.544
1.410
0.822
0.31
0.396
0.764
0.415
0.633
0.548
+Ours
1.433
0.949
4.676
1.453
1.249
0.794
0.288
0.38
0.736
0.399
0.576
0.511
promotion
7.54%
8.95%
11.41%
7.09%
3.66%
10.42%
Autoformer
0.613
0.539
3.006
1.161
0.324
0.367
0.227
0.337
0.627
0.378
0.337
0.381
+Ours
0.504
0.499
2.738
1.108
0.315
0.359
0.217
0.326
0.616
0.367
0.318
0.368
promotion
17.78%
8.91%
2.77%
4.40%
1.75%
5.63%
Reformer
1.620
1.023
4.724
1.445
1.479
0.915
0.337
0.422
0.740
0.421
0.802
0.655
+Ours
1.275
0.907
4.398
1.378
1.443
0.897
0.318
0.397
0.711
0.394
0.585
0.551
promotion
21.29%
6.90%
2.43%
5.63%
3.91%
27.05%
The Electricity dataset was acquired at https://archive.ics.uci.edu/ml/datasets/Electric-ityLoadDiagrams201120142 The ETT dataset was acquired at https://github.com/zhouhaoyi/ETDataset 3 The Exchange dataset was acquired at https://github.com/thuml/Autoformer
The ILI dataset was acquired at https://gis.cdc.gov/grasp/fluview/fluportaldashboard.html
The Traffic dataset was acquired at http://pems.dot.ca.gov/6 The Weather dataset was acquired at https://www.bgc-jena.mpg.de/wetter/
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| {'fraction_non_alphanumeric': 0.05057977858229708, 'fraction_numerical': 0.045577767284047783, 'mean_word_length': 4.692751891676623, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 13, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 6, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Time series forecasting is a long-standing challenge due to the real-world information is in various scenario (e.g., energy, weather, traffic,economics, earthquake warning). However some mainstream forecasting model forecasting result is derailed dramatically from ground truth. we believe it's the reason that models' lacking ability of capturing frequency information which richly contains in real world datasets. At present, the mainstream frequency information extraction methods are Fourier transform(FT) based. However,use of FT is problematic due to Gibbs phenomenon. If the values on both sides of sequences differ significantly, oscillatory approximations are observed around both sides and high frequency noise will be introduced. Therefore We propose a novel frequency enhanced channel attention that adaptively modelling frequency interdependencies between channels based on Discrete Cosine Transform which would intrinsically avoid high frequency noise caused by problematic periodity during Fourier Transform, which is defined as Gibbs Phenomenon. We show that this network generalize extremely effectively across six real-world datasets and achieve stateof-the-art performance, we further demonstrate that frequency enhanced channel attention mechanism module can be flexibly applied to different networks. This module can improve the prediction ability of existing mainstream networks, which reduces 35.99% MSE on LSTM, 10.01% on Reformer, 8.71% on Informer, 8.29% on Autoformer, 8.06% on Transformer, etc., at a slight computational cost,with just a few line of code. Our codes and data are available at https://github.com/Zero-coder/FECAM.", 'arxivid': '2212.01209', 'author': ['Maowei Jiang [email protected] ', 'Pengyu Zeng [email protected] ', 'Kai Wang [email protected] ', 'Huan Liu [email protected] ', 'Wenbo Chen [email protected] ', 'Haoran Liu [email protected] ', 'Maowei Jiang ', 'Pengyu Zeng ', 'Kai Wang ', 'Huan Liu ', 'Wenbo Chen ', 'Haoran Liu ', '\nShenyang Institute of Automation\nChinese Academy of Sciences Shenyang\nChina\n', '\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China\n', '\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China, China, China\n', '\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China\n', '\nUniversity of Chinese Academy of Sciences\nBeijingChina\n'], 'authoraffiliation': ['Shenyang Institute of Automation\nChinese Academy of Sciences Shenyang\nChina', 'Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China', 'Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nShenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China, China, China', 'Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang\nUniversity of Chinese Academy of Sciences\nBeijingChina, China', 'University of Chinese Academy of Sciences\nBeijingChina'], 'corpusid': 254220748, 'doi': '10.48550/arxiv.2212.01209', 'github_urls': ['https://github.com/Zero-coder/FECAM.', 'https://github.com/Zero-coder/FECAM.', 'https://github.com/Zero-coder/FECAM.', 'https://github.com/zhouhaoyi/ETDataset', 'https://github.com/thuml/Autoformer'], 'n_tokens_mistral': 17806, 'n_tokens_neox': 15025, 'n_words': 8105, 'pdfsha': 'c97c1691f5ac8deba248c8bcde9cf9899a2bb552', 'pdfurls': ['https://export.arxiv.org/pdf/2212.01209v1.pdf'], 'title': ['FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting', 'FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting'], 'venue': ['FECAM: Frequency Enhanced Channel Attention Mechanism for Time Series Forecasting. PVLDB']} |
arxiv |
Equivariance with Learned Canonicalization Functions
Sékou-Oumar Kaba
Arnab Kumar Mondal
Yan Zhang
Yoshua Bengio
Siamak Ravanbakhsh
Equivariance with Learned Canonicalization Functions
Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce a canonical representation of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for some groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis, supported by our empirical results, is that learning a shallow neural network to perform canonicalization is better than using predefined heuristics. Our results show that learning the canonicalization function is competitive with existing techniques for learning equivariant functions across many tasks, including image classification, N -body dynamics prediction, point cloud classification and part segmentation, while being consistently faster across the board.
Introduction
The problem of designing machine learning models that properly exploit the structure and symmetry of the data is becoming more important as the field is broadening its scope to more complex problems. In multiple applications, the transformations with respect to which we require a model to be invariant or equivariant are known and provide a strong inductive bias (e.g., Bronstein et al., 2021;Bogatskiy et al., 2022;van der Pol et al., 2020;Mondal et al., 2020;Celledoni et al., 2021).
As is often the case, taking a step back and drawing analogies with human cognition is fruitful here. Human pattern recognition handles some symmetries with relative ease. predictions. In this example, the task is to restyle an MNIST digit in a rotation equivariant way. We propose a class of models that falls in the single-view-plus-transformation framework.
When data is transformed in a way that preserves its essential characteristics, we precisely know if and how we should adapt our response.
One context in which this has been particularly well-studied in cognitive science is visual shape recognition. Experiments have shown that subjects can accurately distinguish between different orientations of an object and actual modifications to the structure of an object (Shepard & Metzler, 1971;Carpenter & Eisenberg, 1978).
There are multiple ways in which this could be achieved. According to Tarr & Pinker (1989), theories of invariant shape recognition broadly fall into three categories: viewpointindependent models, in which object representations depend only on invariants features, multiple-view models in which an object is represented as a set of representations corresponding to different orientations, and single-view-arXiv:2211.06489v2 [cs.LG] 5 Feb 2023 plus-transformation models in which an object is converted to a canonical orientation by a transformation process.
Correspondingly, similar ideas have been explored in deep learning to achieve equivariance; see Figure 1. Models that impose equivariance through constraints in the architecture (Shawe-Taylor, 1989;Cohen & Welling, 2016a;Ravanbakhsh et al., 2017) or that only use invariants as inputs (Villar et al., 2021) can be seen as belonging to the viewpoint-independent type since the dependence of the model on symmetry transformations is trivial. The multipleview approach includes models that symmetrize the input by averaging over all the transformations or a subset of them (Manay et al., 2006;Benton et al., 2020;Yarotsky, 2022;Puny et al., 2022). By contrast, the transformation approach has seen less interest. This is all the more surprising considering that evidence from cognitive science suggests that this approach is used in human visual cognition (Shepard & Metzler, 1971;Carpenter & Eisenberg, 1978;Hinton & Parsons, 1981). When presented with a rotated version of an original pattern, the time taken by humans to do the association is proportional to the rotation angle, which is more consistent with the hypothesis that we perform a mental rotation.
Present work We introduce a systematic and general method for equivariant machine learning based on learning mappings to canonical samples. We hypothesize that among all valid canonicalization functions, some will lead to better downstream performance than others. Rather than trying to hand-engineer these functions, we may as well let them be learned in an end-to-end fashion with a prediction neural network. Our method can readily be used as an independent module that can be plugged into existing architectures to make them equivariant to a wide range of transformation groups, discrete or continuous. Our approach enjoys similar expressivity advantages to methods like frame averaging by Puny et al. (2022), but has several added benefits. It is simpler, more efficient, and replaces hand-engineered frames for each group by a systematic end-to-end learning approach.
Our contributions are as follows:
• Novel Framework: We introduce a general framework for equivariance to arbitrary groups based on mappings to canonical samples. This framework can be plugged into any existing non-equivariant architecture.
• Theoretical Guarantees: We prove that in some settings, such models are universal approximators of equivariant functions.
• Efficient Implementations: We provide multiple variants of efficient implementations of this framework to specific domains.
• Practical Performance: We perform experiments that show that the proposed method achieves excellent results on images, physical dynamical systems and point clouds. We also support our hypothesis that learning the canonicalization function is a better strategy than designing it by hand.
Related Works
Methods based on heuristics to standardize inputs have been around for a long time (Yüceer & Oflazer, 1993;Lowe, 2004). However, these approaches require significant handengineering and are difficult to generalize. An important early work is the Spatial Transformer Network (Jaderberg et al., 2015) which learns input transformations to facilitate processing in a downstream vision task. PointNet (Qi et al., 2017a) also proposed to learn an alignment network to encourage invariance for point cloud analysis. However, these approaches are closer to regularizers and provide no equivariance guarantees. The works of (Esteves et al., 2018b;Tai et al., 2019) provided equivariant versions of the Spatial Transformer using an approach based on canonical coordinates. One limitation of this approach is that it does not exactly handle equivariance to groups that are larger in dimension than the dimension of the data grid. Some recent works have proposed using learned coordinate frames for point clouds (Kofinas et al., 2021;Luo et al., 2022;Du et al., 2022). We provide theoretical and experimental evidence that the neural networks for canonicalization can be made much shallower and simpler without affecting performance. Finally, simultaneously published papers (Winter et al., 2022;Vadgama et al., 2022) have proposed to use canonicalization for equivariance in a different autoencoding setup.
Canonicalization Functions
Problem Setting
We are interested in learning functions φ : X → Y with inputs x ∈ X and outputs y ∈ Y belonging to arbitrary finite-dimensional vector spaces. We will consider a set of linear symmetry transformations T ⊂ GL (X ), where GL (X ) is the set of invertible matrices over the vector space X . This is described by a group representation ρ : G → T , where G is an abstract group. Without loss of generality, we can assume that ρ is a group isomorphism. Therefore,
the inverse ρ −1 : T → G is defined. A function φ is G-equivariant if φ (ρ (g) x) = ρ (g) φ (x) , ∀ g, x ∈ G × X ,(1)
where the group action ρ on the input and the group action ρ on the output will be clear from the context. In particular, when ρ (g) = I, we say that φ is invariant. We use ρ (H)
to denote the image of the subset H under ρ.
We call ρ (G) x = {ρ (g) x | ∀ g ∈ G} the orbit of the element x. It is the set of elements to which x can be mapped by the group action. The set of orbits, denoted X /G forms a partition of the set X .
General Formulation
The invariance requirement on a function φ amounts to having all the members of a group orbit mapped to the same image by φ. It is thus possible to achieve invariance by appropriately mapping all elements to a canonical sample from their orbit before applying any function. For equivariance, elements can be mapped to a canonical sample and, after a function is applied, transformed back according to their original position in the orbit. This can be formalized by writing the equivariant function φ in canonicalized form as
φ (x) = h (x) f h (x) −1 x(2)
where the function f : X → Y is called the prediction function and the function h :
X → ρ (G) is called the canonicalization function. Here h (x) −1 is the inverse of the representation matrix and h (x) = ρ ρ −1 (h (x)) is the counterpart of h (x) on the output.
Equivariance in Equation (2) is obtained for any prediction function if the canonicalization function is itself G-
equivariant 1 , h (ρ (g) x) = ρ (g) h (x) ∀ g, x ∈ G × X .
It may seem like the problem of obtaining an equivariant function has merely been transferred in this formulation. This is, however, not the case: in Equation (2), the equivariance and prediction components are effectively decoupled. The canonicalization function h can therefore be chosen as a simple and inexpressive equivariant function, while the heavy-lifting is done by the prediction function f .
A more general condition can be formulated, such that the decoupling is partial. This enables us to impose part of the symmetry constraint on the prediction network and use canonicalization for "additional" symmetries. This could, for example, be used to imbue a translation equivariant architecture, like a CNN, with rotation equivariance.
Theorem 3.1. For some subgroup K ≤ G, if ∀ g, x ∈ G × X there exists a k ∈ K such that h (ρ (g) x) = ρ (g) h (x) ρ (k)(3)
1 Symmetric inputs in X pose a problem if we use the standard definition of equivariance for the canonicalization function. We explain this in Appendix A and introduce a relaxed version of equivariance that solves this issue. However, because the subset of symmetric inputs is often of measure zero, using standard equivariance is not expected to be a problem in practice. and the prediction function f is K-equivariant, then φ defined in Equation (2) is G-equivariant.
The proof follows in Appendix C. This is equivalent to saying that the canonicalization function should output a representation of a coset in G/K in an equivariant way, the applied transformation being chosen arbitrarily within the coset.
This can be simplified when the group factors into a semidirect product using the following result.
Theorem 3.2. If K is a normal subgroup such that G J K, condition Equation (3) can be realized with a canonicalization function with image ρ (J), and that is Jequivariant and K-invariant.
The proof follows in Appendix D. Going back to the example of using rotation canonicalization with a CNN, Theorem 3.1 says that the canonicalization function should output an element of the Euclidean group transforming equivariantly under rotations of the input. However, since the translation subgroup is normal, Theorem 3.2 can be used to guarantee that the canonicalization network can always simply output a rotation.
In general, when K = {e}, only the canonicalization function is constrained, which is the case described at the beginning of the section. In the image domain, this would correspond to canonicalizing with respect to the full Euclidean group and using an MLP as a prediction function. The other extreme, given by K = G, corresponds to transforming the input in an arbitrary way and constraining the prediction function as is usually done in equivariant architectures like G-CNNs (Cohen & Welling, 2016b). These are, respectively, the single-view-plus-transformation and the viewpoint-independent implementations described in the introduction. Subgroups {e} < K < G offer intermediary options; the lattice of subgroups of G therefore defines a family of models. Since equivariance to a smaller group is less constraining for the prediction function, set inclusion in the subgroup lattice corresponds to increased expressivity.
Universality Result
We can now introduce a more formal result on the expressivity of equivariant functions obtained with canonicalization functions. A parameterized function φ is a universal approximator of G-equivariant functions if for any Gequivariant continuous function ψ, any compact set K ⊆ X and any > 0, there exists a choice of parameters such that ψ (x) − φ (x) < ∀ x ∈ K. We make the additional assumption that the set K is closed under the group action. The proof follows in Appendix E. The following corollary is especially relevant. Corollary 3.4. A G-equivariant parameterized function φ written as Equation (2) with a G-equivariant canonicalization function and a multilayer perceptron (MLP) as a prediction function is a universal approximator of G-equivariant functions.
This result can significantly simplify the design of universal approximators of equivariant functions since a non-universal equivariant architecture for the canonicalization function can be combined with an MLP. In particular, notice that universality of this scheme does not hinge on the expressivity of the canonicalization network.
Model Design for Canonicalization Functions
The canonicalization function can be chosen as any existing equivariant neural network architecture with the output being a group element; we call this the direct approach (figure 2a). For permutation groups and Lie groups, an equivariant multilayer perceptron (Shawe-Taylor, 1989;Finzi et al., 2021) can be used. We provide examples of implementations in the next section.
We also introduce an alternative method, which we call the optimization approach (Figure 2b). The canonicalization function can be defined as
h (x) ∈ arg min ρ(g)∈ρ(G) s (ρ (g) , x)(4)
where s : ρ (G) × X → R can be a neural network. In general, a set of elements minimize s, so one is chosen arbitrarily. The function s has to satisfy the following equivariance condition
s (ρ (g) , ρ (g 1 ) x) (5) = s ρ (g 1 ) −1 ρ (g) , x , ∀g 1 ∈ G
and has to be such that argmin is a subset of a coset of the stabilizer of x. This last condition essentially means that the minimum should be unique in each orbit up to some input symmetry. In Appendix B, we prove that these are sufficient conditions for Equation (4) to be a suitable canonicalization function.
The equivariance condition on s can be satisfied using an equivariant architecture. Remarkably, it can also be satisfied using a non-equivariant function u : X → R and by defining
s (ρ (g) , x) = u ρ (g) −1 x(6)
(a) Direct approach (b) Optimization approach Figure 2: Two general approaches to canonicalization. In the direct approach, an equivariant neural network outputs the transformation. In the optimization approach, a function of the input is minimized to obtain the canonical sample.
Intuitively, the function u then represents a distance between the input and the canonical sample of the orbit and is therefore minimized when ρ (g) is the transformation that maps to the canonical sample.
This implementation presents a close analogy with the mental rotation phenomenon described in the introduction, as humans try to minimize the distance between their representation of an object and the canonical one. As such, it is expected that the optimization process will take more iterations when the input sample is farther away in orbit from the canonical sample. This is consistent with the experimental evidence for mental rotation (Shepard & Metzler, 1971;Carpenter & Eisenberg, 1978).
Next, we elaborate on how suitable canonicalization functions can be obtained in different settings.
Euclidean Group
The Euclidean group E (d) describes rotation, translation, and reflection symmetry. Domains in which this type of symmetry is especially relevant include image processing, point cloud modelling and physics applications. Below we give the design principles to obtain equivariant models for image and point cloud inputs.
Image Input. Elements of the Euclidean group can be written as (O, t), where O ∈ R n×n is an orthogonal matrix and t ∈ R n is an arbitrary translation vector. We consider the space of image inputs I ∈ X as given by a 2D signal I : R 2 → R C , where C is the number of input channels. We adopt a continuous description to facilitate exposition, but in practice, all the operations are discretized using interpolation (Riba et al., 2020). The representation on image inputs is defined by the following linear operator
[ρ (O, t) · I](p) = I(O −1 (p − t)) ∀p ∈ R 2 ,
where p is pixel position. The canonicalization function should output an element of the E (2). It should additionally
be E (2)-equivariant, such that h (ρ (O, t) · I) = ρ (O, t) · h (I).
This condition can be satisfied by using a G-CNN (Cohen & Welling, 2016a) and the optimization approach described above. To do this, we define the function to be optimized as s : O (2) × R 2 × X → R. This can be reinterpreted as s : X → R O(2)×R 2 , which means where the first dimension, a.k.a. the fiber, encodes rotation angles and R 2 is associated with pixel positions. If s is a G-CNN, it correctly satisfies the condition Equation (5), as image rotations act on the fiber and Euclidean transformations on the pixel positions. The canonicalization is then obtained by taking the arg min over pixel positions and fibers
h (x) ∈ arg min (O,t)∈E(2) s (x) (O,t)(7)
This approach can be further simplified if we use a translation equivariant prediction network, such as a CNN-based architecture. As the translation group T (2) is a normal subgroup of the Euclidean group E(2), using Theorem 3.1, we only require the canonicalization function to be equivariant to O(2). This means we can eliminate the spatial dimension in the output feature map of the canonicalization function and only need to take an arg min along the rotation fibre dimension to identify the correct orientation of the image.
There are two potential problems with this approach. First, extending G-CNNs to higher-order rotations is computationally expensive and it leads to artifacts due to the finer rotation of filters. Second, we cannot backpropagate through the canonicalization function as the arg max operation is not differentiable.
We can avoid the first problem by using a shallower network with a larger filter size. We empirically show why this is a sound choice for canonicalization function in Section 5. We use the straight-through gradient estimator (Bengio et al., 2013) to solve the second problem. Appendix I contains a PYTORCH code snippet to perform the canonicalization function of images in a differentiable way using a G-CNN.
Since CNNs are universal approximators of T (2)equivariant functions (Yarotsky, 2022), it follows from Theorem 3.3 that a CNN augmented with an O(2) equivariant canonicalization function is a universal approximator of E(2)-equivariant functions.
Point Cloud Input. The n+1 dimensional representation of the Euclidean group (defined by concatenating a constant 1 to the original vectors) is defined in the following way
ρ(O, t) = O t t T 1(8)
We seek to define an E (d)-equivariant canonicalization function for point clouds. This can be done by defining it as
h (x) = ρ h O (x) , h t (x)
, where the function h O : X → R n×n outputs the rotation and reflection and h t : X → R n the translation. Since the product of elements of E (n) is given by
(O 1 , t 1 ) (O 2 , t 2 ) = (O 1 O 2 , O 2 t 1 + t 2 )
, the equivariance condition requires that we have
h O (ρ(O, t)x) = Oh O (x) (9) h t (ρ(O, t)x) = Oh t (x) + t(10)
This means that h O must be O (d)-equivariant and translation invariant, and that h t must be E (d)-equivariant. These constraints can be satisfied by using already existing equivariant architectures. Since most of the work will be done by a prediction function that can be very expressive, like Pointnet (Qi et al., 2017a), a simple and efficient architecture can be used for the canonicalization function, for example Vector Neurons (Deng et al., 2021). The output of h O can be made an orthogonal matrix by having it output n vectors and ortho-normalizing them with the Gram-Schmidt procedure, which is itself equivariant (Appendix F).
Using Deep Sets (Zaheer et al., 2017) as a backbone architecture would result in a universal approximator of E (d) and permutation equivariant functions, following Theorem 3.3 and Theorem 1 of (Segol & Lipman, 2020).
Symmetric Group
The symmetric group S n over a finite set of n elements contains all the permutations of that set. This group captures the inductive bias that input order should not matter. Domains for which S n -equivariance is desirable to include object modelling and detection, graph representation learning, and applications in language modelling.
S n -equivariant canonicalization functions can be obtained with a direct approach using existing optimal transport solvers (Villani, 2009). For example, the Sinkhorn algorithm (Sinkhorn, 1964;Mena et al., 2018) solves the entropyregularized optimal transport problem (Cuturi, 2013), which results in convex combinations of permutations (doublystochastic matrices) that are equivariant. Obtaining a permutation can also be framed as an optimization problem, which makes our optimization approach in Equation (4) relevant; problems like sorting (Blondel et al., 2020) and optimal transport (Blondel et al., 2018) are often formulated like this, which shows that this is a powerful paradigm.
Experiments
Image classification
We first perform an empirical analysis of the proposed framework in the image domain. We selected the Rotated MNIST dataset (Larochelle et al., 2007), which is used as a benchmark dataset that uses a classification task to test equivariant architectures in prior work (Cohen & Welling, 2016a). In Table 1, we compare our method with different CNN and G-CNN baselines. We denote the group of n discrete rotations as pn, and the networks equivariant to it by putting it with the network's name (e.g. G-CNN (p4)). The training and architecture details are provided in Appendix G.1. For the canonicalization function, we choose a shallow G-CNN with three layers. We start with a lifting layer with a filter that is the same size as the input image, followed by ReLU nonlinearity and group equivariant layers with 1 × 1 filters.
We consider two variants: an untrained canonicalization function with frozen weights and a canonicalization function learned end-to-end with the CNN as the prediction function. We call them CN(p4 & frozen)-CNN and CN(p4)-CNN. For a pure G-CNN-based baseline, we provide the value reported by Cohen & Welling (2016a) and design a variant which has similar architecture to CNN (base) while matching the number of parameters of our CN(p4)-CNN. We call this G-CNN (p4 & = params). We also consider a model with fixed canonicalization, which is performed by finding the orientation of the digits using Principal Component Analysis (PCA) and refer to it as CN(PCA)-CNN. Lastly, we implement Equation (6) as CN(OPT)-CNN. Our u converts the input image into a point cloud representation, which is fed into a PointNet that produces a score. We use gradient descent to optimize this score with respect to the input rotation for a small number of steps. This procedure is visualized in Figure 2b.
Results We see that using a fixed canonicalization function technique like PCA or canonicalization function with frozen parameters improves performance over the CNN baseline. However, learning the canonicalization function provides a significant performance improvement. Our approach outperforms all the CNN-based baselines and is comparable to G-CNNs. We also perform an ablation study to understand the role of expressivity of the canonicalization network and order of rotation in improving performance and report it in Appendix H.1. We show that the order of rotation contributes significantly more to the performance than the expressivity of the canonicalization network. We also visual-ize the canonicalized samples for different canonicalization techniques and order of rotation in Appendix H.1.
Next, we compare the inference time of our model with pure G-CNN-based architectures. For this experiment, we take the CNN architecture of our predictor network and replace the convolutions with group convolutions. As increasing the rotation order in G-CNN requires more copies of rotated filters in the lifting layer and more parameters in the subsequent group convolution layers, we decreased the number of channels to keep the number of parameters the same as our model. Figure 3 shows that although G-CNN's performance is slightly better for the p4 group, increasing the order of discrete rotations improves our model's performance significantly compared to G-CNN. In addition to performance gain, our model's inference speed remains more or less constant while encoding invariance to higher-order rotations due to the shallow canonicalization network. This makes our approach more suitable for building equivariance for bigger groups and network architectures.
N -body dynamics prediction
Simulation of physical dynamics is an important class of E(3)-equivariant problems due to the symmetry of physical laws under rotations and translations. We evaluate our framework in this setting with the N -body dynamics prediction task proposed by (Kipf et al., 2018) and (Fuchs et al., 2020). In this task, the model has to predict the future positions of 5 charged particles interacting with Coulomb force given initial positions and velocities. We use the same version of the dataset and setup as (Satorras et al., 2021).
For this experiment, our architecture uses a simple 2-layer Vector Neurons version of the Deep Sets architecture for the canonicalization function (Deng et al., 2021;Zaheer et al., 2017). The prediction function is a 4-layer Graph Neural Network (GNN) with the same hyperparameters as the one used in (Satorras et al., 2021), and (Puny et al., 2022) for a fair comparison. The architecture of the prediction network was, therefore, not optimized. The canonicalization network is much smaller than the prediction GNN, with around 20 Results Table 2 shows that we obtain state-of-the-art results. The improvement with respect to Frame Averaging is significant, showing that learning the canonicalization provides an important advantage. Our approach also does better than all the intrinsically equivariant (or viewpointindependent to use the classification introduced earlier) baselines both in accuracy and efficiency. This shows that canonicalization can be used to obtain equivariant models with high generalization abilities without sophisticated architectural choices.
ABLATION STUDY
We also test variants of the model. First, we test on a variant of the model where the canonicalization is only learned for the O (3) part of the transformation and where the translation part is given by the centroid. Since for this system all the masses are identical this is the same as the center of mass of the system. The result is reported in Table 2 as CN-GNN-O(3). We obtain only marginally worse performance compared to the fully trained canonicalization function. This shows that, in this setting, the centroid provides an already suitable canonicalization function, which is expected given the physical soundness of choosing the center of mass as the origin of the reference frame. Since the learned translation canonicalization performs on par with this physically motivated canonicalization, this also validates the method.
Second, we compare with a version of the model where the weights of the canonicalization function are frozen at initialization. This canonicalization still provides E(n)equivariance and, as expected, provides a significant improvement of more than 20% with respect to the GNN prediction function alone. However, the learned canonicalization function provides a close to 50% improvement in performance compared to this fixed canonicalization.
Point cloud classification
We use the ModelNet40 (Wu et al., 2015) and ShapeNet (Chang et al., 2015) datasets for experiments on point clouds. The ModelNet40 dataset consists of 40 classes of 3D models, with a total of 12,311 models. 9,843 models were used for training, and the remaining models were used for testing in the classification task. The ShapeNet dataset was used for part segmentation with the ShapeNet-part subset, which includes 16 categories of objects and more than 30,000 models. In the classification and segmentation task, the train/test rotation setup adhered to the conventions established by (Esteves et al., 2018a) and adopted by (Deng et al., 2021). Three settings were implemented: z/z, z/SO (3), and SO(3)/SO(3). The notation z denotes data augmentation with rotations around the z-axis during training, while SO(3) represents arbitrary rotations. The notation x/y denotes that transformation x is applied during training and transformation y is applied during testing. (Wu et al., 2015) in three train/test scenarios. This table is borrowed from (Deng et al., 2021). z here stands for aligned data augmented by random rotations around the vertical axis, and SO(3) indicates data augmented by random 3D rotations. (Zhang et al., 2019b) 93.1 19.9 87.8 PointNet++ (Qi et al., 2017b) 91.8 28.4 85.0 PointCNN 92.5 41.2 84.5 Spherical-CNN (Esteves et al., 2018a) 88.9 76.7 86.9 a 3 S-CNN (Liu et al., 2018) 89.6 87.9 88.7 SFCNN (Rao et al., 2019) 91.4 84.8 90.1 TFN (Thomas et al., 2018) 88.5 85.3 87.6 RI-Conv (Zhang et al., 2019a) 86.5 86.4 86.4 SPHNet (Poulenard et al., 2019) 87.7 86.6 87.6 ClusterNet (Chen et al., 2019) 87.1 87.1 87.1 GC-Conv (Zhang et al., 2020) 89.0 89.1 89.2 RI-Framework (Li et al., 2020) 89 We design our Canonicalization Network (CN) using layers from Vector Neurons (Deng et al., 2021), where the final output contains three 3D vectors that are obtained by pooling over the entire point cloud. We then orthonormalize the three vectors using the Gram-Schmidt orthonormalization process to define a 3D ortho-normal coordinate frame or a rotation matrix. We canonicalize the point cloud by acting on it using this rotation matrix. We use a two-layered Vector Neuron network followed by global pooling, which we call CN(NL). To prove our hypothesis that the canonicalization function can be inexpressive, we use a single linear layer of Vector neuron followed by pooling and call this model CN(L). Furthermore, to understand the significance of learning canonicalization, we freeze the weights of the CN and call this variant CN(frozen). We use PointNet and DGCNN as our prediction networks in our experiments.
Results Table 3 Table 4. In particular, we observe three trends in our point cloud results: 1) Learning canonicalization slightly improves the performance, 2) Using shallow linear canonicalization is enough, and 3) The performance of the prediction network bottlenecks the model's performance. This verifies our hypothesis that decoupling the equivariance using a shallow canonicalization network results in a better and more expressive non-equivariant prediction network to improve the performance of the task while still being equivariant. In Table 5, we also show that the inference time of our algorithm is dominated by the prediction network's inference time. The overhead of canonicalization is negligible, which makes our method faster than existing methods that modify the entire architecture like Vector Neurons (Deng et al., 2021).
Conclusion
In this work, we propose using a learned canonicalization function to obtain equivariant machine learning models. These canonicalization functions can conveniently be plugged into existing architectures, resulting in highly expressive models. We have described general approaches to obtain canonicalization functions and specific implementations for the Euclidean group (for images and point clouds) and the symmetric group.
We performed experimental studies in the image, dynamical systems and point cloud domains to test our hypotheses. First, we show that our approach achieves comparable or better performance than baselines on invariant tasks. Importantly, learning the canonical network is a better approach than using a fixed mapping, either a frozen neural network or a heuristic approach. Our results also show that the canonicalization function can be realized with a shallow equivariant network without hindering performance. Finally, we show that this approach reduces inference time and is more suitable for bigger groups than G-CNNs on images. 49.3 78.6 VN-PointNet (Deng et al., 2021) 72.4 72.8 VN-DGCNN (Deng et al., 2021) 81.4 81.4 PointCNN 34.7 71.4 PointNet++ (Qi et al., 2017b) 48.3 76.7 ShellNet (Zhang et al., 2019b) 47.2 77.1 RI-Conv (Zhang et al., 2019a) 75.3 75.3 TFN (Thomas et al., 2018) 76.8 76.2 GC-Conv (Zhang et al., 2020) 77.2 77.3 RI-Framework (Li et al., 2020) 79 Multiple extensions of this framework are possible. Future work could include experimentation on canonicalization for the symmetric group. Other ways to build canonicalization functions could also be investigated, such as using steerable networks for images. The function would output an orientation fibre that transforms by the irreducible representation of the special orthogonal group. Understanding how design choices for canonicalization functions affect downstream performance would also be a potentially fruitful research direction. Finally, making large pretrained architectures equivariant using this framework could be an exciting extension.
I Algorithm for Image Inputs 19
A Symmetric inputs and relaxed equivariance
We say that an input x ∈ X is symmetric if its stabilizer subgroup G x = {g ∈ G | ρ (g) x = x} is non-trivial.
Given any g 1 , g 2 ∈ G, a necessary and sufficient condition for
ρ (g 1 ) x = ρ (g 2 ) x(11)
is that g 1 and g 2 are part of the same coset for the stabilizer, e.g. g 1 , g 2 ∈ gG x . This follows from the well-known relation between orbits and stabilizers. Therefore, symmetric inputs are always fixed by multiple group elements.
Symmetric inputs are problematic when using the standard definition of equivariance for the canonicalization function because for g 1 , g 2 ∈ gG x , we have
h (ρ (g 1 ) x) = h (ρ (g 2 ) x) (12) ρ (g 1 ) h (x) = ρ (g 2 ) h (x)(13)
For general group actions, there will not exist a h (x) ∈ ρ (G) such that the last equality is satisfied.
A relaxed version of equivariance can be defined to address this issue.
Definition A.1 (Relaxed equivariance). Given group representations ρ : G → GL (X ) and ρ : G → GL (Y), a function h : X → Y satisfies the relaxed equivariance condition if ∀g 1 , x ∈ G × X there exists a g 2 ∈ g 1 G x such that
h (ρ (g 1 ) x) = ρ (g 2 ) h (x)(14)
This is a generalization of the idea of multiset-equivariance introduced by Zhang et al. (2022) to arbitrary group representations. When G x = {e}, standard equivariance is recovered. Canonicalization functions satisfying this condition do not suffer from the aforementioned problem. In addition, this condition is sufficient to obtain relaxed equivariance for canonicalized functions (Eq. 2). This is because, for g 2 ∈ g 1 G x :
φ (ρ (g 1 ) x) = h (ρ (g 1 ) x) f h (ρ (g 1 ) x) −1 ρ (g 1 ) x (15) φ (ρ (g 1 ) x) = ρ (g 2 ) h (x) f h (x) −1 ρ (g 2 ) −1 ρ (g 1 ) x (16) φ (ρ (g 1 ) x) = ρ (g 2 ) h (x) f h (x) −1 ρ g 2 −1 g 1 x(17)
Using the fact that g 2
−1 g 1 ∈ G x , φ (ρ (g 1 ) x) = ρ (g 2 ) h (x) f h (x) −1 x (18) φ (ρ (g 1 ) x) = ρ (g 2 ) φ (x)(19)
The fact that we obtain relaxed equivariance for the canonicalized function is a feature rather than a bug. This captures the desideratum that a function should be able to output asymmetric outputs from symmetric inputs, which is not the case for standard equivariant functions. Figure 4 shows a simple example of a function that cannot be learned by a translation or rotation equivariant neural network.
Assuming that the input to the neural network is an image with uniform pixel values (fully symmetric), the equivariant neural network cannot learn to output any image with less symmetry. Yet, it is clear that in many situations such functions should not be excluded for symmetry reasons.
B Optimization approach to canonicalization
In this appendix, we provide a more formal description of canonicalization functions obtained with the optimization approach of section 4.
We prove a theorem providing a sufficient condition for a canonicalization function to satisfy the relaxed equivariance condition A.1.
Theorem B.1. Let h (x) ∈ arg min ρ(g)∈ρ(G) s (ρ (g) , x) for some s : ρ (G) × X → R. If the conditions 1. ∀g, g 1 ∈ G, ∀x ∈ X , s (ρ (g) , ρ (g 1 ) x) = s ρ (g 1 ) −1 ρ (g) , x 2. ∀x, ∃g 1 ∈ G, such that arg min ρ(g)∈ρ(G) s (ρ (g) , x) ⊆ ρ (G x g 1 )
where G x is the stabilizer subgroup of x are satisfied then h (x) satisfies the relaxed equivariance condition A.1.
Proof. Let us introduce the shorthand notation ρ (H x ) = arg min ρ(g)∈ρ(G) s (ρ (g) , x). We define H x as a subset of G such that its image under ρ is the argmin.
We have
ρ H ρ(g1)x = arg min ρ(g)∈ρ(G) s (ρ (g) , ρ (g 1 ) x)(20)
Using condition 1, we have
ρ H ρ(g1)x = arg min ρ(g)∈ρ(G) s ρ (g 1 ) −1 ρ (g) , x(21)
We can use the fact that left multiplication by ρ (g 1 ) of the elements of ρ (H x ) will give the argmin in the previous equation. Therefore we have ρ H ρ(g1)x = ρ (g 1 H x ).
Next, using condition 2, there exists a g 3 ∈ G such that ρ (H
x ) ⊆ ρ (G x g 3 ) and ρ (g 1 H x ) ⊆ ρ (g 1 G x g 3 ).
Finally, we can show that for any h (x) ∈ ρ (G x g 3 ) and h (ρ (g 1 ) x) ∈ ρ (g 1 G x g 3 ), there is a g 2 ∈ g 1 G x such that
h (ρ (g 1 ) x) = ρ (g 2 ) h (x)(22)
The left-hand side can be expressed as h
(ρ (g 1 ) x) = ρ (g 1 ) h (x), where h (x) ∈ ρ (G x g 3 ).
We then find
ρ (g 2 ) = ρ (g 1 ) h (x) h (x) −1(23)
Since h (x) and h (x) are part of the same coset of the stabilizer, h (x) h (x) −1 must be part of the stabilizer. This completes the proof. Now let us discuss how the conditions of Theorem B.1 can be met.
One way to satisfy the first condition is by using an equivariant function. Notice that the function s : ρ (G) × X → R can be reinterpreted as s : X → R ρ(G) . Therefore s can be seen as a function of the input outputting a vector for which the components index the group representation. This vector should transform equivariently for condition 1 to be satisfied, e.g.
s (ρ (g 1 ) x) ρ(g) = s (x) ρ(g) −1 1 ρ(g) .
Another way to satisfy condition 1 is to define s (ρ (g) , x) = u ρ (g) −1 x . It is easy to verify that this will indeed satisfy the condition.
Finally, as stated in the main text, condition 2 amounts to having a unique minimum in each orbit up an element of the stabilizer of the input. We will not show formally how this can be satisfied, but this is not expected to be a problem in practice. We conjecture that under weak assumptions, following the result of (Cox, 2020), neural network functions can be obtained such that this condition is satisfied almost surely. In addition, for continuous groups, optimization can be made easier by making these neural network functions convex, which can be done using the framework of ICNN (Amos et al., 2017).
C Proof of Theorem 3.1
We prove Theorem 3.1 which shows equivariance for a general subgroup K.
Proof. We have
φ (ρ (g) x) = h (ρ (g) x) f h (ρ (g) x) −1 ρ (g) x(24)
If equation 3 is satisfied, then ∀ g, x ∈ G × X there is a k ∈ K such that
φ (ρ (g) x) = ρ (g) h (x) ρ (k) f ρ (g) h (x) ρ (k) −1 −1 ρ (g) x (25) φ (ρ (g) x) = ρ (g) h (x) ρ (k) f ρ (k) −1 h (x) −1 ρ (g) −1 ρ (g) x(26)
Using the K-equivariance of f , we obtain
φ (ρ (g) x) = ρ (g) h (x) ρ (k) ρ (k) −1 f h (x) −1 x (27) φ (ρ (g) x) = ρ (g) h (x) f h (x) −1 x(28)
D Proof of Theorem 3.2
Proof. We consider the special case where K is a normal subgroup of G such that the group can be taken to be isomorphic to a semidirect product G K J. Then, group elements can be written as g = (k, j), where k ∈ K and j ∈ J. The product is defined as g 1 g 2 = (k 1 , j 1 ) (k 2 , j 2 ) = (k 1 ϕ [j 1 ] (k 2 ) , j 1 j 2 ), where ϕ : J → Aut (K) is a group homomorphism. Setting k 2 = e and j 1 = e, we get any group element as (k 1 , e) (e, j 2 ) = (k 1 , j 2 ).
If the canonicalization function is J-equivariant and K-invariant, we have
h (ρ (k, j) x) = h (ρ (k, e) ρ (e, j) x) (29) h (ρ (k, j) x) = ρ (e, j) h (x)(30)
We then show that there is a k ∈ K such that equation 3 is satisfied.
Multiplying by ρ (e) = ρ (k, e) ρ (e, j) h (x) h (x) −1 ρ (e, j) −1 ρ (k, e) −1 on the left, we have ρ (e, j) h (x) = ρ (k, e) ρ (e, j) h (x) h (x) −1 ρ (e, j) −1 ρ (k, e) −1 ρ (e, j) h (x)(31)
Using the fact that conjugation of an element of K by an element of G preserves K membership, we define
ρ (k , e) = h (x) −1 ρ (e, j) −1 ρ (k, e) −1 ρ (e, j) h (x) ρ (e, j) h (x) = ρ (k, e) ρ (e, j) h (x) ρ (k , e)(32)
which shows that equation 3 is satisfied.
Finally, we show that in this case, the image of h can be chosen to be ρ (J). We first remark that in each orbit X /G of the group action, the canonical samplex can be obtained from any orbit member x, asx = h (x) −1 x. For the canonical sample, we must have a k ∈ K such that
h h (x) −1 x = h (x) −1 h (x) ρ (k, e)(33)
If we impose k = e to satisfy this condition, we have h (x) = ρ (e, e).
Since any orbit member can conversely be written as x = ρ (k, j)x for some k ∈ K and j ∈ J, if the canonicalization function is J-equivariant and K-invariant, we have
h (x) = h (ρ (k, j)x) (34) h (x) = ρ (e, j) h (x) (35) h (x) = ρ (e, j)(36)
which completes the proof.
E Proof of Theorem 3.3
Proof. The proof is inspired by the symmetrization approach of (Yarotsky, 2022).
Let ψ be an arbitrary G-equivariant function, and φ be defined by equation 2. By the equivariance of ψ, we have
ψ (x) − φ (x) = h (x) ψ h (x) −1 x − h (x) f h (x) −1 x(37)
Since Y is finite-dimensional, we know that linear operators in GL (Y) are bounded. This means that for every representation matrix there exists a positive number c that bounds the induced operator norm, e.g. ∀g ∈ G, ∃c > 0, ρ (g) ≤ c.
We therefore obtain
ψ (x) − φ (x) ≤ h (x) ψ h (x) −1 x − f h (x) −1 x (38) ψ (x) − φ (x) ≤ c ψ h (x) −1 x − f h (x) −1 x(39)
where c > 0.
If f is a universal approximator of K-equivarant functions, then it is also a universal approximator of G-equivariant functions. We therefore have
ψ (x) − f (x) ≤ , ∀ x ∈ K(40)
Using the fact that K is closed under the group action, we obtain the desired result
ψ (x) − φ (x) ≤ , ∀ x ∈ K(41)
F Proof of Theorem 5
Theorem F.1. The Gram-Schmidt process is O (n)-equivariant.
Proof. Given n linearly independent input vectors v 1 , . . . , v n , the Gram-Schmidt process first produces the orthogonal vectors u 1 , . . . , u n , with
u i (v 1 , . . . , v n ) = v i − i−1 j=1 u j · v i u j 2 u j(42)
The orthonormal basis e 1 , . . . , e n is then given by
e i (v 1 , . . . , v n ) = u i u i(43)
We wish to prove that ∀i ≤ n, O ∈ O (n), we have
e i (Ov 1 , . . . , Ov n ) = Oe i (v 1 , . . . , v n )(44)
We first prove equivariance of (42) by strong induction. Consider the base case with i = 1. We have u 1 (v 1 , . . . , v n ) = v 1 , which is trivially equivariant. Then, we make the induction hypothesis that (42) is equivariant for 1 ≤ i ≤ k. We can show that this implies equivariance for i = k + 1. We have
u k+1 (v 1 , . . . , v n ) = v k+1 − k j=1 u j · v k+1 u j 2 u j(45)
Using the induction hypothesis, we obtain u k+1 (Ov 1 , . . . , Ov n ) = Ov k+1 − k j=1
Ou j · Ov k+1
Ou j 2 Ou j
Since the dot product and the Euclidean norm are O (n)-invariant, we obtain
u k+1 (Ov 1 , . . . , Ov n ) = Ov k+1 − k j=1 u j · v k+1 u j 2 Ou j (47) u k+1 (Ov 1 , . . . , Ov n ) = O v k+1 − k j=1 u j · v k+1 u j 2 u j (48)
which completes the induction.
We finally see that by O (n)-invariance of the Euclidean norm, (43) is also equivariant. Since the composition of equivariant functions is equivariant, we find that the Gram-Schmidt process is equivariant and this completes the proof.
G Implementation details G.1 Image classification experiments
Training details. In all our image experiments, we train the models by minimizing the cross entropy loss for 100 epochs using Adam (Kingma & Ba, 2014) with a learning rate of 0.001. We perform early stopping based on the classification performance of the validation dataset with a patience of 20 epochs.
CNN architecture. For CNN (base), we choose an architecture with 7 layers where layer 1 to 3 has 32, 4 to 6 has 64, and layer 7 has 128 channels, respectively. Instead of pooling, we use convolution filters of size 5 × 5 with a stride 2 at layers 4 and 7. The remaining convolutions have filters of size 3 × 3 and stride 1. All the layers are followed by batch-norm (Ioffe & Szegedy, 2015) and ReLU activation with dropout(p=0.4) only at layers 4 and 7.
G-CNN architecture. We took the same CNN architecture as above and replaced the standard convolutions with group convolutions (Cohen & Welling, 2016b).
G.2 N -body dynamics prediction experiments
Training details. We train on mean square error (MSE) loss between predicted and ground truth using the Adam optimizer. We train for 10.000 epochs and use early stopping. We use weight decay 10 −8 and dropout in the canonicalization function with p = 0.5.
Canonicalization network architecture. We use a Vector Neurons version of the Deep Sets architecture for the canonicalization network in this task. The network has two layers with hidden dimension size of 32.
Prediction network architecture. The GNN prediction network uses the same architecture as (Satorras et al., 2021).
H Additional results
H.1 Image classification Table 6: Impact of the number of layers in canonicalization function network and order of the discrete rotations to which it is equivariant on the performance.
#lyrs
Order of the discrete rotation group p4 p8 p16 p32 p64 1 2.52 ± 0.12 2.37 ± 0.09 2.20 ± 0.08 2.05 ± 0.15 2.01 ± 0.09 2 2.44 ± 0.06 2.31 ± 0.05 2.16 ± 0.09 2.00 ± 0.07 2.02 ± 0.12 3 2.41 ± 0.11 2.28 ± 0.09 2.11 ± 0.06 1.98 ± 0.09 1.99 ± 0.10 First, we vary the number of layers of the canonicalization network and the number of rotations it is equivariant to. For this, we extend the layers of G-CNN to any arbitrary rotations. As we noticed that using a larger filter leads to better performance for higher order rotations, we stick to architecture with a lifting layer with image-sized filters followed by 1 × 1 filters. From Table 6, we notice that adding equivariance to higher order rotation in the canonicalization function leads to significant performance improvement compared to adding more layers. Figure 5 shows the canonical orientation resulting from the learnt canonicalization function with a single lifting layer on 90 randomly sampled images of class 7 from the test dataset. This suggests that having a shallow network is enough to learn the correct canonicalization function with a sufficiently high order of discrete rotations. For p64, we see that all the similar-looking samples are aligned in one particular orientation. In contrast, although techniques like PCA or freezing parameters of the canonicalization function find the correct canonicalization function for simple digits like 1, they struggle to find stable mappings for more complicated digits like 7.
*
Equal contribution 1 School of Computer Science, McGill University, Montréal, Canada 2 Mila -Quebec Artficial Intelligence Institute, Montréal, Canada 3 Samsung -SAIT AI Lab, Montréal, Canada 4 DIRO, Université de Montréal, Montréal, Canada. Correspondence to: Sékou-Oumar Kaba <[email protected]>. Copyright 2023 by the author(s).
Figure 1 :
1A classification of different frameworks for equivariant
Theorem 3.3. G-equivariant parameterized function φ of Equation(2), which satisfies Equation(3)with K ≤ G, is a universal approximator of G-equivariant functions, if the prediction function is a universal approximator of Kequivariant functions.
Figure 3 :
3Inference time comparison of our method with G-CNN with increasing order of rotations.
contains the results of the ShapeNet experiment, showing the classification accuracy for different augmentation strategies during training and evaluation: z/z, z/SO(3), and SO(3)/SO(3). Our method, which includes CN(frozen)-PointNet, CN(L)-PointNet, CN(NL)-PointNet, CN(frozen)-DGCNN, CN(L)-DGCNN, and CN(NL)-DGCNN, demonstrates competitive results across all rotation types. We achieve similar results in the ShapeNet part segmentation task as presented in
Figure 4 :
4Example of task that cannot be performed by an equivariant function.
Table 1 :
1Comparison with the existing work for Rotated-MNIST.Method
Error % ↓
CNN (base)
4.90 ± 0.20
G-CNN (p4)
2.28 ± 0.00
G-CNN (p4 & = params)
2.36 ± 0.15
G-CNN (p64 & = params) 2.28 ± 0.10
Ours
CN(PCA)-CNN
3.35 ± 0.21
CN(p4 & frozen)-CNN
3.91 ± 0.12
CN(p4)-CNN
2.41 ± 0.10
CN(p64)-CNN
1.99 ± 0.10
CN(OPT)-CNN
3.35 ± 0.00
Table 2 :
2Test MSE for the N-body dynamics prediction task.Method
MSE
Linear
0.0819
SE(3) Transformer
0.0244
TFN
0.0155
GNN
0.0107
Radial Field
0.0104
EGNN
0.0071
FA-GNN
0.0057
CN-GNN
0.0043 ± 0.0001
CN-GNN-O(3)
0.0045 ± 0.0001
CN-GNN (frozen)
0.0085 ± 0.0002
times fewer parameters. This allows us to test the hypothesis
again that only a simple canonicalization function is neces-
sary to achieve good performance. Appendix G.2 contains
more details on the architecture and training setup.
Table 3 :
3Test classification accuracy of different pointcloud models on the ModelNet40 dataset
Table 4 :
4ShapeNet part segmentation results. Overall aver-
age category mean IoU over 16 categories in two train/test
scenarios are reported. z here stands for aligned data aug-
mented by random rotations around the vertical axis, and
SO(3) indicates data augmented by random 3D rotations
Methods
z/SO(3)
SO(3)/SO(3)
Point / mesh inputs
PointNet (Qi et al., 2017a)
38.0
62.3
DGCNN
Table 5 :
5Inference time (in seconds) of the networks for
ModelNet40 classification test split in 1 A100 and 8 CPUs
with a batch size of 32. Vanilla denotes no modification to
the base network, while Vector Neuron and Canonicalization
denote that the base network is redesigned/enhanced with
them to be equivariant.
Base Network Vanilla Vector Neuron Canonicalization
PointNet
18s
30s
20s
DGCNN
23s
39s
25s
AcknowledgementsWe thank Erik J. Bekkers, Pim de Haan, Aristide Baratin, Guillaume Huguet, Sébastien Lachapelle and Miltiadis Ko-Equivariance with Learned Canonicalization Functions finas for their valuable comments. This project is in part supported by the CIFAR AI chairs program and NSERC Discovery. S.-O. K.'s research is also supported by IVADO and the DeepMind Scholarship. Computational resources were provided by Mila and Compute Canada.Appendix
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| {'fraction_non_alphanumeric': 0.06689047530585707, 'fraction_numerical': 0.03367745017382766, 'mean_word_length': 4.222299216963481, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, 'https://': 13, 'lorem ipsum': 0, 'www.': 3, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce a canonical representation of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for some groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis, supported by our empirical results, is that learning a shallow neural network to perform canonicalization is better than using predefined heuristics. Our results show that learning the canonicalization function is competitive with existing techniques for learning equivariant functions across many tasks, including image classification, N -body dynamics prediction, point cloud classification and part segmentation, while being consistently faster across the board.', 'arxivid': '2211.06489', 'author': ['Sékou-Oumar Kaba ', 'Arnab Kumar Mondal ', 'Yan Zhang ', 'Yoshua Bengio ', 'Siamak Ravanbakhsh '], 'authoraffiliation': [], 'corpusid': 253510474, 'doi': '10.48550/arxiv.2211.06489', 'github_urls': [], 'n_tokens_mistral': 23622, 'n_tokens_neox': 20295, 'n_words': 11920, 'pdfsha': 'e67024ee99942aeaa84f539a5ecec2a9edf49311', 'pdfurls': ['https://export.arxiv.org/pdf/2211.06489v2.pdf'], 'title': ['Equivariance with Learned Canonicalization Functions', 'Equivariance with Learned Canonicalization Functions'], 'venue': []} |
arxiv |
Finding a Burst of Positives via Nonadaptive Semiquantitative Group Testing
Yun-Han Li [email protected]
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign
USA
Ryan Gabrys [email protected]
.Naval Information Warfare Center
San Diego
Jin Sima [email protected]
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign
USA
Ilan Shomorony [email protected]
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign
USA
Olgica Milenkovic [email protected]
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign
USA
Finding a Burst of Positives via Nonadaptive Semiquantitative Group Testing
arXiv:2304.01365v1 [cs.IT] 3 Apr 2023
Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing [1] as well as work on semiquantitative group testing (SQGT)[2]. Specifically, we study the setting where the burst-length ℓ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds η1, . . . , ηs. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold ηs. In this context, our main result is an order-optimal test scheme that can recover any burst of length ℓ using roughly ℓ 2ηs + log s+1 (n) measurements. This suggests that a large saturation level ηs is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.
Abstract-Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing [1] as well as work on semiquantitative group testing (SQGT) [2]. Specifically, we study the setting where the burst-length ℓ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds η1, . . . , ηs. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold ηs. In this context, our main result is an order-optimal test scheme that can recover any burst of length ℓ using roughly ℓ 2ηs + log s+1 (n) measurements. This suggests that a large saturation level ηs is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.
I. INTRODUCTION
Group testing (GT) is a protocol for identifying relatively small subsets of marked elements, referred to as positives, within a larger collection of entities termed test subjects. The gist of the approach is to group subjects into carefully selected subpools and test the subjects in each subpool jointly so as to reduce the number of tests compared to that needed for individual testing. The first GT scheme comprising two stages of testing was described by Dorfman [3] in the context of finding individuals with venereal diseases. His scheme also represents the first instance of adaptive testing, where measurements from one round of testing can be used to inform the test selections in subsequent rounds. Unlike adaptive testing, nonadaptive GT requires that all tests be designed and conducted simultaneously. Since Dorfman's work, GT has been extended and generalized in many different directions and has found numerous applications in search systems, experimental and circuit design and computational biology. For comprehensive surveys, the interested reader is referred to [4], [5].
In [1], Colbourn considered a specialized GT technique for identifying one single burst of consecutive positives of length ≤ ℓ within an ordered list of n elements. For nonadaptive techniques, Colbourn showed that the order-optimal number of measurements equals ℓ + log(n). Follow-up works focused on improving some aspects of the scheme [6]- [8], extending the results to include new adaptive protocols [9], and generalizing the approach to handle multiple bursts [10].
However, in many real-life scenarios, such as testing for infections with viral pathogens based on quantitative PCR (quantitative polymerase chain reaction, qPCR), the outcomes are realvalued and usually confined to an interval such as [10,45]. A measurement is known as the C t (cycle threshold) value and it conveys information about how likely an individual is to be infected. For example, a C t value close to 40 is highly indicative of a negative subject, while a value below 20 is a strong sign that the individual is highly virulent. One can therefore quantize the C t values using a carefully selected collection of s thresholds η = (η 1 , . . . , η s ) so that each quantization bin provides an estimate of the viral load in the pool and, consequently, an estimate of the number of positives in the pool. This type of GT approach is known as semiquantitative GT (SQGT) [2], [11]. Furthermore, whenever testing is done on large populations in which individuals that cohabitate are naturally adjacent in the order used for testing [12] (for example, families, dorm-mates etc.), bursty positive models are appropriate and can result in significant savings compared to classical GT approaches [1].
Given the additional quantitative information and the assumption regarding consecutive orderings of positives, one can easily envision performing SQGT for bursty positive that quantizes the C t values into quantization bins that indicate the level of the viral load, or an estimate of the number of infected individuals in the population [13], [14]. Here, for the first time, we study the reduction in the number of group measurements achievable in such a setting. In particular, we investigate two new bursty SQGT models [2], one in which the length of the burst is known and fixed to ℓ (henceforth referred to as the fixed-length burst model, B(n, ℓ, η)); and another, in which the length of the burst is known to be upper-bounded by ℓ (henceforth referred to as the bounded-length burst model, B(n, ≤ ℓ, η)).
Our main contributions include 1) Order-optimal constructions (i.e., constructive lower and upper bounds that differ by a constant factor of 2) for the B(n, ℓ, η) setting with quantization thresholds η for which ℓ = Ω (η s log 2 (η s )). 2) Order-optimal constructions (i.e., lower and upper bounds that differ by a constant factor of 4) for the B(n, ≤ ℓ, η) setting with SQGT thresholds η = (1, . . . , s) corresponding to the so-called saturation model [13], [15]. Two important comments are in place. Semiquantitative measurements significantly decrease the number of tests needed for the B(n, ≤ ℓ, η) setting (the improvement is linear in the number of thresholds s). Somewhat surprisingly, for the B(n, ℓ, η) setting the number of tests is basically determined by the value of the largest threshold η s rather than by the total number of thresholds s. These findings may have interesting consequences for test schedules and quantization schemes used for practical quantitative PCR protocols.
The paper is organized as follows. Section II introduces the notation and provides the formal problem formulation. Section III contains the results for the lower bounds, while Section IV contains the main results of the work, pertaining to upper bounds on the number of SQGT burst identification models for a fixed and upper-bounded length of the burst.
II. PROBLEM FORMULATION
We start by introducing the relevant notation as well as the fixed-length and bounded-length single burst identification problems under SQGT measurements.
Let
(so that R (M) ( * , i) = M( * , w M − i − 1), a c-fold horizontal concate- nation of M (i.e, [M, . . . , M]
with c constituent matrices), and a horizontal concatenation of matrices M such that w M∞ becomes a value specified during the construction process, respectively. Finally, we use M(i, j) to denote the entry in M in row i and column j.
The single burst of positives problem requires introducing the following notions.
Bursts: A burst is denoted by a binary n × 1 column vector b, and is specified by a head and tail
h b ≤ t b , which dictate its length ℓ b = t b − h b + 1. It comprises consecutive positives: b(i) = 0, 0 ≤ i < h b , 1, h b ≤ i ≤ t b , 0, t b < i ≤ n.
When ℓ b is fixed, b i denotes the burst with h b = i, and the distance between two burst b i , b j is defined as the difference of their head position |i − j|.
SQGT measurements: An SQGT measurement is described by a 1 × n binary vector m such that
m (i) = 1, ith element is included in the test, 0, otherwise,
and a set of integer-valued quantized thresholds
η = (η 1 , . . . , η s ) with 0 < η 1 < . . . < η s ≤ n,
such that the SQGT measurement outcomes equal
η(mb) = 0, 0 ≤ mb < η 1 , i, η i ≤ mb < η i+1 , s, η s ≤ mb ≤ n.
Definition 2.1: When η = (1, 2, . . . , s), we refer to this specialized SQGT scheme as the saturation SQGT model.
Correct burst detection: for any hidden burst b, the estimate generated by the detection algorithm, denoted byb, should equal b.
Definition 2.2: B(n, ℓ, η) and B(n, ≤ ℓ, η) are used to denote the fixed-length and bounded-length burst problem with burstlengths = ℓ and ≤ ℓ, respectively, and with n test elements and SQGT quantized thresholds η.
A nonadaptive SQGT testing scheme with m measurements on n elements is represented by a m×n binary measurement matrix M with each row corresponding to a single SQGT measurement. We say M solves the B(n, ℓ, η) (or the B(n, ≤ ℓ, η) ) problem if and only if
∀b = b ′ allowed by the B(n, ℓ, η) (B(n, ≤ ℓ, η) ) problem, one has η(Mb) = η(Mb ′ ).
The smallest possible number of measurements possible to meet this requirement, among all nonadaptive SQGT schemes is denoted by m * B(n, ℓ, η) and m * B(n, ≤ ℓ, η) . Our constructions will make use of Gray codes and generalizations thereof. We say that G s,h ∈ {0, . . . , s} h×s h represents an s-ary Gray code with length h if it satisfies the following two conditions:
1) Any two consecutive columns differ in exactly one position, and the difference has magnitude one. 2) G s,h includes all possible s h codewords exactly once. Example 2.1: The following matrix has columns that constitute a 3-ary Gray code of length two: 0 0 0 1 1 1 2 2 2 0 1 2 2 1 0 0 1 2 .
Fact 2.1: The Gray code G s,h can be constructed by first recursively constructing paired Gray code matrices P s,h := [G s,h , R (G s,h )] using the rule below and then removing half of the columns from the right side:
P s,1 = [0, . . . , s − 1, s − 1, . . . , 0] , P s,i = P s,1 ⊗ 1 s i−1 P s s,i−1 .(1)
Here, ⊗ stands for the Kronecker product while 1 a is a row vector of 1s. Example 2.2: The following matrix P 3,2 is constructed recursively using (1). The left half, as claimed, equals G 3,2 and was illustrated in Example 2.1: We also make use of the following property of binary Gray codes. Fact 2.2: G 2,h ((i, * )) contains 2 i−1 runs of 1s for all i except i = 0, which contains only one run of 1s. Consequently, the matrix contains a total of h−1 i=1 2 i−1 + 1 = 2 h−1 runs of 1s within its rows. This is illustrated by the following example for G 2,3 , with a total number of 2 3−1 = 4 runs of 1s.
0 0 0 0 [1 1 1 1] 0 0 [1 1 1 1] 0 0 0 [1 1] 0 0 [1 1] 0 .
III. LOWER BOUNDS
We first provide lower bounds for the smallest number of measurements needed for the m * B(n, ℓ, η) and m * B(n, ≤ ℓ, η) settings. The proofs mostly use ideas from [1].
Theorem 1: We have m * B(n, ℓ, η) ≥ max log s+1 (n − ℓ + 1) , ⌈ ℓ 2ηs ⌉ , m * B(n, ≤ ℓ, η) ≥ max log 2 (n) , ⌈ ℓ ηs ⌉ . The proof technique used for m * B(n, ≤ ℓ, η)
is similar to that for m * B(n, ℓ, η) ; hence, we only provide the proof for m * B(n, ℓ, η) . We prove the first bound by establishing each of the bounds on the right-hand side separately and combining them via maximization.
1) The bound log s+1 (n − ℓ + 1) follows from a simple counting argument: there are a total of n − ℓ + 1 different head positions and a total of s + 1 possible outcomes for each measurement.
2) The bound ⌈ ℓ 2ηs ⌉: we show that even if we only require to discriminate among the first ℓ + 1 bursts (i.e., bursts b i with 0 ≤ i ≤ ℓ), we still need ⌈ ℓ 2ηs ⌉ measurements. For any measurement m, let m 1 and m 2 denote the first and second block of ℓ bits of m. Only the last η s nonzero bits in m 1 and the first η s nonzero bits in m 2 are relevant. For simplicity, we only provide a proof for the m 2 case. Let ℓ ≤ i 1 < i 2 < . . . < 2ℓ be the elements included in m 2 . Since ℓ b = ℓ and h b ≤ ℓ, if i j is included in the burst b then i 1 , . . . , i j−1 must also be included. Therefore, if j > η s , by observing that η s is the largest threshold, one can remove i j from m 2 without changing the outcome. Hence one can only retain the first η s nonzero bits in m 2 and still arrive at the same outcome. As a result, it suffices to only consider those m for which
IV. MAIN RESULTS
In Section IV-A, we describe and order-optimal construction of measurement matrices for the B(n, ℓ, η) problem pertaining to two different cases, the case of general SQGT thresholds with ℓ = Ω(η s log(η s )) and the saturation model with ℓ ≤ η s = s. It is interesting to note that for the first case, m * B(n, ℓ, η) basically depends only on the largest threshold η s . In other words, as long as ℓ = Ω(η s log 2 (n)) with sufficiently large constant, there is no benefit of using multiple thresholds (SQGT) compared to threshold group testing (TGT) with the single biggest threshold η s . In Section IV-B, we describe an order-optimal scheme (within an approximation constant 4) for B(n, ≤ ℓ, η) problem and the saturation model.
A. The B(n, ℓ, η) Model
Since for this case the burst length is fixed, one only needs to locate the position of the head h b ∈ [0, n − ℓ] of the burst b. Vaguely speaking, the near-optimal construction follows a two-step sketch-and-refine procedure. The first part, referred to as the General Sketch Scheme, uses a measurement matrix K (Theorem 2) that distinguishes bursts separated by > ℓ + 1 positions. The second part, referred to as the General Refinement Scheme, uses a measurement matrix R (Theorem 3) that distinguishes bursts separated by < 2ℓ positions. Stacking the two measurement matrices leads to the result reported in Theorem 4.
Theorem 2: For B(n, ℓ, η) , the measurement matrix K described in Section IV-A1 can distinguish all bursts at distances > ℓ + 1 using ⌈log s+1 n−ℓ+1 ℓ ⌉ measurements. Theorem 3: For B(n, ℓ, η) with parameters η s = 2 h−1 + 2 and ℓ = c2 h + 1, where c, h ∈ N and c > 2(h + 1), the measurement matrix R described in Section IV-A2 can distinguish all bursts at distances < 2ℓ using roughly ℓ 2ηs measurements. The scheme depends only on the largest threshold η s .
Theorem 4: Combining the General Sketch matrix of Theorem 2 and the General Refinement matrix of Theorem 3, leads to the measurement matrix [K ⊺ , R ⊺ ] ⊺ which can be used to solve B(n, ℓ, η) using roughly ℓ 2ηs + log s+1 n−ℓ+1 ℓ measurements. This number of measurement is at most twice the number of measurement from the lower bounds reported in Theorem 1.
Remark 4.1: Note that scheme from Theorem 3 only uses the largest threshold η s . Therefore, if we only make use of η s in the General Sketch Scheme of Theorem2, the resulting measurement matrix has height roughly ℓ 2ηs + log 2 n−ℓ+1 ℓ and depends on one threshold, η s . When ℓ = Ω(η s log 2 (n)) with sufficiently large constant, ℓ 2ηs + log s+1 n−ℓ+1 ℓ
= Ω (log 2 (n)) = ℓ 2ηs + log 2 n−ℓ+1 ℓ . Therefore, in this parameter regime, there is no benefit from using multiple thresholds.
1) Proof of Theorem 2: We start with some relevant notation. Let K hK×n be the measurement matrix. We say that . The key idea is to first construct K with w K ≥ n such that the outcome matrix satisfies
O = G s+1,hK ⊗ 1 ℓ+1 ,(2)
and then truncate it to n columns. By the definition of O, K can identify all bursts at distance ≥ ℓ + 1 if and only if all columns of G s+1,hK are different; that this is true follows from the fact that Gray codes include all vectors {0, . . . , s} hK exactly once. We need the following lemma for our subsequent derivations.
b i , where i ∈ [0, (c − 1) ℓ]. ∀i, η x c b i+1 = η x c b i + x c ( * , i + ℓ) − x c ( * , i) = η x c b i
We are now ready to present our construction. Let M [i] be the measurement matrix recursively constructed as follows:
M [1] = m (1) , M [i] = m (ℓ+1)(s+1) i−1 −1 ℓ M [i − 1] s+1 .
(3)
Note that (ℓ+1)(s+1) i−1 −1 ℓ may not be an integer. We therefore first focus on the special case s = ℓ (therefore (ℓ+1)(s+1
) i−1 −1 ℓ = (ℓ+1) i −1 ℓ
∈ N) and then generalize the result for s < ℓ through slight modifications of the argument.
⊗ 1 (ℓ+1) i P ℓ+1 ℓ+1,i−1 ⊗ 1 ℓ+1 ∞ = P ℓ+1,i ⊗ 1 ℓ+1 ∞ .
By Lemma 4.2, M [h K ] truncated to (ℓ+1) hK+1 +ℓ−1 columns from the right results in the outcome matrix G ℓ+1,hK ⊗1 ℓ+1 , and can consequently distinguish all bursts within distance > ℓ+1. It is not hard to show that a single measurement 0 ℓ 1 ℓ+1 0 ℓ+1 ∞ results in the outcome matrix (0, . . . , ℓ, ℓ, . . . , 0) ∞ . Hence we have the following theorem. Theorem 5: For the saturation SQGT model with ℓ thresholds
η = (1, . . . , ℓ), M ⌈log ℓ+1 (n − ℓ + 1) − 1⌉ ∞ 0 ℓ 1 ℓ+1 0 ℓ+1 ∞ truncated to
n columns on the right can be used as the test matrix for the B(n, ℓ, η) model with ⌈log ℓ+1 (n − ℓ + 1)⌉ measurements. For the case s < ℓ, some modifications in the recursion given by (3) are required. The modification involves truncating a certain number of columns from the left, right, or both sides of M ′ [i] at each stage of recursion i:
M ′ [1] = m (1) , M ′ [i] = 0 α mod ℓ m ⌊ α ℓ ⌋ 0 α mod ℓ M ′ r [i − 1] M ′ lr [i − 1] s−1 M ′ l [i − 1] , where α = w M ′ [i−1] 2 − 1, and M ′ r [i − 1] , M ′ l [i − 1] , M ′ lr [i − 1] denotes M ′ [i − 1]
truncate α mod ℓ columns from the right, left, and both sides, respectively.
By using a similar proof as the one described above and some simple but tedious calculations, one can show that M ′ ⌈log s+1 n−ℓ+1 ℓ ⌉ truncated to n columns from the right can be used as K. Hence, K can distinguish all bursts at distance > ℓ + 1 using ⌈log s+1 n−ℓ+1 ℓ ⌉ measurements. 2) Proof of Theorem 3: We now focus our attention on the General Refinement Scheme. Let B to denote the cyclic shift of columns in B one position to the left so that B( * , i) = B( * , i + 1 mod ℓ). We need the following lemma. Then the following measurement matrix can be used to distinguish all bursts b = b ′ at distance < 2ℓ:
R := R − R + R − R + . . . ,(4)
where R − denotes the "negative" part of B − B (obtained by setting 1s to 0s and −1s to 1s), while R + denotes the positive part of B−B (obtained by setting −1s to 0s), and the last column is changes from 0 hB×1 to 1 hB×1 (note that the last column of B − B before the modification is B( * , 0) − B( * , ℓ − 1) = −B( * , ℓ − 1), which implies that the positive part is zero). Proof: Since R is a repeated horizontal concatenation of R − and R + , it suffices to show that ∀0 ≤ i = j < 2ℓ, η Rb i = η Rb j . (5) In particular, we prove that
Rb i = (η s − 1) 1 hB×1 + B( * , i) 0 ≤ i < ℓ, η s 1 hB×1 − B( * , i) ℓ ≤ i < 2ℓ.(6)
Note that each entry of Rb i is either η s −1 or η s . By condition 1, all Rb i are different. Consequently, all η Rb i are different as well. Therefore, R can distinguishes all bursts at distance < 2ℓ using h B measurements; only the largest threshold η s is relevant. Next we prove (6). For 0 ≤ i < ℓ,
Rb i − Rb 0 = i j=1 Rb j − Rb j−1 = i j=1 R b j − b j−1 = i j=1 R + ( * , j − 1) − R − ( * , j − 1) = i j=1 R( * , j − 1) (1) = B( * , i). For ℓ ≤ i < 2ℓ, Rb i − Rb ℓ = i j=ℓ+1 Rb j − Rb j−1 = i j=ℓ+1 R b j − b j−1 = i j=ℓ+1 R − ( * , j − 1) − R + ( * , j − 1) = − i−ℓ j=1 R( * , j − 1)(1)
= −B( * , i).
The equalities (1) = follows from Condition 2. Finally, by Condition 3 and the fact that we changed the last column of R + from 0 hB×1 to 1 hB×1 , Rb 0 = (η s − 1) 1 hB×1 and Rb ℓ − Rb 0 = 1 hB×1 ⇒ Rb ℓ = η s 1 hB×1 . It remains to construct a matrix B that satisfies Conditions 1-3 in Lemma 4.3 with h B roughly equal to ℓ 2ηs . Let G h×2 h 2,h be the code matrix of a binary Gray code of length h such that
2(h + 1) < h B . We constructḠ hB×2 h 2,h,i as G ⊺ 2,h,i = 0 2 h ×i 1 2 h ×1 G ⊺ 2,h 0 (2 h ×hB−i−h−1) ⊺ .
Then, B is constructed as follows:
B = 0 hB×1Ḡ 2,h,
• Condition 1: we demonstrate a 2-step procedure for recovering the index of each column based on its content which establishes that all columns are different. The idea is first to use 1 1×2 h to recover i (the index ofḠ 2,h,i ) and then use the following h bits from the Gray code to locate the exact column. To do so, in the first step, we need an additional constraint h B ≥ 2(h + 1). In a nutshell, we can recover i by looking at the first 1 after a length-h + 1 burst of 0 in each column. Moreover, by this constraint, each column in B has more zeros than ones. Consequently, all B( * , i) and B( * , i) must be distinct. • Condition 2: this condition is easy to verify. Finally, we set ηs = 2 h−1 + 2. Then, ℓ hB = hB2 h +1 hB > 2 h = 2ηs − 4 with the minor restriction that h B > 2(h + 1).
B. The Saturation SQGT Model for B(n, ≤ ℓ, η)
For the bounded-length burst problem B(n, ≤ ℓ, η) , one needs to recover both h b and t b in order recover the burst. We describe next an order-optimal scheme for B(n, ≤ ℓ, η) restricted to the saturation SQGT model. First, for η s = s ≥ l, we describe an optimal scheme termed the Integer code. Then, for η s = s < l, by adapting the bursty GT scheme from [1], we arrive at an order-optimal scheme (within a constant factor of 4).
Theorem 6: For B(n, ≤ ℓ, η) and the saturation SQGT model for which η = (0, . . . , s) and s ≥ ℓ, there exists and orderoptimal scheme N that solves B(n, ≤ ℓ, η) using ⌈log 2 (n)⌉ + 1 measurements.
Proof: The matrix N is a vertical concatenation of a ⌈log 2 (n)⌉×n Index matrix and an 1 1×n . Note that the ith column of the Index matrix is the binary representation of i. Since ℓ ≤ s, η (Nb) = Nb. We then treat the outcome vector as a binary representation of an integer k equal to
k : = ⌈log 2 (n)⌉ i=0 2 i (Nb) (i, 0) = ⌈log 2 (n)⌉ i=0 2 i t b j=h b N(i, j) = t b j=h b ⌈log 2 (n)⌉ i=0 2 i N(i, j) = t b j=h b j = h b + t b 2 ℓ b .
We can deduce ℓ b from the outcome corresponding to 1 1×n . Hence, N recover the burst b using ⌈log 2 (n)⌉+1 measurements.
Theorem 7: For B(n, ≤ ℓ, η) under the saturation SQGT model with η = (0, . . . , s) and s < ℓ, there exists an order-optimal scheme (within a constant factor 4) C that solves B(n, ≤ ℓ, η) using ≤ 2ℓ s + 2 log 2 (n) + 3 measurements. Proof: The matrix C is as follows:
C = C ⊺ 1 C ⊺ 2 C ⊺ 3 ⊺ ,
where C 1 is the Phase 1 matrix from Theorem 3.2 of [1]. By an argument described in [1], C 1 can distinguish all bursts b within distance ≥ 2(ℓ − 2). Next, we use C 2 := I ⌈ 2ℓ s ⌉ ⊗ 1 s ∞ truncated to n columns from right. The outcome vector corresponding to C 2 is a single run (in a circular sense) of non-zero of the form o h , s, . . . , s, o t , where o h , o t ∈ {1, . . . , s}. Then, 1) For ℓ b > s: since ℓ b s > 1 and ⌈ 2ℓ s ⌉ ≥ ⌈ 2ℓ b s ⌉ ≥ ⌈ ℓ b s ⌉ + 1, h = t. Therefore, one can use o h , o t to recover h b mod 2ℓ, t b mod 2ℓ. Consequently, C 2 can distinguish all bursts at distance < 2ℓ such that ℓ b > s. 2) For ℓ b ≤ s: from the outcome of C 2 , one can recover ℓ b .
If ℓ b ≤ s, then by Theorem 6, C 3 := N can distinguish all bursts with ℓ b ≤ s. Therefore C can be used to solve B(n, ≤ ℓ, η) using h C1 +h C2 + h C3 ≤ log 2 (n) + ⌈ 2ℓ s ⌉ + ⌈log 2 (n)⌉ + 1 ≤ 2ℓ s + 2 log 2 (n) + 3 measurements, which is at most 4 times the lower bound of Theorem 1.
h M , w M , M ( * , i) , M (j, * ) denote the number of rows (height), number of columns (width), i-th column and j-th row of the matrix M hM×wM , respectively. Our row indices lie in [0, h M − 1], while the column indices are confined to [0, h M − 1]. In addition, R (M) , M c , M ∞ are used to denote a matrix obtained from M by reversing the column order
m(j) ≤ 2η s . Let M hM×2ℓ be our measurement matrix restricted to the first 2ℓ columns. Suppose that h M < ℓ 2ηs ; then M(i, j) ≤ 2η s h M < ℓ. This implies that there exists a 0 ≤ j < ℓ such that M( * , j) = M( * , j + ℓ) = 0 hM×1 . Then η Mb i+1 = η Mb i + M( * , j + ℓ) − M( * , j) = η Mb i . Therefore m * B(n, ℓ, η) ≥ ⌈ ℓ 2ηs ⌉.
K hK×n results in the outcome matrix O hK×n−ℓ+1 if O = η Kb 0 . . . η Kb n−ℓ+1represents the collection of outcomes for all length-ℓ bursts b i when using the measurement matrix K.Next, let B ℓ (x) := 0 ℓ−x 1 x and B ℓ (x) := 1 x 0 ℓ−x , for all x ∈ {0, . . . , ℓ}. Also, let B ℓ (0) i , B ℓ (0) i stand for the horizontal concatenation of i copies of B ℓ (x) and B ℓ (x).
Lemma 4 . 1 :
41The following measurementm (c) := B ℓ (0) c 0 B ℓ (η 1 ) c . . . 0 B ℓ (η s ) c 1 B ℓ (ℓ) c 1 B ℓ (η s − 1) c . . . 1 B ℓ (η 1 − 1) c 0 ∞ resultsin the outcome [0, . . . , s, s, . . . , 0] ⊗ 1 cℓ+1 ∞ . Proof: The case c = 1 can be proved easily and is illustrated by the following example. For η = (1, 2, 4) and ℓ = 6, For c > 1, and any length-ℓ row-vector x, η x c b i remains unchanged for all
Lemma 4 . 2 :[ 0 ,
420For s = ℓ, M [i] ∞ results in the outcome matrix P ℓ+1,i ⊗ 1 ℓ+1 ∞ . Where P ℓ+1,i is the ℓ+1-ary length-i paired gray code matrix. Proof: The proof is by induction. 1) For i = 1: by Lemma 4.1, M [i] ∞ = m (1) ∞ results in the outcome matrix [0, . . . , ℓ, ℓ, . . . , 0] ⊗ 1 ℓ+1 ∞ = P ℓ+1,1 ⊗ 1 ℓ+1 ∞ . 2) For i > 1: Suppose that the claim holds for i − 1. Then M [i] ∞ = m (ℓ+1) . . . , ℓ, ℓ, . . . , 0]
Lemma 4 . 3 :
43Suppose that a binary matrix B hB×ℓ satisfies the following three conditions: 1) All columns and their binary complement {B( * , i),B( * , i)} ℓ−1 i=0 are distinct. 2) The first column is the zero vector, B( * , 0) = 0 hB×1 . 3) Each row of B − B has η s − 1 elements equal to −1.
•
Condition 3: by the construction from (IV-A2), each row B (i) is a horizontal cyclic shift of v := 1 2 h , G 2,h (0, * ), . . . , G 2,h (h − 1, * ) with an additional 0 appended at the left. In Example 4.1, we have v = [111100110110]. By this construction, the number of −1s in each row of B − B equals the number of runs of 1s in B(i, * ), which equals the sum of the number of runs of 1s in 1 1×2 h and each row of the Gray code G 2,h (i, * ). By Fact 2.2, the total number of runs of 1s in each B(i,
hB−1 . . .Ḡ 2,h,0 .Example 4.1: The matrix B is constructed using G 2,2 and for h B = 7:
0 0011 0110
1111
0 0110
1111 0011
0
1111 0011 0110
0
1111 0011 0110
0
1111 0011 0110
0
1111 0011 0110
0 1111 0011 0110
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| {'fraction_non_alphanumeric': 0.07981960275538257, 'fraction_numerical': 0.031662187625490376, 'mean_word_length': 3.494516173816465, 'pattern_counts': {'":': 0, '<': 28, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing [1] as well as work on semiquantitative group testing (SQGT)[2]. Specifically, we study the setting where the burst-length ℓ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds η1, . . . , ηs. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold ηs. In this context, our main result is an order-optimal test scheme that can recover any burst of length ℓ using roughly ℓ 2ηs + log s+1 (n) measurements. This suggests that a large saturation level ηs is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.', 'arxivid': '2304.01365', 'author': ['Yun-Han Li [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA\n', 'Ryan Gabrys [email protected] \n.Naval Information Warfare Center\nSan Diego\n', 'Jin Sima [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA\n', 'Ilan Shomorony [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA\n', 'Olgica Milenkovic [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA', '.Naval Information Warfare Center\nSan Diego', 'Department of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA', 'Department of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA', 'Department of Electrical and Computer Engineering\nUniversity of Illinois Urbana-Champaign\nUSA'], 'corpusid': 257921247, 'doi': '10.48550/arxiv.2304.01365', 'github_urls': [], 'n_tokens_mistral': 11324, 'n_tokens_neox': 10139, 'n_words': 5984, 'pdfsha': 'b89979e42a87f5890f5753d35baf4d28ff5d63ee', 'pdfurls': ['https://export.arxiv.org/pdf/2304.01365v1.pdf'], 'title': ['Finding a Burst of Positives via Nonadaptive Semiquantitative Group Testing', 'Finding a Burst of Positives via Nonadaptive Semiquantitative Group Testing'], 'venue': []} |
arxiv |
Restriction and induction for supercharacters of finite groups of triangular type
16 Oct 2016
A N Panov
Restriction and induction for supercharacters of finite groups of triangular type
16 Oct 2016
It is proved the restriction of any supercharacter of a finite group of triangular type on its subgroup is a sum of supercharacters with nonnegative integer coefficients. We define a superinduction and prove the analog of Frobenius reciprocity formula for supercharacters. * The paper is produced by financial support of the Ministry of education and science of the Russian Federation within the basic part of the state task (project No. 204). The work is supported by RFBR grants 16-01-00154-a and 14-01-97017-Volgaregion-a.The following statement is important for constructing supercharacter theories. Proposition 2.2 [1, Lemma 2.1]. Suppose that we have a system of disjoint characters Ch = {χ 1 , . . . , χ m } and a partition K = {K 1 , . . . , K m } of the group G. Assume that each character χ i is constant on each K j . Denote by ν − µ = eνe − eµe = e(ν − µ)e ∈ eJµe = eJeµ.That is ν − µ ∈ J e µ, and ν ∈ N e µ, where N e = 1 + J e . It is also true the opposite: if ν ∈ N e µ, then ν| B = µ ′ | B .Since J ′ is a H × H-invariant ideal in J, the group G acts on J ′ (and J ′ * ) by right ans left multiplications. The equality ν = aµ, where a ∈ N e , admits the restriction on J ′ * in the form ν ′ = aµ ′ . This implies that J ′ ν ′ ,right = J ′ µ ′ ,right = A. Therefore Ind(ξ θ,ν , H(e) + A, G ′ ) = Ind(ξ θ,ν ′ , H(e) + J ′ ν ′ ,right , G ′ ). According to Lemma 4.4, this character is a sum of supercharacters. This proves the statement in Item 2.1. Item 2.2. Suppose that W ⊆ J 12 . Then J 11 = J ′ 11 , J 21 = J ′ 21 , J 22 = J ′ 22 , and J 12 = W ⊕ J ′ 12 . As in the previous item, the restriction Res χ α is calculated by the formula (12). The subspace J λ,right + J ′ is invariant with respect to H(e) × H(e). Denote by W 0 the H(e) × H(e)-invariant subspace in J 12 such that
Introduction
The problem of classification of irreducible characters (representations) is the main problem in the representation theory of finite groups. It is well known that a character is constant on conjugacy classes, the number of irreducible characters is equal to the number of conjugacy classes. The values of irreducible characters on conjugacy classes form the quadratic table which is called a character table. However, for some finite groups, like the unitriangular group UT(n, F q ), the problem of classification of irreducible characters is a very complicated, "wild" problem. In the paper, [1] P.Diaconis and I.Isaaks proposed the concept of a supercharacter theory. Roughly speeking a supercharacter theory is a first approximation of the theory of irreducible characters. Together with supercharacters there are defined superclasses; the values of supercharacters are constant on superclasses and form the quadratic table, which is called a supercharacter table. Originally, any finite group have several supercharacter theories; preference is given to the supercharacter theory that produces better approximation of the theory of irreducible characters.
Historically, the first nontrivial example of a supercharacter theory was the theory of basic characters of UT(n, F q ) constructed in the series of papers of K.Andre [2,3,4,5] (see also [6]). In the paper [1], P.Diaconis and I.Isaaks constructed the supercharacter theory for algebra groups (see example 2.5) which generalizes the Andre theory. Many papers were devoted to investigating different supercharacter theories. Highlight a few of them: supercharacter theory for abelian groups with application to the number theory [13,14], superinduction for algebra groups [15,16,17], application to the problem of random walking on groups [11], supercharacter theory for semidirect products [12]. There is the bibliography on this topic in the paper [10].
Notice that for algebra groups the restriction of a supercharacter on its algebra subgroup is a sum of supercharacters with nonnegative integer coefficients. At the same time, in general, the induced character from a supercharacter is not a sum of supercharacters of group (even with complex coefficients). But if you change induction by superinduction, then the constructed character will be the sum of supercharacters (with nonnegative rational coefficients), and there is an analog of Frobenius reciprocity theorem (see [1]).
In the papers [7,8], there were constructed a supercharacter theory for finite groups of triangular type that generalizes the theory of P.Diaconis and I.Isaaks. A goal of this paper is to transfer the mentioned statements on restriction and superinduction for algebra groups to the case of finite groups of triangular type. The main results are formulated in theorems 4.5 and 5.1.
Overview of supercharacter theory
Let G be a group, 1 ∈ G be the unit element, Irr(G) be the set of irreducible characters (representations) of the group G. Suppose that we have two partitions
Irr(G) = X 1 ∪ · · · ∪ X m , X i ∩ X j = ∅,(1)G = K 1 ∪ · · · ∪ K m , K i ∩ K j = ∅.(2)
Notice that the number of components of these partitions are common. To each X i we correspond the character of group G by the formula
σ i = ψ∈X i ψ(1)ψ.(3)
X i the support of the character χ i (i.e., the set of all irreducible components of χ i ). Then the following conditions are equivalent: 1) {1} ∈ K, 2) the system of subsets X = {X i } form a partition of Irr(G); two partitions X and K defines a supercharacter theory of the group G. Moreover, each χ i is up to a constant factor equal to σ i . Simplifying language, we refer to χ i as supercharacters.
Example 2.3. The system of irreducible characters {χ i } and the system of conjugacy classes {K i }. Example 2.4. Two supercharacters χ 1 = 1 G , χ 2 = ρ − 1 G (here ρ is a character of the regular representation) and two superclasses K 1 = {1},
K 2 = G \ {1}.
The supercharacter table has the form
χ 1 χ 2 K 1 1 |G| − 1 K 2 1 -1
Example 2.5. The supercharacter theory for algebra groups [1] (see also [9]). By definition, an algebra group is a group G = 1 + J, where J is an associative finite dimensional nilpotent algebra over the finite field k = F q . The superclass of the element 1 + x is defined as 1 + ω, where ω is a left-right G × G-orbit of the element x ∈ J. There are also right and left actions of the group G on the dual space J * defined by the formulas λg(x) = λ(gx) and gλ(x) = λ(xg). Let G λ,right be the stabilizer of λ ∈ J * with respect to the right action of G on J * . Fix a nontrivial character t → ε t of the additive group F q with values in the multiplicative group C * . The function ξ : G λ,right → C * defined as
ξ λ (g) = ε λ(g−1)
is a linear character (one dimensional representation) of the group G λ,right . A supercharacter of the algebra group G is the induced character
χ λ = Ind(ξ λ , G λ,right , G).(4)
The systems of supercharacters {χ λ } and superclasses {1 + GxG}, where λ and x run through the systems of double G × G-orbits in J * and J respectively, define a supercharacter theory for the group G. Let us observe the problems of restriction and induction in the theory supercharacters for algebra groups. Let G = 1 + J be an algebra group. If J ′ is an arbitrary subalgebra in J, then the subgroup G ′ = 1 + J ′ is called an algebra subgroup in G.
It is proved in the paper [1] that the restriction of supercharacter χ λ on the algebra subgroup G ′ is a sum of supercharacters of the subgroup G ′ with nonnegative integer coefficients.
Suppose that φ is a superclass function on G ′ (i.e., the function constant on superclasses of G ′ ). Extend φ to the functionφ on G letting it equal to zero outer G ′ . By definition, a superinduction of φ is a function SInd φ on the group G defined as follows
SInd φ(1 + x) = 1 |G| · |G ′ | a,b∈Gφ (1 + axb).
Easy to see that SInd φ(1 + x) is a superclass function on the group G. There is the standard scalar product on the group G (and G ′ ) defined as
(f 1 , f 2 ) = 1 |G| g∈G f 1 (g)f 2 (g).
The following theorem is an supercharacter analog of the Frobenius reciprocity theorem for algebra groups. Theorem 2.6 [1]. Let φ be a superclass function on G ′ , and ψ be a superclass function on G. Then (SInd φ, ψ) = (φ, Res ψ).
Supercharacters of finite groups of triangular type
In this section, we highlight the supercharacter theory for finite groups of triangular type constructed by the author in the papers [7,8].
Let H be a group, and J is an associative algebra over a field k. Suppose that there defined the left h, x → hx and the right h, x → xh linear actions of the group H on J. Assume that for every h ∈ H and x, y ∈ J the following conditions are fulfilled:
1. h(xy) = (hx)y (xy)h = x(yh), 2. x(hy) = (xh)y.
On the set
G = H + J = {h + x : h ∈ H, x ∈ J}
we define an operation of multiplication
g 1 g 2 = (h 1 + x 1 )(h 2 + x 2 ) = h 1 h 2 + h 1 x 2 + x 1 h 2 + x 1 x 2 .(5)
If J is a nilpotent algebra over the field k, then G is a group with respect to operation (5). If the group H is finite, the field k is finite, and J is finite dimensional algebra over the field k, then the group G is finite. Definition 3.1. Under all these conditions we call the group G a finite group of triangular type if the group H is abelian and char k does not divide |H|. Let G = H + J be a finite group of triangular type. The group algebra kH is commutative and semisimple (according to Maschke's theorem). Therefore, kH is a sum of fields. There exists a system of primitive idempotents {e 1 , . . . , e n } such that
kH = k 1 e 1 ⊕ . . . ⊕ k n e n ,(6)
where k 1 , . . . , k n are field extensions of k. Any idempotent in kH is a sum of primitive idempotents. The direct sum A = kH ⊕ J has an algebra structure with respect to the multiplication (5). The group G is a subgroup of the group A * of invertible elements of A, which is considered in the Example 3.4. Observe that the group G decomposes into a product G = HN of the subgroup H and the normal subgroup N = 1 + J that is an algebra group.
Example 3.2. Algebra group G = 1 + J. Example 3.3. G = a b 0 1 : a, b ∈ F q , a = 0 .
Example 3.4. Let A be an associative unital finite dimensional algebra of reduced type over the finite field F q of q elements [18, §6.6]. By definition, the algebra A is reduced if its factor algebra with respect to the radical J = J(A) is a direct sum of division algebras. According to Wedderburn's theorem [18, §13.6], any division algebra over a finite field is commutative. Then the algebra A/J is commutative. There exists a semisimple subalgebra S such that A = S ⊕ J (see [18, §11.6]). In our case, S is commutative. The group G = A * on the invertible elements of A is a finite group of triangular
type G = H + J, where H = S * . If A is the algebra of triangular matrices, then G = T(n, F q ) is the triangular group. Let G = H + J be a finite group of triangular type. Consider the group G of triples τ = (t, a, b), where t ∈ H, a, b ∈ N , with operation (t 1 , a 1 , b 1 ) · (t 2 , a 2 , b 2 ) = (t 1 t 2 , t −1 2 a 1 t 2 a 2 , t −1 2 b 1 t 2 b 2 )
. The group G acts on J as follows
ρ τ (x) = taxb −1 t −1 .
In the dual space J * a representation of the group G is defined as usual
ρ * τ λ(x) = λ(ρ(τ −1 )(x)
). In the space J * there are also left and right linear actions of the group G by the formulas bλ(x) = λ(xb) and λa(
x) = λ(ax). Then ρ τ (λ) = tbλa −1 t −1 .
For any idempotent e ∈ kH we denote by A e the subalgebra eAe. The subalgebra J e = eJe ⊂ J is a radical in A e . Denote e ′ = 1 − e. There is a the Pierce decomposition
J = eJe ⊕ eJe ′ ⊕ e ′ Je ⊕ e ′ Je ′ .
The dual space J * e is identified with the subspace in J * that consists of all linear forms equal to zero on all components of the Pierce decomposition except the first one. Observe that since the group H is abelian, we have he = eh = ehe for all h ∈ H. The subset H e = eHe is a subgroup in the algebra of invertible elements of the algebra A e . The group G e = eGe = H e + J e is a group of triangular type; and it is associated with the algebra A e in the same way as G is associated with A. The map h → he is a homomorphism of the group H onto H e , its kernel is the subgroup
H(e) = {h ∈ H : he = e}.(7)
The following definition and the one of [7,8] differs in form, but equivalent. 2) H(e) = H λ,right ∩ H λ,left for any regular (with respect to H e ) element λ ∈ J * e . It is proved in the papers [7,8] that for any G-orbit O in J the following statements are true. Proposition 3.7. 1) The intersection O ∩ J e is an G e -orbit in J e for any idempotent e ∈ kH.
2) There exists a unique idempotent e ∈ kH such that O ∩ J e is a regular G e -orbit in J e (with respect to H e ). Similar statements are true for G-orbits in J * .
Well known that for any finite transformation group of finite dimensional linear space V defined over a finite field, the numbers of orbits in V and V * are equal [1,Lemma 4.1]. By this statement and above properties of G-orbits, the numbers of regular (singular) G-orbits in J and J * are equal [7, Proposition 2.11].
Turn to a definition of superclasses in the group G. For each g ∈ G and (t, a, n) ∈ G consider the element
R τ (g) = 1 + ta(g − 1)b −1 t −1(8)
in the algebra A = kH + J. If g = h + x, then R τ (g) = h mod J. Hence R τ (g) ∈ G. The formula (8) determines the action of the group G on G. Definition 3.8. We say that G-orbits in G are superclasses.
The group G splits into superclasses. Denote by B the set of triples β = (e, h, ω), where e is an idempotemt in kH, h ∈ H(e), and ω is a regular (with respect to H e ) G e -orbit in J e . All element h + ω belong to a common superclass [8, 3.2] denoted by K β . Theorem 3.9 [8, 3.3]. The correspondence β → K β is a bijection of the set of triples B onto the set of superclasses in G.
Denote by A the set of triples α = (e, θ, ω * ), where e is an idempotent in kH, θ is a linear character (one dimensional representation) of the subgroup H(e), and ω * is a regular (with respect to H(e)) G e -orbit in J * e . Since the subgroup H(e) is abelian, the number of its linear characters is equal to the number of its elements. The number of regular (with respect to H e ) G e -orbits in J e and J * e are equal. Therefore |A| = |B|. Turn to a construction of supercharacters. Let α = (e, θ, ω * ) ∈ A, choose λ ∈ ω * . Consider the subgroup G α = H(e) · N λ,right , where N λ,right is a stabilizer for λ of the right action of N = 1 + J on J * . Any element g ∈ G α can be presented in the form g = h + x, where hλ = λh = λ and λ(xJ) = 0. Hence G α = H(e) + J λ,right .
Fix a nontrivial character t → ε t of the additive group F q with values in multiplicative group C * . By given a triple α = (e, θ, ω * ) and λ ∈ ω * , define a linear character of the subgroup G α by the formula
ξ θ,λ (g) = θ(h)ε λ(x) ,(10)
where g = h + x, h ∈ H(e) x ∈ J λ,right . The induced character
χ α = Ind(ξ θ,λ , G α , G)(11)
is called a supercharacter.
Restriction and superinduction for finite groups of triangular type
Let G = H + J be a finite group of triangular type. Let H be a subgroup of H, and J ′ be a subalgebra of J invariant with respect to left-right action of H ′ × H ′ on J. Then G ′ = H ′ + J ′ is a subgroup in G, which will be called a subgroup of triangular tipe in G. Our goal is to verify that the restriction of a supercharacter on a subgroup of triangular type is a sum of supercharacters of the subgroup with nonnegative integer coefficients.
A n = (B + A 2 ) n ⊆ B n + A n+1 ⊆ B + A n+1 .
Hence B + A n ⊆ B + A n+1 . Since A is a nilpotent algebra, A n+1 = 0 for some n. Then
A = B + A 2 ⊆ B + A 3 ⊆ . . . ⊆ B + A n+1 = B. ✷ Corollary 4.2. If J ′ is maximal H × H-invariant subalgebra in J, then J 2 ⊆ J ′ . In particular, J ′ is a ideal in J.
Proof. Since J ′ ⊆ J ′ + J 2 ⊆ J, and J ′ is maximal H × H-invariant subalgebra in J, then either J ′ = J ′ +J 2 , or J ′ +J 2 = J. The second equality implies J ′ = J, this contradicts to the assumption. Therefore J ′ = J ′ + J 2 . Hence J 2 ⊆ J ′ . ✷ Notice that the algebra J, as the algebra group N = 1 + J, acts on J * by left and right transformations λx(y) = λ(xy) and xλ(y) = λ(yx). The set Jλ is a left orbit of J on J * . Let λ ∈ J * and H 0 be a subgroup in H such that h 0 λ = λh 0 = λ for every h 0 ∈ H 0 . Let θ 0 be a linear character of H 0 . Then G 0 = H 0 + J λ,right is a subgroup in G, and the following formula
ξ θ 0 ,λ (g) = θ 0 (h 0 )ε λ(x) , g = h 0 + x
defines a linear character of the subgroup G 0 . Denote χ θ 0 ,λ = Ind(ξ θ 0 ,λ , G 0 , G). Lemma 4.4. The induced character χ θ 0 ,λ is a sum of supercharacters.
Proof. Let f be the least idempotent in kH 0 such that λ ∈ J * f . There exists the idempotent e f such that the intersection of the G f -orbit of λ in J * f is a regular G e -orbit in J * e (see Proposition 3.6). Choose λ ′ in J * e lying in the same G f -orbit of λ. Then χ θ 0 ,λ = χ θ 0 ,λ ′ (see [7,8,Proposition 4.2]). The subgroup H(e) is contained in H 0 . Decompose Ind(θ 0 , H 0 , H(e)) into a sum of linear characters
Ind(θ 0 , H 0 , H(e)) = m i=1 θ i .
Then χ θ 0 ,λ is a sum of supercharacters constructed by (ē, θ i , ω ′ * ), where ω ′ * is an orbit of λ ′ with respect to G e . ✷ Let us state and prove the main theorem of this paper. Theorem 4.5. The restriction of a supercharacter of the group G to its triangular type subgroup G ′ is a a sum of supercharacters of the subgroup G ′ with nonnegative integer coefficients. Proof. There exists a chain of H ′ × H ′ -invariant subalgebras J ′ = J 1 ⊆ . . . ⊆ J k = J such that each subalgebra J i is a maximal H ′ × H ′ -invariant subalgebra in J i+1 . We complete the chain of subgroups G ′ = G 1 ⊆ . . . ⊆ G k = H ′ + J by G k+1 = G = H + J. It is sufficient to verify the statement of the theorem in the case of restriction of supercharacter of the group G i to G i−1 ; that is in the following two cases: 1) G = H + J and G ′ = H ′ + J, and 2) G = H + J and
G ′ = H + J ′ , where J ′ is a maximal H × H-invariant subalgebra in J.
Let χ α be a supercharacter of the group G = H + J constructed by the triple α = (e, θ, ω). Res
χ α = s Ind(ξ (s) θ,λ , G (s) α ∩ G ′ , G ′ ),(12)
where G (s)
α = sG α s −1 , ξ (s) α is a character of subgroup G (s) α ∩ G ′ defied by the formula ξ (s) θ,λ (k) = ξ θ,λ (s −1 ks),
and s runs through the set of representatives of double cosets G ′ \G/G α . It is sufficient to prove that each character included in the sum (12)
where J 11 = J e = eJe,
J 12 = eJe ′ , J 21 = e ′ Je, J 22 = e ′ Je ′ , e ′ = 1 − e.
Similarly for J ′ . The Pierce components are H × H invariant, therefore, W belongs to a unique the Pierce component. We can consider the representative s of G ′ \G/G α runs through 1 + W ⊂ N e . Hence for any h ∈ H(e) we obtain sh = (1 + ewe)h = h + eweh = h + ewe = h + hewe = h(1 + ewe) = hs. Therefore sH(e)s −1 = H(e). Then G (s) α = sG α s −1 = H(e) + J sλs −1 ,right . Denote µ = sλs −1 . As λ, the element µ belongs to J * e . Likewise (10), the subgroup G (s) α = H(e) + J µ,right has the character ξ θ,µ . We obtain
G (s) α ∩ G ′ = H(e) + B,(14)
where
B = J µ,right ∩ J ′ . The character ξ (s) α of the subgroup H(e) + B is calculated ξ (s) α (h + x) = θ(h)ε µ(x) = ξ θ,µ (h + x).
where µ ∈ J * e runs through {sλs −1 : s = 1 + W }.
It is sufficient to prove that each summand in (15) is a sum of supercharacters of G ′ . Denote by µ ′ the natural projection of µ on J ′ * . As J 2 ⊂ J ′ , we have B = {y ∈ J ′ : µ ′ (yJ) = 0}. The subalgebra B is contained in the subalgebra A = J ′ µ ′ ,right = {y ∈ J ′ : µ ′ (yJ ′ ) = 0}. Since µ ∈ J * e and µ ′ ∈ J ′ * e , the subalgebras A and B are invariant with respect to left and right multiplication by H(e) and graded with respect to the Pierce decomposition (13).
The subalgebras A and B may differ only by their components in J e = J 11 . Indeed, the components e ′ J ′ = J ′ 21 ⊕ J ′ 22 are contained in both A and B. If the element x 12 ∈ J 12 belongs to A, then µ(x 12 J ′ ) = 0. On the other hand, as W ∈ J 11 , we have x 12 W = 0. Hence µ(x 12 J) = µ(x 12 W ) + µ(x 12 J ′ ) = 0, and x 12 ∈ B.
Since Decompose the character φ θ,µ ′ = Ind(ξ θ,µ , H(e)+B, H(e)+A) into irreducible components. Let F stands for the set of allν ∈ A * such thatν| B = µ| B . Observe that anyν ∈ F annihilates each Pierce component of algebra A except the first one (since this components belong to B, and µ ∈ J e annihilates them). As H(e) acts trivially on the first Pierce component, we verifyν(hy) =ν(yh) = ν(y) for each h ∈ H(e).
Let us show that A 2 ⊆ B. Indeed, if y 1 , y 2 ∈ A, then µ ′ (y 1 J ′ ) = µ ′ (y 2 J ′ ) = 0. Hence µ(y 1 y 2 J) = µ ′ (y 1 (y 2 J)) = µ ′ (y 1 J ′ ) = 0. This proves A 2 ⊆ B. Moreover, for anyν ∈ F , we obtainν(y 1 y 2 ) = µ(y 1 y 2 ) = µ ′ (y 1 y 2 ) = 0.
The following formula ξ θ,ν (h + y) = θ(h)εν (y) , h ∈ H(e), y ∈ A
defines a linear character of the subgroup H(e) + A. Really, let g 1 = h 1 + y 1 and g 2 = h 2 + y 2 , where h 1 , h 2 ∈ H(e) and y 1 , y 2 ∈ A. Then ξ θ,ν (g 1 g 2 ) = ξ θ,ν (h 1 h 2 + h 1 y 2 + y 1 h 2 + y 1 y 2 ) = θ(h 1 h 2 )εν (h 1 y 2 ) εν (y 1 h 2 ) εν (y 1 y 2 ) = θ(h 1 )θ(h 2 )εν (y 2 ) εν (y 1 ) = ξ θ,ν (g 1 )ξ θ,ν (g 2 ).
Any character ξ θ,ν being restricted on H(e) + B coincides with ξ θ,µ . By the Frobenius reciprocity theorem, each ξ θ,ν ,ν ∈ F is included in φ θ,µ ′ . The number of elements of F coincides with the degree of character φ θ,µ ′ . Therefore φ θ,µ ′ = ν∈F ξ θ,ν .
Substituting in (16), we obtain Ind(ξ θ,µ , H(e) + B, G ′ ) = ν∈F Ind(ξ θ,ν , H(e) + A, G ′ ).
Let us show that each summand of this sum is a sum of supercharacters. Since B = J µ,right ∩ J ′ ⊂ A ⊂ J ′ , for anyν ∈ F there exists ν ∈ J * that is equal toν being restricted to A, and is equal to µ being restricted to J µ,right . As µ andν are equal on all Pierce components except the first one, we can choose ν ∈ J e . The difference ν − µ annihilates on J µ,right . The Lemma 4.3 implies that ν − µ ∈ Jµ. Since ν, µ ∈ J e , we have
Definition 3 . 5 .
35We say that a ρ G -orbit O is singular (with respect to H) if O ∩ J e = ∅ for some idempotent e = 1 in kH. Otherwise, the orbit O is called regular (with respect to H). Elements of singular (regular) orbits are called singular (regular). Similarly defined singular and regular orbits and elements in J * .The subgroup H(e) admits the following characterization.Proposition 3.6 [8, Lemma 2.5]. 1) H(e) = H y,right ∩ H y,left for any regular (with respect to H e ) element y ∈ J e .
Theorem 3.10 [7, 8, Propositions 4.1, 4.4]. 1) Supercharacters {χ α : α ∈ A} are pairwise disjoint;2) Each supercharacter χ α is constant on each superclass K β ; 3) {1} is a superclass K(g) for g = 1.Applying Proposition w 2.2, we conclude.Theorem 3.11 [8, Theorem 4.5]. The systems of supercharacters {χ α | α ∈ A} and superclasses {K β | β ∈ B} determines a supercharacter theory of the group G.
Lemma 4 .1 [ 1 ,
41Lemma 6.1]. If A is a nilpotent associative algebra, B is a subalgebra in A, and B + A 2 = A, then B = A. Proof. For any positive integer n we have
Lemma 4.3 [1, Lemma 4.2]. J ⊥ λ,right = Jλ for any λ ∈ J * . Proof. If x ∈ J λ,right , then λ(xJ) = 0. Hence Jλ(x) = 0. This verifies the inclusion Jλ ⊂ J ⊥ λ,right . To prove equality it is sufficient to show that the dimensions of subspaces Jλ and J ⊥ λ,right are equal. The dimension of the first one equals to dim J − dim J λ,left , and the second one to dim J − dim J λ,right . The coincidence of dimensions of left and right stabilizers in J can be verified from the fact that the bilinear form λ(xy) provides a pairing of J/J λ,left and J/J λ,right . ✷
Case 1 .
1G = H + J and G ′ = H ′ + J. According to the well known theorem on restriction of induced representation on subgroup [19, Theorem 44.2], we have
of Case 1, we consider that s belongs to H and runs through the set of representatives of the cosets of the subgroup generated by H(e) and H ′ . The subgroup G (s)α ∩ G ′ coincides with (H(e) ∩ H ′ ) + J sλs −1 ,right . Choose λ ∈ ω * . The character ξ λ (h + x) = θ(h)ε sλs −1 (x) .Letē be the least idempotent in kH ′ that e. Show that the subgroup H(e) ∩ H ′ coincides withH ′ (ē) = {h ′ ∈ H ′ : h ′ē =ē}. From Proposition 3.6 we see that H(e) ∩ H ′ = H ′ λ,right ∩ H ′ λ,left .Letω be an orbit of λ in J * e with respect to the group Gē. Taking into account Proposition 3.6, to prove of equality H ′ (ē) = H(e) ∩ H ′ it is sufficient to show thatω is a regular with respect to H ′ e . Assume the contrary. Letω ∩ J * f = ∅ for some idempotent f ∈ kH ′ and f <ē. Recall that λ ∈ J * e , and henceω ∩ J * e = ∅. Thenω ∩ J * ef = ∅ (see[7, Lemma 2.5]). Since ef e and the orbit ω is regular in J * e , we have ef = e. This implies e f <ē. It contradicts to minimality condition forē. The equality H ′ (ē) = H(e) ∩ H ′ is proved.The orbits sω * s −1 (as the orbit ω * ) are regular in Jē (with respect to H ′ e ). Letθ be the restriction of the character θ on H ′ (ē). Then Ind(ξ G ′ , G ′ ) coincides with the supercharacter χᾱ of the subgroup G ′ , whereᾱ = (ē,θ, sω * s −1 ).Case 2. G = H +J and G ′ = H +J ′ , where J ′ is a maximal H ×H-invariant subalgebra in J.The representation of H × H in J is completely reducible. Since J 2 ∈ J ′ (seeCorollary 4.2), every subspace of the form J ′ + W , where W in an invariant with respect to H × H, is a subalgebra (moreover, an ideal). It follows from maximality of J ′ that J = J ′ + W , where W is an irreducible with respect to H × H subspace. For J we have the Pierce decomposition J = J 11 ⊕ J 12 ⊕ J 21 ⊕ J 22 ,
Item 2. 1 .
1Let W ⊆ J 11 . Then J 12 = J ′ 12 , J 21 = J ′ 21 , J 22 = J ′ 22 , and J 11 = W ⊕ J ′ 11 . Apply decomposition(12) to Res χ α . Let us calculate G(s) α ∩ G ′ . Recall that G α = H(e) + J λ,right , where H(e) = {h ∈ H : he = eh = e} and J λ,right = {y ∈ J : λ(yJ) = 0}.
The formula (12) has the form Res χ α = µ Ind(ξ θ,µ , H(e) + B, G ′ ),
B ⊂ A, the subgroup H(e)+B is contained in the subgroup H(e)+A. Then Ind(ξ θ,µ , H(e) + B, G ′ ) = Ind(Ind(ξ θ,µ , H(e) + B, H(e) + A), H(e) + A, G ′ ). (16)
We consider that G ′ \G/G α is represented by the elements s = 1 + z, where z ∈ W 0 . Let us calculate sG α s −1 ∩ G ′ . Let g ∈ G α . Then g = h + x, where h ∈ H(e) and x ∈ J λ,right . Let us show that sgs −1 belongs to G ′ if and only if h commutates with s (equivalently, with z) and sxs −1 ∈ J ′ . Indeed, s(h + x)s −1 = shs −1 + sxs −1 . Observe that since J 2 12 = 0, we have s −1 = 1 − z. The elementbelongs to h+W 0 . The element x is represented in the form x = x 12 +y, where x 12 ∈ J 12 , and y is a sum of all other Pierce components of x. By conditions of Item 2.2, y ∈ J ′ . Since J ′ is an ideal in J, the element sys −1 belongs to J ′ . The subalgebra J λ,right is graded with respect to the Pierce decomposition. As x ∈ J λ,right , we have x 12 ∈ J λ,right . Since J 2 12 = 0, we calculate sx 12 s −1 = x 12 . Then sxs −1 = sx 12 s −1 + sys −1 = x 12 + sys −1 ∈ J λ,right + J ′ . Summing up, we conclude that sgs −1 can be written in the form(19). Therefore, sgs −1 ∈ G ′ if and only if [z, h] = 0 and sxs −1 ∈ J ′ .Denote by F the set of all h ∈ H(e) that commutes with s. Then sG α s −1 ∩ G ′ consists of all elements of the form s(h + x)s −1 , where h ∈ F , and sxs −1 belongs to sJ λ,right s −1 ∩ J ′ . That isLet us show that J sλs −1 ,right ∩ J ′ = J ′ sλ ′ s −1 ,right . Obviously, the left hand side of this equality is contained in the right hand side. Let us show the opposite inclusion. If y ∈ J ′ sλ ′ s −1 ,right , then sλs −1 (yJ ′ ) = 0. Recall that J = J 12 + J ′ . To prove y ∈ J sλs −1 ,right it is sufficient to show that sλs −1 (yJ 12 ) = 0. Indeed, sλs −1 (yJ 12 ) = J 12 sλs −1 (y). As J 2 12 = 0, we have J 12 s = J 12 . We obtain sλs −1 (yJ 12 ) = J 12 λs −1 (y). Finally,The equality (12) takes the formwhere µ ′ runs through all sλ ′ s −1 , s = 1 + z, z ∈ W 0 . It follows from Lemma 4.4 that each summand in the sum(20)is a sum of supercharacters. This proves the statement in Item 2.2.The restriction Res χ α coincides with Ind(ξ θ,λ , G α ∩ G, G ′ ). As in Item 2.1,According to Lemma 4.3, this character is a sum of supercharacters.Suppose that W ⊆ J 21 . As in Item 2.2, since B ⊂ A, the subgroup H(e)+B is contained in H(e)+A, and the formula (16) holds. Decompose the character φ θ,λ ′ = Ind(ξ θ,λ ′ , H(e) + B, H(e) + A) into irreducible components.Denote by F the set of allν ∈ A * such thatν| B =λ| B . Each elementν ∈ F we extend to ν ∈ J * that is equal to λ being restricted to J λ,right . Then ν − λ is equal to zero on J λ,right . Applying Lemma 4.3, we obtain ν − λ ∈ Jλ. Linear forms of J ′ λ annihilate both A and J λ,right . Since J = W ⊕ J ′ , we have ν − λ ∈ W λ.The set F is a projection of elements (1 + W )λ on A. This argues that the group H(e) trivially act on F by right multiplication and nontrivially by left multiplication (this differs with Item 2. Applying the Frobenius reciprocity theorem, easy to verify that χ i includes in decomposition of φ θ,λ ′ into a sum of irreducible characters. Let us calculate the sum of degrees of these characters:From this we concludeSubstituting in(16), we obtainIt remains to show that the characters Ind(ξ i , H i + A, G) are sums of supercharacters.Recall that A = J ′ λ ′ ,right . Extend eachν i ∈ F to ν i ∈ J * that equals to λ on J λ,right . As we saw above, ν i = (1 + w i )λ. Restricting to J ′ , we get ν ′ i = (1 + w i )λ ′ . Hence, the right stabilizers of λ ′ and ν ′ coincides and. Using Lemma 4.4, we conclude that this character is a sum of supercharacters of the subgroup G ′ . ✷ Corollary 4.6. The product of supercharacters of finite group of triangular type is a sum of supercharacters with nonnegative integer coefficients. Proof. Let G be a finite group of triangular type, and χ 1 , χ 2 are its supercharacters. Then G × G = H × H + J ⊕ J is also a finite group of triangular type, and χ 1 ⊗ χ 2 ∈ Fun(G × G) is its supercharacter. The statement is verified using the restriction of χ 1 ⊗χ 2 on the diagonal subgroup {(g, g) : g ∈ G}. ✷The Frobenius theorem for supercharactersLet G = H + J be a finite group of triangular type, G ′ = H ′ + J ′ is its subgroup of triangular type. The scalar product on G is defined as usualLet φ be a superclass function (i.e., a complex valued function constant on superclasses) on G ′ . Denote byφ the function on G equal to φ on G ′ and zero out G ′ . We define superinduction as follows:Easy to see that SInd φ is a superclass function on G. Theorem 5.1. Let ψ be a superclass function on G. Then (SInd φ, ψ) = (φ, Res ψ).Proof. We obtain the proof by direct calculations:Let {χ α } be the system of supercharacters of finite group of triangular type G = H + J, and {φ η } be the system of supercharacters of its subgroup of triangular type G ′ = H ′ + J ′ . By Theorem 4.5, Res χ α = η m α,η φ η , m α,η ∈ Z + .The system of supercharacters form a basis in the subspace of superclass functions (see[1]). Since SInd φ η is a superclass function on G, we obtain SInd φ η = α a η,α χ α .The Theorem 5.1 implies a η,α = m α,η (φ η , φ η ) (χ α , χ α ) .Corollary 5.2. For any supercharacter φ of subgroup of triangular type G ′ , the superinduction SInd φ is a sum of supercharacters of the group G with nonnegative rational coefficients.
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| {'fraction_non_alphanumeric': 0.08735781958357086, 'fraction_numerical': 0.027309498601949127, 'mean_word_length': 3.2153564481061907, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 45, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'It is proved the restriction of any supercharacter of a finite group of triangular type on its subgroup is a sum of supercharacters with nonnegative integer coefficients. We define a superinduction and prove the analog of Frobenius reciprocity formula for supercharacters. * The paper is produced by financial support of the Ministry of education and science of the Russian Federation within the basic part of the state task (project No. 204). The work is supported by RFBR grants 16-01-00154-a and 14-01-97017-Volgaregion-a.The following statement is important for constructing supercharacter theories. Proposition 2.2 [1, Lemma 2.1]. Suppose that we have a system of disjoint characters Ch = {χ 1 , . . . , χ m } and a partition K = {K 1 , . . . , K m } of the group G. Assume that each character χ i is constant on each K j . Denote by ν − µ = eνe − eµe = e(ν − µ)e ∈ eJµe = eJeµ.That is ν − µ ∈ J e µ, and ν ∈ N e µ, where N e = 1 + J e . It is also true the opposite: if ν ∈ N e µ, then ν| B = µ ′ | B .Since J ′ is a H × H-invariant ideal in J, the group G acts on J ′ (and J ′ * ) by right ans left multiplications. The equality ν = aµ, where a ∈ N e , admits the restriction on J ′ * in the form ν ′ = aµ ′ . This implies that J ′ ν ′ ,right = J ′ µ ′ ,right = A. Therefore Ind(ξ θ,ν , H(e) + A, G ′ ) = Ind(ξ θ,ν ′ , H(e) + J ′ ν ′ ,right , G ′ ). According to Lemma 4.4, this character is a sum of supercharacters. This proves the statement in Item 2.1. Item 2.2. Suppose that W ⊆ J 12 . Then J 11 = J ′ 11 , J 21 = J ′ 21 , J 22 = J ′ 22 , and J 12 = W ⊕ J ′ 12 . As in the previous item, the restriction Res χ α is calculated by the formula (12). The subspace J λ,right + J ′ is invariant with respect to H(e) × H(e). Denote by W 0 the H(e) × H(e)-invariant subspace in J 12 such that', 'arxivid': '1610.04846', 'author': ['A N Panov '], 'authoraffiliation': [], 'corpusid': 119738323, 'doi': '10.1070/sm8778', 'github_urls': [], 'n_tokens_mistral': 13563, 'n_tokens_neox': 11839, 'n_words': 7163, 'pdfsha': '2101c0a15abe6fc513691ab37d71b3849050ca67', 'pdfurls': ['https://arxiv.org/pdf/1610.04846v1.pdf'], 'title': ['Restriction and induction for supercharacters of finite groups of triangular type', 'Restriction and induction for supercharacters of finite groups of triangular type'], 'venue': []} |
arxiv |
QUANTUM ANNEALING: FROM VIEWPOINTS OF STATISTICAL PHYSICS, CONDENSED MATTER PHYSICS, AND COMPUTATIONAL PHYSICS
13 Apr 2012 May 10, 2014
Shu Tanaka [email protected]
Ryo Tamura [email protected]
Department of Chemistry
Institute for Solid State Physics
University of Tokyo
7-3-1, Bunkyo-ku113-0033Hongo, TokyoJapan
International Center for Young Scientists
University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi277-8501ChibaJapan
National Institute for Materials Science
1-2-1, Sengen, Tsukuba-shi305-0047IbarakiJapan
QUANTUM ANNEALING: FROM VIEWPOINTS OF STATISTICAL PHYSICS, CONDENSED MATTER PHYSICS, AND COMPUTATIONAL PHYSICS
13 Apr 2012 May 10, 2014arXiv:1204.2907v1 [cond-mat.dis-nn] 17:35 WSPC -Proceedings Trim Size: 9in x 6in Tanaka˙KLN˙main 1Quantum annealingQuantum informationIsing modelOptimiza- tion problem
In this paper, we review some features of quantum annealing and related topics from viewpoints of statistical physics, condensed matter physics, and computational physics. We can obtain a better solution of optimization problems in many cases by using the quantum annealing. Actually the efficiency of the quantum annealing has been demonstrated for problems based on statistical physics. Then the quantum annealing has been expected to be an efficient and generic solver of optimization problems. Since many implementation methods of the quantum annealing have been developed and will be proposed in the future, theoretical frameworks of wide area of science and experimental technologies will be evolved through studies of the quantum annealing.
Introduction
Optimization problems are present almost everywhere, for example, designing of integrated circuit, staff assignment, and selection of a mode of transportation. To find the best solution of optimization problems is difficult in general. Then, it is a significant issue to propose and to develop a method for obtaining the best solution (or a better solution) of optimiza-tion problems in information science. In order to obtain the best solution, a couple of algorithms according to type of optimization problems have been formulated in information science and these methods have yielded practical applications. Furthermore, since optimization problem is to find the state where a real-valued function takes the minimum value, it can be regarded as problem to obtain the ground state of the corresponding Hamiltonian. Thus, if we can map optimization problem to well-defined Hamiltonian, we can use knowledge and methodologies of physics. Actually, in computational physics, generic and powerful algorithms which can be adopted for wide application have been proposed. One of famous methods is simulated annealing which was proposed by Kirkpatrick et al. 1,2 In the simulated annealing, we introduce a temperature (thermal fluctuation) in the considered optimization problems. We can obtain a better solution of the optimization problem by decreasing temperature gradually since thermal fluctuation effect facilitates transition between states. It is guaranteed that we can obtain the best solution definitely if we decrease temperature slow enough. 3 Then, the simulated annealing has been used in many cases because of easy implementation and guaranty.
The quantum annealing was proposed as an alternative method of the simulated annealing. [4][5][6][7][8][9][10][11] In the quantum annealing, we introduce a quantum field which is appropriate for the considered Hamiltonian. For instance, if the considered optimization problem can be mapped onto the Ising model, the simplest form of the quantum fluctuation is transverse field. In the quantum annealing, we gradually decrease quantum field (quantum fluctuation) instead of temperature (thermal fluctuation). The efficiency of the quantum annealing has been demonstrated by a number of researchers, and it has been reported that a better solution can be obtained by the quantum annealing comparison with the simulated annealing in many cases. Figure 1 shows schematic picture of the simulated annealing and the quantum annealing. In optimization problems, our target is to obtain the stable state at zero temperature and zero quantum field, which is indicated by the solid circle in Fig. 1.
Recently, methods in which we decrease temperature and quantum field simultaneously have been proposed and as a result, we can obtain a better solution than the simulated annealing and the simple quantum annealing. [12][13][14] Moreover, as an another example of methods in which we use both thermal fluctuation and quantum fluctuation, novel quantum annealing method with the Jarzynski equality 15,16 was also proposed, 17 which is based on nonequilibrium statistical physics.
Quantum field
Temperature Simulated annealing Quantum annealing In this paper, we review the quantum annealing method which is the generic and powerful tool for obtaining the best solution of optimization problems from viewpoints of statistical physics, condensed matter physics, and computational physics. The organization of this paper is as follows. In Sec. 2, we review the Ising model which is a fundamental model of magnetic systems. The realization method of the Ising model by nuclear magnetic resonance is also explained. In Sec. 3, we show a couple of implementation methods of the quantum annealing. In Sec. 4, we explain two optimization problems -traveling salesman problem and clustering problem. The quantum annealing based on the Monte Carlo method for the traveling salesman problem is also demonstrated. In Sec. 5, we review related topics of the quantum annealing -Kibble-Zurek mechanism of the Ising spin chain and order by disorder in frustrated systems. In Sec. 6, we summarize this paper briefly and give some future perspectives of the quantum annealing.
Ising Model
In this section we introduce the Ising model which is a fundamental model in statistical physics. A century ago, the Ising model was proposed to explain cooperative nature in strongly correlated magnetic systems from a microscopic viewpoint. 18 The Hamiltonian of the Ising model is given by
H Ising = − i,j J ij σ z i σ z j − N i=1 h i σ z i , σ z i = ±1,(1)
where the summation of the first term runs over all interactions on the defined graph and N represents the number of spins. If the sign of J ij is positive/negative, the interaction is called ferromagnetic/antiferromagnetic interaction. Spins which are connected by ferromagnetic/antiferromagnetic interaction tend to be the same/opposite direction. The second term of the Hamiltonian denotes the site-dependent longitudinal magnetic fields. Although the Ising model is quite simple, this model exhibits inherent rich properties e.g. phase transition and dynamical behavior such as melting process and slow relaxation. For instance, the ferromagnetic Ising model with homogeneous interaction (J ij = J for ∀i, j) and no external magnetic fields (h i = 0 for ∀i) on square lattice exhibits the second-order phase transition, whereas no phase transition occurs in the Ising model on onedimensional lattice. Onsager first succeeded to obtain explicitly free energy of the Ising model without external magnetic field on square lattice. 19 After that, a couple of calculation methods were proposed. Furthermore, these calculation methods have been improved day by day, and the new techniques which were developed in these methods have been applied for other more complicated problems. Since the Ising model is quite simple, we can easily generalize the Ising model in diverse ways such as the Blume-Capel model, 20,21 the clock model, 22,23 and the Potts model. 24,25 By analyzing these models, relation between nature of phase transition and the symmetry which breaks at the transition point has been investigated. Then, it is not too much to say that the Ising model has opened up a new horizon for statistical physics. The Ising model can be adopted for not only magnetic systems but also systems in wide area of science such as information science. Optimization problem is one of important topics in information science. As we mention in Sec. 4, optimization problem can be mapped onto the Ising model and its generalized models in many cases. Then some methods which were developed in statistical physics often have been used for optimization problem. In Sec. 2.1, we show a couple of magnetic systems which can be well represented by the Ising model. In Sec. 2.2, we review how to create the Ising model by Nuclear Magnetic Resonance (NMR) technique as an example of experimental realization of the Ising model.
Magnetic Systems
In many cases, the Hamiltonian of magnetic systems without external magnetic field is given bŷ
H = − i,j J ijσi ·σ j = − i,j J ij σ x i ·σ x j +σ y i ·σ y j +σ z i ·σ z j ,(2)
whereσ α i denotes the α-component of the Pauli matrix at the i-th site. The form of this interaction is called Heisenberg interaction. The definitions of Pauli matrices arê
σ x := 0 1 1 0 ,σ y := 0 −i i 0 ,σ z := 1 0 0 −1 ,(3)
where the bases are defined by
|↑ := 1 0 , |↓ := 0 1 .(4)
In this case, magnetic interactions are isotropic. However, they become anisotropic depending on the surrounded ions in real magnetic materials. In general, the Hamiltonian of magnetic systems should be replaced bŷ
H = − i,j J ij c xσ x iσ x j + c yσ y iσ y j + c zσ z iσ z j .(5)
When |c x |, |c y | > |c z |, the xy-plane is easy-plane and the Hamiltonian becomes XY-like Hamiltonian. On the contrary, when |c z | > |c x |, |c y |, the zaxis is easy-axis and the Hamiltonian becomes Ising-like Hamiltonian. Such anisotropy comes from crystal structure, spin-orbit coupling, and dipoledipole coupling. Moreover, even if there is almost no anisotropy in magnetic interactions, magnetic systems can be regarded as the Ising model when the number of electrons in the magnetic ion is odd and the total spin is halfinteger. In this case, doubly degenerated states exist because of the Kramers theorem. These states are called the Kramers doublet. When the energy difference between the ground states and the first-excited states ∆E is large enough, these doubly-degenerated ground states can be well represented by the S = 1/2 Ising spins. Table 1 shows examples of the magnetic materials which can be well represented by the Ising model on one-dimensional chain, two-dimensional square lattice, and three-dimensional cubic lattice.
Nuclear Magnetic Resonance
In condensed matter physics, Nuclear Magnetic Resonance (NMR) has been used for decision of the structure of organic compounds and for analysis of the state in materials by using resonance induced by electromagnetic wave. The NMR can create the Ising model with transverse fields, which is expected to become an element of quantum information processing. In this processing, we use molecules where the coherence times are long compared with typical gate operations. Actually a couple of molecules which have nuclear spins were used for demonstration of quantum computing. [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75] In this section we explain how to create the Ising model by NMR. The setup of the NMR spectrometer as a tool of quantum computing is as follows. We first put molecules which contain nuclear spins under the strong magnetic field B 0 . Next we apply radio frequency ω (rf) magnetic field which is perpendicular to the strong magnetic field B 0 . For simplicity, we here consider a molecule which contains two spins. We also assume that the considered molecule can be well described by the Heisenberg Hamiltonian. Then the Hamiltonian of this system is given bŷ density matrix can be calculated by
i dρ dt = [Ĥ,ρ].(11)
The density matrix on the rotating frame is given bŷ
ρ (R) :=Û (R)ρÛ (R) † .(12)
To be the same form as Eq. (11) on the rotating frame, the Hamiltonian on the rotating frame should bê
H (R) =Û (R)ĤÛ (R) † − iÛ (R) dÛ (R) † dt .(13)
Here we decompose the Hamiltonian on the rotating frame aŝ
H (R) =Ĥ (R) mol +Ĥ (R)(rf) 1 +Ĥ (R)(rf) 2 ,(14)
where the three terms are defined bŷ
H (R) mol :=Û (R)Ĥ molÛ (R) † − iÛ (R) dÛ (R) † dt ,(15)H (R)(rf) 1 :=Û (R)Ĥ(rf) 1Û (R) † ,(16)H (R)(rf) 2 :=Û (R)Ĥ(rf) 2Û (R) † .(17)
The intramolecular magnetic interaction Hamiltonian on the rotating framê H (R) mol can be calculated aŝ
H (R) mol = J 0 0 0 0 0 0 e i(h2−h1)t 0 0 e −i(h2−h1)t 0 0 0 0 0 0 − Jσ z 1σ z 2 ≃ −Jσ z 1σ z 2 .(18)
The approximation is valid when |h 2 − h 1 |τ ≫ 1, where τ is a characteristic time scale since the exponential terms are averaged to vanish. The radio frequency magnetic field Hamiltonian on the rotating frameĤ
(R)(rf) 1 under the resonance condition ω (rf) = h i can be calculated aŝ H (R)(rf) 1 = −Γ 1 0 0 e −iφ1 0 0 0 0 e −iφ1 e iφ1 0 0 0 0 e iφ1 0 0 + γ ′ 0 a −− 0 0 a ++ 0 0 0 0 0 0 a −− 0 0 a ++ 0 , where a −− := e −i(h2−h1)t+φ1 + e −i(H (R)(rf) 1 = −Γ 1 (cos φ 1σ x 1 + sin φ 1σ y 1 ).(19)
In the same way, the HamiltonianĤ
(R)(rf) 2
can be calculated aŝ
H (R)(rf) 2 = −Γ 2 (cos φ 2σ x 2 + sin φ 2σ y 2 ).(20)
By taking the rotation operators on the individual sites, we can rewrite the HamiltoniansĤ
(R)(rf) 1 andĤ (R)(rf) 2
by only the x-component of the Pauli matrix:
e iφ1σ z 1Ĥ (R)(rf) 1 e −iφ1σ z 1 = −Γ 1σ x 1 ,(21)e iφ2σ z 2Ĥ (R)(rf) 2 e −iφ2σ z 2 = −Γ 2σ x 2 .(22)
Then, the total Hamiltonian can be represented by the Ising model with site-dependent transverse fields:
H (R) = −Jσ z 1σ z 2 − Γ 1σ x 1 − Γ 2σ x 2 .(23)
It should be noted that the above procedure is not restricted for two spin system. Then, the NMR technique can be create the Ising model with sitedependent transverse fields in general.
Implementation Methods of Quantum Annealing
As stated in Sec. 1, the quantum annealing method is expected to be a powerful tool to obtain the best solution of optimization problems in a generic way. The quantum annealing methods can be categorized according to how to treat time-development. One is a stochastic method such as the Monte Carlo method which will be shown in Sec. 3.1. Other is a deterministic method such as mean-field type method and real-time dynamics. We will explain the mean-field type method and the method based on real-time dynamics in Secs. 3.2 and 3.3. Although in the Monte Carlo method and the mean-field type method, we introduce time-development in an artificial way, the merit of these methods is to be able to treat large-scale systems. The methods based on the Schrödinger equation can follow up real-time dynamics which occurs in real experimental systems. However, these methods can be used for very small systems and/or limited lattice geometries because of limited computer resources and characters of algorithms. Each method has strengths and limitations based on its individuality. Then when we use the quantum annealing, we have to choose implementation methods according to what we want to know. In this section, we explain three types of theoretical methods for the quantum annealing and some experimental results which relate to the quantum annealing.
Monte Carlo Method
In this section we review the Monte Carlo method as an implementation method of the quantum annealing. In physics, the Monte Carlo method is widely adopted for analysis of equilibrium properties of strongly correlated systems such as spin systems, electric systems, and bosonic systems. Originally the Monte Carlo method is used in order to calculate integrated value of given function. The simplest example is "calculation of π". Suppose we consider a square in which −1 ≤ x, y ≤ 1 and a circle whose radius is unity and center is (x, y) = (0, 0). We generate pair of uniform random numbers (−1 ≤ x i , y i ≤ 1) many times and calculate the following quantity:
number of steps when
x 2 i + y 2 i ≤ 1 is satisfied number of steps .(24)
Hereafter we refer to the denominator as Monte Carlo step. The quantity should converge to π/4 in the limit of infinite Monte Carlo step. This is a pedagogical example of the Monte Carlo method. We first explain how to implement and theoretical background of the Monte Carlo method which is used in physics.
In equilibrium statistical physics, we would like to know the equilibrium value at given temperature T . The equilibrium value of the physical quantity which is represented by the operator O is defined as
O (eq) T := Tr Oe −βH Tr e −βH ,(25)
where Tr means the trace of matrix and β denotes the inverse temperature β = (k B T ) −1 . Hereafter we set the Boltzmann constant k B to be unity. For small systems, we can obtain the equilibrium value by taking sum analytically, on the contrary, it is difficult to obtain the equilibrium value for large systems except few solvable models. Then in order to evaluate equilibrium value of the physical quantity, we often use the Monte Carlo method.
We consider the Ising model given by
H Ising = − i,j J ij σ z i σ z j − N i=1 h i σ z i , σ z i = ±1.(26)
The Ising model without transverse field can be expressed as a diagonal matrix by using "trivial" bit representation |↑ and |↓ which were introduced in Sec. 2. Then, in this case, we can easily calculate the eigenenergy once the eigenstate is specified.
We can use the Monte Carlo method for obtaining the equilibrium value defined by Eq. (25) as well as the calculation of π:
Σ O(Σ)e −βE(Σ) Σ e −βE(Σ) → O (eq) T ,(27)
where O(Σ) and E(Σ) denote the physical value of O and the eigenenergy of the eigenstate Σ, respectively. Here the eigenstate Σ is generated by uniform random number and Σ 1 is equal to Monte Carlo step. In the limit of infinite Monte Carlo step, LHS of Eq. (27) should be converge to the equilibrium value. Equilibrium statistical physics says that the probability distribution at equilibrium state can be described by the Boltzmann distribution which is proportional to e −βE(Σ) . In this case, since we know the form of the probability distribution, it is better to use the distribution function to generate a state according to the Boltzmann distribution instead of uniform random number. This scheme is called importance sampling. When we use the importance sampling, we can obtain the equilibrium value as follows:
Σ O(Σ) Σ 1 → O (eq) T .(28)
In order to generate a state according to the Boltzmann distribution, we use the Markov chain Monte Carlo method. Let P (Σ a , t) be the probability of the a-th state at time t. In this method, time-evolution of probability distribution is given by the master equation:
P (Σ a , t + ∆t) = b =a P (Σ b , t)w(Σ a |Σ b ) + P (Σ a , t)w(Σ a |Σ a ) − b =a P (Σ a , t)w(Σ b |Σ a ) ∆t,(29)
where w(Σ a |Σ b ) represents the transition probability from the b-th state to the a-th state in unit time. The transition probability w(Σ a |Σ b ) obeys
Σa w(Σ a |Σ b ) = 1 (∀ Σ b ).(30)
For convenience, let P(t) be a vector-representation of probability distribution {P (Σ a , t)}. Then the master equation can be represented by
P(t + ∆t) = LP(t),(31)
where L is the transition matrix whose elements are defined as
L ba := w(Σ b |Σ a )∆t,(32)L aa := 1 − b =a L ba = 1 − b =a w(Σ b |Σ a )∆t.(33)
Here the matrix L is a non-negative matrix and does not depend on time.
Then this time-evolution is the Markovian. If the transition matrix L is prepared appropriately, which satisfies the detailed balance condition and the ergordicity, we can obtain the equilibrium probability distribution in the limit of infinite Monte Carlo step regardless of choice of the initial state because of the Perron-Frobenius theorem.
We can perform the Monte Carlo method easily as following process.
Step 1 We prepare a initial state arbitrary.
Step 2 We choose a spin randomly.
Step 3 We calculate the molecular field at the chosen site in Step 2. The molecular field at the chosen site i is defined as
h (eff) i := j ′ J ij σ z j + h i ,(34)
where the summation takes over the nearest neighbor sites of the i-th site.
Step 4 We flip the chosen spin in Step 2 according to a probability defined by some way.
Step 5 We continue from Step 2 to Step 4 until physical quantities such as magnetization converge.
In this Monte Carlo method, we only update the chosen single spin, and thus we refer to this method as single-spin-flip method. There is an ambiguity how to define w(Σ a |Σ b ) in Step 4. Here we explain two famous choices of w(Σ a |Σ b ) as follows. Transition probability in the heat-bath method is given by
w HB (σ z i → −σ z i ) = e −βh (eff) i σ z i 2 cosh(βh (eff) i ) .(35)
Transition probability in the Metropolis method is given by
w MP (σ z i → −σ z i ) = 1 (h (eff) i σ z i < 0) e −2βh (eff) i σ z i (h (eff) i σ z i ≥ 0) .(36)
Since both two transition probabilities satisfy the detailed balance condition, the equilibrium state can be obtained definitely in the limit of infinite Monte Carlo step a . It is important to select how to choice the transition probability since it is known that a couple of methods can sample states in an efficient fashion. [76][77][78][79][80][81][82][83] So far we considered the Monte Carlo method for systems where there is no off-diagonal matrix element. To perform the Monte Carlo method, in a precise mathematical sense, we only have to know how to choice the basis or appropriate transformation so as to diagonalize the given Hamiltonian. However, it is difficult to obtain equilibrium values of physical quantities of quantum systems, since we have to calculate the exponential of the given Hamiltonian e −βĤ in general. If we know all eigenvalues and the corresponding eigenvectors of the given Hamiltonian, we can easily calculate e −βĤ by the unitary transformation which diagonalizes the Hamiltonian H. In contrast, if we do not know all eigenvalues and eigenvectors, we have to calculate any power of the HamiltonianĤ m since the matrix exponential is given by
e = ∞ m=0 1 m! m .(37)
It is difficult to calculate the matrix exponential in general. Then we have to consider the following procedure in order to use the framework of the Monte Carlo method for quantum systems. In many cases, the Hamiltonian of quantum systems can be represented asĤ
=Ĥ c +Ĥ q .(38)
Hereafter we refer toĤ c andĤ q as classical Hamiltonian and quantum Hamiltonian, respectively. The classical HamiltonianĤ c is a diagonal matrix. Here we assume thatĤ q can be easily diagonalized b . This is a key of the quantum Monte Carlo method as will be shown later. SinceĤ c andĤ q cannot commute in general: [Ĥ c ,Ĥ q ] = 0, then e −βĤ = e −βĤc e −βĤq . We a Recently, the algorithm which does not use the detailed balance condition was proposed. 76,77 It should be noted that the detailed balance condition is just a necessary condition. This novel algorithm is efficient for general spin systems. b This fact does not seem to be general. However we can prepare the matrices which can be easily diagonalized by the decomposition asĤq = ℓĤ (ℓ) q in many cases.
decompose the matrix exponential by introducing large integer m,
exp − β mĤ = exp − β m (Ĥ c +Ĥ q ) = exp − β mĤ c exp − β mĤ q + O β m 2 .(39)
This is a concrete representation of the Trotter formula. 84 From now on, we refer to m as Trotter number. By using this relation, we can perform the Monte Carlo method for quantum systems. To illustrate it, we consider the Ising model with longitudinal and transverse magnetic fields. The considered Hamiltonian is given aŝ
H = − i,j J ijσ z iσ z j − N i=1 h z iσ z i − Γ N i=1σ x i =Ĥ c +Ĥ q ,(40)H c := − i,j J ijσ z iσ z j − N i=1 h z iσ z i ,Ĥ q := −Γ N i=1σ x i ,(41)
where optimization problems often can be expressed by this classical Hamil-tonianĤ c . The partition function of the Hamiltonian at temperature T (= β −1 ) is given by
Z = Tr e −βĤ = Σ Σ e −β(Ĥc+Ĥq) Σ .(42)
Using Eq. (39) we obtain
Z = lim m→∞ {Σ k },{Σ ′ k } Σ 1 e −βĤc/m Σ ′ 1 Σ ′ 1 e −βĤq/m Σ 2 × Σ 2 e −βĤc/m Σ ′ 2 Σ ′ 2 e −βĤq/m Σ 3 × · · · × Σ m e −βĤc/m Σ ′ m Σ ′ m e −βĤq/m Σ 1 ,(43)
where |Σ k represents the direct-product space of N spins:
|Σ k := |σ z 1,k ⊗ |σ z 2,k ⊗ · · · |σ z N,k ,(44)
where the first and the second subscripts of |σ z i,k indicate coordinates of the real space and the Trotter axis, respectively. Here |σ z i,k = |↑ or |↓ . Equation (42) consists of two elements Σ k |e −βĤc/m |Σ ′ k and Σ ′ k |e −βĤq/m |Σ k+1 . Since the classical HamiltonianĤ c is a diagonal matrix, the former is easily calculated:
Σ k e −βĤc/m Σ ′ k = exp β m i,j J ij σ z i,k σ z j,k + N i=1 h i σ z i,k N i=1 δ(σ z i,k , σ ′z i,k ),(45)
where σ z i,k = ±1. On the other hand, the latter Σ ′ k e −βĤq/m Σ k+1 is calculated as
Σ ′ k e −βĤq/m Σ k+1 = 1 2 sinh 2βΓ m N/2 exp 1 2 ln coth βΓ m N i=1 σ ′z i,k σ z i,k+1 .(46)
Then the partition function given by Eq. (43) can be represented as
Z = lim m→∞ A {σ z i,k =±1} exp m k=1 i,j βJ ij m σ z i,k σ z j,k + N i=1 βh i m σ z i,k + N i=1 1 2 ln coth βΓ m σ z i,k σ z i,k+1 ,(47)
where A is just a parameter which does not affect physical quantities. It should be noted that the partition function of the d-dimensional Ising model with transverse fieldĤ is equivalent to that of the (d + 1)-dimensional Ising model without transverse field H eff which is given by
H eff = − i,j m k=1 J ij m σ z i,k σ z j,k − N i=1 m k=1 h i m σ z i,k − 1 β N i=1 m k=1 1 2 ln coth βΓ m σ z i,k σ z i,k+1 .(48)
The coefficient of the third term of RHS is always negative, and thus the interaction along the Trotter axis is always ferromagnetic. This ferromagnetic interaction becomes strong as the value of Γ decreases. This is called the Suzuki-Trotter decomposition. 84,85 So far we explained the Monte Carlo method as a tool for obtaining the equilibrium state. However we can also use this method to investigate stochastic dynamics of strongly correlated systems, since the Monte Carlo method is originally based on the master equation. In terms of optimization problem, our purpose is to obtain the ground state of the given Hamiltonian. Then we decrease transverse field gradually and obtain a solution. There are many Monte Carlo studies in which the quantum annealing succeeds to obtain a better solution than that by the simulated annealing. 5,8-10,12,14,86
Deterministic Method Based on Mean-Field Approximation
In the previous section, we considered the Monte Carlo method in which time-evolution is treated as stochastic dynamics. In this section, on the other hand, we explain a deterministic method based on mean-field approximation according to Refs. [87,88]. Before we consider the quantum annealing based on the mean-field approximation, we treat the Ising model with random interactions and site-dependent longitudinal fields given by
H Ising = − i,j J ij σ z i σ z j − N i=1 h i σ z i .(49)
When the transverse field is absent, the molecular field of the i-th spin is given by Eq. (34). Then an equation which determines expectation value of the i-th spin at temperature T (= β −1 ) is given by
m z i = e βh (eff) i − e −βh (eff) i e βh (eff) i + e −βh (eff) i = tanh(βh (eff) i ).(50)
In the mean-field level, we approximate that the state σ z j is equal to the expectation value m z j in Eq. (34), and we obtain
m z i = tanh β j ′ J ij m z j + h i ,(51)
which is often called self-consistent equation.
We can obtain equilibrium value in the mean-field level by iterating the following equation until convergence:
m z i (t + 1) = tanh(βh (eff) i (t)), h (eff) i (t) = j ′ J ij m z j (t) + h i . (52)
In order to judge the convergence, we introduce a distance which represents difference between the state at t-th step and that at (t+1)-th step as follows:
d(t) := 1 N N i=1 |m z i (t + 1) − m z i (t)| .(53)
When the quantity d(t) is less than a given small value (typically ∼ 10 −8 or more smaller value), we judge that the calculation is converged. We summarize this method:
Step 1 We prepare a initial state arbitrary.
Step 2 We choose a spin randomly.
Step 3 We calculate the molecular field given by Eq. (34) at the chosen site in Step 2.
Step 4 We change the value of the chosen spin in Step 2 according to the obtained molecular field in Step 3.
Step 5 We continue from Step 2 to Step 4 until the distance d(t) converges to small value.
The differences between the Monte Carlo method and this method are
Step 4 and Step 5. We can perform the simulated annealing by decreasing temperature and using the state obtained in Step 5 as the initial state in
Step 1 at the time changing temperature c . Next we explain a quantum version of this method. Here we apply transverse field as a quantum field. We consider the Hamiltonian given bŷ
H = − i,j J ijσ z iσ z j − N i=1 h iσ z i − Γ N i=1σ x i .(54)
The density matrix of the equilibrium state iŝ
ρ = exp(−βĤ) Tr exp(−βĤ) = 2 N n=1 exp(−βǫ n ) |λ n λ n | 2 N n=1 exp(−βǫ n ) ,(55)
where ǫ n and |λ n denote the n-th eigenenergy and the corresponding eigenvector. The density matrix satisfies the variational principle that minimizes free energy:
F = min ρ Tr (Ĥ + β −1 lnρ)ρ ,(56)
where the logarithm of the matrix is defined by the series expansion as well as the definition of the matrix exponential (see Eq. (37)). Since it is difficult to obtain the density matrix, we have to consider alternative strategy as follows.
c If we want to decrease temperature rapidly, we choose not so small value for judgement of convergence.
A reduced density matrix is defined aŝ
ρ i := Tr ′ρ = 1 2 Î + m z iσ z + m x iσ x ,(57)
where Tr ′ indicates trace over spin states except the i-th spin. The values m z i and m x i are calculated by
m z i = Tr (σ z iρ ), m x i = Tr (σ x iρ ).(58)
The reduced density matrix satisfies the following relations:
Tr (ρ i ) = 1, Tr (σ z iρi ) = m z i , Tr (σ x iρi ) = m x i .(59)
Here we assume that the density matrix can be represented by direct products of the reduced density matrices:
ρ ≃ N i=1ρ i ,(60)
which is mean-field approximation (in other words, decoupling approximation). Then, the free energy is expressed as
F ≃ min {ρi} F ({ρ i }),(61)F ({ρ i }) = − i,j J ij m z i m z j − N i=1 h i m z i − Γ N i=1 m x i +β −1 N i=1 Tr (ρ i lnρ i ).(62)
From the variation of F ({ρ i }) under the normalization condition, we obtain the following relations:
ρ i = exp(−βĤ i ) Tr [exp(−βĤ i )] ,(63)H i = −h i − ′ j J ij m z j −Γ −Γ +h i + ′ j J ij m z j .(64)
Then the reduced density matrix is represented by using the n-th (n = 1, 2) eigenvalues ǫ (i) n and the corresponding eigenvectors |λ (i) n ofĤ i :
ρ i = exp(−βǫ (i) 1 ) |λ (i) 1 λ (i) 1 | + exp(−βǫ (i) 2 ) |λ (i) 2 λ (i) 2 | exp(−βǫ (i) 1 ) + exp(−βǫ (i) 2 ) .(65)
We can also obtain the equilibrium values of physical quantities as well as the case for Γ = 0:
m z i (t + 1) = Tr(σ z iρ i (t)), m x i (t + 1) = Tr(σ x iρ i (t)),(66)ρ i (t) = exp(−βĤ i (t)) Tr exp(−βĤ i (t)) ,(67)H i (t) = −h i − ′ j J ij m z j (t) −Γ −Γ +h i + ′ j J ij m z j (t) .(68)
We continue the above self-consistent equation until the following distance converges:
d(t) := 1 2N N i=1 (|m z i (t + 1) − m z i (t)| + |m x i (t + 1) − m x i (t)|) .(69)
If the temperature is zero, the reduced density matrix should bê
ρ i = |λ (i) 1 λ (i) 1 | ,(70)
where we consider the case for ǫ
(i) 1 < ǫ (i) 2 . Note that if and only if −h i − ′ j J ij m z j = Γ = 0, ǫ (i) 1 = ǫ (i) 2
is satisfied. Then if we perform the quantum annealing at T = 0, we have to know only the ground state of the local HamiltonianĤ i . The procedure is the same as the case for finite temperature. By using the method, we can obtain a better solution than that obtained by the simulated annealing for some optimization problems. Recently, other type of implementation method based on mean-field approximation was proposed. 13 The method is a quantum version of the variational Bayes inference. 89 We can also obtain a better solution than the conventional variational Bayes inference.
Real-Time Dynamics
In Sec. 3.1 and Sec. 3.2, we considered artificial time-development rules such as the Markov chain Monte Carlo method and mean-field dynamics. In this section, we explain real-time dynamics which is expressed by the time-dependent Schrödinger equation:
i ∂ ∂t |ψ(t) =Ĥ(t) |ψ(t) ,(71)
whereĤ(t) and |ψ(t) denote the time-dependent Hamiltonian and the wave function at time t, respectively. The solution of this equation is given by
|ψ(t) = exp −i t 0Ĥ (t ′ )dt ′ |ψ(t = 0) .(72)
If we use the time-dependent Hamiltonian including time-dependent quantum field, we can perform the quantum annealing by decreasing the quantum field gradually. To obtain the solution, it is necessary to decide the initial state for Eq. (72). Since our purpose is to obtain the ground state of the given Hamiltonian which represents the optimization problem, we have no way to know the preferable initial state that leads to the ground state definitely in the adiabatic limit. However, in general, we often use a "trivial state" as the initial state. Actually, it goes well in many cases. For instance, when we consider the Ising model with time-dependent transverse field which is given bŷ
H(t) = − i,j J ijσ z iσ z j − Γ(t) N i=1σ x i ,(73)
we set the ground state for large Γ as the initial state, hence the initial state is set as
|ψ(t = 0) = |→→ · · · → ,(74)
where |→ denotes the eigenstate ofσ x :
|→ := 1 √ 2 (|↑ + |↓ ).(75)
In real-time dynamics, in order to obtain the ground state by using given initial condition, it is important whether there is level crossing. If there is no level crossing, the system can necessarily reach the ground state by the quantum annealing in the adiabatic limit. To show this fact, we first consider a single spin system under time-dependent longitudinal magnetic field. The Hamiltonian is given bŷ
H single (t) = −h(t)σ z = −h(t) 0 0 h(t) .(76)
Suppose we set |ψ(0) = |↓ as the initial state. For arbitrary sweeping schedules, the state at arbitrary positive t is obtained by
|ψ(t) = exp −i t 0Ĥ single (t ′ )dt ′ |ψ(0) = |↓ .(77)
This is because the state |↓ is the eigenstate of the instantaneous Hamiltonian for arbitrary time t. In general, when there is a good quantum number and the initial state is set to be the corresponding eigenstate, the good quantum number is conserved. Then when we perform the quantum annealing method based on the real-time dynamics, we should take care of the symmetries of the considered Hamiltonian. From this, we can obtain the ground state of the considered system in the adiabatic limit if there is no level crossing. In practice, however, since we change magnetic field with finite speed, a nonadiabatic transition is inevitable. To show this fact, we consider a single spin system under longitudinal and transverse magnetic fields. The Hamiltonian of this system is given bŷ
H single = −hσ z − Γσ x = −h −Γ −Γ h .(78)
Since the eigenenergies are ǫ ± = ± √ h 2 + Γ 2 , the smallest value of the energy difference between the ground state and the excited state is 2Γ at h = 0 as shown in Fig. 2.
Suppose we consider the single spin system under time-dependent longitudinal magnetic field and fixed transverse magnetic field. The Hamiltonian is given byĤ
single (t) = −h(t)σ z − Γσ x = −vt −Γ −Γ vt ,(79)
where we adopt h(t) = vt as time-dependent longitudinal field. Here we set t = −∞ as the initial time. The initial state is set to be the ground state of the Hamiltonian at the initial time |ψ(t = −∞) = |↓ . The ground state at t = +∞ in the adiabatic limit is |ψ (ad) (t = +∞) = |↑ . Then a characteristic value which represents the nature of this dynamics is a probability of staying in the ground state at t = +∞ which is defined by
P stay = ψ (ad) (t = +∞) exp −i +∞ −∞Ĥ single (t ′ )dt ′ ψ(t = −∞) = ↑ exp −i +∞ −∞Ĥ single (t ′ )dt ′ ↓ .(80)
The probability of staying in the ground state should depend on the sweeping speed v and the characteristic energy gap and can be obtained by the Landau-Zener-Stückelberg formula: 90-92
P stay = 1 − exp − π(∆E) 2 4v∆m ,(81)
where ∆E and ∆m represent the energy gap at the avoided level-crossing point and the difference of the magnetizations in the adiabatic limit, respectively. In this case ∆E = 2Γ and ∆m = 2. In many cases, typical shape of energy structure can be approximated by simple systems such as the single spin system. Then the knowledge of the simple transitions such as the Landau-Zener-Stükelberg transition and the Rosen-Zener transition 93 is useful to analyze the efficiency of the quantum annealing based on the real-time dynamics.
Experiments
Transverse field response of the Ising model has been also established in experimentally. [94][95][96][97][98][99][100][101][102][103] A dipolar-coupled disordered magnet LiHo x Y 1−x F 4 has easy-axis anisotropy and can be represented by the Ising model. 104,105 If we apply the longitudinal magnetic field (in other words, the magnetic field is parallel to the easy-axis), phase transition does not take place. 106,107 However, when we apply the transverse magnetic field (in other words, the magnetic field is perpendicular to the easy-axis), phase transitions occur and interesting dynamical properties shown in Ref. [ 6] were observed. In the phase diagram of this material, there are three phases. The ferromagnetic phase appears at intermediate temperature and low transverse magnetic field, whereas at low temperature and low transverse magnetic field, the glassy critical phase 108 appears. The paramagnetic phase exists at the other region. The glassy critical phase exhibits slow relaxation in general. It found that the characteristic relaxation time obtained by ac field susceptibility for quantum cooling in which we decrease transverse field after temperature is decreased is lower than that for temperature cooling case. 6 From this result, it has been expected that the effect of the quantum fluctuation helps us to obtain the best solution of the optimization problem.
Optimization Problems
Optimization problems are defined by composition elements of the considered problem and real-valued cost/gain function. They are problems to obtain the best solution such that the cost/gain function takes the minimum/maximum value. In general, the number of candidate solutions increases exponentially with the number of composition elements in optimization problems. Although we can obtain the best solution by a brute force in principle, it is difficult to obtain the best solution by such a naive method in practice. Then we have to invent an innovative method for obtaining the best solution in a practical time and limited computational resource. Optimization problems can be expressed by the Ising model in many cases. Once optimization problems are mapped onto the Ising model, we can use methods that have been considered in statistical physics and computational physics such as the quantum annealing.
In the anterior half of this section, we explain the correspondence between the Ising model and the traveling salesman problem which is one of famous optimization problems. We demonstrate the quantum annealing based on the quantum Monte Carlo simulation for this problem. In the posterior half, we explain the clustering problem as the example expressed by the Potts model which is a straightforward extension of the Ising model.
Traveling Salesman Problem
In this section, we consider the traveling salesman problem which is one of famous optimization problems. The setup of the traveling salesman problem is as follows:
• There are N cities.
• We move from the i-th city to the j-th city where the distance between them is ℓ i,j . • We can pass through a city only once.
• We return the initial city after we pass through all the cities.
The traveling salesman problem is to find the minimum path under above conditions. The length of a path is given by
L := N a=1 ℓ ca,ca+1 ,(82)
where c a denotes the city where we pass through at the a-th step. In the traveling salesman problem, the length of a path is a cost function. From the fourth condition, the following relation should be satisfied:
c N +1 = c 1 .(83)
In terms of mathematics, the traveling salesman problem is to find {c a } N a=1 so as to minimize the path L under the above four conditions. If the number of cities N is small, it is easy to obtain the shortest path by a brute force. We can easily find the best solution of the traveling salesman problem for N = 6 shown in Fig. 3. Figure 3 (a) and (b) represent a bad solution and the best solution where the length of the path L is minimum, respectively. As the number of cities increases, the traveling salesman problem becomes seriously difficult since the number of candidate solutions is (N − 1)!/2. Then if we want to deal with the traveling salesman problem with large N , we have to adopt smart and easy practical methods such as the simulated annealing instead of a brute force. To use the simulated annealing, we map the traveling salesman problem onto the Ising model with a couple of constraints as follows.
We consider N × N two-dimensional lattice. Let n i,a be the microscopic state which represents the state at the i-th city at the a-th step. The value of n i,a can be taken either 0 or 1. If we pass through the i-th city at the
Furthermore, since it is obvious that we can pass through only one city at the a-th step, this constraint is expressed by
N i=1 n i,a = 1 (for ∀a).(85)
Then the length of the path L can be rewritten as
L = N a=1 i,j ℓ i,j n i,a n j,a+1 = 1 4 N a=1 i,j ℓ i,j σ z i,a σ z j,a+1 + const.,(86)
where the Ising spin variable σ z i,a = ±1 is defined by
σ z i,a := 2n i,a − 1.(87)
Here we used the following relation derived by Eqs. (84) and (85):
N a=1 i,j ℓ i,j σ z i,a = const.(88)
Then the length of the path can be represented by the Ising spin Hamiltonian on N × N two-dimensional lattice. In general, it is difficult to obtain the stable state of the Ising model with some constraints regarded as some kind of frustration which will be shown in Sec. 5.2.
Monte Carlo Method
We explain how to implement the Monte Carlo method in the traveling salesman problem. We cannot use the single-spin-flip method which was explained in Sec. 3.1 because of existence of two constraints given by Eqs. (84) and (85). The simplest way of transition between states is realized by flipping four spins simultaneously as shown in Fig. 4. Suppose we consider the case that we pass through at the i-th city at the a-th step and pass through at the j-th city at the a ′ -th step, which is described as The trial state generated by flipping four spins is as follows:
σ z i,a = +1, σ z j,a = −1, σ z i,a ′ = −1, σ z j,a ′ = +1.(89)σ z i,a = −1, σ z j,a = +1, σ z i,a ′ = +1, σ z j,a ′ = −1.(90)
The heat-bath method and the Metropolis method can be adopted for the transition probability between the present state and the trial state. In Fig. 4, i = 3, j = 6, a = 2, and a ′ = 5.
It should be noted that without loss of generality the initial condition can be set as
σ 1,1 = +1, σ i,1 = −1 (i = 1),(91)
and thus we can fix the states at the first step (a = 1) during calculation. The number of interactions in which we try to flip all spins in each Monte Carlo step is (N − 1)(N − 2)/2.
Quantum Annealing
In order to perform the quantum annealing, we introduce the transverse field as the quantum fluctuation effect as shown in Sec. 3. The quantum Hamiltonian is given bŷ
H = 1 4 N a=1 i,j ℓ i,jσ z i,aσ z j,a+1 − Γ N a=1 N i=1σ x i,a ,(92)
where the first-term corresponds to the length of path and the second-term denotes the transverse field. We can map this quantum Hamiltonian on N × N two-dimensional lattice onto N × N × m three-dimensional Ising model as well as the case which was considered in Sec. 3.1. The effective classical
Hamiltonian derived by the Suzuki-Trotter decomposition is written as
H eff = 1 4m N a=1 i,j m k=1 ℓ i,j σ z i,a,k σ z j,a+1,k − 1 β N a=1 N i=1 m k=1 1 2 ln coth βΓ m σ z i,a,k σ z i,a,k+1 , σ z i,a,k = ±1.(93)
In the quantum annealing procedure, we have to take care of the constraints given by Eqs. (84) and (85) as stated before. Then the simplest way of changing state is to flip simultaneously four spins on the same layer (m is fixed) along the Trotter axis.
Comparison with Simulated Annealing and Quantum Annealing
In order to demonstrate the comparison with the simulated annealing and the quantum annealing, we perform the Monte Carlo simulation for the traveling salesman problem. As an example, we consider N = 20 cities depicted in Fig. 5 (a). The positions of these cities were generated by pair of uniform random numbers (0 ≤ x i , y i ≤ 1). The time schedules of temperature T (t) for the simulated annealing and transverse field Γ(t) for the quantum annealing are defined as
T (t) := T 0 + T 1 1 − t τ ,(94)Γ(t) := Γ 0 + Γ 1 1 − t τ ,(95)
where T 0 and Γ 0 are temperature and transverse field at the final time (t = τ ), and T 0 + T 1 and Γ 0 + Γ 1 are temperature and transverse field at the initial time (t = 0). The value of τ −1 indicates the annealing speed, and the annealing speed becomes slow as the value of τ increases. In our simulations, we adopt T 0 = Γ 0 = 0.01 and T 1 = Γ 1 = 5. Furthermore, we fix the transverse field as Γ = 0 during the simulation in the simulated annealing and the temperature as T = 0.01 during the simulation in the quantum annealing. We execute 100 independent simulations of simulated annealing based on the heat-bath type Monte Carlo method where each initial state generated by the uniform random number is different. To compare the efficiency of the simulated annealing and quantum annealing in an equitable manner, in the quantum annealing, the Trotter number is putted as m = 10, and we execute 10 independent simulations. We also calculate the minimum length of path L min (t) := min{L(t ′ )|0 ≤ t ′ ≤ t}. It should be noted that L min (t) is a monotonic decreasing function. The upper panel of Fig. 6 shows the time dependence of minimum length of path L min (t) for various τ . From the upper panel of Fig. 6, we can see that the convergence of minimum length of path in the quantum annealing is faster than that in the simulated annealing. We also show the sweeping time τ dependence of the minimum length of path at the final state L min (τ ) in the lower panel of Fig. 6. This figure indicates that the obtained solution in the quantum annealing is always better than that in the simulated annealing. Figure 5 (b) shows the obtained best solution in both the simulated annealing and the quantum annealing with slow schedule.
In this way, we can obtain a better solution (in this case, the best solution) by both annealing methods with slow schedule. Moreover, in our calculation, the convergence of solution in the quantum annealing is faster than that in the simulated annealing, and the obtained solution in the quantum annealing is better than that in the simulated annealing regardless of sweeping time τ . Thus, we can say that the quantum annealing method is appropriate as the annealing method for the traveling salesman problem in comparison with the simulated annealing. This fact has been confirmed in some researches. 86
Clustering Problem
In Sec. 4.1, we explained the traveling salesman problem which can be mapped onto the Ising model with some constraints. Many optimization problems can also be mapped onto the Ising model. However, there are a number of optimization problems that can be described by the other models which are straightforward extensions of the Ising model. In this section, we review the concept of clustering problem as such an example. Clustering problem is also one of important optimization problems in information science and engineering. [12][13][14] We need to categorize much data in the real world according to its contents in various situations. For instance, suppose we play stock market. In order to see the socioeconomic situation, we want to extract efficiently important information related to stock market from an enormous quantity of information in news sites and newspapers. In this case, it is better to categorize many articles in news sites and newspapers according to their contents. This is an example of clustering problem which is adopted for many applications in wide area of science such as cognitive science, social science, and psychology. The clustering problem is to divide the whole set into a couple of subsets. Here we refer to the subsets as "cluster". Figure 7 shows schematic picture of the clustering problem. Suppose we consider much data in the whole set which represents the square frame in Fig. 7 (a). The points in Fig. 7 denote individual data. In the clustering problem, our target is to find which the best division is. Figure 7 (b), (c), and (d) represent typical clustering states Σ 1 , Σ 2 , and Σ * , respectively. The states Σ 1 and Σ 2 are an unstable solution and a metastable solution, respectively. The state Σ * denotes the best solution of clustering problem.
In order to consider how to implement the quantum annealing, the clustering problem can be described by the Potts model with random interac-
(a) (b) (c) (d)H Potts = − i,j J ij δ σi,σj , σ i = 1, · · · , Q,(96)
where the summation runs over all pairs of the i-th and j-th data. The spin variable σ i represents individual data. Here the value of Q represents the number of clusters. When σ i = σ j , the i-th and j-th data are in the same cluster. It is natural to adopt ferromagnetic/antiferromagnetic interaction between data in the same/different cluster. It should be noted that the Potts model is a straightforward extension of the Ising model since the Potts model is equivalent to the Ising model if Q = 2. Then the clustering problem is a problem to obtain the ground state of the Hamiltonian of the Potts model with given random interactions. Here we assume that the number of clusters is fixed. Next we explain how to introduce quantum field in order to perform the quantum annealing. In optimization problems which can be represented by the Ising model, we can use transverse field as the quantum fluctuation which is represented as −Γ i σ x i . However, we cannot use this transverse field −Γ i σ x i for the clustering problem directly, since the matrix which represents the state is Q × Q matrix. Thus, we generalize the x-component of the Pauli matrix of the Ising model as follows:
τ x := E Q − I Q = 0 −1 −1 · · · −1 −1 0 −1 · · · −1 −1 −1 0 . . . −1 . . . . . . . . . . . . . . . −1 −1 −1 · · · 0 ,(97)
where E Q and I Q represent the Q × Q unit matrix and the Q × Q matrix whose all elements are unity. By using this generalized Pauli matrix, we can apply the quantum annealing for clustering problem. [12][13][14] Here we consider the following Hamiltonian: where N is the number of individual data. As well as the case for the Ising model, we can calculate the partition function of the Hamiltonian:
H =Ĥ Potts +Ĥ (Potts) q ,Ĥ (Potts) q := −Γ N i=1τ x i ,(98)Z Potts = Tr e −βĤ = Σ Σ e −β(ĤPotts+Ĥ (Potts) q ) Σ = lim m→∞ {Σ k },{Σ ′ k } Σ 1 e −βĤPotts/m Σ ′ 1 Σ ′ 1 e −βĤ (Potts) q /m Σ 2 × Σ 2 e −βĤPotts/m Σ ′ 2 Σ ′ 2 e −βĤ (Potts) q /m Σ 3 × Σ m e −βĤPotts/m Σ ′ m Σ ′ m e −βĤ (Potts) q /m Σ 1 ,(99)
where |Σ k represents the direct-product space of N spins:
|Σ k = |σ 1,k ⊗ |σ 2,k ⊗ · · · |σ N,k .(100)
There are two elements Σ k |e −βĤPotts/m |Σ ′ k and Σ ′ k |e −βĤ (Potts) q /m |Σ k+1 . These factors are calculated as follows:
Σ k e −βĤPotts/m Σ ′ k = exp β m i,j J ij δ σ i,k ,σ j,k N i=1 δ σ i,k ,σ ′ i,k ,(101)Σ ′ k e −βĤ (Potts) q /m Σ k+1 = N i=1 e − βΓ m δ σ ′ i,k σ i,k+1 + 1 Q e − βΓ m (1−Q) − 1 .(102)
By using the above expressions, we can perform the quantum Monte Carlo simulation as well as the Ising model with transverse field. If the spin variable is not S = 1/2 Ising spin as in the case just described, we can implement the quantum annealing by considering appropriate quantum field. There are some studies that the quantum annealing succeeds to obtain the better solution than the simulated annealing for clustering problems. 12-14
Relationship between Quantum Annealing and Statistical Physics
In the preceding sections we explained the Ising model, a couple of implementation methods of the quantum annealing, and the optimization problems. There are a couple of studies that clarify the efficiency and feature of the quantum annealing in terms of statistical physics. In this section we take two examples which display relationship between quantum annealing and statistical physics focusing on the thermal fluctuation effect and the quantum fluctuation effect for ordering phenomena. In the first half, we review the Kibble-Zurek mechanism which characterizes the efficiency of the quantum annealing for systems where a second-order phase transition occurs comparing with the efficiency of the simulated annealing. In the last half, we show similarities and differences between thermal fluctuation and quantum fluctuation for frustrated Ising spin systems.
Kibble-Zurek Mechanism
In statistical physics, it has been an important topic to investigate the ordering process in systems where a phase transition takes place. [110][111][112][113][114][115][116] Especially, dynamical properties during changing control variables such as temperature and external fields are interesting. 111,113,115 Recently, the Kibble-Zurek mechanism has been drawing attention not only in statistical physics and condensed matter physics but also for the quantum annealing. In this section, we explain the Kibble-Zurek mechanism relating to a dynamics which passes across a second-order phase transition point. The Kibble-Zurek mechanism can make clear what happens in systems where the second-order phase transition occurs during the simulated annealing and the quantum annealing from a viewpoint of statistical physics. Before we consider the efficiency of the quantum annealing comparing with the simulated annealing by using the Kibble-Zurek mechanism, we show the general feature of the Kibble-Zurek mechanism.
As an example, we consider the Kibble-Zurek mechanism in the ferromagnetic system where the second-order phase transition occurs at finite temperature. At the second-order phase transition point, the correlation length diverges in the equilibrium state, and thus the relaxation time should be infinite. Hence, the system cannot reach the equilibrium state, when we decrease temperature to the transition temperature with finite speed. Furthermore, since the relaxation time is long around the transition temperature, it is difficult to equilibrate the system. Here, we assume that growth of correlation length stops at the temperature where the system is less able to reach the equilibrium state. If we decrease temperature slow enough, the system can reach the equilibrium state even near the transition point. Thus, it is expected that the value of stopped correlation length because of the long relaxation time depends on the annealing speed. As we will see below, the value of stopped correlation length can be scaled by the annealing speed.
To consider the second-order phase transition at finite temperature in the ferromagnetic systems, we define the dimensionless temperature g as
g := T − T c T c ,(103)
where T c is the phase transition temperature. When the absolute value of g is small, it is believed that the scaling ansatz is valid. By the scaling ansatz, the temperature-dependent correlation length ξ(g) is given as 117
ξ(g) ∝ |g| −ν ,(104)
where ν is one of the critical exponents. Moreover, the relaxation time τ rel is scaled by the following relation: 117
τ rel (g) ∝ [ξ(g)] z ∝ |g| −zν ,(105)
where z is the dynamical critical exponent. Here, we decrease the temperature T (t) against the time t as following schedule:
T (t) = T c 1 − t τ Q (−∞ < t ≤ τ Q ).(106)
The value of τ −1 Q corresponds to the annealing speed. When the value of τ Q is large/small, the system is annealed to low temperature slowly/quickly. At t = 0, the temperature is the phase transition temperature (T (0) = T c ), and the temperature is zero (T (τ Q ) = 0) at t = τ Q . From Eq. (106), the dimensionless temperature g becomes the time-dependent function as follows:
g(t) = T (t) − T c T c = − t τ Q .(107)
In the Kibble-Zurek mechanism, we assume the following situation:
τ rel (g(t)) < |t| : system can reach equilibrium state τ rel (g(t)) > |t| : system cannot reach equilibrium state ,
where |t| is a remaining time to transition temperature. That is, when a remaining time |t| is longer/shorter than the relaxation time τ rel (g(t)), the system can/cannot reach the equilibrium state. Note that the value of considered t should be negative since the relaxation time diverges before the temperature reaches the transition temperature (t = 0). From this assumption, the timet at which the system is less able to reach the equilibrium state is defined by following relation:
τ rel (g(t)) = |t|.(109)
Furthermore, since we have assumed that the growth of correlation length stops at t =t, the value of correlation length is always ξ(g(t)) below T (t) as shown in Fig. 8. Moreover, the dimensionless temperature att is expressed as
g(t) = |t| τ Q = τ rel (g(t)) τ Q ∝ |g(t)| −zν τ Q .(110)
From this relation, g(t) is scaled by the annealing speed, and from Eqs. (104) and (110), the correlation length at t =t is scaled as follows:
g(t) ∝ τ − 1 1+zν Q , ξ(g(t)) ∝ τ ν 1+zν Q .(111)
Furthermore, the density of domain wall n(t) is written as
n(t) ∝ ξ(g(t)) −d ,(112)
where d is the spatial dimension, and n(t) at t =t is scaled as follows:
n(t) ∝ τ − dν 1+zν Q .(113)
For instance, in the ferromagnetic Ising model on two-dimensional lattice (d = 2, ν = 1) when we adopt the Monte Carlo dynamics based on the single-spin-flip method (z = 2.132), 118 the correlation length and the density of domain wall at t =t are naively obtained as
ξ(g(t)) ∝ τ 0.319 Q , n(t) ∝ τ −0.639 Q .(114)
In this way, in the dynamics which passes across the second-order phase transition point at finite temperature, the correlation length and the density of domain wall (topological defect) are scaled by the annealing speed. (g(t)). τ −1 Q is annealing speed and τ Q 1 > τ Q 2 > τ Q 3 . We defineT i := Tc(1 + |t|/τ Q i ) andξ i := ξ(|t|/τ Q i ). The dotted curve represents correlation length in the equilibrium state.
This argument is called the Kibble-Zurek mechanism. Since the Kibble-Zurek mechanism explains the creation of topological defects induced by cooling of the system which takes place the second-order phase transition, this relates to the evolution of cosmic strings by spontaneous symmetry breaking in the Big Bang theory. [119][120][121] The Kibble-Zurek mechanism can also describe the creation of topological defects in magnetic models, 122,123 superfluid helium systems, 124,125 and Bose-Einstein condensations. 126,127 Next we consider the efficiency of the simulated annealing and the quantum annealing using the Kibble-Zurek mechanism by taking examples which can be treated analytically.
Efficiency of Simulated Annealing and Quantum Annealing
Next, we consider the efficiency of the simulated annealing and the quantum annealing according to the Kibble-Zurek mechanism. As an example, we treat the case where the non-domain wall state is the best solution. In this case, the value of n(t) approximately represents the difference between the obtained solution and the best solution. Thus, by using the Kibble-Zurek mechanism, we can compare the efficiency of annealing methods from the behavior of n(t) against the annealing speed. Suppose we solve optimization problems by using annealing methods, we would like to obtain a better solution as fast as possible, in other words, as small τ Q as possible. Then, the comparison obtained by the Kibble-Zurek mechanism is expected to become an useful information for the optimization problems.
As an example, we consider the efficiency of the simulated annealing and the quantum annealing for the random ferromagnetic Ising chain in terms of the Kibble-Zurek mechanism according to Refs. [128,129].
Simulated Annealing for Random Ferromagnetic Ising Chain
The model Hamiltonian of the random ferromagnetic Ising chain is given as
H = − i J i σ z i σ z i+1 , σ z i = ±1,(115)
where J i is the interaction between the i-th site and the (i + 1)-th site. The value of J i is given by the uniform distribution between 0 < J i ≤ 1. The distribution function P (u) (J i ) is given by
P (u) (J i ) := 1 for 0 < J i ≤ 1 0 otherwise .(116)
Since the interaction J i is always positive value, the ground state spin configuration is the all-up spin state or the all-down spin state. In this model, the ferromagnetic transition occurs at zero temperature. The correlation function between two sites where the distance is r is written as
[ σ i σ i+r ] av = 1 β ln cosh β r ,(117)
where · · · and [· · · ] av denote the thermal average and the random average. Physical quantities should depend on the specific spatial pattern of the random interactions {J i }. Then, these averages are defined by
O({J i }) := Tr O({J i })e −βH Tr e −βH ,(118)[O({J i })] av := i dJ i P (u) (J i )O({J i }),(119)
respectively. We omit the argument ({J i }) for simplicity. The relationship between the correlation function and the correlation length ξ is given by
[ σ i σ i+r ] av = e −r/ξ .(120)
Here we mainly focus on the low-temperature limit, since the correlation length grows as temperature decreases. Then the correlation length is given as
ξ = − 1 ln(β −1 ln cosh β) ≃ β ln 2 .(121)
Here, we adopt the Glauber dynamics 130 as the time development, and thus the relaxation time τ rel can be written as
τ rel = 1 1 − tanh 2β ≃ 1 2 e 4β = 1 2 e 4ξ ln 2 .(122)
As we can see, in this model, the correlation length ξ and the relaxation time τ rel are not the power function of temperature unlike the case of the systems where the second-order phase transition occurs at finite temperature (Eqs. (104) and (105)). This is because properties are different between phase transition which exhibits at finite temperature and that occurs at zero temperature. We decrease temperature T (t) against the time t as following schedule:
T (t) = − t τ Q (−∞ < t ≤ 0).(123)
Here T c = 0 in this system. According to the Kibble-Zurek mechanism, we definet by following relation:
τ rel (T (t)) = |t|,(124)
and, we obtain
T (t) = |t| τ Q = τ rel (T (t)) τ Q .(125)
By using Eqs. (121) and (122), low-temperature limit of Eq. (125) is written
as 1 ξ(T (t)) ln 2 ≃ 1 2τ Q e 4ξ(T (t)) ln 2 ,(126)
and, we obtain
ξ(T (t)) = ln τ Q + ln 2 − ln(ξ(T (t)) ln 2) 4 ln 2 ∝ ln τ Q 4 ln 2 .(127)
The approximation of RHS is valid in the case of τ Q ≫ 1 which indicates very slow annealing speed. Thus, we can estimate the density of domain wall n SA (t) at t =t as follows:
n SA (t) ∝ 4 ln 2 ln τ Q .(128)
Quantum Annealing for Random Ferromagnetic Ising Chain
We study the Kibble-Zurek mechanism for the random ferromagnetic Ising chain with transverse field Γ. The model Hamiltonian is given aŝ
H = − i J iσ z iσ z i+1 − Γ iσ x i ,(129)
where the value of J i is given by the uniform distribution between 0 < J i ≤ 1 as well as the case of simulated annealing. In this model, the quantum phase transition from the paramagnetic phase to the ferromagnetic phase occurs at Γ c = exp([ln J i ] av ). 131 Here, we define the dimensionless transverse field g as
g := Γ − Γ c Γ c .(130)
When |g| ≪ 1, it has been known that the correlation length obtained by the renormalization group analysis 132 is scaled by the following relation:
ξ(g) ∝ |g| −ν (ν = 2).(131)
Moreover, a coherence time τ coh is scaled by
τ coh (g) ∝ [ξ(g)] z ∝ |g| −νz (ν = 2),(132)
where the dynamical exponent z is scaled as
z ∝ 1 |g| ,(133)
which is also obtained by the renormalization group analysis. 132 This means that the dynamical exponent diverges at the transition point, and this behavior is a qualitative difference between the random system and the pure system (z = 1). From this fact, τ coh cannot be expressed by the power function of g unlike the case of the second-order phase transition at finite temperature.
We decrease transverse field Γ(t) against the time t as following schedule:
Γ(t) = Γ c 1 − t τ Q (−∞ < t ≤ τ Q ).(134)
According to the Kibble-Zurek mechanism, we definet by following relation:
τ coh (g(t)) = |t|,(135)
and we obtain
g(t) = |t| τ Q = τ coh (g(t)) τ Q .(136)
By using Eqs. (131), (132), and (133), Eq. (136) is written as
1 ξ(g(t)) ∝ 1 τ Q |ξ(g(t))| z ∝ 1 τ Q |ξ(g(t))| √ ξ(g(t)) ,(137)
and, we obtain
ξ(g(t)) + 1 2 ln ξ(g(t)) ∝ ln τ Q .(138)
In the limit of τ Q ≫ 1, since the value of ξ(g(t)) is very large,
ξ(g(t)) + 1 2 ≃ ξ(g(t)),(139)
and we obtain 133
ξ(g(t)) ∝ ln τ Q ln ξ(g(t)) 2 .(140)
Moreover, since the change of ln ξ(g(t)) is gradual in comparison with that of ξ(g(t)), we neglect ln ξ(g(t)) and obtain
ξ(g(t)) ∝ (ln τ Q ) 2 .(141)
From this relation, we can estimate the density of domain wall n QA (t) at t =t as follows:
n QA (t) ∝ (ln τ Q ) −2 .(142)
Comparison between Simulated and Quantum Annealing Methods
We have shown analysis of the domain wall density in the random ferromagnetic Ising chain during the simulated annealing and the quantum annealing by the Kibble-Zurek mechanism. The obtained densities of domain wall are
n SA (t) ∝ (ln τ Q ) −1 : simulated annealing,(143)
n QA (t) ∝ (ln τ Q ) −2 : quantum annealing.
From these relations, it is clear that the decay of n QA (t) is faster than that of n SA (t) against the value of τ Q . Thus, from the Kibble-Zurek mechanism, it is concluded that the quantum annealing method is appropriate as the annealing method for the random ferromagnetic Ising chain in comparison with the simulated annealing method. Suppose we consider the ferromagnetic Ising chain with homogeneous interaction (J i = 1 for all i). In this case, both the domain wall density in the simulated annealing and that in the quantum annealing are obtained as
n(t) ∝ 1 √ τ Q .(145)
This relation for the simulated annealing can be obtained by a simple calculation as well as the case of the random Ising spin chain. On top of that, the relation for the quantum annealing can be derived by Eq. (113). Here the critical exponent ν of the transverse Ising chain with homogeneous interaction is ν = 1 and the dynamical exponent of this system is z = 1. Then there is no difference between the simulated annealing and the quantum annealing in the case of the homogeneous ferromagnetic Ising chain. However, since the optimization problem has some kind of randomness, the abovementioned result encourages that the quantum annealing is better than the simulated annealing for optimization problems.
In general, the existence of the phase transition in optimization problems negatively influences performance of annealing methods. Here, we have introduced the Kibble-Zurek mechanism relating to the dynamics which passes across the second-order phase transition point. As the specific example, we have analyzed the efficiencies of the simulated annealing and the quantum annealing for the random ferromagnetic Ising chain according to the Kibble-Zurek mechanism. For this model, the efficiency of the quantum annealing is better than that of the simulated annealing. Of course, since the efficiency of annealing methods depends on the details of optimization problems, it is not to say that the quantum annealing is always appropriate as the annealing method for general optimization problems in comparison with the simulated annealing. Moreover, we have to develop a theory based on the Kibble-Zurek mechanism itself, 134 since we assume the growth of the correlation length stops at t >t. For example, if we adapt the Kibble-Zurek mechanism to two-or three-dimensional models and more complicated models, it is difficult to estimate the correlation length analytically, and thus we should execute numerical simulations such as the Monte Carlo simulation. For example, in the two-dimensional Ising model with random interactions, it has been shown that the efficiency of the quantum annealing is better than that of the simulated annealing by Monte Carlo simulation. 129 Although the efficiency of annealing methods for a number of optimization problems has been clarified by the Kibble-Zurek mechanism, it remains to be an open problem to investigate when to use the quantum annealing exhaustively.
In the above-mentioned argument, the phase transition under consideration is of the second order. What happens if we adapt the same argument for the other type phase transitions such as first-order phase transition and Kosterlitz-Thouless (KT) transition? In these phase transitions, the behaviors of correlation length are different from that in systems where a secondorder phase transition occurs: the finite-correlation length at the first-order phase transition point and the quasi-long-range correlation length at the KT transition point. Thus, it is an interesting problem to clarify relationship between behaviors of correlation length and the generalized Kibble-Zurek mechanism. By considering dynamical nature of the optimization problems in terms of non-equilibrium statistical physics in a deeper way, we believe that the quantum annealing method will become a central part of practical method for optimization problems.
Frustration Effects for Simulated Annealing and Quantum Annealing
In many cases optimization problems can be represented by the Ising model with random interactions and magnetic fields as mentioned before. The Hamiltonian of this system is given by
H = − i,j J ij σ z i σ z j − N i=1 h i σ z i , σ z i = ±1.(146)
When all interactions are ferromagnetic as the previous example in Sec. 5.1, the ground state is the all-up or the all-down states. However, if there are antiferromagnetic interactions in the system, the situation becomes different. In order to show the difference between ferromagnetic interaction and antiferromagnetic interaction, we first consider three spin system on triangle cluster as shown in Fig. 9. In this section, we treat the case for h i = 0 for all i. The dotted and solid lines in Fig. 9 represent ferromagnetic and antiferromagnetic interactions, respectively. The considered Hamiltonian is written as
H triangle = −J(σ z 1 σ z 2 + σ z 2 σ z 3 + σ z 3 σ z 1 ).(147)
Here we set the all interactions are the same value for simplicity. The ground states for positive J (ferromagnetic interaction) are the all-up or the alldown states shown in Fig. 9 (a). In these states, all spins between all interactions are energetically favorable states. In the case of negative J (antiferromagnetic interaction), while on the other hand, six states shown in Fig. 9 (b) are ground states. These ground states have unfavorable interactions (a) (b) indicated by the crosses in Fig. 9 (b). This situation is called frustration. In the homogeneous antiferromagnetic Ising spin systems on lattices based on triangle such as triangular lattice and kagomé lattice, frustration appears in all triangles. Since such frustration comes from lattice geometry, this is called geometrical frustration. It should be noted that the homogeneous antiferromagnetic Ising spin systems on square lattice and hexagonal lattice have no frustration. Since these systems are bipartite systems which can be decomposed by two sublattices, these systems can be transformed on the ferromagnetic systems by local gauge transformation of all spins belonging to one of the sublattices. Frustration appears in also inhomogeneous systems as shown in Fig. 10. The squares pointed by stars in Fig. 10 represent frustration plaquettes which are satisfied following relation:
κ k := i,j∈ k J ij < 0,(148)
where k indicates the smallest square plaquette at the position k. If κ k for all k is positive, the system is not frustrated. In general, frustration prevents the system from conventional magnetic ordering such as ferromagnetic order and Néel order, since there is no state where all interactions are satisfied energetically in frustrated systems. Frustration makes peculiar density of states which induces unconventional phase transition and slow dynamics. 112,115,[135][136][137][138][139][140][141][142][143][144] Although many optimization problems can be represented by the Ising model with random interactions and magnetic fields, here we focus on the frustration effect which comes from non-random interactions. In terms of statistical physics, this is a firststep study to investigate similarities and differences between thermal fluctuation and quantum fluctuation for frustrated systems. Furthermore, it is important topic for the optimization problems to consider the thermal fluctuation and quantum fluctuation effects for frustrated systems. To obtain the ground state of frustrated systems is to find how to put the unsatisfied bonds represented by the crosses. Since the unsatisfied bonds are regarded as some kind of constraints, this situation is similar with the traveling salesman problem in which there are some constraints as mentioned before. We explain two topics in this section. In the first half, we consider the order by disorder effect in fully-frustrated systems. In the last half, we explain non-monotonic dynamics in decorated bond systems.
Thermal Fluctuation and Quantum Fluctuation Effect of Geometrical Frustrated Systems
In general, there are many degenerated ground states in geometrical frustrated systems such as triangular antiferromagnetic Ising spin systems and kagomé antiferromagnetic Ising spin systems. In these cases, non-zero residual entropy which is entropy at zero temperature exists. Typical configurations of ground states of the triangular antiferromagnetic Ising spin systems are shown in Fig. 11. The residual entropy per spin of this system is S (tri) [145][146][147][148] where k B is the Boltzmann constant. Since the total entropy per spin is k B ln 2 ≃ 0.693k B , 46.6% of the total entropy remains even at zero temperature. In other words, there are macroscopic degenerated ground states in this system. In the antiferromagnetic Ising spin system on kagomé lattice, there are also macroscopic degenerated ground states. The residual entropy per spin of this system is S (kag) res ≃ 0.502k B , which is 72.4% of the total entropy. 149 Suppose we apply the simulated annealing or the quantum annealing with slow schedule for geometrical frustrated spin systems. Since there are macroscopically degenerated ground states in these systems, our purpose is to clarify whether all ground states are obtained with the same probabilities or biased probabilities. We first consider the obtained ground states in the case of the simulated annealing with slow schedule. If we decrease temperature slow enough, the obtained state should satisfy the equilibrium probability distribution. When the temperature is k B T ≪ |J|, the equilibrium probabilities of the ground states are dominant and that of any excited states can be neglected. The principle of equal weight which is the keystone in the equilibrium statistical physics says that if the eigenenergies of the microscopic state Σ A and Σ B are the same, the equilibrium probability of Σ A and that of Σ B are also the same. Then we obtain all macroscopic degenerated ground states with the same probability after the simulated annealing with slow schedule.
res ≃ 0.323k B ,
Next we consider the obtained ground states in the case of the quantum annealing where the transverse field decreases slow enough. Here we assume that the initial state is set to be the ground state of the Hamiltonian at the initial time. In order to capture the feature of the ground states in a graphical way, it is convenient to introduce the concept of free spin where the molecular field is zero. The molecular field at the i-th site is given by
h (eff) i := J j ′ σ z j ,(149)
where the summation runs over the nearest-neighbor sites of the i-th site. For instance, in Fig. 11, spins indicated by dotted circles are free spins.
Here, the transverse field is expressed as
−Γ iσ x i = −Γ i (σ + i +σ − i ),(150)
whereσ + i andσ − i denote the raising and lowering operators at the i-th site, respectively. They are defined bŷ
σ + := 0 1 0 0 ,σ − := 0 0 1 0 .(151)
The x-component of the Pauli matrix corresponds to the operator which flips the considered spin:
σ x |↑ = |↓ ,σ x |↓ = |↑ .(152)
From this, the states which have large number of free spins are expected to become stable at the limit of Γ → 0+ and T = 0. Actually, in the adiabatic limit, the amplitudes of the states which have the maximum number of free spins are larger than the others. [150][151][152][153][154] When we decrease the transverse field slow enough, the state at each time can be well approximated by the ground state of the instantaneous Hamiltonian. Then we obtain specific ground states with high probability after the quantum annealing with slow schedule.
In this section, we considered the thermal fluctuation effect and the quantum fluctuation effect in the adiabatic limit. The simulated annealing can obtain all the ground states with the same probability, while on the other hand, the quantum annealing can obtain specific ground states in this limit. The biased probability distribution can be explained by the character of the quantum Hamiltonian. The selected states should depend on how to choice the quantum Hamiltonian. When we adopt the exchange type interaction as the quantum field, the states that have the maximum value of the "free spin pair" should be selected. Moreover, it is an interesting topic to investigate differences between the simulated annealing and the quantum annealing with finite speed not only in terms of the quantum annealing but also in nonequilibrium statistical physics and condensed matter physics. At the present stage, to consider dynamic phenomena in strongly correlated systems is difficult, since a small number of theoretical methods for obtaining dynamic phenomena have been developed. If the technology of the artificial lattices develops more than ever, real-time dynamics and time-dependent phenomena of frustrated spin systems can be observed in real experiments.
Non-Monotonic Behavior of Correlation Function in Decorated Bond System
In the ferromagnetic Ising spin systems, the correlation function behaves monotonic against the temperature and transverse field. However, the behavior of the correlation function is non-monotonic as a function of temperature in some frustrated spin systems. As an example of non-monotonic correlation function, we introduce equilibrium properties of the correlation function in decorated bond systems in which the frustration exists. The Hamiltonian of the decorated bond systems where the number of system spins is two shown in Fig. 12 is given by
H = −J dir σ z 1 σ z 2 − J N d i=1 s z i (σ z 1 + σ z 2 ),(153)
where σ z i = ±1 and s z i = ±1 are, respectively, called system spins and decorated spins, and N d is the number of decorated spins. The circles and the squares in Fig. 12 represent the system spins and the decorated spins, respectively.
When the direct interaction between system spins J dir is zero and the decorated bond J is positive, the correlation function between system spins σ z 1 σ z 2 is always positive and monotonic decaying function against the temperature. When the direct interaction between system spins J dir is negative and the decorated bond J is zero, on the other hand, the correlation function σ z 1 σ z 2 is always negative and monotonic increasing function against the temperature. From this, the correlation function σ z 1 σ z 2 is expected to behave non-monotonic in some cases for negative J dir and positive J or positive J dir and negative J. In order to obtain temperature dependence of the correlation function between system spins, we trace over spin states except the system spins:
Tr {s z i } e −βH = Ae K eff σ z 1 σ z 2 ,(154)
where A is just a constant which does not affect any physical quantities and the effective coupling K eff is given by
K eff = N d 2 ln cosh(2βJ) + βJ dir .(155)
Temperature dependence of the correlation function between system spins N d = 4). The circles and squares represent system spins and decorated spins, respectively. The dotted and solid lines indicate the direct interaction between system spins and the decorated bonds, respectively.
is represented by using K eff :
C (c) (T ) := σ z 1 σ z 2 = Tr σ z 1 σ z 2 e −βH Tr e −βH = Tr σ z 1 σ z 2 e K eff σ z 1 σ z 2 Tr e K eff σ z 1 σ z 2 = tanh K eff .(156)
Hereafter we set J as the energy unit and J is positive. In order to compare the effect of the direct interaction J dir fairly, we assume the form such as J dir = −xN d J. This is because the effective coupling K eff is proportional to the number of decorated spins N d under the assumption. Figure 13 shows temperature dependence of correlation function between the system spins for N d = 1 and N d = 10 for several x. For small x and large x, the correlation function C (c) (T ) is monotonic decreasing and increasing functions, respectively, against the temperature. However, the correlation function C (c) (T ) behaves non-monotonic as a function of temperature for intermediate x. At the temperatures where the effective coupling K eff is larger than the critical value of the ferromagnetic Ising spin system on square lattice 19 K (square) c = 1 2 ln(1 + √ 2), ferromagnetic phase appears. On the other hand, at the temperature where K eff is less than −K (square) c
, antiferromagnetic phase appears. In this case, successive phase transitions such as paramagnetic → antiferromagnetic → paramagnetic → ferromagnetic phases occur. Such phase transitions are called reentrant phase transitions which are sometimes appeared in frustrated systems. 115,139,[155][156][157][158][159][160] We consider transverse field response of the decorated bond systems in the ground state. The Hamiltonian of the decorated bond system with transverse field is expressed aŝ
H = −J dirσ z 1σ z 2 − J N d i=1ŝ z i (σ z 1 +σ z 2 ) − Γ(σ x 1 +σ x 2 + N d i=1ŝ x i ),(157)
whereŝ α i denotes the α-component of the Pauli matrix of the i-th decorated spin. Here we consider transverse-field dependence of the correlation function in the ground state given by
C (q) (Γ) := ψ (gs) (Γ)|σ z 1σ z 2 |ψ (gs) (Γ) ,(158)
where |ψ (gs) (Γ) denotes the ground state at the transverse field Γ. Figure 14 shows transverse-field dependence of C (q) (Γ) for N d = 1 and N d = 10 for several x. For small x and large x, the correlation function C (q) (Γ) behaves monotonic decreasing and increasing, respectively as a function of transverse field, whereas for intermediate x, transverse-field dependence of the correlation function behaves nonmonotonic as well as the case of thermal fluctuation. Then, the reentrant phase transition also occurs by changing the transverse field. However there is a difference between the thermal fluctuation effect and the quantum fluctuation effect for decorated bond system. The temperature where C (c) (T ) = 0 is satisfied is the same when we change the number of decorated spins N d , whereas the transverse field at C (q) (Γ) = 0 is different when N d is changed.
The thermal fluctuation and the quantum fluctuation have similar properties for the phase transition phenomena in general. Indeed, the reentrant phase transitions occur by changing the thermal fluctuation and also the quantum fluctuation as shown in this section. However as described in Sec. 5.1, in order to obtain the best solution of optimization problems, it is better to erase phase transition. By dealing with thermal and quantum fluctuation effects for frustrated systems exhaustively, we can construct the best form of the adding fluctuation which erases phase transition e .
Conclusion
In this paper, we described some aspects of the quantum annealing from viewpoints of statistical physics, condensed matter physics, and computational physics. Originally, the quantum annealing has been proposed as a method which can solve efficiently optimization problems in a generic way. Since many optimization problems can be mapped onto the Ising model or generalized Ising model such as the clock model and the Potts model, it has been considered that we can obtain a better solution by using methods which were developed in computational physics. For instance, we can obtain a better solution by decreasing temperature (thermal fluctuation) gradually in the simulated annealing which is one of the most famous practical methods. In the quantum annealing, we decrease an introduced quantum field (quantum fluctuation) instead of temperature (thermal fluctuation). In many studies, it was reported that a better solution can be obtained e It is not necessary that the adding fluctuation is restricted in quantum physics. From a viewpoint of optimization problems, we can arbitrary form adding term. Furthermore, it has studied that other novel fluctuation which may be able to erase phase transition as an alternative to thermal and quantum fluctuations. 14,116,161,162 Of course, if we want to realize experimentally, it is better that the added fluctuation term should be some kind of quantum fluctuation. by the quantum annealing efficiently in comparison with the simulated annealing as we explained in Sec. 4. Thus, the quantum annealing method is expected to be a generic and powerful solver of optimization problems as an alternative to the simulated annealing.
The quantum annealing has become a milestone of some related fields under the situation in which the quantum annealing itself has been studied exhaustively. Since we use the quantum fluctuation in the quantum annealing with ingenuity, to obtain a better solution by using the quantum annealing is a kind of quantum information processing. Thus, many implementation methods of the quantum annealing in theoretical and experimental ways have been proposed by many researchers. A number of theoretical implementation methods are proposed based on knowledge of statistical physics. As we shown in Sec. 5, question of what are differences between the simulated annealing and the quantum annealing and question of which is efficient in the given optimization problem are catalysts to investigate differences between the thermal fluctuation and the quantum fluctuation in a deeper way. On top of that, studies on the quantum annealing are expected to open the door to consider equilibrium and nonequilibrium statistical physics. Recently, preparation methods of intended Hamiltonian have been established in some experimental systems such as artificial lattices and nuclear magnetic resonance because of recent development of experimental techniques. As long as we use classical computer and our present knowledge, there are a huge number of problems where to obtain the best solution is difficult without any and every approximation in theoretical methods. However if we prepare the Hamiltonian which expresses our intended problem, we can calculate experimentally the stable state of the prepared Hamiltonian in near future.
The quantum annealing transcends just a method for obtaining the best solution of optimization problems and it will make a development in wide area of science. Although it seems that studies on the quantum annealing itself have been well established, we believe that the quantum annealing plays a role as a bridge with the abovementioned area of science and the quantum information.
Issei Sato, Sei Suzuki, Eric Vincent, and Yoshihisa Yamamoto for their valuable comments. S.T. acknowledges Keisuke Fujii, Yoshifumi Nakada, and Takahiro Sagawa for their useful discussion during the lecture. S.T. is partly supported by Grand-in-Aid for JSPS Fellows (23-7601). R.T. is partly supported financially by National Institute for Materials Science (NIMS). The computation in the present work was performed on computers at the Suprecomputer Center, Institute for Solid State Physics, University of Tokyo.
Fig. 1 .
1Schematic picture of the simulated annealing and the quantum annealing. Our purpose is to obtain the ground state at the point indicated by the solid circle.
Fig. 2 .
2Eigenenergies of the single spin system under longitudinal and transverse magnetic fields for Γ = 0.5 (left panel) and Γ = 1 (right panel). The dotted lines represent eigenenergies for Γ = 0.
Fig. 3 .
3Traveling salesman problem for N = 6. Thin lines and thick lines denote the permitted paths and selected paths, respectively. (a) Bad solution. (b) The best solution in which the length of the path is minimum.a-th step, n i,a is unity whereas n i,a = 0 if we do not pass through the i-th city at the a-th step. The third condition can be represented by N a=1 n i,a = 1 (for ∀i).
Fig. 4 .
4The simplest way of flipping method in traveling salesman problem. Transition between the state depicted in (a) and that depicted in (b) occurs. In this case, i = 3, j = 6, a = 2, and a ′ = 5.
Fig. 5 .
5Traveling salesman problem for N = 20. (a) Positions of cities. (b) The best solution in which the length of the path is minimum.
Fig. 6 .
6(Upper panel) Time dependence of minimum length of path L min (t) for τ = 10, 100, and 1000 obtained by the simulated annealing (SA) and the quantum annealing (QA). (Lower panel) Sweeping-time τ dependence of minimum length of path at the final state L min (τ ) obtained by the simulated annealing indicated by squares and the quantum annealing indicated by circles.
Fig. 7 .
7Schematic pictures of clustering problem. The points represent data and the square denote the whole set. (a) Data set. (b) Unstable solution Σ 1 . (c) Metastable solution Σ 2 . (d) The best solution Σ * .
tions d . TheHamiltonian of the Potts model is given by
Fig. 8 .
8Schematic of the annealing speed dependence of correlation length ξ
Fig. 9 .
9Three spin system on triangle cluster. The dotted and solid lines represent ferromagnetic and antiferromagnetic interactions, respectively. The open and solid circles are the +1-state and the −1-state, respectively. The crosses indicate the positions of unfavorable interactions. (a) Ground states for ferromagnetic case. (b) Ground states for antiferromagnetic case.
Fig. 10 .
10A ground state of the Ising spin system with random interactions. The dotted and solid lines represent ferromagnetic and antiferromagnetic interactions, respectively. The open and solid circles are the +1-state and the −1-state, respectively. The stars and crosses indicate frustration plaquettes and unfavorable interactions, respectively.
Fig. 11 .
11Typical configurations of ground states of antiferromagnetic Ising spin system on triangular lattice. The open and solid circles are the +1-state and the −1-state, respectively. The dotted circles indicate free spin where the molecular field is zero.
Fig. 12 .
12Decorated bond system where the number of system spins is two and the number of decorated spins is four (
Fig. 13 .
13The correlation function between system spins C (c) (T ) as a function of temperature for N d = 1 (left panel) and for N d = 10 (right panel) in the cases of x = 0.1, 0.2, 0.5, 1.0, and 2.0.
Fig. 14 .
14The correlation function between system spins C (q) (Γ) as a function of transverse field for N d = 1 (left panels) and for N d = 10 (right panels) in the cases of x = 0.1, 0.2, 0.5, 1.0, and 2.0.
Table 1 .
1Examples of magnetic materials which can be represented by the Ising model on chain (one-dimension),
square lattice (two-dimension), and cubic lattice (three-dimension).
Material
Spatial dimension Total spin
Type of interaction
J/k B
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1
2
Antiferromagnetic
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26-28
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−2.69 K
56-59
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May 10, 2014 17:35
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| {'fraction_non_alphanumeric': 0.06916665945459423, 'fraction_numerical': 0.03644404441482687, 'mean_word_length': 3.7877682936935444, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 49, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we review some features of quantum annealing and related topics from viewpoints of statistical physics, condensed matter physics, and computational physics. We can obtain a better solution of optimization problems in many cases by using the quantum annealing. Actually the efficiency of the quantum annealing has been demonstrated for problems based on statistical physics. Then the quantum annealing has been expected to be an efficient and generic solver of optimization problems. Since many implementation methods of the quantum annealing have been developed and will be proposed in the future, theoretical frameworks of wide area of science and experimental technologies will be evolved through studies of the quantum annealing.', 'arxivid': '1204.2907', 'author': ['Shu Tanaka [email protected] ', 'Ryo Tamura [email protected] ', '\nDepartment of Chemistry\nInstitute for Solid State Physics\nUniversity of Tokyo\n7-3-1, Bunkyo-ku113-0033Hongo, TokyoJapan\n', '\nInternational Center for Young Scientists\nUniversity of Tokyo\n5-1-5, Kashiwanoha, Kashiwa-shi277-8501ChibaJapan\n', '\nNational Institute for Materials Science\n1-2-1, Sengen, Tsukuba-shi305-0047IbarakiJapan\n'], 'authoraffiliation': ['Department of Chemistry\nInstitute for Solid State Physics\nUniversity of Tokyo\n7-3-1, Bunkyo-ku113-0033Hongo, TokyoJapan', 'International Center for Young Scientists\nUniversity of Tokyo\n5-1-5, Kashiwanoha, Kashiwa-shi277-8501ChibaJapan', 'National Institute for Materials Science\n1-2-1, Sengen, Tsukuba-shi305-0047IbarakiJapan'], 'corpusid': 117511230, 'doi': '10.1142/9789814425193_0001', 'github_urls': [], 'n_tokens_mistral': 38255, 'n_tokens_neox': 32881, 'n_words': 20091, 'pdfsha': '2ee236d574dcbd84b3ffe1b845c2689f0c481879', 'pdfurls': ['https://arxiv.org/pdf/1204.2907v1.pdf'], 'title': ['QUANTUM ANNEALING: FROM VIEWPOINTS OF STATISTICAL PHYSICS, CONDENSED MATTER PHYSICS, AND COMPUTATIONAL PHYSICS', 'QUANTUM ANNEALING: FROM VIEWPOINTS OF STATISTICAL PHYSICS, CONDENSED MATTER PHYSICS, AND COMPUTATIONAL PHYSICS'], 'venue': []} |
arxiv |
Revisiting the Distortion of Distributed Voting
9 Jan 2023
Aris Filos-Ratsikas
School of Informatics
University of Edinburgh
UK
Alexandros A Voudouris
School of Computer Science and Electronic Engineering
University of Essex
UK
Revisiting the Distortion of Distributed Voting
9 Jan 2023
We consider a se ing with agents that have preferences over alternatives and are partitioned into disjoint districts. e goal is to choose one alternative as the winner using a mechanism which first decides a representative alternative for each district based on a local election with the agents therein as participants, and then chooses one of the district representatives as the winner. Previous work showed bounds on the distortion of a specific class of deterministic plurality-based mechanisms depending on the available information about the preferences of the agents in the districts. In this paper, we first consider the whole class of deterministic mechanisms and show asymptotically tight bounds on their distortion. We then initiate the study of the distortion of randomized mechanisms in distributed voting and show bounds based on several informational assumptions, which in many cases turn out to be tight. Finally, we also experimentally compare the distortion of many different mechanisms of interest using synthetic and real-world data.
Introduction
Voting is a ubiquitous method for making decisions with a large number of applications, such as electing political representatives, deciding how to split a public budget between projects, or choosing which services (restaurants, hotels, etc) to recommend to new users based on past user experiences. As such, it has been at the epicenter of research within multiple disciplines including political sciences, economics and computer science [Brandt et al., 2016]. e most prominent question in this research agenda is to identify the best voting rule to use to collectively aggregate the preferences of agents over alternative options into a single winning alternative, with most of the earlier literature focusing on axiomatic properties that good voting rules should have. An alternative way to tackle this question that has been proposed in computer science is through the distortion framework [Anshelevich et al., 2021] which allows to compare different voting rules based on how well they approximate the optimal choice as measured in terms of a social objective function like the utilitarian social welfare. Procaccia and Rosenschein [2006], the distortion framework has been applied to several utilitarian social choice se ings (e.g., [Boutilier et al., 2015, Anshelevich et al., 2018, Gkatzelis et al., 2020). e lion's share of previous work has focused on centralized models with a single pool of agents whose preferences are directly given as input to a voting rule, which thus can utilize all the given information at once to make a decision. However, there are many applications with multiple pools of agents which make independent local decisions that can be thought of as recommendations for the final decision. To give a concrete example, in most political election systems, the citizens are partitioned into districts based on geographic or other criteria, and vote within their districts to propose the candidate (party) they would like to be selected as the winner.
Since its inception in 2006 by
Inspired by situations like the one described above, Filos-Ratsikas et al. [2020] initiated the study of the distortion of mechanisms in a distributed single-winner se ing where a set of n agents with cardinal preferences over a set of m alternatives are partitioned into k disjoint districts. e authors focused on deterministic mechanisms of the form P f , which first choose a representative alternative for each district according to some rule f , by holding a local election with the agents of the district as the voters, and then picking the winner to be the alternative that is representative of the most districts (i.e., using the P rule). Filos-Ratsikas et al. considered mechanisms for which the rule f can be cardinal or ordinal, i.e., it may use the actual numerical information about the preferences of the agents within the districts or just consistent rankings. e authors showed that, when the districts are symmetric (that is, each of them contains the same number of agents), the distortion of a cardinal mechanism, namely P R V is O(km), and provided an asymptotically matching lower bound of Ω(km) on the distortion of any P f mechanism. For ordinal mechanisms, they showed that P P achieves a distortion of O(km 2 ), and that this is asymptotically best among all ordinal P f mechanisms.
Revisiting the distortion of distributed voting
A first observation about the results of Filos-Ratsikas et al. [2020] is that there is a-priori no reason to restrict our a ention to only mechanisms in the class P f , as using other over-districts rules could potentially lead to be er distortion. Indeed, follow-up work considered distributed social choice se ings with metric preferences [Anshelevich et al., 2022, Filos-Ratsikas and without such restrictions on the over-districts rule. In addition, all of the previous work on these se ings only considered deterministic mechanisms that use deterministic in-district and over-districts rules. Randomization has proven out to be a very useful tool in achieving be er (expected) distortion bounds in the centralized se ing (see Boutilier et al. [2015], Ebadian et al. [2022]), so it is only natural to consider randomized mechanisms in the distributed se ing as well. Finally, an important question is how the distortion bounds are affected in case the participants act selfishly, and whether there are strategyproof mechanisms with good distortion bounds.
is question has been considered in the centralized se ing [Filos-Ratsikas and Miltersen, 2014, Ebadian et al., 2022 and also in the distributed metric se ing ; we consider it in the context of the normalized se ing of Filos-Ratsikas et al. [2020] as well.
Our Contributions
We consider the class of all mechanisms for distributed voting in the se ing of [Filos-Ratsikas et al., 2020]. In particular, we consider the f over -of-f in class of mechanisms, where f in is an in-district rule that takes as input the preferences of the agents within each district and outputs a representative alternative for the district, while f over is a rule that takes as input the representative alternatives of all districts and chooses one of them as the overall winner. We consider several different cases depending on the nature of f over and f in (deterministic or randomized), and the type of information they can utilize (cardinal or ordinal). We show the following results; see Table 1 for an overview.
Deterministic Mechanisms. When f over and f in are both deterministic and the districts are symmetric, we show that the best possible distortion is Θ(km) when the valuation functions of the agents are accessible (cardinal mechanisms), and is Θ(km 2 ) when only ordinal information about the preferences of the agents is available (ordinal mechanisms). e upper bounds were shown by Filos-Ratsikas et al. [2020] and here we provide asymptotically tight lower bounds.
ese results show that for general, unstructured (normalized) valuations, employing different over-district rules in fact does not result in improvements on the distortion. We present these results in Section 3.
Randomized Mechanisms. In Section 4, we consider for the first time the distortion of randomized mechanisms in distributed voting. We first prove a simple composition theorem, which shows that using an in-district rule with known distortion δ in the centralized se ing and then selecting the winner uniformly at random from the set of representatives, defines a distributed mechanism with distortion O(kδ). Using this, complemented with new lower bounds, we show that the best possible distortion for cardinal unanimous mechanisms is Θ(k); in fact, this is true even when the districts are asymmetric and when f over is randomized but f in is deterministic.
For ordinal mechanisms, we consider two cases: (a) mechanisms that use deterministic in-district rules f in , and (b) fully-randomized mechanisms, where both f over and f in are randomized rules. For (a), we show that the best possible distortion is Θ(km 2 ). e upper bound follows from the bound on P P proven in [Filos-Ratsikas et al., 2020]; here, we provide an asymptotically matching lower bound assuming a natural universal tie-breaking rule. For (b), we prove a simple but very interesting result: For a well-studied class of randomized centralized voting rules called pointvoting schemes (e.g., see Gibbard [1977], Barbera [1978]), there exists a distributed implementation so that there is no effect on the induced probability distribution, even for asymmetric districts. Simply put, using such rules it is possible to escape the ill effects of districts in terms of the distortion, even when the districts are asymmetric. From this result, it follows that there exists a distributed implementation of a well-known mechanism of Boutilier et al. [2015] that achieves distortion O( √ m log m), almost matching the best possible lower bound of Ω( √ m).
Strategyproof Mechanisms. For strategyproof mechanisms, which are resilient to strategic manipulation, we show that a best-possible distortion of Θ(nm) for deterministic mechanisms (and more generally mechanisms with a deterministic in-district rule) is easy to achieve by a variation of a dictatorship rule. For randomized mechanisms, since point-voting schemes are strategyproof, the bound O( √ m log m) carries over to this class as well. Results about deterministic strategyproof mechanisms are presented in Section 3, and about randomized strategyproof mechanisms in Section 4.
Experiments. Finally, in Section 5, we perform experiments using real-world data and synthetic data to evaluate the effect of distributed decision making to the distortion in se ings closer to practice. e main conclusions of our experimental results mirror that of our theoretical results in Sections 3 and 4.
Further Related Work
e distortion literature is by now rather extensive, including topics such as single-winner voting [Boutilier et al., 2015, Anshelevich et al., 2018, Gkatzelis et al., 2020, Kizilkaya and Kempe, 2022, multi-winner voting [Caragiannis et al., 2017[Caragiannis et al., , 2022, matching problems , Amanatidis et al., 2022a, and participatory budgeting [Benadè et al., 2017]. Generally speaking, most works can be categorized as either studying a normalized utilitarian se ing (e.g., [Procaccia and Rosenschein, 2006, Boutilier et al., 2015, Benadè et al., 2017, Ebadian et al., 2022) or a metric preference se ing (e.g., [Anshelevich and Sekar, 2016, Anshelevich et al., 2018, Gkatzelis et al., 2020, Caragiannis et al., 2022, Kizilkaya and Kempe, 2022). Some more recent works have also studied the interplay between information and distortion [Amanatidis et al., 2021, 2022a,b, Mandal et al., 2019, Abramowitz et al., 2019, and there have also been several works on strategyproofness in the context of distortion [Filos-Ratsikas and Miltersen, 2014, Ebadian et al., 2022. We refer the reader to the survey of Anshelevich et al. [2021] for a detailed overview of the related literature.
Besides the aforementioned works on distributed voting, Borodin et al. [2019] studied a related two-stage se ing in which the voters participate in a central election, but the candidates themselves come from local elections within the political parties' electorates. Beyond distortion, in the context of district-based elections, there have also been other works that have considered the degree of deviation from proportional representation (e.g., see [Bachrach et al., 2016] and references therein), and some works that have studied the complexity of manipulation (e.g., see [Elkind et al., 2021, Lewenberg et al., 2017, Lev and Lewenberg, 2019, Borodin et al., 2018).
Preliminaries
An instance I of our problem is given by a tuple I = (N, A, v, D). ere is a set N of n agents (or voters) that have preferences over a set A of m alternatives (or candidates). e preferences of each agent i ∈ N are captured by a valuation function v i : A → R ≥0 that maps every alternative a ∈ A to a real non-negative value v i (a) = v ia . Following previous work, we assume that the valuation functions are normalized such that a∈A v ia = 1 for every i ∈ N (unit-sum assumption). Let v = (v i ) i∈N be the valuation profile consisting of the valuation functions of all agents. e agents are also partitioned into a set D of k disjoint districts.
For every district d ∈ D, let N d be the set of agents it contains, such that d∈D N d = N . In the symmetric case, each district d contains exactly λ = n/k agents. In the asymmetric case, each district d contains a number n d of agents. All our lower bounds follow by instances consisting of symmetric districts, whereas our upper bounds in Section 4 hold for asymmetric districts.
Mechanisms
Our goal is to choose an alternative to satisfy several criteria of interest.
is choice must be done using a distributed mechanism that uses an in-district voting rule f in and an over-districts voting rule f over to implement the following two independent steps:
• Step 1: For each district d, choose a representative alternative a d ∈ A by holding a local election based on f in .
•
Step 2: Choose a district representative as the winner based on f over by considering the districts as voters and their representatives as the candidates they approve.
For simplicity we refer to such mechanisms as f over -of-f in . Different choices of f in and f over lead to different distributed mechanisms. Note that the in-district rule can in general use various types of information about the preferences of the agents. For instance, it may be able to use exact cardinal information about the valuation functions, or only ordinal information that is induced by the values (i.e., rankings of alternatives that are consistent to the values of the agents for them). In the la er case, we will use σ i to denote the preference ranking of agent i ∈ N so that σ i (a) is the rank of alternative a ∈ A in the ranking of i,
and σ i (a) < σ i (b) if v i (a) ≥ v i (b)
; let σ = (σ i ) i∈N be the ordinal profile consisting of the preference rankings of all agents. To be concise in the definitions below, let δ(I) be the information about the preferences of the agents in instance I = (N, A, v, D) that is used by a mechanism; that is, δ(I) = v in case of cardinal information, or δ(I) = σ in case of ordinal information.
We will focus on different classes of distributed mechanisms depending on the available information about the preferences of the agents at the district level (cardinal or ordinal), and also on whether their decision is deterministic or randomized (that is, they choose the district representatives or final winner based on probability distributions).
Social Welfare and Distortion
Given an instance I, the social welfare of an alternative a ∈ A is the total value that the agents have for a, that is, SW(a|I) = i∈N v ia . So, the expected social welfare achieved by a randomized distributed mechanism M that chooses alternative a ∈ A as the winner w with probability
Pr M [w = a] is E[SW(M (I))] = a∈A Pr M [w = a] · SW(a|I).
e efficiency of a distributed mechanism is measured by the notion of distortion. e distortion of a distributed mechanism M is the worst-case ratio (over all possible instances with n agents, m alternatives, and k districts) of the maximum social welfare achieved by any alternative over the (expected) social welfare of the alternative chosen by the mechanism as the winner w, that is,
dist(M ) = sup I max a∈A SW(a|I) E[SW(M (δ(I))] .
Clearly, dist(M ) ≥ 1. When the denominator in the definition of the distortion tends to 0, we will say that the distortion is infinite or unbounded. Our goal is to identify the best possible distributed mechanisms in terms of distortion.
Strategyproofness
Another important property that we would like our mechanisms to satisfy is that of strategyproofness. A strategyproof mechanism makes decisions such that providing false information never leads to the selection of an alternative that an agent prefers over the alternative chosen when the agent provides truthful information. In particular, for any instance I, it must be the case that v i (M (δ(I))) ≥ v i (M (δ(I ′ ))) for any agent i ∈ N , where I ′ is the instance obtained when only agent i reports information different than that in I.
Some useful observations and properties
Before we present our technical results, let us briefly discuss some useful properties.
Locality of distributed mechanisms: First, observe that any distributed mechanism f over -of-f in satisfies a locality property in the following sense. A district d (that is, the preferences of a number of agents) appears in different instances if in each of these instances there is a district with the same number of agents and the same information about theirs preferences as in d (depending on what is required by the mechanism). Since the information is the same, the in-district rule f in must decide the same alternative as the representative of the district in all these instances. Similarly, in all instances where the mechanism has decided the same set of district representatives, the over-districts rule f over must decide the same final winner.
Distortion of distributed vs centralized: Another useful observation is that the distortion of a distributed mechanism f over -of-f in is at least as much as the distortion of the in-district centralized voting rule f in . Indeed, when k = 1, there is only one representative alternative chosen by f in , and thus this alternative must be chosen as the winner by f over ; this is also true for instances with k ≥ 2 districts which are all copies of one district. Consequently, the distortion of f in is a lower bound on the distortion of f over -of-f in .
Strategyproofness: Observe that for a distributed mechanism f over -of-f in to be strategyproof it is necessary that the in-district rule f in is strategyproof. is again follows by how the mechanism would work in instances with a single district, in which case the over-districts rule f over does not play any role in the selection of the final winner.
Unanimity: A few of our results will require the in-district rules f in to be unanimous. Unanimity stipulates that if all of the agents have the same alternative as the top preference, that alternative must be selected (with probability 1). Unanimity is a very natural property of "reasonable" voting rules, especially deterministic ones. For randomized rules, there might be reasons to consider nonunanimous choices, e.g., see Gibbard [1977], Filos-Ratsikas and Miltersen [2014].
Deterministic mechanisms
We start with deterministic distributed mechanisms and focus explicitly on the case of symmetric districts in this section (that is, the size of each district is λ). When full information about the valuations of the agents is known at the district level, Filos-Ratsikas et al. [2020] showed that the mechanism P R V , which chooses the representative of each district to be the alternative with maximum social welfare for the agents in the district, has distortion O(km). We show that this mechanism is asymptotically best possible over all possible deterministic distributed mechanisms that use unanimous in-district rules (but may not use P as the over-districts rule).
eorem 3.1. e distortion of any deterministic distributed mechanism with a unanimous in-district rule is Ω(km).
Proof. Let M be some deterministic distributed mechanism with a unanimous in-district rule. Without loss of generality, whenever there are k distinct district representatives {a 1 , . . . , a k }, we assume that M chooses a 1 as the overall winner. Let ε > 0 be some positive infinitesimal and consider the following instance with k districts {d 1 , . . . , d k } and m > k alternatives:
• In district d 1 , all agents have value 1/m + ε for alternative a 1 , and value 1/m − ε/(m − 1) for any other alternative.
• For any ℓ ∈ {2, . . . , k}, in district d ℓ , all agents have value 1/2 + ε for alternative a ℓ , value 1/2 − ε for alternative x, and value 0 for any other alternative.
Since the in-district rule is unanimous, the district representatives are alternatives {a 1 , . . . , a k }, and the overall winner is thus a 1 . e social welfare of alternative a 1 is approximately λ/m, whereas the social welfare of alternative x is approximately k · λ/2, leading to distortion Ω(km).
When only ordinal information about the preferences of the agents is available, Filos-Ratsikas et al. [2020] showed that P P , which chooses the favorite alternative of most of the agents in a district as its representative and then the alternative that represents the most districts as the winner, has distortion O(km 2 ). We show that this mechanism is asymptotically best possible among all ordinal distributed mechanisms (without any restrictions), thus improving upon the result of Filos-Ratsikas et al. [2020] who showed that P P is best possible only within the class of mechanisms they studied.
We first prove an easy but important lemma showing that when only ordinal information is available, to achieve finite distortion, it is necessary the representative of each district to be some alternative that is the favorite of at least one agent in the district.
Lemma 3.2. e representative of any district must be some top-ranked alternative, otherwise the distortion is infinite.
Proof. Let d be a district and let T be the set of top-ranked alternatives. Suppose that the representative of d is chosen to be some alternative x ∈ T . en, in any instance consisting of copies of d, the winner must be x. However, the valuation profile might be such that all agents have value 1 for their favorite alternative and 0 for any other alternative. Consequently, the social welfare of x might be 0, whereas the social welfare of any top-ranked alternative is positive, leading to infinite distortion.
We say that a district is divided if its λ agents are partitioned into m/2 equal-sized sets such that all the 2λ/m agents in each set rank the same alternative first and different sets of agents have different top-ranked alternatives. By Lemma 3.2, the representative of such a district must be one of the topranked alternatives. e following lemma shows that choosing the representative of a divided district as the winner is, under some circumstances, a bad choice.
Lemma 3.3. Suppose that some alternative y 1 is chosen as the winner by a deterministic ordinal distributed mechanism when the set of representatives is {y 1 , . . . , y k }. If there exists a divided district that is represented by y 1 , then there are k − 1 districts with representatives y 2 , . . . , y k , and altogether these k districts define an instance such that the distortion of the mechanism is Ω(km 2 ).
Proof. Let M be a deterministic ordinal distributed mechanism that selects y 1 as the winner when the set of representatives is {y 1 , . . . , y k }, and let d be the divided district that is represented by y 1 . Consider the following k districts:
• e first district is a copy of d.
• For every ℓ ∈ {2, . . . , k}, the ℓ-th district is such that all agents therein rank y ℓ first, x ∈ {y 1 , . . . , y k } second, and then all other alternatives. By Lemma 3.2, M must choose y ℓ as the representative of the ℓ-th district, as this is the only top-ranked alternative.
So, indeed the set of representatives is {y 1 , . . . , y k } and M chooses y 1 as the winner by assumption. One possible valuation profile is the following:
• In the first, divided district, the 2λ/m agents that rank y 1 first have value 1/m for all alternatives, and the remaining agents all have value 1 for their favorite alternative.
• For every ℓ ∈ {2, . . . , k}, all agents in the ℓ-th district have value 1/2 for their two favorite alternatives (y ℓ and x).
Consequently, the social welfare of y 1 is λ/m 2 whereas the social welfare of x is approximately k ·λ/2, and thus the distortion is Ω(km 2 ).
Lemma 3.3 shows that deterministic ordinal distributed mechanisms with distortion o(km 2 ) must not output the representative of a divided district as the winner when it is given a set of districts with different representatives. However, as we show in the proof of the next theorem, there are instances where such a choice is inevitable, and thus the distortion is Ω(km 2 ). Let d 1 be a divided district with set of top-ranked alternatives {a 1 , b 1 , . . . , b m/2−1 }. By Lemma 3.3, if a 1 is the representative of d 1 , then there exists an instance such that the distortion of M is Ω(km 2 ). So, suppose that the representative of d 1 is some other top-ranked alternative, say b 1 . Again by Lemma 3.3, if b 1 is chosen as the winner whenever she is part of a representative set consisting of k distinct alternatives, then the distortion of M would be Ω(km 2 ). So, let us assume that when the district representatives are {b 1 , a 2 , . . . , a k }, the winner is an alternative different than b 1 , say a 2 .
We can now repeat this argument step by step for each alternative a ℓ , ℓ ∈ {2, . . . , k}. In particular, let d ℓ be a divided district with top-ranked alternatives {a ℓ , b ℓ , . . . , b m/2+ℓ−2 } (note that alternatives b 1 , . . . , b ℓ−1 do not appear as top-ranked alternatives in d ℓ ). By Lemma 3.3, if a ℓ is the representative of d ℓ then the distortion of M is Ω(km 2 ), so the representative is some other alternative from the set {b ℓ , . . . , b m/2+ℓ−2 }, say b ℓ . Again by Lemma 3.3, if b ℓ is chosen as the winner whenever she is part of a representative set consisting of k distinct alternatives, then the distortion of M would be Ω(km 2 ). So, when the district representatives are {b 1 , . . . , b ℓ , a ℓ+1 , . . . , a k }, the winner is an alternative not in {b 1 , . . . , b ℓ }, say a ℓ . e last step of this repeated argument leads to the lower bound of Ω(km 2 ): We have reached an instance with set of representatives {b 1 , . . . , b k } all of whom are representative of some divided district, and thus no ma er who of them is chosen as the winner, by Lemma 3.3 there exists an instance that includes the corresponding divided district and k − 1 unanimous districts (like in the proof of the lemma) such that the distortion is Ω(km 2 ).
Finally, let us discuss the case of deterministic strategyproof distributed mechanisms. showed that the distortion of any deterministic centralized strategyproof voting rule (including those that have access to the valuation functions) is Θ(nm). From the discussion Section 2.4, we directly obtain a lower bound of Ω(nm) for the distributed se ing as well. A tight upper bound is also not hard to derive by considering the straightforward F F mechanism which works as follows:
• For each district d, choose the favorite alternative of the first agent therein as the representative.
• Choose the representative of the first district as the winner. eorem 3.5. F F is strategyproof and achieves an asymptotically best possible distortion of Θ(nm) within the class of deterministic strategyproof distributed mechanisms.
Proof. e mechanism is clearly strategyproof since the winner is the favorite alternative of the first agent of the first district who acts as a dictator. Since the winner is ranked first by an agent, the social welfare of the mechanism is at least 1/m. e maximum possible social welfare is n, and thus the distortion is O(nm).
Randomized mechanisms
We start our discussion on randomized distributed mechanisms by analyzing a general class of mechanisms that we call U δ A . A mechanism M in this class works as follows:
• For each district d, M chooses the representative a d according to some centralized voting rule f in that has distortion at most δ.
• M chooses the winner uniformly at random from the set of representatives.
Picking the winner uniformly at random from the representatives that have been selected seems to be the most natural choice as there is not much information about the preferences of the agents in the districts, and essentially all we can do is assign higher proportional probability to an alternative that is representative of more districts. We have the following result.
eorem 4.1. e distortion of any U δ A mechanism is O(kδ).
Proof. Consider an arbitrary instance. Let o be the optimal alternative, a d the representative of district d, and w the final winner. Denote by SW d (x) the social welfare of alternative x only from the agents in d; clearly, SW(x) = d∈D SW d (x). e expected social welfare of the mechanism is
E[SW(M )] = a∈A Pr[w = a] · SW(a) = 1 k a∈A d∈D Pr[a d = a] SW(a) = 1 k d∈D a∈A Pr[a d = a] · SW(a) = 1 k d∈D E[SW(a d )] ≥ 1 k d∈D E[SW d (a d )]
Since a d is chosen based on a voting rule with distortion at most δ, we have that E[SW(a d )] ≥ 1 δ · SW d (o). Combining this together with the fact that SW(o) = d∈D SW d (o), and using the linearity of expectation, we obtain
E[SW(M )] ≥ 1 k d∈D E[SW d (a d )] ≥ 1 k d∈D 1 δ · SW d (o) = 1 kδ · SW(o).
Hence, the distortion of the mechanism is at most kδ. eorem 4.1 is a simple composition theorem, analogous to the one presented by Anshelevich et al. [2022] for the metric se ing. Based on it, we can define randomized distributed mechanisms with proven distortion guarantees by appropriately choosing the in-district rule. Before we continue, observe that the sizes of the districts do not appear in the proof of eorem 4.1, and thus the distortion of any U δ A mechanism is O(kδ) even if the districts are asymmetric. So, the distortion of the mechanism depends on the number of agents only if the distortion δ of the in-district rule depends on the number of agents.
If cardinal information is available at the district level, by using R V with δ = 1 as the in-district rule, we obtain the following.
Corollary 4.2. e distortion of U R V is O(k).
If only ordinal information about the preferences of the agents is given at the district level, then we can use P with δ = O(m 2 ) and the randomized rule S L mechanism of Ebadian et al. [2022] with δ = O( √ m) as the in-district rule to obtain the following results. eorem 4.5. e distortion of any randomized distributed mechanism with a unanimous in-district rule is Ω(k).
Proof. Let ε > 0 be a positive infinitesimal. Consider an instance with the following k symmetric districts: For any ℓ ∈ [k], in district d ℓ , all λ agents therein have value 1/2 + ε for alternative a ℓ , 1/2 − ε for alternative x, and 0 for any other alternative. Since, the in-district rule is unanimous, the representative of district d ℓ must be a ℓ with probability 1. Hence, no ma er what the probability of choosing a district representative as the winner is, the expected social welfare of the mechanism is λ · (1/2 + ε). However, the social welfare of alternative x is k · λ · (1/2 − ε), and thus the distortion is Ω(k).
If we consider non-unanimous in-district rules, but require the in-district rule to be deterministic, then we can show a weaker lower bound of Ω( √ k); notice that the theorem also implies the same bound for fully deterministic distributed mechanisms without unanimous in-district rules. eorem 4.6. e distortion of any randomized distributed mechanism with a deterministic in-district rule is Ω( √ k).
Proof. Consider a district d ℓ in which all agents have value 1/2 for alternative a ℓ , value 1/(2 √ k) for each alternative in {b 1 , . . . , b √ k }, and 0 for any other alternative. If the representative of this district is not a ℓ , then in instances consisting of copies of this district, the distortion is at least √ k; in particular, it is at least that much if some alternative in {b 1 , . . . , b √ k } is chosen and infinite if any other alternative is chosen. So, suppose that the representative of d ℓ is a ℓ .
Next, consider an instance with k symmetric districts d 1 , . . . , d k . By the above discussion, for any ℓ ∈ [k], the representative of d ℓ is alternative a ℓ with social welfare λ/2 (note that only the agents of d ℓ have positive value, equal to 1/2, for a ℓ ). Hence, no ma er which district representative is chosen as the winner (or the probability distribution over the representatives), the (expected) social welfare of the mechanism is λ/2. In contrast, the social welfare of any alternative in {b 1 , . . . , b √ k } is k · λ/(2 √ k) = √ k · λ/2, and thus the distortion is √ k.
Next, we show that U P is the best possible among ordinal randomized distributed mechanisms with deterministic in-district rules, assuming an arbitrary but fixed ordering of the alternatives. is is quite surprising, as it shows that randomization over the districts is not be er than just choosing an arbitrary alternative that is representative of the most districts (i.e., not be er than P P ).
eorem 4.7. e distortion of any ordinal distributed mechanism with a deterministic in-district rule is Ω(km 2 ), when there exists an arbitrary but fixed tie-breaking ordering of the alternatives.
Proof. Without loss of generality, suppose that the tie-breaking ordering of the alternatives is a 1 ≻ . . . ≻ a k ≻ b 1 ≻ . . . ≻ b m/2−1 ≻ x ≻ c 1 ≻ . . . ≻ c m/2−k ; the naming of the alternatives is arbitrary but is assumed to be known and can be exploited. For simplicity, for any set of alternatives X, denote by [X] an arbitrary ordering of the alternatives in X.
Consider an instance with k symmetric districts such that in district d ℓ there is a set of 2λ/m agents with preference ordering a ℓ ≻ x ≻ [A \ {a ℓ , x}], a set of 2λ/m agents with preference ordering b 1 ≻ x ≻ [A \ {b 1 , x}], . . ., and a set of 2λ/m agents with preference ordering b m/2−1 ≻ x ≻ [A \ {b m/2−1 , x}]. By Lemma 3.2, the representative of d ℓ must be one of the top-ranked alternatives (otherwise the distortion of the mechanism would be infinite). Since a ℓ is ranked above the other alternatives in the tie-breaking ordering, she chosen as the representative of d ℓ . Hence, the set of representatives is {a 1 , . . . , a k }, and the winner is chosen according to some probability distribution over this set. e valuation profile may be such that the 2λ/m agents in district d ℓ that rank a ℓ first have value 1/m for all alternatives, while all other agents in d ℓ have value 1/2 for their two favorite alternatives. Consequently, the social welfare of alternative a ℓ is 2λ/m 2 , and thus the social welfare of the mechanism is also this much, no ma er the probability distribution over the district representatives. In contrast, the social welfare of x is approximately kλ/2, leading to a distortion of Ω(km 2 ). When randomization at the district level can be leveraged by ordinal distributed mechanisms, then we achieve distortion much be er than what is implied by Corollary 4.4, while also achieving strategyproofness. In particular, there are several centralized voting rules that can be implemented as distributed mechanisms, in the sense that they define the same probability distribution over the alternatives. One such important class of voting rules is that of point-voting schemes, which is part of a larger class of strategyproof mechanisms [Barbera, 1978, Hylland, 1980, Gibbard, 1977 and includes rules with almost best possible distortion guarantees [Boutilier et al., 2015, Ebadian et al., 2022.
Point-voting schemes
A point-voting scheme chooses an agent uniformly at random and then outputs her t-th favorite alternative with probability p t , where p 1 ≥ . . . ≥ p m ≥ 0 and m t=1 p t = 1. Hence, the probability according to which the point-voting scheme using the probability vector p = (p 1 , . . . , p m ) chooses alternative a ∈ A as the winner w is Pr[w = a] = 1 n i∈N p σ i (a) , where σ i (a) is the position that i ranks a in her preference ranking σ.
ere are many point-voting schemes of interest. For every positional scoring rule using the scoring vector s = (s 1 , . . . , s m ), we can define a point-voting scheme f (s) by normalizing the scoring vector, that is, define p t = s t / j∈[m] s j for every t ∈ [m] so that the winning probability of alternative a is
Pr[w = a] = 1 n i∈N s σ i (a) j∈[m] s j = i∈N s σ i (a) n · j∈[m] s j .
Another important point-voting scheme is the rule that chooses each alternative uniformly at random; in this case, we have p t = 1/m for every t ∈ [m] so that Pr[w = a] = 1 n i∈N 1 m = 1 m . For any point-voting scheme f that uses a probability vector p, we consider the distributed mechanism P f P V , which works as follows:
• For every district d, choose the representative a d to be alternative a ∈ A with probability 1 λ i∈N d p σ i (a) .
• Choose the winner to be the representative of district d with probability n d /n. eorem 4.8. P f P V defines the same probability distribution as the pointvoting scheme f .
Proof.
e probability that alternative a is chosen as the winner by P f P V is
Pr[w = a] = d∈D Pr[w = a d ] · Pr[a d = a] = d∈D n d n · 1 n d i∈N d p σ i (a) = 1 n i∈N p σ i (a) ,
that is, P f P V chooses a with the same probability as f . eorem 4.8 shows that P f P V achieves the same distortion bound as the point-voting scheme f it uses as the in-district rule, and also that it inherits its strategyproofness property. is is extremely useful, as there are centralized voting rules that are based on point-voting schemes and achieve almost the best possible distortion. Boutilier et al. [2015] considered a voting rule that is a convex combination of two point-voting schemes: With probability 1/2 choose an alternative uniformly at random, and with probability 1/2 run the point-voting scheme defined by normalizing the harmonic scoring rule H = (1, 1/2, . . . , 1/m). We will refer to this mechanism as BCHLPS. Boutilier et al. [2015] showed that this voting rule has distortion O( √ m log m). An important property of point-voting schemes is that any rule that is a convex combination of point-voting schemes is also a point-voting scheme. e following lemma is similar to lemmas proved before in the literature (e.g., see Filos-Ratsikas and Miltersen [2014], Barbera [1978]); we provide a proof for completeness.
Lemma 4.9. Let f 1 , . . . , f κ be point-voting schemes defined by the probability vectors p 1 , . . . , p κ . For any non-negative numbers q 1 , . . . , q κ such that j∈[κ] q j = 1, the voting rule f that chooses the outcome of f j with probability q j is a point-voting scheme.
Proof. Let σ be an arbitrary preference profile. For any j ∈ [κ], denote the t-th coordinate of p j as p j,t , and let P j (a) = Pr[a = f j (σ)] be the probability of choosing a as the winner according to point-voting scheme f j . en, the voting rule f chooses alternative a as the winner w with probability
Pr[w = a] = j∈[κ] q j · P j (a) = j∈[κ] q j · 1 n i∈N p j,σ i (a) = 1 n i∈N j∈[κ] q j · p j,σ i (a) .
Hence, f is a point-voting scheme defined by the probability vector p with p t = j∈[κ] q j · p j,t .
Consequently, by eorem 4.8 and Lemma 4.9, we can construct a randomized ordinal distributed mechanism based on the point-voting scheme of Boutilier et al. [2015] that achieves the same distortion bound and is strategyproof. is distortion bound is almost best possible as the lower bound of Ω( √ m) for randomized centralized rules holds trivially for distributed mechanisms by considering single-district instances.
Experiments
In this section, we perform experiments with real and synthetic datasets, aiming to identify pa erns in the distortion of several well-known voting rules and examine whether these support our theoretical findings. It is well-documented in the literature (e.g., see [Boutilier et al., 2015, Filos-Ratsikas et al., 2020) that when working with real or realistic preferences, it o en is the case that the distortions bounds are small numbers quite close to 1. In this sense, our goal is not primarily to demonstrate the distortion bounds themselves, but rather the dependence of these bounds on the distributed decisionmaking process, in particular the number of districts, as well as the use of randomization. We perform two main experiments, one with real-world preferences and valuation data, and one with synthetic data. All our experiments are with symmetric districts.
Experiments with the Jester Dataset
For our first experiment, we use the Jester Joke Dataset [Goldberg et al., 2001]. e dataset contains ratings for 100 different jokes in the range [−10, 10], provided by 70000 users. We chose to work with this dataset as it has also been employed by Boutilier et al. [2015] in the context of centralized distortion bounds, and also by Filos-Ratsikas et al. [2020] for the distortion of deterministic distributed mechanisms that use plurality as the over-district rule.
Following the methodology developed in these works, we construct inputs consisting of ratings for the 8 most-rated jokes. In particular, we perform 1000 random runs in which we sample 100 users from the set of all users that have provided rankings for all eight jokes, and then partition them into k equal-sized districts uniformly at random, for k ∈ {1, 2, 5, 10, 20, 25}. Clearly, the case of k = 1 corresponds to the centralized se ing and will be used as a reference point. We interpret the ratings of the jokes as cardinal valuations: to be consistent with our se ing (and with the experiments of [Boutilier et al., 2015, Filos-Ratsikas et al., 2020), we add 10 to each user's rating vector, to ensure that the values are positive and then apply the unit-sum normalization. For these inputs, we compute the average distortion of a set of 20 voting rules over the 1000 runs of the experiment. In particular, we consider distributed mechanisms f over -of-f in , where for f over we use P or U , whereas for f in we have:
Deterministic Rules: We use simple voting scoring rules, namely P (PL), V , B and H , as well as R V (RV), which in the case of k = 1 finds the optimal alternative.
Randomized Rules: Here we use several natural point-voting schemes with probability vectors that are proportional to the aforementioned scoring rules (recall the definition from Section 4), namely
• P P S (P PL);
• P B S (P B );
• P V S (P V );
• P H S (P H ). We also use the rule of Boutilier et al. [2015] (we refer to it as BCHLPS in the following); recall that this is a point-voting scheme that with probability 1/2 selects an alternative at random and with probability 1/2 runs the PropHarmonic rule defined above. As established in Corollary 4.10 (and the discussion before the statement of the corollary), this is best possible in terms of the worst-case distortion.
e results of our experiments can be seen in Table 2. In the table we only present the results where as f over , we used P for deterministic rules and U for randomized rules. is is in accordance to our approach in the theoretical results in previous sections. e bounds for the cases not shown are quite similar, and slightly larger in general. For each of the randomized rules, we perform 300 runs and calculate their expected social welfare, which we then use to calculate the distortion.
From the results of Table 2 we observe that, as expected, the existence of multiple districts has an adverse effect on the distortion of deterministic mechanisms, which becomes worse compared to the centralized case k = 1. For these rules, we can also observe that the distortion generally increases as k increases. In contrast, the distortion of randomized rules remains virtually unchanged for any value of k. is is in complete accordance with our theoretical findings, where we established that these rules induce the same probability distribution. e experiments showcase that this does not only hold in expectation, but also in practice (given sufficiently many runs).
Another crucial observation is that, in terms of the absolute distortion numbers, randomization does not seem to help; if anything, it makes the distortion bounds worse! is can be justified by the fact that real-world instances like those from the Jester dataset display a large degree of homogeneity, which results in the simple deterministic rules performing quite well. On the other hand, randomization o en leads to suboptimal choices even on such "well-behaved" instances, demeaning the distortion bounds on average. Surprisingly, among ordinal voting rules, B seems to perform best across the board even though the theoretical distortion of B is in fact unbounded.
Experiments with Synthetic Datasets
We also perform experiments with datasets that are generated from probability distributions. In particular, and to be consistent with the Jester experiment presented above, we create instances with 100 agents and 8 alternatives, by first drawing the values of the agents from a certain distribution, and then constructing the induced ordinal preference profile from those values. We use the following distributions:
• Uniform distribution in [1,100]. is is the simplest case, where all possible values are equally likely.
• Beta distribution with α = 1/10 and β = 1/10. is distribution has a symmetric convex pdf function centered around a mean of 1/2, assigning higher probabilities to values very close to 1 or 0.
• Exponential distribution with exponent 4, i.e., the pdf is f (x) = 4e 4 for x ≥ 0 and f (x) = 0 otherwise. is distribution generates values close to 0 with high probability, and as the values increase, the probability of them being generated decreases exponentially.
For the rest of the experiment, we perform similar steps as in the case of the Jester dataset: We normalize the values to sum up to 1, and run the set of mechanisms described above. For each randomized mechanism we now perform 150 individual runs and calculate its expected welfare. We calculate the average distortions over 500 runs of the experiment for k symmetric districts, where k ∈ {1, 2, 5, 20, 25}. Note that the number of runs and the number of district sizes is slightly smaller in this experiment, because it is more computationally intensive (as we need to calculate bounds for 3 different distributions). Again, we use P as f over for deterministic and U for randomized mechanisms; the results for the other cases were similar and are not reported. e results can be found in Table 3. Similarly to the Jester experiment, it is evident that the distortion of the deterministic mechanisms becomes worse for k ≥ 2, whereas it remains pre y much the same for randomized mechanisms. Again, we observe that randomization results in worse distortion bounds overall, and that B performs best among deterministic mechanisms. Interestingly, contrary to the Jester dataset, here we do not see a clear pa ern of the distortion increasing as k increases for deterministic mechanisms (other than the jump from k = 1 to k = 2). is is probably due to the fact that the synthetic instances are highly homogeneous, and with uniform random district partitions, the districts end up being quite uniform, regardless of their number and size. e role of unit-sum. We remark here that normalizing the values to sum up to 1 effectively makes the Uniform and Exponential distributions pre y similar, and this is reflected in the results. To get a sense of the effect of normalization, we also ran the experiments without it. We observe that the distortions for the exponential distribution are now larger than those of the uniform distribution. In general, the distortion bounds still lie in the range [1.03, 1.15] for all distributions, but their average values (over all documented distortion bounds) are larger for all distributions except Uniform. It is also the case that for the Beta distribution, the bounds of deterministic mechanisms are much closer to those of randomized ones. e distortion of randomized mechanisms is still almost the same for any number of districts.
Open Problems
From our results, an interesting technical challenge is to remove the requirement for a consistent tiebreaking ordering from the statement of eorem 4.7. Similarly, we could a empt to remove unanimity from the lower bound of eorem 3.1; although unanimity is usually pre y natural, removing it would make the theorem stronger. More interestingly, our result about point-voting schemes in eorem 4.8 crucially does not depend on the normalization of the valuations, and hence also could be applied verbatim to the metric distributed social choice se ing studied by Anshelevich et al. [2022], where randomized mechanisms have never been considered; this seems like a natural starting point for such an investigation.
eorem 3.4. e distortion of any deterministic ordinal distributed mechanism is Ω(km 2 ).Proof. Let M be a deterministic ordinal distributed mechanism. We focus on instances with k districts and sets of alternatives A ∪ B ∪ C ∪ {x}, where A = {a 1 , . . . , a k }, B = {b 1 , . . . , b m/2+k−1 }, and C = {c 1 , . . . , c m−2k }. Without loss of generality, suppose that when the district representatives are {a 1 , . . . , a k }, M chooses a 1 as the overall winner.
Corollary 4 .
410. ere exists a randomized ordinal strategyproof distributed mechanism with distortion O( √ m log m).
Table 1 :
1An overview of our results. Specific details can be found in the appropriate sections.
An important question to ask next is under what circumstances the aforementioned upper bounds of Corollaries 4.2, 4.3 and 4.4 are tight. First, we show that U R V is the best among mechanisms with unanimous in-district rules which may even use cardinal information.Corollary 4.3. e distortion of U
P
is O(km 2 ).
Corollary 4.4. e distortion of U
S
L
is O(k
√ m).
Table 2 :
2Distortion bounds of various voting rules based on valuations defined by the provided scores of the Jester dataset and random district partitions.RV
PL
V
B
H
P
PL P
V
P
B
P
H
BCHLPS
k = 1
Uniform
1
1.038 1.045
1.006
1.019
1.079
1.087
1.085
1.085
1.087
Beta
1
1.086 1.105
1.029
1.050
1.140
1.152
1.147
1.147
1.150
Exponential
1
1.032 1.096
1.018
1.013
1.118
1.137
1.132
1.131
1.134
k = 2
Uniform
1.026 1.052 1.056
1.030
1.039
1.079
1.087
1.084
1.084
1.086
Beta
1.044 1.111 1.118
1.064
1.080
1.140
1.152
1.147
1.147
1.150
Exponential 1.039 1.062 1.115
1.055
1.051
1.118
1.136
1.132
1.130
1.135
k = 5
Uniform
1.031 1.050 1.057
1.029
1.038
1.076
1.084
1.081
1.081
1.084
Beta
1.052 1.113 1.125
1.074
1.094
1.143
1.155
1.151
1.150
1.154
Exponential 1.039 1.069 1.110
1.055
1.056
1.119
1.137
1.133
1.131
1.134
k = 20
Uniform
1.031 1.055 1.077
1.039
1.042
1.077
1.085
1.082
1.082
1.084
Beta
1.055 1.105 1.145
1.073
1.084
1.141
1.154
1.149
1.149
1.152
Exponential 1.047 1.069 1.123
1.060
1.058
1.115
1.133
1.128
1.127
1.129
k = 25
Uniform
1.031 1.056 1.071
1.036
1.044
1.077
1.085
1.082
1.0824
1.084
Beta
1.054 1.124 1.149
1.084
1.094
1.148
1.155
1.150
1.150
1.151
Exponential 1.042 1.069 1.129
1.060
1.054
1.116
1.134
1.129
1.128
1.131
Table 3 :
3Distortion bounds of various voting rules based on valuations defined according to several probability distributions and random district partitions. Results for deterministic mechanisms are presented at the le of the bold vertical line, and results for randomized mechanisms are at the right of the bold vertical line.
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Eigentaste: A constant time collaborative filtering algorithm. information retrieval. Ken Goldberg, Dhruv Eresa Roeder, Chris Gupta, Perkins, 4Ken Goldberg, eresa Roeder, Dhruv Gupta, and Chris Perkins. Eigentaste: A constant time collabo- rative filtering algorithm. information retrieval, 4(2):133-151, 2001.
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Plurality veto: A simple voting rule achieving optimal metric distortion. Erdem Fatih, David Kizilkaya, Kempe, Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI). the 31st International Joint Conference on Artificial Intelligence (IJCAI)Fatih Erdem Kizilkaya and David Kempe. Plurality veto: A simple voting rule achieving optimal metric distortion. In Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI), pages 349-355, 2022.
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Divide and conquer: Using geographic manipulation to win district-based elections. Yoad Lewenberg, Omer Lev, Jeffrey S Rosenschein, Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems. the 16th Conference on Autonomous Agents and MultiAgent SystemsYoad Lewenberg, Omer Lev, and Jeffrey S Rosenschein. Divide and conquer: Using geographic manip- ulation to win district-based elections. In Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems, pages 624-632, 2017.
Efficient and thri y voting by any means necessary. Debmalya Mandal, Ariel D Procaccia, Nisarg Shah, David P Woodruff, Proceedings of the 32nd Annual Conference on Neural Information Processing Systems (NeurIPS). the 32nd Annual Conference on Neural Information Processing Systems (NeurIPS)Debmalya Mandal, Ariel D. Procaccia, Nisarg Shah, and David P. Woodruff. Efficient and thri y vot- ing by any means necessary. In Proceedings of the 32nd Annual Conference on Neural Information Processing Systems (NeurIPS), pages 7178-7189, 2019.
Optimal communication-distortion tradeoff in voting. Debmalya Mandal, Nisarg Shah, David P Woodruff, Proceedings of the 21st ACM Conference on Economics and Computation (EC). the 21st ACM Conference on Economics and Computation (EC)Debmalya Mandal, Nisarg Shah, and David P. Woodruff. Optimal communication-distortion tradeoff in voting. In Proceedings of the 21st ACM Conference on Economics and Computation (EC), pages 795-813, 2020.
e distortion of cardinal preferences in voting. Ariel D Procaccia, Jeffrey S Rosenschein, International Workshop on Cooperative Information Agents (CIA). Ariel D. Procaccia and Jeffrey S. Rosenschein. e distortion of cardinal preferences in voting. In International Workshop on Cooperative Information Agents (CIA), pages 317-331, 2006.
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arxiv |
DISTORTION IN OPHTHALMIC OPTICS: A REVIEW OF THE PRINCIPAL CONCEPTS AND MODELS
ODJean-Marie Hanssens
Université de Montréal
École d'optométrie CP 6128
Succursale Centre-Ville Montréal
QuébecCanada
Bernard Bourdoncle Ing
-Essilor International, R&D Vision Science Director
57 rue de Condé94106Saint-Maur CedexFrance
Jacques Gresset
Université de Montréal
École d'optométrie CP 6128
Succursale Centre-Ville Montréal
QuébecCanada
PhDJocelyn Faubert
Université de Montréal
École d'optométrie CP 6128
Succursale Centre-Ville Montréal
QuébecCanada
Pierre Simonet
O D Phd
Université de Montréal
École d'optométrie CP 6128
Succursale Centre-Ville Montréal
QuébecCanada
DISTORTION IN OPHTHALMIC OPTICS: A REVIEW OF THE PRINCIPAL CONCEPTS AND MODELS
1 *Emeritus professor
AbstractAlthough all members of the ophthalmic community agree that distortion is an aberration affecting the geometry of an image produced by the periphery of an ophthalmic lens, there are several approaches for analyzing and quantifying this aberration. Various concepts have been introduced: ordinary distortion, stationary distortion and central static distortion are associated with a fixed eye behind the ophthalmic lens, whereas rotatory distortion, peripheral distortion, lateral static distortion, and dynamic distortion require a secondary position of gaze behind the lens. Furthermore, concept definitions vary from one author to another. The goal of this paper is to review the various concepts, analyze their effects on lens design and determine their ability to predict the deformation of an image as perceived by the lens wearer. These entities can be classified within 3 categories: the concepts associated with an ocular rotation, the concepts resulting from an optical approach, and the concepts using a perceptual approach. Among the various concepts reviewed, it appears that the Le Grand-Fry approach for analyzing and displaying distortion is preferable to others and allows modeling of the different possible types of distortions affecting the periphery of an ophthalmic lens.Key words: distortion, image deformation, image geometry, visual perception, optical aberrations, spectacle lens design. .2 PurposeAmong the off-axis aberrations affecting the optics of ophthalmic lenses, distortion alters the geometry of the image and not its sharpness as in the case of oblique astigmatism, mean oblique power error or transverse chromatic aberration. For this reason, lens designers as well as clinicians have not given distortion the full attention it deserves given it is a factor in modifying the visual perception of the lens wearer.Even though the majority of authors initially described the qualitative visual effect caused by distortion, the quantitative evaluation of this aberration and its analysis in terms of the deformation of images perceived by the lens wearer has generated several approaches among the scientific community. (1)(2)(3)This article aims at defining the different concepts included within the umbrella term of "distortion", at analyzing the proposed methods for the correction or the reduction of this aberration, and at examining with a critical eye, in terms of the perception of the lens wearer, the accuracy of the concepts used.A critical analysis of the scientific publications concerning distortion separates the term into three categories based on the different approaches developed to define and qualify this aberration. First, we will distinguish the concepts associated with ocular rotation, then that derived from the optical approach and, finally, the one related to a perceptual approach.Approach associated with ocular rotationHistorically, Tscherning (3) was the first to analyze the phenomena of distortion induced by an ophthalmic lens and to attempt to correct this disorder:"Distortion is an error which often affects optical images: straight image lines set in the periphery of the field, are reproduced, in the image, by curves, which sometimes turn their concavity towards the center,
(barrel deformation) and sometimes their convexity (pincushion deformation)".
Tscherning (3) stated:
"Distortion is due to the fact that the magnification varies with the distance from the axis. It occurs because the lens, which forms the image, is not aplanatic in relation to the distance at which the diaphragm is located".
In his calculations for correcting distortion, Tscherning (3) set the diaphragm in the center of rotation of the eye to obtain lenses without spherical aberration between this point and its conjugate point through the lens. The solution he obtained resulted in meniscus lenses with a high curvature, which he called orthoscopic lenses. He found for each power between +13 and -20 diopters and for a center of rotation located 28 mm from the cornea, two forms of lenses which fulfil the conditions imposed by his calculations. In fact, these results were already published without demonstration since 1904, together with the results outlining the conditions for correction of oblique astigmatism. Tscherning (4) considered that lenses having the Wollaston form for the correction of oblique astigmatism were almost similar to the less curved orthoscopic lenses, which led him to produce and favor the use of such lenses. (5,6) It should be noted that in his initial work, Tscherning (4) did not use the term distortion, but orthoscopic error. Moreover, beside the correction of oblique astigmatism proposed in his publication, Tscherning (4) also tried to correct the mean oblique power error, but with no success since the solution he proposed was wrong. The same situation occurred for distortion, since Le Grand (7) indicated that Tscherning's (3,4) approach does not prevent dynamic deformation of the image of a vertical line when the lens wearer moves his visual axis behind the lens from the optical center towards the periphery.
Therefore, by choosing the center of rotation as a reference point in calculating distortion, Tscherning (3,4) ruled out the analysis from perceptual and physiological basis, since other authors, following his example, will adopt an approach of distortion which will remain linked to ocular rotation.
Henker (2) and optical engineers from Zeiss suggested that the origin of distortion is the displacement of the conjugate of the center of rotation of the eye along the optical axis of the lens when the ocular rotation increases.
According to Henker (2) , the Airy condition (the relation between the angle of the ocular rotation behind the lens u', and the angle of eccentricity in the field of fixation u (Figure 12), such as the ratio (tan u'/ tan u) remains constant, which is the condition for the absence of distortion in optical instruments) is not satisfied for ophthalmic lenses because of the displacement of the conjugate of the center of rotation of the eye through the lens as u' increases.
Figure 1
In the distortion linked to ocular rotation, the principal ray intersects the optical axis of the lens at the center of rotation of the eye (CR). The ocular rotation behind the lens is angle u' and angle u is the eccentricity of the fixation point M in the field of fixation.
Henker (2) evaluated the distortion as a percentage for a given angle of rotation u', by considering the discrepancy between the angle u, assessed from the paraxial image of the center of the rotation, and the angle ur, assessed from the actual position of the conjugate of the center of rotation given by refraction through the periphery of the lens. According to Bennett, (8,9) Henker would have proposed the following equation for distortion:
Equation 1: D (%) = 100 (W-w)/W with W = tan u' / tan ur and w = tan u' / tan u In the aphakic patient, Henker (2) Emsley (1, 10) investigated the distortion linked to rotation of the eye and considered the amplitude of ocular rotation when the eye explores the field of fixation. Emsley (1) called rotatory distortion, the variations of the ocular rotation when equal extents of the field of fixation are explored as the eye rotates. Emsley (1) calculated the area of the field of fixation (du) subtended by an ocular rotation of 1-degree (du') and identified the ratio du'/du as "field angle magnification" (Figure 13). For a given base curve of the lens, the angular extent of the field of fixation (du) linked to a 1-degree rotation (du') changes when the eye moves from the center to the periphery of the ophthalmic lens. For convex lenses, the ratio du'/du increases as the eye rotates. This change in the field angle magnification (du'/du) across the field of fixation represents rotary distortion, according to Emsley. (1)
Figure 2
Rotatory distortion, according to Emsley (1) , is associated with the change in the angular extent of the field of fixation (du) linked to 1 degree ocular rotation (du') while the eye moves from the center to the periphery of the ophthalmic lenses.
Although, according to Le Grand (11) , this concept of rotatory distortion is more related to the gauging linearity of the field of fixation than to the deformation of lines in the image, it would be of interest to know if this type of distortion can be corrected. The correction of rotatory distortion implies that the field angle magnification remains constant across the whole lens, even though the spherical aberration affects the conjugate of the center of rotation. By using third-order calculations, Emsley (10) obtained a relation between u and u', when the following parameters of the lens are known: the lens power (F'), its refractive index (n) and the vergence of the center of rotation of the eye calculated with respect to back surface of the lens (Z). This relation is given by:
Equation 2: u/u' = [(1-2F')/Z] -u' 2 x (3.336/106) F' [24D2 2 + D2 (900 -36F') + 18F' 2 -675F' + 7105]
The ratio u/u' is a constant [(1-2F')/Z] if the rest of the relation is equal to zero. This condition corresponds to a quadratic equation for D2 (power of the back surface of the lens), and so it is directly related to the curvature of the lens. This indicates that the correction of rotatory distortion is possible: there are two base curves which allow the ratio u/u' to be constant for lens powers between -21.00 D and +10.00 D. According to Emsley, (10) this quadratic equation, which modifies the constancy of the ratio u/u', would constitute the expression of rotatory distortion. Emsley (1,10) was also looking into the conditions leading to correction of rotatory distortion through exact trigonometrical ray tracing. Depending on the refractive power, the base curve of the lens required for correction of this type of distortion would be steeper than that required for the correction of oblique astigmatism when vision is at infinity. The difference is 2 diopters for a refractive power of +10 D, and reaches10 diopters for a power of -21 D.
Bennett (8,9) viewed distortion quite differently from Emsley, (1) because he proposed a quantitative evaluation of this aberration inspired by the approach advocated by Henker. (2) Like the previous German lens designers, Bennett called w, the ratio between the tangent of two angles, (u') the angle of ocular rotation and (uo) the angle of eccentricity in the field of fixation, this latter angle being calculated from the paraxial conjugate of the center of rotation of the eye. Bennett (8) considered the ratio w as analogous to the paraxial magnification existing at the entrance pupil, and he proposed to express it relative to the center of rotation of the eye, in paraxial conditions:
w = tan u' / tan uo = {1/ [1-t/n)D1]} [1/ (1-zF')]
with z corresponding to the radius of the vertex sphere, D1 to the power of the front surface and t to the central thickness of the lens.
On the other hand, Bennett (8) defined the quantity W (W = tan u' / tan u1), as the ratio between the tangent of the angle of the ocular rotation (u') and the tangent of the real angle of eccentricity in the field of fixation (u1). The angle u1 is calculated relative to the conjugate of the center of rotation of the eye, whose exact position is calculated trigonometrically by ray tracing, considering the spherical aberration linked to ocular rotation. Bennett (8) suggested expressing the distortion in percentage by an equation which differs from Henker's:
Equation 3: D (%) = 100 [(W -w)/w]
For an ocular rotation of 35 o and for lenses of +6 and -6 D, he showed that distortion, as expressed from Equation 3, decreases if the base curve of the lens becomes steeper. Bennett (9) suggested that the minimum distortion of an ophthalmic lens for a given ocular rotation is observed when the effect of deviation associated with refraction of the lens periphery is divided equally between the two sides of the lens. He deduced by approximation, based on Prentice's formula, the condition that has to be fulfilled by the front surface The criticisms directed towards Henker's approach also apply to Bennett's approach (8,9) , since this concept of distortion, and the quantification of this aberration, do not correspond to the lens wearer's perception of image distortion when he explores the lens periphery.
Bennett and Edgar (12) correctly concluded that all questions concerning distortion caused by ophthalmic lenses were not solved because certain important factors arise from the domain of visual perception. In the absence of experimental data relating to these factors, Bennett and Edgar (12) formulated the following assumptions, as a logical approach from the point of view of lens designers: on one hand, the eye aims to have the advantage of foveal vision; consequently, it scans the image it observes. On the other hand, the homogenous geometrical transformation that scales similarity between the object and the image should have, as a center of reference, the center of ocular rotation to consider the eye and the ophthalmic lens in conjunction.
With such premises, Bennett and Edgar (12) considered the apparent position of an object seen through the lens after an ocular rotation (u'), while the object would be seen by the naked eye with an ocular rotation (uo) in a frontal plane. Thus, the parameter (uo) represents a different quantity than the one identified with the same symbol by Bennett. (8) Bennett and Edgar (12) defined the quantity W as the ocular rotation factor, as such: W = tan u' / tan uo ( Figure 14). According to Bennett and Edgar, (13) distortion results from the fact that this ocular rotation factor is not constant in all directions of gaze, and they compute this aberration using equation 3 and expressing the distortion in percentage for a given ocular rotation, generally 30 o . In this equation, w represents the ratio tan u' / tan uo obtained when u' is low and when the paraxial calculation can be used.
Figure 3
The ocular rotation factor W (Bennett and Edgar (12,13) ) is the ratio tan u' / tan uo.
M" is the apparent position of an object M seen through the lens after an ocular rotation u' while the object M is seen by the naked eye with an ocular rotation equal to uo. According to these authors, distortion results from the fact that W is not constant in all directions of gaze.
According to Bennett and Edgar. (13) With u, angular eccentricity of the object in the field of fixation given by the lens, while x1 and y1 are the co-ordinates of the incidence point on the front surface of the lens, the origin of the co-ordinates being at the apex of the lens. In the case when the object is at infinity, Equation 4 and 5 become respectively:
w = {1/[1 -(e/t)D1]} [1/(1 -zF')]
and W = tan u' / tan u Bennett and Edgar (12,13) , provided the results of their calculations of distortion for powers between +8 and -24 D, for u' = 30 degree and z = 27 mm for lenses with equi, plano or meniscus shapes. For these results, the authors also calculated the minimal value of this type of distortion and the base curve required in order to obtain this minimal distortion. It is interesting to note that these base curves are substantially different than the ones proposed by Emsley. (1) However, they remain quite similar to the ones predicted by Bennett. (9) In conclusion, authors such as Tscherning, (3,4) who associated distortion with ocular rotation, adopt an approach which does not allow the definition and quantification of distortion in terms of image deformation in the same way a lens wearer perceives it. Furthermore, Le Grand (7) does not see the theoretical nor practical interest of keeping the field angle magnification constant as in Emsley's (1) approach of rotatory distortion. As for the approach of Henker (2) , Bennett (8,9) or Bennett and Edgar (12,13), the distortion they defined is based on the comparison of two ocular rotation factors, one for a position of gaze remaining in the paraxial domain, and the other for a secondary position of gaze through the periphery of the ophthalmic lens. The validity of this concept is questionable because the subjective perception of the deformation of a line in the lens periphery is not linked to the comparison of the two conditions, but rather to the direct observation of an image. The premises proposed by Bennett and Edgar (12) for the definition of distortion are certainly not the ones used by the ametrope as he perceives the deformations of the image caused by this aberration. However, Bennett and Edgar (12) recognized that the distortion caused by ophthalmic lenses involved perceptual factors.
Optical approach to distortion
Emsley (1) was the first to consider the distortion affecting a stationary eye, distinguishing it from distortion associated to an ocular rotation. Emsley (1) called ordinary distortion the distortion existing when the eye is fixed and when the entrance pupil of the eye, rather than the center of rotation of the eye, forms the aperture stop for the ophthalmic lens.
This type of distortion corresponds exactly to the classical approach in optical instrumentation. Therefore, if the object is at infinity, distortion is defined in the plane of the paraxial secondary focal point and expressed by the difference of position between the principal ray passing, after refraction, through the center of the entrance pupil and a paraxial ray passing through the nodal points of the spectacle lens with the incidence, u. This incidence corresponds to the eccentricity of the object within the visual field of the corrected ametrope ( Figure 15).
Figure 4
In ordinary distortion Emsley (1) ; Jalie (14) ; Katz (15) ) or in stationary distortion (Atchison and Smith (16) ), the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of the entrance pupil of the eye. Distortion is quantified by M'M1, the difference of position in the plane of the paraxial secondary focal point between the principal ray and the paraxial ray. This type of distortion corresponds to the classical approach in optical instrumentation.
Under these conditions, the correction of this type of distortion requires that Airy's condition is fulfilled, and the pupil is free of spherical aberrations, which seems to be impossible for an ophthalmic lens with regards to the position of the pupil with respect to the lens. However, this type of distortion may be Jalie (14) used the concept of ordinary distortion proposed by Emsley (1) to define the distortion observed when the center of the entrance pupil is considered as a reference stop, however, he also introduced a concept of rotatory distortion that, however, differed from Emsley's. Jalie (14) applied the concept of distortion previously used for the center of the entrance pupil to the center of rotation of the eye. For Jalie (14) , rotatory distortion represented, in the paraxial image plane, the difference in position between the paraxial image and the real image when the center of rotation is the reference stop ( Figure 16).
Figure 5
In rotatory distortion, according to Jalie (14) and to Atchison and Smith (16) , the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of rotation of the eye (CR).
Distortion is quantified by M'M1, the difference of position in the plane of the paraxial secondary focal point between the principal ray and the paraxial ray. Jalie (14) demonstrated that cancelling rotatory distortion requires that:
tan u' / tan u = {1/[1 -(t/n)D1]} [1/(1 -zF')]
This condition is only possible within the paraxial approximation, that is when the values u' and u in Figure 16 are small. Jalie (14) expressed rotatory distortion as percentage rates:
D (%) = 100 [(W -w)/w]
where W and w equal the ratio tan u' / tan u respectively for a given ocular rotation and for the paraxial condition. Thus, rotatory distortion, as defined by Jalie (14) , is like the variation of the ocular rotation factor proposed by Bennett and Edgar. (12) Jalie (14) has calculated rotatory distortion for a lens of +10.00 D and has demonstrated that this aberration cannot be corrected.
However, it may be minimized with adequate lens curvature. In order to derive the equation for finding the optimal curvature as a function of lens power, Jalie (14) adopted the same approach as Bennett. (9) Katz (15) maintained the same approach as Emsley (1) concerning ordinary distortion but introduced a new concept about rotatory distortion, as defined by Jalie. (14) He proposed the evaluation of this aberration by considering both the intersections of the paraxial ray and of the ray passing through the center of rotation of the eye with the far-point sphere ( Figure 17) and no longer with the paraxial secondary focal plane, that is a plane perpendicular to the optical axis and therefore perfectly flat.
Figure 6
In rotatory distortion, according to Katz (15) , the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of rotation of the eye (CR). Distortion is quantified by M'M1, the difference of position on the far-point sphere between the principal ray and the paraxial ray.
In order to correct this type of rotatory distortion, the following condition is required:
tan u' / tan u = [f -r (1 -cos u' )] / r cos u'
where f represents the secondary focal length measured from the principal plane and where r equals the radius of the far-point sphere. Katz (15) demonstrated that this condition is not satisfied for lenses corrected for oblique astigmatism. When rotatory distortion is evaluated on a flat focal plane, Katz (15) suggested the use of the following formula:
D (%) = 100 [(Yf -yf)/ yf]
where Yf and yf are the intersections of two rays with the paraxial secondary focal plane. According to Katz (15) , this gives identical values to those obtained from equation 3.
When rotatory distortion is expressed with respect to the far-point sphere, Katz (15) For lenses with powers varying between +6.60 D and -24.65 D and for which oblique astigmatism is corrected, Katz (15) demonstrated that rotatory distortion is less when calculated at the far-point sphere (variation from +0.85 to -13.54%) than when calculated for the paraxial secondary focal plane (variation from + 4.45 to -28.31%). Moreover, Katz (15) established that the minimum rotatory distortion for lenses, whose curvature was previously determined to minimize this effect, is shown to be less if the calculation is made in relation to the far-point sphere rather than to the flat secondary focal plane. According to Katz (15) , the reduction of rotatory distortion occurring when the far-point sphere is used as a reference surface would explain why this type of distortion is hardly perceived by lens wearers.
Katz (17) put forward the importance of aspheric surfaces for negative powered lenses in order to correct simultaneously, rotatory distortion, oblique astigmatism and mean oblique power error, even though rotatory distortion was expressed in relation to a flat secondary focal plane. However, from his computation with aspheric surfaces, it appears that the lens still presents an excessively steep base curve, which is incompatible with conventional methods of surfacing or with their wear.
With third-order calculations, Atchison and Smith (16) have determined the distortion in thin lenses having an aspheric conicoidal surface. Atchison and Smith (16) called peripheral distortion or stationary distortion, the aberration in which the entrance pupil is considered as the reference stop. In fact, they are repeating Emsley's (1) conceptualization of ordinary distortion. They have also studied rotatory distortion defined in relation to the center of rotation of the eye and to a secondary focal paraxial plane, in a similar way to Jalie (14) .
Atchison and Smith (16) indicated that both types of distortion obtained by third-order calculations cannot be corrected by using spherical surfaces and hence, they derived the curvature of the back surface of the lens in order to obtain the minimum distortion. They found a linear relationship that differs from those proposed by previous authors. (8) By introducing to a lens an aspheric conicoidal surface, Atchison and Smith (16) demonstrated that the third-order expression for both types of distortion becomes a cubic function for the paraxial curvature of the lens. For a given power and asphericity, this signifies that there exists at least one lens curvature correcting each type of distortion. The calculations of Atchison and Smith (16) prove that, for a given asphericity, the correction of ordinary distortion requires a greater lens curvature than rotatory distortion.
Furthermore, for a given asphericity, the paraxial curvature associated with the correction of rotatory distortion is steeper than the one required for the correction of oblique astigmatism. It remains possible to correct rotatory and oblique astigmatism simultaneously with an aspheric surface. However, the surface must have a paraxial base curve of very high value that renders its According to Atchison and Smith (16) , the simultaneous correction of oblique astigmatism and stationary distortion is theoretically possible in the third-order approximation if conicoidal aspheric surfaces are employed, but this correction requires a steeper lens curvature than the one used to correct rotatory distortion and oblique astigmatism simultaneously. Atchison and Smith (16) have demonstrated that the correction of both types of distortion in lenses which present high powers and curvatures acceptable from a cosmetic point of view, leads to pronounced errors in the peripheral power.
In the case of high-power convex lenses, asphericities required to correct power aberrations within the periphery of the lens tend to significantly decrease rotatory distortion but have no effect on stationary distortion.
In conclusion, the optical approach to distortion stems from instrumental optics and compares, once again, two images. One is a paraxial image and the other is an image given by refraction through the periphery of the lens. This approach remains valid from a theoretical optics standpoint, if one considers an ophthalmic lens as an independent optical system and does not refer to a lens used in combination with both the eye and the visual system. Consequently, it is far from clear if this is an appropriate mode of analysis and quantification for evaluating distortion as perceived by an ophthalmic lens wearer.
Perceptual approach to distortion
The approach adopted by Le Grand (11) can be considered original and interesting because the distortion introduced by ophthalmic lenses is analyzed in terms of the perception of the wearer and in terms of the actual deformation of images. Le Grand's (11) work was published in French and remained ignored.
Le Grand (11) Central static distortion (Le Grand (11) ; Fry (18) ) occurs in the following conditions: the visual axis is superimposed with the optical axis of the lens and the fixation point, shown by a black spot, is at the center of the frontal plane (coordinates 0, 0). Point A is the upper right corner of a square with coordinates x, y in the frontal plane, this point is perceived in peripheral vision by the eye. At the center of the entrance pupil (P), the orientation of the refracted ray originating from A, gives the direction along which the image is projected in the frontal plane with coordinates X"A" and Y"A". In the presence of distortion, the projected image A"B" of the vertical line AB is not a straight line but a curve. As this projected line A"B" is seen in peripheral vision, its curvature is not necessarily perceived by the lens wearer. The horizontal component of central static distortion can be expressed by the sagitta of the curved line, the linear value ( X"A" -x"B") or by the ratio 100
[(X"A" -x"B") / (x"B" -x)].
Secondly, dynamic distortion is that label given by Le Grand (11) within the concept of lateral static distortion is certainly Le Grand's (11) primary contribution to the study of this aberration. Le Grand (11) concluded that lateral static distortion is the most disturbing distortion in practice since the perception of deformations implies foveal function without having to explore the vertical line in its entirety. In fact, as the subject must assess the deformation of a line passing through the fovea, he finds himself in the situations where hyperacuity is demonstrable. Therefore, lateral static distortion is more likely to be perceived.
Figure 9
Lateral static distortion (Le Grand (11) ) occurs when the visual axis is not The direction along which the image of point A is projected in the frontal plane (A") depends on the orientation of the refracted ray at the center of the entrance pupil (P). Even though A" is seen in peripheral vision, the curvature of the projected image A"B" is analysed by foveal vision in an hyperacuity task, therefore lateral static distortion can be perceived and can be a disturbing aberration. Le Grand (11) attempted to quantify the different types of distortions existing for lenses of powers of +6.00 and -10.00 D. However, the calculations he used for ray tracing induced him to determine the opposite of the distortion caused by a lens in a normal environment: he computed the curvature required for a line in the object space so that, seen through a lens, its image is a vertical line. Le Grand (11) made calculations for a line offset by 30 degree with respect to the optical axis of the lens. By comparing the values of the curvatures obtained by computation to the thresholds of detection of a curve established by Bourdon, (19) Le Grand (11) concluded that central static distortion was perceived minimally whereas dynamic and lateral distortion, were more likely to be noticed, and had almost similar values.
Independently from Le Grand (11) , Fry (18) used a concept of distortion which corresponds to central static distortion and established the appropriate method to represent the deformation of images caused by this distortion.
Fry (18) suggested the use of a skew ray tracing in order to find both the direction of an incident ray coming from a point at the periphery of the frontal plane, and the direction of the corresponding refracted ray aiming the center of the entrance pupil. According to Fry (18) , the portion of the ray passing through the center of the entrance pupil represents the direction in which the image of the point formed by the periphery of the lens is viewed by the ametropic subject (Figure 18 and Figure 20). The projection in this direction helps to determine the intercept of this ray to the frontal plane and hence, the co-ordinates of the projected image in this reference plane. Therefore, it is possible to determine the deformation of the image as perceived by the lens wearer.
For a conventional lens of +6.00 D, Fry (18) put forward a representation of the deformation associated with central static distortion. For a given meridian, Fry (18) suggested that if s represents the distance between a given object point and the origin of the co-ordinates in the reference frontal plane, and s' represents the corresponding distance for the projected image, then the ratio ds'/ds considered as a function of s constitutes a means to represent distortion. Fry (18) was for the most part interested in central static distortion, and he did not apply the representation method he proposed to the other types of distortion defined by Le Grand. (11) The value of each type of distortion can be determined by analyzing the deformation a square grid of known linear or angular dimensions when seen through a lens (Figure 21a and 10b). Simonet et al. (20) showed that the coordinates of the corresponding projected point in the reference plane can be determined from the coordinates of any given point in the object frontal plane using the skew ray tracing method. Le Grand's (11) concepts combined with Fry's (18) Simonet et al. (20) compared both types of static distortions for various corrective powers and different lens designs. They have established that, for a lens with a given design, lateral static distortion is slightly higher than central static distortion and that, for a given dioptric value, these two aberrations are of greater importance for plus lenses than minus lenses.
Moreover, they calculated that both central static distortion and lateral static distortion could only be cancelled out on low negative powered lenses (<1.00 D) during the observation of a square grid presenting an eccentricity of ± 30 degrees in the visual field for its lateral margins. These lenses with spherical surface design can also be corrected for oblique astigmatism.
Furthermore, Simonet et al. (20) showed that, for a given lens power, both static distortions increase as the lens base curve becomes flatter, but the use of aspheric surfaces reduces the static distortions occurring with flatter lenses.
Le Grand-Fry's approach for modeling distortion can be used to report the anamorphic distortion induced by abnormal pantoscopic angle or the deformation of an image due to a progressive power lens. (21) Figure 22 illustrates the distortion induced by a positive pantoscopic angle of 20 degrees, a value is at least twice that expected, for a +5 D lens where the optical center is located 4 mm below the center of the entrance pupil. In case of anamorphic distortion, the quantification of the aberration by a single parameter is not possible and the display of the complete distorted grid is necessary for the assessment of the perceptual effect.
Experimental measurements of distortion induced by ophthalmic lenses, other than afocal prisms, are scarce. Of these, the study by Fowler (22) reporting subjective evaluation of distortion through aphakic ophthalmic lenses seems to correspond to the measure of lateral static distortion or of dynamic distortion. The results demonstrate that, on aphakic lenses, this distortion can be easily observed and quantified.
Conclusion
Distortion, the aberration introduced by the periphery of an ophthalmic lens, is worthy of study in physiological optics only if it can be analyzed and quantified with respect to the perception of the lens wearer.
The concept initially developed by Tscherning (3,4) has proved inappropriate because it is associated with only ocular rotation and seems more related to the problem of gauging the fixation field than distortion (Le Grand (7) ). All subsequent similar approaches (Henker, (2) Emsley, (1, 10) Bennett (8) ) associated with either field angle magnification or an ocular rotation factor present the same shortcoming and do not allow one to determine the exact deformation of the image of a straight line viewed through the periphery of the lens.
The approach which consists of applying the concept of distortion originating from instrumental optics is theoretically valid (Emsley, (1, 10) Jalie, (14) ; Katz, (15) Atchison and Smith (16) ); however, ordinary distortion, stationary distortion, Finally, the perceptual approach to distortion developed by Le Grand (11) and by Fry (18) appears to be the preferred concept to analyze, quantify and model the deformation of images introduced by the periphery of ophthalmic lenses (23) . We recommend that a researcher or clinician who examines the distortion introduced by ophthalmic lenses should use a Le Grand-Fry approach. Figure 12 In the distortion linked to ocular rotation, the principal ray intersects the optical axis of the lens at the center of rotation of the eye (CR). The ocular rotation behind the lens is angle u' and angle u is the eccentricity of the fixation point M in the field of fixation.
Acknowledgments
Figures captions
Figure 13
Rotatory distortion, according to Emsley (1) , is associated with the change in the angular extent of the field of fixation (du) linked to 1 degree ocular rotation (du') while the eye moves from the center to the periphery of the ophthalmic lenses.
Figure 14
The ocular rotation factor W (Bennett and Edgar (12,13) ) is the ratio tan u' / tan uo.
M" is the apparent position of an object M seen through the lens after an ocular rotation u' while the object M is seen by the naked eye with an ocular rotation equal to uo. According to these authors, distortion results from the fact that W is not constant in all directions of gaze.
Figure 15
In ordinary distortion Emsley (1) ; Jalie (14) ; Katz (15) ) or in stationary distortion (Atchison and Smith (16) ), the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of the entrance pupil of the eye. Distortion is quantified by M'M1, the difference of position in the plane of the paraxial secondary focal point between the principal ray and the paraxial ray. This type of distortion corresponds to the classical approach in optical instrumentation.
Figure 16
In rotatory distortion, according to Jalie (14) and to Atchison and Smith (16) , the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of rotation of the eye (CR).
Distortion is quantified by M'M1, the difference of position in the plane of the paraxial secondary focal point between the principal ray and the paraxial ray.
Figure 17
In rotatory distortion, according to Katz (15) , the principal ray, parallel to the paraxial ray having an incidence u, intersects the optical axis of the lens, after refraction, at the center of rotation of the eye (CR). Distortion is quantified by M'M1, the difference of position on the far-point sphere between the principal ray and the paraxial ray.
Figure 18
Central static distortion (Le Grand (11) ; Fry (18) ) occurs in the following conditions: the visual axis is superimposed with the optical axis of the lens and the fixation point, shown by a black spot, is at the center of the frontal plane (coordinates 0, 0). Point A is the upper right corner of a square with coordinates x, y in the frontal plane, this point is perceived in peripheral vision by the eye. At the center of the entrance pupil (P), the orientation of the refracted ray originating from A, gives the direction along which the image is projected in the frontal plane with coordinates X"A" and Y"A". In the presence of distortion, the projected image A"B" of the vertical line AB is not a straight line but a curve. As this projected line A"B" is seen in peripheral vision, its curvature is not necessarily perceived by the lens wearer.
The horizontal component of central static distortion can be expressed by the sagitta of the curved line, the linear value ( X"A" -x"B") or by the ratio 100 [(X"A"x"B") / (x"B" -x)].
Figure 19
Dynamic distortion (Le Grand (11) ), occurs when the eye moves behind the periphery of the lens and scans in foveal vision the projected image A"B". In this case, the orientation at the center of rotation (CR) of the refracted ray originating from point A (x, y) gives the direction along which the image of this point is projected (A") in the frontal plane. As the center of rotation (CR) rather than the center of the entrance pupil (P) is the reference point for the skew ray tracing, the various coordinates of the projected images will differ from the values observed with central static distortion.
Figure 20
Lateral static distortion (Le Grand (11) ) occurs when the visual axis is not superimposed with the optical axis of the lens during the fixation of point B. The direction along which the image of B is projected in the frontal plane (B") depends on the orientation of the refracted ray at the center of the rotation of the eye (CR).
The direction along which the image of point A is projected in the frontal plane (A") depends on the orientation of the refracted ray at the center of the entrance pupil (P). Even though A" is seen in peripheral vision, the curvature of the projected image A"B" is analysed by foveal vision in an hyperacuity task, therefore lateral static distortion can be perceived and can be a disturbing aberration.
perfectly described the perception of barrel deformation of a vertical line observed during rotation of the head when the individual focuses his attention on the middle of the vertical line and performs an ocular rotation behind the lens in the opposite direction to the rotation of the head as his visual axis moves from the center towards the periphery of the lens. However, the concept of distortion quantified by Equation 1 does not correspond to the one associated with the description of the perceptual effect. In fact, Henker's (2) calculations are related only to the shift in the frontal plane, from a paraxial position, of the middle of the vertical line during the modification of ocular rotation. The distortion associated with the perception of curvature of the vertical line for the lens wearer, as the eye rotates behind the lens, results from the displacement of the extremities of the line relative to its central part, this aberration cannot be calculated according to the model proposed by Henker. (2)
to the equation of a straight line tangential to the Wollaston branch of Tscherning's ellipse at the level of a zero-power lens. Therefore, minimum distortion could be obtained for lenses that are highly curved such that, for a vertex sphere of 27 mm and a refractive index of 1.5, D1 would be approximately +19.75 D and +17.25 D for lenses of +5 and -5 D respectively.
being the vergence of the object relative to the front surface L2, being the vergence of the object relative to the back surface L'1, being the vergence of the image relative to the front surface L'2, being the vergence of the image relative to the back surface t, being the central thickness z, being the radius of the vertex sphere
minimized by selecting the base curve of the lens for which the power, the refractive index, and the position with respect to the pupil are previously known. For a 30 degree ocular rotation, Emsley (1) determined the base curve by using trigonometrically ray tracing for lenses of +4.00 D and -5.00 D. The values obtained are too high to be realistic since the lens of +4.00 D should have had a back surface of -27.00 D., whereas for the -5.00 D lens, we would have found a value of -41.00 D. Within the interval from -20.00 D to +22.00 D, Emsley (1) used the third-order approximation to determine the curvature of the back surface of lenses minimizing this type of distortion.
, Ys and xs, ys are respectively the coordinates of the intersections of the far-point sphere with the principal ray and the paraxial ray. The origin of these coordinates is at the secondary focal point, that is, the apex of the far-point sphere.
usage very unlikely. For example, a lens with a vertex power of -12.00 D and an asphericity Q of +0.51 D requires a paraxial back surface D2 of -40.43 D, whereas a lens with a vertex power of +12.00 D and asphericity of -0.16 D would require a back surface of -18.83 D.
distinguished three types of distortion. First, central static distortion is the aberration affecting the eye in primary position of gaze when the lens wearer's visual axis is superimposed with the optical axis of the ophthalmic lens during a fixation at the center of the frontal plane (Figure 18). In this case, the image of a vertical line AB located in the periphery of the frontal plane is given by refraction through the edge of the ophthalmic lens, and the center of the entrance pupil of the eye (P) is the point on which the rays originating from the vertical line lie after being refracted by the periphery of the ophthalmic lens (Figure 18). The orientation, at the center of the entrance pupil, of a refracted ray originating from point A of the vertical line gives the direction along which the image of this point is projected in the frontal plane (A"). In the presence of distortion, the image of the vertical line is not a straight line; rather it is projected as a curve with its convexity or concavity turned toward the fixation point, depending on whether the lens is convex or concave. However, this curve is seen with the peripheral retina and therefore, the subject does not necessarily perceive the deformation of the vertical line. The central static distortion has to be evaluated in the frontal plane by calculating the curvature of the projected image (A"B") formed after refraction in the periphery of the lens by rays originating from the straight line AB located in the frontal plane, at a given eccentricity (Figure 18).
Figure 7
7Figure 7
to the distortion viewed when the subject rotates his eye from the primary position of gaze towards the edge of the lens to fix the foot B of a vertical line BA at the periphery of the frontal plane and then scans, in foveal vision, the image of the line BA seen through the periphery of the lens. In this case, the center of rotation of the eye (CR) is the reference point on which the visual axis lies when it explores the image of the vertical line (Figure 19). This is also the point on which the rays originating from the vertical line lie after being refracted by the periphery of the ophthalmic lens. The center of rotation of the eye is the reference point for finding the projection (B"A") of the image in the frontal plane (Figure 19).
Figure 8
8Dynamic distortion (Le Grand(11) ), occurs when the eye moves behind the periphery of the lens and scans in foveal vision the projected image A"B". In this case, the orientation at the center of rotation (CR) of the refracted ray originating from point A (x, y) gives the direction along which the image of this point is projected (A") in the frontal plane. As the center of rotation (CR) rather than the center of the entrance pupil (P) is the reference point for the skew ray tracing, the various coordinates of the projected images will differ from the values observed with central static distortion.Finally, Le Grand(11) identifies the lateral static distortion as the aberration inducing a deformation of the image observed in a secondary position of gaze when the subject fixates with his fovea the foot B of the vertical line BA situated in the periphery of the frontal plane (Figure 20) and perceives the curve of the image given by the periphery of the lens. In this position of gaze behind the lens, the subject's visual axis passes through the edge of the lens and is no more superimposed with the optical axis. The foot B of the vertical line is observed in foveal vision and the center of rotation of the eye (CR) is the point of reference aligned with the visual axis when the fixation point is B. However, the extremity A of the vertical line BA is viewed in peripheral vision and the center of the entrance pupil (P) is the point on which lie the luminous rays originating from A and from all points of the vertical line, except the ray coming from point B (Figure 20). The use of two different reference points (the center of rotation of the eye and the center of the entrance pupil)
superimposed with the optical axis of the lens during the fixation of point B. The direction along which the image of B is projected in the frontal plane (B") depends on the orientation of the refracted ray at the center of the rotation of the eye (CR).
Figure 21a and 10b illustrate respectively central static distortion and lateral static distortion for a +5 D lens with a Tscherning design (refractive index is 1.61, approximate power of the front surface power is +10.75 D and lens diameter is 70 mm). The object square grid is located in the frontal plane at 5 m from the entrance pupil of the uncorrected subject. The center of the grid is aligned with the optical axis of the lens, the angular eccentricities of the vertical and horizontal lines at the margins of the grid are ± 30 degrees from the center of the grid. The skew ray tracing computation assumed a vertex distance of 14 mm, a value of 27 mm for the radius of the vertex sphere and a center of the entrance pupil (P) located at 10 mm in front of the center of rotation of the eye (CR).
Figure 10 A
10-Central static distortion. B -Lateral static distortion. Both distortions are computed for a +5 D lens with a Tscherning design (n = 1.61, approximate power of the front surface = +10,75 D, diameter = 70 mm). The square object grid is represented by grey lines in the figure. The coordinates are reported in a frontal plane located at 5 m from the entrance pupil of the uncorrected subject. The center of the square grid is aligned with the optical axis of the lens, the angular eccentricities of the vertical and horizontal margins of the grid are ± 30 degrees from the center of the grid. The skew ray tracing computation assumed a vertex distance of 14 mm, a value of 27 mm for the radius of the vertex sphere and a center of the entrance pupil (P) located at 10 mm in front of the center of rotation of the eye (CR). For an object square grid having an upper right corner which coordinates are x and y in the frontal plane, distortion will bend the margins of the projected grid such as X" and Y" will be the coordinates of the projected image of the upper right corner of the distorted grid, whereas x" will be the horizontal coordinate of the projected image of the middle of the vertical margin of the distorted grid and y" will be the vertical coordinate of the projected image of the middle of the horizontal margin. Distortion could then be expressed by the sagitta of the curved image of the lateral margin of the grid. DHoriz. = (X"-x) -(x"-x) = (X"-x") and Dvert. = (Y"-y) -(y"-y) = (Y"-y") This sagitta can be presented as a linear measure or, better yet, as a value in visual angle which can be compared to hyperacuity thresholds. In the case of lateral static distortion, the reference margin of the grid is that fixated by the eye. Distortion could also be expressed as a percentage by the ratio between the sagitta of the curved margin and the coordinate of the projected image of the middle of the lateral margin: Equation 6: D (%)Horiz. = 100 [(X"-x") / x"] Equation 7: D (%)Vert. = 100 [(Y" -y") / y"]or, in an alternative approach, by the ratio between the sagitta of the curved margin and the distance separating the projected image of the middle of this margin (x") from the middle of the marginal side of the object square grid (two last equations consider the size of the object grid, they seem to be a better expression for quantifying distortion in order to compare the values of this aberration for different correcting power or different lens designs.
representation allow modeling of different types of distortions affecting the periphery of ophthalmic lenses. The display of central or lateral static distortions gives the deformation of the grid as it is presented to the lens wearer, whereas the display of dynamic distortion represents the variation of the actual prismatic effect occurring on the visual axis.
Figure 11 Anamorphic
11central static distortion induced for a +5 D lens by a pantoscopic angle of +20 degrees and by a location of the optical center of the lens 4 mm below the center of the entrance pupil. The characteristics of the lens, the size of the object square grid and the position of the frontal plane are similar to the conditions of figure 10.
peripheral distortion or rotatory distortion, either evaluated on the flat secondary focal plane or on the far-point sphere, remain of little interest if one seeks to analyze distortion in terms of the perception of the lens wearer. In fact, there is no physiological basis for the comparison between the position of two images, one paraxial and one resulting from an oblique gaze position, whether either the center of the entrance pupil or the center of rotation of the eye is considered as the reference point. The perception of the deformation of an image is obtained from observing a single image not from the simultaneous comparison of two images.
Figure 21 A
21-Central static distortion. B -Lateral static distortion. Both distortions are computed for a +5 D lens with a Tscherning design (n = 1.61, approximate power of the front surface = +10,75 D, diameter = 70 mm). The square object grid is represented by grey lines in the figure. The coordinates are reported in a frontal plane located at 5 m from the entrance pupil of the uncorrected subject. The center of the square grid is aligned with the optical axis of the lens, the angular eccentricities of the vertical and horizontal margins of the grid are ± 30 degrees from the center of the grid. The skew ray tracing computation assumed a vertex distance of 14 mm, a value of 27 mm for the radius of the vertex sphere and a center of the entrance pupil (P) located at 10 mm in front of the center of rotation of the eye (CR).
Figure 22
22Figure 22
Anamorphic central static distortion induced for a +5 D lens by a pantoscopic angle of +20 degrees and by a location of the optical center of the lens 4 mm below the center of the entrance pupil. The characteristics of the lens, the size of the object square grid and the position of the frontal plane are similar to the conditions of figure 10.
Supported by the NSERC-Essilor Industrial Research Chair to Jocelyn Faubert and NSERC operating grants OGP0105658 to Pierre Simonet and OGP0121333 to Jocelyn Faubert. The authors thank Denis Latendresse for his technical help with the figures.
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| {'fraction_non_alphanumeric': 0.03510904057794882, 'fraction_numerical': 0.017301330092498818, 'mean_word_length': 4.622034541658759, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'AbstractAlthough all members of the ophthalmic community agree that distortion is an aberration affecting the geometry of an image produced by the periphery of an ophthalmic lens, there are several approaches for analyzing and quantifying this aberration. Various concepts have been introduced: ordinary distortion, stationary distortion and central static distortion are associated with a fixed eye behind the ophthalmic lens, whereas rotatory distortion, peripheral distortion, lateral static distortion, and dynamic distortion require a secondary position of gaze behind the lens. Furthermore, concept definitions vary from one author to another. The goal of this paper is to review the various concepts, analyze their effects on lens design and determine their ability to predict the deformation of an image as perceived by the lens wearer. These entities can be classified within 3 categories: the concepts associated with an ocular rotation, the concepts resulting from an optical approach, and the concepts using a perceptual approach. Among the various concepts reviewed, it appears that the Le Grand-Fry approach for analyzing and displaying distortion is preferable to others and allows modeling of the different possible types of distortions affecting the periphery of an ophthalmic lens.Key words: distortion, image deformation, image geometry, visual perception, optical aberrations, spectacle lens design. .2 PurposeAmong the off-axis aberrations affecting the optics of ophthalmic lenses, distortion alters the geometry of the image and not its sharpness as in the case of oblique astigmatism, mean oblique power error or transverse chromatic aberration. For this reason, lens designers as well as clinicians have not given distortion the full attention it deserves given it is a factor in modifying the visual perception of the lens wearer.Even though the majority of authors initially described the qualitative visual effect caused by distortion, the quantitative evaluation of this aberration and its analysis in terms of the deformation of images perceived by the lens wearer has generated several approaches among the scientific community. (1)(2)(3)This article aims at defining the different concepts included within the umbrella term of "distortion", at analyzing the proposed methods for the correction or the reduction of this aberration, and at examining with a critical eye, in terms of the perception of the lens wearer, the accuracy of the concepts used.A critical analysis of the scientific publications concerning distortion separates the term into three categories based on the different approaches developed to define and qualify this aberration. First, we will distinguish the concepts associated with ocular rotation, then that derived from the optical approach and, finally, the one related to a perceptual approach.Approach associated with ocular rotationHistorically, Tscherning (3) was the first to analyze the phenomena of distortion induced by an ophthalmic lens and to attempt to correct this disorder:"Distortion is an error which often affects optical images: straight image lines set in the periphery of the field, are reproduced, in the image, by curves, which sometimes turn their concavity towards the center,', 'arxivid': '2301.07194', 'author': ["ODJean-Marie Hanssens \nUniversité de Montréal\nÉcole d'optométrie CP 6128\n\nSuccursale Centre-Ville Montréal\nQuébecCanada\n", 'Bernard Bourdoncle Ing \n-Essilor International, R&D Vision Science Director\n57 rue de Condé94106Saint-Maur CedexFrance\n', "Jacques Gresset \nUniversité de Montréal\nÉcole d'optométrie CP 6128\n\nSuccursale Centre-Ville Montréal\nQuébecCanada\n", "PhDJocelyn Faubert \nUniversité de Montréal\nÉcole d'optométrie CP 6128\n\nSuccursale Centre-Ville Montréal\nQuébecCanada\n", 'Pierre Simonet ', "O D Phd \nUniversité de Montréal\nÉcole d'optométrie CP 6128\n\nSuccursale Centre-Ville Montréal\nQuébecCanada\n"], 'authoraffiliation': ["Université de Montréal\nÉcole d'optométrie CP 6128", 'Succursale Centre-Ville Montréal\nQuébecCanada', '-Essilor International, R&D Vision Science Director\n57 rue de Condé94106Saint-Maur CedexFrance', "Université de Montréal\nÉcole d'optométrie CP 6128", 'Succursale Centre-Ville Montréal\nQuébecCanada', "Université de Montréal\nÉcole d'optométrie CP 6128", 'Succursale Centre-Ville Montréal\nQuébecCanada', "Université de Montréal\nÉcole d'optométrie CP 6128", 'Succursale Centre-Ville Montréal\nQuébecCanada'], 'corpusid': 255998454, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15286, 'n_tokens_neox': 13293, 'n_words': 9632, 'pdfsha': 'ef33a272af392dcac6685e74301a342f0414f2d9', 'pdfurls': ['https://export.arxiv.org/pdf/2301.07194v1.pdf'], 'title': ['DISTORTION IN OPHTHALMIC OPTICS: A REVIEW OF THE PRINCIPAL CONCEPTS AND MODELS', 'DISTORTION IN OPHTHALMIC OPTICS: A REVIEW OF THE PRINCIPAL CONCEPTS AND MODELS'], 'venue': []} |
arxiv |
Chernoff-type Concentration of Empirical Probabilities in Relative Entropy
January 25, 2022
F Richard Guo
Department of Statistics
Department of Statistics
University of Washington
University of Washington
Thomas S Richardson [email protected]
Department of Statistics
Department of Statistics
University of Washington
University of Washington
Chernoff-type Concentration of Empirical Probabilities in Relative Entropy
January 25, 2022
We study the relative entropy of the empirical probability vector with respect to the true probability vector in multinomial sampling of k categories, which, when multiplied by sample size n, is also the log-likelihood ratio statistic. We generalize a recent result and show that the moment generating function of the statistic is bounded by a polynomial of degree n on the unit interval, uniformly over all true probability vectors. We characterize the family of polynomials indexed by (k, n) and obtain explicit formulae. Consequently, we develop Chernoff-type tail bounds, including a closed-form version from a large sample expansion of the bound minimizer. Our bound dominates the classic method-of-types bound and is competitive with the state of the art. We demonstrate with an application to estimating the proportion of unseen butterflies.
Introduction
Consider a multinomial experiment on an alphabet of size k ≥ 2 (X 1 , . . . , X k ) ∼ Mult(n; (p 1 , . . . , p k )), (1) where (p 1 , . . . , p k ) belongs to the unit simplex ∆ k−1 . The empirical measure is identified with the probability vector (p 1 , . . . ,p k ) = (X 1 /n, . . . , X k /n). We are interested in its entropy relative to the true probability vector p, namely
D(p p) = k i=1p i log(p i /p i ),(2)
where conventions 0 · log(0) = 0 and 0 · log(0/0) = 0 are adopted. The quantity D(p p) is also known as the Kullback-Leibler divergence of p fromp. By the law of large numbers, D(p p) → 0 as n → ∞ almost surely.
Note that
n D(p p) = k i=1 X i logp i p i = log n X 1 ,...,X k k i=1p X i i n X 1 ,...,X k k i=1 p X i i
is also the log-likelihood ratio statistic (without the usual extra factor of 2). By standard asymptotic arguments (see, e.g., van der Vaart [1, Example 16.1]), for fixed k and n → ∞, it holds that n D(p p) → d χ 2 k−1 /2 = d Ga((k − 1)/2, 1),
which is a gamma distribution with shape (k − 1)/2 and rate one. We are interested in upper bounding the probability that n D(p p) exceeds a given threshold. Tail bounds of this type are of interest to many problems in probability, statistics and machine learning, including Sanov's theorem in large deviations [2, §11.4], goodness-of-fit tests [3,4], construction of non-asymptotic confidence regions [5,6] and the performance guarantee of various learning algorithms [7,8].
The classic bound of this type is
P (n D(p p) > t) ≤ exp(−t) n + k − 1 k − 1 , t > 0(4)
obtained by the "method of types" [9, Lemma II.1]. For fixed k and t, this bound is asymptotically tight as n → ∞, in the sense that the exponent exp(−t) matches the rate of the asymptotic gamma distribution in Eq. (3). Nevertheless, the bound above is far from optimal. There are recent developments in the literature that provide sharper results. In particular, Mardia et al. [10] and Agrawal [11] provide significant improvements over the method-of-types result by gaining tighter control for the binomial case (k = 2), and a reduction from multinomial (k > 2) to binomial, although their approaches are different. Additionally, bounds on the moments of D(p p) have been studied; see Jiao et al. [12], Mardia et al. [10], Paninski [13]. On a side note, by Pinsker's inequality, a tail bound on the relative entropy implies a bound on the total variation. For bounds on the latter, see also van der Vaart and Wellner [14, Appendix A.6], Devroye [15] and Biau and Gyorfi [16].
Bounding the moment generating function
In a vein similar to that of Agrawal [11], we develop bounds with Chernoff's method, a classic workhorse for deriving exponential tail bounds; see, e.g., Vershynin [17, §2.3].
The key is to upper bound the moment generating function (MGF) of n D(p p), which is defined as ϕ k,n (λ, p) := E exp (λn D(p p)) ,
where the expectation is taken over Mult(n, p = (p 1 , . . . , p k )).
It follows that
ϕ k,n (λ, p) = X 1 ,...,X k n X 1 , . . . , X k k i=1 p X i i n X 1 ,...,X k k i=1p X i i n X 1 ,...,X k k i=1 p X i i λ = X 1 ,...,X k n X 1 , . . . , X k k i=1p X i i λ k i=1 p X i i 1−λ ,(6)
where X 1 , . . . , X k are non-negative integers that sum to n.
Definition 1. For k ≥ 1, n ≥ 1, p ∈ ∆ k−1 and λ ∈ [0, 1], define G k,n (λ, p) := X 1 ,...,X k n X 1 , . . . , X k k i=1 [λX i /n + (1 − λ)p i ] X i ,(7)
where the summation is over non-negative integers that sum to n.
By definition, G k,n (λ, p) is a polynomial in λ of degree at most n. For the trivial case of k = 1, it is easy to see that G 1,n (λ) ≡ 1.
The multinomial probability in Eq. (6) is log-concave in (p 1 , . . . , p k ). For 0 ≤ λ ≤ 1, by Jensen's inequality, we have
ϕ k,n (λ, p) ≤ G k,n (λ, p), p ∈ ∆ k−1 .
The obvious obstacle here is to obtain a bound on the RHS that does not depend on the true probability vector p.
Family of G k,n (λ)
First comes a surprising fact noticed by Agrawal [11] in the k = 2 case. Proposition 1. G k,n (λ, p) does not depend on p = (p 1 , . . . , p k ).
Proof. This is true for k = 1. Fix any k ≥ 2, we prove by induction on n that G k,n (λ, p) does not depend on p. For the base case,
G k,1 (λ, p) = k i=1 (λ + (1 − λ)p i ) = kλ + 1 − λ,
which does not depend on p.
Suppose G k,m (λ, p) ≡ G k,m (λ) for m ≤ n − 1. We now show that G k,n (λ, p) does not depend on p. Since p k = 1 − p 1 − · · · − p k−1 , it suffices to verify that ∂G k,n (λ, p)/∂p i ≡ 0 for i = 1, . . . , k − 1. Further, by symmetry, it suffices to show ∂G k,n (λ, p)/∂p 1 ≡ 0.
Replacing p k with (1 − p 1 − · · · − p k−1 ), we have G k,n (λ, p) = X 1 ,...,X k n X 1 , . . . , X k k−1 j=2 [λX j /n + (1 − λ)p j ] X j × [λX 1 /n + (1 − λ)p 1 ] X 1 [λX k /n + (1 − λ)(1 − p 1 − · · · − p k−1 )] X k , and ∂G k,n (λ, p) ∂p 1 = X 1 ,...,X k n X 1 , . . . , X k k−1 j=2 [λX j /n + (1 − λ)p j ] X j (1 − λ)X 1 [λX 1 /n + (1 − λ)p 1 ] X 1 −1 [λX k /n + (1 − λ)p k ] X k − (1 − λ)X k [λX 1 /n + (1 − λ)p 1 ] X 1 [λX k /n + (1 − λ)p k ] X k −1 .
Hence, it suffices to show
X 1 ,...,X k n X 1 , . . . , X k k−1 j=2 [λX j /n + (1 − λ)p j ] X j × X 1 [λX 1 /n + (1 − λ)p 1 ] X 1 −1 [λX k /n + (1 − λ)p k ] X k ≡ X 1 ,...,X k n X 1 , . . . , X k × k−1 j=2 [λX j /n + (1 − λ)p j ] X j X k [λX 1 /n + (1 − λ)p 1 ] X 1 [λX k /n + (1 − λ)p k ] X k −1 .
We first simplify the LHS. Clearly, those summands with X 1 = 0 are zero and can be dropped. For X 1 ≥ 1, X 1 n X 1 ,...,X k = n n−1 X 1 −1,X 2 ,...,X k . Let λ := λ(n − 1)/n. For j = 2, . . . , k, by setting p j := 1−λ 1−λ p j < p j , we have
λX j /n + (1 − λ)p j = λ X j /(n − 1) + (1 − λ )p j .
Further, letting p 1 := 1 − k j=2 p j it is easy to see that
λ X 1 − 1 n − 1 + (1 − λ )p 1 = λ X 1 n + (1 − λ)p 1 .
Therefore, by introducing X 1 = X 1 − 1, we have
LHS = n X 1 ,X 2 ,...,X k n − 1 X 1 , X 2 , . . . , X k λ X 1 /(n − 1) + (1 − λ )p 1 X 1 × k j=2 λ X j /(n − 1) + (1 − λ )p j X j = nG k,n−1 (λ , p ),
where the summation is over non-negative integers X 1 , X 2 , . . . , X k summing to n − 1.
For the RHS, similarly, let q j = 1−λ 1−λ p j for j = 1, . . . , k − 1 and q k = 1 − k−1 j=1 q j . With
X k = X k − 1, it follows that RHS = n X 1 ,...,X k−1 ,X k n − 1 X 1 , . . . , X k−1 , X k k−1 j=1 λ X j /(n − 1) + (1 − λ )q j X j × λ X k /(n − 1) + (1 − λ )q k X k = nG k,n−1 (λ , q ).
Finally, by the induction hypothesis, LHS = nG k,n−1 (λ , p ) = nG k,n−1 (λ , q ) = RHS.
In view of this fact, we shall write G k,n (λ) in place of G k,n (λ, p). The set of polynomials {G k,n (λ)} are characterized by the following recurrence.
Proposition 2. For 0 ≤ λ ≤ 1, it holds that G k,n (λ) = G k−1,n (λ) + λG k,n−1 n − 1 n λ , k ≥ 2, n ≥ 1(8)
with G 1,n (λ) ≡ 1 and G k,0 (λ) := 1.
By Proposition 1, we have the freedom to choose p in the definition to evaluate G k,n (λ). In particular, by choosing p k = 0 and p 1 + · · · + p k−1 = 1, we can decompose G k,n (λ) into G k−1,n (λ) and a remainder. By a similar manipulation used in the previous proof, the remainder can be expressed in terms of G k,n−1 . We leave the detailed proof to the Appendix.
Theorem 1. For k ≥ 2, n ≥ 0 and 0 ≤ λ ≤ 1, it holds that G k,n (λ) = n m=0 n! n m (n − m)! m + k − 2 k − 2 λ m .(9)
Proof. We prove by induction. For the base case, the formula gives G k,0 (λ) ≡ 1 for k ≥ 2, which matches the value imposed by Proposition 2. First, supposing the formula holds for G 2,n−1 , we show that it also holds for G 2,n . By Proposition 2, it is easy to check that
G 2,n (λ) = G 1,n (λ) + λG 2,n−1 (λ(n − 1)/n) = 1 + n−1 m=0 (n − 1)! n m (n − m − 1)! λ m+1 = n m=0 n! n m (n − m)! λ m .
Now, for any k ≥ 3 and n ≥ 1, suppose the formula holds for G k−1,n and G k,n−1 . We show that it also holds for G k,n . By Proposition 2, we have
G k,n (λ) = G k−1,n (λ) + λG k,n−1 (λ(n − 1)/n) = n m=0 n! n m (n − m)! m + k − 3 k − 3 λ m + n−1 m=0 (n − 1)! n m (n − m − 1)! m + k − 2 k − 2 λ m+1 = n m=0 n! n m (n − m)! m + k − 3 k − 3 λ m + n m=1 n! n m (n − m)! m + k − 3 k − 2 λ m = n m=0 n! n m (n − m)! m + k − 2 k − 2 λ m ,
where in the last step the addition formula n l = n−1
l + n−1 l−1 is used [18, §5.1].
Remark 1. G k,n (λ) can be rewritten as
G k,n (λ) = n m=0 n m (k − 1) (m) (λ/n) m , where x (m) = x(x + 1) . . . (x + m − 1) is the rising factorial. Lemma 1. For k ≥ 2, G k,n (λ) = 1 (k − 2)! d k−2 dλ k−2 λ k−2 G 2,n (λ) . Proof. 1 (k − 2)! d k−2 dλ k−2 λ k−2 G 2,n (λ) = n m=0 n m m! n m (k − 2)! d k−2 dλ k−2 λ m+k−2 = n m=0 n m m! n m (k − 2)! (m + k − 2)! m! λ m = n m=0 n m (k − 1) (m) (λ/n) m = G k,n (λ). Remark 2. For k ≥ 2, G k,n (λ) is not a moment generating function of some distribution. To see this, suppose G k,n (λ) is the MGF of some random variable Y . Since G k,n (λ) is a polynomial of degree n, then E Y 2n = G (2n)
k,n (0) = 0, which implies Y is zero almost surely. However, the MGF of zero is identically one.
A few polynomials G k,n (λ) are listed in Table 1. Table 1: Polynomials G k,n (λ)
n = 1 n = 2 n = 3 n = 4 k = 2 1 + λ 1 + λ + 1 2 λ 2 1 + λ + 2 3 λ 2 + 2 9 λ 3 1 + λ + 3 4 λ 2 + 3 8 λ 3 + 3 32 λ 4 k = 3 1 + 2λ 1 + 2λ + 3 2 λ 2 1 + 2λ + 2λ 2 + 8 9 λ 3 1 + 2λ + 9 4 λ 2 + 3 2 λ 3 + 15 32 λ 4 k = 4 1 + 3λ 1 + 3λ + 3λ 2 1 + 3λ + 4λ 2 + 20 9 λ 3 1 + 3λ + 9 2 λ 2 + 15 4 λ 3 + 45 32 λ 4
Asymptotic properties
We consider the asymptotic behaviors of G k,n (λ), which can inform how well it captures the right dependence on k and n.
2.2.1 n → ∞ under fixed k Lemma 2. For k ≥ 2, G k,n (λ) increases in n.
Proof. By Theorem 1, it suffices to show that
n! n m (n − m)! ≥ (n − 1)! (n − 1) m (n − m − 1)!
for m = 0, . . . , n. By canceling factors from both sides, this is equivalent to (1 − 1 n ) m ≥ 1 − m n , which holds by Bernoulli's inequality. Proposition 3. For 0 ≤ λ < 1 and any fixed k ≥ 2, we have
G k,n (λ) G k,∞ (λ) := (1 − λ) −(k−1) , as n → ∞.(10)
Proof. For k = 2 and λ ∈ [0, 1),
G 2,n (λ) = n m=0 n! n m (n − m)! λ m ≤ n m=0 λ m → 1 1 − λ , where we used n! n m (n − m)! = n × (n − 1) × . . . (n − m + 1) n × · · · × n ≤ 1.
Further, by Lemma 2, G 2,n (λ) must converge as n → ∞ for λ ∈ [0, 1). Suppose the limit is G 2,∞ (λ). Clearly, G 2,∞ (λ) = lim n G 2,n (λ) = sup n G 2,n (λ) is lower-semicontinuous. Taking limits on both sides of Eq. (8), we have
G 2,∞ (λ) = 1 + λG 2,∞ (λ − ),
where we note n−1 n λ λ. Meanwhile, by Theorem 1, G 2,n (λ) is increasing in λ. Hence, we have G 2,∞ (λ − ) = G 2,∞ (λ) by lower-semicontinuity and monotonicity of G 2,∞ (λ). It follows that G 2,∞ = (1 − λ) −1 . Applying the same reasoning to k = 3, we have
G 3,∞ (λ) = G 2,∞ (λ) + λG 3,∞ (λ), and hence G 3,∞ = (1 − λ) −2 .
Iterating this process, we get G k,n (λ)
(1 − λ) −(k−1) for λ ∈ [0, 1) and k ≥ 2. Note that G k,∞ (λ) = (1 − λ) −(k−1) is the moment generating function of Ga(k − 1, 1). Further, n D(p p) → d Ga((k − 1)/2, 1)
. This means, for fixed k and n → ∞, G k,n (λ) is asymptotically tight in the exponent (rate parameter of gamma), but loose by a factor of 2 in the polynomial term (shape parameter of gamma).
k → ∞ under fixed n
In the following, for two sequences a ν and b ν that are positive for large enough ν, we
write a ν b ν (or b ν a ν ) as ν → ∞ if there exists c > 0 such that a ν ≥ cb ν for all ν large enough. If a ν b ν and a ν b ν , we write a ν b ν as ν → ∞. Proposition 4. For fixed 0 < λ ≤ 1 and n ≥ 1, as k → ∞ we have log G k,n (λ) n log k.(11)
Proof. By Theorem 1, for fixed n and λ, the leading term should be the largest term
of { m+k−2 k−2 : 0 ≤ m ≤ n}, which is when m = n. To see this, note that m+k−2 k−2 = (k − 1) × k × · · · × (k + m − 2)/m!. It then follows from log n+k−2 k−2 n log k.
The following shows that, as k → ∞, the logarithmic dependence on k for an upper bound on the logarithm of MGF is also necessary. Proposition 5. Suppose H k,n (λ) ≥ ϕ k,n (λ; p) for all p and all λ ∈ (0, 1). For fixed 0 < λ ≤ 1 and n ≥ 1, we have lower bound log H k,n (λ) λn log k as k → ∞.
Proof. Let p = (1/k, . . . , 1/k). It follows from Eq. (6) that ϕ k,n (λ, p) = n −λn
X 1 ,...,X k n X 1 , . . . , X k k i=1 X λ i k 1−λ X i .
We claim that ϕ k,n (λ, p) k λn . Consider the configurations of (X 1 , . . . , X k ) such that n of them are one and the rest are zero. As k → ∞, the sum over these configurations becomes
n −λn k n n! 1 k 1−λ n k λn .
Remark 3. Agrawal [11] uses the upper bound G 2,∞ (λ) on G 2,n (λ) to further bound G k,n (λ) for k > 2, by appealing to the chain rule of relative entropy [2, §2.5]. This leads to the following bound:
ϕ k,n (λ) ≤ (1 − λ) −(k−1) = G k,∞ (λ) (0 ≤ λ < 1).(12)
However, observe that for fixed n and large k, the logarithm of the above bound above grows linearly in k. In contrast, as we have shown via a direct approach, the bound log G k,n (λ) has the right logarithmic dependence.
Chernoff bound
To highlight the dependence on (k, n), letp k,n denote the empirical probability vector under k categories and n samples. For any λ ∈ [0, 1], we have
P (n D(p k,n p) > t) ≤ exp(−λt)G k,n (λ).(13)
Exact bound
Minimizing the above over λ ∈ [0, 1] yields the tightest bound.
Theorem 2. For k ≥ 2, n ≥ 1, letp k,n be the empirical probability vector from Mult(p, n) for p ∈ ∆ k−1 . For t > 0, it holds that
P (n D(p k,n p) > t) ≤ min λ∈[0,1] exp(−λt)G k,n (λ).(14)
Proposition 6. The bound in Theorem 2 is meaningful (RHS < 1) if t > min(log G k,n (1), k− 1).
Proof. Let f k,n (λ, t) := exp(−λt)G k,n (λ). Let ψ k,n (t) := min λ∈[0,1] f (λ, t) be the RHS of Eq. (14). First, suppose t > min(log G k,n (1), k − 1) and we show that ψ k,n (t) < 1.
Clearly, either t > log G k,n (1) or t > k − 1. If t > log G k,n (1), then ψ k,n (t) ≤ f k,n (1, t) = exp(−t)G k,n (1) < 1. If t > k − 1, ψ k,n (t) ≤ ψ k,∞ (t) by Proposition 3. One can show that ψ k,∞ (t) = 0, t ≤ k − 1 exp(k − 1 − t) t k−1 k−1 , t > k − 1 .
Writing t = k − 1 + δ for δ > 0, it follows that
ψ k,n (t) ≤ ψ k,∞ (k − 1 + δ) = exp (k − 1) log(1 + δ k − 1 ) − δ < 1 by log(1 + x) < x for x > 0.
We have verified that the converse also holds at least for k ≤ 500. Let λ k,n (t) be the minimizer in Theorem 2. Unfortunately, in general, λ k,n (t) does not permit a closed-form solution. In fact, finding λ k,n (t) is a non-convex problem and exp(−λt)G k,n (λ) can have more than one local minima on the unit interval. In the following, we develop a simple closed-form approximation to λ k,n (t) that leads to a bound that is only slightly looser than Theorem 2, when n is relatively large compared to k.
Large n expansion of the minimizer
Note that λ → −λt − (k − 1) log(1 − λ) is convex. The previous display is uniquely minimized at λ k,∞ (t) = 1 − k − 1 t , for t > k − 1.(15)
Plugging in λ k,∞ (t) into Eq. (13) yields the following bound.
Corollary 1 (without correction). For t > k − 1, it holds that
P (n D(p k,n p) > t) ≤ e −t e k−1 G k,n 1 − k − 1 t .(16)
λ k,∞ (t) is the zeroth-order large n approximation to λ k,n (t). Yet, the bound can be significantly tightened by a further correction.
Proposition 7. Suppose k ≥ 2 and t > k − 1. As n → ∞, we have
λ k,n (t) = λ k,∞ (t) + k k − 1 t − k + 1 n + o(n −1 ).(17)
Proof. Fix k ≥ 2 and t > k − 1. Let f k,n := exp(−λt)G k,n (λ). First, we claim that there exists N (k, t) such that f k,n (λ k,n ) = 0 for n ≥ N (k, t) at the minimizer λ k,n . To see this, note that asymptotically λ k,n cannot be 0 or 1. In particular, (i) λ k,n = 0 would imply RHS = 1 for Eq. (14), and (ii) λ k,n → 1 would imply RHS → ∞ for Eq. (14) -both contradict Proposition 6. Given f k,n (λ k,n ) = f k,n (λ k,∞ ) + f k,n (λ k,∞ )(λ k,n − λ k,∞ ) + o(|λ k,n − λ k,∞ |), it follows that
λ k,n = λ k,∞ − f k,n (λ k,∞ ) f k,n (λ k,∞ ) + o(|λ k,n − λ k,∞ |).
Since f k,n → f k,∞ = exp(−λt)G k,∞ (λ), it is easy to check that
f k,n (λ k,∞ ) = f k,∞ (λ k,∞ ) + o(1) = (k − 1)e −λ k,∞ t (1 − λ k,∞ ) −(k+1) + o(1),
where the limit (k − 1)e −λ k,∞ t (1 − λ k,∞ ) −(k+1) is non-zero and finite. Meanwhile, we have
f k,n (λ k,∞ ) = e −λ k,∞ t G k,n (λ k,∞ ) − tG k,n (λ k,∞ ) .
Using λ k,∞ = 1 − (k − 1)/t, it follows that
λ k,n = λ k,∞ + e −(t−k+1) k−1 1−λ k,∞ G k,n (λ k,∞ ) − G k,n (λ k,∞ ) (k − 1)e −(t−k+1) k−1 t −(k+1) + o(1) + o(|λ k,n − λ k,∞ |).
It is easy to check that the proof is complete given the following lemma.
Lemma 3. For k ≥ 2 and λ ∈ (0, 1), it holds that
n k − 1 1 − λ G k,n (λ) − G k,n (λ) → k(k − 1)λ (1 − λ) k+2 , as n → ∞.(18)
The proof relies on asymptotic expansions of the incomplete gamma function and is left to the Appendix.
Remark 4. The correction in Proposition 7 can be viewed as a one-step Newton's iteration based on λ k,∞ (t). Figure 1: The ideal correction lim n→∞ n(λ k,n (t) − λ k,∞ (t)) (dots, fitted from numerical values) and the theoretical first-order correction k(t − k + 1)/(k − 1) (lines), both plotted against the deviation t.
In Fig. 1, we compare the correction term (the n −1 term) from Proposition 7 to the numerical values. The numerical value corresponding to a pair (t, k) is obtained by numerically finding λ k,n (t) for a sequence of n varying from 200 to 2 × 10 4 , then fitting log(λ k,n − λ k,∞ ) against − log n in least squares, and finally taking the intercept and exponentiating.
Plugging the correction into Eq. (13) yields the following bound.
Corollary 2 (with correction). Letλ k,n := min 1 − k−1 t + k k−1 t−k+1 n , 1 . For n ≥ 1, k ≥ 2 and t > k − 1, it holds that
Comparison
We briefly compare the bounds for several sample sizes under k = 6 in Figure 2; see Fig. B.1 in the Appendix for k = 20. First, our bound is always tighter than Agrawal [11], since Agrawal [11] uses Chernoff bound based on G k,∞ , which upper-bounds G k,n . Second, in the settings plotted, our bound is tighter than that of Mardia et al. [10] for t smaller than some T k,n and vice versa for t > T k,n -an explanation for this phenomenon is provided in the following section. Third, the closed-form correction-based bound is significantly tighter than the bound without correction, and is in fact very close to the exact bound, with the difference between the two only noticeable when both n and t are small. Figure 2: Comparison of probability bounds on P(n D(p k,n p) > t) for k = 6 and t > min(log G k,n (1), k − 1). The y-axis is in logarithmic scale. The methods compared include: "exact" (Theorem 2 from numerical minimization), "correction" (Corollary 2), "w/o corr." (Corollary 1), Agrawal [11, Theorem 1.2], Mardia et al. [10,Theorem 3], and the asymptotic bound that is the exact probability when n → ∞. Note that "asymp." might not be a valid bound and is for reference only.
Combinatorial scaling
Recently Mardia et al. [10] consider a bound of the form
P (n D(p k,n p) > t) ≤ C(k, n) exp(−t),(20)
where C(k, n) captures the combinatorial dependence on k and n. This is motivated by the classic method-of-types inequality Eq. (4), which holds with
C T (k, n) = n + k − 1 k − 1 .
Note that C T (k, n) is the number of ways that {1, . . . , n} can be partitioned into k groups, and hence counts the "types" of possible empirical distributions. Mardia et al. [10] showed that C T (k, n) can be improved to
C M (k, n) = 12 π k−2 i=0 K i−1 e √ n 2π i , where K i = π(2π) i/2 2×4×···×i (i is even) (2π) (i+1)/2 1×3×···×i (i is odd) , K −1 = 1
are constants. It can be shown that C M (k, n) is smaller than C T (k, n) for all k, n ≥ 2 [10, §1.2] Since the choice of λ that tightens our bound depends on t, the bounds presented in the previous section do not take the form of Eq. (20). For comparison, we use the following bound from setting λ = 1 in Eq. (13), which is not the tightest bound except for very large t.
Corollary 3. For n ≥ 1, k ≥ 2 and t > 0, it holds that P (n D(p k,n p) > t) ≤ G k,n (1) exp(−t).
Like C M (k, n) the resulting combinatorial factor G k,n (1) is also uniformly smaller than the method-of-types combinatorial factor C T (k, n).
Proposition 8. For k ≥ 2, n ≥ 1, G k,n (1) < C T (k, n).
Proof. By Theorem 1,
G k,n (1) = n m=0 n × (n − 1) × · · · × (n − m + 1) n m m + k − 2 k − 2 < n m=0 m + k − 2 k − 2 = n + k − 1 k − 1 ,
where the last equality follows from the "parallel summation" [18,Eq. (5.9)].
In fact, the improvement can be significant when n is large.
Proposition 9. For fixed k ≥ 2, as n → ∞, log G k,n (1) log C T (k,n) → 1/2.
This basically says, in the regime of fixed k and large n, G k,n (1) is a square-root improvement over the method-of-types combinatorial factor. We leave its proof to Appendix A.3. In fact, C M (k, n) achieves the same rate of improvement in the same regime; see Mardia et al. [10, §1.2]. For other regimes, we do not have an explicit comparison. Instead, in Fig. 3 we graphically compare the combinatorial factors for a few (k, n). We observe: (i) log G k,n (1) and log C M (k, n) scale quite closely; (ii) for a fixed k, one can check that G k,n (1) < C M (k, n) for small n, and vice versa for large n. Note that (ii) explains why in Fig. 2 the bound of Mardia et al. [10] becomes tighter than our bound for very large deviations when n ∈ {100, 200, 500} -the tightening λ k,n (t) = 1 for t large enough and the exact bound reduces to Corollary 3. Finally, we stress that the improved combinatorial factors are by no means optimal. To see this, note that as n → ∞, G k,n (1) → ∞ for any fixed k ≥ 2 and C M (k, n) → ∞ for any fixed k ≥ 3, which would render the bound in the form of Eq. (20) meaningless (for fixed k and t). However, by Proposition 3 because G k,∞ (λ) only diverges at λ = 1, our bounds stated in Theorem 1, Corollaries 1 and 2 do not suffer from this problem. Nevertheless, we expect future improvements on C(k, n) such that C(k, ∞) < ∞ for k ≥ 2.
Application
The bound developed can be used to obtain a conservative non-asymptotic critical value for the multinomial likelihood ratio. The bound in Theorem 2 can be determined numerically by searching for the minimizer over the unit interval, which is a non-convex but smooth, univariate optimization. Further given a level α ∈ (0, 1) (e.g., α = 0.05), by a binary search, a critical value t k,n (α) can be determined such that the bound at t k,n (α) evaluates to α. The critical value on the likelihood ratio can be inverted to form a convex confidence region on p, which is guaranteed to contain p with probability at least (1 − α). This can be applied to the cases where k is comparable to n, and the standard large-sample χ 2 approximation is unlikely to be accurate (see Frydenberg and Jensen [19]). We demonstrate with the following example. Table 2 shows the famous dataset [20] that naturalist Alexander Steven Corbet presented to Ronald Fisher in the 1940's. Corbet spent two years trapping butterflies in Peninsular Malaysia, and his intriguing question to Fisher was how many new species would he discover had he spent another two years on the islands. Corbet's original question led to the fruitful investigation of estimating the number of unseen species; see Fisher et al. [21], Good and Toulmin [22], Orlitsky et al. [20]. However, here we pose a different question -what percentage of butterflies in Malaya belonged to the species that Corbet had not seen? That is, we want to estimate the proportion of butterflies from all the unseen species. Clearly, the MLE is zero based on the sample. Instead, we ask for an upper bound with 95% confidence. Let k = 435 + 1, where 435 is the number of species observed by Corbet. Letp = (q, 0), whereq is the empirical distribution corresponding to Table 2. The sample size is n = 2, 029 and the corresponding critical value is t k,n (α) = 481.20. The upper bound is given by the convex program max p k s.t. p ∈ ∆ k−1 , n D(p p) ≤ t k,n (α), which evaluates to 21.1%. See also Robbins [23], Bickel and Yahav [24] related to this problem.
Proportion of the unseen butterflies
Conclusion
We have shown that for a multinomial experiment with alphabet size k and sample size n, the moment generating function of the entropy of the empirical distribution relative to the true distribution (scaled by n) can be uniformly bounded by a degree-n polynomial G k,n (λ) over the unit interval. We generalize Agrawal's result [11] on k = 2 and characterize the family of G k,n (λ). The result gives rise to a one-sided Chernoff bound on the relative entropy for deviations t > min(log G k,n (1), k − 1). The bound significantly improves the classic method-of-types bound and is competitive with the state of the art [10]. Further, since the tightest Chernoff bound does not permit a closed-form, we have developed a first-order large-n expansion of the minimizing λ, which provides a good approximation to the tightest bound in closed form. On a technical note, our approach directly constructs bounds for a generic k, in contrast to some other approaches [10,11] that are based on a reduction from multinomial to binomial via the chain rule of relative entropy.
A.2 Proof of Lemma 3
We will use the following two properties of the incomplete gamma function Proof of Lemma 3. We first express G 2,n (λ) in terms of the incomplete gamma function: = n −n λ n n! n m=0 (n/λ) m m! = n −n λ n e n/λ Γ(n + 1, n/λ),
Γ(a, z) := ∞ z t a−1 e −t dt.
where we used the fact [25, Eq. 8.4.8] that Γ(n + 1, z) = n!e −z n k=0 z k k! , n = 0, 1, 2, . . . .
Using Lemmas A.1 and A.2, as n → ∞, we have G 2,n (λ) = (λ/n) n e n/λ nΓ(n, n/λ) + (n/λ) n e −n/λ = 1 + (λ/n) n e n/λ nΓ(n, n/λ) = 1 + (λ/n) n e n/λ n(n/λ) n e −n/λ n −1 Comparison of probability bounds on P(n D(p k,n p) > t) for k = 20 and t > min(log G k,n (1), k − 1). The y-axis is in logarithmic scale. The methods compared include: "exact" (Theorem 2 from numerical minimization), "correction" (Corollary 2), "w/o corr." (Corollary 1), Agrawal [11, Theorem 1.2], Mardia et al. [10,Theorem 3], and the asymptotic bound that is the exact probability when n → ∞. Note that "asymp." might not be a valid bound and is for reference only.
λ −1 − 1 − λ −1 (λ −1 − 1) 3 n −2 + O(n −3 ) = 1 1 − λ − λ 2 (1 − λ) 3 n −1 + O(n −2 ).
By
Proposition 3, when n → ∞ we have exp(−λt)G k,n (λ) → exp(−λt)(1 − λ) −(k−1) = e −λt−(k−1) log(1−λ) .
logFigure 3 :
3Gk,n(1) log CM(k, n) log CT(k, n) Comparison of combinatorial scaling factors G k,n (1) (ours), C M (k, n)[10] and C T (k, n) (method of types).
Lemma A. 1 (
1DLMF [25, §8.8]). It holds that Γ(a + 1, z) = aΓ(a, z) + z a e −z , n is a non-negative integer.Lemma A.2 (DLMF [25, §8.11(iii)]). For fixed γ > 1, as a → ∞, it holds that Γ(a, γa) = (γa) a e −γa n k=0 (−1) k b k (γ) (γ − 1) 2k+1 a −k−1 + o(|a| −n−1 ) , where b 0 (γ) = 1, b 1 (γ) = γ, b 2 (γ) = γ(2γ + 1), and for k = 1, 2, . . . , b k (γ) = γ(1 − γ)b k−1 (γ) + (2k − 1)γb k−1 (γ).
Figure B. 1 :
1Figure B.1: Comparison of probability bounds on P(n D(p k,n p) > t) for k = 20 and t > min(log G k,n (1), k − 1). The y-axis is in logarithmic scale. The methods compared include: "exact" (Theorem 2 from numerical minimization), "correction" (Corollary 2), "w/o corr." (Corollary 1), Agrawal [11, Theorem 1.2], Mardia et al. [10, Theorem 3], and the asymptotic bound that is the exact probability when n → ∞. Note that "asymp." might not be a valid bound and is for reference only.
Table 2 :
2Butterflies recorded by CorbetFrequency 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Species
118 74 44 24 29 22 20 19 20 15 12 14 6
12 6
P (n D(p k,n p) > t) ≤ exp(−λ k,n t)G k,n (λ k,n ).(19)4 DiscussionIn this section, we discuss the behavior of our bound and compare to bounds previously proposed in the literature.
AcknowledgmentF. Richard Guo thanks Jon Wellner for helpful comments. The research was supported by ONR Grant N000141912446.A ProofsA.1 Proof of Proposition 2Proof. By Proposition 1, G k,n (λ) = G k,n (λ, p) for p k = 0 and p 1 + · · · + p k−1 = 1. By Eq. (7), we split G k,n (λ) = A + B, where A sums over those X with X k = 0, and B sums over those with X k ≥ 1. Clearly,where the summation is over non-negative integers X 1 , . . . , X k−1 such that they sum to n. Further, (p 1 , . . . , p k−1 ) forms a probability vector. Hence, A = G k−1,n (λ). Now we evaluateUsing the fact thatwhere X k := X k − 1 ∈ {0, . . . , n − 1} and the summation is over (X 1 , . . . , X k−1 , X k ) such that they sum to n − 1. Let λ := n−1 n λ andHence, by Eq. (7) and Proposition 1, Eq. (21) becomesPutting A and B together, we have G k,n (λ) = G k−1,n (λ) + λG k,n−1 n−1 n λ .where we used the relation (twice)which can be shown via induction.Finally, we haveand henceA.3 Proof of Proposition 9
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. R F Schneider, C W Boisvert, B R Clark, B V Miller, H S Saunders, Cohl, M. A. McClainSchneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. [Online]. Available: http://dlmf.nist.gov/
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arxiv |
APPROXIMATING CLASS APPROACH FOR EMPIRICAL PROCESSES OF DEPENDENT SEQUENCES INDEXED BY FUNCTIONS
Herold Dehling
Olivier Durieu
Marco Tusche
APPROXIMATING CLASS APPROACH FOR EMPIRICAL PROCESSES OF DEPENDENT SEQUENCES INDEXED BY FUNCTIONS
We study weak convergence of empirical processes of dependent data (X i ) i≥0 , indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class F is different from the class of functions f for which we have good properties of the observables (f (X i )) i≥0 . We introduce a new bracketing number to measure the size of the index class F which fits this setting. Our results apply to the empirical process of data (X i ) i≥0 satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, e.g. to ergodic torus automorphisms.Date: December 21, 2013.
Introduction
Let (X i ) i≥0 be a stationary stochastic process of R-valued random variables with marginal distribution µ. We denote the empirical measure of order n by µ n = 1 n n i=1 δ X i . The classical empirical process is defined by U n (t) = √ n(µ n ((−∞, t]) − µ((−∞, t])), t ∈ R. In the case of i.i.d. processes, the limit behavior of the empirical process was first investigated by Donsker (1952), who proved that (U n (t)) t∈R converges weakly to a Brownian bridge process. This result, known as Donsker's empirical process central limit theorem, confirmed a conjecture of Doob (1949) who had observed that certain functionals of the empirical process converge in distribution towards the corresponding functionals of a Brownian bridge. Donsker's empirical process CLT has been generalized to dependent data by many authors. One of the earliest results is Billingsley (1968), who considered functions of mixing processes, with an application to the empirical distribution of the remainders in a continued fraction expansion.
Empirical processes play a very important role in large sample statistical inference. Many statistical estimators and test statistics can be expressed as functionals of the empirical distribution. As a result, their asymptotic distribution can often be derived from empirical process limit theorems, combined with the continuous mapping theorem or a functional delta method. A well-known example is the Kolmogorov-Smirnov goodness-of-fit test, which uses the test statistic D n := sup t∈R √ n|µ n ((−∞, t]) − µ 0 ((−∞, t])| in order to test the null hypothesis that µ 0 is the marginal distribution of X 1 . Under the null hypothesis, the limit distribution of D n is given by the supremum of the Gaussian limit of the empirical process. Another example are Von-Mises-statistics, also known as V-statistics. These are defined as V n := 1 n 2 1≤i,j≤n h(X i , X j ), where h(x, y) is a symmetric kernel function. Specific examples include the sample variance and Gini's mean difference, where the kernel functions are given by (x − y) 2 /2 and |x − y|, respectively. V -statistics can be expressed as integrals with respect to the empirical distribution function, namely V n = h(x, y)dµ n (x)dµ n (y). The asymptotic distribution of V n can then be derived via a functional delta method from an empirical process central limit theorem; see e.g. Beutner and Zähle (2012) for some recent results.
Empirical process CLTs for R d -valued i.i.d. data (X i ) i≥0 have first been studied by Dudley (1966), Neuhaus (1971), Bickel and Wichura (1971) and Straf (1972). These authors consider the classical d-dimensional empirical process √ n(µ n ((−∞, t])−µ((−∞, t])), where (−∞, t] = {x ∈ R d : x 1 ≤ t 1 , . . . , x d ≤ t d }, t ∈ R d , denotes the semi-infinite rectangle in R d . Philipp and Pinzur (1980), Philipp (1984) and Dhompongsa (1984) studied weak convergence of the multivariate empirical process in the case of mixing data. Dudley (1978) initiated the study of empirical processes indexed by classes of sets, or more generally by classes of functions. This approach allows the study of empirical processes for very general data, not necessarily having values in Euclidean space. CLTs for empirical processes indexed by classes of functions require entropy conditions on the size of the index set. For i.i.d. data, Dudley (1978) obtained the CLT for empirical processes indexed by classes of sets satisfying an entropy condition with inclusion. Ossiander (1987) used an entropy condition with bracketing to obtain results for empirical processes indexed by classes of functions. For the theory of empirical processes of i.i.d. data, indexed by classes of functions, see the book by van der Vaart and Wellner (1996). Limit theorems for more general empirical processes indexed by classes of functions have also been studied under entropy conditions for general covering numbers, e.g. by Nolan and Pollard (1987) who investigate empirical U -processes.
In the case of strongly mixing data, Andrews and Pollard (1994) were the first to obtain CLTs for empirical processes indexed by classes of functions. Doukhan, Massart, and Rio (1995) and Rio (1998) study empirical processes for absolutely regular data. Borovkova, Burton, and Dehling (2001) investigate the empirical process and the empirical U -process for data that can be represented as functionals of absolutely regular processes. For further results, see the survey article by Dehling and Philipp (2002), the book by Dedecker, Doukhan, Lang, León R., Louhichi, and Prieur (2007), as well as the paper by .
A lot of research has been devoted to the study of statistical properties of data arising from dynamical systems or from Markov chains. A very powerful technique to prove CLTs and other limit theorems is the spectral gap method, using spectral properties of the Perron-Frobenius operator or the Markov operator on an appropriate space of functions; see Hennion and Hervé (2001). When the space of functions under consideration contains the class of indicator functions of intervals, standard tools can be used to establish the classical empirical process CLT. Finite-dimensional convergence of the empirical process follows from the CLT for n i=1 1 (−∞,t] (X i ), and tightness can be established using moment bounds for n i=1 1 (s,t] (X i ). Collet, Martinez, and Schmitt (2004) used this approach to establish the empirical process CLT for expanding maps of the unit interval.
The situation differs markedly when the CLT and moment bounds are not directly available for the index class of the empirical process, but only for a different class of functions. Recently, Dehling, Durieu, and Volný (2009) developed techniques to cover such situations. They were able to prove classical empirical process CLTs for R-valued data when the CLT and moment bounds are only available for Lipschitz functions. Dehling and Durieu (2011) extended these techniques to R d -valued data satisfying a multiple mixing condition for Hölder continuous functions. Under this condition, they proved the CLT for the empirical process indexed by semi-infinite rectangles (−∞, t], t ∈ R d . The multiple mixing condition is strictly weaker than the spectral gap condition. E.g., ergodic torus automorphisms satisfy a multiple mixing condition, while generally they do not have a spectral gap. Dehling and Durieu (2011) proved the empirical process CLT for ergodic torus automorphisms. Durieu and Tusche (2012) provide very general conditions under which the classical empirical process CLT for R d -valued data holds.
The above mentioned papers study exclusively classical empirical processes, indexed by semi-infinite intervals or rectangles. It is the goal of the present paper to extend the techniques developed by Dehling et al. (2009) to empirical processes indexed by classes of functions. Let (X , A) be a measurable space, let (X i ) i≥0 be a stationary process of X -valued random variables, and let F be a uniformly bounded class of real-valued functions on X . We consider the F-indexed empirical process ( 1
√ n n i=1 (f (X i ) − Ef (X 1 ))) f ∈F .
As in the above mentioned papers, we will assume that there exists some Banach space B of functions on X such that the CLT and a moment bound hold for partial sums n i=1 g(X i ), for all g in some subset of B; see Assumptions 1 and 2. These conditions are satisfied, e.g. when the Perron-Frobenius operator or the Markov operator acting on B has a spectral gap. Again, if the index class F is a subset of B, standard techniques for proving empirical process CLTs can be applied. In many examples, however, B is some class of regular functions, while F is a class of indicators of sets. It is the goal of the present paper to provide techniques suitable for this situation.
Empirical process invariance principles require a control on the size of the index class F, as measured by covering or bracketing numbers; see e.g. van der Vaart and Wellner (1996). In this paper, we will consider coverings of F by B-brackets, i.e. brackets bounded by functions l, u ∈ B. Because of the specific character of our moment bounds, we have to impose conditions on the B-norms of l and u. We will thus introduce a notion of bracketing numbers by counting how many B-brackets of a given L s -size and with a given control on the B-norms of the upper and lower functions are needed to cover F. The main theorem of the present paper establishes an empirical process CLT under an integral condition on this bracketing number. This paper is organized as follows: Section 2 contains precise definitions as well as the statement of the main theorem. In Section 3, we will specifically consider the case when B is the space of Hölder continuous functions. We will give examples of classes of functions which satisfy the bracketing number assumption. In Section 4, we will give applications to ergodic torus automorphisms which extend the empirical process CLT of Dehling and Durieu (2011) to more general classes of sets. Section 5 contains the proof of our main theorem, while proofs of technical aspects of the examples can be found in the appendix.
Main Result
Let (X , A) be a measurable space, and let (X i ) i∈N be an X -valued stationary stochastic process with marginal distribution µ. Let F be a uniformly bounded class of real-valued measurable functions defined on X . If Q is a signed measure on (X , A), we use the notation Qf = X f dQ. We define the map F n : F → R, induced by the empirical measure,
F n (f ) = 1 n n i=1 f (X i ).
The F-indexed empirical process of order n is given by
U n (f ) = √ n(F n (f ) − µf ) = 1 √ n n i=1 (f (X i ) − µf ), f ∈ F.
We regard the empirical process (U n (f )) f ∈F as random element on ∞ (F); this holds as F is supposed to be uniformly bounded. ∞ (F) is equipped with the supremum norm and the Borel σ-field generated by the open sets. It is well known that, in general, (U n (f )) f ∈F is not measurable and thus the usual theory of weak convergence of random variables does not apply. We use here the theory which is based on convergence of outer expectations; see van der Vaart and Wellner (1996). Given a Borel probability measure L on ∞ (F), we say that (U n (f )) n≥1 converges in distribution to L if
E * (ϕ(U n )) → ϕ(x)dL(x),
for all bounded and continuous functions ϕ : ∞ (F) → R. Here E * denotes the outer integral. Note that E * (X) = E(X * ), where X * denotes the measurable cover function of X; see Lemma 1.2.1 in van der Vaart and Wellner (1996).
In what follows, we will frequently make two assumptions concerning the process (f (X i )) i∈N , where f : X → R belongs to some Banach space (B, · B ) of measurable functions on X , respectively to some subset G ⊂ B. The precise choice of B, as well as of G, will depend on the specific example. Often, we take B to be the space of all Lipschitz or Hölder continuous functions, and G the intersection of B with an ∞ (X )-ball.
Assumption 1 (CLT for B-observables): For all f ∈ B, there exists a σ 2 f ≥ 0 such that 1 √ n n i=1 (f (X i ) − µf ) D −→N (0, σ 2 f ), (2.1)
where N (0, σ 2 ) denotes the normal law with mean zero and variance σ 2 .
Assumption 2 (Moment bounds for G-observables): For some subset G ⊂ B, s ≥ 1, and a ∈ R, for all p ≥ 1, there exists a constant C p > 0 such that for all f ∈ G − G := {g 1 − g 2 : g 1 , g 2 ∈ G},
E n i=1 (f (X i ) − µf ) 2p ≤ C p p i=1 n i f i s log 2p+ai ( f B + 1), (2.2)
where f s = ( X |f | s dµ) 1/s denotes the L s -norm of f .
Both Assumption 1 and Assumption 2 have been established by many authors for a wide range of stationary processes. Concerning the CLT, see e.g. the three-volume monograph by Bradley (2007) for mixing processes, for so-called weakly dependent processes in the sense of Doukhan and Louhichi (1999), and Hennion and Hervé (2001) for many examples of Markov chains and dynamical systems. Durieu (2008) proved 4th moment bounds of the type (2.2) for Markov chains or dynamical systems for which the Markov operator or the Perron-Frobenius operator acting on B has a spectral gap. It was generalized to 2p-th moment bounds by Dehling and Durieu (2011). More generally, they gave similar moment bounds for processes satisfying a multiple mixing condition, i.e. assuming that there exist a θ ∈ (0, 1) and an integer d 0 ∈ N such that for all integers p ≥ 1, there exist an integer and a multivariate polynomial P of total degree smaller than d 0 such that
Cov(f (X i 0 ) · · · f (X i q−1 ), f (X iq ) · · · f (X ip )) ≤ f s f B P (i 1 − i 0 , . . . , i p − i p−1 )θ iq−i q−1 (2.3)
holds for all f ∈ B with µf = 0 and f ∞ ≤ 1, all integers i 0 ≤ i 1 ≤ . . . ≤ i p and all q ∈ {1, . . . , p}. See Theorem 4 and the examples in Dehling and Durieu (2011). Note that this multiple mixing condition implies the moment bound (2.2) with for G = {f ∈ B : f ∞ ≤ 1} and a = d 0 − 1. Further, the spectral gap property leads to the multiple mixing condition with d 0 = 0, and thus to the moment bound (2.2) with a = −1, see Dehling and Durieu (2011) Section 4.
We will derive a general statement about weak convergence of the empirical process (U n (f )) f ∈F under the two assumptions (2.1) and (2.2). Empirical process central limit theorems require bounds on the size of the class of functions F, usually measured by the number of ε-balls required to cover F. Here we will introduce a covering number adapted to the fact that (2.1) and (2.2) hold only for f ∈ B or f ∈ G, respectively, and that both the B-norm as well as the L s (µ)-norm enter on the right hand side of the bound (2.2). In our approach, we use B-brackets to cover the class F, which leads to the following definition.
Definition. Let (X , A) be a measurable space, and let µ be a probability measure on (X , A). Let B be some Banach space of measurable functions on X , G ⊂ B and s ≥ 1. (i) Given two functions l, u : X → R satisfying l(x) ≤ u(x), for all x ∈ X , we define the bracket [l, u]
:= {f : X → R : l(x) ≤ f (x) ≤ u(x), for all x ∈ X }. Given ε, A > 0, we call [l, u] an (ε, A, G, L s (µ))-bracket, if l, u ∈ G and u − l s ≤ ε u B ≤ A, l B ≤ A,
where · s denotes the L s (µ)-norm.
(ii) For a class of measurable functions F, defined on X , we define the bracketing number N (ε, A, F, G, L s (µ)) as the smallest number of (ε, A, G, L s (µ))-brackets needed to cover F.
Our definition is close to the definition of bracketing numbers given by Ossiander (1987), but different. In Ossiander (1987), no assumptions are made on the upper and lower functions of the bracket other than that they are close in L 2 . Here, the moment bound (2.2) forces us to require the extra condition that u and l belong to the space B and that their B-norms are controlled. Obviously, our bracketing numbers are always larger than the ones defined in Ossiander (1987), and naturally our condition on the size of F are stronger. On the other hand, our results apply to dependent data, while Ossiander (1987)
studies i.i.d. data.
We can now state the main theorem of the present paper. The proof will be given in Section 5.
Theorem 2.1. Let (X , A) be a measurable space, let (X i ) i≥1
be an X -valued stationary stochastic process with marginal distribution µ, and let F be a uniformly bounded class of measurable functions on X . Suppose that for some Banach space B of measurable functions on X , some subset G ⊂ B, a ∈ R, and s ≥ 1, Assumptions 1 and 2 hold. Moreover, assume that there exist constants r > −1, γ > max{2 + a, 1} and C > 0 such that
1 0 ε r sup ε≤δ≤1 N 2 (δ, exp(Cδ −1/γ ), F, G, L s (µ))dε < ∞.
(2.4)
Then the empirical process (U n (f )) f ∈F converges in distribution in ∞ (F) to a tight Gaussian process (W (f )) f ∈F .
Remark 2.2. (i) Note that the bracketing number N (δ, exp(Cδ −1/γ ), F, G, L s (µ)) might not be a monotone function of δ. This is the reason why we take the supremum in the integral (2.4).
(ii) The proof of Theorem 2.1 shows that the statement also holds if condition (2.2) is only satisfied for some integer p satisfying
p > (r + 1)γ γ − max{2 + a, 1} . (iii) If for some r ≥ 0, N (ε, exp(Cε −1/γ ), F, G, L s (µ)) = O(ε −r ), as ε → 0, condition (2.4) is satisfied for all r > 2r − 1.
In the next section, we will present examples of classes of functions satisfying condition (2.4). Among the examples are indicators of multidimensional rectangles, of ellipsoids, and of balls of arbitrary metrics, as well as a class of monotone functions. In Section 4, we give applications to ergodic torus automorphisms, indexed by various classes of indicator functions.
Examples of Classes of Functions
In many examples that satisfy Assumptions 1 and 2, the Banach space B is the space of Lipschitz or Hölder continuous functions, see examples in Dehling et al. (2009), Dehling and Durieu (2011), or Durieu and Tusche (2012. Thus, in this section, we will restrict our attention to the case where B is a space of Hölder functions and give several examples of classes F which satisfy the entropy condition (2.4).
In this section, we consider a metric space (X , d). Let α ∈ (0, 1] be fixed. We denote by H α (X ) the space of bounded α-Hölder continuous functions on X with values in R. This space is equipped with the norm
f α := sup x∈X |f (x)| + sup x,y∈X x =y |f (x) − f (y)| d(x, y) α .
For this section we chose B = H α (X ). As the approximating class we use the subclass
G = H α (X , [0, 1]) := {f ∈ H α (X ) : 0 ≤ f ≤ 1} of B.
Except in Example 3.5, in all examples we will consider the case where X is a subset of R d equipped with the Euclidean norm denoted by | · |, where d ≥ 1 is some fixed integer.
In most of the examples, we will use the transition function given in the following definition which uses the notations
d A (x) := inf a∈A d(x, a)
and
d(A, B) := inf a∈A,b∈B d(a, b),
for any element x ∈ X and sets A, B ⊂ X , where we define inf ∅ = +∞.
Definition. Let A, B be subsets of X such that d(A, B) > 0. We define the transition function T [A, B] : X → R by T [A, B](x) := d B (x) d B (x) + d A (x) ,ifT [A, B] α ≤ 1 + 3 d(A, B) α .
This lemma is proved in the appendix.
We also use the following notations: For a non-decreasing function F from R to R, F −1 denotes the pseudo-inverse function defined by
F −1 (t) := sup{x ∈ R : F (x) ≤ t} where sup ∅ = −∞. The modulus of continuity of F is defined by ω F (δ) = sup{|F (x) − F (y)| : |x − y| ≤ δ}.
Constants that only depend on fixed parameters p 1 , . . . , p k will be denoted with these parameters in the subscript, such as c p 1 ,...,p k . Furthermore the notation
f (x) = O p 1 ,...,p k (g(x)) as x → 0 or x → ∞ means that there exists a constant c p 1 ,...,p k such that f (x) ≤ c p 1 ,...,p k g(x)
for all x sufficiently small or large, respectively.
3.1. Example 1: Indicators of Rectangles. Here, we consider X = R d . In its classical form, the empirical process is defined by the class of indicator functions of left infinite rectangles, i.e. the class
{1 (−∞,t] : t ∈ R d }, where (−∞, t] denotes the set of points x such that 1 x ≤ t.
Under similar assumptions as in the present paper, this case was treated by Dehling and Durieu (2011). We will see that Theorem 2.1 covers the results of that paper.
The following proposition gives an upper bound for the bracketing number of the larger class
F = {1 (t,u] : t, u ∈ [−∞, +∞] d , t ≤ u}, where (t, u]
denotes the rectangle which consists of the points x such that t < x and x ≤ u.
Proposition 3.2. Let s ≥ 1, γ > 1, and let µ be a probability distribution on R d whose distribution function F satisfies
ω F (x) = O(| log(x)| −sγ ) as x → 0.
(3.1)
Then there exists a constant C = C F > 0 such that
N (ε, exp(Cε − 1 γ ), F, G, L s (µ)) = O d ε −2ds as ε → 0, where G = H α (R d , [0, 1]).
Proof. Let ε ∈ (0, 1) and m = 6dε −s + 1 . For all i ∈ {1, . . . , d} and j ∈ {0, . . . , m}, we define the quantiles
t i,j := F −1 i j m where F −1 i is the pseudo-inverse of the marginal distribution function 2 F i . Now, if j = (j 1 , . . . , j d ) ∈ {0, . . . , m} d , we write t j = (t 1,j 1 , . . . , t d,j d ).
In the following definitions, for convenience, we will also denote by t i,−1 or t i,−2 the points t i,0 and by t i,m+1 the points t i,m . We introduce the brackets [l k,j , u k,j ], k ∈ {0, . . . , m} d , j ∈ {0, . . . , m} d , k ≤ j, given by the α-Hölder functions
l k,j (x) := T [t k+1 , t j−2 ], R d \ [t k , t j−1 ] (x), and u k,j (x) := T [t k−1 , t j ], R d \ [t k−2 , t j+1 ] (x),
where we have used the convention that [s, t] = ∅ if s t and that the addition of an integer to a multi-index is the addition of the integer to every component of the multi-index. For each k ≤ j, we have
l k,j − u k,j s s ≤ µ ([t k−2 , t j+1 ] \ [t k+1 , t j−2 ]) ≤ d i=1 |F i (t i,k i +1 ) − F i (t i,k i −2 )| + |F i (t i,j i +1 ) − F i (t i,j i −2 )| ≤ 2 3d m , and thus l k,j − u k,j s ≤ ε. Moreover, since for a < b < b < a , d([b, b ], R d \ [a, a ]) = inf i=1,...,d inf {|a i − b i |, |a i − b i |} ,
using Lemma 3.1 and (3.1), we have
l k,j α ≤ 1 + 3 α inf i=1,...,d inf {|t i,k i − t i,k i +1 |, |t i,j i −1 − t i,j i −2 |} −α ≤ 1 + 3 α inf x > 0 : ∃i ∈ {1, . . . , d}, ∃t, F i (t + x) − F i (t) ≥ 1 m −α ≤ 1 + 3 α inf x > 0 : c F |log(x)| −sγ ≥ 1 m −α ≤ 1 + 3 α exp α(c F m) 1 sγ ,
where c F is given by (3.1). The same bound holds for u k,j α .
Thus, there exists a new constant C F > 0 such that for all k ≤ j ∈ {0, . . . , m} d , [l k,j , u k,j ] is an (ε, exp(C F ε − 1 γ ), G, L s (µ))-bracket.
It is clear that for each function f ∈ F there exists a bracket of the form [l k,j , u k,j ] which contains f . Further, we have at most (m + 1) 2d such brackets, which proves the proposition.
Notice that under the assumptions of the proposition, condition (2.4) is satisfied and therefore Theorem 2.1 may be applied to empirical processes indexed by the class of indicators of rectangles, taking B to be the class of bounded Hölder functions.
Corollary 3.3. Let (X i ) i≥0 be an R d -valued stationary process. Let F be the class of indicator functions of rectangles in R d and let G = H α (R d , [0, 1]). Assume that, for some s ≥ 1, a ∈ R, and γ > max{2 + a, 1}, Assumptions 1 and 2 hold, and that the distribution function of the X i satisfies (3.1). Then the empirical process (U n (f )) f ∈F converges in distribution in ∞ (F) to a tight Gaussian process.
F := {1 B(x,r) : x ∈ [0, 1] d , r ≥ 0}
where B(x, r) = {y ∈ [0, 1] d : |x − y| < r}. We have the following upper bound.
Proposition 3.5. Let µ be a probability distribution on [0, 1] d with a density bounded by some B > 0 and let s ≥ 1. Then there exists a constant C = C d,B > 0 such that
N (ε, Cε −αs , F, G, L s (µ)) = O d,B ε −(d+1)s as ε → 0, where G = H α ([0, 1] d , [0, 1]).
Note that the second argument in the bracketing number is different from the one appearing in the condition (2.4). In this situation we have a stronger type of bracketing number than in (2.4).
Proof. Let ε > 0 be fixed and m = ε −s . For all i = (i 1 , . . . , i d ) ∈ {0, . . . , m} d , we denote by c i the center of the rectangle
[ i 1 −1 m , i 1 m ] × · · · × [ i d −1 m , i d m ]
. Then we define, for i ∈ {1, . . . , m} d and j ∈ {0, . . . , m}, the functions
l i,j (x) := T B c i , j − 2 m √ d , [0, 1] d \ B c i , j − 1 m √ d (x)
and
u i,j (x) := T B c i , j + 2 m √ d , [0, 1] d \ B c i , j + 3 m √ d (x),
where we use the convention that a ball with negative radius is the empty set. By Lemma 3.1, these functions are α-Hölder and, since d(B(x, r), R d \ B(x, r )) = r − r, we have
l i,j α ≤ 1 + 3m √ d α ≤ 1 + 3ε −sα .
The same bound holds for u i,j α . Since µ has a bounded density with respect to Lebesgue measure, we also have
l i,j − u i,j s s ≤ µ B c i , j + 3 m √ d \ B c i , j − 2 m √ d ≤ Bc d j + 3 m √ d d − j − 2 m √ d d ,
where c d is the constant π d/2 Γ(d/2+1) (Γ is the gamma function). Hence, Now, if f belongs to F, then f = 1 B(x,r) for some x ∈ [0, 1] d , and 0 ≤ r ≤ √ d. Thus, there exist some i = (i 1 , . . . , i d ) ∈ {0, . . . , m} d and j ∈ {0, . . . , m} such that
l i,j − u i,j s ≤ c 1/s d,B ε as ε → 0, where c d,x ∈ i 1 − 1 m , i 1 m × · · · × i d − 1 m , i d m and j m √ d ≤ r ≤ j + 1 m √ d.
We then have l i,j ≤ f ≤ u i,j . Thus, the (m + 1)m d brackets [l i,j , u i,j ], i ∈ {1, . . . , m} d and j ∈ {0, . . . , m}, cover the class F. Therefore, N (c
1/s d,B ε, 4ε −αs , F, G, L s (µ)) = O d,B (ε −(d+1)s ) as ε → 0, which implies that there exists a constant C d,B > 0, for which N (ε, C d,B ε −αs , F, G, L s (µ)) = O d,B (ε −(d+1)s ) as ε → 0.E(x, r) := y ∈ R d : d i=1 (y i − x i ) 2 r 2 i ≤ 1 .
We denote by F the class of indicator functions of these ellipsoids, i.e.
F := {1 E(x,r) : x ∈ [0, 1] d , r ∈ [0, D] d }.
We have the following upper bound.
Proposition 3.6. Let µ be a probability distribution on R d with a density bounded by some B > 0. Then there exists a constant C = C d,B,D > 0 such that
N (ε, Cε −2αs , F, G, L s (µ)) = O d,B ε −2ds as ε → 0, where G = H α (R d , [0, 1]).
Proof. Let ε > 0 be fixed and m = ε −s . For all i = (i 1 , . . . , i d ) ∈ {0, . . . , m} d , we denote by
I i the rectangle [ i 1 −1 m , i 1 m ] × · · · × [ i d −1 m , i d m ].
Then, for i ∈ {1, . . . , m} d and j = (j 1 , . . . , j d ) ∈ {0, . . . , Dm − 1} d , we define the sets
U i,j = x∈I i E x, j m = y ∈ R d : min x∈I i d k=1 (y k − x k ) 2 j 2 k ≤ 1 m 2 and L i,j = x∈I i E x, j m = y ∈ R d : max x∈I i d k=1 (y k − x k ) 2 j 2 k ≤ 1 m 2 .
We introduce the bracket [l i,j , u i,j ] given by
l i,j (x) := T L i,j−1 , R d \ L i,j (x) and u i,j (x) := T U i,j+1 , R d \ U i,j+2 (x),
where we use the convention that an ellipsoid with one negative parameter is the empty set. By Lemma 3.1, these functions are α-Hölder. Further, we have the following lemma which is proved in Appendix:
Lemma 3.7. For all j ∈ {0, . . . , Dm − 1} d , x ∈ R d , we have d E x, j m , R d \ E x, j + 1 m ≥ D −1 m −2 .
As a consequence we infer that the distance between U i,j and R d \U i,j+1 is at least D −1 m −2 and the distance between L i,j and R d \ L i,j+1 is at least D −1 m −2 . Thus, by Lemma 3.1, we have l i,j α ≤ 1 + 3 α D α m 2α ≤ 1 + 3Dε −2αs , and the same bound holds for u i,j α . Now, to bound u i,j − l i,j s we need to estimate the Lebesgue measures of U i,j and L i,j . Recall that, if j = (j 1 , . . . , j d ) ∈ R d + and x ∈ R d , the Lebesgue measure of the ellipsoid E(x, j) is given by
λ(E(x, j)) = c d d k=1 j k ,
where c d is the constant π d/2 Γ(d/2+1) . The set U i,j can be seen as the set constructed as follows: start from an ellipsoid of parameters j/m centered at the center of I i , cut it along its hyperplanes of symmetry, and shift each obtained component away from the center by a distance of 1/2m in every direction; U i,j is then the convex hull of these 2 d components (see Figure 1 for the dimension 2). Let us denote by V i,j the set that has been added to the 2 d components to obtain the convex hull. We can bound the volume of U i,j by the volume of the ellipsoid plus a bound on the volume of V i,j , that is
λ(U i,j ) ≤ c d d k=1 j k m + d k=1 1 m l =k 2j l + 1 m .
The set L i,j can be seen an the intersection of the 2 d ellipsoids of parameters j/m centered at each corner of the hypercube I i (see Figure 2 for the dimension 2). Its volume is larger than the volume of an ellipsoid of parameters j/m minus the volume of V i,j . We thus have
λ(L i,j ) ≥ c d d k=1 j k m − d k=1 1 m l =k 2j l + 1 m .
Since µ has a bounded density with respect to Lebesgue measure, we have
l i,j − u i,j s s ≤ µ (U i,j+2 \ L i,j−1 ) ≤ Bλ(U i,j+2 ) − Bλ(L i,j−1 )
We infer l i,j − u i,j s = c 1/s d,B (ε), as ε → 0, where the constant c d,B only depends on d and B. Now, if f belongs to F, then f = 1 E(x,r) for some x ∈ X , and r ∈ [0, D] d . Thus, there exist some i = (i 1 , . . . , i d ) ∈ {0, . . . , m} d and j ∈ {0, . . . , Dm − 1} d such that
x ∈ i 1 − 1 m , i 1 m × · · · × i d − 1 m , i d m and for each k = 1, . . . , d, j k m ≤ r k ≤ j k + 1 m .
We then have l i,j ≤ f ≤ u i,j . Thus, the D d m 2d brackets [l i,j , u i,j ], i ∈ {1, . . . , m} d and j ∈ {0, . . . , Dm − 1} d , cover the class F. Therefore, there exists a C d,B,D > 0, such that N (ε, C d,B,D ε −αs , F, G, L s (µ)) = O d,B (ε −2ds ), as ε → 0.
Example 4: Indicators of Uniformly Bounded Multidimensional Ellipsoids.
In Example 3, we only considered indicators of ellipsoids centered in a compact subset of R d , namely the unit square. The following lemma will allow us to extend such results to indicators of sets in the whole R d , at the cost of a moderate additional assumption and a marginal increase of the bracketing numbers.
Lemma 3.8. Let µ be a measure with continuous distribution function F , and s ≥ 1. Furthermore let F := {1 S : S ∈ S}, where S is a class of measurable sets of diameter not larger than D ≥ 1, and G = H α (R d , [0, 1]). Assume that there are constants p, q ∈ N, C > 0, and a function f : R + → R + , such that for any K > 0 we have
N (ε, f (ε), F K , G, L s (µ)) ≤ CK p ε −q , (3.2) for sufficiently small ε, where F K := {1 S : S ∈ S, S ⊂ [−K, K] d }. If there are some constants b, β > 0 such that µ({x ∈ R d : |x| > t}) ≤ bt − 1 β , (3.3)
for all sufficiently large t, then
N ε, max f (ε), 4 √ d(ω −1 F (2 −(d+1) ε s )) −α , F, G, L s (µ) = O β,b,C,D,p (ε −(βps+q) ) as ε → 0,
where ω F is the modulus of continuity of F .
The proof is postponed in Appendix.
Proposition 3.9. Let F denote the class of indicators of ellipsoids of diameter uniformly bounded by D > 0, which are aligned with coordinate axes (and arbitrary centers in the whole space R d ). If µ is a measure on R d with a density bounded by B > 0 and if furthermore (3.3) holds for some β > 0 and b > 0, then there exists a constant C = C d,B,D > 0 such that
N ε, Cε −2αs , F, G, L s (µ) = O β,b,d,B,D,s ε −(βs+2)ds as ε → 0, where G = H α (R d , [0, 1]).
Proof. In the situation of Example 3 change the set of the centers of the ellipsoids [0, 1] d to [−K, K] d and apply Lemma 3.8. Following the proof of Proposition 3.6 we can easily see that condition (3.2) holds for p = ds, q = 2ds and f (ε) = C d,B,D ε −2αs . Note that since we have a bounded density, we have ω F (x) ≤ Bx and therefore 4
√ d(ω −1 F (2 −(d+1) ε s )) −α ≤ 4 √ d(2 d+1 B) α ε −αs ≤ C d,B,D ε −2αs for sufficiently small ε.
Remark 3.10. In the situation of Proposition 3.9 for the class F of indicators of balls in R d with uniformly bounded diameter, we can obtain the slightly sharper bound N (ε, Cε −αs , F , G, L s (µ)) = O β,b,d,B,D,s (ε −((β+1)ds+1)s ) as ε → 0 for some C = C d,B > 0 by applying Lemma 3.8 directly to the situation in Example 2 and using the same arguments as in the previous example.
Example 5: Indicators of Balls of an Arbitrary Metric with Common Center.
Let (X , d) be a metric space and fix x 0 ∈ X . A x 0 -centered ball is given by
B(t) := {x ∈ X : d(x 0 , x) ≤ t}.
We have the following bound on the bracketing numbers of the class F := {1 B(t) : t > 0}.
Proposition 3.11. Let s ≥ 1 and γ > 1. If for the probability measure µ on X the modulus of continuity ω G of the function G(t) := µ(B(t)) satisfies
ω G (x) = O(| log x| −sγ ) as x → 0,(3.
4)
then there is a constant C = C G > 0 such that
N (ε, exp(Cε − 1 γ ), F, G, L s (µ)) = O(ε −s ) as ε → 0, where G = H α (X , [0, 1]).
Remark 3.12. Note that in the case that X = R 2 , dµ(t) = ρ(t)dt, the metric d is given by the Euclidean norm, and x 0 = 0, an equivalent condition to (3.4) is
sup r≥0 r+x r t 2π 0 ρ(te iϕ ) dϕ dt = O(| log x| −sγ ) as x → 0.
Proof of Proposition 3.11. Fix ε > 0 and choose m = 3ε −s + 1 . Let G −1 denote the pseudo-inverse of G and set for i ∈ {1, . . . , m}
r i := G −1 i m , B i := B(r i ).
For convenience set B −1 , B 0 := ∅ and B m+1 = X . Define
l i (x) := T [B i−2 , X \ B i−1 ] (x) and u i (x) := T [B i , X \ B i+1 ] (x)
The system {[l i , u i ] : i ∈ {1, . . . , m}} is a covering for F. Obviously
u i − l i s s ≤ µ(B i+1 \ B i−2 ) ≤ 3 m ≤ ε s .
By Lemma 3.1, we have
u i α ≤ 1 + 3 α d(B i , X \ B i+1 ) α ≤ 1 + 3 α (r i+1 − r i ) α .
Since by condition (3.4)
r i+1 − r i ≥ inf x > 0 : ∃t ∈ R such that G(t + x) − G(t) ≥ 1 m ≥ inf x > 0 : ∃t ∈ R such that ω G (x) ≥ 1 m ≥ exp(−c G m 1 sγ )
for some constant c G > 0, there is a constant C G > 0 such that
u i α ≤ 1 + 3 α exp(αc G m 1 sγ ) ≤ exp(C G m 1 sγ ) ≤ exp(C G ε − 1 γ ).
Analogously, we can show that l i α ≤ exp(C G ε − 1 γ ). This implies that all [l i , u i ] are (ε, exp(C G ε − 1 γ ), F, G, L s (µ))-brackets and thus the proposition is proved.
3.6. Example 6: A Class of Monotone Functions. In this example, we choose X = R.
We consider the case of a one-parameter class of functions F = {f t : t ∈ [0, 1]}, where f t are functions from R to R with the properties:
(i) for all t ∈ [0, 1] and x ∈ R, 0 ≤ f t (x) ≤ 1; (ii) for all 0 ≤ s ≤ t ≤ 1, f s ≤ f t ; (iii) for all t ∈ [0, 1], f t is non-decreasing on R.
Note that all the sequel remains true if in (iii), non-decreasing is replaced by non-increasing. Further, for a probability measure µ on R, we define G µ (t) = µf t and we say that
G µ is Lipschitz with Lipschitz constant λ > 0 if |G µ (t) − G µ (s)| ≤ λ|t − s|, for all s, t ∈ [0, 1].
Empirical processes indexed by a 1-parameter class of functions arise, e.g. in the study of empirical U-processes; see Borovkova et al. (2001). The empirical U-distribution function with kernel function g(x, y) is defined as
U n (t) = 1 n 2 1≤i<j≤n 1 {g(X i ,X j )≤t} .
Then, the first order term in the Hoeffding decomposition is given by
n i=1 g t (X i ),
where g t (x) = P (g(x, X 1 ) ≤ t). For this class of functions, conditions (i) and (ii) are automatically satisfied. Condition (iii) holds, if g(x, y) is monotone in x. This is e.g. the case for the kernel g(x, y) = y − x, which arises in the study of the empirical correlation integral; see Borovkova et al. (2001).
Proposition 3.13. Let s ≥ 1 and γ > 1. Let µ be a probability measure on R such that its distribution function F satisfies (3.5) and such that G µ is Lipschitz with Lipschitz constant λ > 0. Then there exists a C = C F > 0, such that
ω F (x) = O(| log(x)| −γ ) as x → 0,N (ε, exp(Cε − 1 γ ), F, G, L s (µ)) = O λ ε −s as ε → 0 where G = H α (R, [0, 1]).
Proof. Let ε > 0 and m = (λ + 4)ε −s + 1 . For i = 0, . . . , m, we set
t i = i m and x i = F −1 i m .
We always have x m = +∞, but x 0 could be finite or −∞. In order to simplify the notation, in the first case, we change to x 0 = −∞. We define, for j ∈ {1, . . . , m}, the functions l j and u j as follows. If k ∈ {1, . . . , m − 1}, we set l j (x k ) = f t j−1 (x k−1 ) and u j (x k ) = f t j (x k+1 ), where we have to understand f (±∞) as lim x→±∞ f (x). If k ∈ {0, . . . , m − 1} and x ∈ (x k , x k+1 ), we define l j (x) and u j (x) by the linear interpolations,
l j (x) = l j (x k ) + (x − x k ) l j (x k+1 ) − l j (x k ) x k+1 − x k , u j (x) = u j (x k ) + (x − x k ) u j (x k+1 ) − u j (x k ) x k+1 − x k , with the exceptions that l j (x) = l j (x 1 ) = f t j−1 (−∞) if x ∈ (−∞, x 1 ) and u j (x) = u j (x m−1 ) = f t j (+∞) if x ∈ (x m−1 , +∞). Then it is clear that for all t j−1 ≤ t ≤ t j , we have l j ≤ f t ≤ u j , i.e. f t belongs to the bracket [l j , u j ].
Further, being piecewise affine functions, l j and u j are α-Hölder continuous functions with Hölder norm
l j α ≤ 1 + max k=1,...,m l j (x k ) − l j (x k−1 ) (x k − x k−1 ) α ≤ 1 + max k=1,...,m 1 (x k − x k−1 ) α ≤ 1 + exp C F m 1 sγ .
Here we have used the condition (3.5) and the same computation as for the class of indicators of rectangles. Analogously, the same bound holds for u j α . Now,
u j − l j s s ≤ u j − l j 1 ≤ u j − f t j 1 + f t j − f t j−1 1 + l j − f t j−1 1 . First, since G µ is Lipschitz, we have f t j − f t j−1 1 ≤ G(t j ) − G(t j−1 ) ≤ λ(t j − t j−1 ) = λ m . For x ∈ [x k−1 , x k ], since f t is nondecreasing, we have u j (x) ≤ f t j (x k+1 ) and u t j (x) ≥ f t j (x k−1 ), thus u j − f t j 1 ≤ m−1 k=1 |f t j (x k+1 ) − f t j (x k−1 )|µ([x k , x k+1 ]) ≤ 1 m m−1 k=1 (|f t j (x k+1 ) − f t j (x k )| + |f t j (x k ) − f t j (x k−1 )|) ≤ 2 m m−1 k=0 |f t j (x k+1 ) − f t j (x k )| ≤ 2 m since, by monotonicity, m−1 k=0 |f t j (x k+1 )−f t j (x k )| ≤ 1.
In the same way we get l j −f t j−1 1 ≤ 2 m and we infer
u j − l j s ≤ λ + 4 m 1/s ≤ ε.
Thus, the number of (ε, exp(C F ε − 1 γ ), G, L s (µ))-brackets needed to cover the class F is bounded by m, which proves the proposition.
Application to Ergodic Torus Automorphisms
We can apply Theorem 2.1 to the empirical process of ergodic torus automorphisms. Let is an automorphism of T d that preserves the Lebesgue measure λ. Thus (T d , B(T d ), λ, T ) is a measure preserving dynamical system. It is ergodic if and only if the matrix A has no eigenvalue which is a root of unity. A result of Kronecker shows that in this case, A always has at least one eigenvalue which has modulus different than 1. The hyperbolic automorphisms (i.e. no eigenvalue of modulus 1) are particular cases of Anosov diffeomorphisms. Their properties are better understood than in the general case. However, the general case of ergodic automorphisms is an example of a partially hyperbolic system for which strong results can be proved. The central limit theorem for regular observables has been proved by Leonov (1960), see also Le Borgne (1999) for refinements. Other limit theorems can be found in Dolgopyat (2004). The one-dimensional empirical process, for R-valued regular observables, has been studied by Durieu and Jouan (2008). Dehling and Durieu (2011) proved weak convergence of the classical empirical process (indexed by indicators of left infinite rectangles). We can now generalize this result to empirical processes indexed by further classes of functions. We can get the following proposition, as a corollary of Theorem 2.1 and the results of the preceding section.
T d = R d /Z d be the torus of dimension d > 1, which is identified with [0, 1] d . If A is
Theorem 4.1. Let T be an ergodic d-torus automorphism and let F be one of the following classes:
• the class of indicators of rectangles of T d ;
• the class of indicators of Euclidean balls of T d ;
• the class of indicators of ellipsoids of bounded diameter of T d ;
Then the empirical process
U n (f ) = 1 √ n n i=1 f • T i − λf , f ∈ F converges in distribution in ∞ (F) to a tight Gaussian process (W (f )) f ∈F .
Proof. Let F be one of the classes of functions and B be the class of α-Hölder functions for some α > 1/2. We set G the subclass of B given by the functions bounded by 1. We consider the T d -valued stationary process X i = T i . Since the distribution of X 0 is the Lebesgue measure on T d , Propositions 3.2, 3.5 and 3.6 show that the condition (2.4) holds for every possible choice of class F. For all f ∈ B, the central limit theorem (2.1) holds; see Leonov (1960) and Le Borgne (1999). Dehling and Durieu (2011), Proposition 3, show that the ergodic automorphisms of the torus satisfy the multiple mixing property (2.3) for functions of the class G, and with the constants = 1 and d 0 the size of the biggest Jordan's block of T restricted to its neutral subspace. Thus the 2p-th moment bound (2.2) holds, and Theorem 2.1 can be applied to conclude.
Proof of the Main Theorem
In the proof of Theorem 2.1, we need a generalization of Theorem 4.2 of Billingsley (1968). Billingsley considers random variables X n , X (m) n , X (m) , X, m, n ≥ 1, with values in a separable metric space (S, ρ) Billingsley (1968) states that then X n D −→X. Dehling et al. (2009) proved that this result holds without condition (b), provided that S is a complete separable metric space. More precisely, they could show that in this situation (a) and (c) together imply the existence of a random variable X satisfying (b), and thus by Billingsley's theorem X n D −→X.
satisfying (a) X (m) n D −→X (m) as n → ∞, for all m ≥ 1, (b) X (m) D −→X as m → ∞ and (c) ∀δ > 0, lim sup n→∞ P (ρ(X (m) n , X n ) ≥ δ) → 0 as m → ∞. Theorem 4.2 of
Here we will generalize this theorem to possibly non-measurable random elements with values in non-separable spaces. Regarding convergence in distribution of non-measurable random elements, we use the notation of van der Vaart and Wellner (1996). In accordance with the terminology of van der Vaart and Wellner (1996), we will call a not necessarily measurable function with values in a measurable space a random element.
Theorem 5.1. Let X n , X (m)
n , X (m) , m, n ≥ 1, be random elements with values in a complete metric space (S, ρ), and suppose that X (m) is measurable and separable, i.e. there is a separable set S (m) ⊂ S such that P (X (m) ∈ S (m) ) = 1. If the conditions
X (m) n D −→ X (m) as n → ∞, for all m ≥ 1, (5.1) lim sup n→∞ P * ρ(X n , X (m) n ) ≥ δ −→ 0 as m → ∞, for all δ > 0 (5.2)
are satisfied, then there exists an S-valued, separable random variable X such that X (m) D −→ X as m → ∞, and X n D −→ X as n → ∞.
(5.
3)
The proof is postponed to the Appendix.
Proof of Theorem 2.1. For all q ≥ 1, there exist two sets of N q := N (2 −q , exp(C2 q γ ), F, G, L s (µ)) functions {g q,1 , . . . , g q,Nq } ⊂ G and {g q,1 , . . . , g q,Nq } ⊂ G, such that g q,i − g q,i s ≤ 2 −q , g q,i B ≤ exp(C2 q γ ), g q,i B ≤ exp(C2 q γ ) and for all f ∈ F, there exists an i such that g q,i ≤ f ≤ g q,i . Further, by (2.4),
q≥1 2 −(r+1)q N 2 q < +∞. (5.4)
For all q ≥ 1, we can build a partition F = Nq i=1 F q,i of the class F into N q subsets such that for all f ∈ F q,i , g q,i ≤ f ≤ g q,i . To see this define F q,1 = [g q,1 , g q,1 ] and F q,i = [g q,i , g q,i ]\(∪ i−1 j=1 F j ). In the sequel, we will use the notation π q f = g q,i and π q f = g q,i if f ∈ F q,i . For each q ≥ 1, we introduce the process
F (q) n (f ) := F n (π q f ) = 1 n n i=1 π q f (X i ); f ∈ F which is constant on each F q,i . Further, if f ∈ F q,i , we have F (q) n (f ) ≤ F n (f ) ≤ F n (π q f ) We introduce U (q) n (f ) := U n (π q f ) = √ n(F (q) n (f ) − µ(π q f )); f ∈ F. Proposition 5.2. For all q ≥ 1, the sequence (U (q) n (f )) f ∈F converges in distribution in ∞ (F) to a piecewise constant Gaussian process (U (q) (f )) f ∈F as n → ∞.
Proof. Since π q f ∈ G and G is a subset of B, by assumption (2.1), the CLT holds and U (q) n (f ) converges to a Gaussian law for all f ∈ F. We can apply the Cramér-Wold device to get the finite dimensional convergence: for all k ≥ 1, for all f 1 , . . . , f k ∈ F, (U (q) n (f 1 ), . . . , U (q) n (f k )) converges in distribution to a Gaussian vector (U (q) (f 1 ), . . . , U (q) (f k )) in R k . Since U (q) n is constant on each element F q,i of the partition, the finite dimensional convergence implies the weak convergence of the process. Indeed, consider the function τ q : R Nq → ∞ (F) that maps a vector x = (x 1 , . . . , x Nq ) to the function F → R, f → x i such that f ∈ F q,i . For f 1 ∈ F q,1 , . . . , f Nq ∈ F q,Nq we have U (q) n = τ q (U (q) n (f 1 ), . . . , U (q) n (f Nq )) and thus the continuous mapping theorem guarantees that U (q) n converges weakly to the random variable U (q) = τ q (U (q) (f 1 ), . . . , U (q) (f Nq )) which is constant on each F q,i . Proposition 5.3. For all ε > 0, η > 0 there exists a q 0 such that for all q ≥ q 0 lim sup
n→∞ P * (sup f ∈F |U n (f ) − U (q) n (f )| > ε) ≤ η. Proof. For a random variable Y let Y denote its centering Y := Y − EY . If for arbitrary random variables Y l , Y, Y u we have Y l ≤ Y ≤ Y u then |Y − Y l | ≤ |Y u − Y l | + E|Y u − Y l |. Using F (q+K) n (f ) ≤ F n (f ) ≤ F n (π q+K f ) and E|F n (π q+K f ) − F (q+K) n (f )| ≤ 2 −(q+K) for all f ∈ F, we obtain |U n (f ) − U (q) n (f )| = K k=1 U (q+k) n (f ) − U (q+k−1) n (f ) + U n (f ) − U (q+K) n (f ) ≤ K k=1 U (q+k) n (f ) − U (q+k−1) n (f ) + U n (π q+K f ) − U (q+K) n (f ) + √ n2 −(q+K) .
In order to assure ε 4 ≤ 2 −(q+K) √ n ≤ ε 2 , for fixed n and q, choose K = K n,q , where K n,q := log 4 √ n 2 q ε log(2) −1 .
For each i ∈ {1, . . . , N q }, we obtain
sup f ∈F q,i |U n (f ) − U (q) n (f )| ≤ Kn,q k=1 sup f ∈F q,i |U (q+k) n (f ) − U (q+k−1) n (f )| + sup f ∈F q,i |U n (π q+Kn,q f ) − U (q+Kn,q) n (f )| + ε 2 .
By taking ε k = ε 4k(k+1) , k≥1 ε k = ε 4 and we get for each i ∈ {1, . . . , N q },
P * sup f ∈F q,i |U n (f ) − U (q) n (f )| ≥ ε ≤ Kn,q k=1 P * sup f ∈F q,i |U (q+k) n (f ) − U (q+k−1) n (f )| ≥ ε k +P * sup f ∈F q,i |U n (π q+Kn,q f ) − U (q+Kn,q) n (f )| ≥ ε 4 .
Notice that the suprema in the r.h.s. are in fact maxima over finite numbers of functions, since the functionals π q and π q (and thus U (q) n ) are constant on the F q,i . Therefore we can work with standard probability theory from this point: the outer probabilities can be replaced by usual probabilities on the right hand side. For each k choose a set F k composed by at most N k−1 N k functions of F in such a way that F k contains one function in each non empty F k−1,i ∩ F k,j , i = 1, . . . , N k−1 , j = 1, . . . , N k . Then, for each i ∈ {1, . . . , N q }, we have
P * sup f ∈F q,i |U n (f ) − U (q) n (f )| ≥ ε ≤ Kn,q k=1 f ∈F q,i ∩F q+k P |U (q+k) n (f ) − U (q+k−1) n (f )| ≥ ε k + f ∈F q,i ∩F q+Kn,q P |U n (π q+Kn,q f ) − U (q+Kn,q) n (f )| ≥ ε 4 .
Now using Markov's inequality at the order 2p (p will be chosen later) and assumption (2.2), we infer
P * sup f ∈F q,i |U n (f ) − U (q) n (f )| ≥ ε ≤ C p Kn,q k=1 f ∈F q,i ∩F q+k 1 ε 2p k p j=1 n j−p π q+k f − π q+k−1 f j s log 2p+aj ( π q+k f − π q+k−1 f B + 1) + C p f ∈F q,i ∩F q+Kn,q 4 ε 2p p j=1
n j−p π q+Kn,q f − π q+Kn,q f j s log 2p+aj ( π q+Kn,q f − π q+Kn,q f B + 1).
At this point, without loss of generality, we can assume that a ≥ −1 (if not, take a larger a) and thus the assumption on γ reduces to γ > 2 + a.
Note that by construction, for each k ≥ 1,
π q+k f − π q+k−1 f s ≤ π q+k f − f s + π q+k−1 f − f s ≤ 3 · 2 −(q+k) π q+k f − π q+k f s ≤ 2 −(q+k) π q+k f − π q+k−1 f B ≤ 2 exp(C2 q+k γ ) π q+k f − π q+k f B ≤ 2 exp(C2 q+k γ ).
Thus,
P * sup f ∈F q,i |U n (f ) − U (q) n (f )| ≥ ε ≤ 2 2p+1 C p p j=1 Kn,q k=1 #(F q,i ∩ F q+k ) (k(k + 1)) 2p ε 2p n j−p 2 −j(q+k) log 2p+aj (2 exp(C2 q+k γ ) + 1),
and if q is large enough,
P * sup f ∈F |U n (f ) − U (q) n (f )| ≥ ε ≤ Nq i=1 P * sup f ∈F q,i |U n (f ) − U (q) n (f )| ≥ ε ≤ D Nq i=1 p j=1 Kn,q k=1 #(F q,i ∩ F q+k ) (k(k + 1)) 2p ε 2p n j−p 2 −j(q+k) 2 (2p+aj) q+k γ ,
where D is a new constant which depends on p, C, and C p . Since (F q,i ) i=1,...,Nq is a partition of F, we have
Nq i=1 #(F q,i ∩ F q+k ) = #(F q+k ) ≤ N q+k−1 N q+k , thus we have P * sup f ∈F |U n (f ) − U (q) n (f )| ≥ ε ≤ D p j=1 n j−p ε 2p Kn,q k=1 N q+k−1 N q+k k 4p 2 (2p+(a−γ)j) q+k γ ≤ D p j=1 n j−p ε 2p 2 (p−j)(γ+2+a) q+Kn,q γ Kn,q k=1 N q+k−1 N q+k k 4p 2 ((−a−γ)p+(2+2a)j) q+k γ ≤ D p−1 j=1 n (j−p) γ−(2+a) 2γ ε 2p+(p−j) γ+2+a γ ∞ k=1 N q+k−1 N q+k k 4p 2 (2+a−γ)p q+k γ + D ε 2p ∞ k=1 N q+k−1 N q+k k 4p 2 (2+a−γ)p q+k γ ,(5.5)
because a ≥ −1 and thus (2 + 2a)j ≤ (2 + 2a)p, and where D and D are positive constants also depending on p, C, and C p . As p 2+a−γ γ → −∞ when p tends to infinity, there exists some p > 1 such that p 2+a−γ γ < −(r + 1) and thus by (5.4),
∞ k=2 N k−1 N k k 4p 2 p(2+a−γ) k γ ≤ ∞ k=2 N 2 k−1 k 4p 2 p(2+a−γ) k γ + ∞ k=2 N 2 k k 4p 2 p(2+a−γ) k γ < +∞.
Therefore the first summand of (5.5) goes to zero as n goes to infinity and the second summand of (5.5) goes to zero as q goes to infinity.
Propositions 5.2 and 5.3 establish for the random elements U n , U
n , U (q) with value in the complete metric space ∞ (F) conditions (5.1) and (5.2) of Theorem 5.1, respectively. Thus Theorem 5.1 completes the proof of Theorem 2.1.
Proof of Lemma 3.1. By the triangle inequality, we have for all x, y ∈ X that
|d B (x) − d B (y)| ≤ d(x, y) d B (x) + d A (y) ≤ d(A, B). Therefore, |T [A, B](x) − T [A, B](y)| = d B (x) − d B (y) d B (y) + d A (y) + d B (y) d B (y) + d A (y) − d B (y) d B (x) + d A (x) d B (x) + d A (x) d B (y) + d A (y) = d B (x) − d B (y) d B (x) + d A (x) + d B (y) d B (y) + d A (y) d B (y) − d B (x) + d A (y) − d A (x) d B (x) + d A (x) ≤ 3 d(x, y) d(A, B)
and thus
T [A, B] α := T [A, B] ∞ + sup x =y T [A, B](x) − T [A, B](y) d(x, y) α ≤ 1 + sup x =y T [A, B](x) − T [A, B](y) d(x, y) α T [A, B](x) − T [A, B](y) 1−α ≤ 1 + 3 d(A, B) α .
Proof of Lemma 3.7. Without loss of generality, assume that x = 0. For v ∈ R d , let D v denote the diagonal d×d-matrix with diagonal entries v 1 , . . . , v d . We define the operator norm of the d×d-matrix A by |A| * := sup y∈R d \{0} |Ay|/|y|. Observe that |D v | * = max i=1,...,d |v i |. We can characterize E(0, j m ) and
R d \ E(0, j m + 1 m ) by E 0, j m = z ∈ R d : D −1 j m z ≤ 1 and R d \ E 0, j m + 1 m = y ∈ R d : D −1 j m + 1 m y > 1
respectively. Thus, for any z ∈ E 0, j m and y ∈ R d \ E 0, j m + 1 m ,
|y − z| ≥ D −1 j m + 1 m −1 * D −1 j m + 1 m y − D −1 j m + 1 m D j m D −1 j m z ≥ D −1 j m + 1 m −1 * D −1 j m + 1 m y − D −1 j m + 1 m D j m * D −1 j m z > D −1 j m + 1 m −1 * 1 − D −1 j m + 1 m D j m * = min i=1,...,d j m + 1 m 1 − max i=1,...,d j i m j i m + 1 m ≤ 1 Dm 2 since j i ∈ {0, . . . , Dm − 1}.
Proof of Lemma 3.8. For any ε > 0, set K ε = sup{K > 0 : µ([−K, K] d ) ≤ 1 − ε}. We will denote the function (0, 1) → R + , ε → K ε by K • . Now, introduce the bracket [L, U ε ], given by
L ≡ 0 and U ε := T R d \[−K ε s /2 , K ε s /2 ] d , [−K ε s , K ε s ] d .
Obviously, we have U ε − L s ≤ U ε − L 1 s 1 ≤ ε. To get a bound for the Hölder-norm of U ε , consider the distribution function
G(t) := µ {x ∈ R d : |x| max ≤ t} on R, where |x| max = max{|x i | : i = 1, . . . , d}.
Observe that the pseudo-inverse G −1 of G is linked to K • by the equality K ε = G −1 (1 − ε). With geometrical arguments we infer
G(t) = j∈{−1,1} d σ(j)F (tj), where σ(j) := d i=1 j i ∈ {−1, 1}. Therefore ω G (x) = sup t∈R {G(t + x) − G(t)} = sup t∈R j∈{−1,1} d σ(j) F ((t + x)j) − F (tj) ≤ j∈{−1,1} d sup t∈R F ((t + x)j) − F (tj) ≤ j∈{−1,1} d ω F ( √ dx) ≤ 2 d ω F ( √ dx).
Now by Lemma 3.1 we obtain
U ε α ≤ 1 + 3 α |G −1 (1 − ε s 2 ) − G −1 (1 − ε s )| α ≤ 1 + 3 α inf x > 0 : ∃t ∈ R such that G(t + x) − G(t) ≥ ε s 2 −α ≤ 1 + 3 α inf x > 0 : ω G (x) ≥ ε s 2 −α ≤ 1 + 3 α sup x ≥ 0 : ω F ( √ dx) ≤ ε s 2 d+1 −α = 1 + (3 √ d) α (ω −1 F (2 −(d+1) ε s )) −α ,
where we used that ω F is continuous here to replace the infimum by the supremum. Then [L, U ε ] is an (ε, 4 √ d(ω −1 F (2 −(d+1) ε s )) −α , G, L s (µ))-bracket for sufficiently small ε. Since [L, U ε ] contains any f ∈ F\F K ε/2 +D , by (3.2) we obtain for all those ε the bound N ε, max f (ε), 4 √ d(ω −1 F (2 −(d+1) ε s )) −α , F, G, L s (µ) ≤ C(K ε s /2 + D) p ε −q + 1.
Let us finally consider the growth rate of K ε s /2 as ε → 0. By assumption (3.3) and since | · | max ≤ | · |, we have 1 − G(t) ≤ bt −1/β for sufficiently large t. Therefore,
G((b/ε) β ) ≥ 1 − ε.
By the definition of K • , we therefore obtain that K ε s /2 ≤ (2b/ε s ) β = O β,b (ε −βs ) which proves the lemma.
Proof of Theorem 5.1. (i) We will first show that X (m) converges in distribution to some random variable X. We denote by L (m) the distribution of X (m) ; this is defined since X (m) is measurable. Moreover, L (m) is a separable Borel probability measure on S. By Theorem 1.12.4 of van der Vaart and Wellner (1996), weak convergence of separable Borel measures on a metric space S can be metrized by the bounded Lipschitz metric, defined by In addition, the theorem states that the space of all separable Borel measures on a complete space is complete with respect to the bounded Lipschitz metric. Thus it suffices to show that L (m) is a d BL 1 -Cauchy sequence. We obtain d BL 1 (L (m) , L (l) ) = sup f ∈BL 1 |Ef (X (m) ) − Ef (X (l) )| ≤ sup f ∈BL 1 |Ef (X (m) ) − E * f (X (m) n )| + |E * f (X (m) n ) − E * f (X n )| + |E * f (X n ) − E * f (X (l) n )| + |E * f (X (l) n ) − Ef (X (l) )| for all n ∈ N. For a Borel measurable separable random element X (m) weak convergence X (m) n D −→ X (m) as n → ∞ is equivalent to sup f ∈BL 1 |Ef (X (m) ) − E * f (X (m) n )| −→ 0; see van der Vaart and Wellner (1996, p.73). Hence by (5.1) we obtain d BL 1 (L (m) , L (l) ) ≤ lim inf
d BL 1 (L 1 , L 2 ) = sup f ∈BL 1 f (x)dL 1 (x) − f (x)dL 2 (x) ,(5.n→∞ sup f ∈BL 1 |E * f (X (m) n ) − E * f (X n )| + |E * f (X n ) − E * f (X (l) n )|.
Using Lemma 1.2.2 (iii) in van der Vaart and Wellner (1996), we obtain
|E * f (X (m) n ) − E * f (X n )| ≤ E(|f (X n ) − f (X (m) n )| * ) and therefore sup f ∈BL 1 |E * f (X (m) n ) − E * f (X n )| ≤ E ρ(X n , X (m) n ) ∧ 2 * = ∞ 0 P * ρ(X n , X (m) n ) ∧ 2 ≥ t dt,(5.7)
where we used the last statement of Lemma 1.2.2 in van der Vaart and Wellner (1996). Now, let ε > 0 be given. By (5.2), there exists an m 0 ∈ N such that for every m ≥ m 0 there is some n 0 ∈ N such that for every n ≥ n 0 we have P * ρ(X n , X (m) n ) ≥ ε/3 ≤ ε/3. Therefore
P * ρ(X n , X (m) n ) ∧ 2 ≥ t ≤ 1, if t < ε 3 ε 3 , if ε 3 ≤ t ≤ 2 0, if 2 < t.
Applying this inequality to (5.7), we obtain lim inf
n→∞ sup f ∈BL 1 |E * f (X (m) n ) − E * f (X n )| ≤ 2 0 ε 3 + 1 {t< ε 3 } dt = ε
for all m ≥ m 0 . Hence for l, m ≥ m 0 we have d BL 1 (L (m) , L (l) ) ≤ 2ε; i.e. (L (m) ) m∈N is a d BL 1 -Cauchy sequence in a complete metric space.
(ii) The remaining part of the proof follows closely the proof of Theorem 4.2 in Billingsley (1968), replacing the probability measure P by the outer measure P * where necessary and making use of the Portmanteau theorem; see van der Vaart and Wellner (1996), Theorem 1.3.4 (iii), and the sub-additivity of outer measures. From part (i), we already know that there is some measurable X such that X (m) D −→ X. Let F ⊂ S be closed. Given ε > 0, we define the ε-neighborhood F ε := {s ∈ S : inf x∈F ρ(s, x) ≤ ε}, and observe that F ε is also closed. Since X n ∈ F } ⊂ {X (m) n ∈ F ε } ∪ {ρ(X (m) n , X n ) ≥ ε}, we obtain P * (X n ∈ F ) ≤ P * (X (m) n ∈ F ε ) + P * (ρ(X (m) n , X n ) ≥ ε), for all m ∈ N. By (5.2) we may choose m 0 so large that for all m ≥ m 0 lim sup n→∞ P * (ρ(X (m) n , X n ) ≥ ε) ≤ ε/2.
As X (m) D −→ X, by the Portmanteau theorem we may choose m 1 so large that for all m ≥ m 1 P (X (m) ∈ F ε ) ≤ P (X ∈ F ε ) + ε/2.
We now fix m ≥ max(m 0 , m 1 ). By (5.1) we have X Since this holds for any ε > 0 and lim ε→0 P (X ∈ F ε ) = P (X ∈ F ), we get lim sup n→∞ P * (X n ∈ F ) ≤ P (X ∈ F ), for all closed sets F ⊂ S. By a final application of the Portmanteau theorem we infer X n D −→ X.
A and B are non-empty, T [A, B] := 0 if A = ∅, and T [A, B] := 1 if B = ∅ but A = ∅. Observe, that we have T [A, B](X ) ⊂ [0, 1], T [A, B](x) = 1 for all x ∈ A and T [A, B](x) = 0 for all x ∈ B.Lemma 3.1. For any subsets A, B of X such that d(A, B) > 0, the transition function T [A, B] is a bounded α-Hölder continuous function and we have
Remark 3. 4 .
4By regarding the class of indicator functions of left infinite rectangles as a sub-class of F, we obtain Theorem 1 of Dehling and Durieu (2011) as a particular case of the preceding corollary. 3.2. Example 2: Indicators of Multidimensional Balls in the Unit Cube. Here, we consider the class F of indicator functions of balls on X = [0, 1] d , i.e.
B is a constant depending only on d and B.
3. 3 .
3Example 3: Indicators of Uniformly Bounded Multidimensional Ellipsoids Centered in the Unit Cube. Set X = R d . Here, we consider the class of ellipsoids which are aligned with the coordinate axes, have their center in [0, 1] d , and their parameters bounded by some constant D > 0. Without loss of generality, we assume that D ∈ N. For x = (x 1 , . . . , x d ) ∈ [0, 1] d and all r = (r 1 , . . . , r d ) ∈ [0, D] d , we set
Figure 1 .
1U i,j in dimension 2 Figure 2. L i,j in dimension 2
a square matrix of dimension d with integer coefficients and determinant ±1, then the transformation T : T d −→ T d defined by T x = Ax mod 1
Borel measures L 1 , L 2 on S. Here, BL 1 := {f : S −→ R : f BL 1 ≤ 1}, where f BL 1 := max sup x∈S |f (x)|, sup x =y∈S f (x) − f (y)ρ(x, y) .
X (m) as n → ∞. Thus an application of the Portmanteau theorem yields lim sup n→∞ P * (X (m) n ∈ F ε ) ≤ P (X (m) ∈ F ε ), lim sup n→∞
On R d , we use the partial order : x ≤ t if and only if x i ≤ t i for all i = 1, . . . , d.2 F i (t) = µ(R × · · · × R × (−∞, t] × R × · · · × R)
P * (X n ∈ F ) ≤ P (X ∈ F ε ) + ε.
AcknowledgmentsThe authors thank the referees and the associate editor for their very careful reading and for their most constructive criticism of an earlier version of this paper that greatly helped to improve the presentation.Appendix
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Germany E-mail address: [email protected] (O. Durieu) Laboratoire de Mathématiques et Physique Théorique UMR-CNRS 7350, Fédération Denis Poisson FR-CNRS 2964. ( H Dehling, univ-tours.fr (M. Tusche) Fakultät für Mathematik. Parc de Grandmont, 37200 Tours, France; Bochum44780Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum ; Université François-Rabelais de Tours ; Ruhr-Universität BochumE-mail address: olivier.durieu@lmpt. Germany E-mail address: [email protected](H. Dehling) Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany E-mail address: [email protected] (O. Durieu) Laboratoire de Mathématiques et Physique Théorique UMR-CNRS 7350, Fédération Denis Poisson FR-CNRS 2964, Université François-Rabelais de Tours, Parc de Grandmont, 37200 Tours, France. E-mail address: [email protected] (M. Tusche) Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany E-mail address: [email protected]
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arxiv |
A Three-Dimensional Babcock-Leighton Solar Dynamo Model: Initial Results with Axisymmetric Flows
March 22, 2016 19 Mar 2016
Mark S Miesch es:[email protected]
High Altitude Observatory
National Center for Atmospheric Research
3080 Center Green Dr80301BoulderCO
Kinfe Teweldebirhan
High Altitude Observatory
National Center for Atmospheric Research
3080 Center Green Dr80301BoulderCO
Physics Department
Faculty of Natural Sciences
Addis Ababa University
Box 1176Addis AbabaEthiopia
A Three-Dimensional Babcock-Leighton Solar Dynamo Model: Initial Results with Axisymmetric Flows
March 22, 2016 19 Mar 2016Preprint submitted to Advances in Space Research(Kinfe Teweldebirhan)solar dynamosolar magnetic activitysolar interior * Corresponding author
The main objective of this paper is to introduce the STABLE (Surface flux Transport And Babcock-LEighton) solar dynamo model. STABLE is a 3D Babcock-Leighton/Flux Transport dynamo model in which the source of poloidal field is the explicit emergence, distortion, and dispersal of bipolar magnetic regions (BMRs). Here we describe the STABLE model in more detail than we have previously and we verify it by reproducing a 2D meanfield benchmark. We also present some representative dynamo simulations, focusing on the special case of kinematic magnetic induction and axisymmetric flow fields. Not all solutions are supercritical; it can be a challenge for the BL mechanism to sustain the dynamo when the turbulent diffusion near the surface is ≥ 10 12 cm 2 s −1 . However, if BMRs are sufficiently large, deep, and numerous, then sustained, cyclic, dynamo solutions can be found that exhibit solar-like features. Furthermore, we find that the shearing of radial magnetic flux by the surface differential rotation can account for most of the net toroidal flux generation in each hemisphere, as has been recently argued for the Sun byCameron and Schüssler (2015).
Introduction
Over the last two decades, Babcock-Leighton (BL) dynamo models have emerged as a leading paradigm for explaining the origin of the solar activity cycle (Dikpati and Gilman, 2009;Charbonneau, 2010;Karak et al., 2014). The defining characteristic of BL models is the critical role of magnetic flux emergence and dispersal in the operation of the dynamo. Emerging flux structures appear in the solar photosphere as bipolar magnetic regions (BMRs) with systematic orientations such that the trailing flux (in the sense of rotational motion) is displaced poleward relative to the leading flux (Stenflo and Kosovichev, 2012;McClintock and Norton, 2013). The subsequent evolution of this flux in response to differential rotation, meridional circulation, and convection, generates a mean poloidal field that can ultimately reverse the Sun's dipole moment, at least at the surface (see §4). The preferential tilt of BMRs is known as Joy's law and this process of poloidal field generation is known as the Babcock-Leighton (BL) mechanism.
Most current BL dynamo models may also be classified as Flux Transport dynamo (FTD) models in which the meridional circulation (MC) regulates the cycle period (Dikpati and Gilman, 2009;Charbonneau, 2010;Karak et al., 2014). The equatorward migration of toroidal flux inferred from the solar butterfly diagram ( §4) is attributed to an equatorward flow of 2-3 m s −1 near the base of the convection zone (CZ) where the progenitor flux for BMRs is thought to originate. Though this presumed equatorward flow at the base of the CZ has not yet been detected, it has long been inferred based on the observed poleward circulation in the upper CZ and the constraint of mass conservation. Recent photospheric observations and helioseismic inversions have called into question this simple single-celled picture of the MC (Hathaway, 2012;Zhao et al., 2013;Jackiewicz et al., 2015;Rajaguru and Antia, 2015). However, FTD models are still viable as long as the circulation at the base of the CZ is equatorward and convection contributes to the transport of poloidal flux across the CZ (Hazra et al., 2014;Belucz et al., 2015). Both conditions are supported by theory and global convection simulations, even when the overall MC profile is multi-cellular (Miesch, 2005;Miesch et al., 2012;Passos et al., 2015).
The BL mechanism has a solid empirical grounding; we observe it operating in the solar photosphere as BMRs continually emerge and disperse. The amount of flux emerging is more than enough to reverse the polar fields, at least at the surface, and there is evidence that the strength of the following cy-cle is correlated with the BL poloidal source term and with the strength of the Sun's polar fields during cycle minimum, as predicted (Schatten et al., 1978;Svalgaard et al., 2005;Dasi-Espuig et al., 2010;Muñoz-Jaramillo et al., 2012). Other observed solar cycle features that are well reproduced by BL/FTD models include the equatorward migration of toroidal flux (solar butterfly diagram), the phase relationship between toroidal and poloidal fields, the phase coherence across grand minima, and the flux budget in active regions (Dikpati and Gilman, 2009;Charbonneau, 2010;Karak et al., 2014;Cameron and Schüssler, 2015).
Though global MHD simulations of convective dynamos have made great strides in recent years (e.g. Charbonneau, 2014), they still cannot capture the full multi-scale complexity of flux emergence and the BL mechanism. Until they do, hybrid approaches are necessary to model the solar cycle with maximum fidelity.
We describe one such hybrid approach here. We call it the STABLE (Surface flux Transport And Babcock-LEighton) solar dynamo model. STA-BLE is an FTD model but unlike most previous FTD models that only address the axisymmetric (2D) magnetic field components, STABLE is fully 3D. So, it can capture the 11-year solar cycle as well as the explicit distortion and dispersal of photospheric BMRs that underlies the BL mechanism. The latter capability in effect makes STABLE a 3D generalization of 2D (latitude/longitude) surface flux transport (SFT) models, which have had notable success in capturing the observed evolution of photospheric fields (DeVore et al., 1984;Wang et al., 1991;Schrijver and DeRosa, 2003;Baumann et al., 2004;Cameron et al., 2010;Upton and Hathaway, 2014;Jiang et al., 2014a,b;Hickmann et al., 2015). A 3D, kinematic FTD/SFT model similar to STABLE was recently described by Yeates and Munoz-Jaramillo (2013). Meanwhile, Lemerle et al. (2015) describe a somewhat different approach to unifying SFT models and FTD models based on retaining the 2D nature of each class of model and coupling them, so that each model provides the source term that sustains the other. There have also been a few 3D mean-field dynamo models based on a turbulent α-effect as opposed to the BL mechanism (e.g. Chan et al., 2004).
One of the main advantages of the 3D approach over 2D FTD models is the potential for a more realistic depiction of flux emergence. Though we acknowledge that this potential has not yet been fully realized, substantial progress in this direction has been made by Yeates and Munoz-Jaramillo (2013) who employ a kinematic flow that lifts and twists toroidal magnetic fields, mimicking the effects of magnetic buoyancy. In the future we will implement this and other flux emergence algorithms into STABLE.
Another promising reason to develop a 3D FTD model is the potential for data assimilation (DA). If solar dynamo models are to be used for the prediction of future solar activity, they must assimilate observational data. If the model is axisymmetric, this data assimilation typically makes use of the mean radial magnetic field at the surface of the sun as a function of latitude and time (e.g. Dikpati and Gilman, 2007;Jiang et al., 2013). This is effectively 1D DA. However, SFT models can exploit the observations more fully, assimilating the full observed radial magnetic field as a function of latitude, longitude and time (2D DA; see references above). Thus, SFT models are able to model the time-evolving surface magnetic field of the Sun with more fidelity. However, since they are not dynamo models, their predictive potential is relatively short-term, spanning less than a decade. STABLE will be capable of assimilating complete 2D magnetograms (2D DA) for use with both short-term and long-term solar activity forecasting. The 2D surface fields will also provide boundary conditions for corona and heliosphere models.
There are many other reasons for developing a 3D, nonlinear, MHD FTD model. Another is turbulent transport. The 3D formulation will allow us to replace turbulent diffusion and magnetic pumping with 3D convective flow fields either computed self-consistently or derived from observations. It will also allow us to capture magneto-shear instabilities in the solar tachocline which may induce non-axisymmetric patterns in flux emergence and may contribute to poloidal field generation via an α-effect (Gilman and Fox, 1997;Dikpati andGilman, 1999, 2001). Other sources of non-axisymmetric activity include converging flows into active regions (Cameron and Schüssler, 2012) and longitudinal variations in the meridional circulation.
First results from STABLE were reported by Miesch and Dikpati (2014), hereafter MD14. In this paper we describe STABLE in somewhat more detail and consider the operation of the dynamo for the special case when the imposed flow fields are kinematic and axisymmetric. In this case the mean (axisymmetric) induction equation decouples from the non-axisymmetric components and behaves essentially as a 2D FTD model. This serves to verify the model and to provide a baseline for future simulations that will include non-axisymmetric flows and Lorentz-force feedbacks. After describing the formulation of the model in §2, we test it against a 2D FTD benchmark in §3. We then give an illustrative example of a solar dynamo simulation in §4
and we discuss the challenge of achieving self-sustained dynamo action in §5.
We close by summarizing our main results in §6.
The STABLE Solar Dynamo Model
Kinematic Induction
The STABLE model solves the kinematic magnetohydrodynamic (MHD) induction equation in a 3D, rotating, spherical shell:
∂B ∂t = ∇× (v×B − η t ∇×B)(1)
where v and B are the velocity and magnetic field in the rotating reference frame and η t (r) is a turbulent diffusion. This equation is solved by means of the Anelastic Spherical Harmonic (ASH) code, which currently serves as the dynamical core for the STABLE model. ASH is a well established, parallel, MHD code that has been used extensively to study convection, instabilities, tachocline confinement, and other aspects of solar and stellar internal dynamics (Clune et al., 1999;Brun et al., 2004;Miesch, 2005;Miesch and Toomre, 2009;Brun, 2010). For the initial stages of STABLE development, we have modified ASH to operate in a kinematic regime (the full MHD system will be considered in future work). Though we will consider 3D and time-dependent flows in the near future, here we focus on steady, axisymmetric mean flows such that v =ρ(r) −1 ∇× ψ(r, θ)φ + λΩ(r, θ)φ
where ψ(r, θ) is the stream function for the meridional mass flux,ρ(r) is the dimensionless density stratification, λ = r sin θ is the cylindrical radius, and Ω(r, θ) is the differential rotation. Note that there is no explicit αeffect; the poloidal field generation needed to sustain the dynamo occurs as a consequence of the spot deposition algorithm described in section 2.2. Note also that the use of an axisymmetric velocity field as given by eq. (2) implies that the axisymmetric (m = 0, where m is the azimuthal wave number) component of equation (1) decouples from the non-axisymmetric components (m = 0). In other words, the mean (longitudinally-averaged) induction equation for the STABLE model is equivalent to a 2D FTD model when v is axisymmetric and kinematic. This allows us to make contact with previous 2D FTD models in the literature before moving on to more general 3D, nonlinear flow fields.
This point can be appreciated by averaging eq. (1) over φ, noting that v, η t , and the curl operator are all independent of φ. Though the SpotMaker algorithm ( §2.2) is distinct from previous representations of the BL mechanism in the literature, one could in principle devise an axisymmetric source term that is equivalent from the perspective of the mean fields. Thus, in terms of the evolution of the mean fields, the models presented here are similar to the 2D FTD models presented by Nandy and Choudhuri (2001) and Munoz-Jaramillo et al. (2010), who used an axisymmetric spot deposition algorithm. However, unlike 2D FTD models, STABLE also provides the corresponding time evolution of the 2D (latitude/longitude) surface field.
In order to build on previous work, we specify v(r, θ) and η t (r) as in previous FTD models. In particular, the Ω profile is taken from Dikpati and Charbonneau (1999)
Ω(r, θ) = Ω c + 1 2 1 + erf 2 r − r c d (Ω s (θ) − Ω c )(3)
where Ω c = 2πν c and Ω s (θ) = 2π (ν eq + a 2 cos 2 θ + a 4 cos 4 θ) .
Here we use ν c = 432.8 nHz, ν eq = 460.7 nHz, a 2 = -62.9 nHz, a 4 = -67.13 nHz, r c = 0.7R, and d = 0.05R, where R is the solar radius. The resulting profile is illustrated in Fig. 1a.
The meridional flow is the same as that used by Dikpati et al. (2010) and Dikpati (2011):
ψ(r, θ) = −ψ 0 λ −1 (θ − θ 0 ) f mc (r) h mc (r, θ) (5) with f mc (r) = sin π(r − r b ) (R − r b ) exp − r − r 0 Γ 2(6)
and
h mc (r, θ) = (1 − exp [−β 1r θ ]) (1 − exp [β 2r (θ − π/2)])(7)
Herer = r/L is a nondimensional radius based on a length scale L = 1.09 × 10 10 cm andR = R/L. The parameters we use here are as follows: ψ 0 = 4.32 × 10 13 cm 2 s −1 , θ 0 = 0, r b = 0.69, r 0 = (R − r b )/5, Γ = 3, β 1 = 0.1, β 2 = 0.3, and = 2 + 10 −8 . The resulting profile is illustrated in Fig. 1b. The nondimensional density stratification is given bỹ
ρ = R r − 0.97 n(8)
with n = 1.5. We use a two-step diffusivity profile, after Dikpati and Gilman (2007)
η = η c + η mid 2 1 + erf 2 r − r da d a + η top 2 1 + erf 2 r − r db d b ,(9)
where η c = 10 9 cm 2 s −1 , η mid = 5×10 10 cm 2 s −1 , r da = 0.725R, d a = 0.0125R, r db = 0.956R, and d b = 0.05R. We consider two values of the turbulent diffusivity in the surface layers: η top = 3 × 10 12 cm 2 s −1 as in Dikpati and Gilman (2007), as well as a lower value of η top = 10 12 cm 2 s −1 . The resulting profile is illustrated in Fig. 1c. Note that the relatively low value of η in the mid CZ, 5×10 10 cm 2 s −1 places our simulations in the so-called advection-dominated regime in which the meridional flow dominates over turbulent diffusion for transporting poloidal magnetic flux across the CZ (e.g. Yeates et al., 2008;Dikpati and Gilman, 2009;Charbonneau, 2010). Though this serves as a good test case, several recent studies have suggested that the diffusion-dominated regime may be more realistic, based on the correlation of the polar field at solar minimum with the strength of the next cycle (Jiang et al., 2007), the Waldmeier effect (Karak and Choudhuri, 2011), and the efficiency of turbulent transport inferred from helioseismic measurements (Miesch et al., 2012). We will consider the diffusion-dominated regime in future work.
When solving eq. (1), we express the magnetic field in terms of poloidal and toroidal components B = ∇× (Ar) + ∇×∇× (Cr) and we impose radial-field boundary conditions at the top (A = ∂C/∂r = 0 at r = R) and perfectly conducting boundary conditions at the bottom (∂A/∂r = C = 0 at r = r b ). All simulations are initiated at t = 0 with a dipolar seed field that grows and saturates as described in sections 2.2.
SpotMaker
The spot deposition algorithm is described in MD14 and we refer the reader to that paper for further details. We call it SpotMaker and it's purpose is to place bipolar magnetic regions (BMRs) on the solar surface in response to the dynamo-generated magnetic field. The subsequent evolution of these BMRs due to differential rotation, meridional circulation, and turbulent diffusion naturally generates a mean poloidal field as originally described by Babcock (1961) and Leighton (1964). SpotMaker can be regarded as a 3D generalization of the double-ring algorithm developed by Durney (1997), Nandy and Choudhuri (2001) and Munoz-Jaramillo et al. (2010). A similar axisymmetric BMR formulation was also used by Jiang et al. (2013) when assimilating sunspot data into a 2D FTD model through the subsurface extrapolation of surface fields.
In SpotMaker, BMRs are placed on the surface based on a spot-producing toroidal field B * (θ, φ, t), which is obtained from B φ (r, θ, φ, t) by first averaging over radius in the lower CZ (0.70-0.71 R) and then applying a mask that excludes latitudes above 40 • ; see eqs.
(2) and (3) in MD14. We refer to the maximum value of B * (θ, φ, t) in the northern and southern hemispheres respectively as B * n (t) and B * s (t). In order for a BMR to be produced in the northern hemisphere (NH), B * n (t) must exceed a threshold value, here taken to be 1 kG. Similarly for the southern hemisphere (SH) and B * s (t). The latitude and longitude of each BMR is chosen randomly from all horizontal The curves in frame (c) represent the log-normal pdf (solid line, τ p = 2 days, τ s = 3 days) and the sawtooth pdf (dashed line) as given by eqs. (10) and (11). Plot symbols represent normalized histograms of the actual BMR lag times in Cases L1 (circles) and S1 (triangles), for the northern (blue) and southern (red) hemispheres. grid points where B * (θ, φ, t) exceeds B * n (t)/2 or B * s (t)/2, depending on the hemisphere.
In addition to the threshold field strength, the timing of BMR creation is governed by a time delay probability density function (pdf) P (∆), where ∆ is the time that has elapsed since the last BMR creation in each hemisphere. For example, suppose that a BMR appeared in the NH at time t 0 . The timing of the next emergence event (BMR creation) in the NH is then given by t 1 = t 0 + ∆ n , where ∆ n is chosen randomly based on the time delay pdf P (∆). Similar records are kept independently for the SH, so the emergence events in each hemisphere are asynchronous.
We consider two forms for P (∆), illustrated in Fig. 2c. The first is a lognormal pdf given by
P ln (∆) = 1 ∆σ √ 2π exp − (ln ∆ − µ) 2 2σ 2 .(10)
In practice we specify the mean and mode of the distribution, τ s and τ p and compute σ 2 = (2/3) [ln(τ s ) − ln(τ p )] and µ = ln τ p + σ 2 . This is similar to the time delay pdf used by MD14 but there it was implemented somewhat differently, based on the cumulative pdf. We also consider a sawtooth pdf that can be approximated as an asymmetric stretched exponential as follows:
P s (∆) = P 0 exp − |∆ − ∆ 0 | n ± σ ± ,(11)
where n ± and σ ± have different values depending on the sign of ∆−∆ 0 . Here we use ∆ 0 =3.1 days, n + = 1, σ + = 1.6 days, n − = 4, σ − = 0.6 days. P 0 is a normalization factor ensuring that ∞ 0 P s (∆)d∆ = 1. Once the timing and location of a BMR is determined, the next step is to specify its spatial structure. This is done by defining a pair of spots on the surface, each specified by a radial magnetic field with circular cross section and a polynomial profile; see eq. (5) in MD14. Distances on the solar surface are computed using the haversine formula. The distance between the two spots of a BMR is given by s a r s , where r s is the radius of each spot (see below), and the trailing spot (in the sense of rotational motion) is displaced poleward relative to the leading spot at an angle that is given by Joy's Law; δ = 32 • .1 cos θ (Stenflo and Kosovichev, 2012).
The subsurface structure of each spot pair is determined by means of a potential field extrapolation below the surface; see eq. (7) in MD14 and Fig. 2a,b. We realize that this is a gross approximation to the true subsurface structure of active regions but it serves to localize the BL poloidal field generation to the upper CZ, as in previous 2D FTD models (Dikpati and Gilman, 2009;Charbonneau, 2010;Munoz-Jaramillo et al., 2010;Karak et al., 2014). The boundary conditions for this subsurface extrapolation ensure that the radial field is equal to the imposed BMR field at the surface (r = R) and vanishes below a specified penetration radius, r = r p . The order of the Laplacian precludes further boundary conditions so the horizontal field is not necessarily zero at the surface. This nominally violates the upper boundary condition but this violation is quickly corrected within a few time steps with the help of explicit and implicit (numerical) diffusion which quickly make the BMR field radial at the surface.
This initial adjustment is illustrated in Fig. 3 for the mean (m = 0) component of B because this is easiest to visualize. This is from a simulation that was initiated with a single BMR at a latitude of 25 • and a penetration depth of r p = 0.90R. The initial magnetic field is identical to that shown in Fig. 2a,b but here the simulation is stepped forward in time, following the evolution of the BMR as it is subject to differential rotation, meridional circulation, diffusion, and the boundary conditions. This latitude gives it a tilt angle of 13.6 • according to the Joy's law expression given above, so it Figure 3: Adjustment of the mean (m = 0) field in a simulation that was initiated with a single BMR. (a) Ratio of horizontal to vertical field strength |B h |/|B r | as a function of radius near the upper boundary. |B h | and |B r | are each averaged over the northern hemisphere before computing this ratio. The three curves correspond to the initial field (t = 0), the field after one time step (t = 0.07 days) and the field after 20 time steps (t = 1.4 days), as indicated. Crosses represent the radial grid points. Frames (b) and (c) show the structure of the mean poloidal field at t = 0 and after 20 time steps (t = 1.4 days) respectively. starts out with a nonzero mean poloidal field component (Fig. 3b). As noted above, the initial field does not satisfy the radial field upper boundary condition (Fig. 3a). However, after one time step (t = 0.07 days), the boundary condition is applied and the horizontal field goes to zero at the surface (Fig. 3a). The turbulent diffusion and the semi-implicit timestepping ensures a smooth transition to the non-zero horizontal field in the interior. By 20 time steps (t = 1.4 days), the adjustment is complete, transitioning to a radial field for r > 0.97R (spanning ∼ 30 grid points; Fig. 3a,c). Note that the time step used for this simulation (0.07 days) is the same as that used for all the simulations reported in this paper. The adjustment is similar for the other field components (m > 0).
The magnetic flux in each BMR is given by
Φ s = 2Φ 0 |B(θ s , φ s , t s )| B q 10 23 1 + B (θ s , φ s , t s )/B q 2 ≈ B s r 2 s .(12)
HereB(θ s , φ s , t s ) is the same as B * (θ s , φ s , t s ) but without the mask that suppresses high latitudes (see above). In short, it is the value of B φ (r, θ, φ, t) taken at the location and time of the BMR (θ s , φ s , t s ), averaged over a thin radial region near the base of the CZ (0.70-0.71 R). B q is a quenching field strength that governs the saturation of the dynamo. Here we use B q = 10 5 G. The parameter Φ 0 regulates the flux budget of each BMR and can be increased to achieve supercritical solutions (see section 5). The normalization in eq. (12) is defined such that for Φ 0 = 1, the strongest BMRs (B = B q ) have a flux of 10 23 Mx, roughly consistent with solar observations. Currently STABLE does not account for the depletion of toroidal flux in the lower CZ/tachocline when a new BMR is created. We will add this capability in future versions of the model. The radius of each spot is determined by its flux content as r s = (Φ s /B s ) 1/2 , where B s is set to 3 kG. Note that we have neglected a factor of order unity (0.3π) in the effective spot area ( B r dA/B s ). Also, we impose a minimum size of r s =16 Mm to ensure that all BMRs are well resolved and a maximum size of r s = 41 Mm for the largest spots. If the above calculation for r s falls outside of these bounds, then r s is set to its maximum or minimum value and B s is readjusted to give the requisite flux: B s = r −2 s Φ s .
A Note on Flux Depletion
We close this section with a note about flux depletion, which is a more sublte issue than it may first appear. To illustrate the problem, consider the process of flux emergence, beginning with a coherent toroidal field near the base of the CZ. For simplicity we can assume that this initial toroidal field is an axisymmetric flux tube with a flux equal to Φ s but relaxing this assumption does not change the essential arguments. Now assume that this flux tube rises and emerges, forming a BMR at the surface with an approximately east-west orientation. Now define the toroidal flux through any meridional plane as
Φ cut (φ, r o , t) = π 0 ro 0 B φ (r, θ, φ, t) rdrdθ ,(13)
where r o is the outer radius of the domain in question. If we set r o = ∞ and if we neglect magnetic diffusion in the flux emergence process described above, then Φ cut (φ, ∞, t) will be independent of φ and t. This follows from the topological properties of B. Now define the emergence time as t e and consider a longitude φ b that bisects the BMR, lying between the two polarities. Again, from the topological properties of B, we can say that the contribution to Φ cut (φ b , ∞, t > t e ) from the emergent field, r > R, must be equal to Φ s . In other words, the toroidal flux at longitude φ b in the solar interior r < R is depleted by an amount Φ s due to the emergence;
Φ cut (φ b , R, t > t e ) = Φ cut (φ b , R, t < t e ) − Φ s .
Now consider the mean toroidal magnetic flux threading through the computational domain of the model
Φ mean (t) = 2π 0 π 0 R r b B φ (r, θ, φ, t) rdrdθdφ .(14)
This will also be depleted by the emergence process. However, the amount it will be depleted will depend on the amount of the tube that has emerged, such that
Φ mean (t < t e ) − Φ s < Φ mean (t > t e ) < Φ mean (t < t e ) .(15)
This is the nature of the flux depletion problem. As stated above, our SpotMaker algorithm can be regarded as a 3D generalization of the axisymmetric double-ring algorithm described by Nandy and Choudhuri (2001) and Munoz-Jaramillo et al. (2010). In those and other papers, the authors take into account flux depletion by subtracting the BMR flux Φ s from the mean toroidal flux near the base of the CZ. However, it should be noted that this corresponds to the maximum depletion limit given by eq. (15), Φ mean (t > t e ) = Φ mean (t < t e ) − Φ s . This limit would only strictly apply if the entire toroidal flux tube were to pass through the solar surface. Since the presence of a BMR requires some portion of the tube to remain below the solar surface, this limit over-estimates the mean flux loss from an emergence event. So, it should be regarded as a conservative upper limit of flux depletion.
To highlight this point further, we go back to the orginal configuration of an axisymmetric toroidal flux tube with flux Φ s . Now assume that during flux emergence, only a small segment of this tube, spanning a longitudinal range ∆φ, rises and exits the CZ, leaving the rest of the tube below the surface. Then the mean toroidal flux Φ mean will be depleted by an amount that is approximately equal to Φ s ∆φ/2π. Observations indicate that most BMRs have a longitudinal extent of less than 10 • (Upton and Hathaway, 2014), implying a mean flux depletion of less than 3% of Φ s . Though the depletion of mean flux from the tachocline is likely more than this, much of this mean flux would be redistributed throughout the CZ and, given our incomplete understanding of flux emergence, we currently have no reliable prescription for how best to redistribute this flux. SpotMaker effectively selects the lower limit of mean flux depletion, Φ mean (t > t e ) = Φ mean (t < t e ), which is valid if only a small segment of the tube emerges (∆φ << 2π). Though our imposed BMR field clearly has a subsurface eastwest component (B φ = 0; see Fig. 2), the topological nature of the field is poloidal, so it does not change the mean toroidal field B φ . However, at the longitude of emergence, the mean flux is indeed depleted in the sense
that Φ cut (φ b , R, t > t e ) ≈ Φ cut (φ b , R, t < t e )
− Φ s , as described above. Unlike previous 2D FTD models, this flux depletion occurs in the upper CZ rather than the lower CZ/tachocline.
A related question is whether or not the flux emergence process conserves magnetic energy. The current SpotMaker algorithm as laid out here does not; the placement of a BMR increases the magnetic energy. If the progenitor toroidal field is re-established quickly by differential rotation, this approximation may be justified. The lifting and twisting of the field during flux emergence can also increase magnetic energy, as in alternative formulations of the Babcock-Leighton mechanism that are based on a local or non-local α-effect (the kinematic α-effect also does not conserve energy). Still, a more careful treatment of the magnetic flux budget and energetics is certainly warranted and will be pursued in the future. The idealized algorithm presented here should be considered as only a starting point, providing a baseline for comparison as we and others develop more sophisticated flux emergence models.
Code Verification
The ASH code has already been verified by comparing it to three other independent codes on carefully selected hydrodynamic and MHD simulations of global convection (Jones et al., 2011). Here we wish to verify the kinematic version of ASH that has been used as a dynamical core for STABLE. As mentioned in §1 and §2, in the special case of kinematic, axisymmetric flow fields, the axisymmetric (m = 0) component of the induction equation (1) decouples from the non-axisymmetric components so it behaves like a 2D FTD model. Thus, though STABLE simulations are explicitly 3D, we can legitimately verify the mean (m = 0) field components by comparing them with an equivalent 2D model.
To perform this verification, we consider the 2D FTD benchmark simulations defined by Jouve et al. (2008). Specifically, we seek to reproduce their case SC , which is an FTD model in which the source term for the mean poloidal field is a non-local α-effect intended to mimic the BL mechanism. Thus, for the purpose of verification, we replace the BMR deposition algorithm described in 2.2 with an explicit poloidal source term as defined in eq. (18) of Jouve et al. (2008):
S(r, θ, t, B φ ) = 1 2 1 + erf( r − r bm d bm ) 1 − erf( r − R d bm ) 1 + B φ (r c , θ, t) B 0 2 −1 cos θ sin θ B φ (r c , θ, t) .(16)
where r bm = 0.95R, d bm = 0.01R and r c = 0.7R, as above. This is implemented by adding a term to the right-hand-side of eq. (1) of the form ∇× Sφ . Note that the presence of the quenching term involving B 0 provides a saturation mechanism for the dynamo, preventing the magnetic energy from growing exponentially without bound. Here we use B 0 = 2×10 5 G.
We also replace the velocity field and turbulent diffusion in eqs. Jouve et al. (2008) for eight independent codes. Note that this simulation is performed in 3D but the fields remain axisymmetric due to the nature of the poloidal source; ME nax as defined in §4.1 is zero. We consider this a successful verification of the kinematic STABLE model.
A Representative Dynamo Simulation
Illustrative results from a typical STABLE dynamo simulation are shown in Figures 6-9. These are all from Case S1, with parameters summarized in Table 1. These and other parameters are defined in §2 and will be discussed further in §5. Here we give a general overview of the self-sustained dynamo solutions achieved with STABLE. Like the other simulations described in §5, this case was done with a resolution of N θ , N φ , N r = 512, 1024, 340.
Overview of Cycle Characteristics
The magnetic cycles are perhaps best demonstrated by the butterfly diagrams in Figs. 6a and b. These show the mean (longitudinally-averaged) radial and toroidal field B r and B φ at r = R and r = 0.71R respectively for the northern hemisphere (NH) and southern hemisphere (SH). Throughout this paper, angular brackets denote averages over longitude. Also shown in Fig. 6 are the time evolution of the (b) polar flux and (d) mean toroidal magnetic flux in each hemisphere, expressed here as an average field strength. Polar field reversals are marked with vertical dotted lines.
As in all advection-dominated FTD models, the equatorward migration of toroidal field at low latitudes in Fig. 6c can be attributed to the equatorward meridional circulation near the base of the CZ. This deep-seated toroidal field is often used as a proxy for sunspots but there is no need for such a proxy with STABLE; BMRs appear at the surface explicitly, and migrate equatorward over the course of the cycle (Fig. 6a) along with the subsurface B φ . The distortion and dispersal of these tilted (Joy's law) BMRs by differential rotation, meridional circulation, and turbulent diffusion gives rise Figure 6: Magnetic cycles in Case S1. (a) B r at the surface (r = R) as a function of latitude and time, highlighting four magnetic cycles. Red and blue denote outward and inward field respectively. Peak amplitudes can exceed 300 G but the color table saturates at ± 100G. (b) B r averaged over the north (blue) and south (red) polar regions, above a latitude of ± 85 • . Vertical dotted lines in this and all other frames mark polar field reversals in the NH (blue) and SH (red). Frames (c) and (d) are similar to frames (a) and (b) but for B φ in the lower CZ (r = 0.71R). However, the averages in (d) are over the entire NH (blue) and SH (red), as opposed to just the polar regions. Red and blue in (c) denote eastward and westward field respectively, with a saturation level for the color table of 50 kG. to a poleward migration of trailing flux that reverses the polar fields (Fig. 6a), as described by the BL mechanism ( §1).
A conspicuous shortcoming of this model (to trained eyes) is the time it takes for mid-latitude flux to migrate poleward and reverse the polar fields, which we'll refer to as τ m . In this model, τ m spans over a decade whereas in the Sun it takes only a few years (e.g. Hathaway, 2010). This can be attributed to the imposed meridional flow, which was originally devised to investigate the impact of high-latitude counter-cells, with diverging flows near the poles Dikpati et al. (2010). Thus, the speed of the poleward MC at high latitudes is slower than in some other FTD and SFT models (Dikpati and Gilman, 2009;Charbonneau, 2010;Karak et al., 2014;Upton and Hathaway, 2014). We have confirmed that the use of different MC profiles can substantially reduce τ m and thus eliminate this apparent shortcoming of the model. Results will be presented in a forthcoming paper. We have also confirmed that τ m is insensitive to the magnitude of the turbulent diffusion at the surface, η top . Simulations with both higher (not shown here) and lower (see cases L2 and L3 in §5) values of η top exhibit similar migration time scales τ m (see Fig. 12). Another apparent shortcoming of the model is the relatively strong polar fields (Fig. 6b). This can be corrected by reducing the parameter Φ 0 ; see §5.
It is interesting to note that the time evolution of the polar fields (Fig. ) show B φ , with red and blue indicating eastward and westward field respectively. Peak field strengths can exceed 100 kG but the color table is clipped at ± 20 kG. Frames (f -g) show the poloidal magnetic potential with a potential-field extrapolation above r = R (to r = 1.25R). Red and blue denote clockwise and counter-clockwise field respectively, with peak values of B r on the order of 800 G.
6b) is nearly sinusoidal whereas the toroidal flux is more asymmetric, with a slower rise and faster decay. Note also the slight phase difference between the northern and southern hemispheres. Though this often persists for several cycles, the dynamo sporadically re-synchronizes, maintaining a dipolar parity (see Fig. 9a). We emphasize again (see §2) that the spot deposition in each hemisphere is asynchronous and that the build-up of the dipole moment is cumulative, with contributions from multiple active regions. So, the crossequator cancellation of surface B r that regulates the polar field strength occurs only in an integrated sense, involving residual flux from many BMRs as in the Sun. This is the origin of the north-south asymmetry. Figure 7 shows the evolution of the surface fields over the course of a magnetic cycle. A close look at each of these snapshots reveals multiple BMRs, in various stages of evolution. Localized, newly formed BMRs obey Joy's Law (increasing tilt with latitude; see §2.2) by construction and Hale's law (oppose orientation in the NH and SH) by virtue of the dipolar nature of the dynamo mode. Axisymmetric bands of radial field at high latitude arise from the distortion and dispersal of tilted BMRs and they migrate poleward due mainly to the MC. A brief cycle overlap can be discerned in Fig. 7e, which shows several BMRs with positive (red) leading polarity at a latitude of about -35 • coexisting with several other BMRs near the equator with negative (blue) leading polarity. Note also the asymmetry apparent in this same Figure: spots for the new cycle appear at mid-latitudes in the SH slightly before they appear in the NH (see also Fig. 6a). The evolution of the mean (longitudinally-averaged) fields during this same magnetic cycle is shown in Fig. 8.
Most of the magnetic energy (over 99%) is in the mean toroidal field, ME tor , which varies by about 15% over the course of a cycle. This is demonstrated in Fig. 9a (see also Table 1), which shows the evolution of the total magnetic energy in the mean and non-axisymmetric fields. Interestingly, the minima of the poloidal magnetic energy ME pol do not coincide with the reversals of the polar fields at the surface. Rather, they occur slightly after. Meanwhile, the polar field reversals occur near the maxima of ME tor , though slightly after, by about 0.5 to 1.5 years. This is suggestive of solar observations in which the poloidal field reversals occur near sunspot maximum. However, in the STABLE model, the magnetic energy in BMRs is reflected mainly by the non-axisymmetric field components, ME nax , which reaches a (global) minimum as the polar fields at the surface are reversing. This is Figure 9: (a) Total magnetic energy in the mean toroidal field, ME tor (blue), the mean poloidal field ME pol (red), and the non-axisymmetric field components ME nax (green) over a time interval spanning 172 years. All quantities are integrated over the entire computational volume (which spans the entire CZ) and normalized for clarity in plotting. Normalization factors are 1.15 × 10 43 erg for ME tor , 1.24 × 10 40 erg for ME nax , and 1.49 × 10 39 erg for ME pol . After normalization, we added one to the ME nax curve and two to the ME tor curve in order to plot all three curves with minimal overlap. Thus, the mean values for the normalized ME tor , ME nax and ME pol curves are 3, 2, and 1 respectively, as indicated by the black dotted lines. (b) Latitudinal positions of the BMR deposition sites in the NH (blue) and SH (red) over the same time interval. Over this interval, 14,452 BMRs were introduced in the NH and 14,411 BMRs were introduced in the SH. Blue and red vertical dotted lines indicated reversal of the polar fields as in Fig. 6. also reflected by the butterfly diagram in Fig. 6a, which suggests that polar fields reverse near a time of minimum sunspot activity. Thus, the phasing of toroidal and poloidal fields is not in good agreement with solar observations. However, it is likely that this aspect of the simulations will improve as we implement different MC profiles that more faithfully capture the poleward migration time scale of trailing magnetic flux τ m . See the discussion above in connection with Fig. 6.
It is also interesting to note that the evolution of ME nax over the course of a cycle is asymmetric, with a fast rise and a slow decline. This is similar to the observed evolution of the solar sunspot number (Hathaway, 2010) but it's opposite to the asymmetry noted above with regard to the integrated toroidal flux in Fig. 6d. Dynamo models often use the subsurface toroidal flux as a proxy for the sunspot number. The differences noted here even for an idealized FTD model such as STABLE suggest that this proxy may not be as reliable as is often assumed. Also, the distribution of the radial field at any instant quickly spreads beyond the emergence sites of BMRs. This can be seen by comparing the butterfly diagram in Fig. 6a with the actual emergence latitudes in Fig. 9b. Note that the corresponding longitudinal locations of the emergent BMRs are random.
The Role of Surface Fields in Dynamo Operation
From the perspective of space weather/space climate forecasting, one of the beneficial aspects of FTD models is the disproportionate role that surface magnetism plays in the operation of the dynamo 1 . If the main source of poloidal magnetic flux is indeed the BL mechanism, then we can observe this occurring and we can use this information to help forecast future magnetic activity. More generally, the generation of the toroidal flux in each hemisphere that is responsible for producing BMRs appears to be linked to the shearing and amplification of the observed poloidal flux that passes through the solar surface. This was demonstrated recently by Cameron and Schüssler (2015), hereafter CS15.
The analysis performed by CS15 begins by averaging the longitudinal (φ) component of the MHD magnetic induction equation (1) over φ and integrating it over the NH. If the velocity field is assumed to be axisymmetric, as it is here, this procedure yields the following expression for the evolution of the mean toroidal flux through the NH
dΦ N H dt = S ∇× [v× B − η t ∇× B ] ·dS = δS [v× B − η t ∇× B ] ·d (17) where Φ N H (t) = π/2 0 r 2 r 1 B φ rdrdθ ≡ S B ·dS .(18)
In eq. (17), d denotes a differential segment of the closed, clockwise, linear circuit δS that encircles the NH, proceeding radially outward at the north pole, equatorward at the solar surface, inward at the equator, and poleward just below the base of the CZ. This astute use of Stokes' theorem links the time evolution of the mean toroidal flux threading through the entire NH to the flows and fields on the boundaries of the CZ. Furthermore, as CS15 demonstrate, the dominant component of this line integral is the shearing of the radial field by the surface DR, which can be written as follows:
S t N H (t) = R 2 π/2 0 B r (Ω(R, θ, t) − Ω eq ) sin θdθ(19)
where Ω(r, θ, t) = Ω 0 + v φ r −1 sin −1 θ is the angular velocity. CS15 chose a reference frame rotating with angular velocity Ω eq , which is the value of Ω at r = R and θ = π/2 (equatorial surface rate). This choice of reference frame, together with the weak radial dependence of the solar rotation rate at the equator inferred from helioseismology (cf. Fig. 1a), implies that the contribution from the DR term (v φφ )× B is small for the equatorial portion of the line integral in eq. (17). Furthermore, the contribution of this DR term to the inner and polar branches of the line integral vanish if there is no mean flux through those boundaries (as here). The impenetrable boundary conditions and the symmetry of the MC (v θ = 0 at the equator) ensure that the MC contributions to the line integral in eq. (17) also vanish. Thus, the only terms other than eq. (19) that contribute significantly to the line integral in eq. (17) are the diffusive terms. Our perfectly-conducting inner BCs preclude diffusion into the deep interior but diffusion along the polar, upper, and equatorial branches of the line integral is in general non-zero. The evolution of S t N H (t) and its counterpart, S t SH (t), are shown in Fig. 10a (dotted lines). These are plotted together with the actual time derivative of Φ N H (t) and Φ SH (t) (solid lines). CS15 model the diffusive terms as an effective drag, inducing an exponential decay of Φ N H,SH in the absence of DR. This effectively decreases the amplitude of dΦ N H,SH /dt during the rising phase of a cycle and leads to a negative phase shift such that reversals and extrema occur earlier than they would without the diffusive terms. However, we find the opposite in our STABLE FTD models. Namely, the presence of the diffusive terms enhances the amplitude of dΦ N H,SH /dt during the rising phase, inducing a positive phase shift such that reversals and extrema occur later than they would otherwise. This is demonstrated by Fig. 10b where the inclusion of the diffusive terms shifts the dotted curve in Fig. 10a to the right by about 3 years, until it lies on top of the actual dΦ N H,SH /dt.
To illustrate why this occurs, consider the NH of Fig 8, focusing on the toroidal field evolution in the upper row. As the flux from the new cycle is building (red), flux from the previous cycle (blue) is pushed equatorward and upward by the MC (Fig. 8b-e). This causes a decay of the flux from the previous cycle due to diffusion first across the equator and then through the upper boundary (Fig. 10c). Meanwhile, the diffusion at the poles is negligible. This diffusive expulsion of the flux from the previous cycle causes the net flux Φ N H (t) to rise, even after the surface DR stops amplifying the flux through S t N H (t) (Fig. 10a). For example, by t = 194.1, shown in Fig. 8e and j, S t N H (t) has already reversed sign, as the surface DR generates negative toroidal flux from the new, clockwise poloidal field at low latitudes. Yet, the net toroidal flux is still growing (dΦ N H /dt > 0; see Fig. 10a) due to the selective removal of opposing flux from the previous cycle by diffusion. This shifts the maximum toward a later time (Fig. 10d). Figure 10d shows the time evolution of the amplitude of Φ N H,SH (t) (solid lines) together with the predicted evolution based on integrating equation (19), shown as dashed lines. The integration constant is chosen such that the zeros of the predicted |Φ N H,SH (t)| curves correspond roughly to the derivative extrema in Fig. 10a (dotted lines). If we were to include diffusion in the predicted |Φ N H,SH (t)| curves, then the result would essentially coincide with the actual |Φ N H,SH (t)| curves (solid lines), as in Fig. 10b.
We emphasize again that CS15 have no information on the structure of the field or the role of diffusion below the solar surface. Instead, they emphasize the importance of the surface DR term, eq. (19), which can be computed based on solar observations. They then model the subsurface diffusion as an exponential decay term that shifts the Φ N H (t) and Φ SH (t) curves to the left, toward earlier times. We confirm the importance of the surface DR term in our idealized FTD model but we find that effect of the subsurface diffusion is instead to shift the Φ N H (t) and Φ SH (t) curves to the right, toward later times. CS15 verify their argument by demonstrating that the predicted peaks of |Φ N H (t)| and |Φ SH (t)| based on S t N H (t) and S t SH (t) correlate with observed maxima in the unsigned surface flux, suggesting that the enhanced surface flux arises from the emergence of greater net subsurface flux. If the subsurface diffusion were to shift the predicted flux curves to the right instead of to the left as suggested by our model, then this would improve the CS15 correlation, substantiating the CS15 argument (see their Fig. 3). However, that said, in our model, there is little variation of the unsigned surface flux over the course of the cycle and what little variation there is appears to be anti-correlated with the predicted flux (Fig. 10d). The positive phase shift due to diffusion improves this correlation slightly but there is still about a four-year delay between the maxima in |Φ N H,SH (t)| and the peak in the unsigned surface flux.
We close this section by noting that the generation of toroidal flux by the surface DR terms S t N H,SH (t) generates enough toroidal flux to account for all of the unsigned flux present on the surface. This is even true for Case S1 (Fig. 10d), in which the flux in BMRs has been artificially enhanced by a factor of five (Φ 0 = 5; see Table 1). CS15 argue that this is also the case for the Sun.
Achieving Self-Sustained Dynamo Action
A shortcoming of the model presented in §4 (Case S1) is that the fields are too strong. For example, the average strength of the polar fields is over 200 G (Fig. 6b). The Sun, by comparison, is about 10 G (Muñoz-Jaramillo et al., 2012). This can be attributed in part to the large value of Φ 0 = 5 used to artificially enhance the flux in BMRs as expressed in eq. (12). We used this large value of Φ 0 in order to ensure that the dynamo is supercritical, meaning that sustained dynamo action can occur despite the inhibiting effects of turbulent diffusion. The main focus of this section is to see if we can achieve supercritical solutions with Φ 0 = 1, thus avoiding artificial amplification of the BMR flux.
However, before proceeding, we note that the quenching field strength, B q , also contributes to our artificially large polar field values (see eq. 12). Here Figure 11: (a) Evolution of the magnetic energy, ME, in Case S6, expressed as an average energy density in erg cm −3 . Solid, dashed and dotted lines correspond to the mean toroidal (ME tor ), mean poloidal (ME pol ), and non-axisymmetric field components (ME nax ) respectively. (b) ME nax for cases S8 (dashed line), L1 (dotted line), L2 (solid line), and L3 (dot-dashed line).
we set B q = 10 5 G and we use the same value for all simulations in order to faciliate a comparison between them. Since the kinematic induction equation is linear apart from this quenching term, scaling down the value of B q should also scale down the poloidal, toroidal, and non-axisymmetric field strengths by the same factor. Thus, it is more meaningful to consider the ratio of toroidal to poloidal field, as argued by Choudhuri (2003). For all of our cases this ratio is similar, at about 85-88 (see Table 1). Scaling down B q to achieve a mean polar field strength of about 10G should thus yield a mean toroidal field strength of about 850-880 G. This is significantly less than the toroidal field strengths inferred from simulations of rising flux tubes, which are in the range 40-100 kG (Choudhuri and Gilman, 1987;Caligari et al., 1995;Fan, 2009;Weber et al., 2011). However, as discussed by Choudhuri (2003), the toroidal field could well be highly intermittent, with strong, buoyantly unstable flux tubes embedded in a more diffuse background field. We also note that a toriodal to poloidal field ratio of under 100 is a common feature of advection-dominated FTD models; see, e.g. Dikpati et al. (2002).
If we start with the case described in §4 (Case S1) and drop the value of Φ 0 from five to unity, the dynamo decays, as shown in Fig. 11a. This is Case S6, from a series of simulations summarized in Table 1. Case S6 was Table 1: Simulation Summary. The S series of simulations uses the sawtooth pdf of eq. (11) while the L series uses the lognormal pdf of eq. (10). Root-mean-square (rms) values listed in columns 8-10 are based on integrals over the entire computational volume and are quoted for the mean toroidal field (B tor ), the mean poloidal field (B pol ), and the non-axisymmetric field component (B nax ), which is mainly composed of BMRs.
Case Φ 0 r s s a η top τ p τ s B tor B pol B nax (cm 2 s −1 ) (days) (days)
(rms) (rms) (rms) S1 5 0.9 1.5 3 × 10 12 --22 kG 250 G 720 G S2 5 0.93 1.5 3 × 10 12 --subcritical --S3 5 0.93 2.5 3 × 10 12 --subcritical --S4 2 0.9 2.5 3 × 10 12 --6.8 kG 79 G 200 G S5 2 0.9 1.5 3 × 10 12 --subcritical --S6 1 0.9 1.5 3 × 10 12 --subcritical --S7 1 0.9 2.5 3 × 10 12 --subcritical --S8 1 0.9 2.5 10 12 --subcritical --L1 1 0.9 2.5 3 × 10 12 2 3 subcritical --L2 1 0.9 2.5 10 12 2 3 11 kG 130 G 300 G L3 1 0.85 4 10 12 1 1.5 72 kG 840 G 960 G started from the same initial conditions as Case S1 (Fig. 9a), provided by a progenitor case with higher Φ 0 (not shown). Though most of the magnetic energy is in the mean toroidal fields, the dynamo cannot operate without the BMRs, which dominate the non-axisymmetric magnetic energy ME nax . As the toroidal field decays due to diffusion, it eventually drops below the threshold field for creating BMRs (see §2). This causes ME nax to drop rapidly beyond t ∼ 57 yrs (Fig. 11a). This is the point of no return; once the dynamo ceases to make BMRs, it will continue to decay indefinitely. The most straightforward way to ensure that the dynamo is supercritical is to artificially boost the flux in BMRs by increasing Φ 0 . However, other parameters also contribute to the efficiency of the dynamo and it is possible to find supercritical solutions with Φ 0 = 1. One such parameter is the penetration depth of the BMRs, r s (see §2). Here, deeper is better. If we take the solution S1 and move the penetration depth up to 0.93R instead of 0.9R (Case S2 in Table 1), the dynamo dies (becomes subcritical).
Another parameter that affects the efficiency of the dynamo is the spacing between spot pairs, s a . Recall from §2 that s a is a nondimensional number that gives the distance between the two opposite polarity components of a BMR relative to the radius of the individual spots. Thus, a large value of s a implies widely spaced spot pairs, which is beneficial for the dynamo because it maximizes the axisymmetric component of the poloidal flux and it minimizes local cancellation, allowing more trailing flux to reach the poles. A comparison of cases S4 and S5 in Table 1 demonstrates that increasing s a from 1.5 to 2.5 can make the difference between a subcritical and a supercritical dynamo. However, this is not always the case; compare also cases S2 and S3 and cases S6 and S7. Since the dynamo must overcome diffusion to achieve supercriticality, a reduction in the diffusion can also be beneficial, particularly in the upper CZ where η is largest (Fig. 1c). This was not enough to revive Case S7; Case S8 is also subcritical even though the value of η top is decreased by a factor of three. However, when combined with more frequent BMR emergence, which we achieved with the lognormal pdf (Fig. 2c), lower η top did yield supercritical solutions, even for Φ 0 = 1; see Cases L2 and L3 in Table 1. The benefit of lower diffusion is demonstrated unambiguously by comparing cases L1 and L2, which both use the same lognormal emergence pdf. The benefit of more frequent BMR emergence is demonstrated unambiguously by comparing cases S8 and L2.
The difference in dynamo efficiency influences not only the growth or decay rate of the dynamo, but also the nonlinear saturation level. This is demonstrated most dramatically by Case L3 (Table 1). This case has all the features that were shown above to be beneficial, including deep penetration of BMRs, wide spacing of BMR polarity components, low diffusion, and frequent BMR emergence, with a mean interval between spot appearances of τ s = 1 day and a mode of τ p = 1.5 days (Table 1). Even though it has the same quenching field strength as all the other cases, B q = 10 5 G, and a relatively low flux amplification factor of Φ 0 = 1, it achieves stronger fields (more magnetic energy) than all of the other cases. This includes case S1 (discussed in §4), which has Φ 0 = 5. Furthermore, though Cases L2 and L3 have the same value of Φ 0 , the latter has much stronger fields. It also has a higher ratio of B pol /B nax , approaching unity. This can be attributed to the large value of s a , which maximizes the mean poloidal field associated with each BMR because the two polarities have minimal overlap in latitude. Fig. 11b shows ME nax in cases S8, L1, L2, and L3 for comparison. All were started from the same initial conditions, obtained from the supercritical Case S4 (Table 1). Note that Case L2 in particular has more magnetic energy than its progenitor, S4 (see also Table 1). This demonstrates that lowering the diffusion by a factor of three more than makes up for lowering Φ 0 by a factor of two. Though the two supercritical cases in this plot, L2 and L3, look somewhat irregular at early times, they both settle into a steady magnetic cycle similar to that described in §4 for Case S1. This is demonstrated in Fig. 12. The most apparent difference between these two cases (Figs. 12a,b) is that case L3 (b) has a smoother distribution of low-latitude poloidal flux. This can be attributed to the more frequent BMR emergence rate, which yields more flux patches at any given time. Furthermore, the lower diffusion in both Cases L2 and L3 relative to Case S1 leads to stronger, more compact fields at low latitudes that more closely track the emergence sites; compare Fig. 12 with Figs. 9b and 6a.
The stronger, more compact fields at low latitudes in Case L3 relative to Case S1 is apparent in Fig. 13. Compare this to Fig. 7. Each individual spot of a BMR takes longer to disperse so at any given time, there are more spots that are strong and localized. The flux distribution in Case L2 (not shown) is similar. Yet, this change in the flux distribution has little effect on the overall characteristics of the magnetic cycles. The cycle period in all three cases is similar: 13.1 years for S1 and L3, and 13.6 years in L2. The other supercritical solution, Case S4, has a cycle period of 13.0 years.
Though a detailed comparison with observations lies outside the scope of this paper, we note that a cursory comparison of our simulations with solar magnetograms and SFT models suggests that cases L2 and L3 produce more realistic surface flux distributions. Their lower diffusion and higher frequency of emergence events tends to produce strong flux patches with a wider range of areas that extends below 5 square degrees, reminiscent of observed magnetograms (Upton and Hathaway, 2014).
As discussed in §2.3, the SpotMaker algorithm effectively assumes that only a small portion of a progenitor toroidal loop emerges through the surface so the depletion of mean flux is minimal. In reality, a significant portion of the mean flux in the lower CZ/tachocline would be redistributed throughout the CZ and outside of the domain. Taking this tachocline flux depletion into account would likely inhibit the dynamo; it is unclear whether or not supercritical solutions could still be found for Φ 0 = 1. It would also reduce the ratio of mean toroidal to poloidal field. These issues are indeed important to investigate but they are subtle, and beyond the scope of this paper.
Summary
We have described a novel 3D Babcock-Leighton/Flux Transport solar dynamo model and we have presented some initial results for axisymmetric, kinematic flow fields. These initial results provide a baseline for subsequent studies that will incorporate 3D flow fields and Lorentz-force feedbacks. The name of the new model is STABLE, reflecting its close ties to both surface flux transport models (ST) and (A) Babcock-Leighton (BLE) dynamo models.
The use of kinematic, axisymmetric flow fields implies that the evolution of mean fields reduces to an equivalent 2D FTD model, though there is no explicit α-effect ( §2). Instead, poloidal field is generated by the BL mechanism in response to the spontaneous appearance of BMRs, as in the 2D FTD model of Munoz-Jaramillo et al. (2010). Our model is also similar to the 3D FTD model of Yeates and Munoz-Jaramillo (2013), although they use a more sophisticated flux emergence algorithm based on imposed rising, helical, flow fields. We plan to implement a similar algorithm in the future. Until then, the current BMR deposition approach may be regarded as the limit in which emergent toroidal flux structures decouple quickly from their roots in the deep CZ. In order to verify the STABLE model ( §3), we provisionally replaced this BMR deposition algorithm with an axisymmetric BL source term and we were able to reproduce the 2D FTD benchmark CS defined by Jouve et al. (2008).
The STABLE model exhibits many promising features that are in good agreement with solar observations. Like other FTD models, it sustains regular magnetic cycles with equatorward propagation of toroidal flux at low to mid latitudes near the base of the CZ. However, unlike many other meanfield and convective dynamo models, there is no need to use this subsurface toroidal flux as a proxy for sunspot number. Instead, sunspots/BMRs appear at the surface of the STABLE model explicitly and, in the special case of axisymmetric flow fields, can be tracked by means of the non-axisymmetric component of the magnetic energy (Fig. 9). The evolution of the radial magnetic field at the surface is similar to SFT models, with the distortion and dispersal of low-latitude, tilted BMRs by DR, MC, and turbulent diffusion producing high-latitude, axisymmetric bands of poloidal flux that migrate poleward and eventually reverse the polar fields. This process is essential for the operation of the dynamo.
As in other FTD models, the period and amplitude of the magnetic cycles are largely determined by the meridional flow speed and the quenching of the poloidal source (through the B q factor in eq. (12)). However, other parameters do play a significant role. For example, though the MC and B q are the same in all simulations presented here (with the exception of the benchmark in §3), the total magnetic energy produced by the dynamo can vary by more than two orders of magnitude (see Table 1). The cycle period is less sensitive to the BMR structure, emergence rate, and diffusion, but it can still vary significantly, from 13.0 years in Case S4 to 13.6 years in Case L2.
Factors that influence the dynamo efficiency include the emergence rate and penetration depth of BMRs, and the spacing between the two polarity components of a BMR. Poloidal field generation is more efficient if each BMR is wider and deeper, and if the number of BMRs is increased by increasing the frequency of emergence (see Fig. 2c). Lower diffusion also enhances the dynamo efficiency and makes the distribution of radial flux at the surface more intermittent (Fig. 13). Any of these factors can make the difference between a subcritical and supercritical solution ( §5).
One deficiency of the model in its current state is the long time it takes for residual flux from mid-latitude BMRs to migrate poleward and reverse the polar fields (Figs. 6 and 12). This does not agree well with solar observations, both in terms of the migration speed and the phasing between low-latitude toroidal flux as traced by sunspots and high-latitude polar field reversals. As discussed in §4.1, this is likely due to a relatively low speed for the highlatitude meridional flow and will be addressed in future versions of STABLE.
Other features of the model can also be calibrated to improve agreement with solar observations, including the time lag pdf of Fig. 2c, which may depend on the phase of the cycle.
Boundary conditions appear to be important for the solar dynamo. In particular, it is clear from solar observations that substantial magnetic flux passes through the surface of the sun and that the shearing of this poloidal flux by the surface DR generates toroidal flux in each hemisphere by means of the Ω-effect. CS15 argue that this is the dominant source of mean toroidal flux in the Sun and can account for all the flux that emerges in BMRs. We find that this is indeed the case for our FTD models. However, we find that the role of turbulent diffusion is somewhat different than that envisioned by CS15 ( §4.2). They modeled turbulent diffusion as an effective drag term that inhibited the generation of net toroidal flux in each hemisphere Φ N H,SH (t). We find instead that it promotes the generation of net flux by selectively dissipating residual oppositely-signed flux from the previous cycle. Thus, instead of inducing a negative phase shift in Φ N H,SH (t) as argued by CS15, we find that diffusion induces a positive phase shift, causing the maxima and reversals in Φ N H,SH (t) to occur later than they would without diffusion.
Future model developments will include flux transport and amplification from 3D convective flow fields, Lorentz force feedbacks, and flows induced by enhanced radiative cooling in active regions, which is expected to play a significant role in dynamo saturation. These developments may produce non-asymmetric dynamo modes, magneto-shear instabilities, and torsional oscillations that will be explored in future papers. Furthermore, we intend to explore the SFT aspects of STABLE and its potential for forecasting future magnetic activity by assimilating data from photospheric magnetograms. We believe this class of 3D BL/FTD models shows great promise as a "Solar Dynamo Frontier".
Figure 1 :
1Model components; Mean flows and turbulent diffusion. (a) Angular velocity Ω/2π, with color table ranging from 350-480 nHz (blue to red/pink). (b) Meridional circulation, shown as streamlines of the mass flux, with red and blue denoting clockwise and counter-clockwise circulation respectively. Horizontal velocity amplitudes are approximately 14 m s −1 and 1.5 m s −1 in the upper and lower CZ. (c) turbulent magnetic diffusivity η t .
Figure 2 :
2(a,b) Subsurface structure of a BMR produced by SpotMaker and (c) time lag pdf of emergence events. The volume rendering in frames (a) and (b) shows magnetic field lines below the solar surface (red) from two different vantage points, (a) east of the BMR looking west and (b) underneath the BMR, looking up. The blue surface represents the surface of the Sun (r = R).
Figure 4 :
4Butterfly diagram for the benchmark case SC of Jouve et al. (2008). (a) Mean radial field B r at the surface (r = R) as a function of latitude and time. Blue and red denote inward and outward polarity respectively. (b) Mean toroidal field B φ near the base of the convection zone (r = 0.70R; blue westward, red eastward). Compare with Fig. 14 in Jouve et al. (2008).
( 3 )
3, (5), and (9) with the corresponding expressions in eqs. (13), (19), and(14)ofJouve et al. (2008). This yields a single meridional circulation cell per hemisphere, qualitatively similar toFig. 1b, directed poleward at the surface, equatorward near the base of the CZ, and vanishing at the bottom boundary r b = 0.65. Thus, it extends a little below the tachocline, which is centered at r c = 0.7R. The boundary conditions are as described in §2.1; perfectly conducting at the bottom of the shell and radial field at the top. For this benchmark, we use a resolution of N θ , N φ , N r = 128, 256, 100.Results are shown in Figs. 4 and 5. These agree well both qualitatively and quantitatively with the results presented in
Figure 5 :
5Quantitative results for benchmark case SC ofJouve et al. (2008). (a) Mean toroidal field B φ , normalized by B 0 , at r = 0.7R and latitude = 60 • . The abscissa is the nondimensional diffusion time τ d = (t − t 0 )η 0 R −2 , where η 0 = 10 11 cm 2 s −1 and t 0 is chosen such that the phase of the cycle corresponds withFig. 15inJouve et al. (2008). (b) Normalized mean radial field B r /B 0 as a function of τ d at r = R and latitude = 30 • . Compare withFig. 15inJouve et al. (2008).
Figure 7 :
7Radial magnetic field B r at the solar surface (r = R) in case S1, plotted in Molleweide projection. Dashed lines denote latitudes of 0 • , ±30 • , and ±60 • . Six snapshots are shown, spanning one magnetic cycle: t = (a) 181.0 (b) 183.6, (c) 186.2, (d) 188.9, (e) 191.5, and (f ) 194.1 years. Red and blue denote radially outward and inward field respectively, with a saturation level on the color table of ± 500 G.
Figure 8 :
8Mean (a-e) toroidal and (f -j) poloidal magnetic fields in case S1. Five snapshots are shown, spanning the same magnetic cycle as inFig. 7: t = (a,f ) 181.0,(b,g) 184.3, (c,h) 187.5, (d,i) 190.8, (e,j) 194.1 years. Frames (a-e
Figure 10 :
10(a) Rate of change of the mean toroidal flux threading the NH dΦ N H /dt (blue solid line) for a selected time interval in Case S1 spanning four magnetic cycles. Red solid lines indicate its counterpart in the SH, dΦ SH /dt. Dotted lines indicate NH (blue) and SH (red) contributions from the surface DR term in eq. (19). (b) As in (a) but including contributions from the diffusive terms in eq. (17). Now the solid and dotted lines coincide.(c) Breakdown of the diffusive contributions to the line integral in eq. (17), including contributions from the upper (blue solid line), equatorial (blue dotted line), and polar (blue dashed line) branches of the closed circuit. The sum of these contributions is plotted as a black solid line. (d) Time evolution of the absolute value of the net toroidal flux in each hemisphere, |Φ N H (t)| (blue solid line) and |Φ SH (t)| (red solid line), together with the predicted evolution obtained by integrating eq. (19) over the NH (blue dashed line) and SH (red dashed line). Dotted lines show the unsigned vertical flux |B r | at the surface integrated over the NH (blue) and SH (red). Compare with Fig. 3 in CS15.
Figure 12 :
12As in Fig. 6a but for Cases (a) L2 and (b) L3, spanning a century of simulated time. Saturation values for the color table are ± 50 G and ± 200 G respectively.
Figure 13 :
13As inFig. 7but for Case L3. The saturation level for the color table is ± 1kG.
By "disproportionate" we do not mean to imply that subsurface fields are not important; they are of course essential to sustain the dynamo. We merely mean that surface fields appear to play a greater role in the dynamo than might be expected given their contribution to the total magnetic energy in the CZ and tachocline, which is thought to be relatively small.
The authors wish to thank the NCAR Advanced Study Program for funding KT's visit to HAO/NCAR in support of this research. We also thank Mausumi Dikpati, Gopal Hazra, Bidya Karak, and Lisa Upton for many enlightening conversations about this project. The computations were performed using resources provided by NASA's High End Computing (HEC) program (Pleiades) and NCAR (Yellowstone). The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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| {'fraction_non_alphanumeric': 0.0538045136573734, 'fraction_numerical': 0.0323889384881395, 'mean_word_length': 4.1252823631624675, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The main objective of this paper is to introduce the STABLE (Surface flux Transport And Babcock-LEighton) solar dynamo model. STABLE is a 3D Babcock-Leighton/Flux Transport dynamo model in which the source of poloidal field is the explicit emergence, distortion, and dispersal of bipolar magnetic regions (BMRs). Here we describe the STABLE model in more detail than we have previously and we verify it by reproducing a 2D meanfield benchmark. We also present some representative dynamo simulations, focusing on the special case of kinematic magnetic induction and axisymmetric flow fields. Not all solutions are supercritical; it can be a challenge for the BL mechanism to sustain the dynamo when the turbulent diffusion near the surface is ≥ 10 12 cm 2 s −1 . However, if BMRs are sufficiently large, deep, and numerous, then sustained, cyclic, dynamo solutions can be found that exhibit solar-like features. Furthermore, we find that the shearing of radial magnetic flux by the surface differential rotation can account for most of the net toroidal flux generation in each hemisphere, as has been recently argued for the Sun byCameron and Schüssler (2015).', 'arxivid': '1511.03613', 'author': ['Mark S Miesch es:[email protected] \nHigh Altitude Observatory\nNational Center for Atmospheric Research\n3080 Center Green Dr80301BoulderCO\n', 'Kinfe Teweldebirhan \nHigh Altitude Observatory\nNational Center for Atmospheric Research\n3080 Center Green Dr80301BoulderCO\n\nPhysics Department\nFaculty of Natural Sciences\nAddis Ababa University\nBox 1176Addis AbabaEthiopia\n'], 'authoraffiliation': ['High Altitude Observatory\nNational Center for Atmospheric Research\n3080 Center Green Dr80301BoulderCO', 'High Altitude Observatory\nNational Center for Atmospheric Research\n3080 Center Green Dr80301BoulderCO', 'Physics Department\nFaculty of Natural Sciences\nAddis Ababa University\nBox 1176Addis AbabaEthiopia'], 'corpusid': 119115668, 'doi': '10.1016/j.asr.2016.02.018', 'github_urls': [], 'n_tokens_mistral': 27021, 'n_tokens_neox': 23255, 'n_words': 14944, 'pdfsha': '19add9b04ac9bbee00bfb0bf68d40f81b34d0bc5', 'pdfurls': ['https://arxiv.org/pdf/1511.03613v2.pdf'], 'title': ['A Three-Dimensional Babcock-Leighton Solar Dynamo Model: Initial Results with Axisymmetric Flows', 'A Three-Dimensional Babcock-Leighton Solar Dynamo Model: Initial Results with Axisymmetric Flows'], 'venue': []} |
arxiv |
Faculty of Physics, Materials and Devices for Electronics and Optoelectronics Research Center
University of Bucharest
P.O. Box MG-11077125Magurele-IlfovRomania
BALLISTIC ELECTRON TRANSPORT IN WRINKLED SUPERLATTICES T.L. Mitran, G.A. Nemnes, L. Ion and Daniela Dragoman
Inspired by the problem of elastic wave scattering on wrinkled interfaces, we studied the scattering of ballistic electrons on a wrinkled potential energy region. The electron transmission coefficient depends on both wrinkle amplitude and periodicity, having different behaviors for positive and negative scattering potential energies. For scattering on potential barriers, minibands appear in electron transmission, as in superlattices, whereas for scattering on periodic potential wells the transmission coefficient has a more complex form. Besides suggesting that tuning of electron transmission is possible by modifying the scattering potential via voltages on wrinkled gate electrodes, our results emphasize the analogies between ballistic electrons and elastic waves even in scattering problems on non-typical configurations.
I. INTRODUCTION
Devices that are able to precisely control charge transport have been one of the main goals of solid state physics, and, presently, of nanoelectronics. One of the methods to accomplish this is the fabrication of nanosize artificial structures, such as superlattices. The engineering of these artificial structures can be paralleled to the development of photonic crystals in optics and phononic crystals in periodic solid state materials, which evolved due to the need to control the propagation of electromagnetic and, respectively, elastic or heat waves at given frequencies. Indeed, due to the well-known analogies between ballistic electrons and light waves [1,2], between electrons and phonons [3] or acoustic waves [4], and between phonons and electromagnetic waves (see, for example, [5] and the references therein), almost any conceptual development or application in one of these domains, i.e. nanoelectronics, optics or phononics, can be adapted to the others. Among many examples, we mention here (besides the emergence of photonic and phononic crystals in analogy to crystalline solids) only the universal conduction fluctuations of light [6], the photonic classical [7] and quantum [8] Hall effects, the photonic [9] and acoustic spin Hall effect [10], and the observation of graphenelike Dirac points in photonic crystals [11].
In this paper, we study an electronic analog of a recently proposed method to control the elastic wave propagation in layered media with interfacial wrinkling [12]. In particular, we are interested in the effect of tunable wrinkled scattering potentials on the transport properties, especially on the transmission coefficient, of two-dimensional semiconductors. The investigated systems consist of a rectangular scattering region connected to two ideal leads, and subject to an oscillating scattering potential parallel to the direction of the leads. In principle, in this configuration, the potential energy, as well as the number, width and periodicity of wrinkles can be modified. Such a wrinkled scattering region can be implemented by a meandering gate, which transforms into a straight gate in the leads, in order to preserve the continuity of the system. The transmission function is studied in all cases for different values of scattering potentials and electron energies by using the scattering formalism of the R-matrix method.
It should be noted that although meandering gates have been used before in ionsensitive field-effect transistors [13], in plasmonic THz detectors [14], to enhance the photoresponse at the second harmonic of the cyclotron resonance in a two-dimensional electron gas [15], or to maximize the active area of graphene-on-MoS 2 capacitors [16], no study on ballistic electron transport in such structures has been performed up to now. Our results suggest that by tuning the amplitude and period of the wrinkles, as well as by modifying the scattering potential, it is possible to adjust the transmission coefficient at the desired value. Such an adjustment is not possible for a straight potential barrier of the same width.
II. THEORETICAL BACKGROUND
The R-matrix formalism was developed by Wigner and Eisenbud in 1947 as a nuclear scattering model [17], but has gained attraction as an efficient numerical means of simulating electron scattering in semiconductors physics [18,19], being applied to the study of nanoscale transistors [20][21][22], the thermoelectrical properties of nanowires [23] and to spin transport [24].
For the present study, it was chosen as the simulation environment for quantum transport because of its numerical efficiency, which is also described in [25]. This numerical speed gain is obtained by separating the problem into an energy-independent part that solves the Wigner-Eisenbud eigenvalue problem and a second one, in which the transmission is computed at each energy value. Such a method is well suited for systems in which the transmission function varies rapidly with the energy, and has narrow peaks. The system is split into a scattering region, with an spatially varying potential in both x and y directions, and leads that act as source and drain, where the potential is varied only perpendicular to the direction of transport (Fig. 1). Electrons originating from the source (left lead) interact with the potential from the scattering region by obeying the stationary Schrödinger equation in the effective mass approximation, Fig. 1. Schematic representation of the scattering region and leads for the 2D system before continuing through to the drain (right lead). The one-particle Hamiltonian in the effective mass approximation is
) ; ( ) ; ( E r E E r H r r Ψ = Ψ ,(1)) ( * 2 2 r W m H r h + Δ − = (2)
where m* denotes the effective mass and ) , ( y x r = r is the position vector.
As seen in Fig. 1, the scattering region contains an oscillating, or wrinkled, potential with a constant width of 1 nm that is continued in the translational invariant leads by a nonoscillating potential of identical width. The variable parameters of the wrinkled potential inside the scattering region are: the geometrical amplitude of the oscillation A, its period λ, and the (positive or negative) value of the applied potential, denoted by . Throughout this paper we consider configurations containing 5 wrinkles, such that the length of the wrinkled potential region is
0 V λ 5 .
Inside the invariant leads, the solution of the Schrödinger equation has the form: The complex coefficients , are related through the scattering S-matrix by . The S-matrix can also be used to obtain the total transmission between the source and drain: , where
) ( ) exp( ) ( ) exp( ) ; ( s i s out s i s in s s y x ik y x ik E r ν ν ν ν ν ν Φ Ψ + Φ − Ψ = Ω ∈ Ψ ∑ ∑ r ,(3)ν ⊥ E in ν Ψ out ν Ψ in out SΨ = Ψ r r ∑ = ' , 2 ' ' | ) ( | ) ( i i ss E S E T νν 2 / 1 2 / 1 − = Sk k S
and the summation is only over open channels. In the R-matrix formalism, the S-matrix can be written as:
Rk m i Rk m i S * 1 * 1 − + − = (4) where ' ' , ) ( νν ν ν ν δ k k = and the R-matrix is expressed as ∑ ∈ − − = ∞ =0 ' * 2 ' ) ( ) ( 2 ) ( ) ( l l l l E E R ν ν νν χ χ h . (5) with ) ( ) ( ) ( s l s s l r y d s Γ ∈ Φ Γ = ∫ Γ r χ χ ν ν ,(6)l ∈ 0 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ Γ s s l x χ .(7)
It is important to note that even though the determination of the Wigner-Eisenbud energies and functions ( l χ , ) and the overlap integrals l ∈ ν χ ) ( l is the most time consuming step, it is only done one time since it is energy independent. After that, the R-matrix can be computed for each energy from eq. (5) by performing a summation over the Wigner-Eisenbud indexes.
The S-matrix is determined by using eq. (4) and finding the inverse of the [1 − i/m*Rk] matrix.
III. RESULTS AND DISCUSSION
The system under investigation is a two-dimensional electron gas in the form of a nanoribbon, with a width w = 20 nm and length L = 120 nm, in which the effective electron mass is m* = 0.0655 m 0 . This effective mass value is characteristic for GaAs, which can form a twodimensional electron gas at the interface with AlGaAs (see, for example, [26]). Note that L is not the length of the wrinkled region (which equals λ [27]. The same appearance of bandgaps upon wrinkling initiations was observed also in the case of elastic waves [12]. A positive V 0 for electrons seems thus to be correlated with a higher stiffness of the deformable interfacial layers in [12].
The application of a scattering potential, which induces a transverse non-homogeneous potential energy distribution, does not modify just the transmission coefficient, but also the wavefunction in the leads. Examples of wavefunction propagation, more precisely of the non- The effect on the propagation of elastic waves of the periodic structure with smaller stiffness of the deformable interfacial layers was not studied in [12]. This situation could correspond to a negative V 0 value, for which the dependence of electron transmission on E and V 0 is quite complex, as seen from Fig. 3. The periodic scattering structure with wells instead of barriers is characterized by interferences not only between quantum wavefunctions scattered of adjacent wells but also by quantum interferences in the wells. The superposition of all such interferences, with different origins, forms the complex pattern in Fig. 3.
To better understand the differences between the effect of periodic barriers and wells on quantum wavefunction scattering in a nanoribbon we performed a simpler simulation of a corresponding one-dimensional scattering problem, in which the wrinkled scattering region was replaced by stripes with the same width and position as encountered by electrons
propagating at different y values in Fig. 1. The one-dimensional configuration is represented in Fig. 6(a)
IV. CONCLUSIONS
In summary, inspired by a similar problem for elastic wave scattering, we have studied the scattering of ballistic electrons on a wrinkled potential energy region. The transmission coefficient behavior on the electron energy and the amplitude of the potential depends on both wrinkle amplitude and periodicity. For scattering on periodic potential barriers, minibands appear in electron transmission, as in any superlattice, whereas for scattering on periodic potential wells the transmission coefficient has a more complex form. For a scattering potential/gate electrode of a specific form, the value of the transmission coefficient can be tuned to a desired value by modifying the applied gate voltage. Such tuning is not possible in a structure with a straight scattering region with the same width.
Apart from studying the effect on electron transmission of a wrinkled scattering potential, our results emphasize once more the analogies between ballistic electrons and elastic waves. In this respect, the similarities between bandgap formation in wrinkled structures have been observed for scattering on potential barriers for electrons and higher stiffness of interfacial layers for elastic waves. The situation in optics would correspond to scattering on regions with a smaller refractive index. The opposite situation, not considered for the elastic wave case, has revealed a complex behavior for ballistic electrons. The corresponding situation in optics, that of scattering on regions with a higher refractive index, although not studied in a configuration similar to Fig. 1, at least to the authors' knowledge, would imply anti-resonant reflections, as in ARROW-like structures (see [28], for example).
where and are complex coefficients corresponding to the lead with index s. The composite index ν = (s, i) denotes the lead index s and the channel index i. The coordinate system was chosen such that the x axis points away from the scattering region. The scattering boundary conditions for an electron in channel ν are found by setting the perpendicular modes obtained from the one-dimensional Schrödinger equation.
are the Wigner-Eisenbud functions and energies, and Ω 0 represents the scattering region. These are the eigenfunctions and eigenvalues of the Hamiltonian in eq. (1) with new and fixed boundary conditions on the boundary of the scattering region with lead s, or Γ s :
0 V 0 VFig. 2 .Fig. 3 .
0023of this region. The constant amplitude of the sinusoidal electrostatic potential can be varied by modifying the gate voltage, and is considered in our simulations to vary between -1 eV and 1 eV. Using the R-matrix formalism, we have calculated the transmission coefficient as a function of the electron energy E and potential , for the cases when the amplitude A and the period λ of the wrinkles vary. The results are plotted in Fig. 2 for the case A = 0 (straight gate) and in Fig. 3 for A = 3, 6, and 9 (upper, central and lower rows, respectively) and λ = 16 nm, 18 nm and 20 nm (left, centre and right columns, respectively). Transmission coefficient in the case of straight scattering potentials (A = 0) as a function of electron energy (E) and scattering potential (V 0 ) In all cases, in the region in the lower right corner of the figures, with white color, electron propagation is prohibited by the transversal constraint/quantum confinement, which imposes a minimum energy value for propagation in the first allowed mode. Note that this minimum energy depends on the confining potential; it increases as V 0 increases and vanishes at V 0 values smaller than about -0.12 eV when the transverse spatial confinement of the electron propagation in the nanoribbon is no longer apparent. Indeed, the application of a potential energy on the 1-nm-wide straight continuation of the wrinkled region inside the leads modifies the effective nanoribbon width; the nanoribbon is effectively wider for negative V 0 values and effectively narrower for positive values of this parameter. Transmission coefficient as a function of electron energy (E) and scattering potential (V 0 ) for different combinations of amplitudes (A) and periods (λ) of the scattering potential From Figs. 2 and 3 it can be seen that the transmission coefficient has a totally different behavior for the cases of a straight and a wrinkled scattering potential. In the first case the transmission increases uniformly with the number of open channels, as expected. On the other hand, for a wrinkled scattering potential the transmission coefficient has a complex dependence on E and V 0 , with a clear difference in behavior for negative and positive values of V 0 . The behavior of the transmission coefficient for positive V 0 can be easily understood in terms of appearance of minibands in the structure containing periodic potential barriers. The width of the allowed minibands decreases and their central position is shifted towards higher energies as the barrier height increases
normalized probability distribution, in the same wrinkle configuration, for the fundamental mode at two different electron energies, are represented in Figs. 4 and 5 for the situation when V 0 = -0.5 eV and 0.5 eV, respectively. These figures illustrate the effect of the narrow and straight continuation of the wrinkled region on the form of the fundamental mode in the leads. For the same energy values of incident electrons, the form of the wavefunction depends on whether the applied potential is attractive (negative V 0 values) or repulsive (positive V 0 ); the applied potential changes the form of wavefunctions since it is not applied on the whole width of the nanoribbon but only in its central part. Also, the wrinkled scattering potential influences the form (not only the y-dependence of the amplitude, but also of the phase) of the transmitted wavefunction. For example, in the upper Fig. 4 constructive interference/wavefunction maxima seems to occur near the center of the nanoribbon before the scattering region, while after scattering destructive interference is observed close to the nanoribbon center. Again, in the upper Fig. 5, the transmitted wavefunction, while having the same general form, has a different oscillation period along the x axis.
Fig. 4 .Fig. 5 .
45Wavefunction representation for an attractive scattering potential Wavefunction representation for a repulsive scattering potential
, where the gray line illustrates the wrinkled region and its straight continuation in the left lead, the vertical thick solid (dashed) black lines represent the positions of scattering regions for electrons propagating along trajectories for which y = 0 ( 0 ≠ y ); these trajectories are illustrated with thin horizontal black lines of the same type. Note that the width of the scattering regions for electrons with 0 ≠ y is wider than for electrons propagating along y = 0, and the position of the scattering regions is not equidistant in a period λ, as is the case for y = 0.
Fig. 6 .From
6Schematic representation of the 1D system (a) and the transmission as a function of electron energy (E) and scattering potential (V 0 ) for trajectories along y = 0 (b), (c), (d) and (e), and the mean value of (b)-(e) in (f) since in the one-dimensional problem the continuation of the wrinkled region in the leads was not taken into account, the white color region representing prohibited electron propagation by the transversal constraint/quantum confinement has the same width, irrespective of the V 0 potential value applied on the transverse scattering region. The width of the white-color-region is the same as that in Figs. 3 for V 0 = 0, i.e. for a lead with a homogeneous (and zero valued) potential distribution. Fig. 6(b) it follows also that the transmission coefficient T has similar minibands for positive and negative V 0 values. However, the effect of barriers and wells (positive and negative values of the scattering potential) on electron propagation is more dissimilar for increasing y values, for which the effective width of the scattering potential region. In particular, for wide enough wells distinct interferences form inside them, which superimpose on the interferences due to adjacent wells, the result being the complex interference pattern in Fig. 6(e). In this figure, the transmission for positive V 0 is inhibited by the wide barriers, while for negative V 0 values the wide transmission bands do not monotonously decrease in width with the applied potential (as in Figs. 6(b)-6(d)) because of the influence of constructive interferences forming inside the wide-enough wells. This simple one-dimensional model can provide a helpful insight in the behavior of the transmission coefficient of the two-dimensional problem. More precisely, assuming that the total transmission coefficient is just a sum of the contributions for different y values, the form of the mean value of T in Figs. 6(b)-6(d), represented in Fig. 6(f), is similar to the dependence of T on E and V 0 in the two-dimensional problem. This similarity is quite remarkable taking into account the drastic simplification of the scattering geometry. Of course, the simple one-dimensional cannot capture all the details of the two-dimensional scattering problem, in particular the dependence of the white region on the value of the scattering potential.
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| {'fraction_non_alphanumeric': 0.05584857507443641, 'fraction_numerical': 0.028030625265844322, 'mean_word_length': 4.176354029062087, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Inspired by the problem of elastic wave scattering on wrinkled interfaces, we studied the scattering of ballistic electrons on a wrinkled potential energy region. The electron transmission coefficient depends on both wrinkle amplitude and periodicity, having different behaviors for positive and negative scattering potential energies. For scattering on potential barriers, minibands appear in electron transmission, as in superlattices, whereas for scattering on periodic potential wells the transmission coefficient has a more complex form. Besides suggesting that tuning of electron transmission is possible by modifying the scattering potential via voltages on wrinkled gate electrodes, our results emphasize the analogies between ballistic electrons and elastic waves even in scattering problems on non-typical configurations.', 'arxivid': '1507.08616', 'author': ['\nFaculty of Physics, Materials and Devices for Electronics and Optoelectronics Research Center\nUniversity of Bucharest\nP.O. Box MG-11077125Magurele-IlfovRomania\n'], 'authoraffiliation': ['Faculty of Physics, Materials and Devices for Electronics and Optoelectronics Research Center\nUniversity of Bucharest\nP.O. Box MG-11077125Magurele-IlfovRomania'], 'corpusid': 117213738, 'doi': '10.1016/j.physe.2016.03.003', 'github_urls': [], 'n_tokens_mistral': 6977, 'n_tokens_neox': 6017, 'n_words': 3791, 'pdfsha': '49a7266949efde040240529d8980b8e6b643ffff', 'pdfurls': ['https://arxiv.org/pdf/1507.08616v1.pdf'], 'title': [], 'venue': []} |
arxiv |
Existence and asymptotic large time behavior of singular solutions of the fast diffusion equation
Of Hong Kong
THE UNIVERSITY
Institute of Mathematical Research Department of Mathematics
Institute of Mathematics
Academia Sinica
Taiwan
ProfessorKin Ming Hui
THE UNIVERSITY
Institute of Mathematical Research Department of Mathematics
Institute of Mathematics
Academia Sinica
Taiwan
Existence and asymptotic large time behavior of singular solutions of the fast diffusion equation
Date: September 26, 2017 (Tuesday) Time: 11:00am -12:00noon Venue: Room 210, Run Run Shaw Bldg., HKU All are welcome | {'fraction_non_alphanumeric': 0.01990049751243781, 'fraction_numerical': 0.028192371475953566, 'mean_word_length': 5.494623655913978, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Date: September 26, 2017 (Tuesday) Time: 11:00am -12:00noon Venue: Room 210, Run Run Shaw Bldg., HKU All are welcome', 'arxivid': '1508.01980', 'author': ['Of Hong Kong \nTHE UNIVERSITY\nInstitute of Mathematical Research Department of Mathematics\nInstitute of Mathematics\nAcademia Sinica\nTaiwan\n', 'ProfessorKin Ming Hui \nTHE UNIVERSITY\nInstitute of Mathematical Research Department of Mathematics\nInstitute of Mathematics\nAcademia Sinica\nTaiwan\n'], 'authoraffiliation': ['THE UNIVERSITY\nInstitute of Mathematical Research Department of Mathematics\nInstitute of Mathematics\nAcademia Sinica\nTaiwan', 'THE UNIVERSITY\nInstitute of Mathematical Research Department of Mathematics\nInstitute of Mathematics\nAcademia Sinica\nTaiwan'], 'corpusid': 119142048, 'doi': '10.3934/dcds.2017258', 'github_urls': [], 'n_tokens_mistral': 179, 'n_tokens_neox': 145, 'n_words': 83, 'pdfsha': '4cf8b2a91b544bebe59e6cbcc227440d3e610e79', 'pdfurls': None, 'title': ['Existence and asymptotic large time behavior of singular solutions of the fast diffusion equation', 'Existence and asymptotic large time behavior of singular solutions of the fast diffusion equation'], 'venue': []} |
arxiv |
Non-Hermitian Hamiltonians for Linear and Nonlinear Optical Response: a Model for Plexcitons
Daniel Finkelstein-Shapiro
Instituto de Química
Universidad Nacional Autónoma de México
CDMX
México
Pierre-Adrien Mante
Division of Chemical Physics and Nanolund
Lund University
Box 124221 00LundSweden
Sinan Balci
Department of Photonics
Izmir Institute of Technology
35430IzmirTurkey
Donatas Zigmantas
Division of Chemical Physics and Nanolund
Lund University
Box 124221 00LundSweden
Tõnu Pullerits
Division of Chemical Physics and Nanolund
Lund University
Box 124221 00LundSweden
Non-Hermitian Hamiltonians for Linear and Nonlinear Optical Response: a Model for Plexcitons
In polaritons, the properties of matter are modified by mixing the molecular transitions with light modes inside a cavity. Resultant hybrid light-matter states exhibit energy level shifts, are delocalized over many molecular units and have a different excited-state potential energy landscape which leads to modified exciton dynamics. Previously, non-Hermitian Hamiltonians have been derived to describe the excited states of molecules coupled to surface plasmons (i.e. plexcitons), and these operators have been successfully used in the description of linear and third order optical response. In this article, we rigorously derive non-Hermitian Hamiltonians in the response function formalism of nonlinear spectroscopy by means of Feshbach operators, and apply them to explore spectroscopic signatures of plexcitons. In particular we analyze the optical response below and above the exceptional point that arises for matching transition energies for plasmon and molecular components, and study their decomposition using double-sided Feynman diagrams. We find a clear distinction between interference and Rabi splitting in linear spectroscopy, and a qualitative change in the symmetry of the lineshape of the nonlinear signal when crossing the exceptional. This change corresponds to one in the symmetry of the eigenvalues of the Hamiltonian. Our work presents an approach for simulating the optical response of sublevels within an electronic system, and opens new applications of nonlinear spectroscopy to examine the different regimes of the spectrum of non-Hermitian Hamiltonians.
INTRODUCTION
The coupling between light modes and radiative transitions of matter creates exciton polariton states, i.e. half-light and half-matter states. These were first demonstrated in atoms inside a microwave cavity and were used to study decoherence and quantum logic operations in atoms [1][2][3][4]. More recently, polaritonic states have also been observed in molecules in microcavities, having both infrared and visible wavelengths. They open new paradigms with which to influence chemical processes [5][6][7][8][9][10][11][12][13][14][15][16]. For example, the delocalization of polaritonic states is advantageous for long-distance energy transfer [11,[17][18][19][20][21][22], polaritonic chemistry [23][24][25][26] and photocatalysis [27,28]. They can also be used for lasing and to study nonequilibrium condensates and phase transitions [29,30]. Using plasmonic nanoparticles instead of cavities confines the light to subwavelength dimensions. This allows creating billions of polaritonic systems in solution [31]. It allows many more interesting arrangements that exhibit strong coupling in the single or few molecule limit [32].
The number of molecules in the interaction volume and the properties of the cavity determine the light-matter coupling strength, while the quality of the cavity, and the molecular dissipative processes determine the coherence lifetime. The coupling strength and coherence lifetimes span a large parameter space that can be broken into different regimes, which are also associated to different phenomena [33,34]. For example, for weak coupling we observe a Purcell enhancement of the fluorescence rate while for strong coupling we see a Rabi splitting, which is reflected in a splitting of the absorption into two peaks termed upper and lower polariton branches. However, such an apparent splitting is also possible in the presence of an interference that appears already in the weak coupling regime (also called electromagnetically induced transparency, or a Fano interference [34]). Finally, the ultrastrong coupling and deep strong coupling regimes are associated to the breaking of the rotating-wave approximation and are difficult to reach using molecules in cavities, but have been observed with plasmonic lattices [35].
It is important to discuss dissipation when considering polariton transitions. While in atoms the sources of dissipation are few and well understood, an intense experimental and theoretical effort has been necessary to assign the timescale and processes of coherent and incoherent dynamics in molecular polaritons [31,[36][37][38][39][40][41]. Briefly, time-resolved spectroscopy experiments have detected coherent dynamics in the form of Rabi oscillations via transient absorption [38], and established the possible relaxation pathways: the upper branch can decay via radiation damping to the ground state [42], inject into the dark plexciton states [43,44], or decay directly into the lower branch [45]. The dark states themselves can decay to the lower polariton branch, with some degree of back-transfer (from lower polariton to dark reservoir) taking place [46]. The long lifetime of the dark states is responsible for the excited state stabilization with decay times for up to several µs [43,47,48]. In systems, where the cavity mode is confined to a plasmonic nanoparticle, the dynamics are strongly influenced by electron-electron and electron-phonon scattering inside the metal nanoparticle. Notably, a direct coupling between the molecular dark states and metal surface states appears to limit the lifetime of the dark states to that of the electron-electron scattering time [41].
Simulations of the optical response are crucial to correctly interpret the experiments, and more so in time-resolved spectroscopy experiments such as two-dimensional coherent spectroscopy (2DCS) or transient absorption. Coupled oscillator models work well to understand linear optical absorption [45]. For transient absorption they can also work as long as the experiment can be simulated as two linear experiments, the bleach of the ground state, and the absorption of an excited state dominated by the signal from the remaining molecules in the ground state. This is true when the main nonlinearity is a Rabi contraction whereby the Rabi splitting of the ground state is reduced because of the molecules that have been excited [38]. However, they are not sufficient for simulating a third-order signal as in general all orders of the field appear together and it is not straightforward to disentangle them. Input-output theory has also been used to compute with a very good agreement the experimental two-dimensional infrared spectroscopy (2DIR) signal of molecules in infrared cavities [25]. The Rabi contraction and softening of the vibrational mode (for 2DIR) have been proposed as the mechanisms of nonlinearity. Similar physics has been found in polaritons formed from molecules adsorbed on plasmonic lattices [38]. In an earlier work, we have simulated the two-dimensional electronic spectroscopy response of molecules coupled to plasmonic nanoparticles using an extension of the response function formalism to nondiagonal non-Hermitian Hamiltonians, successfully reproducing the spectra at early times using a Rabi contraction nonlinearity and at late times using a thermal expansion of the nanoparticle [41]. Gu et al. have also employed a non-Hermitian Hamiltonian to calculate several nonlinear experiments, finding significant modifications of the energy levels as well as the selection rules [49]. Models of disordered polaritons have suggested the existence of exceptional points in these non-Hermitian Hamiltonians which vary according to the degree of disorder [50] and these have been measured experimentally in infrared cavities [51]. Non-Hemiticity of the Hamiltonian can arise from a number of different scenarios, not only due to energy relaxation but also from particle decay for example in photoionization. Moiseyev et al. has proposed a more realistic model of molecules including the ionized continuum states, and studied their modification inside a cavity [52].
There are two difficulties related to the description of the dynamics of hybrid plasmonic systems. First, the description of the optical response in terms of non-diagonal non-Hermitian Hamiltonians is not typical in the approach for calculating 2D spectra in molecular systems. Usually an excitonic basis is used, where diagonal fluctuations induced by the bath are added and result in a lineshape function after bath-averaging [53]. Second, plasmonic systems add a rich dimension of dynamics in terms of the non-equilibrium distribution of hot electron-hole pairs, formed inside the metallic band structure after plasmon decay. The dynamics inside the metallic band observed in transient absorption experiments has been beautifully demonstrated in pure metals [54], however the description has not yet been extended to mixed molecule-metal systems. One complication is the co-existence of coherent effects related to the Rabi oscillation between two discrete transitions along with the incoherent scattering of continuous non-equilibrium distribution of electrons in the metal. The latter phenomena is described by the dielectric constant of an electron gas whose temperature is described by the two-temperature model [55] and has not been adapted to handle the molecular states.
In this work, we consider the linear and nonlinear optical response of non-diagonal non-Hermitian Hamiltonians. We begin by providing a derivation of the non-Hermitian Hamiltonian and discuss its regime of validity. We then describe the decomposition of the linear and third-order optical response using double-sided Feynman diagrams. We conclude with a discussion of the polariton branch energy structure in the weak and strong coupling regimes, and their distinctive spectral signatures.
THEORETICAL DESCRIPTION
Effective operators. Non-Hermitian Hamiltonians have been derived as effective operators for molecules coupled to plasmonic nanoparticles from ab initio considerations [56].
The non-Hermiticity arises from an implicit inclusion of certain degrees of freedom of the system. Regardless of the procedure to obtain the effective operators, one is left with a 2 × 2 Hamiltonian describing a plasmon excitation coupled to a molecular excitation where each of these two transitions is associated to a separate decay channel. In the site basis, consisting of the molecular bright state and the plasmon transition, the Hamiltonian is written as (setting = 1)
H eff = ω J − iγ J g g ω P − iγ P (1)
where g is the coupling between the bright molecular transition and the plasmon transition, the P and J indices denote plasmon or molecular J-aggregate, and ω i , γ i relate to the transition frequencies and dephasing rates, respectively. We impose γ J < γ P . The Hamiltonian of equation (1) admits an exceptional point when ω J = ω P ≡ ω 0 and g = ∆γ 2 where ∆γ = γ P − γ J [57], and which sets the limit between weak and strong coupling.
We can easily solve for the eigenvalues ω ± and eigenvectors v ± of the Hamiltonian as:
ω ± = ω 0 − i γ J + γ P 2 ± g 2 − (∆γ/2) 2 ≡ ω 0 − iγ ± ξ(2)
and v ± = 1
(±ξ + i∆γ/2) 2 + g 2 ±ξ + i∆γ/2 g (3)
where we will denote by + and − the upper and lower polariton branches, respectively.
The transition dipole moments are calculated as
µ ±g = v T ± µ J µ P
where µ P and µ J are the transition dipole moments of the plasmon and bright molecular transition, respectively. We assume that the coupling g is real. ξ can be real or imaginary and the transition from one to the other appears at the exceptional point. Given that the matrix in Eq. (A.10) is symmetric, we have µ ±g = µ g± ≡ µ ± . For µ P = 0, µ J = 0, we have µ ± = µ P g √ (±ξ+i∆γ/2) 2 +g 2 . The denominator can be rewritten as g (−iδ ± √ 1 − δ 2 ) 2 + 1 where δ = ∆γ 2g . Doing a perturbative expansion on either side of the exceptional point δ = 1 ± for small , we have to leading order µ ± ≈ ± √ 2 in the weak coupling regime, making one transition dipole moment purely real, and the other purely imaginary. In the strong coupling limit, to leading order we have µ ± ≈ 2 ∓ i √ 2 which leads to the relation µ 2 + = (µ 2 − ) * . These relations will be important when deriving the symmetries of the signals. The limits presented here, either µ J = 0 or µ P = 0 should be physically understood as µ J µ P or µ P µ J since otherwise there would be no dipolar coupling between molecules and plasmons.
The Hamiltonian from Eq. (1) can be derived from classical arguments [57] or be obtained from the quantization of the modes of the electromagnetic field sustained by a spherical metallic particle [56]. In Appendix A we show a derivation suitable for the parameters of plexcitons explored earlier [41] while in Appendix B we show a derivation obtained by projecting out continuum states. However, the effective non-Hermitian operator remains an incomplete description for time-resolved experiments as it cannot capture the dynamics occurring in the implicit degrees of freedom, i.e. the degrees of freedom that when removed induce the non-Hermiticity. In our case these correspond to the excited electron-hole pairs in the metallic conduction band. The non-Hermitian can be used for calculating the linear response, but it is not justified for higher-order signals. The dephasing of the optical coherence can give a width to the lineshape, however when this width arises for instance from decay into a different excitation (i.e. electron-hole states), these will contribute to the excited-state absorption in a way that cannot be handled by the non-Hermiticity alone. We provide a justification for its use under restricted conditions in the next section, and derive the expressions applicable to the general case.
Optical response. The density matrix to n−th order in the light-matter interaction ρ (n) (t) is given by (omitting the hat symbol for operators [58]):
ρ (n) (t) = i n dt n dt n−1 · · · dt 1 × E(t − t n )E(t − t n − t n−1 )...E(t − t n − ... − t 1 ) × [µ(t n−1 + ... + t 1 ), [µ(t n−2 + ... + t 1 ), ...[µ(0), ρ(−∞)]] B(4)
where µ(t) = e +iH 0 t µe −iH 0 t is the transition dipole moment operators in the interaction picture, H 0 is the field-free Hamiltonian, E(t) is the electric field from the impinging radiation and B is the average over the realizations of the bath. We define the Feshbach operators P and Q for the bright and dark partitions, respectively. An illustration of the transformation from molecular and plasmonic partitions to bright and dark partitions is shown in Figure 1 (details are found in Appendix A).
To remove the explicit description of the dark modes in Eq. (4), we derive the elements needed to calculate the nested commutators in terms of effective operators. Because P delim- After diagonalization. Details are found in Appendix A its the bright partition, the optical response function will be contained in this partition. The needed terms of the n-th order response are of the form P µ(t n )...µ(t 1 )P ρ(t 0 )P µ(t 1 )...µ(t n )P which involve calculating terms of the form P µ(t n )...µ(t 1 )P . We only need to calculate explicitly the first two terms (and the rest follow straightforwardly). Recognizing that µ = P µP (i.e. that µ is entirely contained in P ):
P µ(t 1 )P = P e iH 0 t 1 P µP e −iH 0 t 1 P P µ(t 2 )µ(t 1 )P = P e iH 0 t 2 P µP e −iH 0 t 2 (P + Q)e iH 0 t 1 P µP e −iH 0 t 1 P
The required evolution operator e −iH 0 t can more easily be calculated in its resolvent form:
e −iH 0 t = −1 2πi dzG(z)e −izt(6)
where G(z) = [z − H 0 ] −1 , then for the terms appearing in Eq. (5):
P e −iH 0 t P = −1 2πi dzP G(z)P e −izt P e −iH 0 t Qe −iH 0 t 1 P = −1 4π 2 dz dz P G(z)Qe −izt QG(z )P e −iz t 1(7)
The resolvent approach allows us to calculate P G(z)P and P G(z)Q of the Hamiltonian in Eq. (A.6) using Lippman-Schwinger series. As we have done before, [59], we express all of the operators in terms of H eff (z) = P H eff (z)P = P H 0 P + P H 0 QG 0 (z)QH 0 P where
P G(z)P = [z − H eff (z)] −1 .
In principle, all of the processes that can occur inside the band structure such as electron-electron scattering, electron-phonon scattering appear in the operators that describe the metallic band QG 0 (z)Q. However, they are difficult to solve explicitly and go beyond the scope of this work, where we will constrain ourselves to the nonlinear response functions at the delay time between pump and probe T = 0. We thus neglect any scattering between two states k and k of the metal. This means that
QG 0 Q = [z − QH 0 Q] −1
is diagonal and thus easily invertible. To first order, we can make an approximation regarding the energy dependence of the states in partition Q referred to as the wideband approximation that involves an infinite flat continuum with energy independent couplings and a linear dispersion [60]. Within these approximations, terms containing P G(z)QG(z )P all vanish, and H eff (z) becomes z−independent. We can then express the optical response exclusively in terms of the effective operators in the bright partition:
µ(t) = P µ(t)P = P e iH 0 t P µP e −iH 0 t P = e iH † eff t µe −iH eff t(8)
In general, the effective operator H eff (z) is nonlinear in that it depends on the frequency parameter z. The physical meaning of this is that the particle density can transfer from the P partition to the Q partition (and back), which can play a role in the dynamics. This goes beyond the scope of the article, although in the description in the wideband approximation will be valid at short enough times. Before understanding the spectral signatures depending on the non-Hermiticity, we isolate the important regimes of Eq. (1) which can correspond to different symmetries of the eigenvalues. As mentioned, non-Hermiticity can arise in a number of settings, and is usually connected with a manifold of states that are not explicitly described. The physical effect of the imaginary part of the energy is to destroy (when negative) or create (when positive) particle density. The linear response of a system is not sensitive to what happens to the particle density that the non-Hermitian part of the Hamiltonian destroys. The fate of the particle density only becomes important in nonlinear response (most easily accessible in time-resolved experiments) when the particle density that leaves the system can come back into it. This can be captured exactly by Eq. 6,7, provided that the effective operators can be calculated exactly.
Phenomenology in different coupling strength regimes. The behavior of polaritonic states can be categorized in regimes according to the strength of the light-matter coupling, the magnitude of the dephasing rates and detuning [33]. We limit ourselves to coupling strengths for which the rotating-wave approximation is valid. For isoenergetic plasmonic and molecular transitions (ω P = ω J ), the non-Hermitian Hamiltonian (see Eq.
(A.1)) has an exceptional point at S = 2g/(γ P − γ J ) = 1 [57], taken as a boundary between the weak (S < 1) and strong (S > 1) coupling regimes. An additional dimensionless parameter, the cooperativity C = g 2 /γ P γ J has been proposed to separate the region of weak coupling C < 1 and the region where interference processes are present C > 1 [34,61]. We will systematically analyze the first and third-order optical response above and below the exceptional point with six parameter families cases marked in Figure 2 As has been pointed out, for example in [57], the behavior is different on either side of the exceptional point. We make this explicit by plotting the eigenvalues of Hamiltonian Feynman diagram formalism [59] in order to analyze the expected spectral signatures.
SIMULATIONS
We consider the n-th order response of the density matrix using Eq. (4) and calculate the signal of two-dimensional electronic spectroscopy and linear absorption in the impulsive limit [58].
Linear absorption. Linear optical absorption is a routine measurement for nanoparticles and can yield a wealth of information at a relatively inexpensive cost, compared to more advanced spectroscopies. In the case of plexcitons, it can reveal the coupling strength, relative magnitude of transition dipole moments and dephasing rates of each individual component. The signal can be obtained from Eq. (9) [41]: dip is only observed if it is the plasmon that couples predominantly to the far-field (as is usually the case). The depth of the minimum is modulated by the molecular dephasing rate, and the degree to which the molecular aggregate also couples to the far-field. The set of points D, E and F show that having the condition S > 1 is not enough to observe a peak splitting. In the region where S > 1 but C < 1 the system where the molecular aggregate couples more strongly to the field does not show a Rabi splitting (see Appendix E, Figure 10). While the features of destructive interference are more easily identified, there will also be regions of constructive interference away from the dip. Thus, both the cooperativity and strength parameters are important for classifying the features of the spectra. The difference between interference and Rabi splitting has also been observed in simulations with disordered molecules [50]. There, the narrower cavity absorption generates an interference pattern in the heterogeneously broadened molecular absorption while in the case presented here the narrow molecular absortpion generates an interference pattern in the homogeneously broadened plasmonic absorption.
S (1) ∝ Re dte iωtt a=± µ 2 a e −iωat = Re a=± −µ 2 a i(ω t − Re(ω a )) + Im(ω a ) = a=± −Re(µ 2 a )Im(ω a ) − Im(µ 2 a )(ω t − Re(ω a )) (ω t − Re(ω a )) 2 + (Im(ω a )) 2
Fano interferences vs. electromagnetically induced transparency. It is appropriate to clarify an important point concerning the interference process. This dip is often referred to either as a Fano interference, or an electromagnetically induced transparency (EIT), and both processes denote different physics and spectral signatures. The simplest system where EIT is observed is a three-level Λ system consisting of a ground state manifold with two levels, and one discrete excited state. Its spectral signature is a frequency region of suppressed absorption. The Fano interference appears in a structure akin to the Λ system where one of the levels becomes a continuous manifold of levels and the Fano lineshape is characterized by a distinctive asymmetric lineshape which also includes a frequency region of suppressed absorption [63]. The plexciton Hamiltonian is more closely related to a Λ system where instead of having two light-fields coupling each ground state with the excited state, we have an external light field coupling the ground state with the plasmon excitation, and radiative coupling connecting the excited plasmon with the excited J-aggregate ( Figure 5.a). Figure 5 shows the absorption spectrum for a Λ system and a Fano system as a function of the asymmetry parameter which is related to the ratio of the different coupling elements [63]. For Fano, the asymmetry parameter is q = µ J πµ P g . We choose the same definition for an effective asymmetry parameter for the Λ system although its physical meaning is not exactly the same [59]. For the Fano model, the limiting cases µ J = 0 (q = 0) and µ J → ∞ (q → ∞) correspond to anti-Lorentzian and Lorentzian lineshapes, respectively. For finite values of µ J (or q), the Fano system shows distinctive asymmetries that reverse sign as the sign of q is reversed (middle panel), while shifting the condition of destructive interference to the left or right of the zero detuning condition. The Λ system (Figure 5.a) shows a similar behavior for the limiting cases q = 0 and q = ∞, if we allow that far away from the resonance condition the absorbance vanishes instead of going to a finite value as in the Fano model. However, for finite values of the effective asymmetry parameter, and unlike the Fano model, we do not find an asymmetry. For finite q, the contrast at the point of destructive interference is reduced and the plots are identical for q and −q. This point will become important in the analysis of nonlinear signals. We mention that real systems often do show asymmetries, although this is due more to the detuning of the plasmonic transition with respect to the molecular bright transition, and not a true Fano asymmetry.
We now turn to a more detailed analysis of the different features that appear in the spectrum of third-order spectroscopies of non-Hermitian Hamiltonians.
Third order response. Two-dimensional electronic spectroscopy (2DES) is the most complete third-order spectroscopy. During the experiment, three pulses interact with a sample and an echo is detected (photon-echo 2DES [64]) or alternatively four-pulses interact with the sample and an excited state observable is detected, most often fluorescence (action-detected 2D [65,66]). Of the three characteristic times between the pulses in action detected 2D, the first and last can be Fourier-transformed to obtain two-dimensional plots of the system that show the correlation between excited and detected states at different population times (in photon-echo 2DES the echo is dispersed by a grating and one Fourier transforms along the first delay). By scanning the population time we can reconstruct the dynamics of the excited state. The complexity of the signal makes simulations crucial for their understanding, in particular for plasmonic-molecule systems where the signals can be unintuitive [59]. The induced polarization can be calculated by the same perturbative and strong cooperativity and strong coupling (c,f).
approach P (t) = Tr(µ(t)ρ (3) (t)) where:
ρ (3) = i 3 dt 1 dt 2 dt 3 E(t − t 3 ) × E(t − t 3 − t 2 ) × E(t − t 3 − t 2 − t 1 ) [µ(t 2 + t 1 ), [µ(t 1 ), [µ(0), ρ(−∞)](10)
The nested commutator implied by Equation (10) Feynman diagrams that are grouped into contributions denoted ground state bleach (GSB), stimulated emission (SE) and excited state absorption (ESA) (see Figure 6). The ground state bleach can be simulated using the fits from the ground state absorption spectrum, while the other two reflect properties of the excited state. The physics captured by the ESA is crucial for the observation of a non-zero nonlinear signal, since it exactly cancels the GSB and SE in linear (harmonic) systems [58].
We address in this article the effects of using a non-Hermitian Hamiltonian and not the physics behind the optical nonlinearities in plexcitons or polaritons which have been addressed by others [25,43]. Consequently, we will focus on the GSB contribution and
Diagrams for t 2 =0
Diagrams for early times analyze its structure for the different regimes exemplified by points A, B and C of Figure 2 and of points D, E and F in the Appendix C. The other pathways SE and ESA, however, share similar features and the conclusions related to the symmetry of the final signal remain unchanged (see Figure 10 in Appendix D for the real part of the total signal in the case of the Rabi contraction nonlinearity shown in Figure 6). In addition, we only consider the case where the plasmon couples predominantly to the far-field so that we can contrast the lineshapes of interferences with that of Rabi splitting. We can write the GSB contribution as We now analyze the effect of having both µ P = 0 and µ J = 0. Figure
S (3) ∝ i,a=± Im(ω a )(Im(µ 2 i µ 2 a )(ω t − Re(ω i )) + Re(µ 2 i µ 2 a )Im(ω i )) ((ω t − Re(ω i )) 2 + Im(ω i ) 2 )((ω τ − Re(ω a )) 2 + Im(ω a ) 2 )(11)
DISCUSSION
General comment on non-Hermitian Hamiltonians. The description of physical phenomena using non-Hermitian Hamiltonians is pervasive and extends far beyond the physical systems described here [67]. These can exist in metamaterials, engineered Floquet states and gain/loss media in general. The treatment described in the previous section applies to all of these systems as long as they have a non-zero nonlinear response. Our findings suggest that it is interesting to consider two-dimensional spectroscopy (or more generally higher orders) as tools that can distinguish between different symmetries of the eigenvalues by encoding them in symmetries of the signal.
Rigorously, a Lindblad operator should be used instead of a non-Hermitian Hamiltonian.
However, the quantum jump operator that restores the particle density back to the ground interference phenomena, and in particular to suggest its use for discerning between interference processes and Rabi splitting. We mention that it is not the only option and previous works that compare scattering and photoluminescence (PL) show a marked difference in the Rabi splitting with significant reductions in the PL spectra compared to the scattered light [34]. For regions of interference, the peak splitting entirely disappears from the PL spectrum while it is visible in the scattered spectrum [62,70], resulting in a diagnos-tic tool which can be applied in the case of few emitters coupled to a plasmonic nanoparticle.
Symmetries of the eigenvalues and symmetries of the signal. One of the findings of the paper is that the third-order signal is qualitatively sensitive to changes in the symmetry of the eigenvalues, unlike the first order signal. The symmetry of the linear and nonlinear signal with respect to reflection along ω τ = ω 0 and ω t = ω 0 can be analytically obtained by the symmetries in the eigenvalues and eigenvectors below and above the exceptional point. We define the detunings with respect to the transition frequencies as δ i = ω i − ω 0 , and the reflection operations that take δ τ → −δ τ and δ t → −δ t as O τ () and O t (), respectively. Below the exceptional point, we have the following properties of the eigenvalues and transition dipole moments: δ + = δ − = 0, µ + is purely real and µ − purely imaginary. It is straightforward to verify that O t (S (1) ) = S (1) . In the strong coupling limit, we have that (1) and the signal is symmetric with respect to the central transition frequency ω 0 .
µ 2 − = (µ 2 + ) * , so that it is also verified that because δ − = −δ + , O t (S (1) ) = S
In a similar fashion we can calculate the symmetries of the third order signal described by equation (11) In this Appendix, we derive the effective Hamiltonian corresponding to a plexciton. We consider similar parameters to a physical sample we have measured previously and base our approximations on its values (see [41] for the parameters).
H = H molecule + H metal + H metal−molecule + H field + H fluctuations (t) H molecule = N i ω 0 a † J,i a J,i + i,j∈n.n. Ka † J,i a J,j H metal = ω P a † P a P + dkω k a † k a k + dk(V k a † k a P + V * k a † P a k ) H metal-molecule = i g 0 (a † J,i a P + a † P a J,i ) H Field = E(t) i µ 0 (a † J,i + a J,i ) + µ P (a † P + a P ) H fluctuation (t) = i ∆ω ii (t)a † J,i a J,i (A.1)
where a † J i (a J i ) are the creation (annihilation) operators for the i-th molecule and K is the nearest neighbor coupling between molecules, a † P (a P ) are the creation (annihilation) operators for the plasmon mode, a † k (a k ) those of the manifold of continuum states in the metal band structure with momentum k, V k the plasmon-electron coupling responsible for Landau decay. g 0 the dipolar coupling between the molecular transitions and the plasmon mode, which we assume to be identical for all molecules. With Eq. (A.1), we model the molecular aggregate as N two-level systems which are coupled to their nearest neighbour with strength K. The plasmon is represented by a boson coupled to each two-level system with a coupling strength g 0 . The plasmon excitation can decay into hot electron-hole pairs that form a continuum we label by its momentum k. The decay is mediated by the plasmon-electron interaction V k [71]. Both the molecules and the plasmon couple to the far-field with transition dipole moments µ 0 and µ P , respectively (see Figure 1). These couplings to the light field are analyzed in the main text for the two limiting cases when µ 0 = 0 and when µ P = 0.
There are three energy scales of the problem: the coupling strength g = g 0 √ N , the nearest neighbor coupling K and the transition energy fluctuation amplitudes ∆ω ii . To derive the current effective Hamiltonian we assume the limit ∆ω ii g, K which is appropriate for the study of interference effects. For simplicity we also assume here the limit of vanishing intermolecular coupling K. The approach is to diagonalize the Hamiltonian for the molecular component, average over fluctuations over the bath, couple to the plasmonic part and project onto the bright states.
In the molecular basis, the transition frequency of molecule i is modulated by the bath by an amount ∆ω ii (t). We transform H from the molecular basis to the molecular exciton basis via the unitary transformation W to obtain a diagonal Hamiltonian D:
D = W HW −1 (A.2)
where W are the eigenvectors of the Hamiltonian without fluctuations. We index the new eigenstates by q and transform the fluctuations into the molecular exciton basis as well:
Then:
H exciton = H − H metal − H metal-molecule − H f ield = N q=0 ω q a † q a q + N q=0 N l=0 δ ql (t)a † q a l (A.3)
The q = 0 is bright and is responsible for coupling to the far field as well as for the coupling to the plasmonic transition. The flucutations in the new basis are given by:
δ ql (t) = W qi ∆ω ii (t)W −1 il (A.4)
We can neglect the off-diagonal terms δ ql , for q = l so that only the diagonal fluctuations of the eigenstates survive [53]. The cumulant expansion of the evolution operator e −i dt H(t ) B = e −i H t can be carried out straightforwardly:
e −i dt H(t ) ≈ e −iHmt−Γt (A.5)
where H m = H exciton − H fluctuations and Γ = q γ q a † q a q and γ q = dt e iωt δ qq (t )δ qq (0) is obtained from the two-time correlation function. Assuming the Markovian limit, γ q is real and each excitonic state dephases with a different time constant. We will only concern ourselves with the q = 0 level corresponding to the bright state.
The new Hamiltonian is:
H = H bright + H dark + H bright-dark + H Field + H fluctuation H bright = ω J a † J a J + ω P a † P a P + i g(a † J a P + a † P a J ) H dark = N q=1 ω q a † q a q + +∞ −∞ dkω k a † k a k H bright-dark = dk(V k a † k a P + V * k a † P a k ) H Field = E(t) µ J (a † J + a J ) + µ P (a † P + a P ) H fluctuation = −i N q=0 γ q a † q a q (A.6)
where g = √ N g 0 Assuming a tight-binding model with identical site energies and identical nearest neighbor coupling K, the dispersion relation for the molecular component is ω q = ω 0 − K 2 (2cos( 2π N q) + 2) [72]. The Hamiltonian (Eq. (A.6)) can be already used to calculate the optical response of the plexcitons, however, the resulting expression for the signal will contain an explicit description of dark states that do not couple to the far-field and so is inefficient. In order to transform to a fully excitonic picture we must also diagonalize the plasmon-metal Hamiltonian. However, it is much more preferable and transparent to obtain effective operators for the bright transitions.
Effective operators. We have obtained the Hamiltonian after bath averaging in the limit of vanishing dipolar interaction and now proceed to calculate the effective operators for the bright and dark parts. We have [59]:
H eff (z) = P H 0 P + P H 0 QG 0 (z)QH 0 P (A.7)
where QG 0 (z)Q = [z − QH 0 Q] −1 . For the evaluation of H eff (z) we assume the wideband approximation where ω k = k n with n a density of states and V k is k−independent. Then, considering that only the metallic dark states are the only ones to affect significantly the optical transitions, we have
P HQG 0 (z)QHP = dk |V k | 2 z − k/n = −inπ |V k | 2 (A.8)
We also include here the calculation of terms needed to calculate the most general form of the optical response:
P G(z)Qe −izt QG(z )P e −iz t 1 = P G(z)P dkP H 0 Q Q z − k/n Q z − k/n e −izt e −iz t 1 QH 0 P G(z )P (A.9)
which vanishes due to the wideband approximation. We then obtain the effective Hamilto-
nian in P H eff = (ω J − iγ J )a † J a J + (ω P − iγ P )a † P a P + g(a † J a P + a † P a J ) (A.10)
where γ P = nπ |V k | 2 and γ J = dte iω J t δ 00 (t)δ 00 (0) . The non-Hermitian model presented accounts for plasmon-electron coupling, as well as fluctuation of the energy levels. It is expected that more complicated relaxation schemes, notably relaxation from upper to lower polariton branch in concert with pure dephasing are not fully described. The effect of these pathways will be explored in future work. (1) and (2) for the plasmon and J-aggregate, respectively. Notice that if two levels are coupled to the same continuum we obtain offdiagonal non-Hermitian terms.
APPENDIX C: EXTENDED PARAMETER SPACE FOR S > 1 BUT C < 1
In this Appendix we explore the region marked by points D, E and F of Figure 2 that are in the strong coupling regime (S > 1) but not in the weak coupling regime with respect to the interference parameter (C < 1). Figure 9 shows that point E shows the presence of Rabi splitting for the case where the plasmon couples to the far field but not for the case where the molecular aggregate couples to the far field. We see here the value of the cooperativity parameter. In the case where the molecular component couples predominantly to the far field, the Rabi splitting is absent even though we are in the strong coupling limit, if we are in the low cooperativity regime. It takes both parameters to be larger than one to observe some the Rabi splitting.
FIG. 1 :
1Energy level structure. a) Before diagonalization of the molecular component b)
FIG. 2 :
2(red) corresponds to the weak coupling regime for C = 0.20 and S = 0.21, Point B (green) corresponds to the interference regime with C = 1.8 and S = 0.63 and Point C (blue) corresponds to the strong coupling with C = 11.25 and S = 1.58. The points D (C = 0.25,S = 0.67), E (C = 0.44,S = 1.6) and F (C = 1.78,S = 3.2) are meant to explorea small region where we can have the condition S > 1 but C < 1 and is outlined in Appendix C. In all cases the molecular dephasing is set at γ J /ω 0 = 0.01. As mentioned previously, we consider two limiting cases where the plasmonic transition couples predominantly to the far field (i.e. µ J = 0) and where the molecular transition couples to the far field (µ P = 0). Main regime for a plexciton systems with isoenergetic transitions ω J = ω P = ω 0 .We show in orange and blue the curves for S = 1 and C = 1 which separate the different regimes. The parameters of coupling strength g and plasmon dissipation strength γ p are normalized to the molecular transition frequency ω 0 . We choose three representative points A, B and C to simulate the linear and nonlinear optical response in the main text. The points D, E (unlabeled point) and F are explored in the Appendix to characterize the region where S > 1 but C < 1) (which is obtained for point E, while D and F are provided as references).
(
A.10) and eigenvectors as a function of the coupling strength (Figure 3). For S < 1 the real part of the eigenvalues (i.e. the transition frequencies) are equal while the imaginary part (i.e. the dissipation) are not. The two polariton branches are either predominantly plasmonic or predominantly molecular and the distinction is carried by the dephasing rate of the branch (imaginary part of the eigenvalue). At the exceptional point both eigenvalues and eigenvectors coalesce and become indistinguishable. For S > 1, the real part of the eigenvalues begins to split (i.e. Rabi splitting) while the imaginary part is now identical for both branches. The two polariton branches are now (and for all stronger couplings) an equal mixture of plasmon and exciton. We analyze in more detail the first and third order optical response of the non-Hermitian Hamiltonian from Eq. (A.1) using the double-sided
|+ , |− are the eigenstates of the non-Hermitian Hamiltonian (Eq. (A.10)) (or upper and lower polariton branches). We can decompose the absorption spectra into each polariton contribution, i.e. plot each and b) eigenvectors of the non-Hermitian Hamiltonian. Vertical dashed lines show the dipolar coupling strength corresponding to the points A, B and C of Figure 2term of the sum in Eq. (9) separately. InFigure 4, we show the total absorption (green, solid) decomposed into the upper (blue, dotted) and lower (yellow, dashed) polaritons, for different values of the coupling corresponding to points A, B and C ofFigure 2. We consider two limiting cases, one where the plasmon predominantly couples to the far field (top row), and one where the J-aggregate predominantly couples to the far field (bottom row). The Euler plane of the insets show the transition dipole moments for each polariton branch, µ − , µ + . We first consider the case where the plasmon couples to the far field (top row). We can see that for all parameters there is the appearance of a peak splitting, however below the exceptional point it arises from the destructive interference of a broad transition and a narrow transition while above the exceptional point it arises from a Rabi splitting. Below the exceptional point (A and B), µ + is purely real while µ − is purely imaginary. The branch with the purely imaginary transition dipole moment will have a negative contribution to the total absorption and cause an interference dip. We note that the individual contributions of the branches are not physical by themselves, so that a negative contribution does not mean a stimulated emission but rather intuitively describes the interference process. The dip is clearly seen for spectra a) and b) where the yellow (dotted) line cancels the plasmonic absorptive contribution. This results in a spectrum that seems to have two apparently separate peaks well below the condition for Rabi splitting.However, while the spectra might look similar above and below the exceptional point, the nature of the excitation and consequently the excited state dynamics are expected to be very different. As the coupling increases, the transition dipole moments are no longer purely real or purely imaginary but have both real and imaginary components (see the Euler plane insets ofFigure 4). The contribution of each branch now amounts to two true peaks separated by Rabi splitting. While qualitatively the final spectra look similar, the nature of the resonances is not. For example, exciting on the right or left of the dip in the interference regime excites the same homogeneously broadened excited state while in the Rabi splitting regime exciting on the right or left addresses a different excited state (upper polariton or lower polariton). This has been recognized theoretically and experimentally in the fluorescence signatures of quantum dots coupled to plasmons[62]. The case where the J-aggregate couples predominantly to the far field is strikingly different. Below the exceptional point we see no evidence of peak splitting since the negative contribution is too broad. Above the exceptional point we see the expected Rabi splitting although with less contrast than when the plasmon couples to the far-field(Figure 4 d,e). The interference
FIG. 4 :
4Decomposition of the absorption spectra into polariton branches contributions (dotted yellow and dotted blue give the total contribution for the total absorption shown in green). We show the cases for µ j = 0 (a,b,c) and µ p = 0 (d,e,f). The transition dipole moments to each branch are displayed in the complex plane. We illustrate the regimes of weak cooperativity and weak coupling (a,d), strong cooperativity and weak coupling (b,e)
FIG. 5 :
5Role of the magnitude of the transition dipole moment of the discrete system µ J for a three-level system (a) and a Fano system (b). Traces have been ofset by 1.2 for clarity. The characteristic asymmetry of the Fano model is clearly missing from the three-level system.
FIG. 6 :
6Feynman diagrams during the coherent evolution (top) and after dephasing of the plasmon transition. The excited state e i,j , i, j ∈ LP, U P correspond to the polariton branches.
Figure 7
7shows the spectrum for weak (7.a) and strong (7.b) coupling along with their decomposition into diagonal elements ((UP,UP) and (LP,LP)) and cross-terms ((UP,LP) and (LP,UP)). For the weak coupling case, the diagonal elements are positive while it is the cross-terms that provide the negative signal and split the main plasmonic resonance into four regions. That the negative features arise from the cross-terms between upper and lower branch is a clearer indication that the dips are a result of an interference process between the plasmonic resonance and the molecular resonance.The strong coupling regime is qualitatively different. Each of the four contributions now appear at a different position in the spectrum because of the Rabi splitting. A notable difference between the GSB signal between the two regimes is that in the interference regime has a D 4 fourfold reflection symmetry while the strong coupling regime possesses a C 2 rotational symmetry (as we show later when both µ J = 0 and µ P = 0 some symmetries are lost).
FIG. 7 :FIG. 8 :
788 shows the GSB for different values of the coupling strength g (columns) and also different values of the ratio µ J /µ P (rows). Along the columns we observe the same qualitative difference between weak and strong coupling, namely the transition in the symmetry of the lineshape. In this case when µ J = 0, we loose some symmetries. Strikingly, while the linear response for different values of µ J /µ P does not show an asymmetry as the effective asymmetry parameter changes, the third-order response shows a clear asymmetry along the detection dimension. Projections along the excitation dimension (lower row of Figure 8) show that this asymmetry will be visible in transient absorption. This suggests that differences in the linear spectrum and the 2D projection onto the detection dimension can reveal information on the Hamiltonian. Asymmetries in the linear response will denote different transition frequencies between the plasmonic and molecular parts, while additional asymmetries in the transient absorption Decomposition of the GSB contribution for the case of weak coupling (top) and strong coupling (bottom). Each panel shows the individual contributions corresponding to the different permutations e i , e j =LP,UP of the diagram inFigure 6will indicate finite values of µ J . Dependence of the spectral asymmetry along the detection dimension as a function of the ratio µ J /µ P = -0.2 (top row), 0 (middle row) and 0.2 (bottom row). The coupling strength increases from left to right.
state can be encoded by a judicious choice of Feynman diagrams.Comparison to other models in the literature. There exist many models to describe the response of emitters coupled to plasmonic resonances which can fit absorption spectroscopy, scattering or photoluminescence data well. For example, the coupled oscillators model can explain the regimes of interference and Rabi splitting and reproduce the measured lineshapes. However, the formalism is not apt to describe nonlinear spectroscopy because of the impossibility of isolating the signals to a certain order in the field. Ab initio approaches involving the solution of Maxwell's equations around a metallic nanoparticle result in Fano-like equations with slightly modified detunings, or in non-Hermitian Hamiltonians for molecules coupled to plasmonic resonances. These expressions are sometimes reflective of scattered signals and not absorptive signals[34,68,69], as such they are suitable for linear response but not nonlinear signals. The use of non-Hermitian Hamiltonians as justified above solves this problem and allows the use of the response function formalism and double-sided Feynman diagrams to describe
using equations(2) and(3). Below the exceptional point, we have that
O t (S (3) ) = O τ (S (3) ) = S(3) , and above it we recover that O t (O τ (S (3) )) = S(3) . This work raises the possibility of using the symmetry of the lineshape of nonlinear spectroscopy to study the symmetry of the eigenvalues of the Hamiltonian. In real systems we expect to have deviations from these idealized Hamiltonian, however we do expect the symmetries of the spectra to be recognizable.CONCLUSION We have derived a non-Hermitian Hamiltonian for the calculation of the linear and nonlinear optical response. Decomposing the Hamiltonian into bright and dark partitions and expressing the linear and nonlinear optical response in terms of effective operators provides a new framework for systems with large manifolds of dark states. An analysis of the linear optical absorption reveals that the exceptional point separating weak from strong coupling regime is clearly illustrated when decomposing the total signal into individual Feynman diagram contributions. Below the exceptional point interference effects are indicated by negative contributions to the absorbance while the Rabi splitting appears above the excep-tional point. The decomposition of the nonlinear signal also provides additional insight. We find that interference and Rabi splitting regimes have different symmetries in the twodimensional maps, an indication that cannot be obtained from linear absorption alone. The connections outlined in this article between the symmetry of the spectral signatures and the nonlinear response can open new ways of thinking for classifying symmetries in the eigenvalue/exciton structure of complex materials. APPENDIX A. DERIVATION OF A NON-HERMITIAN HAMILTONIAN FOR PLEXCITONS
H Field is the light-matter interaction with an external field E, where µ 0 is the transition dipole moment of a molecule and µ P that of the plasmon. H fluctuation (t) represents the fluctuations induced by the bath, where we have only considered the modulation of the transition frequency of the i-th molecule with amplitude ∆ω ii (t). The Hamiltonian is depicted in Figure 1.a. There are two main line-broadening mechanisms. On one hand, the optical coherence of the molecular component dephases with fluctuations of the transition frequency (with a time constant of 40 fs[41] for TDBC, which is much faster than the decay induced dephasing into molecular dark states). To capture this dephasing mechanism, it is enough to consider the stochastic fluctuation of the transition frequency of the molecules. The plasmon coherence, on the other hand, dephases via Landau damping whereby the plasmon decays into neutral hot electron-hole states inside the metal. For this process, we can consider the coupling of the plasmon transition to a a continuum of electron-hole states via a plasmon-electron coupling element[71]. These two are very different mechanisms. The lifetime induced broadening due to decay of the plasmon into hot electron-hole states is not a thermal process and has negligible temperature dependence, in stark contrast to line broadening in molecules induced by fluctuations of the bath which have strong temperature dependence and can be significantly reduced at low temperatures.
-Hermitian Hamiltonian of Eq. (A.10) appears only under some limiting conditions of the magnitudes of molecular line broadening induced by fluctuations of the bath,the decay rate into continuum states, and dipolar coupling. More general cases quickly become complicated and untractable analytically[73]. However, other extended Hamiltonians reduce to the same expression, the simplest of them being two coupled excited states, each one coupled to its own set of continuum states. In general, for a set of N discrete states coupled to each other and to M continua via couplings V coupling (including radiative coupling) between level i and the continuum (a) and which reduces to equation (A.10) with γ P = nπ V labeled the individual continua as
FIG. 9 :Figure 2 .FIG. 10 :
9210Decomposition of the absorption spectra into polariton branches contributions (dotted yellow and dotted blue give the total contribution for the total absorption shown in green). We show the cases for µ j = 0 (a,b,c) and µ p = 0 (d,e,f), for points D, E and F of We illustrate the regimes of weak cooperativity and weak coupling (a,d), weak cooperativity and strong coupling (b,e) and strong cooperativity and strong coupling (c,f).APPENDIX D: INCLUSION OF ESAWe can simulate the full spectrum by choosing a model for the excited state absorption (ESA). If the excited state absorption is dominated by the molecules remaining in the ground state after the pump pulse pair, we can simulate the full spectrum by the Feynman diagrams shown inFig. 6. The differences found with experimental measurements can be ascribed to the finite bandwidth of the pulses. Dependence of the spectral asymmetry for GSB and ESA along the detection dimension as a function of the ratio µ J /µ P = -0.2 (top row), 0 (middle row) and 0.2 (bottom row). The coupling strength increases from left to right.
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arxiv |
Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation
Karl Bringmann [email protected]
Max Planck Institute for Informatics
Saarland Informatics Campus
Saarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics
Saarland University
Saarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany
André Nusser [email protected]
Max Planck Institute for Informatics
Saarland Informatics Campus
Saarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics
Saarland University
Saarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany
Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation
and phrases Hausdorff Distance Under TranslationFine-Grained Complexity TheoryLower Bounds
Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible.Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size n and m, the Hausdorff distance under translation can be computed in timẽ O(nm) for the L1 and L∞ norm [Chew, Kedem SWAT'92] andÕ(nm(n + m)) for the L2 norm [Huttenlocher, Kedem, Sharir DCG'93].As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of (nm) 1−o(1) for L1 and L∞ (and all other Lp norms) assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of n 2−o(1) for L2 in the imbalanced case of m = O(1) assuming the 3SUM Hypothesis.ACM Subject ClassificationTheory of computation → Problems, reductions and completeness 1 There is a directed and an undirected variant of the Hausdorff distance, see Section 2. In this introduction, we do not differentiate between these two, since all our statements hold for both variants. 2 ByÕ-notation we ignore logarithmic factors in n and m.
Introduction
As data sets become larger and larger, the requirement for faster algorithms to handle such amounts of data becomes increasingly necessary. One very common type of data that is created during measurements is point sets in the plane, for example when recording GPS trajectories or describing shapes of objects, in medical image analysis, and in various data science applications. A fundamental algorithmic tool for analyzing point sets is to compute the similarity of two given sets of points. There are several different measures of similarity in this setting, for example Hausdorff distance [21], geometric bottleneck matching [18], Fréchet distance [3], and Dynamic Time Warping [25]. Among these measures, the Hausdorff distance is arguably the most basic and intuitive: It assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of all assigned pairs of points. 1 For a discussion of the other previously mentioned distance measures, see Section 1.1.
While these similarity measures are of great practical relevance, for some applications it is a drawback that they are not translational invariant, i.e., when translating one of the point sets, the distance can -and in most cases will -change. This is unfavorable in applications that ask for comparing the shape of two objects, meaning that the absolute position of an object is irrelevant. Examples of this task arise for example in 2D object shape similarity, medical image analysis [19], classification of handwritten characters [10], movement patterns of animals [12], and sports analysis [17].
Fortunately, any point set similarity measure has a canonical translational invariant version, by minimizing the similarity measure over all translations of the two given point sets. For the Hausdorff distance this variant is known as the Hausdorff distance under translation, see Section 2 for a formal definition. Given two point sets in the plane of size n and m, the Hausdorff distance under translation can be computed in time O(nm log 2 nm) for the L 1 and L ∞ norm [16], and in time O(nm(n + m) log nm) for the L 2 norm [22]. We are not aware of any lower bounds for this problem, not even conditional on a plausible hypothesis. The only results in this direction are Ω(n 3 ) lower bounds on the arrangement size [16] and on the number of connected components of the feasible translations [28] (for the decision problem on points in the plane with n = m). However, these bounds also hold for L 1 and L ∞ , where they are "broken" by the O(nm log 2 nm)-time algorithm [16], so apparently these bounds are irrelevant for the running time complexity.
In this paper, we approach the Hausdorff distance under translation from the viewpoint of fine-grained complexity theory [29]. For two problem settings, we show that the known algorithms are optimal up to lower order factors assuming standard hypotheses:
1.
We show an (nm) 1−o (1) lower bound for all L p norms -and in particular L 1 and L ∞ , matching the O(nm log 2 nm)-time algorithm from [16] up to lower order factors, see Section 3. This result holds conditional on the Orthogonal Vectors Hypothesis, which states that finding two orthogonal vectors among two given sets of n binary vectors in d dimensions cannot be done in time O(n 2−ε poly(d)) for any ε > 0. It is well-known that the Orthogonal Vectors Hypothesis is implied by the Strong Exponential Time Hypothesis [30], and thus our lower bound also holds assuming the latter [23]. These two hypotheses are the most standard assumptions used in fine-grained complexity theory in the last decade [29]. 2. We show an n 2−o(1) lower bound for L 2 in the imbalanced case m = O(1), matching the O(nm(n + m) log nm)-time algorithm from [16] up to lower order factors, see Section 4. Previously, an n 2−o(1) lower bound was only known for the more general problem of computing the Hausdorff distance under translation of sets of segments in the case that both sets have size n (a problem for which the best known algorithm runs in time 2 O(n 4 )) [6]. Our result holds conditional on the 3SUM Hypothesis, which states that deciding whether, among n given integers, there are three that sum up to 0 requires time n 2−o(1) . This hypothesis was introduced by Gajentaan and Overmars [20], is a standard assumption in computational geometry [24], and has also found a wealth of applications beyond geometry (see, e.g., [1,2,4,26]).
Our lower bounds close gaps that have not seen any progress over 25 years. Furthermore, note that our second lower bound shows a separation between the L 2 norm and the L 1 and L ∞ norms, as in the imbalanced case m = O(1) the latter admits aÕ(n)-time algorithm [16] while the former requires time n 2−o(1) assuming the 3SUM Hypothesis. We leave it as an open problem whether for L 2 the balanced case n = m requires time n 3−o(1) .
Related work
Our work continues a line of research on fine-grained lower bounds in computational geometry, which had early success with the 3SUM Hypothesis [20] and recently got a new impulse with the Orthogonal Vectors Hypothesis (or Strong Exponential Time Hypothesis) and resulting lower bounds for the Fréchet distance [7], see also [13,11]. Continuing this line of research is getting increasingly difficult, although there are still many classical problems from computational geometry without matching lower bounds. In this paper we obtain such bounds for two settings of the classical Hausdorff distance under translation.
Besides Hausdorff distance, there are several other distance measures on point sets, including geometric bottleneck matching [18], Fréchet distance [3], and Dynamic Time Warping [25]. The geometric bottleneck matching minimizes the maximal distance in a perfect matching between the two given point sets. Fréchet distance and Dynamic Time Warping additionally take the order of the input points into account. They both consider the same class of traversals of the input points, and the Fréchet distance minimizes the maximal distance that occurs during the traversal, while Dynamic Time Warping minimizes the sum of distances.
Let us discuss the canonical translational invariant versions of these distance measures. For geometric bottleneck matching under translation, Efrat et al. designed anÕ(n 5 ) algorithm [18]. The discrete Fréchet distance under translation has anÕ(n 4.66... )-time algorithm and a conditional lower bound of n 4−o(1) [9], see also [10] for algorithm engineering work on this topic. While Dynamic Time Warping is a very popular measure (in particular for video and speech processing), no exact algorithm for its canonical translational invariant version is known in L 2 since it contains the geometric median problem as a special case [5].
Further work on the Hausdorff distance under translation includes an O((n + m) log nm)time algorithm for point sets in one dimension [27]. For generalizations to dimensions d > 2 see [16,15].
Preliminaries
In this paper we consider finite point sets which lie in R 2 . For any p ∈ R 2 , we use p x and p y to refer to its first and second component, respectively. For a point set A ⊂ R 2 and a translation τ ∈ R 2 , we define A + τ := {a + τ | a ∈ A}. To denote index sets, we often use [n] := {1, . . . , n}. Given a point q ∈ R 2 , its p-norm is defined as
q p := (|q x | p + |q y | p ) 1 p .
We now introduce several distance measures, which are all versions of the famous Hausdorff distance. First, let us define the most basic version. Let A, B ⊂ R 2 be two point sets. The Note that, intuitively, the directed Hausdorff distance measures the distance from A to B but not from B to A, and it is not symmetric. A symmetric variant of the Hausdorff distance, the undirected Hausdorff distance, is defined as
δ T H (A, B) := min τ ∈R 2 δ H (A, B + τ ). Again, it holds that δ T H (A, B) ≤ δ T H (A, B).
Naturally, for all of the above distance measures, the decision problem is defined such that we are given two point sets A, B and a threshold distance δ, and ask if the distance of A, B is at most δ.
For the Hausdorff distance on point sets (without translation) the undirected distance is at most as hard as the directed distance, because the undirected distance can be calculated using two calls to an algorithm computing the directed distance. 3 However, note that for the Hausdorff distance under translation, we cannot just compute the directed distance twice and then obtain the undirected distance as we have to take the maximum for the same translation.
OV based (mn) 1−o(1) lower bound for L p
We now present a conditional lower bound of (mn) 1−o(1) for the Hausdorff distance under translation -first for L 1 and L ∞ , and then we discuss how to generalize this bound to L p . We present the first lower bound only for the L 1 case, as the same construction carries over to the L ∞ case via a rotation of the input sets by π 4 . Our lower bound is based on the hypothesized hardness of the Orthogonal Vectors problem.
Definition 1 (Orthogonal Vectors Problem (OV)). Given two sets X, Y ⊂ {0, 1} d with |X| = m, |Y | = n, decide whether there exist x ∈ X and y ∈ Y with x · y = 0.
A popular hypothesis from fine-grained complexity theory is as follows. This hypothesis is typically stated and used for the balanced case n = m. However, it is known that the hypothesis for the balanced case is equivalent to the hypothesis for any unbalanced case n = m α for any fixed constant α > 0, see, e.g, [8, Lemma 2.1]. See Figure 1 for an overview of the reduction. Intuitively, the first dimension of the translation chooses the vector y ∈ Y while the second dimension of the translation chooses the vector x ∈ X. An alignment of the Vector Gadgets within distance 1 is then possible if and only if x and y are orthogonal. Alignments that can circumvent this orthogonality check are not possible as we restrict the translations to a small set of candidates by placing dummy Vector Gadgets on the right side and by including a Translation Gadget.
Gadgets
We now describe the gadgets in detail. Let ε > 0 be a sufficiently small constant, e.g., ε = 1 20mnd . Recall that the distance for which we want to solve the decision problem is δ = 1. Furthermore, we denote the ith component of a vector v by v[i] and we use 0 d and 1 d to denote the d-dimensional all-zeros and all-ones vector, respectively.
Vector Gadget
We define a general Vector Gadget, which we then use at several places by translating it. Given a vector v ∈ {0, 1} d , the Vector Gadget consists of the points p 1 , . . . , p d ∈ R 2 :
p i = (ε 2 , iε), if v[i] = 0 (0, iε), if v[i] = 1 V ((1, 1, 1, 0, 1, 0, 0)) V ((0, 1, 0, 1, 0, 0, 1)) p d p 1 q d q 1 1 2 1 0 0 1 Figure 2
A depiction of the two types of Vector Gadgets and how they are placed to check for orthogonality.
We denote the Vector Gadget created from vector v by V (v). Additionally, we define a mirrored version of the gadget V as
V (v) := V (v),
wherev is the inversion of v, i.e., each bit is flipped.
Lemma 3. Given two vectors
v 1 , v 2 ∈ {0, 1} d and corresponding Vector Gadgets V 1 = V (v 1 ) and V 2 = V (v 2 ) + (1, 0), we have δ H (V 1 , V 2 ) ≤ 1 if and only if v 1 · v 2 = 0. Proof. Let the points of V 1 (resp. V 2 ) be denoted as p 1 , . . . , p d (resp. q 1 , . . . , q d ). First, note that p i − q j 1 = 1 + |i − j|ε + (v 1 [i] + v 2 [j] − 1)ε 2 > 1 for i = j. Thus,V 1 = V (v 1 ) and V 2 = V (v 2 ) + (1, 0), we have δ H (V 1 , V 2 ) ≤ 1 if and only ifv 1 ·v 2 = 0, wherev 1 ,v 2 are the inversions of v 1 , v 2 .
For any x, y, D ∈ R, we call Vector Gadgets
V 1 = V (v 1 )+(x, y) and V 2 = V (v 2 )+(x+D, y)
vertically aligned, or more precisely, vertically aligned at distance D.
Translation Gadget
To ensure that B cannot be translated arbitrarily, we introduce a gadget to restrict the translations to a restricted set of candidates. The Translation Gadget T consists of two translated Vector Gadgets of the zero vector:
T := (V (1 d ) − (2 − nε, 0)) ∪ (V (0 d ) + (2 + 2ε, 0)).
We show that restricting the coordinates of the points of the other set involved in the Hausdorff distance under translation instance, already restricts the feasible translations significantly.
Lemma 5. Let P ⊂ [−1 − 1 2 ε, 1 + 1 2 ε] × R be a point set and T the Translation Gadget. If δ T H (T, P ) ≤ 1, then τ * x ∈ [−(n + 1 2 )ε − ε 2 , − 3 2 ε], where τ * is any translation satisfying δ H (T, P + τ * ) ≤ 1.
Proof. We show the contrapositive. Therefore, assume the converse, i.e., that τ *
x is not contained in [−(n+ 1 2 )ε−ε 2 , − 3 2 ε]. If τ * x < −(n+ 1 2 )ε−ε 2 , then −1− 1 2 ε−(−2+nε+ε 2 +τ * x ) > 1
and thus the left part of T cannot contain any point of P at distance at most 1. If τ * x > − 3 2 ε, then 2 + 2ε + τ * x − (1 + 1 2 ε) > 1 and thus the right part of T cannot contain any point of P at distance at most 1. Thus, δ T H (T, P ) > 1.
Undirected Gadget
To ensure that each point in A can be matched to a point in B within distance 1, we add auxiliary points to B. The Undirected Gadget is defined by the point set
U := {(− 1 2 , 0), ( 1 2 , 0)}.
Lemma 6. Given a set of points
P ⊂ [−1− 1 2 ε, 1+ 1 2 ε]×[− 1 8 , 1 8 ], it holds that δ H (P, U +τ ) ≤ 1 for any τ ∈ [−(n + 1 2 )ε − ε 2 , (n + 1 2 )ε + ε 2 ] × [− 1 8 , 1 8 ].
Proof. By symmetry, we can restrict to proving that the distance of the point set
P = P ∩ [0, 1 + 1 2 ε] × [− 1 8 , 1 8 ] to ( 1 2 , 0) + τ is at most 1. For any p ∈ P , we have |p x − ( 1 2 + τ x )| ≤ 1 2 + (n + 1 2 )ε + ε 2 ≤ 1 2 + 1 10 , where the last inequality follows from plugging in ε = 1 20mnd , and also |p y − τ y | ≤ 1 4 . Thus, p − (( 1 2 , 0) + τ ) 1 ≤ 3 4 + 1 10 < 1.
Reduction and correctness
We now describe the reduction and prove its correctness. We construct the point sets of our Hausdorff distance under translation instance as follows. The first set, i.e., set A, consists only of Vector Gadgets:
A := i∈[m] V (x i ) + (−1 − 1 2 ε, i · 2dε) ∪ i∈[m] V (1 d ) + (1 + 1 2 ε, i · 2dε)
The second set, i.e., set B, consists of Vector Gadgets, the Translation Gadget, and the Undirected Gadget: Figure 1 for a sketch of the above construction. To reference the vector gadgets as they are used in the reduction, we use the notation
B := j∈[n] V (y j ) + (jε, 0) ∪ T ∪ U SeeV r (x i ) := V (x i ) + (−1 − 1 2 ε, i · 2dε) and V r (y j ) := V (y j ) + (jε, 0).
We can now prove correctness of our reduction. In the reduction, we return some canonical positive instance, if the 0 d vector is contained in any of the two OV sets. This allows us to drop all 1 d vectors from the input, as they cannot be orthogonal to any other vector. Thus, we can assume that all vectors in our input contain at least one 0-entry and at least one 1-entry. Proof. Recall that we only have to consider the L 1 case. We first prove that there is a pair of orthogonal vectors x ∈ X and y ∈ Y if and only if δ T H (A, B) ≤ 1. To prove the theorem for the directed and undirected Hausdorff distance under translation at the same time, it suffices to show "⇒" for the undirected version and "⇐" for the directed version.
⇒: Assume that there exist x i ∈ X, y j ∈ Y with x i · y j = 0. Then consider the translation τ = (−(j + 1 2 )ε, i · 2dε) which vertically aligns the Vector Gadgets V r (x i ) and V r (y j ) + τ at distance 1. As x i and y j are orthogonal, it follows from Lemma 3 that δ H (V r (y j )+τ, A) ≤ 1. We now show that all of the remaining points of B + τ have a point of A at distance at most 1. The Vector Gadgets V r (y j ) + τ with j < j are strictly to the left of V r (y j ) + τ and are thus also in Hausdorff distance at most 1 from V r (x i ). If j = n, then we are done with the Vector Gadgets. Otherwise, consider the Vector Gadget V r (y j+1 ) + τ . We claim that each point of it is at distance at most 1 from V (1 d ) + (1 + 1 2 ε, i · 2dε). As the two gadgets are vertically aligned, we just have to check their horizontal distance, which is
1 + 1 2 ε − ((j + 1)ε − (j + 1 2 )ε) = 1.
Thus, by Lemma 3, we have δ H (V r (y j+1 ) + τ, A) ≤ 1. Now, by the same argument as above, all gadgets V r (y j ) + τ with j > j + 1 are in directed Hausdorff distance at most 1 from A.
As the points of the Undirected Gadget U + τ are closer by a distance of almost 1 2 to A than the Vector Gadgets in B + τ , also δ H (U + τ, A) ≤ 1 holds. Finally, we have to show that the Translation Gadget T + τ is at distance at most 1 from A. As the left part of T and V r (x i ) are aligned vertically, we only have to check the horizontal distance. The horizontal distance is
−1 − 1 2 ε − (−2 + nε − (j + 1 2 )ε) = 1 − (n − j)ε ≤ 1
for any j ∈ [n]. Similarly, the distance of the right part of the Translation Gadget from the vertically aligned
V (1 d ) in A is 2 + 2ε − (j + 1 2 )ε − (1 + 1 2 ε) = 1 − (j − 1)ε ≤ 1
for any j ∈ [n]. Thus, by Lemma 3 and Lemma 4, it holds that δ H (T + τ, A) ≤ 1. As ⇐: Now, assume that δ T H (A, B) ≤ 1 and let τ be any translation for which δ H (B + τ, A) ≤ 1. Note that we used the directed Hausdorff distance in the previous statement on purpose, as we prove hardness for both versions. Lemma 5 implies that τ x ∈ [−(n + 1 2 )ε − ε 2 , − 3 2 ε]. Let V r (y j ) + τ, V r (y j+1 ) + τ be the Vector Gadgets such that V r (y j ) + τ has directed Hausdorff distance at most 1 to the left Vector Gadgets of A and V r (y j+1 ) + τ has directed Hausdorff distance at most 1 to the right Vector Gadgets of A. This is well-defined as the left Vector Gadgets of A and the right Vector Gadgets of A are at distance at least 2 + ε − ε 2 from each other, and thus no Vector Gadget of B + τ can be at distance at most 1 from both sides. Furthermore, as τ x ≤ − 3 2 ε, the Vector Gadget V r (y j ) + τ has directed Hausdorff distance at most 1 to the left Vector Gadgets of A, as
τ ∈ [−(n + 1 2 )ε − ε 2 , − 3 2 ε] × [− 1 8 , 1 8 ],jε − 3 2 ε − (−1 − 1 2 ε) = 1 + (j − 1)ε ≤ 1
for j = 1. If j = n, then V r (y j+1 ) + τ is undefined.
As δ H (B + τ, A) ≤ 1, we know that V r (y j ) + τ has directed Hausdorff distance at most 1 to a gadget V r (x) for some x ∈ X. We claim that this distance cannot be closer than 1 as V r (y j+1 ) + τ must have a directed Hausdorff distance at most 1 from the right side of A or, in case j = n, due to the restrictions imposed by the Translation Gadget. Let us consider the case j = n first. Any translation τ which places V r (y j+1 ) + τ in directed Hausdorff distance at most 1 from the right side of A needs to fulfill
1 + 1 2 ε − ((j + 1)ε + τ x ) ≤ 1
and thus τ x ≥ −(j + 1 2 )ε, using the fact that each vector in Y contains at least one 0-entry. This, on the other hand, implies that V r (y j ) + τ is in Hausdorff distance at least
jε − (j + 1 2 )ε − (−1 − 1 2 ε) = 1 from V r (x)
. Now consider the case j = n. As by Lemma 5 we have τ x ≥ −(n + 1 2 )ε − ε 2 , it follows that V r (y n ) + τ is in Hausdorff distance at least
nε − (n + 1 2 )ε − (−1 − 1 2 ε) = 1
from V r (x), using the fact that each vector in Y contains at least one 0-entry (this is the reason why the ε 2 disappears). By the arguments above, the two gadgets V r (y j ) + τ and V r (x) have to be horizontally aligned as required by Lemma 3. They also have to be vertically aligned as a vertical deviation would incur a Hausdorff distance larger than 1 for the pair of points in the two gadgets that are in horizontal distance 1. Then, applying Lemma 3, it follows that x and y j are orthogonal.
It remains to argue why the above reduction implies the lower bound stated in the theorem. Assume we have an algorithm that computes the Hausdorff distance under translation for L 1 or L ∞ in time (mn) 1−γ for some γ > 0. Then, given an Orthogonal Vectors instance X, Y with |X| = m and |Y | = n, we can use the described reduction to obtain an equivalent Hausdorff under translation instance with point sets A, B of size |A| = O(md) and |B| = O(nd) and solve it in time O((mn) 1−γ poly(d)), contradicting the Orthogonal Vectors Hypothesis.
Generalization to L p
We can extend the above construction such that it works for all L p norms with p = ∞ by changing the spacing between 0 and 1 points of the Vector Gadgets and also set ε accordingly. More precisely, we can set ε = 1 40pmnd (instead of 1 20mnd ) and use ε 2p as spacing (instead of ε 2 ), i.e., the Vector Gadget for a vector v ∈ {0, 1} d then consists of the points p 1 , . . . , p d ∈ R 2 :
p i = (ε 2p , iε), if v[i] = 0 (0, iε), if v[i] = 1
We prove that these modifications suffice in the remainder of this section. To this end, first note that in the proof of Theorem 7, the proof for "⇒" for L p already follows from the L 1 case as the L 1 norm is an upper bound on all L p norms. Thus, we only have to modify the proof of "⇐". To show "⇐", note that the only place where we use the L 1 norm in the proof is in the invocation of Lemma 3. Otherwise, we only argue via distances with respect to a single dimension, which carries over to L p as (x, 0) p = |x|. Thus, we now prove Lemma 3 for the general L p case.
Proof of Lemma 3 for L p .
To adapt the proof of Lemma 3 to the L p case, we only have to argue that we cannot match any p i , q j for i = j, as the remaining arguments merely argue about distances in a single dimension. We have that
p i − q j p = (|i − j|ε) p + (1 − (v 1 [i] + v 2 [j] − 1)ε 2p ) p 1/p ≥ ε p + (1 − ε 2p ) p 1/p ,
which is greater than 1 if ε p + (1 − ε 2p ) p > 1, which we obtain by using Bernoulli's inequality:
ε p + (1 − ε 2p ) p ≥ ε p + 1 − pε 2p ≥ 1 + 1 40pmnd p − p 1 40pmnd 2p > 1.
The remainder of the proof is analogous to the remainder of the proof of Lemma 3.
By all of the above arguments, the following theorem follows.
3Sum based n 2−o(1) lower bound for m ∈ O(1)
We now present a hardness result for the unbalanced case of the directed and undirected Hausdorff distance under translation. We base our hardness on another popular hypothesis of fined-grained complexity theory: the 3Sum Hypothesis. Before stating the hypothesis, let us first introduce the 3Sum problem. 4
Definition 9 (3Sum). Given three sets of positive integers X, Y, Z all of size n, do there exist x ∈ X, y ∈ Y, z ∈ Z such that x + y = z?
The corresponding hardness assumption is the 3Sum Hypothesis.
Definition 10 (3Sum Hypothesis). There is no O(n 2−ε ) algorithm for 3Sum for any ε > 0.
There are several equivalent variants of the 3Sum problem. Most important for us is the convolution 3Sum problem, abbreviated as Conv3Sum [26,14].
Definition 11 (Conv3SUM). Given a sequence of positive integers
X = (x 0 , . . . , x n−1 ) of size n, do there exist i, j such that x i + x j = x i+j ?
This problem has a trivial O(n 2 ) algorithm and, assuming the 3Sum Hypothesis, this is also optimal up to lower order factors. As 3Sum and Conv3Sum are equivalent, a lower bound conditional on Conv3Sum implies a lower bound conditional on 3Sum. Therefore, given a Conv3Sum instance defined by the sequence of integers X with |X| = n, we create an equivalent instance of the directed Hausdorff distance under translation for L 2 by constructing two sets of points A and B with |A| = O(n) and |B| = O(1) and providing a decision distance δ. We provide some intuition for the reduction in the following. See Figure 3 for an overview. Intuitively, we define a low-level gadget from which we build three separate high-level gadgets by rotation and scaling. Recall that in the Conv3Sum problem we have to find values i, j which fulfill the equation x i + x j = x i+j . Intuitively, we encode the choice of these two values into the two dimensions of the translation: the horizontal translation chooses the pair (i, x i ) in the first high-level gadget and the vertical translation chooses the pair (j, x j ) in the second high-level gadget. The third high-level gadget then allows for a Hausdorff distance below the threshold iff the chosen i and j fulfill the Conv3Sum constraint x i + x j = x i+j . To make this construction also work for the directed Hausdorff distance under translation, we add a simple gadget that restricts translations. In the remainder of this section, we present the details of our reduction and prove that it implies the claimed lower bound.
Construction
Given a Conv3Sum instance with X ⊂ [M ] where n = |X|, we now describe the construction of the Hausdorff distance under translation instance with point sets A, B and threshold distance δ. We use a small enough ε, e.g., ε = (4M n 2 ) −4 , as value for microtranslations. Furthermore, we set δ = 1 + 4n 2 ε 2 . The additional 4n 2 ε 2 term compensates for the small variations in distance that occur on microtranslations due to the curvature of the L 2 -ball.
Low-level gadget
We use a single low-level gadget, which is then scaled and rotated to obtain high-level gadgets. This gadget consists of two point sets A l and B l . The point set A l contains what we call number points p 1 i , p 2 i and filling points q i for 0 ≤ i < n. The set B l just contains two points: r 1 and r 2 . The number points p 1 i , p 2 i encode the number x i , while the filling points make sure that no other translations than the desired ones are possible. See Figure 4 for an overview. All of the points in this gadget are of the form (x, 0). The number points are
p 1 i = 2iε + x i ε 1.5 , 0 , p 2 i = p 1 i + (ε, 0)
Figure 4
The A set of the low-level gadget of the 3Sum reduction, which is used to build the high-level gadgets. We just show the leftmost part of the gadget, but the remainder is similar.
for 0 ≤ i < n. The filling points are
q i = 2i + 3 2 ε, 0 for 0 ≤ i < n.
The points in B l should introduce a gap to only allow alignment of the number gadgets such that the microtranslations (i.e., those in the order of ε 1.5 ) correspond to the number of the gap in the number gadget. To this end, B l contains the points
r 1 = (−1, 0), r 2 = (1 + ε, 0).
Before we prove properties of the low-level gadget, we first prove that the error due to the curvature of the L 2 -ball is small.
Lemma 12. Let (p x , p y ), (q x , q y ) ∈ R 2 be two points with |p x − q x | ∈ [ 1 2 , 2] and p y = q y . For any τ ∈ [0, (2n − 1)ε] 2 , we have |p x − (q x + τ x )| ≤ p − (q + τ ) 2 ≤ |p x − (q x + τ x )| + 4n 2 ε 2 .
Proof. As each component is a lower bound for the L 2 norm, the first inequality follows. Thus, let us prove the second inequality. We first transform
p − (q + τ ) 2 = (p x − (q x + τ x )) 2 + τ 2 y = |p x − (q x − τ x )| 1 + τ 2 y /(p x − (q x + τ x )) 2 . As √ 1 + x ≤ 1 + x 2 for any x ≥ 0, we have p − (q + τ ) 2 ≤ |p x − (q x − τ x )| + τ 2 y /(2|p x − (q x − τ x )|).
As τ y ≤ 2(n − 1)ε and |p x − (q x − τ x )| ≥ 1 2 , we obtain the desired upper bound.
An analogous statement holds when swapping the x and y coordinates. Note that the 4n 2 ε 2 term also occurs in the value of δ that we chose, as this is how we compensate for these errors in our construction. While we have to consider this error in the following arguments, it should already be conceivable that it will be insignificant due to its magnitude.
To this end, we use a compact notation to denote a value being in a certain range around a value. More concretely, for any y, r ∈ R, let x = y ± r denote x ∈ [y − r, y + r]. We now state two lemmas which show how the Hausdorff distance under translation decision problem is related to the structure of the low-level gadget.
Lemma 13. Given a low-level gadget A l , B l as constructed above and the translation being restricted to τ ∈ [0, (2n − 1)ε] 2 , it holds that if δ H (A l , B l + τ ) ≤ δ, then ∃i ∈ N : τ x = 2iε + x i ε 1.5 ± 4n 2 ε 2 .
Proof. Let τ ∈ [0, (2n − 1)ε] 2 and assume δ H (A l , B l + τ ) ≤ δ. Then all points in A l are at distance at most δ from one of the two points in B l . Furthermore, both points in B l + τ also have at least one close point in A l , as
r 1 +τ −p 1 0 2 ≤ 1−τ x +4n 2 ε 2 ≤ δ and r 2 +τ −q n−1 2 ≤ 1+τ x −(2n− 3 2 )ε+4n 2 ε 2 < δ,
using that n ≥ 1 and Lemma 12.
The gaps between neighboring points in A l either have width close to 1 2 ε, if the gap is between a number point and a filling point (p 1 i and q i−1 , or p 2 i and q i ), or they have a width of ε, if the gap is between two number points (p 1 i and p 2 i ). Furthermore, the two points in B l have distance 2 + ε, so there is an ε − 8n 2 ε 2 gap between their δ-balls. Thus, there is an i such that p 1 i has distance at most δ to r 1 , and p 2 i has distance at most δ to r 2 . This alignment of the gadgets can only be realized by a translation τ for which
τ x = 2iε + x i ε 1.5 ± 4n 2 ε 2 ,
which completes the proof.
Lemma 14.
Given a low-level gadget A l , B l as constructed above and the translation being
restricted to τ ∈ [0, (2n − 1)ε] 2 , it holds that if ∃i ∈ N : τ x = 2iε + x i ε 1.5 , then δ H (A l , B l + τ ) ≤ δ. Proof. Let i ∈ N and let τ x = 2iε + x i ε 1.5 . Consider any translations τ ∈ {τ x } × [0, 2(n − 1)ε].
Due to the restricted translation and Lemma 12, we can disregard the error terms that arise from the vertical translation τ y as they are compensated for by δ. Then all the points in A l before and including p 1 i are at distance at most δ from r 1 ∈ B l + τ and all the points afterwards are at distance at most δ from r 2 ∈ B l + τ . Clearly, both points in B l + τ then also have points from A l at distance δ, and thus δ H (A l , B l + τ ) ≤ δ.
High-level gadgets
This construction is inspired by the hard instance that was given in [28]. We want to obtain a grid of translations of spacing ε with some microtranslations in the O(ε 1.5 ) range. We already defined the low-level gadget above, and we now define the high-level gadgets.
Column Gadget
The column gadget induces columns in translational space, i.e., it enforces that valid translations have to lie on one of these columns. The column gadget is actually the low-level gadget we already described above. You can see a sketch of this gadget in Figure 5a. To semantically distinguish it from the low-level gadget, we refer to the point sets of the column gadget as A c and B c .
Row Gadget
The row gadget induces rows in translational space, i.e., it enforces that valid translations have to lie on one of these rows. We obtain the row gadget by rotating all points in the low-level gadget around the origin by π/2 counterclockwise. You can see a sketch of this gadget in Figure 5b. We call the point sets of the row gadget A r and B r .
Diagonal Gadget
The diagonal gadget induces diagonals in translational space, i.e., it enforces that valid translations have to lie on one of these diagonals. As opposed to the column and row gadget, the diagonal gadget also has to be scaled. We scale the sets A l and B l separately. We scale A l such that the gap between the number point pairs p 1 i , p 2 i becomes 1 √ 2 ε. And we scale B l such that the gap between the points becomes 2 + 1 √ 2 ε. After scaling, we rotate the points counterclockwise around the origin by π/4. You can see a sketch of this gadget in Figure 5c. We call the point sets of the diagonal gadget A d and B d .
Translation Gadget
To restrict the translations for the directed Hausdorff distance under translation, we introduce another gadget. The first set of points A t contains z l := (−1 + (2n − 1)ε, 0), z r := (1, 0), z b := (0, −1 + (2n − 1)ε), z t := (0, 1).
The second point set B t only contains the origin z c := (0, 0). We want to make sure that this gadget behaves well in a certain range.
Lemma 15. Given τ ∈ [0, (2n − 1)ε] 2 , it holds that δ H (A t , B t + τ ) ≤ δ.
Proof. As B t has a point on all sides, clearly δ H (B t + τ, A t ) ≤ δ. Furthermore,
z l − (z c + τ ) 2 ≤ 1 + 4n 2 ε 2 ≤ δ and z r − (z c + τ ) 2 ≤ δ,
using Lemma 12. Analogous statements hold for z b and z t . Thus, also δ H (A t , B t +τ ) ≤ δ.
Complete construction
To obtain the final sets of the reduction, we now place all four described high-level gadgets (i.e., column gadget, row gadget, diagonal gadget, and translation gadget) far enough apart. The far placement ensures that the two point sets of the respective gadgets have to be matched to each other when the Hausdorff distance under translation is at most delta δ.
Proof of correctness
First, we want to ensure that everything relevant happens in a very small range of translations.
Lemma 16. Let τ ∈ R 2 . If δ H (A, B + τ ) ≤ δ, then τ ∈ [0, (2n − 1)ε] 2 .
Proof. Note that for a Hausdorff distance at most δ, the sets A c and B c have to matched to each other and analogously for A r , B r , and A d , B d , and A t , B t . To show the contrapositive, assume τ / ∈ [0, (2n − 1)ε] 2 . For simplicity, we refer to the points in the high-level gadgets with the notation of the low-level gadget. Due to the translation gadget, we have
z l − (z c + τ ) 2 > δ for τ x > (2n − 1)ε + 4n 2 ε 2 , and z r − (z c + τ ) 2 > δ for τ x < −4n 2 ε 2 .
We now show that under these restricted translations and as δ H (A, B + τ ) ≤ δ, both points r 1 , r 2 in B c have at least one point of A c at distance δ. In the column gadget for τ x ∈ [−4n 2 ε 2 , 0), we have
(r 1 + τ ) − p 1 0 2 ≥ |−1 − (p 1 0 ) x + τ x | > δ and (r 2 + τ ) − p 1 0 2 ≥ 1 + ε − O(ε 1.5 ) > δ
for small enough ε and as x 0 > 0 and thus there is a component of order ε 1.5 . On the other hand, for τ x ∈ ((2n − 1)ε, (2n − 1)ε + 4n 2 ε 2 ], we have r 2 +τ −p 2 n−1 2 ≥ 1+ε+τ x −(2n−1)ε > δ and r 1 +τ −p 2 n−1 2 ≥ 1+O(ε 1.5 )−4n 2 ε 2 > δ for small enough ε. An analogous argument holds for the row gadget and τ y , as the row gadget is just a rotated version of the column gadget and the translation gadget is symmetric with respect to these gadgets.
We can now prove the main result of this section.
Theorem 17.
Computing the directed or undirected Hausdorff distance under translation in L 2 for two sets of size n and 7 cannot be solved in time O(n 2−γ ) for any γ > 0, unless the 3Sum Hypothesis fails.
Proof. We construct a Hausdorff under translation instance in this proof from a Conv3Sum instance as described previously in this section, and then show that they are equivalent. We first consider how to apply Lemma 13 and Lemma 14 to the diagonal gadget. More precisely, we consider which translations align the gaps of A d and B d as is used in these two lemmas. Consider the constraint τ x = 2kε + x k ε 1.5 ± 4n 2 ε 2 that is encoded by the low-level gadget. Recall that we scale this gadget by 1 √ 2 and rotate it by π 4 , i.e., we apply the transformation matrix
1 √ 2 · 1 −1 1 1 · 1 √ 2 0 0 1 √ 2 = 1 2 · 1 −1 1 1
to the right side of the constraint. Thus, for any α ∈ [0, (2n − 1)ε], the diagonal gadget encodes the constraints τ x τ y = 1 2 · 1 −1 1 1 · 2kε + x k ε 1.5 ± 4n 2 ε 2 α = 1 2 · 2kε + x k ε 1.5 ± 4n 2 ε 2 −α 2kε + x k ε 1.5 ± 4n 2 ε 2 α .
By adding up the two constraints, we obtain τ x + τ y = 2kε + x k ε 1.5 ± 4n 2 ε 2 .
We now show correctness of the reduction. ⇐: Assume X is a positive Conv3Sum instance. Then there exist x i , x j such that x i + x j = x i+j . Consider τ = (2iε + x i ε 1.5 , 2jε + x j ε 1.5 ) as translation. Due to Lemma 14, we have that δ H (A c , B c + τ ) ≤ δ and analogously δ H (A r , B r + τ ) ≤ δ. By the initial observation, we can also apply Lemma 14 to the diagonal gadget, and thus δ H (A d , B d + τ ) ≤ δ. Finally, by Lemma 15, we also have that δ H (A t , B t + τ ) ≤ δ for the given τ .
⇒: Assume δ T H (A, B) ≤ δ. From Lemma 16, it follows that τ ∈ [0, (2n − 1)ε] 2 . Then, due to Lemma 13 and the initial observation about the diagonal gadget, we have that there exist i, j, k that fulfill τ x = 2iε + x i ε 1.5 ± 4n 2 ε 2 , τ y = 2jε + x j ε 1.5 ± 4n 2 ε 2 , τ x + τ y = 2kε + x k ε 1.5 ± 4n 2 ε 2 .
It follows that 2iε + x i ε 1.5 + 2jε + x j ε 1.5 ± 8n 2 ε 2 = 2kε + x k ε 1.5 ± 4n 2 ε 2 , and thus i + j = k and x i + x j = x k .
It remains to argue why the above reduction implies the lower bound stated in the theorem. Assume we have an algorithm that computes the Hausdorff distance under translation in L 2 in time O(n 2−γ ) for some γ > 0. Then, given a Conv3Sum instance X with |X| = n, we can use the described reduction to obtain an equivalent Hausdorff under translation instance with point sets A, B of size |A| = O(n) and |B| = 7 and solve it in time O(n 2−γ ), contradicting the 3Sum Hypothesis.
Conclusion
In this work, we provide matching lower bounds for the running time of two important cases of the fundamental distance measure Hausdorff distance under translation. These lower bounds are based on popular standard hypotheses from fine-grained complexity theory. Interestingly, we use two different hypotheses to show hardness. For the Hausdorff distance under translation for L p , we show a lower bound of (nm) 1−o(1) using the Orthogonal Vectors Hypothesis, while for the imbalanced case of m = O(1) in L 2 , we show an n 2−o(1) lower bound using the 3Sum Hypothesis. We leave it as an open problem whether Hausdorff distance under translation for the balanced case admits a strongly subcubic algorithm or if conditional hardness can be shown.
directed Hausdorff distance is defined as δ H (A, B) := max a∈A min b∈B a − b p .
δ
H (A, B) := max{δ H (A, B), δ H (B, A)}. Note that, by definition, δ H (A, B) ≤ δ H (A, B). Both of the above distance measures can be modified to a version which is invariant under translation. The directed Hausdorff distance under translation is defined as δ T H (A, B) := min τ ∈R 2 δ H (A, B + τ ), and the undirected Hausdorff distance under translation is defined as
Definition 2 (
2Orthogonal Vectors Hypothesis (OVH)). The Orthogonal Vectors problem cannot be solved in time O((nm) 1−ε poly(d)) for any ε > 0.
Figure 1
1Sketch of the reduction from OV to the undirected Hausdorff distance under translation. We now describe a reduction from Orthogonal Vectors to Hausdorff distance under translation. To this end, we are given two sets of d-dimensional binary vectors X = {x 1 , . . . , x m } and Y = {y 1 , . . . , y n } with |X| = m and |Y | = n, and we construct an instance of the undirected Hausdorff distance under translation defined by point sets A and B and a decision distance δ = 1. First, we describe the high-level structure of our reduction. The point set A consists only of Vector Gadgets, which encode the vectors of X using 2md points. The point set B consists of three types of gadgets: Vector Gadgets: They encode the vectors from Y , very similarly to the Vector Gadgets of A. Translation Gadget: It restricts the possible translations of the point set B. Undirected Gadget: It makes our reduction work for the undirected Hausdorff distance under translation by ensuring that the maximum over the directed Hausdorff distances is always attained by δ H (B + τ, A).
for the Hausdorff distance to be at most 1, we have to match p i to q i for all i ∈ [d]. This is possible if and only if v 1 [i] = 0 or v 2 [i] = 0, as p i and q i are only at distance larger than 1 for v 1 [i] = 1 and v 2 [i] = 1. See Figure 2 for an example. Note that if we swap both gadgets and invert both vectors (i.e., flip all their bits), the Hausdorff distance does not change and thus an analogous version of Lemma 3 holds in this case, as we are just performing a double inversion.
Lemma 4 .
4Given two vectors v 1 , v 2 ∈ {0, 1} d and corresponding Vector Gadgets
Theorem 7 .
7Computing the directed or undirected Hausdorff distance under translation in L 1 or L ∞ for two point sets of size n and m in the plane cannot be solved in time O((mn) 1−γ ) for any γ > 0, unless the Orthogonal Vectors Hypothesis fails.
we know by Lemma 6 that δ H (A, B + τ ) ≤ 1 and thus also δ T H (A, B) ≤ 1.
Theorem 8 (
8Theorem 7 for L p ). Computing the directed or undirected Hausdorff distance under translation in L p for two point sets of size n and m in the plane cannot be solved in time O((mn) 1−γ ) for any γ > 0, unless the Orthogonal Vectors Hypothesis fails.
Figure 3
3Sketch of the reduction from Conv3Sum to the directed and undirected Hausdorff distance under translation in the Euclidean plane.
Figure 5
5Three of the high-level gadgets. The points of A are all in the low-level gadgets, while the points in B are explicitly shown including their δ-ball.
More explicitly, the point sets A, B of the Hausdorff distance under translation instance are defined as A := A c ∪ (A r + (10, 0)) ∪ (A d + (20, 0)) ∪ (A t + (30, 0)) and B := B c ∪ (B r + (10, 0)) ∪ (B d + (20, 0)) ∪ (B t + (30, 0)).
Actually, the directed Hausdorff distance is also at most as hard as the undirected Hausdorff distance (thus, they are equally hard), as δ H (A, B) = δ H(A ∪ B, B).
Note that we do not explicitly restrict the universe of the integers here. In the WordRAM model, we use the standard assumption that each integer in the input has bit complexity O(log n). In the RealRAM model, we can perform the common arithmetic operations on reals in constant time, so there is no need to restrict the universe. With these conventions, our reduction works in both models.
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| {'fraction_non_alphanumeric': 0.06641874945948284, 'fraction_numerical': 0.04600882124016259, 'mean_word_length': 3.710061099796334, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 27, 'https://': 2, 'lorem ipsum': 0, 'www.': 3, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 44, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible.Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size n and m, the Hausdorff distance under translation can be computed in timẽ O(nm) for the L1 and L∞ norm [Chew, Kedem SWAT'92] andÕ(nm(n + m)) for the L2 norm [Huttenlocher, Kedem, Sharir DCG'93].As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of (nm) 1−o(1) for L1 and L∞ (and all other Lp norms) assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of n 2−o(1) for L2 in the imbalanced case of m = O(1) assuming the 3SUM Hypothesis.ACM Subject ClassificationTheory of computation → Problems, reductions and completeness 1 There is a directed and an undirected variant of the Hausdorff distance, see Section 2. In this introduction, we do not differentiate between these two, since all our statements hold for both variants. 2 ByÕ-notation we ignore logarithmic factors in n and m.", 'arxivid': '2101.07696', 'author': ['Karl Bringmann [email protected] \nMax Planck Institute for Informatics\nSaarland Informatics Campus\nSaarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics\nSaarland University\nSaarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany\n', 'André Nusser [email protected] \nMax Planck Institute for Informatics\nSaarland Informatics Campus\nSaarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics\nSaarland University\nSaarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany\n'], 'authoraffiliation': ['Max Planck Institute for Informatics\nSaarland Informatics Campus\nSaarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics\nSaarland University\nSaarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany', 'Max Planck Institute for Informatics\nSaarland Informatics Campus\nSaarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics\nSaarland University\nSaarland Informatics CampusSaarbrücken, SaarbrückenGermany, Germany'], 'corpusid': 231639233, 'doi': '10.4230/lipics.socg.2021.18', 'github_urls': [], 'n_tokens_mistral': 20068, 'n_tokens_neox': 16691, 'n_words': 10351, 'pdfsha': 'b595de7f7632cfcddca760c2b7804fcda8bc50ae', 'pdfurls': ['https://arxiv.org/pdf/2101.07696v5.pdf'], 'title': ['Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation', 'Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation'], 'venue': []} |
arxiv |
Cylindrically polarized Bessel-Gauss beams
7 Oct 2014 (Dated: October 8, 2014)
Daena Madhi
Marco Ornigotti
Andrea Aiello
Max Planck Institute for the Science of Light
Günther-Scharowsky-Straße 1/Bau2491058ErlangenGermany
Erlangen Graduate School in Advanced Optical Technologies (SAOT)
Institute of Applied Physics
Friedrich-Schiller University
Paul-Gordan-Straße 6, Max-Wien Platz 191052, 07743Erlangen, Jena, JenaGermany, Germany
Max Planck Institute for the Science of Light
Günther-Scharowsky-Straße 1/Bau2491058ErlangenGermany
Institute for Optics, Information and Photonics
University of Erlangen-Nuernberg
Staudtstraße 7/B291058ErlangenGermany
Cylindrically polarized Bessel-Gauss beams
7 Oct 2014 (Dated: October 8, 2014)arXiv:1410.1359v2 [physics.optics]
We present a study of radially and azimuthally polarized Bessel-Gauss beams in both the paraxial and nonparaxial regimes. We discuss the validity of the paraxial approximation and the form of the nonparaxial corrections for Bessel-Gauss beams.We show that, independently from the ratio between the Bessel aperture cone angle ϑ 0 and the Gauss beam divergence θ 0 , the nonparaxial corrections are always very small and therefore negligible. Explicit expressions for the nonparaxial vector electric field components are also reported.
because of their peculiar properties such, e.g., the ability of producing a smaller focus [1].
Such beams demonstrated to be useful in various fields of research such as spectroscopy [2], microscopy [3], optical tweezing [4], material processing [5], propagation of linear and nonlinear waves in crystals [6][7][8] and quantum information [9]. This vast plethora of applications motivated the development of several different experimental techniques to generate such beams [10][11][12][13][14]. A detailed theoretical analysis of the properties of these beams and their application in the paraxial case can be found in Ref. [15].
Motivated by these many applications, different groups have then tried in the last years to provide a suitable extension of these beams to the nonparaxial case, by exploring the field of a strongly focused beam [16], using complex dipole sources [17], elegant Laguerre-Gauss beams in the nonparaxial regime [18] and vector Bessel beams [19]. Recently we also contributed to this subject by proposing a direct and simple generalization of the formalism introduced by Holleczek et al. [15], based on the use of Bessel beams to generate Hermite-Gaussian-like beams with zero total angular momentum [20].
Although Bessel beams are exact solutions of the Helmohltz equation, they are not physical states of the electromagnetic field, as they carry infinite energy [21]. Bessel-Gauss beams, on the other side, are also an exact solutions of the Helmholtz equation, but with a finite energy spectrum [22][23][24][25], a feature that makes it possible to realize such beams experimentally [26,27].
It is then the aim of this work to extend the results of Ref. [20] to the case of Bessel-Gauss beams by deriving the expressions for the electric field of cylindrically polarized beams of light both in the paraxial and nonparaxial case. Since Bessel-Gauss beams can be nowadays easily generated in an optical laboratory with the help of a suitably programmed spatial light modulator [28,29], we believe that the present work could serve as a toolbox to extend the framework of radially and azimuthally polarized states of light to the nonparaxial domain straightforwardly.
This work is organized as follows: in Sect. 2 we briefly revise the paraxial and nonparaxial form of Bessel-Gauss (BG) beams. These results are then used in Sect. 3 to generate the cylindrically polarized vector fields in the paraxial regime, according to the method presented in Ref. [20]. In Sect. 4, we briefly discuss the various regimes of BG beams and how strong is the influence of nonparaxial correction in all these regimes. In Sect. 5, the explicit expression of the vector electric and magnetic fields of cylindrically polarized Bessel-Gauss beams is given. Finally, conclusions are drawn in Sect. 6.
II. PARAXIAL AND NONPARAXIAL BESSEL-GAUSS BEAMS
As it is well known, Bessel beams carry infinite energy, and therefore they do not represent physically meaningful solutions of the Helmholtz equation [21]. This peculiar characteristic is intimately related to the fact that the support of the angular spectrum of such beams is a circle of zero thickness whose radius is given by K 0 = k 0 sin ϑ 0 (being ϑ 0 the characteristic cone angle of the Bessel beam) represented by the Dirac delta δ(K − K 0 ), a highly singular function. A more realistic description of such beams is represented by Bessel-Gauss beams, that can be thought as the equivalent of Bessel beams where the Dirac-delta circle in Fourier space is replaced by a finite Gaussian distribution. w 0 [22]. Another possible interpretation of BG beams is that they are given as a superposition of tilted Gaussian beams with waist w 0 whose axes of propagation are uniformly distributed on a surface of a cone of half aperture ϑ 0 [30]. In contrast with pure Bessel beams, however, BG beams are not diffractionless anymore, even if they maintain their diffractionless character up to a maximal distance D = w 0 / sin ϑ 0 [22], after which their Gaussian character dominates over the nondiffracting one given by the the Bessel part. Bessel-Gauss beams are, however, still an exact solution of the Helmholtz equation, i.e.,
(∇ 2 + k 2 0 )ψ ℓ (x, y, z) = 0,(1)
where k 0 = 2π/λ is the vacuum wave number. If we write the previous equation in cylindrical coordinates, BG solutions at z = 0 can be found according to Gori et al. [30] to be as follows:
ψ ℓ (R, ϕ, 0) = J ℓ (K 0 R) e − R 2 w 2 0 e iℓϕ ,(2)where K 0 = k 0 sin ϑ 0 , R = x 2 + y 2 , J l (x) isψ ℓ (K, φ) = 1 2π d 2 R ψ ℓ (R, ϕ, 0)e −iK·R = w 2 0 2i ℓ I ℓ w 2 0 KK 0 2 e − w 2 0 4 (K 2 +K 2 0 ) e iℓφ ,(3)
where d 2 R = dxdy, K = k 2 x + k 2 y , K x = K cos φ, K y = K sin φ, R = xx + yŷ and I ℓ (x) is the modified Bessel function of the first kind [31]. From the previous equation one can easily see that in the limit w 0 → ∞, Eq. (2) gives the traditional Bessel beam, as the Gaussian envelope goes to 1. Correspondingly, the angular spectrum defined in Eq. (3) becomes
lim w 0 →∞ψ ℓ (K, φ) = lim w 0 →∞ w 2 0 e iℓφ 2i ℓ I ℓ KK 0 2/w 2 0 e − K 2 +K 2 0 2/w 2 0 = e iℓφ i ℓ K 0 δ(K − K 0 ),(4)
where in order to calculate the limit we used the following asymptotic expression of the modified Bessel function of the first kind in the vicinity of infinite [31]:
I ν (z) ≃ e z √ 2πz 1 − 4ν 2 − 1 2 8z + · · · ,(5)
Equation (4) is therefore the correct limit that leads to the angular spectrum of a Bessel beams. To find the expression of the BG beam in the generic plane z > 0, we now propagate Eq. (3) according to the propagation rule of the angular spectrum [32], thus obtaining
ψ ℓ (R, z) = 1 2π d 2 Kψ ℓ (K, φ)e −iK·R e iz √ k 2 0 −K 2 = N ∞ 0 dK Ke − K 2 4/w 2 0 I ℓ − w 2 0 KK 0 2 J ℓ (KR)e izk 0 √ 1−K 2 /k 2 0 ,(6)
where
d 2 K = dk x dk y and N = (w 2 0 /2) exp [iℓφ − K 2 0 /(4/w 2 0 )
]. This expression is still exact but cannot be calculated analytically, due to the presence of the square root at the exponent of the last exponential function. However, in the paraxial limit one has that K/k 0 ≪ 1 and a Taylor expansion of the square root around K/k 0 = 0, i.e.,
1 − K 2 /k 2 0 ≃ 1 − 1 2 K k 0 2 + O K k 0 4 ,(7)
allows us to rewrite the angular spectrum propagator in the approximate form
e ik 0 z √ 1−K 2 /k 2 0 ≃ e ik 0 z e − izK 2 2k 0 ,(8)
where the quadratic phase factor is the so-called Fresnel propagator and it is responsible for the paraxial propagation [32]. With this in mind, we can now calculate from Eq. (6) the form of the BG beam in the paraxial limit and retrieve the nonparaxial corrections as higher order correction to the paraxial limit. In order to do so, we first need to isolate the Fresnel term from the exact propagator
e izk 0 √ 1−K 2 /k 0 = e izk 0 exp −iz K 2 2k 0 e izk 0 √ 1−K 2 /2k 0 e izk 0 exp −iz K 2 2k 0 ,(9)
and then perform a Taylor expansion of the nonparaxial part of the propagator (the one in square brakets in the previous equation), thus obtaining
e izk 0 √ 1−K 2 /k 0 e izk 0 exp −iz K 2 2k 0 ≃ 1 − ik 0 z 8 K k 0 4 − ik 0 z 16 K k 0 6 + · · · .(10)
By inserting this result into Eq. (6) we can then write the exact form of the BG beam in a series form as follows:
ψ ℓ (R, z) ≃ N ∞ 0 dK Ke −K 2 1 4/w 2 0 +i z 2k 0 I ℓ − w 2 0 KK 0 2/w 2 0 J ℓ (KR) × 1 − ik 0 z 8 K k 0 4 − ik 0 z 16 K k 0 6 ... , = e ik 0 z ψ (0) ℓ (x, y, z) + ψ (1) ℓ (x, y, z) + ψ (2) ℓ (x, y, z) + ... ,(11)
This expression allows us to evaluate all the expansion terms, the lowest one being the paraxial approximation and the higher ones being the nonparaxial corrections.
The paraxial Bessel-Gauss beam is then given by:
ψ (0) ℓ (R, ϕ, z) = N ∞ 0 dK Ke −K 2 1 4/w 2 0 +i z 2k 0 I ℓ − w 2 0 KK 0 2/w 2 0 J ℓ (KR) = e iℓϕ 1 + iζ exp − 1 1 + iζ (ρ 2 + iζΘ 2 ) J ℓ 2ρΘ 1 + iζ ,(12)
where ρ = R/w 0 , Θ = sin ϑ 0 /θ 0 and ζ = z/z R , with w 0 = 2z R /k 0 and θ 0 = 2/(k 0 w 0 ) being the waist and the angular aperture of the beam, respectively. This equation should be compared with Eq. (12) in Ref. [22]: an explicit calculation shows that Eq.(12) in Ref. [22] is incorrect as it does not satisfy the paraxial equation. This is the first result of our paper.
According to Eq. (11), the first nonparaxial correction can be written in the following simple compact form:
ψ (1) ℓ (R, ϕ, z) = −izN (2k 0 ) 3 ∞ 0 dK K 5 e −K 2 1 4/w 2 0 +i z 2k 0 I ℓ − w 2 0 KK 0 2 J ℓ (KR) = iz 2k 0 ∂ 2 ∂z 2 ψ (0) ℓ (x, y, z) ,(13)
The explicit expression of Eq. (13) evaluated for arbitrary ℓ is quite cumbersome and, for the sake of clarity, it will not be reported here. However, in the present work we are interested in the circumstances ℓ = ±1 solely and in these cases the formulas are much simpler:
ψ (1) ℓ (R, ϕ, z) ℓ=±1 = ∓ ζθ 2 0 (1 + iζ) 5 e − 1 1+iζ (ρ 2 +iζΘ 2 )±iϕ ρΘ 3 2 (1 + iζ) − (ρ 2 − Θ 2 ) J 0 2ρΘ 1 + iζ − 1 2 (1 + iζ)(ρ 2 − Θ 2 ) − 1 2 (ρ 4 − 6ρ 2 Θ 2 + Θ 4 ) J 1 2ρΘ 1 + iζ .(14)
III. CYLINDRICALLY POLARIZED PARAXIAL BESSEL-GAUSS BEAMS Now that we have correctly calculated the exact form of a paraxial BG beam and its nonparaxial corrections at all orders (each of them can be simply evaluated analytically thanks to the Gaussian form of the integrals), we can now build the Hermite-Gaussian-like BG beams, by combining the paraxial solutions with ℓ = 1 and ℓ = −1 as follows:
φ 10 (R, ϕ, z) = 1 √ 2 ψ (0) 1 (R, ϕ, z) + ψ (0) −1 (R, ϕ, z) = i √ 2 1 + iζ e − 1 1+iζ (ρ 2 +iζΘ 2 ) J 1 2ρΘ 1 + iζ sin ϕ,(15a)φ 01 (R, ϕ, z) = −i √ 2 ψ (0) 1 (R, ϕ, z) − ψ (0) −1 (R, ϕ, z) = − i √ 2 1 + iζ e − 1 1+iζ (ρ 2 +iζΘ 2 ) J 1 2ρΘ 1 + iζ cos ϕ,(15b)
where ψ
1 (R, ϕ, z) and ψ (0) −1 (R, ϕ, z) are defined by the Eq.(12) for ℓ = ±1 respectively. A sketch of the functon φ 10 (R, ϕ, z) in z = 0 and its comparison with the Hermite-Gaussian beam HG 10 (x, y) is reported in Fig. 1. As can be noted, the two functions have the same cartesian symmetry. Moreover, Fig. 1 also shows that unlike the case of real Bessel beams [20] [ Fig. 1(c)], BG beams do not present any rings outside the paraxial region. This is a consequence of the fact that their angular spectrum is tailored with a Gaussian function, instead of being a simple Dirac delta function.
In analogy with Ref. [15], we can then build a four dimensional space spanned by the basis formed by the Cartesian product of {φ 10 , φ 01 } mode basis defined above and the polarization vectors {x,ŷ}, namely Radially (û R ) and azimuthally (û A ) polarized beams can be then easily obtained as linear combinations of these four modes as follows:
{φ 10 , φ 01 } ⊗ {x,ŷ} = {φ 10x , φ 10ŷ , φ 01x , φ 01ŷ }.(16)u ± R = 1 √ 2 (±φ 10x + φ 01ŷ ),(17a)u ± A = 1 √ 2 (∓φ 01x + φ 10ŷ ),(17b)
where the ± sign refers to co-rotating and counter-rotating modes respectively [15]. The polarization patterns and the intensity profile of these paraxial modes are shown in Fig. 2 and 3.
IV. NONPARAXIAL CORRECTIONS
A Bessel-Gauss beam is characterized by two competing parameters: the Bessel cone angle ϑ 0 and the width w 0 of the Gaussian beam composing the spectrum or, alternatively, its angular spread θ 0 = 2/(k 0 w 0 ). Depending on the relative weight of these two parameters, according to Ref. [30] we can define three different regimes that are schematically represented in Fig. 4. The first of these regimes corresponds to ϑ 0 /θ 0 > 1 [Fig 4(a)]. In this regime, the Gaussian beam components are well separated and the spot size of each single component diffracts during the propagation along z. However, up to a distance D defined as the distance from z = 0 at which a Gaussian beam component has receded from the z-axis by a quantity w 0 [30], the beam remains diffractionless.
The second regime that we can analyze is given by ϑ 0 /θ 0 < 1, with ϑ 0 ≪ 1. In this case, as it is reported in detail in Ref. [30] for the fundamental BG beam, we expect that the central region of the beam (whose radius is approximately ξ m /K 0 , being ξ m the first zero of the function J m (ξ)) closely resembles the central part of a Gaussian beam, as the beam waist w 0 of the component gaussian beams is less than the central radius of the BG beam.
This correspond to the most paraxial situation. We therefore expect that in this case l (x, z) l=1 . As can be seen, the intensity of the first nonparaxial order of Eq. (11) is of the order of 10 −6 and it can be therefore neglected.
Although regarding the first case one could intuitively say that the contributions of higher order nonparaxial terms in Eq. (11) are higher than the second one, Fig. 4(d) shows that also in this case the nonparaxial corrections are negligible with respect to the paraxial part of the beam, having an intensity 10 6 times smaller that their paraxial counterpart.
For the sake of completeness, we present also the intermediate case ϑ 0 /θ 0 ≃ 1, where the component Gaussian beams overlap strongly during propagation [ Fig. 4(b)]. Also in this case, however, as it appears clear from Fig. 4(e), the effects of the nonparaxial corrections to Eq. (11) are negligible.
In all three regimes, ϑ 0 and θ 0 are in the paraxial regime. This is why the nonparaxial corrections contribution is negligible.
V. ELECTRIC AND MAGNETIC FIELDS
The modes obtained from Eqs. (17) and depicted in Figs. 2 and 3 are strictly paraxial.
As already explained in Ref. [20], however, sinceû R,A are paraxial modes, they are not exact solutions of the Helmholtz equation (1). In order to fix this problem, in principle, all the nonparaxial corrections to Eq. (13) must be take into account. Once the nonparaxial modes U ± R,A have been calculated by substituting Eq. (11) into the definition of the Hermite-Gauss-like beams given by Eq. (15) instead of ψ (0) ±1 , they can be used as Hertz vectors to determine the correct form of the electric and magnetic fields, according to the following equations [20]:
E(r, t) = ∇ × [∇ × Π(r, t)] , B(r, t) = 1 c 2 ∂ ∂t [∇ × Π(r, t)] .(18)
where Π(r, t) =Û ± R,A exp(−iωt) depending on which kind of polarization one wants to attribute to the fields. However, as we discussed in the previous section, the nonparaxial corrections are always very small and they can be neglected irrespectively on the relative weight between the two characterizing parameters of a BG beam, namely ϑ 0 and θ 0 . It is therefore sufficient to use the paraxial modes u ± R,A given by Eq. (17) as Hertz potentials to generate the nonparaxial electric and magnetic fields. Here we report the explicit expression of the components (in normalized cylindrical coordinates {ρ, φ, ζ}) of the electric field for all the four cylindrically polarized modes, as deriving from Eq. (18), whose explicit expression reads as follows:
E ρ R+ (r, t) = i (ζ − i) 5 i −2ρ 2 iζ + 3Θ 2 + 1 + Θ 2 2iζ + Θ 2 + 2 + ρ 4 I 1 2Θρ −i + ζ − 2Θρ 3iζ + 2Θ 2 − 2ρ 2 + 3 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) ,(19a)E φ R+ (r, t) = 0, (19b) E ζ R+ (r, t) = 1 (ζ − i) 4 2ρ iζ + 3Θ 2 − ρ 2 + 1 I 1 2Θρ −i + ζ − 2iΘ 2iζ + Θ 2 − 3ρ 2 + 2 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) ,(19c)
for the co-rotating radially polarized electric field,
E ρ R− (r, t) = − i cos(2φ) (ζ − i) 5 ρ 2 i 2ρ 4 −iζ − 3Θ 2 − 1 + ρ 2 2(1 + iζ)Θ 2 − 4i(ζ − i) 3 + Θ 4 + 4(ζ − i) 4 + ρ 6 I 1 2Θρ −i + ζ − 2Θρ ρ 2 3iζ + 2Θ 2 + 3 + 2(−1 − iζ) 3 − 2ρ 4 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) (20a) E φ R− (r, t) = − i sin(2φ) (ζ − i) 5 ρ 2 i 2ρ 4 −2ζ 2 + 5iζ + 3Θ 2 + 3 + ρ 2 2(−3 + ζ(2ζ − 5i))Θ 2 − 4i(ζ − i) 3 − Θ 4 + 4(ζ − i) 4 − ρ 6 I 1 2Θρ −i + ζ + 2Θρ ρ 2 ζ(−4ζ + 11i) + 2Θ 2 + 7 − 2i(ζ − i) 3 − 2ρ 4 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) (20b) E ζ R− (r, t) = 2 cos(2φ) (ζ − i) 4 ρ −3iζ − 3Θ 2 + ρ 2 − 3 I 1 2Θρ −i + ζ + iΘ Θ 2 − 3ρ 2 I 2 2Θρ −i + ζ e χ(ρ,Θ,ζ,t)(20c)
for the counter-rotating radially polarized electric field,
E ρ R− (r, t) = 0, (21a) E φ R− (r, t) = i (ζ − i) 5 i Θ 2 2ζ(−2ζ + 5i) − 6ρ 2 + 6 + ρ 2 2ζ(2ζ − 5i) + ρ 2 − 6 + Θ 4 I 1 2Θρ −i + ζ − 2Θρ ζ(−4ζ + 11i) + 2Θ 2 − 2ρ 2 + 7 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) (21b) E ζ R− (r, t) = 0,(21c)
for the co-rotating azimuthally polarized electric field
E ρ A− (r, t) = sin(2φ) (ζ − i) 5 ρ 2 2ρ 4 −iζ − 3Θ 2 − 1 + ρ 2 2(1 + iζ)Θ 2 − 4i(ζ − i) 3 + Θ 4 + 4(ζ − i) 4 + ρ 6 I 1 2Θρ i − ζ − 2iΘρ ρ 2 3iζ + 2Θ 2 + 3 + 2(−1 − iζ) 3 − 2ρ 4 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) (22a) E φ A− (r, t) = i cos(2φ) (ζ − i) 5 ρ 2 − i 2ρ 4 −2ζ 2 + 5iζ + 3Θ 2 + 3 + ρ 2 2(−3 + ζ(2ζ − 5i))Θ 2 − 4i(ζ − i) 3 − Θ 4 + 4(ζ − i) 4 − ρ 6 I 1 2Θρ −i + ζ − 2Θρ ρ 2 ζ(−4ζ + 11i) + 2Θ 2 + 7 − 2i(ζ − i) 3 − 2ρ 4 I 0 2Θρ −i + ζ e χ(ρ,Θ,ζ,t) (22b) E ζ A− (r, t) = 2 sin(2φ) (ζ − i) 4 ρ 3iζ + 3Θ 2 − ρ 2 + 3 I 1 2Θρ −i + ζ − iΘ Θ 2 − 3ρ 2 I 2 2Θρ −i + ζ e χ(ρ,Θ,ζ,t)(22c)
for the counter-rotating azimuthally polarized electric field. In all these expressions
χ(ρ, Θ, ζ, t) = −ζΘ 2 + iρ 2 ζ − i − iωt,(23)
VI. CONCLUSIONS
In this work we have theoretically investigated the cylindrically polarized modes associated to Bessel-Gauss beams. We have derived the correct paraxial form of a BG beam in a plane z = 0 by propagating the angular spectrum and we have expressed the full nonparaxial BG field as a paraxial contribution ψ (0) ℓ (R, ϕ, z) plus a series of nonparaxial corrections and we have analyzed their role in three different regimes defined by the ratio ϑ 0 /θ 0 . We have shown that independently on the considered regime (corresponding to how much nonparaxial the BG beam is), the nonparaxial corrections are always very small and therefore their contribution can be neglected.
the Bessel function of the first kind and (R, ϕ, z) are the usual cylindrical coordinates defined with respect to the main axis of propagationẑ. The angular spectrum at z = 0 is then obtained by taking the 2D Fourier transform of Eq. (2), namelỹ
FIG. 1 :
1(a) Contour plot of the scalar function φ 10 (R, ϕ, 0) close to the propagation axes. (b) Contour plot of the Hermite-Gauss beam HG 10 (x, y, 0). (c) Contour plot of the scalar function Ψ 10 (R, ϕ, 0) (as defined in Ref. [20]) analogous to φ 10 (R, ϕ, 0) but defined by using Bessel beams instead of BG beams. A direct comparison between panels (a) and (b) shows that in the paraxial domain φ 10 correctly reproduces the behavior of the Hermite-Gauss beam HG 10 . A comparison between panels (a) and (c) show that the introduction of a Gaussian envelope in the angular spectrum of a BG beam makes the nonparaxial ring to disappear, resulting in a beam carrying finite energy but preserving the same symmetry of Ψ 10 .
FIG. 2 :
2Non-uniform polarization patterns of (a) co-rotating radially polarized paraxial modeu + R and (b) counter-rotating radially polarized paraxial mode u − R , underlayed with the doughnut shaped intensity distribution. The axes of both span the interval [-2,2] in units of beam waist w 0 .
[Fig. 4(c)] the contribution of the nonparaxial corrections would be negligible. To show this, in FIG. 3: Complex polarization patterns of (a) co-rotating azimuthally polarized paraxial mode u + A and (b) counter-rotating azimuthally polarized paraxial mode u − A , underlayed with the donut shaped intensity distribution. The axes of both span the interval [-2,2] in units of beam waist w 0 .
Fig. 4 (
4f) we report a section (along the plane y = 0) of the scalar first order correction ψ(1)
FIG. 4 :
4Upper row: Schematic representation of the three regimes of BG beams considered in Sect. 4 depending on the ratio between the Bessel cone angle ϑ 0 and the Gaussian beam divergence θ 0 . (a) ϑ 0 /θ 0 > 1, (b) ϑ 0 /θ 0 ≃ 1 and (c) ϑ 0 /θ 0 < 1. Lower row: Three dimensional plot of the (x, z)-section intensity of the first nonparaxial correction ψ three cases (d) ϑ 0 /θ 0 = 10, (e) ϑ 0 /θ 0 = 1 and (f) ϑ 0 /θ 0 = 0.1. In all the figures of the lower panel, the x-direction has been normalized to the beam waist w 0 while the z-direction to the correspondent Rayleigh range z R . The vertical scale, in units of 10 −6 , is arbitrary, but the same for all the three plots.
r
= {ρr + φφ + ζẑ} and I l (x) are the modified Bessel functions of the first kind, which are related with the usual Bessel functions J l (x) by the relations I l (x) = (−i) l J l (ix) [31]. The calculation of the explicit expression of the components of the magnetic field is left to the reader.
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arxiv |
Relativistic Chiral Kinetic Theory from Quantum Field Theories
17 Apr 2017
Yoshimasa Hidaka
Theoretical Research Division
Nishina Center
RIKEN
351-0198WakoSaitamaJapan
Shi Pu
Department of Physics
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Di-Lun Yang
Theoretical Research Division
Nishina Center
RIKEN
351-0198WakoSaitamaJapan
Relativistic Chiral Kinetic Theory from Quantum Field Theories
17 Apr 2017(Dated: April 18, 2017)arXiv:1612.04630v2 [hep-th]
The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with sidejump phenomena further influenced by background fields and collisions.Introduction.-Novel quantum transport processes in Weyl fermionic systems have been widely investigated, in particular for the so-called chiral magnetic and vortical effects such that charged currents are induced by magnetic and vortical fields [1-3]. These effects associated with quantum anomaly have been studied from different theoretical approaches including relativistic hydrodynamics[4][5][6][7], lattice simulations[8][9][10][11][12], and gauge/gravity duality[13][14][15]. These effects might be (in-)directly observed in heavy ion collisions [16] and in condensed matter systems such as Weyl semimetals[17].From both theoretical and experimental perspectives, it is imperative to understand these anomalous effects in non-equilibrium conditions. One promising approach is kinetic theory, which can delineate non-equilibrium transport of a particle when the interaction and background fields are sufficiently weak. Nevertheless, it is hard to incorporate anomalous effects through the standard Boltzmann equations[6]. The chiral kinetic theory (CKT), which describes anomalous transport of Weyl fermions, has been thus developed from the path-integral[18],Hamiltonian [19], and local-equilibrium quantum kinetic approaches[20,21]. In such formalism, the effective velocity and forces for a single particle are modified by the Berry curvature Ω p = p/(2|p| 3 ), where p represents the spatial momentum of the particle, which originates from the Berry phase in an adiabatic process[22]. Further generalization to massive Dirac fermions can be found in Ref.[23]. In order to bridge the semi-classical approaches[18,19]and quantum field theories, the CKT is also derived from Wigner functions in the high-density effective theory[24](see also Ref. [25] for relevant study of the on-shell effective field theory.)However, there still exist potential issues in the chiral kinetic equation. First, the field-theory derivation in Refs.[24] and[20,21]are subject to a predominant chemical potential and local equilibrium, respectively. The derivation for more general systems beyond local equilibrium is thus needed. Second, the non-manifestation of Lorentz invariance in the chiral kinetic equation has been recently discussed in Refs.[26,27]from the semi-
The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with sidejump phenomena further influenced by background fields and collisions.
Introduction.-Novel quantum transport processes in Weyl fermionic systems have been widely investigated, in particular for the so-called chiral magnetic and vortical effects such that charged currents are induced by magnetic and vortical fields [1][2][3]. These effects associated with quantum anomaly have been studied from different theoretical approaches including relativistic hydrodynamics [4][5][6][7], lattice simulations [8][9][10][11][12], and gauge/gravity duality [13][14][15]. These effects might be (in-)directly observed in heavy ion collisions [16] and in condensed matter systems such as Weyl semimetals [17].
From both theoretical and experimental perspectives, it is imperative to understand these anomalous effects in non-equilibrium conditions. One promising approach is kinetic theory, which can delineate non-equilibrium transport of a particle when the interaction and background fields are sufficiently weak. Nevertheless, it is hard to incorporate anomalous effects through the standard Boltzmann equations [6]. The chiral kinetic theory (CKT), which describes anomalous transport of Weyl fermions, has been thus developed from the path-integral [18], Hamiltonian [19], and local-equilibrium quantum kinetic approaches [20,21]. In such formalism, the effective velocity and forces for a single particle are modified by the Berry curvature Ω p = p/(2|p| 3 ), where p represents the spatial momentum of the particle, which originates from the Berry phase in an adiabatic process [22]. Further generalization to massive Dirac fermions can be found in Ref. [23]. In order to bridge the semi-classical approaches [18,19] and quantum field theories, the CKT is also derived from Wigner functions in the high-density effective theory [24] (see also Ref. [25] for relevant study of the on-shell effective field theory.)
However, there still exist potential issues in the chiral kinetic equation. First, the field-theory derivation in Refs. [24] and [20,21] are subject to a predominant chemical potential and local equilibrium, respectively. The derivation for more general systems beyond local equilibrium is thus needed. Second, the non-manifestation of Lorentz invariance in the chiral kinetic equation has been recently discussed in Refs. [26,27] from the semi-classical approach. The authors propose that the ordinary Lorentz transformation is modified by "side-jump" phenomena necessary to ensure angular-momentum conservation in collisions, while the same scenario is not fully understood in quantum field theories. Furthermore, it is more systematic to incorporate collisions in the fieldtheory framework.
In this letter, we address the aforementioned issues related to Weyl fermions in quantum field theories. It turns out that these issues are in fact connected. By solving Dirac equations, nontrivial side-jump terms coupled to background fields and self-energy from collisions naturally appear in the perturbative solution for Wigner functions up to O( ) in terms of expansion (equivalent to the gradient expansion as the long-wavelength approximation), which allude to modified Lorentz transformation of coordinates and momenta for distribution functions. The findings based on field theories support the modified Lorentz transformation proposed in [26,27] and further incorporate the effects from background fields and collisions simultaneously. For free fermions, we further show that the side-jumps are related to reparametrization of distribution functions. From field theories, a self-consistent expression of the CKT with collisions and background fields is systematically derived.
Side-jumps from Wigner functions.-We start from Dirac equations for non-interacting right-handed Weyl fermions,
/ D x S <(>) (x, y) = S <(>) (x, y) / D † y = 0,(1)
where / D = σ µ D µ with D µ = (∂ µ + iA µ / ) and σ µ = (1, σ) for σ i being Pauli matrices. Here S < (x, y) = ψ † (y)ψ(x) and S > (x, y) = ψ(x)ψ † (y) correspond to lesser and greater propagators in position space and hereafter we focus on just S < (x, y). In this Letter, we include the electric charge into the definition of the gauge field. We take the mostly negative Minkowski metric for convention. By taking Y = x − y and X = (x + y)/2, we carry out the Wigner transformatioǹ
S < (q, X) = d 4 Y e iq·Y S < (x, y),(2)
where the gauge link is implicitly embedded and q µ denotes the canonical momentum. The Dirac equations after Wigner transformation up to O( ) become
σ µ ( ∆ µ − 2iq µ )S < = 0, ( ∆ µ + 2iq µ )S < σ µ = 0, (3)
where ∆ µ = ∂ µ + F νµ ∂/∂q ν and ∂ µ = ∂/∂X µ [28]. Adding and subtracting two equations above give the difference equations:
σ µ , ∆ µS < − 2i q µ σ µ ,S < = 0, σ µ , ∆ µS < − 2i q µ σ µ ,S < = 0,(4)
where
{A, B} = AB + BA and [A, B] = AB − BA.
Note that these equations are up to all orders of for constant background fields.
The task now is to solve Eq. (4) perturbatively including quantum corrections up to O( ). We may takè
S < =σ µS< µ ,S < µ =S (0)< µ + δS < µ ,(5)
whereσ µ = (1, −σ). It is trivially to show that the leading-order solution readsS
(0) µ = (2π)q µ δ(q 2 )f and q µ ∆ µ f = 0 up to O(1), where f (q, X) denotes the distri- bution function.
For simplicity, we first consider the case that f = 1 as a constant. The first equation in Eq. (4) then results in
(F ij q 0 q j − F 0i |q| 2 + F j0 q i q j ) ∂δ(q 2 ) ∂q 2 = ǫ ijk q j δS < k 2π ,(6)
where we apply q 2 ∂δ(q 2 )/∂q 2 = −δ(q 2 ) in the derivation. From Eq. (6), one finds the O( ) correction for constant f takes the form,
S c< µ = (2π) q µ δ(q 2 ) + ǫ µναβ q ν F αβ ∂δ(q 2 ) 2∂q 2 f,(7)
where ǫ µναβ represents the Levi-Civita tensor with ǫ 0123 = ǫ 123 = 1. The superscript c denotes that the spectral density (the part aside from f ) is Lorentz covariant. Nevertheless, when f is non-constant, this is not the complete solution.
Coming back to the Dirac equations in Eq. (4), it is then more convenient to separate the trace and traceless part (linear to σ i ), which yields
∆ µS µ< = 0, q µS< µ = 0,(8)
and
∆ [iS < 0] − 2ǫ ijk q jS < k = 0, ǫ ijk ∆ jS < k + 2q [iS < 0] = 0.(9)
Now the perturbative solution is actually solved from the traceless part in Eq. (9), while q µS< µ = 0 gives the constraint and ∆ µS <µ = 0 yields the kinetic theory. It turns out that Eq. (9) results in the side-jump term on top of Eq. (7). From Eq. (9), one finds [29]
S < µ =S c< µ + δS f < µ ,(10)δS f < µ = 2πδ µi ǫ ijk δ(q 2 )q j 2q 0 ∆ k f,(11)
where the superscript f here denotes that the spectral density of the side-jump term δS f < µ is not Lorentz covariant (frame dependent) [30]. One may recognize that δS f < µ corresponds to the magnetic-moment coupling in the absence of background fields in Refs. [24,26] or the vorticity-related term in local equilibrium [20,21]. The solution in Eq. (10) is not unique: We may further introduce arbitrary terms proportional to q µ δ(q 2 ), while they could be absorbed into f . This degeneracy is actually pertinent to Lorentz covariance of Wigner functions, which will be discussed later.
Chiral kinetic theory.-By inserting the perturbative solution up to O( ) in Eq. (10) into Eq. (8), q µS <µ = 0 is trivially satisfied, whereas ∆ µS <µ = 0 gives the CKT. In the following, we present some critical steps of the derivation. Now, ∆ µS <µ = 0 leads to
δ(q 2 )q µ + ǫ µναβ q ν F αβ ∂δ(q 2 ) 2∂q 2 ∆ µ f + ǫ 0βνµ F αβ ∂ ∂q α q ν δ(q 2 ) 2q 0 ∆ µ f + ǫ 0βνµ q ν δ(q 2 ) 2q 0 (∂ β F αµ ) ∂ ∂q α f = 0.(12)
For simplicity, we consider only the particle with positive energy. Performing some computations, we obtain
δ q 2 + B · q q 0 q µ ∆ µ − ∂ k (B · q) 2q 0 ∂ ∂q k + ǫ ijk E i q j 2q 2 0 ∆ k f = 0,(13)
where E i ≡ F i0 and B i ≡ −ǫ ijk F jk /2 (equivalently F ij = ǫ ijk B k ), and we utilize ∇ · B = 0. The on-shell condition q 2 + B · q/q 0 = 0 now implies the shift of energy,
q 0 = ǫ q = |q| 1 − B · q 2|q| 3 .(14)
We hereby introduce an effective velocitỹ
v = ∂ǫ q ∂q = q |q| − B 2|q| 2 + (B · q)q |q| 4 .(15)
Eventually, we reproduce the CKT in Ref. [24],
(1 + B · Ω q ) ∂ t + (ṽ + E × Ω q + (ṽ · Ω q )B) · ∇ + Ẽ +ṽ × B + (Ẽ · B)Ω q · ∂ ∂q f = 0,(16)
whereẼ = E − ∇ǫ q . As a cross check, we may also evaluate the currents from the perturbative solution in Eq. (10). The particle number density is given by
J 0 = d 4 q (2π) 4 Tr S < = d 3 q (2π) 3 (1 + B · Ω q ) f.(17)
Also, the spatial current reads
J = d 4 q (2π) 4 Tr σS < (18) = d 3 q (2π) 3 ṽ + E × Ω q − ǫ q B Ω q · ∂ ∂q − ǫ q Ω q × ∇ f,(19)
where we take the integration by part. The current incorporates the anomalous Hall effect, chiral magnetic effect, and side-jump as shown in Ref. [24]. We notice that the expression in our formalism is valid up to O( ). Lorentz invariance.-Although we derive the CKT and anomalous effects from the perturbative solution in Eq. (10), one may be concerned about the Lorentz covariance of Wigner functions due to the side-jump term. Since the Wigner function should be Lorentz covariant, the existence of the side-jump term suggests that f is not a Lorentz scalar [26,27]. To understand the modifications of Lorentz transformation on f , it is instructive to manifest the frame (in)dependence of the side-jump term. Accordingly, we may generalize Eq. (10) as [31]
δS f < µ = 2πδ(q 2 )ǫ µναβ q α u β 2q · u ∆ ν f (u) ,(20)
where u µ is the four velocity of a frame. The solution in Eq. (10) corresponds to u µ = (1, 0). Now, one may consider a Lorentz transformation X ′µ = Λ µ ν X ν and q ′µ = Λ µ ν q ν , which is in fact equivalent to the transformation between frames u ′µ = (Λ −1 ) µ ν u ν . From Eq. (20), by taking f ′ (u) (q ′ , X ′ ) = f (u) (q, X) + δf (u) (q, X), we find
(Λ −1 ) ν µS ′< ν (q ′ , X ′ ) −S < µ (q, X) (21) = 2πδ(q 2 ) q µ δf (u) + ǫ µναβ q α u ′β 2q · u ′ − q α u β 2q · u ∆ ν f (u) .
Since Wigner functions are Lorentz covariant, we should have (Λ −1 ) ν µS ′ < ν −S < µ = 0, which is equivalent to the frame-independence of currents. Making contraction with u µ , Eq. (21) gives rise to
f ′(u) (q ′ , X ′ ) = f (u) (q, X) + N µ uu ′ ∆ µ f (u) (q, X) ,(22)
where
N ν uu ′ = − ǫ µναβ q α u ′ β u µ 2(q · u ′ )(u · q) .(23)
This suggests that a particle makes following side-jumps in the phase space, X µ → X µ + N µ uu ′ and q µ → q µ + N ν uu ′ F µν , under the Lorentz transformation. These sidejumps take the same form as those in the path-integral approach [26,27].
To better understand the origin of side-jumps and frame dependence of distribution functions, we consider a free fermions with positive energy in quantum field theory. To discuss the side-jump, we consider a Lorentz transformation Λ of the wave function for massless particles, which non-trivially transforms with an extra phase [32]: |p, λ → e −iΦ(p,Λ) |Λp, λ , where λ represents helicity. The Lorentz transformation of the wave function of a particle with positive energy thus takes the form,
v + (Λp) = e iΦ(p,Λ) U (Λ)v + (p),(24)
where v + (p) = |p|+p 3
p 1 +ip 2 √ (|p|+p 3 )(25)
with the relativistic normalization v † + (p)v + (p) = 2|p|. One can accordingly write down the second quantization of a field operator as
ψ(x) = d 3 p (2π) 3 2|p| e −ip·x v + (p)a p ,(26)
where a ( †) p correspond to annihilation(creation) operators. For simplicity, we drop anti-fermions, which carry negative energy.
Considering
G µ (p ′ , p) = v † + (p ′ )σ µ v + (p)
, which is proportional toS <µ , the Lorentz transformation leads to
G µ (p ′ , p) → e i(Φ(p,Λ)−Φ(p ′ ,Λ)) Λ µ ν G ν (p ′ , p). (27)
Therefore, G µ (p ′ , p) is not a vector. Nonetheless, the extra phase does not contribute to any physical observables. For example, perturbation theory in thermal equilibrium. A free propagator has p ′ = p, so that the phase cancels. In contrast, in non-equilibrium, the phase cancels with the nontrivial transformation of the distribution function, i.e., N (p ′ , p) ≡ a † p ′ a p transforms as N (p ′ , p) → e −i(Φ(p,Λ)−Φ(p ′ ,Λ)) N (p ′ , p), due to the Lorentz covariance of field operators.
To make N (p ′ , p) a scalar, we introduce a phase field in momentum space as φ(p) such that φ(p) → φ ′ (Λp) = φ(p)−Φ(p, Λ) under the Lorentz transformation. We can always introduce such a phase using gauge degrees of freedom associated with a transformation that keeps v + (p)a p invariant: v + (p) → e iφ(p) v + (p) and a p → e −iφ(p) a p . We hereby define the Lorentz-scalar distribution functioň
N (p ′ , p) ≡ e −i(φ(p)−φ(p ′ )) N (p ′ , p).(28)
Using Eq. (26) and carrying out the Wigner transformation, we find
S < µ (q, X) = (2π)θ(q 0 )δ(q 2 ) q µ 1 − (∂ ν q φ − a ν )∂ ν + δ µi ǫ ijk q j 2|q| ∂ k f (q, X) ,(29)
where a ν ≡ ic † + (q)∂ ν q c + (q) denotes the Berry connection with c + (q) = v + (q)/ 2|q| from non-relativistic normalization and we defině (30) as a Lorentz scalar, wherep 0 =p · q/|q|. Nonetheless, for the CKT, we apply the parametrization f (q, X) = f q µ , X µ − ∂ µ q φ(q) + a µ . The distribution function f (q, X) is apparently not Lorentz invariant.
f (q, X) ≡ d 3p (2π) 3Ň q −p 2 , q +p 2 e −ip·X
Collisions.-In quantum field theories, we may systematically incorporate collisions (see e.g. Ref. [33]). Based on the Dyson-Schwinger equation and taking integration along the Schwinger-Keldysh contour, the Dirac equations are given by
i / D x − Σ δ (x) S < (x, y)(31)= ∞ −∞ d 4 z Σ R (x, z)S < (z, y) − Σ < (x, z)S A (z, y) , S < (x, y) −i / D † y − Σ δ (y) = ∞ −∞ d 4 z S < (x, z)Σ A (z, y) − S R (x, z)Σ < (z, y) ,
where Σ <(>) represent the self-energy and Σ δ denotes the one-particle potential and the subscripts R/A correspond to retarded/advanced propagators, which are de- where we set Σ = σ µ Σ µ without the loss of generality and denote < (>) explicitly since the greater propagator is also involved. On the other hand, the traceless part is given by
fined as S R (x, y) ≡ iθ(x 0 − y 0 )S + (x, y) and S A (x, y) ≡ −iθ(y 0 −x 0 )S + (x, y) with S + (x, y) = S > (x, y)+S < (x, y).∆ [iS < 0] − 2ǫ ijk q jS < k = Σ < [iS > 0] − Σ > [iS < 0] ,(33)ǫ ijk ∆ jS < k + 2q [iS < 0] = ǫ ijk Σ < jS > k − Σ > jS < k .
We may now solve for the perturbative solution. Compared to the collisioneless solution in Eq. (10), we find thatS c< µ is unchanged, whereas the side jump term becomes
δS f < µ = (2π)δ µi ǫ ijk δ(q 2 ) q j 2q 0 (∆ k f − C k ) ,(34)
where C β [f ] = Σ < βf − Σ > β f withf denoting the distribution function of outgoing particles.
Since the side-jump term is altered by collisions, we investigate the modified Lorentz transformation of distribution functions. We may generalize the side-jump term as
δS f < µ = 2πδ(q 2 )ǫ µαβν q α u β 2q · u ∆ ν f (u) − C ν [f (u) ] . (35)
The Lorentz covariance of Wigner functions yields the modified Lorentz transformation on the distribution function as
f ′(u) = f (u) + N µ uu ′ ∆ µ f (u) − C µ [f (u) ] . (36)
From Eqs. (32) and (34), we also obtain the CKT with collisions, for u µ = (1, 0), (37) where ∆ <(>) = ∆ + Σ <(>) and CKT 0 corresponds to the left-hand side of the collisionless CKT in Eq. (16).
CKT 0 − (1 + B · Ω q ) C 0 +(ṽ + E × Ω q + (ṽ · Ω q )B) · C − ǫ q Ω q · f (∆ > × Σ < ) − f (∆ < × Σ > ) = 0,
2-2 scattering without background fields.-Practically, it is more useful to make further approximations of the self-energy and analyze specific collisional processes. For simplicity, here we present the leading-order 2-2 Coulomb scattering between right-handed fermions with positive energy in the absence of background fields as an example. In this particular case, we will show that the center of mass frame corresponds to a "no-jump frame" as proposed in Ref. [27]. Now, the perturbative solution reduces tõ
S < µ = 2πδ(q 2 ) q µ f − 2q · u ǫ µναβ u ν q α ∂ β f +Σ >β f − Σ <βf ,(38)
where we write down the frame dependence for the spectral density. The self-energy in the leading contribution corresponding to the Coulomb scattering can be expressed as
Σ < µ = q ′ ,k,k ′ P(q ′ , k, k ′ )S > µ (q ′ ) S < (k) ·S < (k ′ ) ,
and similar for Σ > µ by exchanging > and <, where
P(q ′ , k, k ′ ) = 4e 4 1 (q − k) 2 + 1 (q − k ′ ) 2 2 ,(39)
and
q ′ ,k,k ′ ≡ d 4 q ′ d 4 kd 4 k ′ (2π) 8 δ (4) (q + q ′ − k − k ′ ).(40)
Although the self-energy takes a complicated form, by choosing the the center of mass frame u µ c ≡ (q + q ′ ) µ / √ s = (k + k ′ ) µ / √ s with s = (q + q ′ ) 2 , one can show that the expression is considerably simplified as
Σ < µ (41) = q ′ ,k,k ′ P(q ′ , k, k ′ )(k · k ′ ) q ′ µf (uc) (q ′ ) − ǫ µναβ q ν q ′α ∂ βf (uc) (q ′ ) 2q · q ′ f (uc) (k)f (uc) (k ′ ), where q ′ ,k,k ′ = d 3 q ′ d 3 kd 3 k ′ (2π) 5 δ (4) (q + q ′ − k − k ′ ) 8E q ′ E k E k ′ .(42)
Since bothS >µ and Σ < µ are frame independent, we can write down the distribution functions of all scattering particles in the the center of mass frame, which yields
S >µ Σ < µ = 2πδ(q 2 ) q ′ ,k,k ′ P(q ′ , k, k ′ )(k · k ′ )(q · q ′ ) ×f (uc) (q)f (uc) (q ′ )f (uc) (k)f (uc) (k ′ ),(43)
where the O( ) correction vanishes and similar for S <µ Σ > µ . Therefore, in the the center of mass frame, the kinetic theory from the first equation in Eq. (32) reduces to
∂ µS µ< = 2πδ(q 2 )q µ C µ f (uc) ,(44)
where q µ C µ f (uc) = 1 4 q ′ ,k,k ′ |M| 2 (45)
× f (uc) (q)f (uc) (q ′ )f (uc) (k)f (uc) (k ′ ) −f (uc) (q)f (uc) (q ′ )f (uc) (k)f (uc) (k ′ )
with |M| 2 = 4P(q ′ , k, k ′ )(k ·k ′ )(q ′ ·q) comprises "no sidejumps" as a standard collisional kernel This result supports that the the center of mass frame corresponds to a "no-jump frame" previously proposed in Ref. [27]. The details of computations incorporating background fields will be presented elsewhere.
Outlook.-Our findings not only provide a solid footing for side-jump phenomena associated with Lorentzsymmetry properties of CKT from field theories, but also raise some new issues. It turns out that a frameindependent distribution function can be introduced, but the trade-off is the modification of the spectral density and the corresponding CKT. However, in such a case, the CKT should be manifestly Lorentz invariant, whereas the derivation is nontrivial due to the presence of background fields, which will be pursued in the future.
Furthermore, the Wigner functions and the CKT influenced by collisions, can be applied to distinct systems with proper approximations of the self-energy, which may intrigue further studies for phenomenological interests.
On the other hand, it is straightforward to include higherorder quantum corrections in our approach, which may reveal novel effects in a self-consistent fashion.
In general, Σ R/A are complex, where the imaginary parts contribute to the scattering cross-section and the real parts result in renormalization of propagators. Since we only focus on the scattering process, we may drop the real parts by setting Σ δ = Re[Σ R/A ] = 0 and take Re[S R ] = 0 for the same reason. After making the Wigner transformation of Eq. (31), the trace of difference equations leads
Acknowledgments.-We thank M. Stephanov for useful discussions. Y.H. was partially supported by JSPS KAKENHI Grants Numbers 15H03652, 16K17716 and the RIKEN interdisciplinary Theoretical Science (iTHES) project. S.P. is supported by JSPS postdoctoral fellowship for foreign researchers. D.Y. was supported by the RIKEN Foreign Postdoctoral Researcher program.
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in order to show the solution satisfies the first equation in Eq. ) satisfies Eqs. (8) and. Also, one can show that Eqin order to show the solution satisfies the first equation in Eq. (9). Also, one can show that Eq. (7) satisfies Eqs. (8) and (9) for constant f .
Note that the gradient expansion implicitly introduces an infrared cutoff for |q|. which has to be larger than the scale of gradient or background fieldsNote that the gradient expansion implicitly introduces an infrared cutoff for |q|, which has to be larger than the scale of gradient or background fields.
One can in fact consider the frame dependence of Eq. (9) and check that the following solution indeed satisfies both equations therein. One can in fact consider the frame dependence of Eq. (9) and check that the following solution indeed satisfies both equations therein.
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| {'fraction_non_alphanumeric': 0.10359836528557663, 'fraction_numerical': 0.06445825165298867, 'mean_word_length': 3.657691117301145, 'pattern_counts': {'":': 0, '<': 81, '<?xml version=': 0, '>': 22, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with sidejump phenomena further influenced by background fields and collisions.Introduction.-Novel quantum transport processes in Weyl fermionic systems have been widely investigated, in particular for the so-called chiral magnetic and vortical effects such that charged currents are induced by magnetic and vortical fields [1-3]. These effects associated with quantum anomaly have been studied from different theoretical approaches including relativistic hydrodynamics[4][5][6][7], lattice simulations[8][9][10][11][12], and gauge/gravity duality[13][14][15]. These effects might be (in-)directly observed in heavy ion collisions [16] and in condensed matter systems such as Weyl semimetals[17].From both theoretical and experimental perspectives, it is imperative to understand these anomalous effects in non-equilibrium conditions. One promising approach is kinetic theory, which can delineate non-equilibrium transport of a particle when the interaction and background fields are sufficiently weak. Nevertheless, it is hard to incorporate anomalous effects through the standard Boltzmann equations[6]. The chiral kinetic theory (CKT), which describes anomalous transport of Weyl fermions, has been thus developed from the path-integral[18],Hamiltonian [19], and local-equilibrium quantum kinetic approaches[20,21]. In such formalism, the effective velocity and forces for a single particle are modified by the Berry curvature Ω p = p/(2|p| 3 ), where p represents the spatial momentum of the particle, which originates from the Berry phase in an adiabatic process[22]. Further generalization to massive Dirac fermions can be found in Ref.[23]. In order to bridge the semi-classical approaches[18,19]and quantum field theories, the CKT is also derived from Wigner functions in the high-density effective theory[24](see also Ref. [25] for relevant study of the on-shell effective field theory.)However, there still exist potential issues in the chiral kinetic equation. First, the field-theory derivation in Refs.[24] and[20,21]are subject to a predominant chemical potential and local equilibrium, respectively. The derivation for more general systems beyond local equilibrium is thus needed. Second, the non-manifestation of Lorentz invariance in the chiral kinetic equation has been recently discussed in Refs.[26,27]from the semi-', 'arxivid': '1612.04630', 'author': ['Yoshimasa Hidaka \nTheoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoSaitamaJapan\n', 'Shi Pu \nDepartment of Physics\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n', 'Di-Lun Yang \nTheoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoSaitamaJapan\n'], 'authoraffiliation': ['Theoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoSaitamaJapan', 'Department of Physics\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan', 'Theoretical Research Division\nNishina Center\nRIKEN\n351-0198WakoSaitamaJapan'], 'corpusid': 119271500, 'doi': '10.1103/physrevd.95.091901', 'github_urls': [], 'n_tokens_mistral': 12169, 'n_tokens_neox': 9933, 'n_words': 5081, 'pdfsha': '1b5f29e1f2f58427e3781f0802af0d54a8e340db', 'pdfurls': ['https://arxiv.org/pdf/1612.04630v2.pdf'], 'title': ['Relativistic Chiral Kinetic Theory from Quantum Field Theories', 'Relativistic Chiral Kinetic Theory from Quantum Field Theories'], 'venue': []} |
arxiv |
Towards End-to-End Semi-Supervised Table Detection with Deformable Transformer
Tahira Shehzadi [email protected]
Department of Computer Science
Technical University of Kaiserslautern
67663KaiserslauternGermany
Mindgarage
Technical University of Kaiserslautern
67663KaiserslauternGermany
German Research Institute for Artificial Intelligence (DFKI)
67663KaiserslauternGermany
− 979x
Khurram Azeem Hashmi
Department of Computer Science
Technical University of Kaiserslautern
67663KaiserslauternGermany
Mindgarage
Technical University of Kaiserslautern
67663KaiserslauternGermany
German Research Institute for Artificial Intelligence (DFKI)
67663KaiserslauternGermany
Didier Stricker]
Marcus Liwicki
Department of Computer Science
Technical University of Kaiserslautern
67663KaiserslauternGermany
Mindgarage
Technical University of Kaiserslautern
67663KaiserslauternGermany
Department of Computer Science
Luleå University of Technology
971 87LuleåSweden
German Research Institute for Artificial Intelligence (DFKI)
67663KaiserslauternGermany
Muhammad Zeshan Afzal
Department of Computer Science
Technical University of Kaiserslautern
67663KaiserslauternGermany
Mindgarage
Technical University of Kaiserslautern
67663KaiserslauternGermany
German Research Institute for Artificial Intelligence (DFKI)
67663KaiserslauternGermany
Towards End-to-End Semi-Supervised Table Detection with Deformable Transformer
Semi-Supervised Learning · Deformable Transformer · Table Analysis · Table Detection
Table detectionis the task of classifying and localizing table objects within document images. With the recent development in deep learning methods, we observe remarkable success in table detection. However, a significant amount of labeled data is required to train these models effectively. Many semi-supervised approaches are introduced to mitigate the need for a substantial amount of label data. These approaches use CNN-based detectors that rely on anchor proposals and post-processing stages such as NMS. To tackle these limitations, this paper presents a novel end-to-end semi-supervised table detection method that employs the deformable transformer for detecting table objects. We evaluate our semi-supervised method on PubLayNet, DocBank, ICADR-19 and TableBank datasets, and it achieves superior performance compared to previous methods. It outperforms the fully supervised method (Deformable transformer) by +3.4 points on 10% labels of TableBank-both dataset and the previous CNN-based semi-supervised approach (Soft Teacher) by +1.8 points on 10% labels of PubLayNet dataset. We hope this work opens new possibilities towards semi-supervised and unsupervised table detection methods.
Introduction
A visual summary is the main aspect of different applications in document analysis, such as recognizing graphical components in the visualization pipeline and summarizing the content of a document. As a result, localizing and detecting graphical items such as tables will be an important action in the analysis and summary of the document. Due to the increase in the number of documents, manually retrieving the table data is no longer practical. Automated processes offer efficient, reliable, and successful solutions for manual tasks. Previously, optical character recognition [1,2] and rule-based [3,4,5] table detection approaches were used to identify and locate them. Then, some automated methods [6,7,8] have been suggested to detect tables. However, these approaches are often rule-based because the documents have a set structure or dimension [9]. Moreover, they cannot generalize to a new table structure, such as borderless tables. Later on, deep learning methods were used by researchers to identify them [10,11,12,13], and shows that machine-learning approaches are more effective than traditional methods [14].
Deep learning approaches [15,16,17,18,19,20] do not rely on rules and can accurately generalize the problem. However, deep learning models take a considerable quantity of labeled data for training. These supervised methods achieve impressive results on public benchmarks, and their performance cannot be translated into industrial applications unless similar large-scale annotated datasets exist in that domain. It is potentially error-prone and time-consuming to generate this data manually or via other pre-processing approaches. Therefore, it is important to develop a semi-supervised approach due to concerns about the availability of labeled training data, which shifts the problem from a supervised to a semi-supervised setting. Recently, semi-supervised learning-based methods are introduced in computer vision containing two detectors. The first detector provides pseudo labels for unlabeled data. The second detector trains using pseudo labels generated by the first detector and a small percentage of label data and provides final predictions. Both detectors update each other during training. This approach has been described in several works, including [21,22,23,24]. In most cases, the first detector is not strong enough, which can negatively impact the pseudo-labeling process. Moreover, previous semi-supervised approaches used CNN-based networks [11] that depend on anchors to generate region proposals and post-processing stages such as Non-Maximal suppression (NMS) to reduce the number of overlapping predictions.
To address these limitations, this paper proposes a semi-supervised table detection approach that employs the deformable transformer [25]. It generates pseudo-labels for unlabeled data and then trains the detector using them and a small quantity of label data in each iteration. This approach aims to improve the pseudo-label generation procedure by iteratively refining the pseudo-labels and the detector. It involves training in two modules. The teacher module contains a pseudo-labeling framework. The student module is the final detection network that uses pseudo-labels and a small quantity of label data. The teacher module is simply an Exponential Moving-Average (EMA) of the student module, which ensures that the pseudo-label generation and detection modules are constantly updating each other. Unlike other pseudo-labeling methods, we propose the idea of employing the deformable transformer that allows completing the pseudo-labeling process without needing object proposals and post-processing steps as Non-maximal suppression (NMS). Another benefit is having a dynamic effective receptive field to adapt fot tables of different sizes and scales in the input image. This allows the network to effectively detect tables of varying sizes and orientations, making it more robust and versatile. Additionally, this framework has a reinforcing effect, providing that the Teacher model consistently monitors the Student model. In this paper, we show through empirical evidence that this semi-supervised table detection approach that uses a deformable transformer can produce results comparable to CNN-based approaches without needing object proposals and post-processing steps such as Non-maximal suppression (NMS). In summary, the main contributions of the paper are as follows:
• We present an end-to-end semi-supervised table detection method that employs the deformable transformer and allows completing the pseudo-labeling process without needing object proposals and post-processing steps such as Non-maximal suppression (NMS). • We formulate the problem of table detection as an object detection problem and leverage the potential of deformable detection transformer for this task. To the best of our knowledge, this work is the first that exploits the transformerbased method in a semi-supervised setting. • We perform an exhaustive evaluation on four different datasets, PubLayNet, DocBank, ICDAR-19 and TableBank, and produce results comparable to CNN-based semi-supervised approaches without needing object proposals process and post-processing steps such as NMS. [26,27,28,29]. Recently, researchers employed statistical methods [30] and deep learning approaches to make the table detection systems more generalizable [15,31,32,33]. This section gives a detailed summary of these techniques and an overview of the CNN-based semisupervised object detection methods.
Related Work
Rule-based Approaches
To the best of our knowledge, Itonori et al. [26] presented a
Learning-based Approaches
Cesarini et al. [39] presented a supervised learning system for detecting table objects in document images. It converts a document image into an MXY tree model and labels the blocks as tables confined in horizontal and vertical lines. Hidden Markov Models [40,41] and the SVM classifier with traditional heuristics [42] are applied to document images for table detection. Though these machine learning approaches performed better than ruled-based approaches on documents, these methods need additional information, such as ruling lines. Deep Learning-based approaches outperformed traditional approaches in accuracy and efficiency. These methods are categorised into object detection, semantic segmentation, and bottom-up approaches. Semantic segmentation-based Approaches. These approaches [43,44,45,46] consider table detection a segmentation task and apply available semantic segmentation networks to generate segmentation masks on the pixel level and then combine and Riba et al. [57] considered text areas as nodes, formed a graph to determine the design per document and then employed graph-neural networks for node-edge classification. These approaches rely on specific assumptions, such as text line boxes as an extra input. Object Detection-based Approaches. Detecting tables from document images can be represented as an object detection task, with table objects treated as natural objects. Hao et al. [58] and Yi et al. [59] applied R-CNN for detecting tables, but the performance of these approaches still relies on heuristic rules as in previous methods. Later, more efficient single-stage object detectors like RetinaNet [60] and YOLO [61] and two-stage object detectors like Fast R-CNN [10], Faster R-CNN [11], Mask R-CNN [62], and Cascade Mask R-CNN [63] were applied for other document objects such as figures and formulas detection in document images [64,65,66,67,68,69,70,15,16,17]. Furthermore, [65,69,71] applied different image transformation approaches, such as coloration and dilation, to improve the results further. Siddiqui et al. [72] combined deformable-convolution and RoI-Pooling [73] into Faster R-CNN to provide a more efficient network for geometrical modifications. Agarwal et al. [70] used a composite network [74] as a backbone with deformable convolution to increase the performance of two-stage Cascade R-CNN. These CNN-based object detectors have a few heuristic stages, like proposals generating step and post-processing steps such as non-maximal suppression (NMS). Our semisupervised approach considers detection a set prediction task, eliminating the anchor generation and post-processing stages such as NMS and providing a simpler and more efficient detection pipeline.
Semi-supervised Object Detection
Semi-supervised learning approaches in object detection are divided into two types: consistency-based approaches [75,76] and pseudo-label generation-based approaches [77,78,79,80,81,82,83]. Our method falls into the pseudo-label type. Previous work [77,78] combined prediction results from varied data augmentation techniques to produce pseudolabels for unlabeled data, while [79] trained a SelectiveNet to generate the pseudo-labels. In [79], a box from unlabeled data was placed onto labeled data and evaluated localization consistency on the labeled images. However, this method requires a very complex detection procedure due to the modification of the image. STAC [82] presented to perform strong augmentation on the data for pseudo-label generation and weak augmentation for model training. We propose an end-to-end semi-supervised table detection method that employs the deformable transformer. Similar to other pseudolabel generation approaches [77,78,79,82,83], it follows a multi-level training mechanism. It effectively avoids the need for anchors generation stage and post-processing steps such as Non-Maximal suppression (NMS).
Methodology
First, we revisit Deformable DETR, a modern transformer-based object detector, in Section 3.1. Later, we explain the proposed semi-supervised learning mechanism and its pseudo-label generation module in Sections 3.2.
Revisiting Deformable DETR
Deformable DETR [25] contains a Transformer encoder-decoder network that considers object detection as a setpredictions task. It uses Hungarian loss and avoids overlapped predictions for ground-truth bounding boxes through bipartite matching. It eliminates the need for hand-crafted elements such as anchors and post-processing stages such as Non-maximal suppression (NMS) used in CNN-based object detectors. Deformable DETR is an extension of the DETR [84] architecture that addresses some of the limitations of DETR, such as slow training convergence and poor performance on small objects. Deformable DETR introduces deformable convolutions into the architecture, which allows for more flexible modeling of object shapes and better handling of objects of varying scales. This can lead to improved performance, particularly on small objects, and faster convergence during training. Here, we provide an overview of the encoder-decoder network, Multi-scale Feature processing and attention mechanism of deformable DETR. Figure 1 shows all modules of the deformable transformer, including multi-scale features and encoder-decoder network. Fig. 1: Illustration of the deformable transformer employed in semi-supervised table detection method. We split the input image into small equal-sized patches, add position embeddings, and feed the resulting patches along with input multi-scale features to the transformer encoder. In the decoder, We use object queries as reference points and provide bounding boxes predictions and class labels as the final output.
Transformer Encoder. The CNN backbone (ResNet-50) extracts the input feature maps f m ∈ R hi×wi×ci . The spatial dimensional feature maps are converted into one-dimensional z m ∈ R hi×wi×d1 feature maps as the transformer encoder network takes input as a sequence. This one-dimensional vector is fed as input along with positional embeddings [85,86] to the transformer encoder network, which further transforms them into features for object queries. Every layer of the encoder module contains an attention network and a feed-forward network (FFN) where query and key values are the pixels of feature maps. Readers can refer to [87] for a detailed explanation of transformer. Transformer Decoder. The decoder network takes the output of the encoder features and N number of object queries as input. It contains two attention types self-attention and cross-attention. The self-attention module finds the connection between object queries. Here both key and query matrics contain object queries. The cross-attention module extracts feature using object queries from the input feature map. Here key matrix contains the feature maps provided by the encoder module, and the query matrix is the object queries fed as input to the decoder. After the attention modules, feed-forward networks (FFN) and linear projection layers are added as the prediction head. The linear projection layer predicts class labels, while FFN provides final bounding-box coordinate values. Deformable Attention Module. The attention module in the DETR network considers all spatial locations of the input feature map, which makes the training convergence slower. However, a deformable DETR can solve this issue using the deformable convolution-based [73,88] attention network and multiscale input features [89,90]. It considers only a few sample pixels near a reference pixel, whatever the size of input features, as illustrated in Figure 2. The query matrix takes only a small set of keys, which resolves the slow training convergence issue of DETR. Readers can refer to [25] for a detailed explanation of Deformable DETR.
Semi-Supervised Deformable DETR
In this subsection, we describe the learning mechanism of our proposed semi-supervised approach that employs the Deformable transformer and then explain the pseudo-labeling strategy. Semi-supervised Deformable-DETR is a unified learning approach that uses fully labeled and unlabeled data for object detection. It contains two modules a student module and a teacher module. The training data has two data types label data and unlabeled data. The student module takes both labeled and unlabeled images as input where strong augmentation is applied on unlabeled data while both (strong and weak augmentation) is applied on label data. The student module is trained using detection losses of labeled and unlabeled data through pseudo-boxes. The unlabeled data contains two groups of pseudo boxes for providing class labels and their bounding boxes. The teacher module only takes unlabeled images as input after applying weak augmentation. Figure 3 presents a summary of proposed pipeline. The teacher module feeds prediction results to the pseudo-labeling framework to get pseudo-labels. Then, the student module uses these pseudo-labels for supervised training. Here, weak augmentation on unlabeled data is used for the teacher module to generate more precise pseudo-labels. Strong augmentation on unlabeled data is used for the student module to have more challenging learning. The student module also takes a small percentage of labeled images with strong and weak augmentation as input. The student module s m is optimized with the total loss as follows:
L sm = n L(x l,sa j , y l,sa j ) + L(x l,wa j , y l,wa j ) + n L(x u,sa j , y tm j )(1)
Where s a represents strong augmentation, w a represents weak augmentation. x l,sa j is the strong augmented input image and its corresponding label is y l,sa j . The term x l,wa j is the weak augmented input image and its corresponding label is y l,wa j . For the labeled images, strong and weak augmentations are also applied for learning, and are fed to the student module. The term x u,sa j represents unlabeled strong augmented image fed to student module and the term y tm j is the pseudo-label from teacher module. Here, L is the weighted sum of classification (class labels) and regression (bounding box) loss as follows:
L = α 1 L reg + α 2 L cls(2)
Where α 1 and α 2 are the weight values, the teacher-student modules are initialized randomly at the start of training. During training, the student module continuously updates the teacher module with an Exponential Moving-Average (EMA) [91] strategy. Pseudo-label generation for image classification tasks is easy, considering probability distribution as Pseudo-labels. In contrast, object detection tasks are more complicated as an image may include numerous objects, and annotation contains object location and class label. The CNN-based object detectors use anchors as object proposals and remove redundant boxes by post-processing steps such as non-maximal suppression (NMS). In contrast, transformers use attention mechanisms and object queries. Figure 4 shows sample points and attention weights from multi-scale deformable attention feature maps for both student and teacher networks. Its training complexity is
O(N q c 2 i + min(h i w i c 2 i , N q kc 2 i ) + 5N q kc i + 3N q c i p s k)
. This takes into account the computation of the sampling coordinate offsets and attention weights, as well as the bilinear interpolation and weighted sum in the attention mechanism. N q is the number of query elements, c i is the channel dimension, k is the kernel size, p s is the number of sampling points, and h i w i is the height and width of the feature map. In our experiments, p s = 8, k ≤ 4 and c i = 256 by default, thus 5k + 3p s k < c i and the complexity is of O(2N q c 2 i + min(h i w i c 2 i , N q kc 2 i )). When used in the DETR encoder with N q = h i w i , the complexity of the deformable attention module is O(h i w i c 2 i ), which scales linearly with the spatial size. When used in the DETR decoder with N q = N (the number of object queries), the complexity becomes O(N kc 2 i ), which is independent of the spatial size as attention is focused on the object queries. Training The semi-supervised network is trained in two steps: a) train the student module independently on labeled data and generate pseudo-labels by teacher module; b) combine training of both modules to provide final prediction results. Pseudo-Labeling Framework We used a simple framework to provide pseudo-labels for unlabeled data at the output of the teacher module, as applied in SSOD [92]. Usually, object detectors give confidence score vector s k ∈ [0, 1] Ci for every provided bounding box b k . A simple approach to provide pseudo-labels is to just thresholding these scores. In a simple pseudo-labeling filter, pseudo-labels can be formed by providing a threshold to the confidence value s c k k of the ground-truth class c k . If the prediction value is not greater than the confidence value for a ground-truth class, the highest prediction value is considered the pseudo-label. Inspired by DETR [84], we develop the pseudo-label assignment task as a bipartite matching task between the teacher module predictions and the generated semi-labels. Specifically, the permutation of K elements is as follows:
σ = arg min σ∈N Ni k L match (y k ,ŷ(k)),(3)
Where L match (y k ,ŷ(k)) is the match-cost between teacher labels and ground-truth semi-labels as follows:
L match (y k ,ŷ(k)) = −1 {c k =φ}pσ(k) (c k ) + 1 {c k =φ} L bbox (b k ,bσ(k))(4)
The Pseudo-Labeling framework is applied to the predictions of teacher moduleŷ(k) whereŷ(k) = {ŷ class ,ŷ bbox } is the prediction, withŷ class andŷ bbox represent the class and box values, respectively. Here,
ŷ cls = [v 1 , ..., v N ] T ∈ R N ×Ci andŷ box = [b 1 , ...,b N ] T ∈ R N ×4 ,
where v N is the output vector (before the softmax) ,b N the related bounding-box prediction, and N is the object queries provided as input to the transformer decoder. y k represents pseudo-labels Fig. 3: Our proposed semi-supervised approach that employs Deformable transformer [25]. (1) The training data has two data types label data and unlabeled data. (2) It contains two modules a student module and a teacher module.
(3) The teacher module only takes unlabeled images as input after applying weak augmentation. (4) After applying strong augmentation on unlabeled data type, the student module takes both labeled and unlabeled images as input. (5) During training, the student module continuously updates the teacher module with an Exponential Moving-Average (EMA) [91] strategy. Fig. 4: Visualization of the sample points and attention weights from multi-scale deformable attention feature maps. Each sample point is denoted as a circle whose color represents its relative attention weight value. The reference points are the object queries taken as input in the encoder, represented by the green plus sign. In the decoder, the final bounding boxes are represented as green rectangles, and the class label and its confidence value are shown on the upper side in black text. generated from confidence-score. The optimal selection is allowed with the Hungarian match mechanism [84,93], giving pseudo-labels {(b k , c k )}. This approach to select matching between the teacher module's prediction and semi-labels generated by providing threshold works in the same way as the heuristic selection rules used for matching proposals [11] or anchors [89] with ground-truth objects in CNN-based object detectors. The main difference is that it determines one-to-one matching without duplicates. The second stage calculates the loss function, the Hungarian loss for all pair matching in the last stage. We define the loss similar to the previous object detector's losses as a linear combination of a negative log-likelihood for class label and a bounding box as follows:
L H (y,ŷ) = N i=1 [−logpσ (k) (c k ) + 1 {c k =φ} L box (b k ,bσ(k))](5)
Here, b k ∈ R 4 is the pseudo-bounding box, and c k is the pseudo-class label.σ is the matching determined in the previous stage. In training, we reduce the weight of log probability by ten times when c k for class imbalance. This mechanism is similar to the Faster R-CNN training strategy to balance proposals by sub-sampling [11]. , we provide results on the modern datasets with an IoU threshold ranging from 0.5-0.9.
Evaluation Criteria
We use some evaluation metrics to analyze the performance of our semi-supervised table detection approach that employs the deformable transformer. This section defines the employed evaluation metrics as precision, Recall, and F1-score. The Precision [96] is the fraction of actual instances as True Positives among the predicted instances as False Positives and True Positives). The Recall [96] is the fraction of actual instances as True Positives that were retrieved (True Positives + False Negatives). The F1-score [96] is the harmonic mean of Precision and Recall. We compute the intersection over union(IoU) by performing the intersection divided by the union for the region of the ground-truth box A g and the formed bounding box A p .
IoU = area(A g ∩ A p ) area(A g ∪ A p )(6)
IoU estimates that either a detected table object is a false positive or a true positive. We find the average precision(AP) by a precision-recall (PR) curve following the context of MS COCO [94] evaluation. It is the area under the PR curve, calculated using the following equation:
AP = N k=1 (Re k+1 − Re k )P intr (Re k+1 )(7)
Where Re1, Re2, . . . , Re k represent the recall parameter. The mean average precision (mAP) is often used to evaluate the performance of detection methods. It is calculated by taking the mean of average precision for all classes in a dataset. The mAP can be affected by changes in the performance of individual classes due to class mapping, which is a limitation of this metric. We set the intersection over union (IoU) threshold values at 0.5 and 0.95. The mAP is calculated as follows:
mAP = 1 S S s=1 AP s(8)
Where S represents total classes.
Implementation Details
We use the Deformable DETR [25] with a ResNet-50 [97] backbone pre-trained on the ImageNet [98] dataset as our detection framework for evaluating the usefulness of the semi-supervised approach. We perform training on PubLayNet, ICDAR-19, DocBank and all three splits of the TableBank dataset. We use 10%, 30% and 50% of labeled data and the rest as unlabeled data. The threshold value for pseudo-labeling is set at 0.7. We set the training epochs to 150 for all experiments with the learning rate reduced by a factor of 0.1 at the 120th epoch. We follow [92,25] to apply strong augmentation as horizontal flip, resize, remove patches, crop, grayscale and Gaussian blur. We use horizontal flipping to apply weak augmentation. The value N for the number of queries to the input of the decoder of Deformable DETR is set to 30 as it gives the best results. Unless otherwise specified, we evaluated the results using the mAP (AP50: 95) metrics. All models are trained with a batch size of 16, using the same hyperparameters as Deformable DETR [25]. The weight α 1 is 2 and α 2 is 5 to balance the classification loss (L cls ) and regression loss (L box ). To make the training faster, we set the height and width of the input image to 600 pixels. We employ the standard size of 800 pixels for comparison with other approaches.
Results and Discussion
TableBank
In this subsection, we provide the experimental results on all splits of the TableBank dataset on different percentages of label data. We also compare the transformer-based semi-supervised approach with previous deep learning-based supervised and semi-supervised approaches. Furthermore, we give results on 10% TableBank-both data split for all IoU threshold values. Table 1 provides the results of semi-supervised approach that employs deformable transformer for TableBank-latex, TableBank-word, and TableBank-both data splits on 10%, 30% and 50% label data and the rest as unlabeled data. It shows that the TableBank-both data split has the highest AP 50 value of 95.8%, TableBank-word has 93.5%, and TableBank-both has 92.5% at 10% label data. The qualitative analysis of semi-supervised learning for the TableBank-both data split is shown in Figure 5. Part (b) of Figure 5 has a matrix with a similar structure as rows and columns, and the network detects the matrix as a table giving false positive detection results. Here, incorrect detection results indicate where the network fails to provide correct detection of table regions. Table 2 gives the results of this semi-supervised approach on different IoU threshold values for all splits of the TableBank dataset on 10% label data and the rest as unlabeled data. A visual comparison of Precision, Recall and F1-Score of semi-supervised network that employs deformable transformer with ResNet-50 backbone on different IoU threshold values on 10% labeled dataset of TableBank-both data split is shown in Figure 6. Comparisons with Previous supervised and semi-supervised approaches Table 3 compares the deep learningbased supervised and semi-supervised networks on the ResNet-50 backbone. We also compare supervised deformable-DETR trained on 10%, 30% and 50% TableBank-both data split label data with our semi-supervised approach that employs deformable transformer. It shows that our attention mechanism-based semi-supervised approach provides comparable results without using proposal generation process and post-processing steps such as Non-maximal suppression (NMS). Table 1: Performance of the semi-supervised approach that employs deformable transformer for TableBank-latex, TableBank-word, and TableBank-both data splits on different percentages of label data. Here, mAP represents mean AP at the IoU threshold range of (50:95), AP 50 indicates AP at the IoU threshold of 0.5, and AP 75 denotes AP at the IoU threshold of 0.75. AR L indicates average recall for large objects.
PubLayNet
In this subsection, we discuss the experimental results on PubLayNet table class dataset on different percentages of label data. We also compare the transformer-based semi-supervised approach with previous deep learning-based supervised and semi-supervised approaches. Furthermore, we give results on 10% PubLayNet dataset for all IoU threshold values. Table 4 provides the results of the semi-supervised approach that employs deformable transformer for PubLayNet table class on the different percentages of label data and rest as unlabeled data. Here, AP 50 value is 98.5%, 98.8%, and 98.8% for 10%, 30% and 50% label data, respectively. Furthermore, our semi-supervised network is trained on different IoU threshold values on 10% of labeled PubLayNet Dataset. Table 5 gives the results of the semi-supervised approach on different IoU threshold values for PubLayNet table class on 10% label data and the rest as unlabeled data. A visual comparison of Precision, Recall and F1-score of the semi-supervised network that employs the deformable transformer network with ResNet-50 backbone on different IoU threshold values on 10% labeled dataset of PubLayNet table class is shown in Figure 6. Here, blue indicates precision results on different IoU threshold values on different IoU threshold values, red shows recall results, and green represents F1-score results on different IoU threshold values. Comparisons with Previous supervised and semi-supervised approaches Table 6 compares the deep learningbased supervised and semi-supervised networks on PubLayNet table class using ResNet-50 backbone. We also compare supervised deformable-DETR trained on 10%, 30% and 50% PubLayNet table class label data with our semi-supervised approach that employs the deformable transformer. It shows that our semi-supervised approach provides comparable results without using proposal and post-processing steps such as Non-maximal suppression (NMS).
DocBank:
In this subsection, we discuss the experimental results on DocBank dataset on different percentages of label data. We compare the transformer-based semi-supervised approach with previous CNN-based semi-supervised approach in Table 7. Furthermore, we also compare our semi-supervised approach on different percentages of label data with previous table detection and document analysis approaches for different datasets TableBank, PubLayNet, and DocBank in Table 8. Although we cannot directly compare our semi-supervised approach with previous supervised document analysis approaches. However, we can observe that even with 50% label data, we achieve comparable results with previous supervise approaches.
ICDAR-19
We also evaluate our method for table detection on the Modern Track A portion of the table detection dataset from the cTDaR competition at ICDAR 2019. We summarize the quantitative results of our approach at different percentages of label data and compare it with previously supervised table detection approaches in Table 9. We evaluate results at higher IoU thresholds of 0.8 and 0.9. For a direct comparison with previous table detection approaches, we also evaluate our approach on 100% label data. Our approach achieved a precision of 92.6% and a recall of 91.3% on the IoU threshold of 0.9 on 100% label data.
Ablation Study
In this section, we validate the key design elements. Unless otherwise stated, all the ablation studies are conducted using a ResNet-50 backbone with 30% labeled images from the PubLayNet dataset. Pseudo-Labeling confidence threshold In Section 3.2, the threshold value (referred to as the confidence threshold) plays an important role in determining the balance between the accuracy and quantity of the generated pseudo-labels. As this threshold value increases, fewer examples will pass the filter, but they will be of higher quality. Conversely, a smaller threshold value will result in more examples passing but with a higher likelihood of false positives. The impact of various threshold values, ranging from 0.5 to 0.9, is presented in Table 10. The optimal threshold value was determined to be 0.7 based on the results. Influence of Learnable queries Quantity In our analysis, we investigate the impact of varying the number of queries fed as input in the decoder of deformable DETR. Figure 7 compares prediction results by varying the number of object queries fed as input in the decoder of deformable DETR. The optimal performance is attained when the number of queries N is set to 30; deviating from this value results in a decrease in performance. Table 11 presents and analyzes the result for varying object query quantities. Choosing a small value for N could result in the model failing to identify particular objects, negatively impacting its performance. On the other hand, selecting a large value for N may cause the model to perform poorly due to overfitting, as it would incorrectly classify certain regions as objects. Moreover, training complexity O(N kc 2 i ) of this semi-supervised self-attention mechanism in the decoder of student- Here, (a) takes N=3 object queries as input, (b) contains N =30 object queries as input, and (c) has N =300 object queries as input. The optimal performance is achieved by selecting the number of queries N to 30; deviating from this value results in a decrease in performance. Here, blue rectangles denote ground truth (GT), green rectangles indicate object class, and red rectangles show background class.
teacher module depends on the number of object queries and is subsequently improved as complexity is reduced by minimizing the number of object queries.
Conclusion
This paper introduces a semi-supervised approach that employs the deformable transformer for table detection in document images. The proposed method mitigates the need of large-scale annotated data and simplifies the process by integrating the pseudo-label generation framework into a streamlined mechanism. The simultaneous generation of pseudo-labels leads to a dynamic process known as the "flywheel effect", where one model continually improves the pseudo-boxes produced by the other model as the training progresses. The pseudo-class labels and pseudo-bounding boxes are improved in this framework using two distinct modules named student and teacher. These modules update each other by the EMA function to provide precise classification and bounding box predictions. The results indicate that this approach surpasses the performance of supervised models when applied to labeling ratios of 10%, 30%, and 50% on TableBank all splits and the PubLayNet training data. Furthermore, when trained on the 10% labeled data of PubLayNet, the model performed comparably to the current CNN-based semi-supervised baseline. In future, we aim to investigate the impact of the proportion of annotated data on the ultimate performance and develop models that function effectively with a minimal quantity of labeled data. Additionally, we intend to employ the transformer-based semi-supervised learning mechanism for table structure recognition task.
Fig. 2 :
2Deformable Attention network. It considers only a few sample pixels near a reference pixel, whatever the size of input features. The query matrix takes only a small set of keys, which resolves the slow training convergence issue of DETR. (image from[25]).
Fig. 5 :
5Semi-supervised table detection results that employs deformable transformer on TableBank-both data split. Green color represents true positives, blue denotes false negatives and red shows false positives. Here, (a) indicates true positive detection results, (b) shows true positive and false positive detection results, and (c) gives false negative detection results.
Fig. 7 :
7Comparison of performance by variation of the number of object queries fed as input in the decoder of deformable DETR.
Table detection
detectionis an essential task for document image analysis. Many researchers have proposed different approaches for detecting tables containing arbitrary structures in document images. Previously, most presented approaches used custom rules or relied on extra meta-data input to deal with table detection tasks
table detection approach for the first time on document images. This method represents the table as a text block that uses specified rules. Later, [28] introduced a table detection approach that works on horizontal and vertical lines. Pyreddy et al.[34] proposed a procedure that extracts tabular regions from the text using custom heuristics. Pivk et al.[35] presented a system that transforms HTML format table documents into logical forms. It introduces an appropriate tabular layout employed for extracting tables. Hu et al.[36] presented a table detection approach that relies on white regions and vertically connected elements in document images. Readers can find a complete overview of these rule-based methods in[3,4,5,37,38]. Though rulebased approaches perform fine on document images with matching table formats, these methods can not provide generic solutions. Therefore, systems with more generalizable abilities are needed to solve table detection tasks on document data.
It also includes a dataset for recognizing the structures of the table. In our experiment, We only used the dataset for table detection from TableBank. PubLayNet: PubLayNet[48] is a large public dataset with 335,703 images in the training set, 11,240 in the validation set, and 11,405 in the test set. It includes annotations such as polygonal segmentation and bounding boxes of figures, lists titles, tables, and text of images from research papers and articles. The dataset was evaluated using the coco analytic technique[94]. In our experiment, we only used 102,514 of the 86,460 table annotations.4 Experimental Setup
4.1 Datasets
TableBank: TableBank [52] is the second-largest dataset in the document analysis domain for the table recognition
problem. The dataset has 417,000 document images annotated through the arXiv database crawling procedure. The
dataset features tables from three categories of document images: LaTeX images (253,817), Word images (163,417),
and a combination of both (417,234). DocBank: DocBank [95] is a large dataset of over 5,000 annotated document images from various sources designed to
train and evaluate tasks such as text classification, entity recognition, and relation extraction. It includes annotations
of title, author name, affiliation, abstract, body text, etc.
ICDAR-19: The competition for Table Detection and Recognition (cTDaR) [47] is organized at ICDAR in 2019. For
the table detection task (TRACK A), two new datasets (modern and historical) are introduced in the competition.
For direct comparison against the prior state-of-the-art [69]
Table 2 :
2The performance comparison of semi-supervised network that employs deformable transformer with ResNet-50 backbone on different IoU threshold values on 10% labeled dataset of TableBank-both data split.Method
IoU
Precision
Recall
F1-score
0.5
95.8
90.5
93.1
Semi-Supervised
0.6
94.6
90.5
92.5
Deformable-DETR+ResNet-50
0.7
93.3
90.3
91.8
10% labels
0.8
91.8
89.8
90.8
0.9
89.1
87.2
88.1
Table 3 :
3Performance comparison of previous supervised and semi-supervised approaches.Supervised Deformable-
Table 4 :
4Performance results for PubLayNet table class
dataset. Here, mAP represents mean AP at the IoU thresh-
old range of (50:95 ), AP 50 indicates AP at the IoU thresh-
old of 0.5 and AP 75 denotes AP at the IoU threshold of
0.75. AR L indicates average recall for large objects.
Dataset
Label-percent mAP AP 50 AP 75 AR L
PubLayNet
10%
88.4
98.5
97.3
91.0
30%
90.3
98.8
97.5
93.2
50%
92.8
98.8
97.3
96.0
Table 5 :
5The performance comparison of semi-supervised
network that employs deformable transformer with
ResNet-50 backbone on different IoU threshold values on
10% PubLayNet labeled Dataset.
Method
IoU Precision Recall F1-score
0.5
98.5
91.0
94.6
Semi-Supervised
0.6
98.1
90.9
94.4
Deformable-DETR 0.7
97.4
90.8
94.0
10% labels
0.8
94.0
90.0
92.0
0.9
89.0
87.0
88.0
Table 6 :
6Performance comparison of previous supervised and semi-supervised approaches. Deformable-DETR and Faster R-CNN trained on just 10%, 30% and 50% table data while semi-supervised networks used 10%, 30% and 50% PubLayNet dataset as labeled and rest as unlabeled data. Here, all results are represented on AP 50 at the IoU threshold of 0.5. The best threshold values are shown in bold.Method
Approach
Detector
10%
30%
50%
Ren et al. [11]
supervised
Faster R-CNN
93.6
95.6
95.9
Zhu et al. [25]
supervised
Deformable DETR
93.9
96.2
97.1
STAC [82]
semi-supervised
Faster R-CNN
95.8
96.9
97.8
Unbiased Teacher [92]
semi-supervised
Faster R-CNN
96.1
97.4
98.1
Humble Teacher [99]
semi-supervised
Faster R-CNN
96.7
97.9
98.0
Soft Teacher [100]
semi-supervised
Faster R-CNN
96.5
98.1
98.5
Our
semi-supervised
Deformable DETR
98.5
98.8
98.8
Table 7 :
7Performance comparison of previous semi-supervised approach and our Deformable-DETR based semisupervised approach on DocBank dataset. Here, all results are represented on mAP (0.5 : 0.95).Method
Approach
Detector
10%
30%
50%
Soft Teacher [100]
semi-supervised
Faster R-CNN
72.3
74.4
81.5
Our
semi-supervised
Deformable DETR
82.5
84.9
87.1
Table 8 :
8Performance comparison of previous supervised approaches for document analysis. Our semi-supervised network uses 10%, 30% and 50% label data and rest as unlabeled data. Here, all results are represented on mAP (0.5 : 0.95).Method
Approach
Labels
TableBank
PubLayNet
DocBank
CDeC-Net [70]
supervised
100%
96.5
97.8
-
CasTabDetectoRS [32]
supervised
100%
95.3
-
-
Faster R-CNN [48]
supervised
100%
-
90
86.3
VSR [101]
supervised
100%
-
95.69
87.6
Our
semi-supervised
10%
84.2
88.4
82.5
Our
semi-supervised
30%
86.8
90.3
84.9
Our
semi-supervised
50%
91.8
92.8
87.1
Table 9 :
9Performance comparison between the proposed semi-supervised approach and previous state-of-the-art results on the dataset of ICDAR 19 Track A (Modern).Method
Approach
IoU=0.8
IoU=0.9
Recall
Precision
F1-Score
Recall
Precision
F1-Score
TableRadar [47]
supervised
94.0
95.0
94.5
89.0
90.0
89.5
NLPR-PAL [47]
supervised
93.0
93.0
93.0
86.0
86.0
86.0
Lenovo Ocean [47]
supervised
86.0
88.0
87.0
81.0
82.0
81.5
CascadeTabNet [69]
supervised
-
-
92.5
-
-
90.1
CDeC-Net [70]
supervised
93.4
95.3
94.4
90.4
92.2
91.3
HybridTabNet [33]
supervised
93.3
92.0
92.8
90.5
89.5
90.2
Our
semi-supervised (50%)
71.1
82.3
76.3
66.3
76.8
71.2
Our
supervised (100%)
92.1
94.9
93.5
91.3
92.6
91.9
Table 10 :
10Performance comparison using different Pseudolabeling confidence threshold values. The best threshold values are shown in bold.Threshold
AP
AP 50
AP 75
0.5
86.9
91.6
90.1
0.6
89.5
98.1
95.7
0.7
90.3
98.8
97.5
0.8
89.4
97.2
95.3
0.9
87.9
96.3
94.5
Table 11 :
11Performance comparison using different numbers of learnable queries to the decoder input. Here, best performance results are shown in bold.N
AP
AP 50
AP 75
3
61.4
69.7
62.6
30
90.3
98.8
97.5
50
89.4
90.3
85.4
100
88.4
89.7
83.9
300
78.5
94.7
90.2
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Evaluation: from precision, recall and f-measure to roc, informedness, markedness and correlation. D M W Powers, CoRR. D. M. W. Powers, "Evaluation: from precision, recall and f-measure to roc, informedness, markedness and correlation," CoRR, vol. abs/2010.16061, 2020. [Online]. Available: https://arxiv.org/abs/2010.16061
Inception-v4, inception-resnet and the impact of residual connections on learning. C Szegedy, S Ioffe, V Vanhoucke, abs/1602.07261CoRR. C. Szegedy, S. Ioffe, and V. Vanhoucke, "Inception-v4, inception-resnet and the impact of residual connections on learning," CoRR, vol. abs/1602.07261, 2016. [Online]. Available: http://arxiv.org/abs/1602.07261
Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, Advances in Neural Information Processing Systems. F. Pereira, C. Burges, L. Bottou, and K. WeinbergerCurran Associates, Inc25A. Krizhevsky, I. Sutskever, and G. E. Hinton, "Imagenet classification with deep convolutional neural networks," in Advances in Neural Information Processing Systems, F. Pereira, C. Burges, L. Bottou, and K. Weinberger, Eds., vol. 25. Curran Associates, Inc., 2012. [Online]. Available: https://proceedings.neurips.cc/paper/2012/file/ c399862d3b9d6b76c8436e924a68c45b-Paper.pdf
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| {'fraction_non_alphanumeric': 0.06753594115680374, 'fraction_numerical': 0.046484692171753356, 'mean_word_length': 4.374382339729192, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 35, 'lorem ipsum': 0, 'www.': 11, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Table detectionis the task of classifying and localizing table objects within document images. With the recent development in deep learning methods, we observe remarkable success in table detection. However, a significant amount of labeled data is required to train these models effectively. Many semi-supervised approaches are introduced to mitigate the need for a substantial amount of label data. These approaches use CNN-based detectors that rely on anchor proposals and post-processing stages such as NMS. To tackle these limitations, this paper presents a novel end-to-end semi-supervised table detection method that employs the deformable transformer for detecting table objects. We evaluate our semi-supervised method on PubLayNet, DocBank, ICADR-19 and TableBank datasets, and it achieves superior performance compared to previous methods. It outperforms the fully supervised method (Deformable transformer) by +3.4 points on 10% labels of TableBank-both dataset and the previous CNN-based semi-supervised approach (Soft Teacher) by +1.8 points on 10% labels of PubLayNet dataset. We hope this work opens new possibilities towards semi-supervised and unsupervised table detection methods.', 'arxivid': '2305.02769', 'author': ['Tahira Shehzadi [email protected] \nDepartment of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nMindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nGerman Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany\n', '− 979x ', 'Khurram Azeem Hashmi \nDepartment of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nMindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nGerman Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany\n', 'Didier Stricker] ', 'Marcus Liwicki \nDepartment of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nMindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nDepartment of Computer Science\nLuleå University of Technology\n971 87LuleåSweden\n\nGerman Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany\n', 'Muhammad Zeshan Afzal \nDepartment of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nMindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany\n\nGerman Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany\n'], 'authoraffiliation': ['Department of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'Mindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'German Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany', 'Department of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'Mindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'German Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany', 'Department of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'Mindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'Department of Computer Science\nLuleå University of Technology\n971 87LuleåSweden', 'German Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany', 'Department of Computer Science\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'Mindgarage\nTechnical University of Kaiserslautern\n67663KaiserslauternGermany', 'German Research Institute for Artificial Intelligence (DFKI)\n67663KaiserslauternGermany'], 'corpusid': 258480298, 'doi': '10.48550/arxiv.2305.02769', 'github_urls': [], 'n_tokens_mistral': 26925, 'n_tokens_neox': 22836, 'n_words': 12156, 'pdfsha': '8f87ad8807a4b149c69bac1f169d9f28d13f8928', 'pdfurls': ['https://export.arxiv.org/pdf/2305.02769v2.pdf'], 'title': ['Towards End-to-End Semi-Supervised Table Detection with Deformable Transformer', 'Towards End-to-End Semi-Supervised Table Detection with Deformable Transformer'], 'venue': []} |
arxiv |
Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling
M Moustapha
Chair of Risk, Safety and Uncertainty Quantification
ETH Zurich
Stefano-Franscini-Platz 58093ZurichSwitzerland
B Sudret
Chair of Risk, Safety and Uncertainty Quantification
ETH Zurich
Stefano-Franscini-Platz 58093ZurichSwitzerland
Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling
Surrogate modelling -non-smooth functions -discontinuities -Dirichlet process mixture models -uncertainty quantification
Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approximate. This assumption is however not always met in reality. For instance in civil or mechanical engineering, some models may present discontinuities or non-smoothness e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these fundamentally different behaviours or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression.The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.Keywords: Surrogate modelling -non-smooth functions -discontinuities -Dirichlet process mixture models -uncertainty quantificationIntroductionComputational models, which allow scientists and engineers to accurately simulate complex systems and predict their behaviour in various contexts, are nowadays a key tool present in 1 arXiv:2211.16909v1 [stat.ML] 30 Nov 2022 virtually all fields of applied sciences and engineering. Cast as computer experiments, they are able to predict with high fidelity the behaviour of the studied system in replacement of, or as a complement to laboratory experiments. The downside of such high-fidelity models is however that they are computationally demanding. This is even more relevant in the context of uncertainty quantification or design optimization, where the models need to be evaluated multiple times.Surrogate models have become paramount in such fields as they allow for an efficient solution of otherwise computationally intractable problems. They are inexpensive proxies that can be used in lieu of expensive computational models. Examples of such surrogates include Gaussian process models also known as Kriging (Santner et al., 2003;Rasmussen and Williams, 2006), polynomial chaos expansions (Xiu and Karniadakis, 2002; Blatman and Sudret, 2011), support vector machines (Vapnik, 1995), polynomial response surfaces (Myers and Montgomery, 2002), etc. These methods have been applied in various problems pertaining to uncertainty quantification or design optimization. The use of surrogate models in such fields are now mature as shown by the recent reviews in reliability analysis (Teixeira et al., 2021; Moustapha et al., 2022), Bayesian inversion (Yan and Zhang, 2017) or design optimization (Chatterjee et al., 2019; Moustapha and Sudret, 2019a).In most of these applications, it is assumed that the computational models to approximate feature some accommodating properties such as smoothness, differentiability or stationarity.Yet there exists cases when these assumptions do not hold. In mechanical engineering, this may happen for instance when solving non-linear problems involving instability such as snap-through or bifurcations in the solution path, e.g., crash simulation. In computational fluid dynamics, simulations of compressive flows that involve shocks also belong to this category. In other cases, the underlying phenomenon may present different localized features or extreme regime variations which are strongly dependent on the inputs.Various methods have been developed in the field of uncertainty quantification to tackle such problems. The first class of methods borrows from digital signal processing and image detection to identify discontinuities or strong gradients of the function to approximate using techniques such as polynomial annihilation(Le Maître et al., 2004;Gorodetsky, 2012). Such approaches however rely on uniformly sampled grids and are often limited to two-dimensional problems.Sargsyan et al. (2012)proposed a technique combining Bayesian inference and polynomial chaos expansions that does not require using a regular grid and hence allowing for a reduced number of samples. However, their approach was also developed for two-dimensional problems and the authors did not investigate how well it scales with dimensionality.Another class of methods relies on Gaussian process (GP) regression where the irregularities on the model to approximate are tackled by introducing non-stationary covariance functions or 2 kernels. Indeed, such kernels allow one to capture heterogeneous variations or heteroscedastic noise while keeping the computational budget low. The direct approach to build such kernels is to consider the noise variance, signal variance and/or characteristic length scale to be input-dependent, such as inPaciorek and Schervish (2003).Heinonen et al. (2016)proposed an approach where all three parameters are considered latent variables and inferred as hyperparameters of the GP. Such an approach has shown increased efficiency compared to vanilla GP but it also comes with an increased inference cost due to the fact that there are no more closedform solution and the hyperparameters need to be calibrated using sampling based techniques (See Rasmussen and Ghahramani(2001)). Furthermore, they do not allow to tackle problems with discontinuities.
A more sensible approach based on non-stationary GP consists in splitting the input space using for instance treed Gaussian processes or a mixture of experts (Tresp, 2000;Rasmussen and Ghahramani, 2001;Meeds and Osindero, 2005). Similarly, it is also possible to define nonstationary Gaussian process models by partitioning the training data into smaller subsets using clustering techniques, such as in Zhang et al. (2019) and Konomi et al. (2019), where K-means and nearest-neighbors clustering are used. Such approaches also have the advantage of offering faster training and testing of the model as the experimental design is divided into smaller and more computationally manageable subsets. Finally, another popular way to define non-stationary kernels is by warping the input, and sometimes the output, space. By doing so, one may find a latent space where the function to approximate is smoother. Examples of such techniques include warped GP (Marmin, 2018) or manifold GP regression (Calandra et al., 2016;Kuleshov et al., 2018).
In this work, we will focus on multi-stage techniques where the problem is solved by using a sequence of well-known machine learning techniques. More specifically, we consider the class of methods based on the following three-stage approach: clustering, classification and regression (Boroson and Missoum, 2017;Dupuis et al., 2018). Basudhar and Missoum (2008); Serna and Bucher (2009) were the first to propose decomposing the problem of identifying multiple failure domains of mechanical systems using support vector machines. However, they do not include the regression step as they are only concerned with an optimization problem where only the state of a sample is of interest (i.e., whether it belongs to the failure domain or not). Moustapha (2016); Moustapha and Sudret (2019b) extended the approach to the prediction of the model responses by building local Kriging surrogates in each identified domain. However in all these approaches, it was assumed that the clusters were identified either using expert knowledge or by only considering the model responses which span different ranges. Niutta et al. (2018) proposed identifying the clusters by detecting jumps in the model responses for relatively close samples. However, this technique works only in low-dimensional problems and when the response of different clusters are disjoint. This is a strong limitation and was to some extent overcome by using joint clustering of both the inputs and outputs in Bernholdt et al. (2019). In that work, they use K-means clustering to identify the clusters and multi-layer perceptrons for classification and regression tasks. The number of clusters is defined here using the elbow approach, which is a visual technique requiring user interaction. Furthermore it is not robust w.r.t. the initialization of the K-means algorithm and noise in the data. More generally, an important limitation in the contributions presented above is that the three steps are disconnected and the prediction uncertainty in one step is not accounted for in the subsequent ones.
In this paper, we propose an approach that aims at solving these two limitations. First, to automatically identify the number of clusters in a robust way, we consider a non-parametric Bayesian technique, namely Dirichlet process mixture models (DPMM). The interest in using DPMM are three-fold: i. they automatically estimate the optimal number of clusters according to patterns identified in the data, ii. they offer a probabilistic framework that allows one to propagate the epistemic uncertainty related to this clustering task to both the subsequent classification and regression steps, and iii. they are flexible enough and their complexity can grow as new data is observed (for instance in an active learning scheme, where new regimes of the model could be identified).
In the remainder of this paper, we first present the three-stage methodology and how the steps are connected in Section 2. In Section 3, we present in details the three methods used in each step, namely, Dirichlet process mixture models, support vector machines for classification and Gaussian process modelling. We finally illustrate the proposed approach in Section 4 using two analytical examples and an engineering application related to the design of a tensile membrane structure (Valdés-Vázquez et al., 2020.
2 Problem set-up and three-stage approach Let us consider a set of N data points (X , Y) where X = x (i) ∈ X ⊂ R M , i = 1, . . . N is a set of M -dimensional inputs and Y are corresponding scalar outputs such that
Y = y (i) = M x (i) ∈ R, i = 1 . . . N .
The model M is assumed black-box, meaning that it is only accessible through an evaluation over a finite set of input points. We further assume in this setting that the model is non-smooth, i.e., it exhibits sharp localized features and, most noticeably, discontinuities. As the model can only be evaluated on a finite set of samples, discontinuities in the current work is assumed when the model presents extreme variations in the outputs for seemingly close input points.
The goal of the analysis is to learn the input-output relationship of the model M through the limited set of training data D = (X , Y), also known as experimental design. This ultimately leads to a cheaper-to-evaluate surrogate model that can be used to predict the response of the model for any new point. Generally, this type of problems is tackled using regression techniques where a class of parameterized models are assumed and then their hyper-parameters are calibrated so as to minimize a generalization error. Such models would however fail when there are discontinuities or heterogeneous variations associated to limited observations.
In this work, we consider tackling this problem by splitting the space along the discontinuities and building local regression models in each of the obtained subdomains. To achieve this, we consider a three-stage framework which is illustrated in Figure 1 and summarized as follows:
Clustering
Classification Regression Figure 1: Illustration of the three-stage approach.
1. Clustering: The first learning step aims at identifying patterns in the data that hint to subdomains separated by discontinuities. To achieve this, we cluster the joint input-output data points. This is an unsupervised learning problem for which numerous techniques have been developed (Pham and Afify, 2017). K-means clustering (Lloyd, 1982) is probably the most common approach thanks to its simplicity. However, it assumes that the number of clusters is known and further fails when the clusters are of disproportionate sizes. Another approach that partially overcomes difficulties related to K-means clustering are Gaussian mixture models which offer a probabilistic framework for clustering (Rokach and Maimon, 2005). They hence allow for a more nuanced clustering of the data by providing soft cluster memberships, i.e., each data point is assigned with a probability of belonging to a given cluster. This feature allows one to solve more complex problems, e.g., when the clusters are partially overlapping. However, similarly to K-means, they assume that the number of clusters is known in advance. In general, trial-and-errors approaches are used to define the optimal number of clusters for such problems, which is not optimal.
We therefore consider in this work a more holistic approach where the number of clusters is also inferred from the data using a non-parametric Bayesian model, more specifically Dirichlet process mixture models (Li et al., 2019) as described in Section 3.1.
At the end of this step, the experimental design is split into K subsets C k , k = 1, . . . , K.
2. Classification: Assuming that the data have been clustered, we can now place labels on them and turn to supervised learning. More specifically, let us assume K clusters are identified in the previous step. We thus define the labels { 1 , . . . , K } and the labelled training
data X × L where each couple x (i) , (i) is defined such that (i) = k if x (i) , y (i) ∈ C k .
The goal of this step is then to partition the input space such that any new sample can be mapped to at least one of the clusters C k . This will ultimately allow us to select the appropriate local regression model(s) to evaluate the new point.
This task is carried out in this work by using support vector machines (SVM) for binary and multi-class classification (Vapnik, 1995). The probabilistic framework is introduced by considering Platt's approach to computing posterior probabilities given a binary SVM prediction (Platt, 2000). For multi-class problems, binary classifiers are appropriately combined to provide both class membership and posterior probabilities.
3. Regression: In this final step, Gaussian process (GP) models (Rasmussen and Williams, 2006) are employed to make the final prediction. We further investigate the use of three different approaches for combining the various GP models built in this stage. In the first two approaches, local surrogate models M k are built for each of the K identified clusters.
When it comes to prediction, the recombination is made as follows:
• Hard recombination: In this approach, the surrogate model which corresponds to the cluster predicted by the classifier is solely used to make the final prediction, i.e.,
M (x) = K k=1 1 C k (x) M k (x) ,(1)
where 1 C k (x) is equal to 1 if x is predicted to belong to the cluster C k , i.e., M SVC (x) = k and 0 otherwise;
• Soft recombination: In this approach, the prediction for each point is obtained as a weighted combination of all the local surrogate models, i.e.,
M (x) = K k=1 w k (x) M k (x) ,(2)
where the weight w k (x) ∈ [0, 1] with K k=1 w k (x) = 1 may be related to the actual probability that the point x belongs to the cluster C k as defined by the classifier.
• Categorical recombination: Contrary to the previous two approaches, a single Gaussian process model is built here. This is achieved by using an additional variable which is a categorical parameter indicating which cluster a given point belongs to, i.e., the training set is the couple {X , L} × Y where L = (i) , i = 1, . . . , N are the labels of the training set identified in the clustering stage. The surrogate model is therefore built on a space of dimension M + 1: M (x) = M cat x = x, (x) , where the categorical variable is given by the SVC prediction, i.e., (x) = M SVC (x).
The following section describes in details each of the ingredients introduced in the proposed framework.
3 Description of the components of the proposed method 3.1 Clustering using Dirichlet process mixture models
Gaussian mixture models
Let us now consider the set of available data W = w (i) , i = 1, . . . , N , where w (i) =
x (i) , y (i) is a vector gathering both inputs and outputs, and let us assume that they are associated to some latent variables z. In a clustering set-up, say using a Gaussian mixture, the latent variables would be z = {π, µ, Σ} where π are mixing coefficients and µ and Σ are the mean and covariance of multivariate normal random variables. The goal is then to find the posterior distribution p(z|w) of the latent variables given the data and using Bayes rules, i.e.,
p(z|w) = p (w, z) p (w) = p (w|z) p (z) p (w) ∝ p (w|z) p (z) ,(3)
where p (w|z) is the data likelihood, p (z) = p (π, µ, Σ) is the prior over the latent variables and p (w) is the evidence.
The prior can be fully factorized into p (π) p (µ) p (Σ) since the three parameters are considered mutually independent. The prior on the mixing coefficients p (π) is usually chosen as a Dirichlet distribution with parameters α/K where α is a positive scaling parameter and K is the predefined number of clusters:
p (π 1 , . . . , π K |α) = Dirichlet (α/K, . . . , α/K) = Γ (α) Γ (α/K) K K k=1 π α/K−1 k ,(4)
where Γ is the Gamma function.
The Dirichlet distribution is chosen precisely because it is the conjugate distribution to the multinomial distribution, which is used for clusters membership assignment, later denoted by c.
The generative model for data derived from a Gaussian mixture model can therefore be cast as
π k ∼ Dirichlet (α/K, . . . , α/K) , k = {1, . . . , K} , c (i) ∼ Multinomial (π 1 , . . . , π K ) , i = {1, . . . , N } , w (i) | c (i) = k ∼ N (µ k , Σ k ) , i = {1, . . . , N } ,(5)
where µ k and Σ k are respectively the mean and covariance parameters of each local Gaussian distribution in the mixture.
It is generally assumed in such a model that K << N , which in other words means that samples from all clusters have been observed. However, there may exist cases when K is in the same order or even larger than N . An alternative view to such cases is that at any moment all clusters have not yet been observed and drawing more data from the generative model will reveal new clusters. This naturally leads to extending this finite mixture model into an infinite one using non-parametric Bayesian models whose complexity can grow as more data are observed.
This is precisely what a Dirichlet process mixture model does. It generalizes the generative model described in Eq. (5) by assuming an infinite number of clusters, i.e., that K → ∞. This corresponds to choosing a Dirichlet process (Ferguson, 1973) as prior for the mixing coefficients, as explained in the sequel.
Dirichlet process
A Dirichlet process (DP) is a distribution over distributions defined by a base distribution G 0 and a positive scaling parameter α. The output from a Dirichlet process is therefore a discrete distribution. It is however not possible to directly draw from G considering the formal definition of a Dirichlet process. Other alternative views such as the Chinese restaurant process (Aldous, 1985), the Pólya urn scheme (Blackwell and MacQueen, 1973) or the stick-breaking representation (Sethuraman, 1994) have been proposed instead.
In this work, we consider the latter approach. More specifically, let us consider an infinite collection of two random variables V k ∼ Beta(1, α) and η * k ∼ G 0 with k = {1, 2, . . .}. The stick-breaking representation of G is then defined as follows:
π k = v k k−1 j=1 (1 − v j ) , G = ∞ k=1 π k (v) δ η * k (η k ) ,(6)
where δ is the Kronecker symbol. This representation is illustrated in Figure 2 where the η * k are location parameters also known as atoms and π k are corresponding weights.
In a DP, there is a countably infinite number of atoms and the weights sum up to 1, making G a discrete distribution. This infinite set of atoms lends itself to modelling priors in infinite mixture models. More specifically, the DP is used in Dirichlet process mixture models as a non-parametric prior in a hierarchical Bayesian model specified as follows (Antoniak, 1974;Blei Figure 2: Illustration of a Dirichlet process: G 0 is the base distribution from which the atoms η * k are sampled, π k are the corresponding weights and G a realization of the DP. and Jordan, 2006):
G| {α, G 0 } ∼ DP (α, G 0 ), η (i) |G ∼ G, W (i) |η (i) ∼ p(w (i) |η (i) ).(7)
Given a dataset W, each data point w (i) is assumed to be generated by first drawing a component label c (i) = {1, 2, . . .} with probability distribution p(c (i) = k|V ) = π k (v) and then drawing w (i) from p(w (i) |η k ). In this work, p is chosen as a distribution from the exponential family for which G 0 is a conjugate prior, which turns out to also belong to the exponential family and hence making inference easier.
Posterior estimation
The latent variables in this setting are therefore z = {v, η, c}. The goal of the analysis is then to find the posterior distribution of these latent variables given the observed data W, which is denoted by p (z|W, θ). There is no closed-form solution to this problem and typical solution schemes rely on Markov Chain Monte Carlo (MCMC). MCMC algorithms allow one to obtain an approximation of the posterior using Markov chains whose stationary distribution is the sought posterior. The usual approach in Dirichlet process mixture models is Gibbs sampling which is particularly suited to this task as one can have access to the conditional distributions of the latent variables analytically (Neal, 2000;Ishwaran and James, 2001). However, the difficulty with MCMC algorithms is that they are expensive, as they require a large number of samples, often generated sequentially, and their convergence is difficult to monitor.
An alternative approach to circumvent these issues is variational inference, where the esti-mation of the posterior is replaced by an optimization problem (Wainwright and Jordan, 2003).
More specifically, the intractable posterior is replaced by a parametric family of variation distributions denoted here by q ν (z|ν). In this paper, we consider the approach proposed by Blei and Jordan (2006) which relies on the mean-field approximation, i.e., the variational distribution is fully factorized (all the latent variables are mutually independent). The optimization problem then consists in finding within the selected family of variational distributions the values of the hyperparameters ν that will minimize the Kullback-Liebler (KL) divergence between the true posterior and its approximation. This quantity reads
KL (q ν (z|ν)||p (z|W, θ)) = ∞ −∞ q ν (z|ν) log q ν (z|ν) p (z|W, θ) dz = E qν [log q ν (z|ν)] − E qν [p (z, W|θ)] + log p (w|θ) .(8)
By noting that the divergence is always positive (or using Jensen's inequality), it can be shown that minimizing Eq. (8) is equivalent to maximizing a lower bound of the marginal log likelihood, also referred to as ELBO and denoted by
log p (w|θ) ≥ E qν [p (z, W|θ)] − E qν [log q ν (z|ν)] .(9)
By appropriately choosing the family of variational distributions for each latent variable, it is possible to make the computation of the ELBO tractable. In the approach proposed by Blei and Jordan (2006) considered here, the factorized variational distribution is cast as
q ν (v, η, z|ν) = T −1 t=1 q γt (v t ) T t=1 q τt (η t ) N k=1 q Φ k (c k ) ,(10)
where q γt (v t ) are Beta distributions, q τt (η t ) are exponential family distributions and q Φ k (c k ) are multinomial distributions. In this equation, the infinite samples is truncated to T terms by setting q(v T = 1) = 1, which implies that π t (v) = 0 for t ≥ T . The solution to this problem is eventually obtained using a coordinate ascent algorithm for which the incremental updates can be computed analytically (Ghahramani and Beal, 2000). The reader is referred to Blei and Jordan (2006) for further details.
3.2 Classification using support vector machines Given this training set, the support vector classifier (SVC) prediction for any yet-to-be observed sample reads (Smola and Schölkopf, 2004)
M SVC (x) = N i=1 α i (i) k x (i) , x; θ + b,(11)
where {α, b} are parameters to calibrate. The coefficients α i , some of which are the so-called support vectors, and the offset parameter b are obtained by solving a quadratic optimization
problem min α 1 2 α T KY Y T α + h T α subject to: α T Y = 0, α i ≥ 0, i = {1, . . . , N } ,(12)
where h = {−1, . . . , −1} is a column vector of size N and K = K + 1/CI N with C > 0 being a penalty term. The matrix K is the so-called Gram matrix built by evaluating the parameterized kernel function on pairs of points of the training data set, such that K ij = k x (i) , x (j) ; θ , i, j ∈ {1, . . . , N }. Multiple kernels have been used in SVM. In this work, we consider the Gaussian kernel defined by
k x (i) , x (j) ; θ = M l=1 exp − 1 2 x (i) l − x (j) l θ 2 l 2 .(13)
The hyperparameters of this model are the penalty term C which controls the penalty in-
Extension to multi-class classification
Let us now consider the case when the classification task aims at categorizing data with a set of K > 2 labels, where each label is defined as (i) = k if the original training pair x (i) , y (i) belongs to the cluster C k .
The most popular approach to tackle this multi-class problem is to reduce it to a series of binary classification problems that can be solved using a standard SVM algorithm. The two most popular approaches are the one-against-all and the one-vs-one decomposition schemes (Hastie and Tibshirami, 1997;Moreira and Mayoraz, 1998). In the former, one binary problem is derived for each class k by assigning one label, say the positive one, to all samples such that (i) = k and the negative label to all the other samples. In the one-vs-one approach, binary classifiers considering all pairs of labels and ignoring all other samples are built. This leads to a total of K(K − 1)/2 classifiers, which is larger than the K classifiers required by the one-against-all approach. However, such classifiers are trained on a noticeably smaller subset of the training samples making the overall procedure computationally efficient despite the larger number of classifiers to build.
Both approaches can be generalized, or somehow combined, using concepts of the error correcting output codes (ECOC) (Dieterich and Bakiri, 1995). The recombination of the binary classifiers into a final one can be achieved either by a simple voting system or by considering the posterior probabilities derived from each classifier. In this work, we consider the one-vs-one approach with a final voting system thanks to its simplicity and efficiency. We note that in case of equal voting between two classes, we heuristically choose the class that was predicted with the classifier that considered the two classes of interest.
Posterior probabilities
As mentioned in Section 2, the soft recombination of the final predictor requires some weights which are proportional to the probability that the sample belongs to a given class. In case of SVM, such weights can be derived by computing posterior probabilities derived from the classifier. In practice, this can be achieved by post-processing the output of the classifier using a sigmoid function as proposed by Platt (2000):
P (x) = 1|M SVC (x) = 1 1 + exp (A M SVC (x) + B) ,(14)
where the coefficients A and B are calibrated by solving a regularized maximum likelihood problem. In this work, we use an efficient numerical implementation proposed by Lin et al.
(2007).
There have been many attempts to extend these probabilities to multi-class problems (Hastie and Tibshirami, 1997;Moreira and Mayoraz, 1998;Wu et al., 2004;Wang, 2008). Let us denote by
p ij = P (x ∈ C i | x ∈ C i ∪ C j )(15)
the posterior probability provided by the classifier that discriminates between the classes C i (positive) and C j (negative). Note however that we are interested in the overall probability of belonging to a class given all possible classes, i.e. p i = P (x ∈ C i ). Moreira and Mayoraz (1998) proposed estimating this probability by combining the partial ones, i.e.,
p i = 2 k(k − 1) K j =i,j=1 p ij(16)
This value is however flawed, as it accounts for spurious probabilities defined by classifiers discriminating two classes, none of which being the true one.
Using Bayes theorem, it can however be noted that
p i = P (x ∈ C i ) = P (x ∈ C i | x ∈ C i ∪ C j ) P (x ∈ C i ∪ C j ) ,(17)
which, by averaging over all combinations of i and j, leads to the following system of equations:
p i = 1 k − 1 K j =i,j=1 p ij (p i + p j ) ,(18)
since P (x ∈ C i ∪ C j ) = (p i + p j ). Wu et al. (2004) noted that this system of equations can be written in a matrix form
p = T p,(19)
where p = {p 1 , . . . , p K } T and T is a K × K matrix whose elements read Wu et al. (2004) then noted that there exists a finite Markov chain whose transition matrix is T , since K j=1 T ij = 1 and 0 ≤ T ij ≤ 1. Further assuming that p ij > 0 for any i, j ∈ {1, . . . , K} implies that T ij > 0, which ensures that the Markov chain is irreducible and aperiodic. In fine, these conditions guarantee that Eq. (19) defines a Markov chain whose stationary distribution exists and is unique.
T ij = 1 k−1 p ij if i = j, 1 k−1 K j =i,j=1 p ij if i = j.(20)
Taking advantage of the fact that T is a transition kernel and p is the stationary distribution of the corresponding Markov chain, we cast Eq. (18) in an iterative scheme
p (t+1) i = 1 k − 1 K j =i,j=1 p ij p (t) i + p (t) j ,(21)
where the initial values p
Regression using Kriging
Basics of Kriging
The final ingredient considered in the proposed framework is Kriging a.k.a. Gaussian process model. It is used here to build local surrogates in the different regions identified by the clustering step.
A Kriging model assumes that the model to approximate is of the form (Santner et al., 2003;Rasmussen and Williams, 2006) M
(x) = p j=1 β j f j (x) + Z (x) ,(22)
where the first summand represents the trend written here in a polynomial form using p regressors f j with corresponding coefficients β j . The second summand is a zero-mean stationary covariance distribution N µ M , σ 2 M where the mean is the actual prediction, while the standard deviation informs about the local accuracy of the prediction. The two quantities respectively read
µ M (x) = f T (x) β + r (x) R −1 Y − F β , σ 2 M (x) = σ 2 1 − r (x) T R −1 r (x) + u (x) T F T R −1 F −1 u (x) ,(24)
where
• u (x) = F T R −1 r (x) − f (x)
has been introduced for convenience,
• β = F T R −1 F −1 F T R −1 Y is the generalized least-square estimate of the regression coefficients β, • σ 2 = 1 N Y − F β T R −1 Y − F β
is the estimate of the process variance,
• F = f j x (i) , j = 1, . . . , p, i = 1, . . . , n 0 is the Vandermonde matrix,
• R is the correlation matrix with R ij = R x (i) , x (j) ; θ ,
• r (x) is a vector gathering the correlation between the unknown sample x and the experimental design points and
• Y = Y (i) = M x (i) , i = 1, . . . ,R (i) , (j) = 1 if (i) = (j) , r if (i) = (j) ,(25)
is considered. The parameter r is computed here by embedding this kernel within a usual auto-correlation function for continuous variables with a tunable parameter θ cat that can be calibrated in the same setting than the continuous parameters. More precisely, we consider a Gaussian kernel which then reads:
R (i) , (j) ; θ cat = exp − 1 2 S (i) , (j) θ cat
where S (i) , (j) = 0 if (i) = (j) and 1 otherwise. The final auto-correlation function is obtained by multiplying the M + 1 one-dimensional auto-correlation functions i.e.,
R x (i) , x j , θ = exp − 1 2 M k=1 x (i) − x (j) θ k 2 − 1 2 S (i) , (j) θ cat 2 ,(27)
where θ = {θ, θ cat } and x (i) = x (i) , (i) .
Examples
The proposed algorithm is illustrated in this section with two analytical toy functions and an engineering problem related to a tensile membrane structure design. To assess its accuracy, we estimate the following two generalization errors using a validation set of size N val = 10 4 :
• Normalized mean-square error:
N M SE = N val i=1 Y i − Y i 2 N val i=1 Y i −Ȳ 2 ,(28)
• Mean absolute error:
M AE = 1 N val N val i=1 Y i − Y i .(29)
Furthermore, each analysis is repeated 20 times in order to assess the robustness of the proposed algorithm with respect to the statistical uncertainty associated with the experimental designs.
Manhattan function
For this first validation example, we consider a two-dimensional function proposed by Rai (2015).
The function consists of three global regions, one of which is a checkerboard, and reads
M (x) = Checker board if x 1 ≥ 0, sin (7x 1 ) · sin (4x 2 ) ; if x 1 ≤ 0 and x 2 ≤ 0, 1 + 2 7 (2x 1 + 1) 2 + (2x 2 + 1) 2 ; if x 1 ≤ 0 and x 2 ≥ 0
The checkerboard is made of smaller rectangular regions alternating the values of 0 and 1 as illustrated in Figure 3.
In this section, we will illustrate each of the three steps of the proposed algorithm. We first start by showing how the clustering algorithm splits the data. Figure 4 shows the clusters identified using three experimental designs of different sizes. The original model is built assuming 10 regions where each of the squares in the checkerboard is considered as one region on its own.
However, regardless of the experimental design, the clustering algorithm reduces the checkerboard into two regions, one with y = 1 and the other with y = 0. This results in disconnected Figure 3: Example 1 -Three-dimensional representation of the Manhattan function.
subdomains but as we will see in the next paragraph, this does not affect the overall prediction capability of the algorithm. Another important observation from the partitions in Figure 4 is that the more data points, the more clusters are identified. For small datasets, the partition is quite sensitive to the data. However, the partition becomes more stable and robust as the data size is increased.
Once the clusters are identified (4 different ones in the case of medium-size experimental design, and in the sequel), the inputs are labelled accordingly and binary classification is performed on each pair of classes. Figure 5 shows the resulting classifiers for one realization of the experimental design. The blue and orange points correspond to the positive and negative labels respectively, while the support vectors are highlighted in green. The thick line is the classifier, whereas the dashed ones delimit the margin. Finally, the gray triangles represent the data points that were ignored by the illustrated classifier. As expected, support vector machines are appropriately calibrated for the problem at hand. However, the choice of the Gaussian kernel may not be appropriate for the classification of C 3 against C 4 (Figure 5f) as it produces smooth boundaries whereas the original boundary results from a checkerboard with sharp edges. This does not substantially affect the results. However, better prediction could have been obtained by including the choice of the kernel in the model selection.
The next step is then to recombine those predictions into a final one. In the hard reconstruction, a vote is carried out and the class that wins is the final prediction. The resulting partition of the input space is shown in Figure 6. Figure 7 shows the soft reconstruction approach where each tile represents the probabilities of a given point to belong to a given class.
The resulting classification is in accordance with the regions defined by the original model except for the boundaries of the checkerboard which present some slight deviations. Also, the boundary between the two regions where M is smooth (i.e. , polynomial or sines) is not exactly the line This partition of the input space is eventually used to build local Kriging surrogates to provide the final prediction. For this example, we repeat the analysis 20 times where each repetition starts with a randomly sampled Sobol' sequence. Figure 8 shows boxplots of the resulting errors for increasing sizes of the experimental design. For any ED size, both recombination techniques yield improved N M SE and M AE. In general, the soft reconstruction also yields better prediction than the hard one. This is even more clear when considering the M AE error. For this example, the prediction with categorical Kriging is not included, since it does not lead to good results. This is due to the fact that each region is fundamentally different from the other, hence using a single Kriging model, even with categorical variables, is not appropriate.
(a) C1 vs. C2
(b) C1 vs. C3 (c) C1 vs. C4 (d) C2 vs. C3 (e) C2 vs. C4 (f) C3 vs. C4
Snap-through instability problem
This example is a mechanical problem related to the snap-through instability of a two-bar truss structure. The structure is loaded at its tip and responds linearly with small displacements until a critical point is reached. Past that point, the structure suddenly snaps through a new equilibrium point and resumes its small displacements. In this example, we consider as quantity of interest the displacement w of the tip of the structure as illustrated in Figure 9 . Figure 9: Illustration of the two-bar truss structure subject to snap-through.
The load at the deformed position can be expressed as a function of the inclination angles at the initial position and deformed one, respectively denoted by α 0 and α, the bars cross-sectional areas A and their constitutive material Youngs's modulus E P = −2EA tan (α) (cos (α 0 ) − cos (α)) .
The corresponding displacement of the tip of the truss can then be computed as follows:
w = l 0 cos (α 0 ) (tan (α 0 ) − tan (α)) .(31)
In this example, we assume that the length of the bar l 0 = 5 m and the initial inclination angle α 0 = 10 • are deterministic. In contrast, the load, the Young's modulus and the cross section areas are assumed random and characterized by the distributions shown in Table 1 We run the analysis using the proposed method and considering three different experimental design sizes and 20 repetitions. The resulting errors are summarized as boxplots shown in Figure 10. The first observation is that the difference between the results obtained by the proposed method and a direct Kriging model (i.e. a single Kriging model built using the entire data set) is much more important than in the previous case, often by orders of magnitude. This is due to the fact that the two regimes of non-linear structure behaviours are prominently different as shown in Figure 11. Furthermore in this example, categorical Kriging performs quite well. It is not clear however which recombination approach is the best. When looking at the normalized mean square error, the hard recombination is slightly better. This is the opposite when looking at the mean absolute error, i.e., the soft and categorical recombination are slightly better.
Tensile fabric structure
In this final example, we investigate a model that simulates the behaviour of a tensile membrane structure (TMS) under extreme loading (Valdés-Vázquez et al., 2020. TMS are flexible lightweight structures made of composite fabric spanning long distances. They have many advantages in terms of architectural sophistication but are yet challenging to design. By their very nature, they are unable to carry out-of-plane moments and shear forces that may result from the extreme wind loads they are expected to withstand. They further require careful pre-stressing to keep a stable form.
Special codes are designed to simulate the response of complex tensile membrane structures.
Comet is one such in-house finite element code developed at the University of Gua (Valdés-Vázquez et al., 2021). In this work, we consider a hypar (hyperbol-paraboloid), which is one of the most common shapes for TMS, designed using Comet and illustrated in Figure 12. The probabilistic model is described using the random variables presented in Table 2. There are various quantities of interest for such a design model. We consider here the maximum reaction forces on the supports of the system (cables or mast). It turns out that according to the boundary conditions, the maximum reaction force occurs in two different locations with entirely different magnitudes. This is shown by the bi-modality of the kernel density estimate of the model response in Figure 13. Mast cross-sectional area (A m -m 2 ) Gaussian 1.7 · 10 −3 0.032
Cable cross-sectional area (A c -m 2 ) Gaussian 7.854 · 10 −5 0.032 building a single surrogate model to account for both leads to inaccurate results. We consider then the three-stage approach proposed in this paper, with an experimental design of size 500 and a validation set of size 1, 000. The experimental design is split into five different subsets of sizes 100, 200, 300, 400 and 500. In each of these, the DPMM clustering rightly identifies that there are two sets of responses. Figure 14 shows the resulting NMSE and MAE for each experimental design size. As ex- Finally, Figure 15 shows PDFs of the responses for different models with ED sizes of 100 and 300. We can see that even for 100 samples, the densities with the hard recombinations are extremely similar to those obtained from the original model. This shows that the reconstructed surrogate models are extremely accurate except for a few outliers which are due to misclassification in the second step of the workflow. The soft recombination puts more mass in the middle of the density support, due to the weighted recombination. This mass reduces as the ED size increases.
Conclusion
Surrogate modelling is now a well-established method that allows one to reduce the computational burden of simulation intensive methods that require multiple evaluations of a costly computational model. Building an accurate surrogate model with limited data generally requires that the functions to approximate are smooth and regular. This is however not always the case in many applications, e.g. crash simulation or computational fluid dynamics.
In this paper, we propose a three-stage approach for the approximation of non-smooth functions for systems exhibiting multiple behaviours and/or discontinuities. The problem is tackled by dividing the task into three complementary parts: i. a joint input-output clustering stage that identifies the different patterns exhibited by the system using a non-parametric Bayesian approach, namely a Dirichlet process mixture model, ii. a partition of the input space according to the identified clusters using support vector machines, and eventually iii. the construction of local surrogates, herein Kriging models, using data from each of the partitions. For any new point, the prediction is made by appropriately recombining the predictions made by each of the Kriging models, according to the assigned class of the new point.
The proposed approach is validated on two analytical examples and an engineering application (FE-based tensile membrane structure). It is shown to be both accurate and efficient compared to a traditional surrogate modelling approach ignoring the non-smoothness.
The three methods selected for each stage all provide probabilistic predictions. While the posterior probabilities of the support vector machines classifiers have been used within the soft reconstruction scheme, the ones provided by the Dirichlet process mixture models have not been exploited yet. However, as seen in the examples, mislabelling the initial data leads to large errors. These could be reduced by accounting for the uncertainties in the clustering stage. In a future work, we intend to account for the latter so as to provide a fully probabilistic prediction scheme that propagates the epistemic uncertainties from one step to the next.
machines are a popular supervised learning algorithm developed by Vapnik(1995). They were developed for binary classification and were later extended to account for multiple classes. Let us first consider the binary case (i.e., assuming only two clusters were identified) and denote the dataset by x (i) , (i) , i = 1, . . . , N where (i) = {−1, 1} are the labels of the training points.
curred for misclassifying a training point and the kernel parameter θ which controls, among others, the smoothness of the separating hyperplane. They are both estimated in this work by minimizing the span estimate of the leave-one-out error(Vapnik and Chapelle, 2000;Chapelle et al., 2002) using the covariance-matrix adaptation evolution scheme (CMA-ES) (See Arnold and Hansen (2012);Moustapha et al. (2018Moustapha et al. ( , 2021 for details).
i
, i = {1, . . . K} using the estimate in Eq. (16) and p ij are the partial probabilities obtained by the binary one-vs-one classifiers using Eq. (14) . This chain eventually converges after a few iterations, generally with t < 100 in our examples, to the posterior probabilities estimates.
Figure 4 :
4Example 1 -Clustering of the data by DPMM considering two repetitions of three experimental designs of increasing sizes.{x 1 ≤ 0, x 2 = 0}.
Figure 5 :
5Example 1 -Pairwise classification of the data (with 4 clusters identified in Step 1 for the medium-size experimental design).
Figure 6 :Figure 7 :Figure 8 :
678Example 1 -Partition of the space in the 4 regions using hard reconstruction. Example 1 -Partition of the space in the 4 regions using soft reconstruction. Example 1: Boxplots of the computed errors for various methods and experimental design sizes.
Figure 10 :
10Example 2: Boxplots of the computed errors for various methods and experimental design sizes.
Figure 11
11shows the original vs. predicted vertical displacement for the four approximations using a random subset of the validation set of size 200. The left panel of this figure shows how a single model (called "direct") spans the entire range between the two regimes of the truss and leads to huge errors. In contrast, the multi-stage approaches properly detect the discontinuities. It is also clear from this figure how the recombination scheme affects the final prediction when there are classification errors. The soft recombination reduces the error for those cases when there is uncertainty in the classification. Note that the same outlier points are observed in Figures 11a and 11b when hard reconstruction and categorical Kriging are used: these outliers only stem from classification error.
Figure 11 :
11Example 2: Original vs. predicted vertical displacement for different approximation techniques.
Figure 12 :
12Hypar structure considered in this study.
Figure 13 :
13Example 3: Kernel smoothing density of the maximum reaction force of the hypar. pected, the error decreases with increasing ED size and our proposed workflow yields more accurate approximations than a global single Kriging model, except for NMSE when N = 100 due to the large weight of misclassification errors. The soft recombination is slightly better than hard recombination and categorical Kriging which have very similar predictions.
Figure 14 :
14Example 3: Computed errors for the hypar structure for increasing experimental design sizes.
Figure 15 :
15Example 3: Computed errors for the hypar structure for increasing experimental design sizes.
n 0 is the vector of available model responses.To account for the categorical variable, the compound symmetry kernel defined by Pelematti
et al. (2020)
.Parameter
Distribution Mean C.o.V.
Load (P in N)
Gumbel
430
0.20
Young's modulus (E in GPa)
Lognormal
210
0.10
Cross sectional area (A in cm 2 )
Gaussian
10
0.05
Table 1: Truss snap-through problem: probabilistic input model.
The underlying mechanisms leading to each of two model response modes are different andParameter
Distribution Mean
C.o.V.
Wind load (V w -m/s)
Gumbel
36.11
0.132
Cable pre-stress (S xx -N/m 2 )
Gaussian
5.09 · 10 8
0.06
Young's modulus (E wf -N/m)
Lognormal
8 · 10 5
0.07
Poisson modulus (ν -)
Gaussian
0.4
0.05
Fabric prestress warp (F w -N/m 2 )
Gaussian
4 · 10 6
0.05
Fabric prestress fill (F f -N/m 2 )
Gaussian
4 · 10 6
0.05
Mast Young's modulus (E m -N/m 2 ) Lognormal
2.1 · 10 11
0.03
Cables Young's modulus (E c -N/m 2 ) Lognormal
2.1 · 10 11
0.03
Table 2 :
2Hypar structure: probabilistic input model.
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| {'fraction_non_alphanumeric': 0.05291354042493514, 'fraction_numerical': 0.023897342402356078, 'mean_word_length': 4.399106534413568, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 23, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approximate. This assumption is however not always met in reality. For instance in civil or mechanical engineering, some models may present discontinuities or non-smoothness e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these fundamentally different behaviours or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression.The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.Keywords: Surrogate modelling -non-smooth functions -discontinuities -Dirichlet process mixture models -uncertainty quantificationIntroductionComputational models, which allow scientists and engineers to accurately simulate complex systems and predict their behaviour in various contexts, are nowadays a key tool present in 1 arXiv:2211.16909v1 [stat.ML] 30 Nov 2022 virtually all fields of applied sciences and engineering. Cast as computer experiments, they are able to predict with high fidelity the behaviour of the studied system in replacement of, or as a complement to laboratory experiments. The downside of such high-fidelity models is however that they are computationally demanding. This is even more relevant in the context of uncertainty quantification or design optimization, where the models need to be evaluated multiple times.Surrogate models have become paramount in such fields as they allow for an efficient solution of otherwise computationally intractable problems. They are inexpensive proxies that can be used in lieu of expensive computational models. Examples of such surrogates include Gaussian process models also known as Kriging (Santner et al., 2003;Rasmussen and Williams, 2006), polynomial chaos expansions (Xiu and Karniadakis, 2002; Blatman and Sudret, 2011), support vector machines (Vapnik, 1995), polynomial response surfaces (Myers and Montgomery, 2002), etc. These methods have been applied in various problems pertaining to uncertainty quantification or design optimization. The use of surrogate models in such fields are now mature as shown by the recent reviews in reliability analysis (Teixeira et al., 2021; Moustapha et al., 2022), Bayesian inversion (Yan and Zhang, 2017) or design optimization (Chatterjee et al., 2019; Moustapha and Sudret, 2019a).In most of these applications, it is assumed that the computational models to approximate feature some accommodating properties such as smoothness, differentiability or stationarity.Yet there exists cases when these assumptions do not hold. In mechanical engineering, this may happen for instance when solving non-linear problems involving instability such as snap-through or bifurcations in the solution path, e.g., crash simulation. In computational fluid dynamics, simulations of compressive flows that involve shocks also belong to this category. In other cases, the underlying phenomenon may present different localized features or extreme regime variations which are strongly dependent on the inputs.Various methods have been developed in the field of uncertainty quantification to tackle such problems. The first class of methods borrows from digital signal processing and image detection to identify discontinuities or strong gradients of the function to approximate using techniques such as polynomial annihilation(Le Maître et al., 2004;Gorodetsky, 2012). Such approaches however rely on uniformly sampled grids and are often limited to two-dimensional problems.Sargsyan et al. (2012)proposed a technique combining Bayesian inference and polynomial chaos expansions that does not require using a regular grid and hence allowing for a reduced number of samples. However, their approach was also developed for two-dimensional problems and the authors did not investigate how well it scales with dimensionality.Another class of methods relies on Gaussian process (GP) regression where the irregularities on the model to approximate are tackled by introducing non-stationary covariance functions or 2 kernels. Indeed, such kernels allow one to capture heterogeneous variations or heteroscedastic noise while keeping the computational budget low. The direct approach to build such kernels is to consider the noise variance, signal variance and/or characteristic length scale to be input-dependent, such as inPaciorek and Schervish (2003).Heinonen et al. (2016)proposed an approach where all three parameters are considered latent variables and inferred as hyperparameters of the GP. Such an approach has shown increased efficiency compared to vanilla GP but it also comes with an increased inference cost due to the fact that there are no more closedform solution and the hyperparameters need to be calibrated using sampling based techniques (See Rasmussen and Ghahramani(2001)). Furthermore, they do not allow to tackle problems with discontinuities.', 'arxivid': '2211.16909', 'author': ['M Moustapha \nChair of Risk, Safety and Uncertainty Quantification\nETH Zurich\nStefano-Franscini-Platz 58093ZurichSwitzerland\n', 'B Sudret \nChair of Risk, Safety and Uncertainty Quantification\nETH Zurich\nStefano-Franscini-Platz 58093ZurichSwitzerland\n'], 'authoraffiliation': ['Chair of Risk, Safety and Uncertainty Quantification\nETH Zurich\nStefano-Franscini-Platz 58093ZurichSwitzerland', 'Chair of Risk, Safety and Uncertainty Quantification\nETH Zurich\nStefano-Franscini-Platz 58093ZurichSwitzerland'], 'corpusid': 254096517, 'doi': '10.1016/j.compstruc.2023.107035', 'github_urls': [], 'n_tokens_mistral': 20135, 'n_tokens_neox': 17355, 'n_words': 11106, 'pdfsha': '7635b5d17f5843178b7fd353d3ac5e4d1dfe5f49', 'pdfurls': ['https://export.arxiv.org/pdf/2211.16909v1.pdf'], 'title': ['Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling', 'Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling'], 'venue': []} |
arxiv |
**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Very Cool Close Binaries
28 Mar 2006
J Scott Shaw
Department of Physics and Astronomy
University of Georgia
30602AthensGAUSA
Mercedes López-Morales
Department of Terrestrial Magnetism
Carnegie Institution of Washington
5241 Broad Branch Rd. NW20015WashingtonDCUSA
**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Very Cool Close Binaries
28 Mar 2006arXiv:astro-ph/0603744v1
We present new observations of cool <6000K and low mass <1M ⊙ binary systems that have been discovered by searching several modern stellar photometric databases. The search has led to a factor of 10 increase in the number of known cool close eclipsing binary systems.
Introduction
For a long time the lower main sequence was neglected due to the difficulty in modeling stars of such low mass and due to scarcity of observations of cool close eclipsing binaries. Recently there has been much progress on the side of theory with models of, among others, Siess et al. (1997), Baraffe et al. (1998), and Yi, S. et al. (2001). However, the observational efforts have lagged behind. Our knowledge of the fundamental parameters stellar masses and radii comes primarily from stars that are eclipsing and double lined spectroscopic binaries. Before 2003 only three systems with M-type components were well studied: YY Gem (Torres & Ribas 2002, Leung & Schneider 1978, CM Dra (Metcalfe et al. 1996) and CU Cnc (Delfosse et. al 1999;Ribas 2003). The low number of known binary systems with cool <6000K and low mass <1M ⊙ has hindered our understanding of the lower main sequence. In the last three years the number of well know systems has tripled to include: OGLE BW5 V038 (Maceroni & (Torres et al, 2006), 2MASS J05162881+2607387 (Bayless & Orosz, priv. comm), NSVS01031772 (López-Morales et al., in prep.). The importance of low mass stars in our understanding of stellar evolution and the contents of the Milky Way is pointed out in our companion paper (López-Morales & Shaw, these proceedings). There we also note the discovery of the challenge posed by the new data to our understanding of the theory of stars in the lower main sequence.
The Search for Low Mass Eclipsing Binaries
Only in the last few years have sky surveys looking for stellar variability attained the faint magnitudes, and number of targets necessary to reveal close 2 Shaw and López-Morales eclipsing binaries in the lower main sequence. To date we have searched five recent databases containing photometry of a large number of stars: ROTSE-I (Akerlof et al. 2000), OGLE (Udalski et al. 1992), NSVS (Wozniak et al. 2004), EXPLORE (Mallén-Ornelas et al. 2003), and ASAS III (Pojmansky 2002). Listed in Table 1 are (1) the name of the survey, (2) an approximate limiting magnitude for the photometry and its approximate color, (3) the number of stars in the survey in thousands, (4) the number of variable stars found of all types, (5) the number of eclipsing binaries, and (6) the number of cool eclipsing binaries. The four variable systems from OGLE-I were first noted by Maceroni & Rucinski (1990). The results of our search are, so far, 41 new detached eclipsing binaries with masses below 1M ⊙ . The search of the NSVS survey is approximately 50% complete. In EXPLORE we have only searched part of the database at southern latitudes.
To identify the candidates the photometric data was searched for periodicities using two distinct methods: the string/rope length method based on the Lafler-Kinman statistic (Clarke 2000) and the analysis of variance method (Schwarzenberg-Czerney 1989). Light curves showing periodicity detected by either method were looked at more closely to see if they were cool, detached systems and if their masses were likely to be below 1M ⊙ . The results are contained in Table 2. The stars are grouped by their survey. The stars in the OGLE-I have names containing BW and MM; the others are self-evident. The columns are (1) name, (2) right ascension (2000), (3) declination (2000), (4) period in days, (5) V magnitude, (6) amplitude of variation, (7) estimated total mass of the system, (8) comments (see below).
We have begun a long-term project to obtain multicolor light curves and radial velocity curves for all of the objects brighter than 14 magnitude. Our progress is shown in the comments column in Table 2: LC=we have light curves, RV=we have radial velocity curves, pLC=we have partial light curves, S=we have a solution and it is included in López-Morales & Shaw (these proceedings) and López-Morales et al. (2006, in prep.)
Conclusions
Recent large stellar photometric surveys are yielding a large number of newly discovered variable stars. Some of these turn out to be the up-to-now rarely known M-type binaries. We estimate that approximately 1 of 1000 stars surveyed Very Cool Close Binaries 3
Montalbn 2004), GU Boo (López-Morales & Ribas 2005), TrES-Her0-07621 (Creevy et al. 2005), EMTSS-6 [RX J0239.1-1028]
Table 1 .
1Surveys Searched for Cool BinariesSurvey
Lim Mag Stars(K) Variables
EB Cool
ASAS-III
I <13
13,000
150,000 50,000
1
NSVS
V=10-15
17,000
16,000
4,000
12
OGLE-I
I=14-18
5,000
2,861
1,650
4
OGLE-II
I= 11-18
10,000
221,801
6,000
20
EXPLORE V<20
98
143
101
4
Table 2 .
2New Low-Mass Eclipsing Binaries are eclipsing binaries and, of those, about 1 of 1000 are M-type systems. Our discoveries have helped increase the number of well-studied M-type systems to three times the number known before the year 2003. Moreover we have three times that number again as many good candidates to study. Future work should insure adequate observations to test stellar models in the lower main sequence.Name
RA(2000)
Dec(2000)
P(days)
V
Amp M1+M2 Comments
ASAS1647
16 47 55
-08 44 30
0.503264
13.0
0.87
1.3-1.4
LC RV
NSVS06507557
01 58 24
+25 21 19
0.51509957
13.1
1.0
0.8-1.0
LC
NSVS01828214
02 13 51
+65 46 26
2.21433074
13.2
0.65
>1.3
NSVS06848235
05 01 59
+34 47 56
1.5383066
13.4
1.2
1.6-1.7
NSVS07394765
08 25 51
+24 27 05
2.26543128
12.8
1.1
1.7
pLC
NSVS02502726
08 44 11
+54 23 48
0.55973867
13.5
1.0
1.1-1.2
LC
NSVS07453183
09 16 12
+36 15 32
0.3669689
13.2
0.8
<0.6
LC RV
NSVS10441882
13 30 25
+13 49 32
0.5166492
12.8
0.8
1.3-1.4
NSVS01031772
13 45 35
+79 23 48
0.36814246
12.6
1.3
0.8
LC RV S
NSVS10653195
16 07 28
+12 13 59
0.56072686
12.6
0.75
1.2
pLC
NSVS01178845
17 45 25
+69 18 22
0.4937332
12.2
1.1
<1.0
LC
NSVS01286630
18 47 09
+78 42 33
0.38390916
13.3
0.9
1.1
NSVS11868841
23 17 58
+19 17 03
0.60176906
14.2
1.1
1.0
LC
BW9 v31
18 00 24.3
-29 51 24.3 1.99655
18.5
0.24
BW5 v149
18 02 37.4
-30 00 27.2 1.31085
19.7
0.93
MM7B v101
18 12 02.2
-26 00 31.4 1.20532
19.5
0.53
BW11 v48
18 01 16.2
-30 17 16.2 3.77077
>19.5 0.24
OGLE432075
17 35 01.6
-27 05 28.5 3.97628808 19.32 0.57
>1.6
OGLE431010
17 35 14.5
-27 19 56.3 1.23358142 19.76 0.50
>1.3-1.4
OGLE271691
17 48 41.7
-35 12 26.4 0.38797888 19.35 0.45
>0.5-0.6
OGLE445255
17 48 55.8
-29 51 12.8 1.59890199 20.88 0.85
0.7
OGLE445095
17 49 05.2
-29 52 04.1 0.77739549 20.26 0.73
0.9-1.0
OGLE441737
17 49 22.7
-30 14 34.5 0.74692839 20.12 0.28
>0.85
OGLE051231
17 49 58.0
-30 13 45.6 0.72695040 20.37 1.20
0.5-0.6
OGLE056610
17 50 14.3
-29 33 00.0 2.25488782 19.70 0.60
>1.5
OGLE400589
17 50 40.2
-33 34 16.8 0.81628138 19.54 0.50
1.2
OGLE401621
17 50 43.8
-33 20 18.3 0.46866554 19.96 0.70
0.85
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17 51 03.6
-33 32 51.5 1.62296712 19.50 0.47
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17 52 22.9
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0.9-1.0
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17 54 31.1
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17 57 50.4
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18 02 17.9
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0.8-0.9
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18 04 59.0
-28 29 19.0 0.76093984 20.58 0.50
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18 16 49.1
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18 22 56.0
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EXPLORE411492 16 25 54.10 -52 38 27.8 1.1024147
EXPLORE720427 16 28 28.01 -52 46 02.0 1.2543001
18.09 0.44
>0.95
EXPLORE523625 16 28 52.22 -53 10 55.1 1.5752375
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EXPLORE632136 16 29 20.22 -52 56 53.5 0.5735012
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Acknowledgments. We would like to thank the following institutions for providing support for this work: Carnegie Institution of Washington, NASA Astrobiology Institute, Southeastern Association for Research in Astronomy (SARA), University of Georgia at Athens, Instituto de Astrofísica de Canarias, National Science Foundation, and the University of North Carolina at Chapel Hill. We also thank our collaborators Jerome A. Orosz (SDSU, USA), Ignasi Ribas (IEEC, Spain), Maria Jesús Arévalo and Carlos Lázaro (IAC, Spain), and Guillermo Torres (HarvardCfA, USA) for their time and efforts towards this project. We also gratefully acknowledge the hard work of student researchers Yelena Pelimskaya (Lehigh University), Cecelia Hedrick (University of Nebraska), Travis McIntyre (Clemson University) who were supported in part by the National Science Foundation grant AST-0097616 to SARA and of student researchers Susan Chung (University of Georgia) and David Hou (Athens Academy). M. L-M. acknowledges research and travel support from the Carnegie Institution of Washington through a Carnegie fellowship.
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arxiv |
Born-Infeld inspired modifications of gravity
17 May 2017
Jose Beltrán Jiménez
Lavinia Heisenberg [email protected]
Gonzalo J Olmo [email protected]
Diego Rubiera-Garcia
Jose Beltrán Jiménez
Institute for Theoretical Studies
Aix Marseille Univ
Université de Toulon
CNRS
MarseilleCPTFrance
Depto. de Física Teórica and IFIC
Departamento de Física
ETH Zurich
Clausiusstrasse 47, Centro Mixto Universidad de Valencia-CSIC, Burjassot-461008092Zurich, ValenciaSwitzerland., Spain
Instituto de Astrofísica e Ciencias do Espaço
Universidade Federal da Paraíba
58051-900João Pessoa, ParaíbaBrazil
Faculdade de Ciencias
Universidade de Lisboa
Campo GrandePT1749-016LisboaPortugal
Born-Infeld inspired modifications of gravity
17 May 2017Preprint submitted to Physics Reports May 18, 2017(Gonzalo J. Olmo), [email protected] (Diego Rubiera-Garcia)Born-Infeld gravityAstrophysicsBlack HolesCosmologyEarly universeCompact objectsSingularities
General Relativity has shown an outstanding observational success in the scales where it has been directly tested. However, modifications have been intensively explored in the regimes where it seems either incomplete or signals its own limit of validity. In particular, the breakdown of unitarity near the Planck scale strongly suggests that General Relativity needs to be modified at high energies and quantum gravity effects are expected to be important. This is related to the existence of spacetime singularities when the solutions of General Relativity are extrapolated to regimes where curvatures are large. In this sense, Born-Infeld inspired modifications of gravity have shown an extraordinary ability to regularise the gravitational dynamics, leading to non-singular cosmologies and regular black hole spacetimes in a very robust manner and without resorting to quantum gravity effects. This has boosted the interest in these theories in applications to stellar structure, compact objects, inflationary scenarios, cosmological singularities, and black hole and wormhole physics, among others. We review the motivations, various formulations, and main results achieved within these theories, including their observational viability, and provide an overview of current open problems and future research opportunities.
Motivations and introduction
General Relativity (GR) is nowadays firmly established as the standard theory to describe the gravitational interaction with the same mathematical framework and physical principles as those used by Einstein more than one hundred years ago. After all this time, it still stands out as the most successful theory able to explain all the gravitational phenomena in a wide range of scales. Direct tests comprise from sub-milimeter to Solar System scales, where the Parameterised Post-Newtonian formalism has allowed to constrain deviations from GR in the weak field limit at the level of ∼ 10 −5 [364]. Moreover, the amazing direct observation of gravitational waves by the LIGO collaboration is also compatible with the prediction of GR for the merging of two black holes, where strong field effects are relevant [2,1]. On the other hand, we have witnessed how the accurate measurements of the CMB anisotropies and galaxy surveys have established ΛCDM as the standard model of cosmology, which is based on a homogeneous and isotropic Universe governed by GR as the theoretical framework for gravity. This picture requires an unobserved cold dark matter source plus a tiny cosmological constant to account for the current accelerated expansion of the Universe. Furthermore, the ΛCDM model needs to be supplemented with the inflationary paradigm so that the primordial perturbations are generated during a short period of accelerated expansion at very early times. For a review on the current status of the ΛCDM model, its challenges and possible alternatives, see Bull et al. [89]. Further observational tests, for instance via the Euclid satellite [16], will hopefully shed light on all the additional elements above and their contributions to fundamental physics.
Despite its observational success, there are strong arguments supporting and/or motivating to seek for theories beyond GR. These arguments are of two kinds. On the theoretical side, GR itself predicts the unavoidable existence of spacetime singularities, i.e., events where our ability to make predictions comes to an end [326]. Such singularities are unavoidably developed during the gravitational collapse of a fuel-exhausted star to form a black hole [223], as well as during the cosmological evolution in the early Universe. In this sense, the requirement that "nothing should cease to exist suddenly" and that "nothing should emerge out of nowhere" should be seen as basic consistency conditions for any physical theory, including GR. The existence of singularities in GR unavoidably leads to the breakdown of these conditions, and gives clear indications that we have pushed the theory beyond its regime of validity. According to the standard lore [90], GR is a good effective field theory up to a scale somewhere near the Planck mass and, therefore, those singular behaviours are regarded as manifestations that the higher order operators should be included. For this reason, quantum gravity is usually expected to regularise such singularities, although it is possible that high energy modifications of GR might allow to classically regularise some of those singularities before reaching the cut-off of the theory without invoking any quantum gravity effects.
On the phenomenological side, the unprecedented experimental precision reached by observational cosmology requires the aforementioned ad hoc extra ingredients in order to account for the observations. While the cosmological constant is fundamental part of the theory and its difficulty resides in its aesthetic value that poses naturalness problem, dark matter and inflation require the introduction of new physics and, as a consequence, a large degeneracy among all the proposed models. This degeneracy is more prominent owing to the lack of experimental signatures from laboratory experiments and particle accelerators, despite the existence of different ongoing galactic [342,5], cosmic rays [23], CMB [7], collider [100] and underground laboratory [9] searches.
In view of the above situation, one may wonder if the difficulties and lack of naturalness faced in GR indicates that a new framework to describe gravity is needed, which would yield different astrophysical and cosmological observational signatures from the ΛCDM model [224]. From a conservative perspective, one may stick to the point of view that gravitation is a manifestation of the curvature of spacetime, but one that is not sufficiently well described by GR. As a matter of fact, the common factor to all the issues discussed above is the extrapolation of GR to regions where it has not been directly well tested and this may introduce significant bias in the interpretation of astrophysical and cosmological observations. The consideration of additional curvature contributions to the Einstein-Hilbert action, usually under the form of curvature invariants, has been used in the literature as a way to enlarge the phenomenology of gravity. This typically involves a number of problems such as higher-order field equations, which usually entail the presence of ghost-like instabilities [341,340,266,107], or the difficulty to make these models compatible with solar system tests due to the existence of new degrees of freedom [270,108] 1 . The arbitrariness in the choice of curvature invariants also implies a strong lack of naturalness in these models. The main references regarding such models and their applications are provided by de Felice and Tsujikawa [130], Capozziello and de Laurentis [95], and Nojiri and Odintsov [261] (see also Faraoni and Sotiriou [338]).
The difficulties with ghost-like instabilities in higher curvature modifications of gravity can be avoided by formulating those theories in the so-called Palatini or metric-affine formalism [272]. Though this approach is sometimes viewed as a shortcut to obtain the field equations of GR (and rightly so for some specific Lagrangians), it actually represents an inequivalent formulation of gravity in which metric and affine structures are regarded as independent geometrical entities. The fact that, when formulatedà la Palatini [163], metric and connection are compatible in the case of GR has spread the view that such condition should always hold regardless of the form of the gravity Lagrangian. However, this is not true in general. In the metric-affine approach, the specific relation between metric and connection is determined by the field equations, not imposed a priori by mathematical conventions. In fact, whether the affine connection is determined by the metric degrees of freedom or not is as fundamental a question as the number of spacetime dimensions or the existence of supersymmetry.
The metric-affine or Palatini approach, therefore, avoids the problems with ghosts that affect extensions of GR in the usual metric formulation. In vacuum configurations, the field equations of these theories boil down to Einstein's equations with an effective cosmological constant [164] which, apparently, supports their compatibility with orbital motion tests (see [269,271] for a discussion). Though this mathematical framework cannot solve on its own the arbitrariness in the choice of gravity Lagrangian, a novel class of extensions of GR with a solid motivation for a high-energy completion of gravity has been proposed and explored with much interest in the last few years. These models are motivated by the Born-Infeld approach to electrodynamics, where a modification of Maxwell's Lagrangian is introduced to set an upper bound on the electromagnetic field intensity [? ], with the result that the divergence of the self-energy of a point-like charge is regularised. This type of high-energy modification is analogous to the transformation that leads from a free particle in Newtonian mechanics to a free relativistic particle, whose maximum speed is bounded by the speed of light. The same Lagrangian structure describes the electromagnetic fields of p-branes in string theories [180,86,94]. It is natural to wonder whether such an approach, now fully defined in terms of geometrical objects, could play a similar role in order to avoid divergences and spacetime singularities in the high-energy/curvature regime and, accordingly, different proposals have been considered in the literature. Indeed, a major reason for the investigation of such models is the fact that, using standard matter sources satisfying the energy conditions, they naturally lead to non-singular cosmologies, inflationary scenarios without the need for scalar fields, and black hole spacetimes without singularities, among other appealing results. Moreover, the physics of these gravity theories has been studied in numerous astrophysical, black hole and cosmological scenarios where high-energy physics is relevant.
In this work we shall refer to this kind of models, which are close to the original spirit of Born-Infeld electrodynamics, as Born-Infeld inspired modifications of gravity. They are defined by the following basic principles:
• Square-root form: Some geometric object(s) appears under a square-root with a determinantal structure in the action which defines the gravitational theory, alongside with some new mass/length scale.
• Consistency: No obvious pathologies are present, among which the absence of ghostlike instabilities is of utmost importance. In turn, this almost unavoidably enforces the use of a metric-affine formulation.
• High-energy modification: The modifications of GR mostly occur in the ultraviolet regime, i.e., in regions of large mass/curvature or short scales. This implies that GR is recovered in the low-energy limit.
Nonetheless, as there are available proposals in the literature for these theories that run away from one (or both) of the two last requirements, for completeness of this work we shall also discuss such proposals. A more precise description and classification of such theories will be presented in section 2, alongside a criticism of each of them.
This review is intended to fill a gap in the recent literature of Born-Infeld inspired modifications of gravity by providing a comprehensible account of the many different scenarios on which these classes of theories have been considered, including the astrophysics and internal structure of compact objects, solar physics constraints, modifications on black hole structure, non-singular black holes and wormholes, early universe and bouncing solutions, inflation, and dark energy, among others. Its aim is to summarise, classify and unify the different theoretical approaches, to clarify the assumptions on which the different approaches to build the theory are formulated, to discuss the numerous theoretical and phenomenological results, to highlight the experimental constraints these theories are subjected to, to clarify some existing misunderstandings, and to provide an overview of the future research opportunities. It is designed to be useful both for pure theorists and for astrophysicists/cosmologists working on alternatives to the ΛCDM (plus inflation) model.
For a review on modified gravity in cosmology mainly focused on infrarred modifications of gravity in connection with late-time solutions (but with little contact with Born-Infeld-inspired theories or the Palatini formalism), see instead Clifton et al. [120]. For additional astrophysical and cosmological observational constraints over different modified theories of gravity deviating from GR predictions, see Berti et al. [67].
Outline
The main content of this review is split in four sections, according to the context on which Born-Infeld-inspired theories of gravity have been investigated.
In section 2 we will briefly review the original Born-Infeld electrodynamics theory from which the motivation for analogue constructions within gravity emerges. After explaining the early attempts that resulted in pathological theories, we will introduce what represents the most extensively studied theory of gravity with the Born-Infeld structure. The slightly different formulations of such a theory will be discussed as well as the main equations. Along the way, we will spend some time discussing the two frames existing in these theories and clarify the physical meaning of the different geometrical objects arising in them. We will end this section with a survey on the different Born-Infeld inspired theories of gravity existing in the literature and we will provide a general mathematical framework for these theories. The general developments introduced in this section will serve as starting points for the practical applications discussed in the subsequent sections.
In section 3 some attempts to place observational constraints on the Born-Infeld theory using stellar models are reviewed. We will make special emphasis on the central role played by the energy density in the modified dynamics of this theory, which affects in a nontrivial way the mass-radius relation and maximum mass limit of compact objects, the energy transport mechanisms and oscillation frequencies of stars, the intensity of neutrino fluxes from the Sun, . . . providing numerous tests to confront the theory with observations. The need for a careful description of the outermost layers of compact objects is also discussed in detail, considering for this purpose some relevant examples in which the peculiarities of metric-affine theories demand additional modeling beyond the canonical approaches of GR.
In section 4 we will review the counterparts of the Schwarzschild and Reissner-Nordström black hole solutions of GR, where a coupling to a Maxwell field is considered. We will spend some time explaining the procedure for derivation of the corresponding solutions, so as to highlight some important subtleties. Then we will explain the main differences of such solutions as compared to the GR ones, in particular, regarding the modifications on the horizon structure, which bear some resemblance to that of black holes supported by Born-Infeld electrodynamics in GR. On the other hand, we will study how these black holes may affect the description of strong gravitational lensing as well as the physics regard-ing mass inflation. An important issue will be the existence of non-singular geometries in these theories, whose nature and properties is tested using different well-established criteria. We also review some wormhole solutions constructed out of anisotropic fluids. Finally, different extensions to higher and lower dimensions, as well as to magnetically charged solutions will be discussed.
The section 5 will be devoted to the effects of Born-Infeld inspired theories of gravity in cosmological scenarios. We will discuss the existence of homogeneous and isotropic solutions free from Big Bang singularities with standard matter sources as well as couplings of these theories to other types of fields. Anisotropic models and inhomogeneous perturbations will also be discussed. Since the Born-Infeld inspired theories are designed to modify gravity in the high curvatures regime, their natural domain of applicability is the early universe. However, there have also been studies where Born-Infeld theories are considered for late time cosmology and we will revisit them.
We will end in section 6 by giving a summary of all the material presented in the core of this review. We will discuss the most outstanding achievements and will make special emphasis on the open questions that remain as well as the prospects for future research within the field.
Preliminaries
In this section we will review some basic ingredients of differential geometry that we will use throughout the different parts of this review. We will assume that the reader is familiarized with the concepts presented here and the main purpose of this section will be to fix the notation and the conventions for the different choices of signs and numerical factors in the definitions of relevant geometrical objects. It does not intend to be an exhaustive and rigorous exposition, but rather it should be regarded as a brief compendium of useful concepts and formulae. For a more detailed treatment we urge the reader to consult her/his favourite book on differential geometry or General Relativity or, in the lack thereof, see e.g. [325,251,359]. One reference particularly useful and with numerous applications in gravitation and gauge theories is [148].
Connection, curvature and torsion conventions
The theories that will be considered throughout the present review will be formulated either in (pseudo-)Riemannian or non-Riemannian geometries. In order to construct the necessary geometrical framework, we first introduce a 4-dimensional manifold M that will eventually constitute our spacetime. In that spacetime we introduce a general connection Γ that defines the covariant derivative of a 1-form A µ as
∇ µ A ν = ∂ µ A ν − Γ λ µν A λ . (1.1)
This definition results in the following covariant derivative for a vector field A µ :
∇ µ A ν = ∂ µ A ν + Γ ν µλ A λ . (1.2)
These expressions can then be easily generalised to arbitrary tensors T µ 1 ···µp ν 1 ···νq so that ∇ α T µ 1 ···µp ν 1 ···νq = ∂ α T µ 1 ···µp ν 1 ···νq − Γ λ αν 1 T µ 1 ···µp λν 2 ···νq − · · · − Γ λ ανq T µ 1 ···µp ν 1 ···ν q−1 λ + Γ µ 1 αλ T λµ 2 ···µp ν 1 ···νq + · · · + Γ µp αλ T µ 1 ···µ p−1 λ ν 1 ···νq . (1. 3) In addition to objects with tensorial transformation properties under changes of coordinates, we will also find objects with other transformation properties throughout this review. In particular, we will encounter vector densities, which pick up some power of the Jacobian under a change of coordinates. If A µ is a vector density of weight w, it transforms as 2Ã µ = det ∂x α ∂x β w ∂x µ ∂x ν A ν . (1.4) This modified transformation property makes necessary to add a piece to the definition of the covariant derivative to maintain its tensorial character, that reads
∇ µ A ν = ∂ µ A ν + Γ ν µλ A λ + wΓ λ µλ A ν . (1.5)
Again, this formula can be generalized for an arbitrary tensorial density T µ 1 ···µp ν 1 ···νq by adding a term wΓ λ αλ T µ 1 ···µp ν 1 ···νq in (1.3). After introducing the connection, we can start computing geometrical objects from the commutator of covariant derivatives acting on different tensorial fields. The first commutator we can compute is that of two covariant derivatives acting on a scalar field, which reads
∇ µ , ∇ ν φ = −T λ µν ∂ λ φ (1.6) with T λ µν ≡ Γ λ µν − Γ λ νµ (1.7)
the torsion tensor. Let us notice that it has tensorial transformation properties because it can be seen as the difference of two connections. The next geometrical important object is obtained by computing the commutator of two covariant derivatives acting on a vector field, which can be written as
∇ µ , ∇ ν A α = R α βµν A β − T λ µν ∇ λ A α (1.8)
where we have introduced the curvature Riemann tensor, defined as (1.9) Out of this general Riemann tensor, we can build two independent traces, namely the Ricci tensor defined as usual R µν = R α µαν and the homothetic tensor given by Q µν = R α αµν . While the Ricci tensor does not have any symmetry (even for a torsion-free connection), the homothetic tensor is antisymmetric. A quantity that we will need to compute field equations is the variation of the Ricci tensor under an infinitesimal displacement of the connection Γ →Γ + δΓ, which reads δR µν =∇ λ δΓ λ νµ −∇ ν δΓ λ λµ +T λ ρν δΓ ρ λµ (1.10) where the bars denote quantities corresponding to the background connectionΓ. This relation reduces to the usual Ricci identity for torsion-free connections.
R α βµν ≡ ∂ µ Γ α νβ − ∂ ν Γ α µβ + Γ α µλ Γ λ νβ − Γ α νλ Γ λ µβ
Metric convention
After setting-up the notation and convention for the objects directly related to the connection, we will turn to the conventions for the metric tensor g µν . This object is assumed to be non-degenerate and its inverse is denoted with upper indices g µν so that g µα g αν = δ µ ν and so on. Furthermore, this object is used to raise and lower indices of arbitrary tensors (i.e. it establishes an isomorphism between the tangent and the cotangent spaces). We will use the mostly plus signature for the metric so that the Minkowski metric is η µν = diag(−, +, +, +). The covariant derivative of the metric defines the nonmetricity tensor Q αµν as ∇ α g µν = Q αµν . (1.11) Notice that the non-metricity is symmetric in the last two indices. This expression can be solved in the usual way to write the connection as Γ α µν = 1 2 g αλ ∂ ν g µλ + ∂ µ g λν − ∂ λ g µν + L α µν (Q) + K α µν (T ) (1.12) where the first term is the standard Levi-Civita piece, the second term depends on the non-metricity and the last term (usually called contorsion) is determined by the torsion. If the non-metricity vanishes and the connection is symmetric (i.e. vanishing torsion), the connection reduces to the Levi-Civita connection given by the Christoffel symbols. With a metric at hand, there is yet a third rank-2 tensor we can construct from the Riemann tensor of the full connection, known as co-Ricci tensor and defined as P α µ ≡ g βν R α βµν . Of course, for the Levi-Civita connection all three objects coincide up to a sign so the only independent trace of the Riemann is the Ricci tensor R µν . Throughout this review we will denote with calligraphic letters R µν , ... the objects corresponding to an arbitrary connection, while the curvature objects associated to the Levi-Civita connection will be denoted with normal characters R µν , ... The determinant of the metric det g µν ≡ g is a tensorial density of weight −2 so that √ −g is a tensorial density of weight −1 whose covariant derivative is given by
∇ µ √ −g = ∂ µ √ −g − Γ λ µλ √ −g.
(1.13)
We can thus use √ −g to tensorialize tensorial densities. For instance, if A µ is a tensorial density of weight w, then A µ ≡ ( √ −g) w A µ has weight zero. Another important use of this object is to construct invariant volume elements. Since dV generates a Jacobian under a change of coordinates, we can compensate for that by adding a factor of √ −g so that √ −gdV will be invariant. Let us notice that this is a choice and actually we could use ϕdV with ϕ being whatever scalar density of weight −1. For instance, det a µν with a µν being an arbitrary rank 2 tensor will do the job.
The totally antisymmetric tensor is defined as
ε µνρσ = √ −g µνρσ (1.14)
with µνρσ the totally antisymmetric Levi-Civita symbol with [0123] = 1. The contravariant version of it is ε µ 1 µ 2 µ 3 µ 4 = g µ 1 ν 1 g µ 2 ν 2 g µ 3 ν 3 g µ 4 ν 4 ε ν 1 ν 2 ν 3 ν 4 = − 1 √ −g µ 1 µ 2 µ 3 µ 4 .
(1.15)
The Levi-Civita tensor allows to introduce the Hodge dual that establishes an isomorphism between 3 p-forms and (D − p)-forms. If F µ 1 ···µp is a p-form, its dual is defined as
F µ 1 ···µ D−p = 1 p! ǫ µ 1 ···µ D−p ν 1 ···νp F ν 1 ···νp . (1.16)
As a specific example that we will use throughout the review, the dual of a 2-form F µν in four dimensions is given byF µν = 1 2 ǫ µναβ F αβ .
(1.17)
For an antisymmetric rank 2 tensor we can introduce the so-called electric E µ and magnetic B µ components relative to an observer with 4-velocity u µ as E µ = F µν u ν and B µ =F µν u ν . (1.18) For an observer with u µ = (1, 0) these definitions reduce to the usual expressions F 0i = E i and F ij = 1 2 ǫ ijk B k .
Tetrads formulation
An alternative language to describe the geometrical framework of gravity theories is provided by the formalism of frames. We start by introducing a set of vectors defined on the tangent space e a = e a µ ∂ µ with a Lorentz index a so that they satisfy the following orthonormality condition e a µ e b ν g µν = η ab .
(1. 19) with respect to the Minkowski metric η ab . These objects receive several aliases in the literature: tetrads, vierbein or frames. The corresponding dual objects e a = e a µ dx µ belonging to the cotangent space are defined in the usual way e a µ e b µ = δ a b . This relation in turns also implies e a µ e a ν = δ ν µ . They are sometimes interpreted as the square root of the metric because g µν can be expressed as e a µ e b ν η ab = g µν .
(1.20)
The vierbein can be used to transform tangent space indices into spacetime indices for arbitrary tensors. All the geometrical objects introduced above thus have their corresponding object in the tetrads formulation. If we introduce the so-called spin connection given by the set of 1-forms ω a µ b , the associated curvature 2-form is given by
R a b = dω a b + ω a m ∧ ω m b (1.21)
where d is the exterior derivative and ∧ stands for the exterior product. The existence of the tetrad allows to define the torsion 2-form as
T a = de a + ω a b ∧ e b .(1.22)
Applying the exterior derivative on this expression we obtain a consistency condition (1.23) that relates all the relevant objects, namely, the tetrads, the spin connection, the torsion and the curvature. Taking a second exterior derivative of this expression will yield the usual Bianchi identities, which we do not need to display here. Instead, let us focus on two special connections that will be of relevance for this review. The first one is defined by the condition of being torsion-free, so it is defined by de a + ω a b ∧ e b = R a b ∧ e b = 0 and it is the relevant one for the usual formulation of General Relativity. The second connection is curvature-free so we have dω a b + ω a m ∧ ω m b = dT a + ω a b ∧ T b = 0 and defines the so-called Weitzenböck space. This is the natural place for the Teleparallel formulation of GR.
dT a + ω a b ∧ T b = R a b ∧ e b
Energy conditions
A perfect fluid can be defined as one in which the energy-momentum tensor is locally seen as isotropic and it is fully determined by its density ρ and its pressure p. According to this definition, the energy-momentum tensor of a perfect fluid as seen by an observer with 4-velocity u µ (u 2 = −1) is given by.
T µν = (ρ + p)u µ u ν + pg µν (1.24) where it is immediate to see that ρ = T µν u µ u ν and p = 1 3 (g µν + u µ u ν )T µν . For a comoving observer with u µ ∝ ∂ t we have that T 0 0 = −ρ and T i j = pδ i j .
For a general energy-momentum tensor, there is a set of conditions known as energy conditions that play an important role in theories of gravity in relation with singularity theorems, instabilities, superluminal propagation or entropy bounds. In the following we list them for future reference:
• Weak Energy Condition (WEC). This condition states that T µν v µ v ν ≥ 0 for every time-like vector v µ (v 2 < 0). For a perfect fluid, it implies the positivity of the energy density ρ ≥ 0 as measured by any observer and ρ + p ≥ 0.
• Dominant Energy Condition (DEC). This condition is satisfied if T µν w µ w ν ≥ 0 for every causal vector w µ (w 2 ≤ 0) and −T µ ν w ν is a future-oriented causal vector. For a perfect fluid, this condition translates into ρ ≥ |p|.
• Strong Energy Condition (SEC). The SEC is satisfied if T µν v µ v ν ≥ − 1 2 T for every time-like vector v µ (v 2 < 0). A perfect fluid satisfies this conditions if ρ + p ≥ 0 and ρ + 3p ≥ 0.
• Null Energy Condition (NEC). The NEC is satisfied if for any null vector n µ (n 2 = 0) the condition T µν n µ n ν ≥ 0 holds. For a perfect fluid this implies ρ + p ≥ 0. This condition is satisfied for all known types of matter and it is saturated by a cosmological constant.
Matrix notation
Given a rank-2 tensor, we will often use a hat to denote the corresponding matrix. Thus, the metric tensor g µν will also appear asĝ and its inverse g µν will be denoted bŷ g −1 and similarly for other objects. The determinant of a matrixM will be explicitly spelled out as det(M ) or will be alternatively denoted as |M | where no confusion with absolute value should occur. In the special case of a metric g µν , we will alternatively use the broadly used notation g for its determinant. Analogously, for the trace of a matrix we will use either the explicit notation Tr(M ) or the more compact notation [M ] where, again, the context should clarify when the square brackets stand for the trace or simply play the role of actual brackets.
A recurrent matrix formula that we will use throughout this review is the expansion valid for an arbitrary n × n matrixM given by det ½ +M = It is useful to notice that the last elementary symmetric polynomial coincides with the determinant ofM . Moreover, ifM is antisymmetric its trace is identically zero and, thus, e 1 and e 3 vanish.
Units and constants
Unless otherwise stated, we will use units with = c = 1. We will mostly use the reduced Planck mass, related to Newton's constant as M −2 Pl = 8πG N . We will also make use of the Einstein's constant κ 2 = 8πG N .
Born-Infeld theories
The class of theories that generally go under the name of Born-Infeld all share the same basic feature of being defined in terms of some square root structure aimed at regularising the presence of divergences. The inception of these theories originated from the pioneering works by Born and Infeld in the 1930's [72,73,74,75] where they assumed a principle of finiteness, according to which physical quantities are always bounded and can never become infinite. The self-energy of the electron, or a general point-like charged particle, is infinite in the classical Maxwell's theory so they searched for a non-linear modification capable of regularising this divergence as to comply with the principle of finiteness, i.e., a non-linear theory where point-like charges had finite self-energy 4 . Motivated by the existence of an upper bound for the velocities of particles in relativistic mechanics, in the summer of 1933 Born proposed to introduce the same square root structure for electromagnetism in order to have an upper bound for the electric fields [72,73]. A few months later Infeld joined Born and together worked on a better version of this construction because they wanted a theoretically better motivated argument for such a theory and, then, they argued that the square root structure should come in from symmetry arguments. In analogy with mechanics where going from Newtonian to relativistic mechanics means upgrading Galilean transformations to the fully relativistic Lorentz group, Born and Infeld assumed that the Lorentz symmetry of Maxwell's theory should be enlarged in the new theory. They considered the new symmetry to be the full group of coordinate transformations which, after imposing the recovery of Maxwell's theory in the appropriate limit, led to the non-linear theory now known as Born-Infeld electromagnetism, expressed as the square root of a certain determinant [74,75]. It is no surprise that the use of symmetries as a guiding principle gave rise to a remarkable theory of non-linear electromagnetism which, not only classically regularises the self-energy of point like charges, but it also shares some interesting features with Maxwell's theory and found a natural arena in the realm of other theoretically appealing theories, like e.g. string theory [307,308,374].
Given the success of Born-Infeld theory to classically regularise divergences in electromagnetism, it is perhaps surprising that the same ideas were applied to resolve the singularities of General Relativity (GR) only in the late 1990's 5 . The first attempt in this direction came about in a work by Deser and Gibbons [140], where they finally took over the idea and tried to apply it to the case of gravity. However, as usual with gravity, things can very quickly go wrong when one tries to modify the Einstein-Hilbert action. The most straightforward application of the Born-Infeld philosophy by introducing a square root structure of a determinant involving the Ricci tensor gives rise to the presence of 4 We should perhaps remark here that, at the time when Born and Infeld developed their theory for electromagnetism, the full machinery of quantum electrodynamics and the renormalization techniques were not available. Today we know that quantum electrodynamics is a renormalizable quantum field theory where physical quantities are finite and, in particular, the charge of a particle acquires radiative corrections at high energies owed to virtual processes. 5 A possible reason for this was the relative lack of interest in these topics until the seminal works by Hawking and Penrose [299,300,197] concerning the singularity theorems in GR. On the other hand, the extraordinary success of quantum field theory perhaps motivated to invoke quantum gravity effects as the most likely mechanism that should regularise gravity in the high curvatures regime.
ghosts owing to the Ostrogradski instability associated to higher order equations of motion [366,367]. In order to resolve the ghost problem, they proposed to add an additional term to remove the ghost order by order so that, when expanding the full action in the curvature, only the corresponding Lovelock term remains. This avoids the problem of the ghost, but the large freedom remaining in the choice of the additional piece and the lack of any guiding principle, makes the construction less appealing than the case of Born-Infeld electromagnetism. An obvious way to get around the ghost problem is to only use the Ricci scalar and apply the Born-Infeld construction to this quantity. This would lie within the class of f (R) theories that contain one extra degree of freedom with respect to GR and, thus, it would deviate from the original Born-Infeld spirit where the theory is modified in some high energy regime by changing the structure of the theory in that regime instead of adding additional modes.
Some years later, Vollick re-considered Born-Infeld type of actions for gravity from a different perspective [356]. Similarly to Deser and Gibbons, Vollick also resorted to a straightforward translation of the Born-Infed action to the case of gravity. However, instead of adopting the metric formalism, he considered the action within a metric-affine approach so that the connection is left arbitrary and promoted to an independent field. Within that formalism, the problem of the ghosts encountered in the metric formalism are avoided and, thus, no additional terms to remove undesired interactions are needed. This approach can actually be seen as a combination of the Born-Infeld ideas together with the original purely affine theory of gravity proposed by Eddington. Later on, Bañados and Ferreira took on Vollick's approach with a slight modification of the original action, that now goes under the name of Eddington-inspired Born-Infeld gravity (EiBI), and showed the existence of non-singular cosmological and black hole solutions. This particular realisation of Born-Infeld gravity theories has since then received a considerable attention and has been extensively explored in different contexts with promising results.
The proposal by Vollick and its relative by Bañados and Ferreira finally succeeded to implement the ideas of Born-Infeld electrodynamics to the case of gravity. However, it is fair to say that this initial proposal merely consisted in obtaining a gravitational actioǹ a la Born-Infeld, but it lacked any underlying guiding principle, based on symmetries like in Born-Infeld electrodynamics or any other equally valid motivation. In fact, it is very simple to come up with more general actions that could also be catalogued as Born-Infeld theories and could be considered on the same footing as EiBI. It does not come as a surprise then that very soon, modifications, extensions or alternative implementations of the Born-Infeld ideas to gravity appeared in the literature.
In this section we will review in detail the developments discussed above that led to the formulation of Born-Infeld gravity theories. We will start by reviewing Born-Infeld electrodynamics as a good starting point to motivate the search for analogous theories within gravitational contexts. We will show how the first attempts formulated in the metric formalism did not succeed due to the presence of ghosts. After that, we will turn to the formulation of Born-Infeld actions for gravity within a metric-affine approach and explain how the ghost issue is avoided. The general properties of these theories will be discussed in detail and, in particular, we will explain the existence of two frames. We will end this section by performing a classification of the different Born-Infeld inspired theories of gravities considered in the literature so far and briefly discuss them.
Born-Infeld electromagnetism in a nutshell
The underlying idea used by Born and Infeld to develop a modification of the Maxwell action as a potential mechanism to regularise some divergences associated to point-like charges was motivated by the appearance of an upper bound for the speed of particles when upgrading Newtonian mechanisms to relativistic mechanics. In that case, the Newtonian Lagrangian for a massive particle of mass m is simply L = 1 2 m 2 v 2 , where v is its velocity and can take any value. When including the principles of relativistic mechanics, the Lagrangian for the massive particle becomes L = −m 2 c 2 1 − (v/c) 2 , where the speed of light c makes its appearance as an upper bound for the velocities due to the square root. Taking inspiration from this, Born came up with the idea of modifying Maxwell's Lagrangian in such a way that the divergences of the Coulomb potential are automatically regularised due to the existence of a natural upper bound in the theory. In [72,73], he followed the most straightforward application of this idea and proposed the following replacement of Maxwell's Lagrangian:
L = − 1 4 F µν F µν → L = b 2 1 − 1 2b 2 F µν F µν − 1 , (2.1)
with b representing the desired upper limit of possible electric fields. Although this simple replacement could do the job of regularising the infinities associated to point-like charges, it is not completely satisfactory from a theoretical point of view since there is no guiding principle for it other than the principle of finiteness. That is the reason that motivated Born, this time in collaboration with Infeld, to look for a more theoretically appealing modification of Maxwell electromagnetism. They noted that, when going from classical mechanics to relativistic mechanics, the symmetry group is enlarged from the Galileo to the Lorentz group and it is precisely this group structure that nicely introduces the desired square root. Born and Infeld embraced this line of reasoning and looked for a non-linear theory of electromagnetism enlarging the group of special relativity as the relevant one. The idea is then that, very much like Newtonian mechanics is the limit of special relativity for small velocities and the Lorentz group reduces to the Galilean transformations, Maxwell electromagnetism should be the limit of some theory with a larger group of symmetries which, in some suitable limit, should reduce to the usual relativistic Lorentz transformations. Motivated by recent developments in gravity where the relevant group was shown by Einstein to be general coordinate transformations, they opted by enlarging the symmetry group of electromagnetism from the Lorentz group to the full group of general coordinate transformations 6 . Then, to have general covariance, the action should be constructed as S = d 4 x det a µν , with a µν some rank-2 covariant tensor whose symmetric part can be identified with the metric tensor and its antisymmetric part is identified with the electromagnetic field strength F µν . After imposing that Maxwell's theory should be recovered for small electromagnetic fields and neglecting some boundary terms, they arrived at the celebrated Born-Infeld action
S BI = −b 2 d 4 x − det η µν + 1 b F µν − 1 . (2.2)
This action has the properties they were after, namely, it introduces the square root structure by means of enlarging the symmetry group of Maxwell's theory. The constant b is the only free parameter of the theory and it precisely gives the maximum allowed value for electric fields. Born and Infeld assumed the value of b to be such that the corrections arise at the electron radius, although that value is now experimentally ruled out (see [171] for a recent review on experimental bounds for non-linear electromagnetism). In order to see the appearance of a maximum value for the electric field, let us notice that the action can be written in several useful ways by expanding the determinant in (2.2) to obtain
S BI = − b 2 d 4 x 1 + 1 2b 2 F µν F µν − 1 16b 4 (F µνF µν ) 2 − 1 (2.3) = − b 2 d 4 x 1 − E 2 − B 2 b 2 − ( E · B) 2 b 4 − 1 ,
withF µν ≡ 1 2 ε µναβ F αβ the dual of the field strength, E and B the corresponding electric and magnetic parts and we have used the matrix identity
det δ µ ν + 1 b F µ ν = 1 + 1 2b 2 F µν F µν − 1 16b 4 F µνF µν 2 . (2.4)
Notice that this implies a Z 2 symmetry F µν → −F µν owed to the property of the determinant det(½ +M ) = det(½ −M ) for an arbitrary matrixM . From the above expression it is straightforward to see that Maxwell's electromagnetism is recovered for electromagnetic fields much smaller than b and that, for configurations without magnetic field, we also recover the first Lagrangian (2.1) considered by Born. Furthermore, written in this way, we can easily understand why the self-energy of point-like charged particles is regularised. Since a particle at rest (or in its own rest frame) only generates electric field, the Lagrangian reduces to
L BI = −b 2 1 − E 2 b 2 (2.5)
and we clearly see that the electric field is bounded by b. Given the gauge character of the theory, we still have the constraint equation generating the gauge symmetry (or the equivalent of Gauss' law) given by
∇ · Π = ρ with Π = ∂L BI ∂ E = E 1 − E 2 b 2 (2.6)
and ρ is the density of electric charge. As usual, for a point-like particle of charge Q we can integrate the equation over a sphere to obtain
∇ · Π d 3 x = Q ⇒ | Π| = Q 4πr 2 , (2.7)
where Q is the total charge enclosed by the sphere Q = ρd 3 x. By inverting the relation (2.6) between Π and E we can obtain the solution for the electric field generated by the particle
E = Π 1 + Π 2 b 2 . (2.8)
As promised, for | Π| ≪ b we have | Π| ≃ | E| ∝ 1/r 2 which is the usual result in Maxwell's electromagnetism, while in the opposite regime with Π ≫ b the electric field saturates to the value | E| = b. This saturation is in turn the responsible for the regularization of the self-energy of the particle, that is given by
U = d 3 xH = b 2 d 3 x 1 + Π 2 b 2 − 1 = 4πb 2 ∞ 0 r 2 dr 1 + Q 4πbr 2 2 − 1 , (2.9)
where we have used the expression for the Hamiltonian density 7 H = Π · E − L and the corresponding solution (2.7). The integral diverges in the case of Maxwell electromagnetism due to the unbounded contributions from the small scales where one has H Maxwell ∼ E 2 ∝ r −4 . In the Born-Infeld case however, the small scales region is modified and we have H BI ∼ Π ∝ r −2 which makes the integral convergent (see Fig. 2.1). The integral can be exactly computed in terms of the gamma function Γ(x) and the total finite result is
U = Γ 2 (1/4) 12π bQ 3 . (2.10)
Let us return to the solution for the electric field given in (2.11) and express it directly in terms of the generating charge as 8
| E| = 1 1 + Q 4πbr 2 2 Q 4πr 2 .
(2.11) 7 For the more careful reader, let us clarify that the Hamiltonian density including the interaction between the electric potential and the charge is H = Π ·˙ A − LBI + A0ρ. However, we can use the definition of the electric field E =˙ A − ∇A0 to express the Hamiltonian density, up to total derivatives giving rise to boundary terms, as H = Π · E − LBI + A0(ρ − ∇ · Π). The term depending on A0 will then be responsible for the gauge constraint giving Gauss' law that vanishes on-shell, so that it will not modify the self-energy of the particle. 8 For the amusement of the reader familiarised with screening mechanisms in modified gravity, let us notice that this solution realises a screening mechanism for the electromagnetic interaction resembling the so called K-mouflage or Kinetic screening of scalar fields. In the left panel we show the profile (as a function of x ≡ 4πb/Qr) for the electric field generated by a point-like charge. We can clearly see the change from the usual 1/r 2 behaviour at large distances to the saturation for the electric field due to the Born-Infeld corrections on small scales. In the right panel we show how this modified behaviour at small scales also regularises the energy density of the particle.
This expression allows for an alternative interpretation of Born-Infeld electromagnetism. Instead of having modified Maxwell equations in the sector of the electromagnetic field, we can equivalently interpret Born-Infeld electromagnetism as a modification in the source term, i.e., the way in which charges generate electric fields is modified on small scales. In other words, we can interpret it as an effective scale-dependent charge, showing a certain formal resemblance with the renormalisation of the charge when radiative corrections are included in standard QED, but here from a purely classical standpoint without any quantum effect. This re-interpretation of Born-Infeld electromagnetism will be useful for the case of gravity where the Born-Infeld inspired modified gravity theories will also admit an interpretation as a modification of the way in which matter gravitates at high energies.
We will conclude by stressing that the resulting theory turned out to have a series of remarkable features that make the Born-Infeld action be very special among all possible non-linear extensions of electrodynamics. Such properties are related to its special structure, giving additional motivation and support to the idea of implementing the principle of finiteness by enlarging the symmetry group of Maxwell theory. This is nothing but another example of the power of using symmetries as guiding principles to formulate physical theories. In order to avoid further delays in entering into the main topic of this review, namely Born-Infeld inspired theories of gravity, we will abstain our desire of going through all the fascinating features of Born-Infeld electromagnetism and we will content ourselves with briefly enumerating some of its more remarkable properties. For more detailed information we refer to [303,181,228] or standard textbooks on string theory where the Born-Infeld Lagrangian naturally appears, as e.g. [307,308,374]:
• The Born-Infeld action arises in string theory from T -duality invariance when describing an open string in an electromagnetic field, i.e., the Born-Infeld action is the appropriate one to couple strings to electromagnetic (or more general gauge) fields.
• Born-Infeld electromagnetism shares with its Maxwellian relative (and other nonlinear theories of electromagnetism) the so-called electric-magnetic self-duality [69,182]. This is a highly non-trivial invariance of the theory corresponding to a dual transformation of the electric and magnetic fields. See [27] for a review on many interesting aspects of duality rotations and theories with duality symmetry.
• Despite the highly non-linear character of the Born-Infeld action, the corresponding equations of motion give rise to causal propagation and avoid the presence of shock waves and birrefringence phenomena.
• The equations of Born-Infeld electromagnetism admit solitonic solutions with finite energy, known as BIons [94,180].
As we can see, the Born-Infeld theory for electromagnetism not only conforms to the task it was devised for, namely the regularisation of divergences associated to point-like charges, but it is kind enough as to also provide a number of additional gifts that were not required a priori. In the remaining of this section we will overview the attempts to apply similar ideas to the case of gravity. In general, we could say that, by the time of writing, there is not a gravitational analogue of Born-Infeld electromagnetism exhibiting all the successes and remarkable properties discussed above, but the search for it has nevertheless yielded very interesting gravitational theoriesà la Born-Infeld, both from a theoretical and a phenomenological points of view. We will start our tour however by reviewing the first attempts in this direction that led to pathological theories.
The Deser-Gibbons proposal: The ghost problem of the metric formalism
The original idea by Born and Infeld to regularise divergences in electromagnetism was taken over by Deser and Gibbons [140] as a potential mechanism to regularise the singularities that commonly appear in General Relativity, like e.g. the divergences at the center of black holes or the original Big Bang singularity. Following the same scheme, they considered an action for the gravitational interaction including the same determinantal and square root structures that appear in Born-Infeld electromagnetism. A straightforward translation of the Born-Infeld action for electromagnetism to the case of gravity would be the naive replacement of field strength F µν by the Ricci tensor R µν so that the first naive tentative action for a gravitational version of Born-Infeld electromagnetism would be
S = d 4 x − det ag µν + bR µν ,(2.12)
where a and b some parameters, g µν the spacetime metric and R µν the Ricci tensor of the corresponding Levi-Civita connection. However, this naive procedure leads to a theory plagued by ghost-like instabilities. The reason is clear from the well-known fact that an arbitrary action containing a non-linear function of the Ricci tensor will give rise to higher order gravitational field equations and, thus, it will be prone to the Ostrogradski instability [366]. In order to avoid the presence of ghosts in the theory, Deser and Gibbons considered instead the action
S DG = d 4 x − det ag µν + bR µν + cX µν ,(2.13)
where the fudge tensor X µν must be tuned in order to get rid of the ghost. The form X µν can be obtained perturbatively to remove the ghost at a given order and its effects are then pushed to higher orders. We can use the identity
det ½ +M = 1 + [M ] + 1 2 [M ] 2 − [M 2 ] + O(M 3 ) = 1 + 1 2 [M ] + 1 8 [M ] 2 − 1 4 [M 2 ] + O(M 3 ) ,(2.14)
valid for an arbitrary matrixM , to expand the action in powers of the curvature as
S DG = d 4 x − det(ag µν ) 1 + bR + cX 2a + bR + cX 2 8a 2 − bR µν + cX µν 2 4a 2 + · · ·
(2.15) where R = g µν R µν is the Ricci scalar and X = X α α . In this expression we can see that, omitting X µν for a moment, we have a cosmological constant at zeroth order, while at first order we recover the usual Einstein-Hilbert term. At higher orders however the appearance of the quadratic terms R µν R µν will lead to higher order equations of motion, thus rendering the theory unstable due to the presence of ghosts. Since we know that, at quadratic order, only the Gauss-Bonnet prevents the appearance of such ghosts, we must use the leading order contribution from X µν in order to remove the undesired terms. We can then assume an expansion in curvatures starting at quadratic order 9 for the fudge tensor of the form X µν = X (2) µν + · · · and choose X (2) µν to satisfy
cX (2)µ µ + b 2 4a R 2 − 2R µν R µν = α R µνρσ R µνρσ − 4R µν R µν + R 2 ,(2.16)
with α some constant. The above choice thus only leaves the Gauss-Bonnet contribution at second order. By iterating this procedure one could remove the ghosts at any desired order. However, we already see at quadratic order that only the trace of X µν is determined and, therefore, a large variety of fudge tensors can do the job (see [188,187] for explicit constructions). In fact, except for some singular actions, one can presumably write almost any gravitational action in the form of (2.13) by means of an appropriate choice of X µν . We can exemplify this by taking the Born-Infeld gravity theory developed by Nieto in [260]. Motivated by the MacDowell-Mansouri formalism, Nieto considered a spacetime manifold endowed with a connection giving rise to a total curvature R a µ that can be split as R a µ = R a µ + λe a µ , where R a µ is the usual curvature of the Levi-Civita connection, e a µ is the vielbein field and λ a constant parameter. For this connection, he then considers a Lagrangian in D dimensions given by
L = det R a µ .
(2.17) 9 We could also add lower order terms for the fudge tensor as, e.g. X
(1) µν = Rgµν, but that will not introduce the discussion other than adding some more terms in the equations.
If we use the previous splitting, we can write the Lagrangian as
L = λ D e det δ µ ν + 1 λ R µ ν = λ D e D n=0 L (n) (R) , (2.18)
where e = det e a µ and we have used the matrix identity det(½ +M ) = D n=0 e n (M ), with e n (M ) denoting the n-th elementary symmetric polynomial of the matrixM (see (1.25)). In the present case, the matrix is the Ricci tensor and its elementary symmetric polynomials are precisely the Lovelock invariants, that we denote by L (n) (R), so that the considered action is nothing but a combination of all the Lovelock terms and, thus, the theory is ghost-free. One can then rewrite this Lagrangian in the Deser-Gibbons form by simply defining a matrixĜ given byĜ = −R 2 so that the Lagrangian can be alternatively written as
L ∝ − det g µν + 2 λ R µν + 1 λ 2 R µα R µ α ,(2.19)
where we have used the commutativity of the determinant and the square root (whenever it exists). This is the form found by Nieto and which he then related to Born-Infeld gravity. However, as we have seen, it is nothing but Lovelock gravity written in an obscure way. Furthermore, no additional work is necessary to know that the theory does not contain any ghosts. This example perfectly illustrates the necessity of a better defined strategy to construct theories of gravityà la Born-Infeld in order not to be deluded with well-known healthy theories in mysterious disguises.
Other proposals in the metric formalism
In the procedure presented in the precedent section, we have been careful to impose that only the Lovelock invariants should remain at a given order in the expansion. This is a crucial requirement for the consistency of the theory, as the presence of ghosts invalidates any background classical solution. The approach followed by Deser and Gibbons can be seen as a way to make sense of the theory by pushing the scale at which the ghost becomes relevant at higher scales, but the lack of any other guiding principle obstructs the construction of an appealing and well-defined full theory.
One might however take a less demanding approach and impose instead a weaker condition without compromising the stability of the theory due to the presence of ghosts, but at the expense of partially abandoning the original Born-Infeld spirit. For instance, instead of using the fudge tensor to only leave Lovelock invariants at each order, one could allow for some arbitrary functions of them. Thus, at quadratic order we could have allowed for terms involving some linear combination of the squares of the Ricci scalar and the Gauss-Bonnet term. This would find motivation in the fact that arbitrary functions of these two scalar quantities are known to be particular cases where the Ostrogradski instability is bypassed. In the end, this would be nothing but a complicated way of rewriting the class of theories described by an arbitrary function f (R, G), with R and G the Ricci scalar and the Gauss-Bonnet term respectively. Although perfectly legitimate, these theories introduce additional scalar degrees of freedom and, thus, they can hardly be considered as genuine Born-Infeld modifications of gravity. Of course, this does not mean that those alternatives are uninteresting, but rather they should be regarded as belonging to another class of theories.
In case one is interested in obtaining gravitational theories with an upper bound for the curvature, then one can simply write a specific model of an f (R) theory where the function f presents a branch cut at some high but finite curvature R 0 . The square root function typical from Born-Infeld would achieve this, but other functions involving e.g. logarithms could serve as well. Feigenbaum et al. [158] explored this route in two dimensions where the curvature is fully determined by the Ricci scalar and they studied some black holes solutions. In a subsequent work [157], Feigenbaum extended the analysis to four dimensions where he considered an action of the following type:
L = R + β 1 − k 1 R µνρσ R µνρσ − k 2 R µν R µν − k 3 R 2 ,(2.20)
with k i and β some constants. Again he studied black hole solutions that we will briefly review in section 4.1. However, the problem of ghosts arising from the explicit dependence on the full Riemann and the Ricci tensors is not discussed. In fact, from the own equations of motion given in [157], one can see that they are fourth order and, thus, it would be expected to have ghosts. This pathology renders the black hole solutions of limited physical interest, as the perturbations around them are likely to be unstable. The same problem applies to the theories considered by Comelli and Dolgov in [123] constructed in terms of the Lagrangian
L = det A(R)g µν + B(R)R µν ,(2.21)
with A and B some given functions of the Ricci scalar. This Lagrangian combines the Deser and Gibbons proposal with f (R)-type of theories, but without taming the presence of ghosts so that the obtained cosmological solutions are again of limited realistic applicability.
A more interesting proposal that is also closer to the Born-Infeld spirit was given by Wohlfarth in [365]. The theory is based on a symmetric object defined as
R AB ≡ R [a 1 a 2 ][b 1 b 2 ] , (2.22) where the indices A ≡ [a 1 a 2 ], B ≡ [b 1 b 2 ]
should be regarded as ordered pairs of indices. He then introduces the new metric and Kronecker delta
g AB ≡ g a 1 b 1 g a 2 b 2 − g a 2 b 1 g a 1 b 2 (2.23) δ A B ≡ δ a 1 b 1 δ a 2 b 2 − δ a 2 b 1 δ a 1 b 2 (2.24)
that are then used in the usual way to manipulate capital indices. Moreover, one has the identity det g AB = (det g ab ) d−1 valid in d dimensions. The proposed Lagrangian within this formalism is
L = √ −g det δ A B + λR A B ζ ,(2.25)
with λ some constant and ζ a parameter with the only restriction to be a fractional number in order to allow for a regularization of curvature divergences. This represents an extension of Deser and Gibbons construction since more general curvature invariants appear in the Lagrangian. However, it shares the same problematic of containing ghosts (typically appearing at the scale determined by λ) which is then resolved in a similar fashion, i.e., the Lagrangian is corrected as
L = √ −g det δ A B + λM A B + λ 2 N A B ζ ,(2.26)
where M A B and N A B are general expressions containing linear and quadratic curvature terms, respectively. The relative parameters among all the terms must be tuned to remove the ghosts at quadratic order, although one would expect to find again the ghost at higher orders. Thus, similarly to the Deser and Gibbons construction, additional requirements are necessary to find a satisfactory Born-Infeld theory of gravity within this formalism.
Another approach that has been taken in the literature consists in choosing the fudge tensor X µν such that some specific gravity theories are recovered in the low curvatures regime. In [185], the authors followed this path to construct a Born-Infeld extension of the so-called New Massive Gravity theory [66], whose action is given by
S NMG = 1 κ 2 d 3 x √ −g −R + 1 m 2 R µν R µν − 3 8 R 2 (2.27)
and describes a massive graviton in 3 dimensions 10 . One can then see that this action is recovered at quadratic order from (2.13) in 3 dimensions by choosing X µν proportional to Rg µν and appropriately tuning the parameters (with the possible addition of a cosmological constant). Interestingly, the resulting action that they consider recovers at cubic order the extension of New Massive Gravity found in [332] by imposing the existence of a c−theorem. The same authors pursued a similar approach in [186] to construct theories that recover Horava's gravity [210,211] in 3 dimensions at quadratic order.
Eddington-Born-Infeld gravity
In the previous sections we have seen that a straightforward implementation of the Born-Infeld idea to the case of gravity is not an obvious task. It is not difficult to convince oneself that the main difficulty is the avoidance of ghosts and this is hardwired in the use of the metric formalism in the action. One can however seek for Born-Infeld inspired modifications of gravity within the realm of affine theories of gravity where the connection is regarded as an independent object. Within this framework, it is very natural to remember the purely affine theory of gravity introduced by Eddington and described by the following action 11 [147]:
S E = d 4 x | det R (µν) (Γ)| ,(2.28)
where R (µν) (Γ) is the symmetric part of the Ricci tensor of an arbitrary connection Γ α µν . In vacuum, this theory is equivalent to GR 12 . This is easy to understand, since this theory can be seen as GR after integrating out the metric tensor. If we start with GR in the presence of a cosmological constant and in the Palatini formalism, we have
S = 1 2 M 2 Pl d 4 x √ −g R(Γ) + 2Λ (2.29)
that gives the Einstein equations
R (µν) − 1 2 Rg µν = Λg µν . (2.30)
We can now take the trace to obtain R = −4Λ, which allows to rewrite the equations as R (µν) = −Λg µν . This relation can be used in the action to remove the dependence on the metric tensor and we then recover the Eddington action. This procedure of integrating out the metric tensor is also valid when including matter fields as long as they couple minimally, i.e., the metric tensor will only enter algebraically. In that case, the resulting action will be more involved, but it allows to write a fully affine theory of gravity, as it was Eddington's original idea. An important consequence of using the connection as a fundamental geometrical object in Eddington's theory is the avoidance of introducing ghosts associated to higher order equations of motion for the metric tensor. This is not a specific feature of Eddington's theory, but it is a general result for theories formulatedà la Palatini. In view of these results, Eddington's action seems to be a better suited starting point to implement the Born-Infeld construction for theories of gravity. This approach was taken by Vollick [356], who considered the action 13
S EBI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R µν (Γ) − − det g µν ,(2.31)
where M BI is a mass scale determining when high curvature corrections are important. The second term is introduced to remove a cosmological constant, thus allowing for Minkowski solutions in vacuum. The above action for a theory of gravity combines the ideas of Eddington's theory with the Born-Infeld construction, resulting in a theory of gravity formulated in a metric-affine approach and incorporating the square root and determinantal structures characteristic of Born-Infeld electrodynamics. 12 The recovery of the GR equations in vacuum is not specific of Eddington's theory and, in fact, it is a general result for any theory of gravity. The generality of this result actually boils down to the covariance of the field equations which imposes that, in vacuum, the Ricci tensor must be proportional to the metric. In theories of gravity with additional degrees of freedom, the extra fields should be regarded as matter fields and the recovery of GR in vacuum also applies. Another complementary way of understanding this general result is provided by the fact that GR is the only Lorentz invariant and unitary theory for a self-interacting massless spin-2 field in 4 dimensions, usually called graviton. Thus, if by gravity we understand a theory for such a particle, we will inevitably find GR in vacuum. Differences can however show up when including matter fields, as we will discuss later. 13 Here we use the dimension 1 parameter MBI as the Born-Infeld scale instead of the constant b used in [356]. The relation between both is b = M −2 BI .
Before entering into further developments, let us check that GR is indeed recovered when the curvature is much smaller than the scale M 2 BI . When taking that limit, the leading order correction is
S EBI (|R µν | ≪ M 2 BI ) ≃ 1 2 M 2 Pl d 4 x √ −gg µν R µν (Γ) (2.32)
thus reproducing the Einstein-Hilbert action in the first order formalism, which is known to coincide with GR on-shell and provided the matter fields couple minimally 14 (see for instance [201,288]). Let us pause here for a moment and seize the opportunity to clarify some subtleties concerning this point which are well-known in the community but are still source of a little confusion in some works (see for instance the discussion at this respect in section 2.3.1 of [120]). When considering the Einstein-Hilbert action in the Palatini formalism in the presence of minimally coupled fields, the field equations of the connection can be recast as a metric compatibility condition for the metric tensor 15 and, thus, a solution of the equations is the Levi-Civita connection of the spacetime metric. An important point to note however is that such a solution represents a solution, but the most general solution for the connection field equations involves an arbitrary 1-form, which can be taken to be the trace of the non-metricity or the trace of the torsion tensor. This is of course nothing but a reflection of the fact that the metric compatibility condition obtained from the connection field equations does not fully determine the connection and the Levi-Civita connection is only obtained after imposing a symmetric condition. It is sometimes stated that such a condition must be supplemented for the Einstein-Hilbert action to give GR in the Palatini formalism. However, one must also notice that the Einstein-Hilbert action has a projective invariance 16 Γ λ µν → Γ λ µν + ξ µ δ λ ν which also involves an arbitrary 1-form ξ µ , and this is precisely the undetermined mode obtained when solving the connection equation. The gauge character of the projective invariance is discussed in great detail in [225,128].
In the case of the action (2.31), the projective invariance is only obtained as a low curvature accidental symmetry, but it is generally broken by higher order interactions, unless the initial theory is defined only in terms of the symmetric part of the Ricci tensor, in which case the projective invariance is a symmetry of the full theory. Considering only the symmetric part of the Ricci tensor is a widely adopted (and very convenient) option in the literature and, in addition, it would be closer to Eddington's original theory. This 14 The equivalence between the metric and the Palatini approaches has also been considered for more general actions in, e.g. [156,76,127]. A particularly interesting result is that the equivalence of both formulations extends to the whole series of Lovelock invariants, among which the Einstein-Hilbert action represents nothing but the lowest order term. 15 See for instance [288] for details. We will also show more details on how this is achieved in section 2.7.1 within the context of more general theories. 16 In section 2.5 we will show that this symmetry is shared by all theories defined in terms of the symmetric part of the Ricci tensor and we will compute the associated conserved current. is the option adopted by Bañados and Ferreira in [45] 17 , where they considered the action
S EiBI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R (µν) (Γ) − λ − det g µν (2.33)
that has now become the standard version of the so-called Eddington-inspired-Born-Infeld gravity (EiBI). In this version, it is customary to let a cosmological constant term be encoded in the parameter λ as Λ = (λ − 1)M 2 BI . An important notational convention that might lead to some misinterpretations but is very common in the community is to use R µν to denote the symmetric part of the Ricci tensor without the explicit symmetrisation. To avoid any confusion, we will always make explicit the corresponding symmetrisation.
Field equations
In the literature there is a number of subtle points in the derivation of the field equations that are sometimes overlooked or omitted, so we will provide a detailed derivation here. The main differences that one can encounter eventually boil down to whether only the symmetric part or the full Ricci tensor is considered and if the connection is assumed to be symmetric a priori or not. The former condition is related to the presence of a projective invariance, while the latter has to do with the presence of torsion. In many practical applications, these differences do not make a huge impact in the results, but one should nevertheless be careful to obtain the correct field equations. Let us then consider the action
S = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R µν (Γ) − λ − det g µν + S M [Ψ, g µν , Γ]
(2.34) where no assumptions are made a priori on the connection and the full Ricci tensor R µν with both its symmetric and its antisymmetric parts. Let us stress again that Vollick [356] used the full Ricci tensor but constrained the connection to be symmetric, while Bañados and Ferreira left the connection fully undetermined but considered only the symmetric part of the Ricci. We have also added general matter fields Ψ that can, in principle, couple to both the metric and the connection. Then, we will detail where the differences arise when making the different assumptions. For later convenience and to comply with standard notation in the literature, let us introduce the notation
q µν ≡ g µν + 1 M 2 BI R µν . (2.35)
Then, the variation of the action (2.34) can be expressed as 18
δS = M 2 BI M 2 Pl 2 d 4 x √ −q q −1 νµ δg µν + 1 M 2 BI δR µν − λ √ −gg µν δg µν +δS M [Ψ, g µν , Γ]
(2.36) where q = detq and we have used the formula
δ − detM = 1 2 − detM Tr M −1 δM (2.37)
valid for an arbitrary matrixM . The field equations for the metric tensor are then immediately seen to be
√ −q q −1 (µν) = √ −g λg µν − 1 M 2 Pl M 2 BI T µν (2.38)
with the energy-momentum tensor of the matter fields defined as
T µν ≡ 2 √ −g δS M δg µν Γ . (2.39)
Notice that this energy-momentum tensor is defined at constant connection. For minimallycoupled bosonic fields this is not relevant and the energy-momentum tensor will have the standard form. However, when considering fermionic and non-minimally coupled bosonic fields, the expression for the energy-momentum tensor will be in general different from the one that would be obtained in a purely metric formalism. It is important to note the symmetrisation of the objectq −1 in the field equations as a consequence of the symmetry of the metric tensor. Had we considered only the symmetric part of the Ricci tensor in the starting action, this symmetrisation would be innocuous. Furthermore, as said before, in most practical applications in cosmological contexts or spherically symmetric solutions, the matrixq is symmetric and then one could omit the symmetrisation, but in the general case it is important to properly include it. We will come back to this point in section 2.7.1 for more general Lagrangians. The derivation of the connection field equations requires a bit more of work. In order to obtain them, we need the variation of the Ricci tensor:
δR µν = ∇ λ δΓ λ νµ − ∇ ν δΓ λ λµ + T λ ρν δΓ ρ λµ ,(2.40)
where T λ ρν = Γ λ ρν − Γ λ νρ is the torsion tensor. Equipped with this relation, we can now proceed to compute the variation with respect to the connection. Leaving aside the variation of the matter sector for a moment, we have
δS Γ = M 2 Pl 2 d 4 x √ −q q −1 νµ δR µν = M 2 Pl 2 d 4 x √ −q q −1 νµ ∇ λ δΓ λ νµ − ∇ ν δΓ λ λµ + T λ ρν δΓ ρ λµ = − M 2 Pl 2 d 4 x ∇ λ √ −q q −1 νµ δΓ λ νµ − ∇ ν √ −q q −1 νµ δΓ λ λµ − √ −q q −1 νµ T λ ρν δΓ ρ λµ + M 2 Pl 2 d 4 x ∇ λ √ −q q −1 νµ δΓ λ νµ − ∇ ν √ −q q −1 νµ δΓ λ λµ . (2.41)
Let us take a moment here to elaborate on the terms in the last line. Usually in (pseudo-)Riemannian geometries without torsion, these terms correspond to total derivatives that can be simply dropped and do not contribute to the equations of motion. However, the divergence of a vector density A µ of weight w = −1 for a general connection is given by
(see equation (1.5)) ∇ µ A µ = ∂ µ A µ + T λ λµ A µ . (2.42)
Since √ −q is indeed a scalar density of weight w = −1, we then see that the usual boundary terms generated when integrating by parts, actually contribute non-trivially to the field equations whenever torsion is present. Let us stress that the crucial element here is the torsion, i.e., even if there is non-metricity, the boundary terms would not contribute to the equations in the absence of torsion. This is in fact one of the important differences arising from considering a torsion-free connection from the beginning. After taking into account these considerations in the variation (2.41) we obtain
δ Γ S = − M 2 Pl 2 d 4 x ∇ λ √ −q q −1 νµ δΓ λ νµ − ∇ ν √ −q q −1 νµ δΓ λ λµ − √ −q q −1 νµ T λ ρν δΓ ρ λµ + M 2 Pl 2 d 4 x √ −q q −1 νµ δΓ β νµ − √ −q q −1 βµ δΓ λ λµ T α αβ . (2.43)
After an appropriate re-shuffling of the indices, the connection field equations can finally be expressed as
∇ λ √ −q q −1 µν − δ µ λ ∇ ρ √ −q q −1 ρν = ∆ µν λ + √ −q T µ λα (q −1 αν + T α αλ (q −1 µν − δ µ λ T α αβ (q −1 βν (2.44)
where, for completeness, we have added the hypermomentum of the matter fields
∆ µν λ ≡ 2 M 2 Pl δS m δΓ λ µν gµν . (2.45)
Analogously to the energy-momentum tensor, the hypermomentum must be computed at constant metric. In most of the cases, we deal with minimally coupled bosonic fields in which case we have ∆ µν λ = 0. However, the standard way of coupling fermionic fields to gravity is by resorting to the vierbeins formalism that allows to generalise the definition of the gamma matrices to curved spacetime. In that formalism, the fermions couple directly to the spin connection and, thus, contributions to the hypermomentum typically arise. We will leave this case aside and will assume vanishing hypermomentum. For this simplified case, we have the full set of equations for the Born-Infeld gravity that we display grouped together here for future reference
√ −q q −1 (µν) = √ −g λg µν − 1 M 2 Pl M 2 BI T µν , (2.46) ∇ λ √ −q q −1 µν − δ µ λ ∇ ρ √ −q q −1 ρν = √ −q T µ λα (q −1 αν + T α αλ (q −1 µν − δ µ λ T α αβ (q −1 βν (2.47) where q µν ≡ g µν + 1 M 2 BI R µν . (2.48)
These will be the fundamental set of equations that need to be solved in Born-Infeld gravity. In most practical situations, the equations are greatly simplified and the general case is rarely required. Thus, instead of tackling the full set of equations directly, let us first first consider a simplified case where most of the results will be sufficient for the astrophysical and cosmological applications discussed in the subsequent sections.
Simplified case: Vanishing torsion and projectively invariant case
We will start by considering the simplest possible case with vanishing torsion a posteriori and where the action is constructed out of the symmetric part of the Ricci tensor solely, and we will postpone the general case for later. The busy reader rushing to explore the different applications of the theory will be able to grasp the essential details in this section, since this simplified scenario is the most extensively considered case in spherically symmetric and cosmological solutions. The thorough reader will hopefully be satisfied with the more detailed discussion provided in section 2.7.1 for more general theories (where in fact we will see that getting rid of the torsion does not represent any limitation for a class of theories among which we find EiBI). Let us notice that the assumption on R µν refers to the own definition of the theory while the torsion-free condition restricts the considered class of solutions within the theory.
The fact that we only consider the symmetric part of the Ricci tensor in the action has two important consequences. On one hand, the object q µν will inherit the symmetry of the Ricci tensor (along with that of the metric g µν ). On the other hand, we are enlarging the symmetries of the theory by introducing a projective invariance and, thus, this condition can be naturally introduced by imposing such a symmetry in the gravitational sector. The projective invariance corresponds to a shift in the connection of the form
Γ λ µν → Γ λ µν + ξ µ δ λ ν (2.49)
for an arbitrary 1-form ξ µ . That this is in fact a symmetry of the theory containing only the symmetric part of the Ricci tensor can be easily seen from (2.40) by taking δ ξ Γ λ µν = ξ µ δ λ ν to obtain that, under a projective transformation, the full Ricci tensor transforms as
δ ξ R µν = ∇ µ ξ ν − ∇ ν ξ µ + T λ µν ξ λ . (2.50)
We can clearly see from here that the variation of the Ricci tensor under a projective transformation of the connection is antisymmetric and, thus, its symmetric part is invariant δ ξ R (µν) = 0. A consequence of this symmetry is that one of the traces of the connection field equations vanishes identically, i.e., the constraint associated to the projective symmetry is
δ λ ν δS δΓ λ µν = 0. (2.51)
Let us stress here that the projective symmetry will not be broken by the presence of minimally coupled fields. Bosonic fields with minimal couplings will only couple to the metric, so the projective invariance is obvious. On the other hand, minimally coupled fermions do couple to the connection, but such a coupling still respects the projective symmetry (see for instance [202]). Finally, it is also interesting to note that the projective invariance is so-called because it is in fact a symmetry of the geodesics equations, since its effect can be re-absorbed into a re-definition of the affine parameter. For minimally coupled fields this is irrelevant because they are only sensitive to the Levi-Civita part of the full connection. The field equations under the conditions at hand now reduce to
√ −q q −1 µν = √ −g λg µν − 1 M 2 Pl M 2 BI T µν (2.52) ∇ λ √ −q q −1 µν − δ µ λ ∇ ρ √ −q q −1 ρν = 0 (2.53)
where we have set T λ µν = 0 and dropped the explicit symmetrization for q −1 µν since it is automatically symmetric. We can check that the trace of the connection equations with respect to λ and ν vanishes identically, as a consequence of the projective symmetry, while the trace with respect to λ and µ gives
∇ λ √ −q q −1 λν = 0. (2.54)
This constraint can then be plugged back into the connection equations to finally obtain
∇ λ √ −q q −1 µν = 0. (2.55)
Since the action only depends on the symmetric part of the Ricci, the object q µν is symmetric and the above equation tells us that the connection must be compatible with the auxiliary metric q µν , i.e., the connection is given by the Levi-Civita connection of the metric q µν . It is important to notice that the metric compatibility condition only determines the symmetric part of the connection and, in general, leaves a vector component of the antisymmetric part undetermined. However, the assumption of a symmetric condition fixes this undetermined part. At this point, one could fairly object that we have not solved the connection yet, as the auxiliary metric q µν is defined in terms of the Ricci tensor, which depends on the connection itself. The resolution to this comes about by going back to the metric field equations (2.52). From there, we can see that the auxiliary metric can be fully expressed in terms of the spacetime metric g µν and the matter content through its energy-momentum tensor 19 T µν , so that the solution for the connection has actually been achieved. An important feature of this procedure that should not go unnoticed is that the connection has been obtained by solving algebraic equations and, therefore, no degrees of freedom are actually associated to it. In other words, there are no additional boundary conditions that we need to provide to solve for the connection, which means that it is nothing but an auxiliary field. This is the reason why the Born-Infeld theory modifies gravity without introducing new degrees of freedom. We will come back to this point later for more general cases. Now that we have the solution for the connection, we can proceed to complete the resolution of the problem. This is not a very difficult task, since the field equations determining the auxiliary metric (that then gives the connection) are simply
R µν (q) = M 2 BI q µν − g µν (2.56)
where we need to remember thatq =q(ĝ, Ψ) is obtained from the metric field equations. However, instead of using these equations directly in this form, it is convenient to work them out a little bit to recast them into a more suitable form for direct applications. Let us begin by introducing some additional notation that is commonly used in the literature and which will allow to make contact with more general theories. We will denote byΩ the deformation matrix relating the auxiliary and the spacetime metrics as q µν = g µα Ω α ν (2.57) or, in matrix notation,q =ĝΩ. In the present case, this matrix is simplyΩ = ½+ 1
M 2 BIĝ −1R ,
obtained from the definition ofq. However, the advantage of introducing this notation is that we can very easily solve the metric field equations (2.52) forΩ. When plugging (2.57) into (2.52), we obtain the relation
Ω −1 = 1 detΩ λ½ − 1 M 2 Pl M 2 BITĝ .
(2.58)
Now, we can multiply (2.56) byq −1 and use (2.57) to obtain
q −1R (q) = 1 M 2 Pl detΩ M 2 Pl M 2 BI detΩ − λ ½ +Tĝ .
(2.59)
This will be the starting point for many of the discussions in the subsequent sections devoted to astrophysical, black holes and cosmological applications. Let us stress that the components ofΩ will be obtained as solutions of the metric field equations now expressed as (2.58). Thus, the solution of the problems is achieved in two steps. One first solves the set of algebraic equations (2.58) to obtainΩ =Ω(Tĝ), i.e., in terms of the metric and the matter content through the combinationTĝ. For some important material contents, this combination does not depend on the metric but only on the energy density ρ and the pressure p. This is the case for instance for a perfect fluid or an electromagnetic field (see sections 4.3, 4.4 and 5.2). In that case, solving (2.58) will yieldΩ =Ω(ρ, p). After obtaining these expressions, one can then complete the resolution of the problem by solving the differential equations (2.59).
To end this section, let us notice that the equations (2.59) admit yet another formulation in terms of the Born-Infeld gravitational Lagrangian defined by means of S BI = d 4 x √ −gL BI . If we restore the components notation, we have
R µ ν (q) = 1 M 2 Pl detΩ L BI δ µ ν + T µ ν ,(2.60)
where we have used the metric q µν to raise the first index of the Ricci tensor, i.e., R µ ν (q) ≡ q µα R αν . The interest of writing the equations in this form is twofold. Firstly, as we will see in section 2.7.1, this form of the field equations is valid not only for the Born-Infeld gravity theory considered here, but also for a large variety of theories formulated in the Palatini formalism. Thus, given a certain specific theory, we can immediately obtain the corresponding field equations by using (2.60) directly. Secondly, this will be the starting point for many of the developments for practical applications that will be discussed in the subsequent sections of this review. Another important feature of (2.60) is that we can directly compare it with the usual Einstein equations of GR written as These equations show even more clearly how the Born-Infeld theory can be seen as usual gravity for the auxiliary metric with a modified source term (let us remember once again that Ω is algebraically related to the matter content through (2.58)). Furthermore, from here we can also easily understand a very distinguishing property of the theory. If we now use the relation between the two metricsq =ĝΩ to expand the Einstein tensor in (2.63) in terms ofĝ-related objects we can immediately see that, since the Einstein tensor contains up to second derivatives of the metric, we will end up with up to second derivatives of the deformation matrixΩ. This deformation matrix depends on the energy momentum tensor through (2.58)) so that the evolution equations for the spacetime metric g µν will contain derivatives of the energy-momentum tensor components 20 . This is a very distinctive feature of these theories that gives rise to new effects and, among others, a dependence of the gravitational potential on the local density and not only on an integrated density as in the usual case (see (3.6)). In fact, this effect has been claimed to lead to very serious drawbacks. We will give a careful discussion about this issue in section 3.1. Finally, this feature will also be the responsible for a dependence of the background cosmology evolution on the sound speed and not only on the equations of state parameter as in ordinary gravitational theories. We will see in section 2.7 that these properties are in fact shared by a large class of theories. This non-standard interplay between the gravitational sector and the matter fields has been noticed and extensively used in the literature. See for instance [138] for a devoted discussion on this point and [297] where it is shown that gravity theories with generic auxiliary fields exhibit these properties.
In the next subsection we will re-obtain this result in a slightly different and complementary way that will allow to clarify the role played by both metrics. Already here we can sense that the auxiliary metric carries physical relevance and it is not simply a mathematical object. We will postpone a thorough discussion about this point for the next subsection. Let us notice now that, very much like for the electromagnetic case, when we take curvatures much smaller than M 2 BI (or, equivalently, densities much smaller than M 2 BI M 2 Pl ), the deformation matrix is approximately the identityΩ = ½+O(R/M 2 BI ) so that q µν and g µν coincide up to corrections O(R/M 2 BI ). In that case, we also have L BI ≃ 1 2 M 2 Pl R and (2.63) reduces to the usual Einstein equations, confirming that the modifications only appear when the curvatures become order one as compared to the Born-Infeld scale M BI . Equivalently, the Born-Infeld modifications will appear when |T µ ν | ∼ M 2 Pl M 2 BI . To end this section, we will give some good news that will appease the less thorough reader. Despite having neglected the torsion, all the results obtained here are completely valid for the general case with torsion provided the projective symmetry is imposed. We will show this explicitly in section 2.7.1 20 One could object that second derivatives of Tµν will give rise to higher than second order derivatives of the matter fields because the energy-momentum tensor typically contains first derivatives and, thus, the system might be prone to the very same Ostrogradski instabilities we claimed to be avoided. However, one should keep in mind that the matter fields will have their own second order field equations so they will in any case propagate the correct number of degrees of freedom.
2.6. The two frames of Born-Infeld gravity and the physical relevance of the auxiliary metric We have seen that Born-Infeld gravity naturally leads to the appearance of two metric tensors, namely the spacetime metric g µν and the auxiliary metric q µν . The former plays the role of the metric to which matter fields couple, while the latter has been introduced as an auxiliary object to solve the equations so that the connection is the one compatible with it. So far, we have not provided this object with any physical meaning and it simply appeared as a mathematical trick to facilitate the resolution of the field equations or, equivalently, it appears as a result of integrating out the connection. The aim of this section will be to clarify the role of this object and unveil its physical significance.
The bi-metric character of the theory can be better appreciated by rewriting the EiBI action in the equivalent form 21
S EiBI = 1 2 M 2 Pl M 2 BI d 4 x √ −q q −1 µν g µν + 1 M 2 BI R (µν) (Γ) − 2 + S M [Ψ, g µν ] (2.64)
where we have introduced an auxiliary field that we have suspiciously called q µν . To see that this is in fact equivalent to the EiBI action we can compute the field equations for this auxiliary field
− 1 2 q −1 µν g µν + 1 M 2 BI R (µν) − 2 q αβ + g αβ + 1 M 2 BI R (αβ) = 0. (2.65)
If we contract this equation with q −1 αβ we obtain the relation
q −1 µν g µν + 1 M 2 BI R (µν) = 4 (2.66)
which can be plugged into the field equation to obtain the solution q µν = g µν + 1 M 2 BI R µν , that justifies our original name for this auxiliary field, since it turns out to be nothing but the auxiliary metric defined above. If we insert the solution into (2.64) we see that we recover the original determinantal form of the EiBI action after integrating out the auxiliary field q µν , proving the equivalence of both representations. The bimetric representation however provides a more orderly arrangement of the two metrics that allows to unveil their role in the theory. The role of the metric tensor is already clear from the beginning as the metric seen by matter fields and, therefore, determining their causal structure. In particular, point-like particles will follow the geodesics of the Levi-Civita connection corresponding to g µν . There is nothing special here as this is a consequence of considering minimally coupled fields, the only difference with respect to the usual case being that the solution for the metric tensor will be different. In order to properly identify the physical role of the auxiliary metric, let us notice two important features in (2.64). The first one is that the spacetime metric g µν only enters the action algebraically, i.e., without any derivatives. This means that g µν is an auxiliary field that can be integrated out. In fact, its equation of motion is given by
√ −q q −1 µν = − 1 M 2 Pl M 2 BI √ −gT µν (2.67)
which allows to obtain g µν algebraically in terms of the matter fields and the auxiliary metric q µν . For some types of matter fields this step might not be possible and, thus, the following discussion would not apply. Barring these singular cases, we can integrate out the spacetime metric and we will end up with an action of the form
S EiBI = 1 2 M 2 Pl d 4 x √ −q q −1 µν R (µν) (Γ) +S M [Ψ, q µν ] (2.68)
whereS M represents the new form of the matter sector after replacing the solution for g µν obtained from (2.67). We thus arrive at an equivalent action with the Einstein-Hilbert term in the Palatini formalism to describe the dynamics of the auxiliary metric q µν , but now the coupling of this auxiliary metric to the matter fields will have a complicated form. As discussed above, the Einstein-Hilbert sector will state that the connection Γ must correspond to the Levi-Civita connection of the auxiliary metric q µν , which is again the result obtained when working with the determinantal form of the action. This version of the action reveals a more profound role for the auxiliary metric since now we can see that the gravitational waves can be straightforwardly interpreted as the tensor part of the perturbations of the auxiliary metric. To further clarify this point, let us assume that we have a background configuration for both metrics given byḡ µν andq µν . In this background geometry, the matter fields will propagate in the metricḡ µν that will determine the corresponding causal structure. In particular, the light cone for photons will be determined by this metric. Furthermore, massive objects will be coupled in the standard way to the gravitational potentials and will follow the geodesics ofḡ µν . On the other hand, gravitational waves will propagate on the background metricq µν and it is this auxiliary metric that determines the causal structure for them so that gravitons will follow the geodesics of the auxiliary metricq µν . Since matter fields couple in a non-standard way to this metric, the interaction of the gravitational potentials encoded in the perturbations of q µν with the matter fields will differ from the usual case. We can then summarise this discussion by saying that the spacetime metric determines the propagation of matter fields and the auxiliary metric determines the propagation of gravitons. The result obtained here and that boils down to the equivalent action (2.68) for EiBI gravity is equivalent to the finding presented at the end of 2.5.1 where the field equations were eventually written as (2.63) in the form of Einstein equations for the metric q µν with a modified source term. This is exactly what the action (2.68) is telling us, since the corresponding field equations will consist of the Einstein tensor obtained from varying the gravitational sector which will then be sourced by the energy-momentum tensor of the matter sector as computed with respect to the metric q µν . In other words, the field equations are 22
G µν = 1 M 2 PlT µν (2.69)
whereT µν is the effective energy-momentum tensor defined as
T µν ≡ 2 √ −q δS M δq µν Γ . (2.70)
We thus recover the field equations for Born-Infeld gravity written in an Einsteinian form as in (2.63) where we need to identify the non-standard source term in the right hand side with the effective energy-momentum tensorT µν , which is non-trivially related to T µν . It is important to notice that both energy-momentum tensors will satisfy their corresponding conservation equations, namely:∇ µT µν = ∇ µ T µν = 0 with∇ and ∇ the covariant derivatives associated to q µν and g µν respectively. The result found here will help explaining why singular solutions like the Big Bang and/or black holes can be regularised without violating the null energy condition, because the object that will need to violate an effective null energy condition is not the standard energy-momentum tensor of the matter fields (see also sections 4.3, 4.4 and 5.2).
A certain familiarity with modified gravity allows to appreciate a close analogy between the above discussion and the existence of two frames in scalar-tensor theories. In the Jordan frame matter fields are minimally coupled to the metric, but gravity is described by a scalar-tensor theory. In the Einstein frame however gravity is described by the Einstein-Hilbert term, but matter fields couple to a conformal metric whose conformal factor depends on the scalar field. In the case of Born-Infeld, the situation is alike, but with the crucial difference that there are no additional propagating degrees of freedom. In the original description of the theory, that we will call the Born-Infeld frame, matter fields couple in the standard way to the metric but the gravitational action is non-standard. In this frame, we have that gravity reacts differently to the presence of matter when the densities are very high and particles follow the geodesics of the metric just as in standard gravity. In the alternative description exposed in this section, that we will call Einstein frame for obvious reasons, gravity has the standard Einstein-Hilbert action, but now the couplings of the matter fields to gravity are not the usual ones, i.e., we cannot simply follow the usual minimal coupling rule from flat spacetime and replace the Minkowski metric by the curved one appearing in the Einstein-Hilbert term.
The existence of the Einstein frame also helps understanding the Born-Infeld inspired gravity theories from a particle physics perspective. The common wisdom says that GR is the only consistent 23 theory for a massless spin 2 field in 4 dimensions and this is usually used to state that modifications of gravity either introduce additional degrees of freedom or they reduce to GR. As we have seen, the Born-Infeld theories modify gravity without introducing additional degrees of freedom so we seem to face an apparent paradox. However, the more precise statement about GR being the unique theory for a massless spin 2 field concerns the IR regime and, thus, it is modifying gravity in the IR what requires the introduction of additional degrees of freedom. This is what usually happens in models of dark energy based on IR modifications of gravity. On the other hand, the high energy regime is not locked by the consistent requirements and, as we understand now from the Einstein frame, the Born-Infeld theories precisely modify this regime of gravity.
The Einstein frame also permits a more clear interpretation of the different regimes that we encounter in Born-Infeld inspired theories of gravity. As we have seen, these theories are characterised by two different scales, namely the Planck mass M Pl and the Born-Infeld scale M BI . These two scales are assumed to satisfy M BI ≪ M Pl and this hierarchy introduces yet another relevant scale in the problem given by their geometrical
meanM BI = √ M Pl M BI .
The introduced hierarchy has the purpose of having Born-Infeld corrections before hitting the quantum gravity regime, that takes place at some scale near M Pl , so that we can have a range of scales betweenM BI and M Pl where gravity behaves differently but the quantum gravity effects can still be safely neglected. From the action in the Einstein frame expressed as 2.68, we see that the Born-Infeld scale M BI can be completely moved to the matter sector and, in combination with M Pl throughM BI , it controls the scale at which the generated non-linear interactions of the matter fields become relevant 24 . Interestingly, even fields that do not interact directly in the Born-Infeld frame will couple in the Einstein frame and the coupling will again be controlled byM BI . The fact that all fields will be generically coupled in the Einstein frame and the coupling constantM BI is universal can be nicely interpreted as a consequence of dealing with a gravitational theory, i.e., as a sort of additional Born-Infeld equivalence principle. In other words, the Born-Infeld inspired theories have the usual equivalence principle, according to which all matter fields couple to gravity with a universal coupling constant M Pl (fully valid on scales belowM BI ), and what we have called Born-Infeld equivalence principle, according to which all the generated couplings in the matter sector come in with another universal coupling constantM BI . Since we have not observed any anomalous interactions beyond those of the standard model at LHC, we can straightforwardly impose the very conservative constraintM BI 10 TeV, which translates into M BI 10 −1 eV so that the Born-Infeld corrections can only have effects in regions of spacetime where the curvature is larger than 10 −2 eV 2 .
The couplings generated in the matter sector bring about one important point that is usually overlooked in the literature and has not been properly addressed yet, namely whether, or under which conditions, the quantum corrections can remain under control in 24 Since the source of gravity in most situations is the energy density ρ, the transition between the usual GR and the Born-Infeld regimes in the gravitational sector is expected to occur when ρ ∼M 4 BI , as we will confirm in the numerous applications studied in the subsequent sections. However, we should point out that this is only true in the simplest scenarios, but, in general, the Born-Infeld corrections will become relevant whenever some interactions reachMBI. To give an example, one could imagine a situation where the densities are small as compared toM 4 BI , but some anisotropic stresses or heat fluxes are of order 1 as compared to the scaleMBI. the Born-Infeld regime. This is not obvious a priori because the couplings generated in the matter sector controlled byM BI will usually contain non-renormalisable operators and, in fact, one would naively expectM BI to play the role of a strong coupling scale and, thus, the effects at that scale will require non-perturbative analysis for large background field configurations. This however does not necessarily mean that the Born-Infeld regime will inevitably face strong coupling problems. We will give here a taste on a possible situation where the Born-Infeld regime can be safe, but a more careful analysis should definitely be performed. If we consider a massless scalar field in the Born-Infeld frame, in the Einstein frame we will have a K-essence type of theory where the interactions will be controlled byM BI . The strong coupling scale in these theories around a trivial background isM BI and one can apply the standard perturbative analysis because the background value of the field is smaller thanM BI . The worry comes when the background field takes values near M BI and non-perturbative effects would be expected to become relevant. However, around these non-trivial backgrounds the vacuum value of the scalar field re-dresses the strong coupling scale so that it can be pushed to values higher thanM BI . This mechanism is at work for instance in theories featuring a K-mouflage/Kinetic or Vainshtein screening (see for instance [135,133,91,136,204,137,85]). In these situations the non-linear classical solutions can be trusted in the Born-Infeld regime. In this case the scalar will also couple to other matter fields throughM BI , but again the coupling scale will be re-dressed by the background value of the scalar field, so that these interactions can also remain small. As we have emphasised, this is only a potential resolution of the strong coupling problems that one would expect in these theories, but one should carefully check whether this is the actual situation.
Let us end this section by noting that the discussion presented here is not particular of the Born-Infeld gravity, but it is a feature of a general class of gravity theories formulated a la Palatini. We will show this explicitly in section 2.7.1
Classes of Born-Infeld inspired gravity.
The Born-Infeld theory of gravity discussed in the previous sections are naturally formulated on a spacetime manifold endowed with a general affine connection. Thus, given the richness offered by this geometrical framework, it is of no surprise that the Eddington-Born-Infeld theory described so far has found extensions in different directions. Unlike the case of Born-Infeld electrodynamics, the Eddington-Born-Infeld theory has not been singled out by resorting to symmetries principles or any other guiding criteria, but rather it originates from a straightforward transcription of the Born-Infeld Lagrangian for electromagnetism to gravity and taking inspiration from the Eddington affine theory. Thus, Born-Infeld inspired gravity is more prone to modifications and extensions than its electromagnetic relative. However, before proceeding to review the existing models and in view of the zoology of Born-Infeld inspired gravity theories found in the literature, we find it convenient to introduce some taxonomic system. We will classify the theories according to their proximity to the original Born-Infeld spirit, consisting in modifying the high curvature regime of gravity without introducing additional fields or pathologies. Furthermore, we will take the EiBI theory as the baseline because it is the most extensively studied model. After these considerations, we have decided to make the following classification:
• Class 0. We start our classification with a class comprising all those early attempts of building gravity theoriesà la Born-Infeld which did not succeed due to the presence of pathologies. Subsequent proposals sharing these pathologies will also be considered to belong to this class.
• Class I. Here we will include the EiBI theory and the modifications that are the closest to the Born-Infeld spirit and do not introduce additional ingredients, be it new degrees of freedom or additional geometrical objects.
• Class II. A next step with respect to the Class I is to allow for more general geometrical objects, but respecting the Born-Infeld philosophy, i.e., only the high curvature regime is modified and no additional degrees of freedom are present.
• Class III. Under this category we will classify those models where the Born-Infeld structure remains but additional degrees of freedom are included.
• Class IV. Finally, in this class we will include theories that, although resemble Born-Infeld theories in some aspect, they could be very well classified within a different class of theories.
The above classification does not intend to be exhaustive nor having sharp edges. For instance, sometimes the presence of additional degrees of freedom might depend on some subtle assumptions on the theory or its solutions so that the same theory can have slightly different versions belonging to different classes. In those cases, we have opted by classifying it according to the most extensively used version in practical applications.
A substantial part of the formal developments and equations for many of the Born-Infeld inspired theories share numerous similarities among them and with the theory discussed so far. For that reason, prior to discussing specific theories we will present a general framework applicable to most of them.
General mathematical framework
In this section we will discuss some features that are common to a large class of theories, that include many of the proposed extensions and which are shared with EiBI gravity. Let us consider a general theory of the form
S = 1 2 M 2 Pl M 2 BI d 4 x √ −gF g µν , R µν (Γ) (2.71)
where F is a function of the inverse of the metric and the Ricci tensor. Notice that we have included a factor √ −g in the measure so that F behaves as a true scalar. For simplicity, we will assume that the function will only depend on the combination P µ ν = g µα R αν /M 2 BI , where we have introduced the scale M BI for dimensional reasons. This is also the usual case in the literature so it will suffice for us 25 . An important consequence of the function being a scalar is that F (Â −1PÂ ) = F (P ) for any non-degenerate transformationÂ. Furthermore, the independent scalars built out ofP can be expressed as traces of powers of [P n ]. By using the Cayley-Hamilton theorem, we can express any power higher than 4 in terms of lower powers so that the action could in principle be written as F (X 1 , X 2 , X 3 , X 4 ) with X n = [P n ]. This is useful to show some general properties of this general class of theories. We will not make extensive use of the advantages introduced by writing the action in this form and we will instead consider the action written as
S = 1 2 M 2 Pl M 2 BI d 4 x √ −gF P . (2.72)
In order to recover GR in the limit |P µ ν | ≪ 1 we need to impose
∂F ∂P µ ν P =0 = δ µ ν . (2.73)
Notice that this is not really a constraint and any analytic function will satisfy it up to a constant factor that can be absorbed into M 2 BI . The Einstein-Hilbert action is recovered for F (P ) = P α α , in which case the above relation is exactly fulfilled for all values ofP and not only atP = 0.
To be completely precise we should say that the above condition will guarantee the existence of one branch of solutions that will be continuously connected with GR at low curvatures. Nevertheless, the non-linearity of the equations can, in general, present several branches and some of them will give a different behaviour for the low curvatures regime. We will encounter specific examples where this situation occurs when studying explicit solutions.
For the general action considered, we can obtain the corresponding field equations by taking with respect to both the metric and the connection, yielding
δS = 1 2 M 2 Pl M 2 BI d 4 x √ −g − 1 2 F g µν δg µν + 1 M 2 BI ∂F ∂P µ α δg µν R να + g µν δR να . (2.74)
For the subsequent developments, it is convenient to write the above variation in matrix notation
δS = 1 2 M 2 Pl M 2 BI d 4 x √ −gTr − 1 2 Fĝδĝ −1 + 1 M 2 BI ∂F ∂PR T δĝ −1 +ĝ −1 ∂F ∂P δR T (2.75)
Now it will be useful to introduce some definitions before proceeding any further. First, let us define √ −qq αν ≡ √ −gg νµ ∂F ∂P µ α (2.76)
or in matrix notation √ −qq −1 ≡ √ −g ĝ −1 ∂F ∂P T . (2.77)
This definition is not an innocent choice and we will see later that q µν will actually play the role of the auxiliary metric determining the connection, as in the EiBI case. We can take determinants in both sides of (2.77) to obtain the relation
g q = 1 detFP (2.78)
where we have introduced the notationFP ≡ ∂F/∂P . Then, we can re-write the definition (2.77) asq (2.79) or, if we invert both sides, we finally obtain an expression for q µν as follows:
−1 = 1 detFP ĝ −1 ∂F ∂P T ,q = detFP ∂F ∂P −1ĝ T . (2.80)
For the Einstein-Hilbert term, the derivative of F gives the identity andq exactly coincides with the spacetime metric, as expected. If we consider f (R) types of theories for which F f (R) = F (P α α ), the derivative gives
∂F f (R) ∂P = F ′ f (R) ½ (2.81)
so we have that q µν = F ′ f (R) g µν , recovering the known result that in these theories the two metrics are conformally related. Finally, in the case of the EiBI action (2.34) we have
F EBI = 2 det ½ +P so its derivative is ∂F EBI ∂P T = det ½ +P ½ +P −1 .
(2.82)
When inserting this expression into the definition (2.77) we obtain
√ −qq −1 = √ −g det ½ +P ½ +P −1ĝ −1 = − det ĝ + 1 M 2 BIR ĝ + 1 M 2 BIR −1
(2.83) and we recover that q µν = g µν + 1 M 2 BI R µν as it should. After this little satisfaction, we can continue with the computation of the field equations. Another useful relation for the variation of the action that we can obtain from the definition of q µν is the following:
√ −g M 2 BI ∂F ∂PR T = √ −q ĝPq −1ĝ T . (2.84)
With the new jargon, we can re-write the variation (2.75) as
δS = 1 2 d 4 xTr √ −gL Gĝ −1 δĝ − M 2 Pl M 2 BI √ −q q −1 TP T δĝ + M 2 Pl √ −qq −1 δR
(2.85) where we have used the ciclic property of the trace and the identityĝδĝ −1 = −δĝĝ −1 . Furthermore we have re-introduced the Lagrangian L G = 1 2 M 2 Pl M 2 BI F . From the last term we can already sense that q µν will be related to the metric generating the connection, since that piece resembles the variation one would obtain from the Einstein-Hilbert action in the Palatini formalism with a metric q µν . A word of caution is necessary though, since q µν does not need to be symmetric at this point. Again, if we assume a projective symmetry so only the symmetric Ricci enters, only the symmetric part of q µν will contribute and, thus, it will exactly be the auxiliary metric. Prior to the discussion of the connection field equations, let us first write the metric field equations:
1 2 M 2 Pl M 2 BI √ −q Pq −1 + Pq −1 T − √ −gL Gĝ −1 = √ −gT (2.86)
where the symmetrization follows from the symmetry of g µν and we have also added the energy-momentum tensor of the matter sector. For the sake of completeness, we will also give the expression of this equation in components
M 2 Pl M 2 BI √ −qq α(µ P ν) α − √ −gL G g µν = √ −gT µν . (2.87)
As one of our favourite exercises, let us check that we recover the expected results when the above equation is particularised to known cases. For the Einstein-Hilbert action, we have already seen that q µν = g µν and it is immediate to see that (2.86) reduces to
R (µν) − 1 2 Rg µν = 1 M 2 Pl T µν .
For the Born-Infeld inspired theory with F EBI = 2 det ½ +P we have also shown thatq reduces to the expected result. In that case, it is easy to see from
g −1q = ½ +P thatPq −1 =ĝ −1 −q −1 .
If we insert this relation into (2.86) and use that
√ −gL G = M 2 Pl M 2 BI √ −q, we can see that the equations reduce to −M 2 Pl M 2 BI √ −qq (µν) = √ −gT µν , in agreement with (2.38) (taking λ = 0).
Let us pause a bit before moving on to the connection field equations to discuss the structure of the metric field equations. In general, the symmetry of the metric results in a set of ten independent equations. The general treatment of theoriesà la Palatini requires the use of these equations to solve for the Ricci tensor (or connection-dependent objects for more general theories) in terms of the metric and the matter fields. This step is algebraic and it is crucial for the subsequent resolution of the connection as the Levi-Civita connection of some auxiliary metric. However, while the Ricci tensor has in general sixteen components, the metric field equations are limited to ten and, therefore, the full Ricci cannot be obtained from them. This means that the method to solve the connection as the Levi-Civita connection will fail. This motivates considering theories with the projective symmetry for simplicity reasons.
Let us know turn to the computation of the connection field equations. By looking at the last piece of (2.85) we can see that it reads exactly the same as the corresponding variation for the Born-Infeld case in (2.41). Hence, the derivation will follow analogously and we can simply use the equations already obtained in (2.44)
∇ λ √ −q q −1 µν − δ µ λ ∇ ρ √ −q q −1 ρν = ∆ µν λ + √ −q T µ λα (q −1 αν + T α αλ (q −1 µν − δ µ λ T α αβ (q −1 βν (2.88)
where we only need to remember that nowq is defined in (2.76) and we have added the hypermomentum of the matter fields defined in (2.45). If only the symmetric part of the Ricci tensor enters in the action, so that we have a projective symmetry Γ α µν → Γ α µν +ξ µ δ α ν , we can see from (2.74) that only the symmetric part of q µν will contribute to the connection field equations. In that case we can easily see that the trace with respect to ρ and ν vanishes identically, as a consequence of the projective invariance. Sometimes, this is regarded as a flaw of these theories because, in case the symmetry is not present in the matter sector, there is no reason to expect to have ∆ µρ ρ = 0 and this would be the source of an inconsistency in the equations. However, there is an obvious way to evade this apparent problem by assuming that matter fields do not couple to the connection directly so that we actually have that the full hypermomentum vanishes. Again, this is the case for minimally coupled bosonic fields, but complications might arise due to fermions. In any case, even if we need to have ∆ µρ ρ = 0, this should be regarded as a constraint in the matter sector and there is no reason a priori to assume that solutions satisfying that constraint cannot be found 26 . For simplicity and to comply with most of the literature we will take ∆ µν ρ = 0 in the following. On the other hand, if there is no projective symmetry in the action, the object q µν will not have, in general, any defined symmetry. In that case, the equations have a formal resemblance with non-symmetric gravity theories [255,129,350,58] so one could try to apply the same techniques to solve the equations. However, the similarities are purely formal and, in fact, there are profound conceptual differences between the non-symmetric gravity theories and the ones under study here, mainly the absence of an actual nonsymmetric metric.
We will manipulate the equations to recast them in more useful forms. We can first take the trace of the equations with respect to µ and λ to obtain that
∇ λ √ −qq λν = 1 2 T λ λα √ −qq αν (2.89)
If we plug this relation back into the equations we obtain
∇ λ √ −qq µν + √ −q 1 3 δ µ λ T α αβ q βν − T µ λα q αν − T α αλ q µν = 0. (2.90)
Now, it is convenient to introduce the shifted connectioñ
Γ α µν = Γ α µν − 1 3 T λ µλ δ α ν (2.91)
that satisfiesΓ α αµ =Γ α µα and it is invariant under a projective transformation of the original connection Γ α µν , i.e., we have thatΓ α µν →Γ α µν when Γ α µν → Γ α µν + ξ µ δ α ν for an arbitrary ξ µ . This will play a crucial role in the following because it means that the connectionΓ will only determine Γ up to a projective transformation and we will see that it isΓ what is determined by the equations. In terms of the shifted connection, the equations (2.90)
read 1 √ −q ∂ λ √ −qq µν +Γ µ αλ q αν +Γ ν λα q µα −Γ α λα q µν = 0. (2.92)
If we take the two possible traces of these equations and subtract them we find
∂ λ √ −qq [µλ] = 0 (2.93)
and, thus, the antisymmetric part of q µν satisfies a Maxwell-like equation. Another useful relation is obtained by multiplying (2.92) by √ −qq µν to obtain
∂ λ log √ −q =Γ α αλ ,(2.94)
which can then be used in (2.92) to finally write the equations as ∂ λ q µν +Γ µ ρλ q ρν +Γ ν λρ q µρ = 0 (2.95) or, if we multiply by q αµ q νβ , in the equivalent way
∂ λ q αβ −Γ µ βλ q αµ −Γ µ λα q µβ = 0 . (2.96)
These equations will determine the connectionΓ in terms of q µν and, thus, the original connection Γ up to the aforementioned projective mode. We can do a bit better by following the usual procedure to compute the connection in terms of the metric, i.e., we subtract appropriate permutations of indices from (2.96) to write it in the following form:
q (µλ)Γ µ αβ = 1 2 ∂ α q βλ + ∂ β q λα − ∂ λ q αβ + q [αµ]Γ µ βλ + q [µβ]Γ µ λα . (2.97)
This expression is crucial to understand many features of the theories under consideration that will in turn determine many of their properties. Let us stress that we have not considered any simplifying assumption, so our result is completely general. This pays our debt to the meticulous reader, who was promised a more thorough analysis in section 2.5.1. The first thing to notice is that, for a symmetric q µν , the solution for the connectionΓ is nothing but the usual Levi-Civita connection of q µν . Of course, the matrix q µν as defined in (2.76) depends on the Ricci and, thus, on the curvature. As usual, the resolution to this is that q µν can be algebraically solved from the metric field equations (2.87). The connectionΓ is thus solved as the Christoffel symbols of q µν and this is howq earns its denomination of auxiliary metric in the general case. All this reasoning is however based on the assumption that q µν is symmetric, but this is not an outrageous wish to ask and, in fact, it will be nicely granted by the projective invariance. To see this, we can express the definition of q µν in terms of derivatives with respect to the Ricci tensor as follows
√ −qq αν ≡ √ −gg νµ ∂F ∂P µ α = √ −gg νµ ∂F ∂R ρσ ∂R ρσ ∂P µ α = M 2 BI √ −g ∂F ∂R να (2.98)
where we have used the definition ofP to compute its derivative with respect toR. From here, we see that the matrix q µν will inherit the symmetries of the Ricci tensor.
In particular, if only the symmetric part of the Ricci enters the action, then q µν will automatically be symmetric and its Levi-Civita connection will be the solution forΓ. Equivalently, if only the symmetric part of the Ricci appears in the action, only the symmetric part of q µν will contribute to the connection field equations. On the other hand, it is also very easy to see that, in that case, the metric field equations permit to obtain q µν (the number of equations will coincide with the number of components ofq) and, thus, the usual procedure giving the connection as the Levi-Civita of the auxiliary metric q µν is fully consistent. This is an appropriate place to make some remarks on these results. The first one is that the connection has only been obtained up to a projective mode. However, this does not represent a flaw and, in fact, rather the opposite for theories based on a symmetric Ricci. For those theories, there is a projective gauge symmetry that will necessary be responsible for the presence of undetermined modes in the solutions. In other words, the apparent undetermined projective mode will be innocuous and can be removed by a simple gauge fixing. This also applies to the case of the Einstein-Hilbert action and it is precisely the discussion we exposed below (2.32). Hence, for theories with the projective symmetry, the whole resolution of the field equations is consistent and, at least formally, achievable.
So far we have discussed the case when q µν is symmetric by definition. Things can be quite different when this condition is abandoned. In that case, we find problems in the two sets of equations, namely the metric and the connection equations. For the metric equations, we find the trouble already discussed above that, while the metric field equations provide ten independent equations, q µν will in general have sixteen independent components and, therefore, it cannot be fully expressed in terms of the spacetime metric and the matter content. Concerning the connection field equations and its polished expression in (2.97), simply obtainingΓ in terms of the non-symmetric q µν is an arduous task. In fact, in theories with non-symmetric metrics, the solution is usually obtained only perturbatively with respect to the antisymmetric part of the metric [255,129,350,58]. This gives further motivation to consider only theories with the projective symmetry, but theories without it will definitely present a much richer structure. In particular, they will likely contain additional degrees of freedom, among which there could be propagating torsion. Additionally, the results obtained here give support to the simplifying assumption of vanishing torsion upon which the results of section 2.5.1 were obtained.
For the projectively invariant theories, we can also make contact with the previous formalism developed in the case of EiBI theories and the definition of the deformation matrix relating the spacetime and the auxiliary metrics. If we remember the relation between both metrics defined in (2.57) asq =ĝΩ we see that we can re-write (2.79) in a similar form by definingΩ
−1 = 1 detFP ∂F ∂P T . (2.99)
As one would require, the condition (2.73) imposed to recover GR in the low curvatures regime implies thatΩ ≃ ½ in that limit, so that both metrics coincide when |P | ≪ 1. An important derived relation is that, in four dimensions, we have detΩ = detFP . Let us also notice that the Lorentzian signature for the auxiliary metric will be guaranteed as long as the derivativeFP is positive definite or, equivalently, if the deformation matrix Ω is positive definitive. In Born-Infeld inspired theories of gravity, this is usually related to the existence of the square root of a matrix characteristic of those theories, which is then imposed as a condition on physical solutions. In the general case, we will need to impose the deformation matrix be positive definite for physical solutions. This will in turn guarantee that detFP is a real quantity. We can now follow the same procedure as we did with the EiBI theory and obtain an algebraic equation for the deformation matrix by introducingq −1 =Ω −1ĝ−1 into (2.86) and multiplying byĝ to obtain
Ω −1P = 1 M 2 BI M 2 Pl detΩ L G ½ +Tĝ , (2.100)
where we have used thatP andΩ commute and the property 27
g −1 (Ω −1P ) Tĝ =Ω −1P .
This is an algebraic equation for the deformation matrix providedP can be expressed in terms ofΩ by inverting (2.99).
Now, if we use thatq −1R = M 2 BIq −1ĝP = M 2 BIΩ −1P
we finally obtain the differential equations satisfied by the auxiliary metric
R µ ν (q) = 1 M 2 Pl detΩ L G δ µ ν + T µ ν (2.101)
in complete analogy with the equations (2.60) obtained for the EiBI case. This proves our claim that those equations are valid for general theories. Furthermore, the same conclusions drawn there are automatically valid for this more general case. In particular, in the low curvatures regime the deformation matrix is the identity and the Lagrangian is L G ≃ 1 2 M 2 Pl R by construction and, thus, we recover the usual Einstein equations. To end this section, let us extend the discussion on the existence of two frames shown for the Born-Infeld case to the more general theories considered here. In view of the discussions so far about the structure of the theories, it should be clear by now that assuming a projective symmetry would be a wise decision on the grounds of simplicity. Very much like we did for EiBI, let us go to a bi-metric representation of the theory by introducing an auxiliary field Σ µν as follows:
S = 1 2 M 2 Pl M 2 BI d 4 x √ −g F (g µν , Σ µν ) + ∂F ∂Σ µν 1 M 2 BI R (µν) − Σ µν + S M [Ψ, g µν ]
(2.102) We can see that Σ µν can be integrated out by solving its own equation of motion and we recover the original action. We have considered the case with projective invariance to 27 Both properties can be easily proven by assuming that F is an analytic function so thatFP and, as a consequence,Ω are analytic matrix functions ofP . If we have an arbitrary analytic functionF ofP we can expand it asF = n cnP n from where it is trivial to see that it commutes withP . Furthermore, we can also show thatĝF (P )
ĝ −1 =F(ĝPĝ −1 ) =F (P T ) =F T (P ), where we have used thatĝPĝ −1 = M −2 BIRĝ −1 =P T
which is valid whenever the Ricci tensorR is symmetric or, as in our case, when only its symmetric part is considered. From this relation we can obtained the desired property by simply takingF =Ω −1P .
simplify the analysis which in turn implies that Σ µν is symmetric. Now we can introduce a field re-definition as
√ −qq µν = √ −g ∂F ∂Σ µν (2.103)
that can be used to obtain Σ µν = Σ µν (ĝ,q) so that the action can be expressed as
S = 1 2 M 2 Pl d 4 x √ −qq µν R µν + √ −gM 2 BI U (ĝ,q) + S M [Ψ, g µν ] (2.104) with U (ĝ,q) = F − ∂F ∂Σ µν Σ µν . (2.105)
In this action, the spacetime metric g µν appears as an auxiliary field (provided the matter fields are minimally coupled) so it can be integrated out. Its equation is simply
∂U ∂g µν − 1 2 U g µν = 1 M 2 Pl M 2 BI T µν (2.106)
which allows to solve algebraically for g µν in terms of q µν and the matter fields, similarly to the case of Born-Infeld. Thus, the original action can be alternatively expressed as
S = 1 2 M 2 Pl d 4 x √ −qq µν R µν (Γ) +S M [Ψ, q µν ] (2.107)
and we see again that the theory is equivalent to GR but with modified couplings to the matter fields. Hence, the same discussion presented in section 2.6 applies to the more general class of theories considered here. The Born-Infeld frame introduced in that section naturally extends to a more general affine frame within the framework of the general class of theories discussed here. This naturally motivates an extension of the Born-Infeld equivalence principle to a more general affine equivalence principle with the same theoretical and phenomenological consequences, in particular the constraint M BI 10 −1 eV obtained by imposing the absence of anomalous interactions at LHC also applies here. Notice however that some exceptions exist where this argument fails, since we are assuming that (2.105) can be inverted to express Σ µν in terms of g µν and q µν and similarly for (2.106) that allows to integrate out g µν . One important example of theories where this argument is not applicable is the case of f (R) theories. In that case, only the trace of Σ enters (2.103) so it is not possible to invert it and obtain Σ µν = Σ µν (ĝ,q). As it is well-known, in that case it is a better idea to add a scalar field in the Legendre transformation instead of Σ µν . As we showed above, for these theories q µν and g µν are conformally related. After the general considerations discussed in this section, let us turn to considering specific examples of extensions of Born-Infeld gravity corresponding to the classification introduced above.
Class 0
We will classify under this category those theories aiming at modifying GR in the high curvature regime with a Born-Infeld type of modification, but which fail in fulfilling some crucial consistency requirement, like the presence of unavoidable ghost-like instabilities. Into this class will go the first attempts towards Born-Infeld gravity explained in the sections 2.2 and 2.3 that were based on the metric formalism. As extensively discussed there, the higher order field equations for the metric arising in those theories compromise their stability due to the presence of ghosts.
As another example of Born-Infeld inspired gravity theories that would belong to this class we can mention theories consisting of a Born-Infeld sector formulated in the affine approach (similar to the EiBI Lagrangian) supplemented with another sector formulated in the metric formalism. This type of action was already considered by Bañados in [44] and some phenomenological consequences were explored in [319,47,46]. In view of the analysis performed in section 2.7.1, it is clear that these theories are generally plagued by ghost-like instabilities, similarly to the original attempts made in the pure metric formalism. The problem with these theories is precisely the presence of the metric sector. We can repeat the same construction leading to (2.104), but now with the additional sector formulated in the metric formalism. If we take such a sector to consist of an Einstein-Hilbert term for the metric 28 g µν , as it was the case considered in [44], we end up with the equivalent action
S = 1 2 M 2 Pl d 4 x √ −gg µν R µν (g)+ √ −qq µν R µν + √ −gM 2 BI U (ĝ,q) +S M [Ψ, g µν ] , (2.108)
so we have a bi-metric theory where both metrics are coupled through U (ĝ,q). If there was no metric sector explicitly making g µν a propagating field, the spacetime metric could be integrated out and we would be left with only one propagating metric, as we obtained in (2.107). However, having an independent Einstein-Hilbert term for the spacetime metric makes it a propagating field and, thus, the action (2.108) shows that we can no longer integrate the metric g µν out. The result of this is that we have a bi-metric theory where the two metrics are dynamical and interact through the potential U (ĝ,q). Unless the interactions encoded in that potential belong to the class of ghost-free bi-gravity type [134,196], the theory will contain the so-called Boulware-Deser ghost [82] and, therefore, the theory will be unstable. In general, the absence of ghosts will then be guaranteed if the following condition holds
U (ĝ,q) = F − ∂F ∂Σ µν Σ µν = 4 n=0 β n e n ĝ −1q , (2.109)
where the terms in the last sum are the massive gravity and bi-gravity potentials written in terms of the the elementary symmetric polynomials e n defined in (1.26). One can easily check that the EiBI action does not fulfill this condition and, thus, the theory will contain the undesired ghostly mode. A construction with auxiliary fields that somehow connect the EiBI Lagrangian with bi-gravity theories as different branches of the same underlying theory was presented in [324], but our discussion here differs from the one given there. We should notice that this is in fact a general result for theories mixing sectors formulated in the metric and in the affine formalism that go under the name of hybrid theories [96]. We see that the hybrid theories containing the Ricci tensor will either be unstable or equivalent to massive bi-gravity if (2.109) holds. As with simple affine theories, actions built out of the Ricci scalar alone do not fall within this general result since, as pointed out below (2.107), in those cases the construction fails. The way to go for those theories is introducing a scalar auxiliary field that makes the two metrics be conformally related. That explains why we will not eventually obtain two propagating metrics so that those particular hybrid theories will avoid the ghost. This was obtained for the perturbative degrees of freedom around relevant backgrounds in [232] (see also [96]), but we can see from our analysis here that this also extends fully non-linearly.
We can conclude that the presence of ghost-like instabilities is a generic pathology of the theories belonging to the class 0 and represents a serious drawback for their phenomenological consequences. In fact, the very existence of such pathologies could be taken as the defining property of this class.
Class I
We identify this class as the one containing the most extensively studied case in the literature, i.e., the EiBI reviewed in the precedent subsections, as well as its extensions. The most immediate class of extensions of the EiBI gravity is to consider some sort of functional extension. As we have extensively seen above, the fundamental object in EiBI gravity is the determinant
det g µν + 1 M 2 BI R µν (Γ) (2.110)
in terms of which the action is written. One of the reasons to introduce the determinant is to guarantee the diffeomorphisms invariance of the volume element because the determinant of a rank-2 covariant tensor transforms as a scalar density of weight w = −2. Thus, in order to introduce functional extensions of the EiBI theory, it is more convenient to rewrite the action in the following form:
S EiBI = M 2 BI M 2 Pl d 4 x √ −g det ½ +P (2.111) with P µ ν ≡ 1 M 2 BI g µα R αν . (2.112)
From here one can straightforwardly perform functional extensions in different directions that are discussed in the following sections. There can be slightly different versions of the theory depending on whether the connection is assumed symmetric a priori or if only the symmetric part of the Ricci tensor is considered, as we have seen above, and this could also be the origin of differences in the formulation of the theories.
• Arbitrary function of the determinant
It is a common practice in modified gravity to generalise theories by introducing arbitrary functions of the defining quantities as it is done for instance in f (R) or f (R, G) theories where arbitrary functions of the Ricci scalar and/or the Gauss-Bonnet term are considered. Thus, probably the first extension one could think of for the EiBI theory is taking an arbitrary function of the defining determinant. This was done in [268] where the following extension of Born-Infeld was considered 29
S = M 2 BI M 2 Pl d 4 x √ −gf X (2.113)
with X = det(½ +P ). The EiBI theory is recovered for f (X ) = X 1/2 . As usual, the function f should be chosen so that we recover GR at small curvatures. The action in that limit can be obtained by expanding the function around X = 1 so we have
S ≃ M 2 BI M 2 Pl d 4 x √ −gf ′ (1) X − 1 , (2.114)
which only differs from the original EiBI theory by the factor f ′ (1) so we need to impose f ′ (1) = 1 to have the correct limit at low curvatures. The generality introduced by considering an arbitrary function of the determinant can be handled in the usual way by introducing a Legendre transformation with an auxiliary field φ as
S = M 2 BI M 2 Pl d 4 x √ −g f (φ) + f φ X − φ , (2.115)
followed by a field redefinition ϕ = f φ so that the action can be written in the equivalent way
S = M 2 BI M 2 Pl d 4 x √ −g ϕX − V (ϕ) . (2.116)
The field equation of the scalar field imposes the constraint
X = V ,ϕ (2.117)
which can be eventually incorporated in the final form of the equations. The procedure presented above for general theories can be straightforwardly applied to this case and one finds that the auxiliary metric reads
q µν = ϕ √ X g µν + 1 M 2 BI R (µν) (2.118)
while the deformation matrixΩ relating the two metrics is given bŷ
Ω = ϕ √ X ½ +P . (2.119)
After some more manipulations along the lines of the general case depicted in the previous section, one can finally write the equations as
R µ ν (q) = 1 2M 2 Pl ϕ 2 X 3/2 M 2 Pl M 2 BI f (X )δ µ ν + T µ ν ,(2.120)
which, as shown above in (2.101), is the standard form for these theories and will permit the direct applications that will be discussed in detail in the next sections.
• Extension to all the elementary symmetric polynomials
A slightly different way of writing the EiBI action leads to another class of extensions. By commuting the square root and the determinant in (2.111), we can alternatively write the action as
S EiBI = M 2 BI M 2 Pl d 4 x √ −g detM (2.121)
where the matrixM has been defined asM ≡ Ω = ½ +P . Now, since the determinant of a matrix is nothing but the invariant elementary symmetric polynomial of highest degree, the EiBI action rewritten as (2.121) calls for a natural extension including the full series of elementary symmetric polynomials of the fundamental matrixM . This is the path taken in [59] that led to the family of Born-Infeld inspired theories described by the following actions
S GBI = M 2 BI M 2 Pl d 4 x √ −ge 2 (M ) = 1 2! [M ] 2 − [M 2 ] , e 3 (M ) = 1 3! [M ] 3 − 3[M ][M 2 ] + 2[M 3 ] , e 4 (M ) = 1 4! [M ] 4 − 6[M ] 2 [M 2 ] + 8[M ][M 3 ] + 3[M 2 ] 2 − 6[M 4 ] . (2.123)
As commented above, the fourth symmetric polynomial coincides with the determinant i.e. e 4 (M ) = detM so that the β 4 term contributes the usual EiBI Lagrangian. The low curvature limit |g µα R αν | ≪ M 2 BI gives
S ≃ M 2 Pl M 2 BI d 4 x √ −g β 0 + 4β 1 + 6β 2 + 4β 3 + β 4 + 1 2M 2 BI β 1 + 3β 2 + 3β 3 + β 4 g µν R µν (Γ) (2.124)
which coincides with the Einstein-Hilbert action in the Palatini formalism supplemented with a cosmological constant term (which can be cancelled by tuning the parameter β 0 ), provided we impose β 1 + 3β 2 + 3β 3 + β 4 = 1. The projective symmetry always appears here as an accidental symmetry of the low curvature action and it will only be a symmetry of the full theory if the elementary symmetric polynomials are constructed in terms of the symmetric Ricci tensor. Let us now consider the high curvature limit where |g µα R αν | ≫ M 2 BI . This means thatM ≃ P and, therefore, the action in this regime turns into a combination of the elementary symmetric polynomials of P . In the presence of all the polynomials, this regime will be dominated by the fourth one and we will recover an Eddington-like action
S ≃ β 4 M 2 Pl M 2 BI d 4 x det R µν (Γ). (2.125)
In the general case, the Born-Infeld regime will be determined by the highest degree polynomial present in the action. The case of e 2 admits an amusing interpretation since its Born-Infeld regime gives
S =m 2 d 4 x √ −g ĝ −1R − ĝ −1R 2 (2.126)
withm some scale. This theory could even be treated in the metric formalism. Now if we interpret the operation of tracing as a type of averaging, the above action can be interpreted as being the variance of ĝ −1R . Despite its amusing interpretation, its physical viability is dubious since it likely gives rise to observational conflicts and a lack of hyperbolicity in the field equations might. However, these issues should be explored before reaching a definite conclusion. Again, we can apply the machinery developed above for the general case to this particular family of theories with the identification where we have made extensive used of the chain rule and dropped the term with n = 0 because that is just a cosmological constant term. Now, we will introduce the notation E k n = ∂e n /∂[M k ], whose explicit form is given by
E k n = e 0 0 0 0 e 1 − e 0 2 0 0 e 2 − e 1 2 e 0 3 0 e 3 − e 2 2 e 1 3 − e 0 4 ,(2.129)
and use that ∂[M k ]/∂M = k M k−1 T and ∂M /∂P = 1 2 M −1 T to finally obtain the auxiliary metric as given in (2.77) for the present case:
√ −qq −1 = √ −g ĝ −1 ∂F ∂P T = √ −g 4 n=1 β n 4 k=1 E k nM k−2 ĝ −1 . (2.130)
This is precisely the result found in [59], where the sums in the brackets corresponds to the matrixŴ defined in that reference. Since the sum over k runs from 1 to 4, the right hand side of (2.130) will contain powers ofM from −1 to 2. This allows to re-write (2.130) in the more useful form
√ −qq −1 = √ −g f 1M −1 + f 2 ½ + f 3M + f 4M 2 ĝ −1 (2.131) with f 1 = β 1 e 0 + β 2 e 1 + β 3 e 2 + β 4 e 3 (2.132) f 2 = −(β 2 e 0 + β 3 e 1 + β 4 e 2 ) (2.133) f 3 = β 3 e 0 + β 4 e 1 (2.134) f 4 = −β 4 e 0 . (2.135)
From these expressions one can now straightforward adapt the general formalism for this family of theories and obtain all the relevant equations, which, of course, coincide with those in [59]. As a particularly simple case, we can take a theory containing only e 1 so that the action reads
S Min = M 2 BI M 2 Pl d 4 x √ −gTr ½ + M −2 BIĝ −1R − ½ . (2.136)
This is the model that was studied in more detail in [59] and subsequently used in [61] to develop an inflationary scenario. For that case, we have that f 2 = f 3 = f 4 = 0 and f 1 = β 1 is a constant, which is set to 1 in order to recover GR at low curvatures. In this very simple case, we can easily compute the deformation matrix from (2.99), which yieldŝ
Ω = 1 detMM . (2.137)
If we use this relation together withP =M 2 − ½ obtained from the definition ofM , the equation for the deformation matrix given in (2.100) can be written in terms ofM aŝ
M −1 −M − Tr M − ½ ½ = 1 M 2 BI M 2 PlTĝ (2.138)
which exactly coincides with the equation found in [59]. This equation will give the matrix M in terms of the matter sector and then one can follow the common procedure to solve the equations. This will be explicitly done in section 5.3, where the cosmology of this model will be studied.
Class II
There is a second class of extensions of the EiBI theories that makes use of additional geometrical objects. Let us remind that the original EiBI theory only utilizes the Ricci tensor and the metric and its natural arena is a non-Riemannian geometry. The metric affine formulation of the theory implies the presence of a completely independent connection and its associated curvature encoded in the Riemann tensor R α βµν . For this general Riemann, we can take three independent traces, namely: the Ricci R βν = R α βαν , the homothetic tensor Q µν = R α αµν and the co-Ricci P α µ = g βν R α βµν . The traces of these three objects are all the same and give the Ricci scalar R = g µν R µν . We can see that the EiBI theory only makes use of the Ricci tensor, but a much larger variety is possible thanks to the rich geometrical structure at our disposal. For instance, the determinantal form of EiBI can be extended to include an arbitrary combination of the three different traces of the Riemann tensor so we could consider actions of the type
S = M 2 Pl M 2 BI d 4 x − det a 1 g µν + 1 M 2 BI a 2 R µν + a 3 Q µν + a 4 P µν (2.139)
where a i can be arbitrary scalar functions of curvature invariants. In the simplest case we could take a i = a i (R), but other scalars like Gauss-Bonnet combinations or R µν R µν could also be envisaged. Obviously, here we encounter once again a similar obstacle as in the Deser and Gibbons construction discussed in section 2.2 (although this time avoiding the ghost problem), namely the lack of a guiding principle. Thus, very much like in that case, one can foreshow that any gravitational theory (except for some singular cases) can be recast in the above form by appropriately tuning the free functions a i . The EiBI theory corresponds to possibly the simplest among the possible theories described by the action (2.139). Let us stress that we always remain within the Born-Infeld spirit, so we leave out here well-known theories, like those written in terms of Lovelock invariants, rewritten in a way that resemble the characteristic square root structure of Born-Infeld theories.
Let us notice that we can consider even more general actions by including generalised determinants for the Riemann tensor itself. Faced with the obstruction of lacking some motivation to select extensions of EiBI within the Class II, people have resorted to the always welcomed principle of simplicity. In this case, it means that extensions along the lines depicted here have predominantly resorted to adding new terms only containing the Ricci scalar. This has been considered in two fashions, either by writing new R-dependent terms outside the EiBI action, as in the Born-Infeld plus f (R) models introduced in [245], or by modifying the determinantal structure with only R-dependent terms, as was considered in the appendix of that same work [245] and in [106] for a specific case.
Another possibility that differs more profoundly from the one sketched so far is to consider other geometrical frameworks. At this respect, an interesting class of theories formulated on a Weitzenböck space was introduced in [166,167]. It is well-known that GR admits a formulation in a Weitzenböck space, where the connection is constrained to have vanishing curvature and all the gravitational effects are encoded in the torsion tensor T λ µν . This construction goes under the name of Teleparallel Equivalent of General Relativity (TEGR) and it has been used as the starting point of some modifications of gravity, among them some Born-Infeld inspired gravity theories that are of interest for us (see section 4.6.4 for applications of this theory on black holes). An extensive and comprehensive review on TEGR can be found in [12]. Even though TEGR is GR in disguise, this mask shows an interesting face for GR as a gauge theory of the inhomogeneous part of the Poincaré group where the vierbeins are precisely the gauge fields of translations. Thus, TEGR provides a very appealing starting point for Born-Infeld modifications of gravity that deserves to be explored.
After discussing some of the different approaches that can be taken to obtain Born-Infeld inspired theories of gravity within the Class II, let us briefly review some specific examples.
• Born-Infeld plus f (R)
A simple extension within this class is to combine the Born-Infeld action with the wellknown f (R) theories in the metric-affine formalism, as considered in [245,244,246,151]. The resulting action adapted to our notation is given by
S = 1 2 M 2 Pl M 2 BI d 4 x 2 − det g µν + α M 2 BI R (µν) + √ −gf (R) = 1 2 M 2 Pl M 2 BI d 4 x √ −g 2 det ½ + αP + f ([P ]) (2.140)
with α some dimensionless constant. Since the small curvature limit of the EiBI sector in the above action already gives the Einstein-Hilbert term, we need to impose α+f [P ] (0) = 0 to recover GR at low curvatures. The above action is simply a combination of the EiBI and the f (R) and, as such, the corresponding solutions are expected to interpolate between these two cases. The general formulae obtained in 2.7.1 can be straightforwardly applied to this case. For instance, the definition of the auxiliary metric given in (2.76) yields
√ −qq −1 = − det ĝ + α M 2 BIR ĝ + α M 2 BIR −1 + √ −gf [P ]ĝ −1 , (2.141)
which coincides with the result found in the literature. A second possibility to extend EiBI gravity by including the Ricci scalar is to include it in the determinantal structure. This was considered in an appendix in [245] where the authors considered an action of the form
S = M 2 Pl M 2 BI d 4 x − det 1 + f (R) g µν + α M 2 BI R (µν) (2.142)
as another example of the addition of an f (R) piece to the EiBI action. In this case, recovering GR at low curvatures requires to have 4M 2 BI f R (0) + α = 0. In [106] this path was considered in more detail and the authors explored the cosmology of the following specific case:
S = M 2 Pl M 2 BI d 4 x − det 1 + β M 2 BI R g µν + α M 2 BI R (µν) (2.143)
with α and β some constants satisfying α + 4β = 0 in order to recover GR in the limit of small curvatures.
• Born-Infeld actions in Weitzenböck spaces.
Another example of extensions that we classify within the Class II, but which take a different direction, are those based on the teleparallel equivalent of GR. In this description of GR, one makes a fundamental use of the vierbein language and Weitzenböck spaces, characterised by having a curvature-free connection so that the torsion is the only relevant object. In terms of the vierbein e a µ and its inverse e µ a , the connection is given by Γ λ µν = e λ a ∂ ν e a µ so that the torsion tensor reads T λ µν = e λ a (∂ ν e a µ − ∂ ν e a µ ). From the torsion we can built a useful quantity called the super-potential and that is given by
S ρ µν = 1 4 T νµ ρ + T νµ ρ − T ρ µν + 1 2 δ µ ρ T αν α − δ ν ρ T αµ α . (2.144)
With this object, we can construct the Weitzenböck invariant defined as
T = S ρ µν T ρ µν . (2.145)
Then, the so-called Teleparallel Equivalent of General Relativity is described by the action
S TEGR = 1 2 M 2 Pl d 4 xeT (2.146)
where e = det e a µ . That this action is equivalent to GR can be seen from the fact that T for the Weitzenböck connection differs from the Ricci scalar of GR by a total divergence, so that both theories give rise to the same equations of motion. The TEGR, however, serves as an alternative starting point to develop modifications of gravityà la Born-Infeld. This was pursued in [166,167,170], where the authors considered a general expression of the form
S BITG = M 2 Pl M 2 BI d 4 x g µν + 1 M 2 BI α 1 S µ λρ T νλρ + α 2 S λµ ρ T λ νρ + α 3 g µν T − λ √ −g ,
(2.147) with α i some parameters that must satisfy α 1 + α 2 + 4α 3 = 1 in order to recover (2.146) in the limit of T ≪ M 2 BI . Similarly to the case of EiBI gravity, the parameter λ controls the presence of a cosmological constant. Unlike the proposals discussed so far, this class of theories must be formulated with the vierbein being the fundamental fields and the general framework presented in section 2.7.1 cannot be applied to this case. The Born-Infeld extensions along these lines are substantially less explored than those based on the EiBI formulation. However, the Born-Infeld theories based on the TEGR also show interesting features and, furthermore, could be seen to be closer to Born-Infeld electromagnetism, since TEGR can be seen as a gauge theory where the vierbeins play the role of gauge fields associated to the translations group. One might be concerned however with the fact that, similarly to what happens in the models belonging to the class 0 formulated in the metric formalism, generic theories described by the action (2.147) will introduce instabilities. In particular, the loss of local Lorentz symmetry when going from TEGR to the action (2.147) will likely introduce additional degrees of freedom. At the time of writing this review, it lacks a full analysis of the fields content and their stability around relevant backgrounds of those theories.
Class III
In this category we will include theories based on the Born-Infed structure but which make use of additional fields. The most natural example of these theories would be to combine EiBI gravity with its electromagnetic predecessor or with a Dirac-Born-Infeld scalar field φ, resulting in actions of the form
S = M 2 Pl M 2 BI d 4 x − det g µν + 1 M 2 BI b 1 R µν + b 2 F µν + b 3 ∂ µ φ∂ µ φ . (2.148)
This type of actions are perhaps the most natural combination of Born-Infeld actions for gravity, electromagnetism and/or scalar fields. Already Vollick considered a combination of this type in [357]. A different approach is to simply add the corresponding Lagrangians and consider actions of the form
S = M 2 Pl M 2 BI d 4 x − det g µν + 1 M 2 BI R µν + c 1 − det g µν + 1 M 2 BI F µν +c 2 − det g µν + 1 M 4 BI ∂ µ φ∂ µ φ . (2.149)
This was considered for instance in [218,219]. More general actions that belong to this class can be obtained from the EiBI action formulated in higher dimensions after a dimensional reduction, for instance by compactifying one extra dimension as done in [159].
Class IV
Besides the extensions or variations around the EiBI theory discussed so far, there are other alternatives that make use of some of the ideas characteristic of Born-Infeld theories, but they could be classified as belonging to other classes of theories. Within this category we could mention some of the early attempts to build a Born-Infeld inspired gravity theory in the metric formalism already discussed in section 2.3. Among them, we could cite here specific f (R) models with a square root or some other bounded function (see also [235,236]). Although the square root structure introduces some resemblance with Born-Infeld theories, those models could be classified as belonging to the f (R) class of theories. The same would apply to theories involving not only the Ricci scalar, but also the higher order Lovelock invariants, in particular, the Gauss-Bonnet term G which is the only relevant one in four dimensions besides the Ricci scalar. It is possible to construct theories of the type f (R, G) that incorporate some square root structure, as it is considered in 30 [122]. These theories would then belong to the general class of f (R, G) theories. The same would apply to theories based on the teleparallel equivalent of GR as for instance in [165] that can be classified as belonging to the f (T ) extensions of teleparallel theories [92]. We will also include in this class theories making use of the determinantal structure characteristic of Born-Infeld theories, but which reduce to other types of theories, either completely or in its regime of validity. This is for instance the case of [260] that secretly describes Lovelock gravity, as discussed in section 2.2.
Final remarks
In this section we have provided the reader with a general framework for the study of Born-Infeld theories, as well as an overview of the different classes of these theories existing in the literature. We have started by briefly reviewing Born-Infeld electromagnetism and surveyed the attempts to adapt the same ideas to the case of gravity as potential mechanisms to regularise the divergences appearing in GR. The first attempts formulated in a metric formalism faced serious shortcomings due to the presence of ghosts. In order to bypass these pathologies, one can introduce higher order corrections to remove the ghosts at a given order, but the large freedom existing in the choice of the counter-terms renders the procedure unappealing. It is fair to say that, to date, there is no compelling theory free from ghosts that comply with the Born-Infeld philosophy in the metric formalism. A step forward was given when considering Born-Infeld types of actions in the affine approach. In that case, the ghost is not present from the beginning and the theory can really be regarded as a proper Born-Infeld theory of gravity, meaning that it modifies the gravitational interaction at high curvatures where a natural bound appears. We would like to remark once again that other attempts that resemble Born-Infeld theories actually contain additional degrees of freedom so that they deviate from what we consider should be the spirit of Born-Infeld theories. At this respect, we have introduced a classification of the different Born-Infeld inspired theories of gravity attending to their closeness to the Born-Infeld realm.
Since the most extensively explored theory within the framework of Born-Infeld extensions of gravity is the EiBI model, we have devoted a substantial effort to showing in detail its main properties, although we have later shown that the same features are shared by a much larger class of theories. We have provided a detailed derivation of the field equations and highlighted the importance of the projective symmetry in the construction of the theories. In particular, we have seen that theories with that invariance can be fully solved in terms of an auxiliary metric and the torsion only enters as an irrelevant projective mode (under some assumptions on the matter sector). Even though this auxiliary metric makes its first appearance as an object allowing to solve the connection, we have seen that it carries physical relevance. This was apparent when we discussed the two frames existing in these theories. From there, we clearly saw that, while the spacetime metric determines the causal structure of matter fields, the auxiliary metric determines the causal structure of the gravitational waves. This in turn implies that while photons travel along null geodesics of the spacetime metric, gravitons move along null geodesics of the auxiliary metric and, thus, even if both particles are massless, their motion will differ in regions where the curvature is large as compared to the Born-Infeld scale.
An issue that remains within the affine formulation of Born-Infeld gravity is the lack of clear guiding principles to select a unique family of theories. Born and Infeld followed a symmetry principle that allowed them to single out their non-linear electrodynamics, which was later shown to have a number of remarkable features and it was even related to string theory. The same is currently lacking for Born-Infeld inspired theories of gravity. In fact, modifications and extensions of Born-Infeld gravity have flourished in several directions. By studying some of the proposed extensions, one can convince oneself that some families of theories seem to lead to much simpler equations than others. While this simplicity principle can be useful to explore the physical consequences of these families of theories, a more profound and appealing principle would be desired.
Astrophysics
A generic feature of extended theories of gravity in which the connection is regarded as independent of the metric (Palatini approach) is the emergence of a dependence of the metric on the local stress-energy densities. This property was soon noticed in the case of f (R) theories [272] and its extensions with Ricci-squared R µν R µν terms [283,275], and is also present in Born-Infield inspired theories of gravity and its known generalisations, see section 2.5. This local dependence on the matter fields may at first appear as something exotic but is such a basic and fundamental issue in metric-affine theories of gravity that it must be properly understood in order to handle these theories correctly and properly define strategies to test their viability.
In this section we will explore situations of astrophysical interest in which the dependence of the metric on the local densities of energy and momentum manifests itself very clearly. In fact, numerous observables of stellar objects are very sensitive to the physical processes taking place in their interiors, whose properties strongly depend on the local density. This is the case, for instance, of the mass-radius relation, the mechanisms of energy transport, the seismic properties of stars, the type and intensity of neutrino fluxes, the speed of sound profile of acoustic waves in the sun, the potential existence of phase transitions in terms of ordered (crystalline) and superfluid phases inside neutron stars, the deconfinement of quarks or the mechanisms of generation of very large magnetic fields. For some reviews on these topics see e.g. [203,184]. This dependence on the local density can thus be used to efficiently test some aspects of this type of modified theories of gravity but it may also lead to unexpected subtleties. In particular, we will see that the fluid approximation and some models regularly used in the context of GR must be handled with care or conveniently adapted in order to avoid fictitious forces induced by the averaging procedure employed in the transition from the microscopic description to the continuous limit. This will be particularly relevant in the discussion of the outer boundaries of some stelar models both in the relativistic and in the non-relativistic limit.
We will begin this section by considering the weak-field, slow-motion limit of the Eddington-inspired Born-Infield (EiBI) theory first introduced in section 2.4 31 , and its implications for non-relativistic stars. This will allow us to visualise in a very simple way where the subtleties of the fluid approximation may arise, which will help us better understand the peculiarities of these theories and identify situations in which an improved description of the matter sector may be necessary in order to construct realistic models. We will then move to consider relativistic stars, their structure, and their observational properties.
A word on the notation of this section
For operational convenience and to make contact with existing literature, both in this section and in the black holes section 4, we shall redefine part of the notation employed in section 2.4 and redefine Born-Infield mass as M BI = 1/ǫ and reintroduce Einstein's constant in the action via M Pl = 1/(8πG) = 1/κ 2 . This way, by dimensional consistency ǫ has dimensions of length squared, while the Einstein-Hilbert action of GR reads
S GR = 1 2κ 2 d 4 x √ −gR.
Newtonian limit and fluid approximation 3.1.1. The modified Poisson equation
To better visualise the local dependence of the spacetime metric on the stress-energy densities, it is useful to study the weak field, non-relativistic limit of EiBI theory given by Eq. (2.33). For this theory one finds that, to leading order in the EiBI parameter ǫ, the right-hand side of the Ricci tensor on the field equations (2.59) takes the form
R µν (q) ≈ κ 2 T µν − 1 2 g µν T + ǫκ 4 S µν − S 4 g µν ,(3.1)
where S µν = T α µ T αν − 1 2 T T µν , while T and S are the trace of T µν and S µν , respectively. This equation indicates that the deviation of the auxiliary metric q µν from the Minkowski metric will be determined by the total amount of energy-momentum appearing on the right-hand side of this equation. For weak sources, therefore, q µν will be given by the Minkowski metric plus corrections which depend on integrals of the elements on the righthand side. Now, since g µν is related to q µν via the deformation matrixΩ as defined in Eq.(2.57), which in the low density limit is given by
Ω µ ν ≈ δ µ ν + ǫκ 2 T µ ν − T 2 δ µ ν (3.2)
the relation between the perturbations in q µν ≈ η µν + t µν and g µν ≈ η µν + h µν turns out to be
t µν = h µν + ǫκ 2 T µν − 1 2 η µν T . (3.3)
The left-hand side of (3.1), once the standard gauge choice
∂ λ (h λ µ − h 2 δ λ µ ) = 0 is made, leads to R µν (η + t) ≈ − 1 2 t µν ,
where is the flat d'Alembertian. For weak sources, therefore, the above equations lead to
− 1 2 t µν = κ 2 T µν − 1 2 η µν T , (3.4)
where only the leading order contributions on the right-hand side have been kept. For the (weak field and slow-motion) Newtonian limit we just focus on the t 00 -component assuming, as usual, a pressureless fluid with T µν ≈ ρu µ u ν , where ρ is the energy density of the fluid. Defining t 00 = −2φ N and h 00 = −2φ N , such thatφ N = φ N − ǫκ 2 4 ρ, the above equation in the non-relativistic limit can be written as
∇ 2 φ N = κ 2 2 ρ + ǫκ 2 4 ∇ 2 ρ , (3.5)
which admits a general solution of the form
φ N (t, x) = κ 2 8π d 3 x ′ ρ(t, x ′ ) | x − x ′ | + ǫκ 2 4 ρ(t, x) . (3.6)
The first term in (3.5) represents the standard Newtonian source, while the second one corresponds to a new source of gravity that involves derivatives of the matter density. Whenever those gradients become important, significant deviations from Newtonian gravity will arise. To estimate the scale at which such deviations occur and the kind of effects one may find, it is illuminating to take the Fourier transform of (3.5) [29], which leads to
k 2φ N ( k) = κ 2 2 ǫk 2 2 − 1 ρ( k) ,(3.7)
whereφ N ( k) andρ( k) are the momentum space counterparts of φ N and ρ. It is clear from this expression that in the GR limit, ǫ → 0, the right-hand side of (3.7) is always negative. For any finite (but positive) ǫ, however, one finds a scale k J = 2/ǫ beyond which the right-hand side of (3.7) flips its sign, thus leading to repulsive rather than attractive gravity. This allows us to interpret the effective Jeans length λ J = 2π/k J as the critical scale below which the collapse of pressureless dust is not possible due to the dominance of repulsive interactions. One obvious consequence of this is that for ǫ < 0 nothing seems to prevent the possibility of complete gravitational collapse for pressureless fluids (within this approximation), which is a significant change of behavior as compared to the case ǫ > 0. Another important consequence that can be derived from the ǫ < 0 case is that the growth of the intensity of the gravitational field at small scales may lead to equality between electric and gravitational forces at length scales ∼ 10 −15 − 10 −14 m unless 8πGǫ < 10 −3 m 5 s −2 kg −1 [29]. Our current understanding of nuclear and particle physics, therefore, requires ǫ < 6 × 10 5 m 2 or, equivalently, √ ǫ 800 m.
Non-relativistic fluids
The application of the modified Poisson equation (3.5) to the study of non-relativistic self-gravitating fluids was first carried out in [294] (see also [295] for more details and some clarifications). For fluids in hydrostatic equilibrium, one must supplement the modified Poisson equation (3.5) with the fluid conservation equation ∇ µ T µ ν = 0 in the appropriate limit. For spherically symmetric systems, the conservation equation boils down to dp/dr = −(ρ + p)Γ r tr (where p is the pressure of the fluid), and for weak sources Γ r tr ≈ − 1 2 ∂ r h tt leads to
p r = − κ 2 8π m(r)ρ r 2 − ǫκ 2 4 ρρ r ,(3.8)
where p r ≡ dp/dr, ρ r ≡ dρ/dr, m(r) = 4π r dxρ(x)x 2 , and an equation of state p = p(ρ) must be specified. An immediate solution of this equation corresponds to the case in which p(r) = 0. Unlike Newtonian gravity, where pressureless solutions cannot be in hydrostatic equilibrium, the above equation yields a nontrivial solution when ǫ > 0. This case simply requires solving the equation m(r) = −2πǫr 2 ρ r . Applying on this equation a radial derivative, it can be cast as the Lane-Emden equation of a polytrope with index n = 1 (recall that polytropes have equation of state p(ρ) = Kρ 1+ 1 n , where K and n are real positive constants, and n is the so-called polytropic index). If the regularity condition ρ(0) = ρ c is imposed at the centre to get rid of the 1/r term, the solution to this equation takes the form
ρ(r) = ρ c sin(k J r) k J r . (3.9)
As is standard in the study of polytropes, the authors in [294,295] restricted the range of validity of this solution to the interval r ∈ [0, π/k J ] to avoid the presence of a negative energy density beyond the first zero at r = π/k J (see Fig.2). Though this restriction is natural and harmless in the usual Newtonian theory, the fact is that it forces a discontinuity in ρ r at r = π/k J , thus causing a divergence on the right-hand side of (3.5). 3.1.3. The issue with the matter profiles at a star surface The example above illustrates an important property of this type of theories of gravity, namely, that the matter fields must satisfy certain differentiability conditions that are not necessary in the context of GR. The matter/energy profiles must be continuous and differentiable up to some degree. This requirement may certainly be inconvenient, because it forces us to pay more attention to the modeling of our energy sources in certain applications, but is not a fundamental problem because matter and radiation are ultimately described in terms of quantum fields, which are sufficiently smooth to comply with the differentiability requirements of these theories. Therefore, the solution (3.9) admits two possible interpretations: 1) that we are dealing with an unconventional fluid or 2) that an improved description of the matter fields (with a different fluid or even beyond the fluid approximation) is necessary near the surface at k = π/k J to avoid undesired or fictitious unphysical effects.
• Regarding option 1), note that in the transition from the (relativistic) weak-field approximation (3.4) to (3.5) we (implicitly) assumed that the stress-energy tensor T µν of the matter fields could be averaged to yield that of a perfect fluid without causing any harm to the theory. In this process, the fluid we had in mind was some distribution of localised particles (or wavepackets) such that when averaged over a certain scale should yield a continuum distribution characterisable by the T µν of a perfect fluid. In particular we expected a positive definite density function ρ(x), which turns out to be in conflict with our solution (3.9) beyond r = π/k J . Our microscopic interpretation of the fluid, therefore, does not fit well with the predictions of this theory, which indicates that we are dealing with an unconventional matter source. Note in this sense that the authors in [294,295] argued in favor of this solution representing some kind of dark matter, which might give plausibility to this result. The effects on the galactic metric of a dark matter density profile of this kind has been studied in detail in [193].
• Regarding option 2), if the fluid is interpreted as made out of standard particles, an improved microscopic description of those particles should be considered near the outer boundary (where the density is close to zero) to get a smooth transition to the exterior region in the neighborhood of r = π/k J . Thus, a refinement of the physics near the surface is necessary to build a complete solution. As mentioned above, this might be inconvenient but is not a fundamental problem. In fact, as shown in [29], different averaging procedures in the transit to the continuum fluid approximation may lead to different (fictitious) acceleration fields associated to the specific weight functions employed in the averaging. The emergence of negative densities in the outer regions of these solutions can thus be interpreted as a manifestation of fictitious effects which should be regarded as unphysical and avoidable by an improved description.
The view that one should go beyond the fluid approximation or consider a suitable transition thick shell in the description of the surface region is further reinforced by the analysis presented in [295] regarding the process of dust collapse. Starting with generic static profiles, it was found that the fate of the system is to reach a universal configuration which oscillates around the pressureless solution (3.9) with a period that coincides with the fundamental mode of proper oscillations of the pressureless case. This means that the configurations provided by (3.9) are not a fine-tuned solution of an exotic matter field but, rather, they are a universal, regular final state for the collapse of reasonable matter sources. The role of the EiBI dynamics is, clearly, to stabilise the object against collapse by generating repulsive gravitational forces at short scales. The problems arising on the surface can be regarded as artifacts of the particular fluid approximation considered.
Limitations and improvements of the polytropic description
Similar problems affecting the exterior boundary of some polytropic stellar models were also found in [296]. Due to the divergence of derivatives of the energy density with respect to the pressure as p → 0 near the surface, quantities such as the Ricci curvature scalar diverge (this also happens in the Newtonian model above if one imposes a discontinuity in ρ r at the surface). This occurs, in particular, for polytropic indices γ = 1 + 1 n > 3/2, which include the case of a gas of degenerate non-relativistic electrons (γ = 5/3) or the case γ = 10/3 used to model the atmosphere of white dwarfs. This result led to claim that these divergences could not even be cured by abandoning the fluid approximation, because a microscopic description of the matter sources would increase the differential order of the field equations in the matter sector, thus making the curvature even more sensitive to sharp variations in the matter fields [296].
A number of objections can be presented to the pessimistic view of [296]. Firstly, the claim that a microscopic description cannot cure these problems was just a conjecture which has never been explicitly proven. In fact, the accumulated evidence so far goes in the opposite direction. For instance, self-gravitating solutions of isolated charged particles in the EiBI theory do not show any pathologies neither at high nor at low curvatures, see black holes section 4.5. If individual particles are well behaved, it is difficult to conceive how a collection of them (a fluid) could develop pathologies in the low density regime, where the interparticle separations are large and the isolated particle description is better justified. Similar results are found in the case of self-gravitating scalar fields, which possess a solitonic structure compatible with the idea of isolated neutral particles [? ]. Since the microscopic constituents are individually well behaved, the curvature divergences on the surface of polytropes are likely to be a manifestation of fictitious accelerations induced by the continuum approximation [29]. Secondly, a careful analysis of the validity of the polytropic equation of state near regions of divergent curvature was carried out in [230]. The idea was not to estimate the corrections due to finite temperature or electromagnetic repulsion between charged particles, as is necessary in realistic models to properly account for the opacities in stellar atmospheres, but to explore how the microscopic definition of pressure could be affected near curvature divergences. By analysing the geodesic deviation equation (4.132), the frequency of the interactions between a particle and a nearby (fictitious) wall 32 was found to increase with respect to the corresponding statistical estimate in flat space-time. This represents an additional pressure which changes the effective equation of state for the case 3/2 < γ < 2 and avoids the original curvature divergence. For the case 2 < γ < 3, which is also problematic, it is found that the fluid is repelled from the surface. It is then argued that such fluids would not be appropriate to describe the surface and that some other type of matter should be necessary. The conclusion is that the fluid reacts as the curvature grows on the surface and that an improved description of the matter there is necessary.
We thus see that the fluid approximation and/or the modeling of certain objects in the EiBI theory of gravity may require some refinements to avoid unphysical effects that arise at the outer boundaries of some solutions. This occurs when the derivatives of the matter density diverge too rapidly as the pressure goes to zero or when the matter profile and its derivatives are abruptly set to zero at some point in order to match with the external (idealised) Schwarzschild solution. By smoothing the behavior of the matter profiles, these problems can, in principle, be overcome. Though this is certainly an inconvenience, it is not that far from what realistic models require. In fact, in order to qualitatively and quantitatively understand numerous observational features of neutron stars, such as their electromagnetic spectra, envelope composition, X-ray bursts, surface temperature profiles, etc, it is not only necessary, it is essential, to carefully describe the microphysics of the outer layers. Some of these layers are very thin as compared to the radius of the star, with a height of ∼ 0.1 − 10 cm and density, ρ ∼ 10 −2 − 10 3 g/cm 3 [209] in the 32 Note that a particle and a wall are necessary to define the pressure microscopically.
photosphere, and densities always below 10 10 g/cm 2 on the outer 10 4 cm of the envelope. The composition of this region is dominated by a gas of (partially) ionized atoms and electrons plus radiation, with the electron equation of state transitioning from an ideal to a degenerate gas as one goes deeper into the star [105,291], which has a crucial effect on the efficiency of the different energy transport mechanisms and, thus, dramatically affects the observable features of the star [310]. We thus see that in these layers, finite temperature, radiation fields, chemical composition, electromagnetic repulsion, magnetic fields, etc, induce significant deviations from the basic polytropic equations of state [231], which are nonetheless very useful to estimate the gross properties of these objects. Though models with this level of refinement have not been yet constructed in the EiBI gravity scenario, as we will see below, the evidence so far indicates that there is no fundamental reason to believe they are not possible.
Non-relativistic stars
From the discussion in the previous section, it is now clear that the external boundary of stars should be modeled in such a way that the matter and pressure as well as its first and second-order derivatives should smoothly vanish to guarantee a correct matching with the exterior empty solution. This refinement should be done if one is really interested in understanding observational features of the models such as electromagnetic spectra, but can be overlooked in situations in which only structural aspects are important. In this sense, the standard approach in which the stellar surface is identified as the region where the pressure is sufficiently low can be retained as valid, as long as one accepts that a thin transition shell should be added to correctly complete the model.
Having understood the peculiarities that the matter profiles should satisfy on the outer boundaries of stars, we now focus on the information that stellar models can provide to test the viability of EiBI gravity and constrain its parameters. The results of [295] establish a limitation for the existence of polytropic solutions with regular boundary condition at the centre, ρ ≈ ρ c + ρ 2 r 2 , which requires
ǫκ 2 > −4K 1 + 1 n ρ −1+ 1 n c .
(3.10)
The reason for this bound in static configurations is related to the monotonicity of ρ(r), which requires ρ 2 < 0. An expansion of (3.8) around the centre puts forward that if the bound (3.10) is not satisfied, then ρ 2 > 0. Going beyond polytropic models, a non-rotating, zero temperature white dwarf model with parametric equation of state
p(x) = πm 4 e c 5 µ e m P 3h 3 x(2x 2 − 3) 1 + x 2 + 3 sinh −1 x , (3.11) ρ(x) = 8πm 3 e c 3 µ e m P 3h 3 x 3 ,(3.12)
was studied for different values of ǫ [295]. It was found that for ǫ > 0, the mass of these objects is not limited by the Chandrasekhar bound M ≈ 1.4M ⊙ . It turns out that the mass can be arbitrarily large while the radius tends to a minimum value which scales as √ ǫ. In the relativistic version of these objects, however, an upper bound for the mass does appear, though it can be much larger than in GR (see Fig.3).
Solar physics constraints
A closer confrontation with observations is certainly possible by considering the effects of the modified Poisson equation on the properties of the Sun [103]. Since the hydrostatic equilibrium and energy transport ultimately depend on this equation, any correction would have an impact on the thermal balance and temperature profile inside the star, which can leave observable traces. In fact, neutrino fluxes are very sensitive to the temperature profile inside the Sun [36,349]. An increase or decrease of the innermost conditions due to a modified Poisson equation will necessarily leave a trace on the amounts of emitted neutrinos, which are relatively well understood observationally. Something similar happens with helioseismic data, which provide very accurate information on the solar acoustic modes, the sound speed profile, and the depth of the convective envelope, see e.g. [116] for a review. In order to extract the necessary information to use solar neutrinos and helioseismic data to test EiBI gravity, the hydrostatic equilibrium equation (3.8) and the continuity equation, dm/dr = 4πr 2 ρ(r), must be supplemented with the conservation of thermal energy equation
dL dm = q(r) − r ds dt , (3.13)
where q(r) represents the rate of heating from nuclear reactions and s is the entropy per unit mass [119], plus the corresponding equation for the convective energy transport, which takes the form
dT dm = − Gm(r) 4πr 4 + ǫκ 2 ρ 4 dρ dm T p τ .
(3.14)
Here τ ≡ d log T /d log p is the temperature gradient, which for adiabatic changes becomes τ = (Γ 2 − 1)/Γ 2 , where Γ 2 is one of the adiabatic exponents [362]. In the radiative zone, the transport energy equation is unmodified
dT dm = − 3λ R 16σT 3 L 16π 2 r 4 , (3.15)
where λ R is the Rosseland mean opacity and σ the Boltzmann constant. Implementing the above equations in CESAM [290], a numerical code for stellar structure and evolution, the authors of [103] constructed a number of models able to fit the solar properties with an accuracy of 10 −5 in the interval −0.032GR 2
⊙ < ǫκ 2 < 0.02GR 2 ⊙ .
For smaller values of ǫκ 2 , no equilibrium stars were found, in agreement with the constraint (3.10) for polytropic models. For ǫκ 2 > 0.02GR 2 ⊙ , solutions do exist but are unable to match simultaneously the observed values for age, radius, mass, luminosity and metallicity of the Sun.
Qualitatively, models with ǫ > 0 show lower central density and temperature than in GR (ǫ = 0), whereas for ǫ < 0 those magnitudes grow due to the larger attractiveness of the modified potential. An increment in the central density and temperature imply a raise in the thermonuclear reactions, which must be followed by a modification in the neutrino emission. In the inner 10% radius, the pp-III chain produces high-energy neutrinos associated to the generation of 8 B with an intensity that scales as φ8 B ∝ T 18 c . This flux is currently measured with high precision by neutrino telescopes: (5.046 ± 0.16) × 10 6 cm −2 s −1 . From the numerics one observes a decay in the neutrino flux for ǫ > 0 and a growth for ǫ < 0, such that with the current data it is possible to conclude that ǫκ 2 −0.024GR 2 ⊙ is ruled out.
The precision with which acoustic modes are currently measured by helioseismic missions allows to probe the solar interior revealing the sound speed and density profiles down to 10% of the solar radius [146]. The separation between the frequencies of modes with different degree l and radial order n, δν n,l = ν n,l − ν n−1,l+2 , is a quantity very sensitive to the temperature gradient. The case of l = 0 is particularly important because it corresponds to waves that traveled through the entire solar radius, carrying valuable information about the density profile. The small uncertainties associated with these quantities allow to rule out the regions ǫκ 2 > 0.016GR 2 ⊙ and ǫκ 2 < −0.01GR 2 ⊙ . On the other hand, the agreement between the sound speed profile and the solar standard model reaches an accuracy better than 1% in most of the solar interior, with larger uncertainties right below the convective envelope. Comparison with this model and the data, one can safely rule out the region ǫκ 2 > 0.012GR 2 ⊙ . Constraints on the depth of the convective envelope and the helium surface abundance, which also follow from helioseismic data, imply that −0.016GR 2 ⊙ < ǫκ 2 < 0.013GR 2 ⊙ and ǫκ 2 > −0.018GR 2 ⊙ , respectively. These examples clearly illustrate the capabilities of solar observations to constrain modifications of gravity with new couplings in the matter sector.
Relativistic stars
White dwarfs and neutron stars are by far the natural scenarios where the highest pressures can be achieved, which offers an excellent opportunity to test our theories about nuclear matter and also the modified dynamics of alternative gravity theories. It is well known that the masses and radii of neutron stars depend critically on the equation of state of dense matter [183,238]. For a given equation of state, a mass-radius relation and a maximum mass can be derived. The so-called stiffness of the equation of state depends on how many bosons are present. Since bosons do not contribute to the Fermi pressure, they tend to soften the equation of state, which leads to low maximum neutron star masses (∼ 1.5M ⊙ ). GR sets a maximum mass ∼ 3.2M ⊙ , and it is expected that the maximum achievable mass in nature is of order ∼ 2.5M ⊙ , but this depends on the stiffness of the equation of state [226] and is thus open to observational scrutiny. The density-dependent modifications induced by the EiBI dynamics can be seen as the effect of a modified source [138] and, for this reason, must also leave an imprint in the mass/radius relation of these compact objects. In this section we consider the efforts carried out so far to understand the impact of the EiBI modified dynamics on the structure and stability of neutron stars as well as some strategies to distinguish its predictions from those of GR.
Stellar structure
In the EiBI gravity scenario, the equations of stellar structure in the full relativistic case have been studied in numerous works [295,329,192,333], and for a line element of the form
ds 2 = −e φ(r) dt 2 + e λ(r) dr 2 + f (r)dΩ 2 ,(3.16)
can be written as
dm dr = r 2 4ǫ 2 − 3 ab + a b 3 (3.17) dp dr = − 2m r 2 + r 2ǫ 1 ab + a b 3 − 2 1 − 2m r 2 ρ+p + ǫ 2 3 b 2 + 1 a 2 c 2 s (3.18) with f (r) = r 2 ab (3.19) a ≡ 1 + ǫκ 2 ρ (3.20) b ≡ 1 − ǫκ 2 p (3.21) c 2 s ≡ dp dρ . (3.22)
Given a barotropic equation of state p = p(ρ) and appropriate boundary conditions, concrete models can be studied. The boundary conditions at the centre typically involve the specification of the central density ρ c (or the central pressure p c ), and the condition m(0) = 0. For rough structural considerations, the radius of the star is defined by the condition p(R) ≤ p 0 , where p 0 is ideally zero but in practice is represented by a small positive number. At that point one assumes that the Schwarzschild solution should take over, provided a sufficiently smooth transition profile is used. This last step is usually assumed to hold and can be omitted (more details on this will come later). From the definitions of the functions a and b above, the constraints
ǫκ 2 p c < 1 , for ǫ > 0 (3.23) |ǫ|κ 2 ρ c < 1 , for ǫ < 0 (3.24)
appear naturally for stellar models. Assuming that ρ c in neutron stars is of the order ρ c ∼ 10 18 kg/m 3 and p c ∼ 10 34 N/m 2 , we get that |ǫκ 2 | 1 m 5 kg −1 s −2 . Numerically one verifies that compact objects only exist if p(r) ≈ p c + p 2 r 2 near the centre has p 2 < 0. This leads to a condition compatible with (3.10).
The case of pressureless relativistic fluids was considered in [295] as an extension of the Newtonian case, finding that solutions always exist if ǫ > 0. These objects have a maximum compactness of GM/R ∼ 0.3 and a maximum mass and radius that linearly grow with ǫ. The fact that the current cosmological model requires a significant component of cold dark matter particles with p = 0 makes these solutions particularly interesting, since they indicate that such particles could aggregate in structures with the typical compactness and mass of most neutron stars.
Models in EiBI gravity with realistic equations of state based on nuclear physics have been studied in detail in [329] (see also [333] and [295]). These equations of state are usually presented in tabulated form and require numerical interpolation for their implementation in the codes. Though this is not a problem in the case of GR, the interpolation method may introduce numerical noise and artificial effects which should be avoided. In [329] this was solved by using smooth analytic functions to model the tabulated equations of state (see Fig.4), while in [295] a piecewise polytropic interpolation was implemented. As a general feature, it is observed that the mass of the solutions increases with the central density until a certain maximum value. This maximum mass is larger than the GR prediction if ǫ > 0 and smaller if ǫ < 0. The maximum appears at a lower central density than in GR if ǫ > 0. As pointed out in [295], the larger mass predicted by models with ǫ > 0 could serve to prevent ruling out some softer equations of state for which the observation of a neutron star with mass M ≈ 1.97M ⊙ was a critical test (see also [312] for updated data on massive pulsars). Different examples were studied in [192], including a model with a causal stiff fluid for which the speed of sound equals the speed of light, a radiation-type equation of state, a polytrope of index n = 3 (relativistic neutrons), and the quark matter equation of state. An exact (and exotic) analytical solution of the relativistic equations was also found there. More recently, the influence of hyperons in the equation of state has been explicitly considered in [312] to illustrate that the "hyperon problem" found in neutron star models within GR may be avoided in the EiBI theory. Their conclusions are in agreement with the previous literature on this topic. Slowly rotating relativistic stars were also considered in [295] using the perturbative approach introduced by Hartle [195].
Stability
A detailed analysis of the stability under radial perturbations of relativistic stars was carried out in [329] and [334], both focusing on the fluid modes and neglecting the spacetime modes. In [329] a fixed, static physical background was assumed but it was noted that the auxiliary metric could develop a non-zero contribution in the t − r sector due to the perturbations in the fluid. The approach of [334] is different, as the author adopts a crude Cowling approximation forcing both the physical metric perturbation, δg µν , and the auxiliary metric perturbation, δq µν , to vanish.
Denoting by ξ the radial Lagrangian displacement andξ its time derivative, the fourvelocity of the fluid is given by u µ = (−e φ/2 , e −φ/2ξ , 0, 0), which to linear order induces an off-diagonal perturbation in T µ ν given by T r t = −(ρ +p)ξ, with the bar denoting background values. Assuming a time dependence e iωt for all the perturbed quantities, the relevant eigenvalue equation for the radial oscillation modes can be written as
χ ′′ = −W 1 χ − W 2 χ ′ ,(3.25)
where χ ≡ r 2 Q 1 (ρ +p)ξ, and the functions W 1 , W 2 , and Q 1 depend on the background fields and the frequency ω 2 (see Appendix A of [329] for the explicit expressions of these functions). The analogous equation for linear radial perturbations in the non-relativistic limit was studied in [295] in the context of polytropic fluids finding a more tractable expression:
− 1 ρ γp r 2 (r 2 ξ) ′ ′ + 4 ρr ξp ′ + ǫκ 2 4 2ξρ ′ r − ξ ′ρ′ − ρ r 2 (r 2 ξ) ′ ′ = ω 2 ξ , (3.26)
where γ defines the adiabatic index of the perturbations. In both cases, the resulting eigenvalue equation must be solved subject to standard regularity condition at the centre, Three different values of the Born-Infield parameter ǫ, denoted as κ in the plots (not to be confused with the Einstein constant, as follows from the notation employed in this section) are considered. The circle on each M -density curve corresponds to the maximum-mass configuration. Plots taken from [329].
ξ(0) = 0 = ξ ′ (0)
, being ξ(R) finite at the surface. An instability corresponds to an eigenmode with ω 2 < 0.
In the relativistic case, the stability of compact stars is investigated using four different equations of state (APR [10], BBB2 [37], FPS [243], and SLy4 [143]). The results confirm in a robust manner that ω 2 remains positive up to the value of the central density at which the stellar mass reaches its maximum (see Fig.5). This critical density sets the onset of a dynamical instability against radial perturbations. At lower densities, the solutions are linearly stable. This behavior is qualitatively identical to that already observed in GR. The numerical results in Fig.5 show that in the EiBI gravity theory the mass of the solutions may attain a local minimum at larger central densities. The location of this extremum coincides with a zero in the frequency square of the first radial harmonic. While the frequency of the first and higher harmonics depend on the specific value of ǫ chosen, for a given mass, the fundamental mode is quite insensitive to ǫ.
For non-relativistic stars, in the presureless case one finds that, for a given ǫ, there is one fundamental mode, which is numerically determined as ω = αρ
1/2 c , where α ≈ 2.
1866 is a factor independent of ǫ [295]. These solutions do not have unstable modes, confirming that they are linearly stable. For polytropic models, the stability is improved as compared to the case of GR. In GR, models with adiabatic index γ = 4/3 are marginally stable for any polytropic index n, whereas in the Born-Infeld theory these models are always stable if ǫ > 0 and unstable for ǫ < 0.
Observational discriminations from GR
The exceptional conditions of matter inside neutron stars, with densities that may be several times above nuclear densities, turn these objects into natural laboratories from which to extract valuable information on the properties and behavior of nuclear matter. The study of their structure, rotation, and magnetic fields can thus offer a powerful window on the microscopic properties of matter, complementing in this way the knowledge obtained from laboratory experiments. In the context of GR, this could help constrain the form of the equation of state of nuclear matter, which has led to the study of empirical relations between the structural properties of neutron stars (mass, radius, moment of inertia, ...) and their equations of state. Alternatively, the potential existence of relations weakly dependent on the equation of state could also be used to constrain the gravitational dynamics.
The possibility of discriminating between the predictions of GR and those of EiBI gravity using a special kind of neutron stars was raised in [333]. As mentioned above, the mass-radius relation of neutron stars is intimately linked to the details of the equations of state and the internal gravitational dynamics. Extracting useful information about these two inputs of the theory requires not only measuring the radii of neutron stars for a broad range of masses but also breaking the potential degeneracies that may arise between the matter and the gravity sectors. In this respect, a normal neutron star with M ∼ 1.4M ⊙ is more sensitive to the high density properties of the equation of state than a low mass star with M ∼ 0.5M ⊙ , and the uncertainties in the corresponding equations of state of each model are very different. The importance of the analysis of [333] lies on the observation that the radii of neutron stars with M ∼ 0.5M ⊙ are strongly correlated with the neutron skin thickness 33 of 208 P b nuclei in a way which is independent of the particular equation of state [101]. This suggests that laboratory measurements of the neutron skin thickness of 208 P b combined with the precise observational determination of the radii of neutron stars with 0.5M ⊙ could provide a direct estimate of ǫ.
Quantitatively, [333] finds that the radius of 0.5M ⊙ neutron stars, R 05 (measured in km), for different equations of state admits a linear parametrisation of the form (see Fig.6)
R 05 = c 0 + c 1 ∆R , (3.27)
where ∆R (measured in fm) represents the neutron skin thickness of 208 P b. It should be noted that the value of ∆R depends on the particular equation of state considered but is independent of ǫ. The constants c 0 and c 1 depend on the value of ǫκ 2 ρ 0 , where κ 2 ρ 0 /8π = 1.99×10 −4 km −2 follows from taking ρ 0 = 2.68×10 14 g/cm 3 (nuclear saturation density). The dependence of these parameters on ǫκ 2 ρ 0 is explored in the range −0.02 ≤ ǫκ 2 ρ 0 ≤ 0.04, finding that .27), as a function of the neutron skin thickness of 208 Pb for ǫκ 2 ρ0 = −0.02, 0, and 0.02, using various EOS as shown in [333]. The curves are insensitive to the EOS but do depend on the Born-Infield parameter ǫ.
c 0 km = 8.21 + 60.3 × (ǫκ 2 ρ 0 ) (3.28) c 1 km/fm = 31.0 − 125.8 × (ǫκ 2 ρ 0 ) . (3.29)
Combining these relations with Eq.(3.27), one finds that
ǫκ 2 ρ 0 = R 05 − 8.21 − 31.0∆R 60.3 − 125.8∆R , (3.30)
where, recall, R 05 is measured in km while ∆R in fm. Using this expression, with observational values of R 05 and ∆R one can, in principle, determine the value of ǫ. In this regard, assuming that R 05 and ∆R had ±10% variances, the uncertainties in the determination of ǫκ 2 ρ 0 would reach ±0.04 for R 05 = 12 km and ±0.06 for R 05 = 14 km even in the case of GR. According to the best current data, the situation is even worse because ∆R = 0.33 +0.16 −0.18 fm does not allow to constrain ǫ even if R 05 were exactly known. Given that the measurement of R 05 is expected to be more difficult than that of ∆R, because 0.5M ⊙ is exceptionally small for a neutron star, the use of this approach to constrain the theory is really challenging. Nonetheless, there is still hope that the observation of neutron stars with ∼ 0.7M ⊙ , for which this qualitative analysis is still valid, could be used in the future (the lowest mass of neutron stars observed so far is (0.87 ± 0.07)M ⊙ [316]).
Phase transitions
Let us now focus our attention on phenomena taking place in the high density family of neutron stars. The potential existence of phase transitions in the nuclear matter of massive neutron stars could have more dramatic effects in Born-Infield theories of gravity than in GR due to the role that matter gradients play in this theory. The relativistic hydrostatic equilibrium equation (3.18) and the study of stellar pulsations put forward the appearance in the equations of terms associated with the sound speed and its first derivative, which are dependent on the first and second-order derivatives of the pressure with respect to the matter density. This, in part, motivated the use of interpolating functions to approximate the tabulated equations of state in order to avoid artificial numerical effects. However, should first-order (or second-order) phase transitions take place in the interior of neutron stars, discontinuities in the matter density (sound speed) would occur. The potential effects of first-order phase transitions have been investigated in [328].
The first thing to note in the case of first-order phase transitions, is that in the limit in which c 2 s → 0, Eq.(3.18) behaves as dp/dr ∝ −c 2 s /ǫ thus implying that dρ/dr = c −2 s dp/dr is finite even when dp/dr vanishes. If ǫ > 0, one finds that in the region of constant pressure dρ/dr is continuous, constant, and negative, generating in that way a discontinuity in the function p = p(ρ). This region of constant pressure is self-supported due to the repulsive gravity generated by the strong density gradient, in much the same way as pressureless solutions are stabilized. The continuity of ρ(r) in this scenario contrasts with the case of GR, where a discontinuity in ρ is unavoidable. If ǫ < 0, both dp/dr and dρ/dr are positive at the c 2 s ≈ 0 region, yielding a completely different qualitative behavior. The relevant term in the hydrostatic equilibrium equation is now of the form dp dr
∝ − 2 ρ + p + κ 2 ǫ 16π 3 b 2 + 1 a 2 c 2 s −1 .
(3.31)
Given that the ρ + p term is positive and that the other one is negative and grows as c 2 s → 0, a divergence in dp/dr is unavoidable, which indicates the impossibility of having equilibrium static solutions when ǫ < 0. For the ǫ > 0 case, one can verify that the metric functions are smooth and finite but the Ricci scalar develops a delta-type divergence due to its dependence on the radial derivative of c 2 s , which is discontinuous at the phase transition. The physical implications of this divergence have not been studied in detail (see the section 4.5.2 on black holes for closely related discussions), though as suggested in [230] they could induce a backreaction able to avoid them. In fact, as acknowledged in [328], there is no evidence whatsoever that compact stars in nature exhibit phase transitions in their interiors. As a final comment, we just note that curvature divergences of this type are common in many physical problems involving thin shells, in which a certain thick boundary is idealised in the form of a hypersurface that separates two regions [257,141,173]. The delta-like divergences are expected to disappear on physical grounds once small perturbations are allowed in the density/pressure profiles.
Universality relations: f -mode and I-Love-Q.
Aside from the mass-radius relation in the low range band of neutron stars, other empirical relations connecting parameters of neutron stars have been proposed. In particular, a correlation between the scaled moment of inertia I/M R 2 and the compactness M/R has been observed [57,239]. Also the frequency and damping rate of the quadrupolar f −mode, associated to internal fluid oscillations, can be related to global properties such as M and I in a way that depends very weakly on the equation of state [18,19,63,64,348,240]. Also, the values of M, R, and the moment of inertia I can be accurately inferred from the f −mode gravitational wave signals [240]. More recently, a universal relation involving QM/J 2 and M/R, where Q is the spin-induced quadrupole moment and J the angular momentum, has been found. Other universal relations between I, Q and the tidal and rotational Love numbers λ tid and λ rot have also been discovered [370,369]. These socalled I-Love-Q relations are relevant for the understanding of gravitational wave signals in neutron stars binary mergers.
The f −mode universality relations and the I-Love-Q relations of [370,369] have been investigated in the context of Born-Infield gravity in [330]. For this purpose, the authors wrote the field equations in GR-like form following the approach of [138] and computed the oscillation frequencies also in that representation. The consistency of this approach to the problem of stellar perturbations with that provided in [329] was also confirmed, within numerical accuracy, putting forward the usefulness of this representation of the field equations.
Let us first focus on the properties of the f -mode. Neutron star oscillations are damped by the emission of gravitational waves, which implies the existence of a complex part in the oscillation eigenfrequencies (quasi-normal modes), ω = ω r + iω i , with ω i representing the damping rate of the oscillation mode. Within the frame of GR, it turns out that ω r and ω i of the f -modes (fluid oscillations) can be related to global parameters of the star according to
M ω r = −0.0047 + 0.133η + 0.575η 2 (3.32) I 2 M 5 ω i = 0.00694 − 0.0256η 2 , (3.33)
where the factor η ≡ M 3 /I is dimensionless (in the appropriate units). These relations are more insensitive to the equation of state than previous relations where the radius R was chosen as a parameter. The motivation for this choice comes from the fact that R is sensitive to the low-density part of the equation of state, while the moment of inertia I measures the mass distribution globally, which is more closely related to the f -mode oscillations of the star. The approach of [330] consisted on writing the Born-Infield field equations in GR-like form [138], solving the stellar structure equations for several nuclear matter equations of state, and then computing perturbations around the different backgrounds obtained to identify the f -mode frequency using well-established methods developed in GR [233]. Considering the cases ǫκ 2 ρ 0 = −0.1, 0.0, 0.1, with ρ 0 ≡ 10 15 g/cm 3 , for which M can change up to 30%, it was found that the relations M ω r (η) and I 2 ω i /M 5 (η) are essentially independent of the chosen equation of state and ǫ, being in excellent agreement with Eqs. (3.32) and (3.33), respectively. The moment of inertia of a star is defined by I ≡ J/Ω, where J and Ω are the angular momentum and the angular velocity of the star, respectively. For a given J, I determines how fast a star can spin and, for this reason, it is expected to be correlated with the spin-induced quadrupole moment Q of the star. Interestingly, in [370,369] it was found a relation between I and Q which is independent of the equation of state. Related to this, the (traceless) quadrupole moment induced on a neutron star by a nearby companion is determined by Q ij = −λ tid E ij , where E ij is the tidal tensor and λ tid is the so-called Love tidal number. Though, in principle, there is no reason to expect an equation-of-state-independent relation between the variablesĪ ≡ I/M 3 ,Q ≡ −Q/(M 3 χ 2 ), andλ tid ≡ λ tid /M 5 , where χ ≡ J/M 2 , it turns out that they are related by an expression of the form
ln y i = a i + b i ln x i + c i (ln x i ) 2 + d i (ln x i ) 3 + e i (ln x i ) 4 ,(3.34)
where the pairs (x i , y i ) represent (λ tid ,Ī), (λ tid ,Q), and (Q,Ī) (see Figs.7) and the coefficients a i , b i , c i , d i , and e i are constant. The numerical analysis puts forward that the I-Love-Q relations for the EiBI theory of gravity are the same as the GR ones for the range of parameters explored, |ǫκ 2 ρ 0 | ≤ 0.1 and, therefore, they cannot be used to observationally discriminate between these two theories. For the sake of completeness, we briefly comment now on the approach of [138] used to study the above universality relations. Following our notation and manipulations, the field equations of the theory can be written in Einstein-like form as (recall Eq.(2.63))
G µ ν (q) = κ 2 |Ω| 1/2 T µ ν − L G + T 2 δ µ ν ,(3.35)
where L G represents the gravity Lagrangian, being L G = (|Ω| 1/2 − λ)/(κ 2 ǫ) in the Born-Infield theory, and |Ω| is the determinant of the deformation matrix that relates q µν and g µν , as defined by Eq.(2.57). In the case of GR one gets L G = R/(2κ 2 ) = −T /2 leaving the T µ ν term alone on the right-hand side of the equations. Note that while the Einstein tensor on the left-hand side is defined in terms of the auxiliary metric q αβ , the matter terms on the right-hand side (including L G and |Ω| 1/2 ) depend on the physical metric g µν . Since the (algebraic) relation between these two metrics depends on T µ ν , which can have some dependence on g µν (typically through kinetic terms), the resulting field equations may become highly nonlinear in the matter variables. The case of a perfect fluid, with T µ ν = (ρ + p)u µ u ν + pδ µ ν , is particularly simple because the metric dependence of T µ ν only appears through the covariant vector u ν ≡ g να u α . For this matter source Eq.(3.35) takes the explicit form
G µ ν (q) = κ 2 (ρ + p) |Ω| 1/2 u µ u ν + 1 |Ω| 1/2 ρ − p 2 + L G δ µ ν ,(3.36)
with |Ω| 1/2 = (λ + ǫκ 2 ρ) 1/2 (λ − ǫκ 2 p) 3/2 . This representation suggests a redefinition of variables such that the right-hand side of (3.36) can be interpreted as an effective perfect fluid coupled to the geometry defined by q µν . The proposal of [138] thus follows naturally, defining
p q = 1 |Ω| 1/2 ρ − p 2 + L G (3.37) ρ q + p q = (ρ + p) |Ω| 1/2 (3.38) v µ v ν = u µ u ν ,(3.39)
where v µ is normalised using the auxiliary metric, q µν v µ v ν = −1, and v ν ≡ q να v α . Using Eq.(3.39) and the fact that in this model q αβ = Ω 0 g αβ + Ω 1 u α u β , with
Ω 0 = (λ + ǫκ 2 ρ)(λ − ǫκ 2 p) (3.40) Ω 1 = ǫκ 2 (ρ + p) λ − ǫκ 2 p λ + ǫκ 2 ρ ,(3.41)
the relation between v µ and u ν can be readily established. In fact, contracting (3.39) with
v ν , one finds v µ = −(v · u)u µ ,
where v · u ≡ v µ u ν g µν is the usual scalar product between vectors. If the contraction is done with u ν , one finds instead v ν = −u ν /(v · u). Using the definition of q µν above to compute u α v α = (v·u)(Ω 0 −Ω 1 ) and the relation v ν = −u ν /(v·u) to get the alternative expression u α v α = 1/(v · u), one finds that (v · u) 2 = 1/(Ω 0 − Ω 1 ), which completely specifies the relation between v µ , v ν and u µ , u ν . These new variables have mapped the EiBI gravity theory into the usual Einstein equations, which can now be manipulated and solved using standard methods. The spacetime metric follows from the relation g µν = (q µν − Ω 1 u µ u ν )/Ω 0 . This approach should, in principle, be applicable to other matter sources as well.
Magnetic fields
Magnetic fields are thought to play an important role in supernova explosions [368], gamma-ray bursts [302], soft gamma repeaters and quasi-periodic oscillations, anomalous X-ray pulsars [345,234], etc, and are also fundamental to understand basic observational features of neutron stars. In particular, it is well known that the spectrum of radiation emergent from a neutron star atmosphere can significantly differ from a blackbody spectrum, and its angular distribution be far from isotropic due to the presence of magnetic fields [372]. In this sense, it is important to note that the radiation properties of neutron stars are strongly conditioned by their superficial layers [289], which can be in a gaseous state (atmosphere) or condensed state (liquid or solid) depending on surface temperature, magnetic field, and chemical composition. A condensed surface, for instance, may arise at low temperatures and very strong magnetic fields (T 10 5 K and B = 10 13 G or T 10 6 K and B = 10 14 G). On the other hand, the strong gravitational field on the surface layers, which is usually regarded as constant and of order g ∼ 10 14−15 cm/s 2 , rapidly sinks the heaviest elements, leaving the lightest available ones at the surface, which will then be responsible for the radiative properties of the atmosphere, critically affecting its spectrum. A thin layer of Hydrogen of just 10 −20 M ⊙ , for instance, is sufficient to condition the whole spectrum. This is so because magnetic fields are able to shift the ionization energy of Hydrogen up to 160 eV if B = 10 12 G (or 310 eV if B = 10 13 G). The intensity of magnetic fields on the neutron star outer layers is thus essential to understand the features of their radiation spectra, polarization, and thermal conductivity. The presence of magnetic fields above B ∼ 10 9 − 10 10 G, therefore, may affect the opacity of the outer layers resulting in a nonuniform surface temperature distribution, which may lead to pulsations of the thermal radiation due to rotation. At lower intensities, however, its impact on the opacity is negligible and can be safely neglected.
The effects of the Born-Infield gravitational dynamics on the magnetic fields of neutron stars have been investigated in [335] focusing on the axisymmetric dipole configurations, which are expected to dominate in old neutron stars, and assuming spherically symmetric configurations. This assumption implies that the magnetic energy in the star is much smaller than the gravitational binding energy, which allows to neglect any deformation induced by the magnetic pressure. The stellar structure is thus determined by the fluid, while the magnetic field is just computed on top of the resulting geometry. The equations governing the magnetic field follow from Maxwell's equations
F [µν;α] = 0 (3.42) ∇ µ F µν = 4πJ µ ,(3.43)
while the coupling between the fluid and the magnetic field result from the conservation equation ∇ µ T µν = 0, which in the ideal magneto-hydrodynamic approximation takes the form
(ρ + p)u ν ∇ ν u µ + (δ ν µ + u ν u µ )∂ ν p = F µν J µ . (3.44)
With the appropriate gauge condition, A µ can be written as A µ = (0, A r , 0, A ϕ ), and expanding A ϕ as A ϕ = a l (r) sin θ∂ θ P l (cos θ), where P l (cos θ) is the Legendre polynomial of order l, the equation describing the dipole magnetic field (l = 1) becomes
a ′′ 1 + (φ ′ − λ ′ ) 2 a ′ 1 + ζ 2 e −φ − 2 f e λ a 1 = −4πe λ j 1 ,(3.45)
where the line element (3.16) has been used, prime denotes radial derivative, j 1 ≡ c 0 f (r)(ρ+ p), with c 0 a constant, and the constant ζ is related to A r = ζe (λ−φ)/2 a l P l . The components of the magnetic field, B µ = ǫ µναβ u ν F αβ /2 can thus be written as
B r = 2a 1 e λ/2 f cos θ (3.46) B θ = −a ′ 1 e −λ/2 sin θ (3.47) B ϕ = −ζa 1 e −φ/2 sin 2 θ ,(3.48)
from which it is apparent that ζ controls the strength of the toroidal magnetic field. Assuming that the exterior geometry is described by the Schwarzschild solution, the external poloidal magnetic field (ζ = 0) is determined by
a (ex) 1 = − 3µ b r 2 8M 3 ln 1 − 2M r + 2M r + 2M 2 r 2 , (3.49)
where µ b is the magnetic dipole moment at infinity. This solution sets the external boundary condition for a 1 and a ′ 1 . From (3.45), one finds that at the centre a 1 (r) ≈ α 0 r 2 +O(r 4 ), with α 0 a constant. The constants α 0 and c 0 (which appear in j 1 ) should be chosen so as to guarantee the continuity of a 1 and a ′ 1 at the surface. The magnetic field strength can thus be written as
B ≡ (B µ B ν g µν ) 1/2 = f −1 [4a 2 1 cos 2 θ + a ′2 1 f e −λ sin 2 θ + ζ 2 a 2 1 f e −φ sin 2 θ] 1/2 . (3.50)
At the stellar centre, one finds that B 0 = 2α 0 1 + ǫκ 2 ρ c 1 − ǫκ 2 p c . The analysis of [335] considered stellar models with M = 1.4M ⊙ , a range of parameters |ǫκ 2 ρ s | < 0.05, with ρ s = 2.68×10 14 g/cm 3 representing the nuclear saturation density, and two different realistic equations of state for nuclear matter, FPS [243] and SLy4 [143]. This choice was necessary in order to compare the effects of the modified dynamics with those of different equations of state in different regions of the star. The magnetic distributions observed in the pure poloidal case, ζ = 0, are qualitatively the same as in GR, with deviations smaller than 10% in some regions and reaching departures of less than 0.5% in the crust. The mixed case, ζ = 0, manifests some peculiar features depending on the value of ζ, but roughly are also very similar to those of GR. Thus, the differences with respect to GR in the internal regions are comparable to the uncertainties due to the equation of state. The magnetic fields on the crust, however, depend very weakly on the coupling constant ǫ, while properties of this region such as its thickness are very sensitive to the equation of state. It was suggested in [335] that this could be used to extract information on the equation of state by exploring physical processes associated to the crust, such as stellar oscillations. However, given that stellar oscillations are also very insensitive to the Born-Infield parameter due to the universality relations discussed in [330], it seems that the magnetic field is a poor probe for this type of theories.
Final remarks
The use of astrophysical objects to constrain the magnitude of the non-linearity parameter in the EiBI theory of gravity has shown that with current data reasonable bounds can be placed on the theory. However, several important degeneracies arise which make it difficult to distinguish the theory from GR or discriminate its effects from those coming from the matter sector. The exploration of other Born-Infeld inspired theories in these scenarios could help better understand whether these degeneracies are proper of the EiBI or are common to a larger family of gravity theories. For all such theories, a realistic and satisfactory modeling of the transient from the top layers of the star to the external (idealized) vacuum solution is still missing.
Black Holes
According to General Relativity (GR), a fuel-exhausted star with a mass exceeding the refined Tolman-Oppenheimer-Volkoff limit, which may raise up to ∼ 2.5M ⊙ , depending on the equation of state for dense matter (see section 3.3) may end up its lifetime collapsing to form a region of spacetime causally disconnected from asymptotic observers, and which is called a black hole [331]. The three-dimensional null hypersurface marking the boundary of this region, which acts as a one-way membrane, is the event horizon. According to the unicity theorems formulated by Israel [215,214], Carter [102] and Hawking [198] and others [318] (together with the no-hair conjecture, see [253]), starting from any initial (non-necessarily symmetric) configuration the final state of the gravitational collapse corresponds to a stationary and axisymmetric object solely described in terms of three parameters: mass, charge and angular momentum, leading to the Kerr-Newman family of solutions [227,258] (see [222] for a review on gravitational collapse). With the recent detection of gravitational waves ascribed to black hole merger processes by the LIGO collaboration [2] 34 , which is added to the classical observations from compact X-ray sources (with Cygnus X-1 as the first historical and most influential example [286]), the astrophysics of compact objects has entered into a golden era, where GR can be tested with an unprecedent precision in new regimes [3].
Black holes have been and still are a very active area of research as they pose a number of challenges to our comprehension of gravitational interaction. These problems are of different kinds. First, it has been convincingly established in the literature that, if one assumes the validity of the Einstein's equations all the way down to the innermost region of a black hole, a spacetime singularity unavoidably develops [299]. Moreover, this result is not due to an artifact of an excessively simplified modelling, but instead grounded on some physically reasonable restrictions [326] (spacetime singularities and non-singular black holes in the context of Born-Infeld inspired modifications of gravity will be extensively discussed in section 4.5). To avoid the breakdown of predictability and determinism, Penrose introduced the cosmic censorship conjecture [300], by which it is assumed that an event horizon covering the singularity is always developed during the gravitational collapse process, and thus a naked singularity cannot be seen from external observers. Second, there is a tension between the classical description of gravitational phenomena provided by GR and the fundamental tenet of quantum mechanics, namely, unitarity, as given by the apparent disappearance of information inside a black hole, known as the black hole information loss problem [200,248]. On the other hand, the very connection between Hawking's radiation [199] and standard thermodynamic systems still calls for an understanding in terms of hypothetical black hole microscopic degrees of freedom, and the controversy about the potential existence of firewalls at the event horizon still goes on [14]. Finally there are apparent counterexamples of solutions with hair when adding the new ingredient of superradiance [205] (see [206] for a recent review), with related intensive searches for observational discriminations from the Kerr solution [98].
As black holes allow to test the strong field limit of GR, determining the deviations of black hole solutions from the Kerr one of GR and comparing them with astrophysical observations has become a major test on the viability of any modification of gravity 35 . Their study could shed new light on the understanding of all the open questions discussed above. In the context of Born-Infeld inspired theories of gravity we have already seen that the vacuum solutions, in Palatini approach, yield the same dynamics of GR with a cosmological constant term. Thus, the class of static, spherically symmetric vacuum black hole solutions of such theories is represented by the Schwarzschild one, characterized by mass M . In order to excite the dynamics contained in the new couplings of this theory one needs to couple it to some matter source. The available literature so far amounts to two such sources, namely, electrovacuum fields 36 and anisotropic fluids. In this section we shall review in detail the corresponding deviations from the GR solutions and their contributions to fundamental and observational issues of black hole physics.
Spherically symmetric solutions with matter in Born-Infeld gravities
Along the years, a number of Born-Infeld inspired actions have been considered in the literature regarding the search for spherically symmetric solutions. A quick review on some of the first proposals will prepare us to deal with the Eddington-inpired Born-Infeld gravity introduced by Bañados and Ferreira [45], for which most of the research on black hole physics in the literature has been carried out. Note that in the original proposal of Deser and Gibbons [140] the field equations of Born-Infeld gravity were derived using a purely metric variation, which results in fourth order equations of motion and presence of ghosts (see section 2.2), rendering the problem of finding exact solutions to such equations almost intractable. Nonetheless, Feigenbaum [157] considered the metric formulation of the four-dimensional, Class-0 action (recall the classification of theories of section 2.7):
S = d 4 x √ −g R + β 1 − k 1 R 2 − k 2 R µν R µν − k 3 R α βµν R α βµν − 1 (4.1)
where β, k 1 , k 2 , k 3 are some constants. Despite the unavoidable trouble with ghosts, Feigenbaum investigated spherically symmetric solutions in the approximation R µν ≃ 0 and with the additional simplification of taking k 1 = k 2 = 0, which imposes the constraint of R α βµν R α βµν ≤ 1 k upon the Kretschman scalar as long as β = 0. Given the limited physical interest of this scenario due to the ghost problem, let us just mention that Feigenbaum obtained analytical solutions (perturbatively to lowest order in ǫ) under the form
ds 2 = −f 2 (r)dt 2 + dr 2 h 2 (r) + r 2 dΩ 2 (4.2) f (r) = 1 − 2M r 1 − 8k 2 M 3 β r 9 8r − 11M r − 2M + O k 3 β 2 M 3 (4.3) h(r) = 1 − 8k 2 M 3 β r 9 36r − 67M r − 2M + O k 3 β 2 M 3 . (4.4)
In this expression M is the total mass of spacetime, as seen from a far away observer.
In the limit of negligible β these solutions reduce to the Schwarzschild black hole of GR. When increasing the constant β the event horizon disappears (the transition value follows from a non-trivial relation between k, β and M that can only be numerically determined) and a kind of "bare mass" objects free of curvature divergences arise. We will see later that the existence of solutions without curvature divergences turns out to be a feature of other Born-Infeld inspired theories of gravity as well.
The explicit addition of matter, and the corresponding search for spherically symmetric black hole solutions, was first explored with some detail by Vollick. In [357] he considered the following action:
S = 1 κβ d 4 x |g µν + βR µν + κβM µν | − |g µν | (4.5)
where β is a constant (whose interpretation shall be clear later), M µν contains the matter contribution and the connection is taken to be symmetric. When M µν = 0, the purely metric variation of this action (Class-0) has been considered by Feigenbaum, Freund and Pigli [158], and Feigenbaum [157]. Working in the Palatini approach (Class-III theories), Vollick finds electrostatic, spherically symmetric solutions. In this case, one takes the matter contribution M µν = αF µν , where α is a constant and F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor of the vector potential A µ . In order to obtain a system of equations that can be solved exactly Vollick assumes sufficiently weak fields and compute the field equations up to quadratic terms in the fields as
G µν (g) = β 8 g µν R 2 − 4RR µν − 2g µν R αβ R αβ + 8R µα R α ν −α 2 κ 2 β F µ α F να − 1 4 g µν F αβ F αβ .
(4.6) Since the last term in brackets in this expression corresponds to the energy-momentum tensor of a Maxwell field, in order to obtain Einstein's equations to lowest order in β one must take α 2 = 1/(κβ), which implies the positivity of β. Now let us consider (electrostatic) spherically symmetric solutions using the gauge A µ = (φ(r), 0, 0, 0), which implies that the only non-vanishing component of the field strength tensor is F tr (r) = 0. After a bit of algebra, this restriction allows to cast the field equations (4.6) as
G µν (g) = κ F µ α F να 1 − F 2 2b 2 − b 2 g µν 1 − 1 − F 2 2b 2 (4.7)
where by convenience we have introduced a new constant as b 2 = 1/(κβ) and F 2 = F αβ F αβ denotes the electromagnetic field invariant. Note that the contribution on the right-handside of these equations is formally similar to that of the energy-momentum tensor of Born-Infeld theory of electrodynamics [75] with a reversed sign in front of second term and inside the square root. The Einstein equations (4.7) have to be compatible with the equations for the matter, which follow from variation of the action (4.5) with respect to A µ as ∇ µ |q| q −1 [µν] = 0, where the object q µν ≡ g µν + βR µν + √ κβF µν . To quadratic order, and in the notation above, these equations become
∇ µ F µν 1 − F 2 2b 2 = 0 (4.8)
which is nothing but the field equations of Born-Infeld electrodynamics with a reversed sign inside the square root. In Vollick's solutions, the mass function is given by (by convenience, we shall absorb here the factor 4π from the integration of the electromagnetic field equations as Q → 4πQ)
dm(r) dr = b 2 r 4 − Q 2 /b 2 − r 2 (4.9)
again with the reversed sign inside the square-root. The main novelty of Vollick's reversed sign solution is that it is only defined beyond a radius r = r c , where r 2 c = |Q|/b. The replacement of the point-like singularity of GR by a finite-size structure is a feature that will re-appear later when discussing electromagnetic geons in section 4.4. In the present case, at the radius r = r c one finds that the curvature scalar behaves as
R = −8b 2 1 − r 4 − r 2 c r 2 r 4 − r 4 c (4.10)
and thus there is a curvature singularity, displaced here from r = 0 to a finite radius. To find the horizons of these solutions one considers the zeros of the metric component g tt , which can be found by solving the equation h(r) = r−2M +2b 2 (|Q|/b) 3/2 ∞ r/rc u 2 − √ u 4 − 1 du = 0. A careful analysis of this equation reveals the presence of charged black holes with either two horizons, a single (degenerate) one or none (and a time-like singularity at r = r c ), or black holes with a single horizon and a time-like, space-like or null singularity, depending on the parameters of the solutions.
Black holes with a cosmological constant λ can also be implemented within this framework via the Class-III action [358] S = 1 κβ d 4 x |g µν + βR µν + κβF µν + βλg µν | − |g µν | . Now Vollick consider both electrostatic, E(r) ≡ F tr , and magnetostic fields, B(r) ≡ F θϕ , via the two field invariants, F 2 = F µν F µν and G 2 = F µνF µν . In analogy with the solutions above, now the Lagrangian, L = L(F, G), corresponding to the energy-momentum on the right-hand-side of the gravitational field equations, is obtained as
L BI = b 2 1 − 1 − F 2 2b 2 − G 2 16b 4 (4.12)
with the same redefinitions as in the asymptotically flat case above. One can still assume a spherically symmetric line element given by Eq.(4.16) and follow a similar procedure to solve the field equations, which yields the expression for the mass function
dm(r) dr = Λ 2 r 2 + b 2 (1 +λ) −1 r 4 −Z 2 /b 2 − r 2 (4.13) where Λ = λ 2+βλ 1+βλ
plays the role of the cosmological constant term, while we have definedλ = λ/(κb 2 ),Z 2 =Q 2 +p 2 , withQ = (1 +λ)Q andp = (1 +λ)p being the re-scaled electric and magnetic charges, respectively. From the computation of the Ricci scalar constructed out of the spacetime metric
R = −4Λ − 8b 2 (1 +λ) −1 1 − r 4 − r 4 c /2 r 2 r 4 − r 4 c (4.14)
it follows that a curvature singularity is still present at the finite radius r = r c = |Z|/b. To close this part, let us briefly mention that spherically symmetric solutions were investigated in the context of f (R) models with a square root (Class-IV), see [235], but only mundane (Anti-)de Sitter solutions were found.
Born-Infeld black holes in General Relativity
There is a remarkable parallelism between the modifications on the structure of horizons for some of the solutions above and those of Born-Infeld electrodynamics coupled to GR. As this parallelism will re-appear later in the literature, it is instructive to consider the spherically symmetric solutions of Born-Infeld electrodynamics. In this sense, the framework of Einstein's gravity coupled to non-linear electrostatic fields has been developed to a great detail in the literature, particularly for Born-Infeld electrodynamics [321,139,132,87,162]. The action is written as
S = d 4 x √ −g R 2κ 2 − L(F 2 ) (4.15)
where the case of Born-Infeld electrodynamics is given by Eq.(4.12) with G = 0. Due to the symmetry of the energy-momentum tensor for electrostatic solutions, T t t = T r r , one can write a line element
ds 2 = − 1 − 2m(r) r dt 2 + 1 − 2m(r) r −1 dr 2 + r 2 dΩ 2 (4.16)
where dΩ 2 = dθ 2 + sin 2 (θ)dϕ 2 is the solid angle element, and the mass function m(r) is determined through the resolution of the Einstein's equations as
dm(r) dr = r 2 T t t (r) = b 2 r 4 + Q 2 /b 2 − r 2 (4.17)
(compare this equation with (4.9)). This can be explicitly integrated (with the constraint of recovering Schwarzschild black hole as r → ∞) as
m(r) = M − 4πb 2 r 3 r 2 − r 4 + Q 2 /b 2 + 2Q 2 b 2 r 2 2 F 1 1 4 , 1 2 , 5 4 , − Q 2 b 2 r 4 (4.18)
where M is the Schwarzschild mass. Due to the finiteness of the self-energy associated to a point-like charge in Born-Infeld electrodynamics, see Eq.(2.10), the behaviour of the metric component g tt at r = 0, with the expressions (4.16) and (4.18), becomes there:
g tt = g −1 rr ≃ r→0 − 1 − 8πbQ − 2(M −Ũ ) r + O(r 2 ) (4.19) whereŨ = 4π 3/2 U , with U defined in Eq.
(2.10) and the factor 4π 3/2 comes from the redefinition Q → 4πQ above. The zeros of g tt in (4.19) set the location of the horizons. In the (asymptotically flat) Reissner-Nordström solution of GR, such horizons are obtained as r ± = M ± M 2 − Q 2 , where the signs ± refer to the outer (event) and inner (Cauchy) horizons, respectively. For these horizons to exist, the inequality M 2 > Q 2 has to be fulfilled (when this bound is saturated, M 2 = Q 2 , one has an extreme black hole with a degenerated horizon), otherwise one ends up into a naked singularity. In the Born-Infeld electrodynamics case, due to the finite character ofŨ 37 , it turns out that the behaviour of the metric at the center determines the existence of three classes of configurations depending on the hierarchy between M and U . In this sense, if M <Ũ the solutions resemble the Reissner-Nordström configurations of GR, in that two, a single (degenerate) horizon or none can be found, while for M >Ũ a single horizon is always found, with similar features to those of the Schwarzschild black hole of GR. Finally, when M =Ũ the metric at the center is finite and equal to −(1 − 8πbQ), which consequently yields either a single horizon or none. This description is depicted in Fig.10 for a particular choice of b = Q = 1/2. In all these cases a curvature divergence is always present at r = 0 and this way Born-Infeld electrodynamics fails to solve the singularity problem within GR. We will see that quite a similar structure of horizons arises when considering Eddington-inspired Born-Infeld gravity in section 4.4, while the issue with singularities will be reviewed in section 4.5.
Eddington-inspired Born-Infeld black hole solutions
We now turn our attention upon to the most widely employed proposal in the literature for Born-Infeld inspired modifications of gravity, and where the influential electrovacuum black hole solutions of Bañados and Ferreira were found [45]. This proposal is defined via the action (2.33), and nowadays is usually known as Eddington-inspired Born-Infeld gravity (EiBI), which is a Class-I action (see section 2.7 for details on this classification). By convenience, let us write this action in the notation employed in this section as
S EiBI = 1 κ 2 ǫ d 4 x − det g µν + ǫR (µν) (Γ) − λ − det g µν + S M (g µν , ψ m ) (4.20)
where ψ m denote the matter fields. A few remarks are in order: for the purpose of this section we shall assume hereafter that the (symmetric) connection Γ is not coupled to the matter sector in the action (4.20), in agreement with Einstein's equivalent principle, that dictates that free-falling particles should follow geodesics of the background geometry g µν (see 4.2.2 for specific details). On the other hand, in vacuum, S M = 0, the equation of motion for g µν implies g µν = ǫ λ−1 R (µν) so that an effective cosmological constant term emerges as Λ = λ−1 ǫ (thus asymptotically flat solutions correspond to λ = 1). This is consistent with the non-relativistic limit described in section 3.1, where post-newtonian corrections only emerge under variations on the energy density of the matter fields.
Geometry and properties
Let us now study spherically symmetric configurations in EiBI gravity sourced by electrovacuum (Maxwell) fields, as given by the action
S M = − 1 16π d 4 x √ −gF µν F µν . (4.21)
The energy-momentum tensor for this source is written as
T µν = 1 4π F µσ F σ ν − 1 4 g µν F σρ F σρ . (4.22)
Bañados and Ferreira considered the spherically symmetric line element for the metric g µν as
ds 2 g = −ψ(r) 2 f (r)dt 2 + dr 2 f (r) + r 2 dΩ 2 (4.23)
and solved the EiBI equations for an asymptotically flat geometry, λ = 1. It is instructive to consider in detail the obtention of the field equations, which is not provided in [45], but derived in detail by Wei et. al. in Ref. [360] for arbitrary λ. This will be useful to understand the different results obtained in similar but slightly different scenarios in EiBI gravity. In many of such scenarios it is much simpler to solve the field equations for the auxiliary metric q µν , and then transform the solution back to the spacetime metric g µν using Eq. (2.57). In the present case one proposes a line element for q µν as
ds 2 q = −G 2 (r)F (r)dt 2 + 1 F (r) dr 2 + H 2 (r)dΩ 2 . (4.24)
The five metric functions {ψ(r), f (r), G(r), F (r), H(r)} are to be determined via the field equations (2.60) and the transformations (2.57). The gravitational field equations form a compatible set with the electromagnetic ones, ∂ r (ψ −1 r 2 E) = 0, which gives the result E(r) = Q r 2 ψ(r), where Q arises as an integration constant associated to the electric charge. Note that there is one redundant equation between the former and the latter, and consequently there are several ways to proceed. Wei et al. [360] choose to replace the expression for the electromagnetic field into the energy-momentum tensor (4.22), and insert the result into the field equations for the auxiliary metric (2.60), which yields
4 G ′ G H ′ H + 2 F ′ F H ′ H + 3 G ′ G F ′ F + 2 G ′′ G + F ′′ F = 1 ǫF 1 λ − ǫQ 2 r 4 − 1 ,(4.25)4 H ′′ H + 2 F ′ F H ′ H + 3 G ′ G F ′ F + 2 G ′′ G + F ′′ F = 2 ǫF 1 λ − ǫQ 2 r 4 − 1 , (4.26) − 1 H 2 F + F ′ F H ′ H + G ′ G H ′ H + H ′2 H 2 + H ′′ H = 1 ǫF 1 λ + ǫQ 2 r 4 − 1 ,(4.27)
where primes stand for derivatives with respect to the radial coordinate r. On the other hand, the transformations (2.57) lead to the relations between the metric functions in each line element
G = ψ λ − ǫQ 2 r 4 ; F = f λ − ǫQ 2 r 4 −1 ; H = r λ + ǫQ 2 r 4 . (4.28)
Now one just needs to solve the field equations (4.25), (4.26) and (4.27) with the relations (4.28), imposing the asymptotic GR limit:
ψ(r → ∞) → 1 ; f (r → ∞) → 1 − 2M r + Q 2 r 2 − Λ r 2 3 (4.29)
which is nothing but the Reissner-Nordström-Anti-de Sitter solution, corresponding to the spacetime geometry outside a spherical distribution characterized by mass M , charge Q, and cosmological constant Λ. Now a bit of algebra yields the following expressions for the metric components and the electromagnetic field in the EiBI case ψ(r) = r 2
r 4 + (ǫ/λ)Q 2 (4.30) f (r) = r ǫQ 2 + λr 4 λr 4 − ǫQ 2 3r 2 − Q 2 − (λ − 1)r 4 /ǫ ǫQ 2 + λr 4 3r 3 + 1 3 Q 3 π √ ǫλ Γ 2 (1/4) + 4 3 iQ 3 √ ǫλ F iarcsinh i Q λ ǫ r , −1 − 2 √ λM (4.31) E(r) = Q r 4 + (ǫ/λ)Q 2 (4.32)
where F (Φ, m) = Φ 0 (1 − m sin 2 θ) −1/2 dθ (with −π/2 < Φ < π/2) is the elliptic integral of first kind. These explicit expressions were given in Ref. [360], and refine that of f (r) appearing under the form of an integral in Bañados and Ferreira paper [45], besides correcting a factor 2 under the square root of the function ψ(r) of the latter. Regarding the horizon structure, one finds the remarkable result that, for any value of the EiBI parameter ǫ, its mere presence induces a change in the causal structure of these black holes (see Fig.9), moving from the two-horizons description of the Reissner-Nordström solution of GR to a configuration with a single horizon (resembling the Schwarzschild solution of GR) or none, depending on the combination of parameters [360].
Exploring further EiBI black holes, the expression for the electric field (4.32) bears a remarkable similarity with that obtained in Born-Infeld electromagnetism [75]. In the present case, despite the finiteness of the electric field everywhere, the metric functions are singular at the finite radius r = r c , where r 2 c = √ ǫQ, which may be hidden or not behind an event horizon. In Ref. [360] Wei et al. compute, for asymptotically flat solutions, λ = 1, the following curvature scalars:
R(g) ≡ g µν R (µν) (g) ∝ 1 (r 4 − r 4 c ) 3 (4.33) R(g, q) ≡ q µν g µν R (µν) (q) = g µν (q µν − g µν )/ǫ = 8ǫ (4.34) R(q) ≡ q µν R (µν) (q) = 8(r 4 + r 4 c / √ 2)(r 4 − r 4 c / √ 2) (r 4 + r 4 c )(r 4 − r 4 c )
.
(4.35)
It is thus immediately seen that the curvature scalar constructed either out of the metric g µν or of q µν blows up as the surface of radius r = r c is approached. However, no interpretation on the nature of such a surface is given, and the presence of divergences on curvature scalars could be interpreted as signal of the breakdown of the geometry and thus of the presence of a physical singularity. To overcome this point, Bañados and Ferreira argue that the geometry (4.23) describes just the exterior of a charged object, so a realistic model should consider the process of gravitational collapse to explore such a question in detail. Nonetheless, we shall see later when discussing non-singular solutions in section (4.5) that EiBI gravity hides some surprises regarding the singularity issue.
Geodesic motion
The new non-trivial gravitational dynamics introduced by EiBI gravity, that modifies the shape of the geometry, necessarily has its impact upon the geodesic behaviour of both null (associated to light rays) and timelike (associated to massive particles) geodesics. As already mentioned, the fact that in EiBI action (2.33) the connection does not couple directly to the matter sector, implies that Einstein's equivalence principle holds (see section 2.6). This way, the equations of motion for a geodesic curve γ µ = x µ (u), where u is some affine parameter, can be derived from the action
S = duL = 1 2 du g µν dx µ du dx ν du (4.36)
from which the geodesic equation, in a coordinate system, follows as [359,104]
d 2 x µ du 2 + Γ µ αβ (g) dx α du dx β du = 0 ,(4.37)
where Γ µ αβ (g) is the affine connection constructed as the Christoffel symbols of the spacetime metric g µν . Eq.(4.37) represents a set of second-order differential equations that provide a unique solution once initial conditions, {x µ (0), dx µ /du| 0 }, are given. Now, replacing the line element (4.23) for g µν into the Lagrangian density of Eq.(4.36) one gets the result 38
L = −ψ 2 (r)f (r)ṫ 2 + f (r) −1ṙ2 + r 2 (θ 2 + sin 2 (θ)φ 2 ) (4.38)
where dots denote derivatives with respect to the affine parameter u. From the Hamiltonian description of the system it follows that there are two conserved quantities, namely, E = ψ(r)f (r)ṫ 2 and L = r 2 sin(θ)φ. For timelike observers these quantities can be interpreted as the energy per unit mass and angular momentum per unit mass, respectively, while for null geodesics we can identify b = L/M as an apparent impact parameter from asymptotic infinity. In addition, due to spherical symmetry one can assume the motion to be confined to a plane, that can be chosen to be θ = π/2 without loss of generality. Now, the equation of the radial motion of a particle in the background geometry (4.23), can be deduced from Eq.(4.37) as
ψ 2 dr du 2 = E 2 − V 2 . (4.39)
This is just the equation of motion of a one-dimensional particle moving in an effective potential of the form
V (r) = f ψ 2 k + L 2 r 2 ,(4.40)
where ψ and f are defined in Eqs.(4.30) and (4.31), respectively, while the causal vector u µ = dx µ /du satisfies u µ u µ = −k, with k = 0(+1) for null (time-like) geodesics. Now, if one considers the circular motion of a test massive particle (k = +1) around an electrically charged EiBI black hole, this implies the constraint dr/du = 0 which, via Eq.(4.39), yields E = V (r). This orbit is realised, indeed, at the minimum of the effective potential V (r).
In [336] Sotami and Miyamoto perform a numerical analysis of such a motion, using fixed values of Q/M and ǫ/M 2 and varying the ratio L/M , depicted in Fig.10 (left). The main 38 In general, imposing a symmetry and obtaining the equations of motion do not commute. The conditions under which these two operations do commute are established by the Palais criticality theorem [293].
result is that as the ratio L/M decreases, the maximum of the effective potential decreases as well, while its minimum gets closer to the centre of the EiBI black hole, in such a way that there is a minimum bound for L/M (depending on ǫ), below which no minimum of the potential occurs. This bound determines the innermost stable circular orbit (ISCO), which is the minimum radius below which no stable circular orbit of a test massive particle can exist around an EiBI black hole. These results are qualitatively similar to those of GR, though the specific quantitative details depend on the particular value of the EiBI constant ǫ.
Strong gravitational lensing
Two works [360,337] have been carried out in the literature to determine the effect of the parameter ǫ of charged EiBI black holes regarding the lensing in a strong gravitational field. Gravitational lensing is indeed a powerful test to determine the nature of a compact object, which may allow to find deviations from GR predictions in the strong field regime [311]. For a massless particle, k = 0, Eq.(4.39) can be conveniently rewritten as
ψ 2 dr du 2 = 1 − b 2 f ψ 2 r 2 (4.41)
where we have redefined u → u/E. To characterise the orbits of photons in the effective potential (4.40) one first establishes the existence of the photon sphere, namely, the innermost region for a photon in orbit around a black hole, which for static, spherically symmetric spacetimes coincides with the unstable circular orbit (UCO) radius. According to the analysis carried out by Virbhadra and Ellis [351,118,352], for a line element of the form (4.23) this radius is simply defined by the equation (ψf 2 ) ′ r = 2ψ 2 f . Explicitly, for the EiBI black hole metric defined by the functions (4.30) and (4.31), the UCO radius, r U CO , corresponds to the solution of the equation (in units 2M = 1, which is equivalent to making dimensionless the black hole parameters as r → r/(2M ), Q → Q/(2M ) and κ → κ/(2M ) 2 ):
8ǫ 5/4 Q 2 r 2 r 2 − Q 2 ǫQ 2 + r 4 = 3ǫ 2 Q 4 + 2ǫQ 2 r 4 + 3r 8 (4.42) × − 4 √ iQ 3/2 rF (iarcsinh( i √ κQ r), −1) − Q 3/2 rΓ 1 4 2 √ π + 4 √ ǫ 3r − 2 ǫQ 2 + r 4 ,
which is consistent with the fact that, when the charge Q = 0, the above equation yields the result r U CO = 3/2 (restoring units, this is the well known result r U CO = 3M ), which corresponds to that of the Schwarzschild black hole. For non-vanishing Q, however, finding analytic solutions to (4.42) is highly non-trivial. One may note instead that the integration of the photon sphere equation above, and comparison with the effective potential (4.41), tells us that the UCO radius r U CO corresponds to the solution of the equation dV /dr = 0 with d 2 V /dr 2 < 0. This way photons will be swallowed by the black hole if V (r U CO ) < 1, be scattered by it at some radius r = r 0 if V (r U CO ) > 1, and move indefinitely around it in absence of perturbation if V (r U CO ) = 1. This can be translated into the condition b 2 ⋚ b 2 c , where b c is a critical number that depends non-trivially on the black hole parameters and the EiBI constant. As a comparison, in the Reissner-Nordström case of GR, ǫ = 0, one has r U CO = 3M (1 Fig.10 (right) the effective potential for the choice ǫ/M 2 = 0.6 for EiBI black holes is depicted, with the presence of the ISCO radius (marked by solid and dashed vertical lines, for EiBI and GR, respectively) and scattering radius r 0 . The dependence of the UCO radius r U CO at fixed charge with the EiBI constant can also be studied numerically, with the result that it monotonically decreases with increasing ǫ [337,360], meaning that it is harder to capture a photon by the EiBI black hole than in the Reissner-Nordström black hole of GR.
+ 1 − 8Q 2 /(9M 2 ))/2 and b 2 c = r 4 U CO /[(r U CO − r + )(r U CO − r − )]. In
Let us now consider the scattering process of a photon by the electrically charged EiBI black hole, which can only take place for b > b c . First, from Eqs.(4.39) and (4.40) we obtain the equation
dϕ dr = bψ r r 2 − f ψ 2 b 2 . (4.43)
We assume a photon that travels from infinity, is scattered at r = r 0 and ϕ = 0 (see Fig.10, right), and returns to infinity. By construction, this turn-around point satisfies dr/dϕ = 0, which implies b 2 = r 2 0 /(f (r 0 )ψ 2 (r 0 )), where the subindex 0 means that functions are being evaluated at r 0 . This way, the integration of (4.43) yields the result
φ(r) − φ(r 0 ) = r r 0 bψ r r 2 − b 2 ψ 2 f dr . (4.44)
With this expression, the deflection angle α(r 0 ) of the photon, which is defined as ∆(ϕ)(r 0 ) = 2φ(∞) − π [353], can be written for the EiBI metric as
∆(ϕ)(r 0 ) = 2b ∞ r 0 ψ r r 2 − b 2 ψ 2 f dr − π . (4.45)
Despite the presence of a pole in the integrand of (4.45) at r = r 0 , this can be isolated and properly handled using the variable z = 1 − r 0 /r, which finally yields a finite result (see [337] for details). This way, for the EiBI black hole the deflection angle can be numerically computed and compared to the GR solution, and the result is plotted in Fig.11. There it is seen that the deflection angle increases as r 0 decreases. As the ratio r 0 /M decreases the deflection angle increases until it reaches the value 2π corresponding to the point where the massless particle completes a loop around the black hole before reaching the asymptotic observer. By decreasing further the ratio r 0 /M one gets subsequent values 2πn (n an integer number) of the deflection angle, which means that the photon performs n loops around the black hole before escaping from it. Indeed, should r 0 be able to reach the UCO radius r U CO , then the deflection angle would diverge, meaning that the photon would turn indefinitely around the EiBI black hole, again, in absence of any perturbation. These light rays passing close to the UCO radius give rise to multiple images on both sides of the optical axis, called relativistic images. The position of such images in this case depends strongly on the value of the EiBI parameter ǫ/M , i.e., on the gravitational theory. Thus, this strong gravitational lensing represents a promising scenario to experimentally test EiBI gravity in the strong field limit. As already mentioned, when r 0 = r U CO the integrand in (4.45) diverges, and it has to be handled with care via the new variable z = 1 − r 0 /r. In both Refs. [360,337] this allows to perform the integration of (4.45) around the region r 0 ≃ r U CO , with the (finite) result
∆ϕ(b) = −a 1 log b b c − 1 + a 2 + O (b − b c ) 1/2 (4.46)
(alternatively one can write this expression in terms of r U CO , as it is done by Wei et al. [360]) where the strong deflection coefficients a 1 and a 2 depend on the EiBI parameter ǫ in a non-trivial way (see [337] for details). For fixed charge, it turns out that increasing (and positive) ǫ implies an increasing of the deflection angle as compared to the Schwarzschild black hole (see Fig.5 of Ref. [360]), that could be used to obtain information on ǫ using strong gravitational lensing.
Next, to investigate the position and magnification of the relativistic images in strong gravitational lensing, one considers the lens geometry, where it is assumed that the black hole lens is located between the source and the observer, which are both required to be far away from the black hole so that the gravitational fields there are weak enough to be described by a flat metric. Under such constraints the form of the lens equation was found by Virbhadra and Ellis [351] as
tan ω = tan θ − D LS D OS [tan(∆(ϕ) − θ) + tan(Θ)] (4.47)
where ω and Θ correspond to the lens/source and the lens/observer angular separation between, respectively, while D LS and D OS stand from the distance between lens and source, and observer and source, respectively. In the strong deflection limit source, lens and observer can be assumed to be highly aligned, i.e., ω ≪ 1 and Θ ≪ 1 (and (∆ϕ n − Θ) ≪ 1, where ∆ϕ n ≡ ∆ − 2πn is the deflection angle when all the loops of photons around the EiBI black hole are removed [83]), and using also that in the lens geometry b ≃ D OL Θ one gets the deflection angle
∆ϕ(Θ) = −a 1 log D OL Θ b c − 1 + a 2 . (4.48)
The relativistic images correspond to ∆ϕ(Θ) = 2πn, which yields
Θ 0 n = b c D OL 1 + exp a 2 − 2nπ a 1 (4.49)
where Θ 0 n is the angle of the nth relativistic image. Due to the exponential contribution the first relativistic image, Θ 0 1 , is the brightest one, while the other are greatly demagnified. In Fig.12 the position of such an image is depicted as a function of the EiBI parameter ǫ/M 2 for several values of the electric charge (set of curves) with assumed values of D OL = 8.5 kpc and M = 4.4 × 10 6 M ⊙ , corresponding to the supermassive black hole at the centre of the Milky Way [176]. From this figure it is clear that the deviation from the GR prediction increases with stronger EiBI coupling ǫ/M 2 (in the range ∼ 3% − 5% for Q/M = 0.5 and |ǫ/M 2 | = 10), which is consistent with the fact that the location of the scattering radius r 0 decreases as the EiBI parameter increases.
There are, in addition, other quantities that can be constructed to be compared with astronomical observations. In order to take the simplest situation for observation, one can assume that the first relativistic image Θ 0 1 can be resolved from the others, that are collectively packed at Θ 0 ∞ [83]. This way, one finds three observables: the position of the relativistic images except the first one, Θ 0 ∞ , and the two quantities
s ≡ Θ 0 1 − Θ 0 ∞ = Θ 0 ∞ exp a 2 − 2π a 1 (4.50) R = exp (2π/a 1 ) (4.51)
corresponding to the angular separation between the first image and all the others, and to the ratio between the flux of the first image and all the others, respectively. The latter defines a more convenient observable, R m = 2.5 log 10 R, which is the relative magnification of the images. This way, given an EiBI parameter ǫ one can numerically compute the strong deflection coefficients a 1 and a 2 and thus the three observables above. By comparing them with astronomical observations one can test the nature of black holes via gravitational lensing and, in particular, put experimental constraints on the value of the EiBI constant ǫ. This has been explored, for ǫ > 0, by Wei et al. [360] by assuming that the EiBI black hole describes the supermassive black hole at the centre of our galaxy, and compare it to the description provided by Schwarzschild black hole [351]. In table 1 of that paper, an explicit computation of these three observables for different values of ǫ has been done.
The main result is that these observables fulfill the inequalities Θ EiBI ∞ < Θ RN ∞ < Θ Sch ∞ , R EiBI < R RN < R Sch and s Sch < s RN < s EiBI (for ǫ < 0 the inequality on s do not necessarily hold for all values of ǫ [337]). For instance, the difference in the observable Θ ∞ between the charged EiBI black hole and the Schwarzschild black hole is of order ∼ 4 µarcsecs, which seems to be far from the reach of current astronomical instruments [84]. On the other hand, the relative magnification R m may significantly deviate from the GR prediction, for instance, with the choice Q/M = 0.5 one obtains a 5.5% − 12.7% deviation with respect to GR for the EiBI parameter choice ǫ/M 2 = ∓10. This way, strong gravitational lensing can complement other techniques for testing deviations from the Kerr solution such as the measurement of the iron Kα line observed in the X-ray fluorescence spectrum produced by the illumination of a cold accretion disk by a hot corona of (stellar-mass or supermassive) black hole candidates [220,221,39].
Mass inflation
The innermost structure of black holes in the presence of accretion has been studied for decades, with the striking result first found by Israel and Poisson [305,306], and further extended by Ori [284] and others, that over the inner (Cauchy) horizon of a rotating black hole there occurs an exponential growth of the local Misner-Sharp mass, which in turns induces un unbounded growth of the curvature, a phenomenon known as mass inflation (see [189] for a review on the topic). It is triggered by the relativistic counter-streaming effects between ingoing and outgoing streams, which occurs not only in the context of GR, but also in black hole solutions of other theories of gravity. In the case of EiBI gravity, this question has been investigated by Avelino [31] using electric charge instead of rotation in order to simplify the problem. The reason for this choice lies on the fact that the interior structure of a charged black hole closely resembles that of rotating black hole, where the negative pressure generated by the electric field yields a gravitational repulsion analog to that produced by the centrifugal force in a rotating black hole.
Since the inner structure of charged EiBI black holes can be drastically affected by the accretion of mass, one has to employ some simplifying assumptions in order to obtain analytic solutions. In particular, the homogeneous approximation assumes the ingoing and outgoing streams to be equal. This implies that all relevant quantities can be written as a function of a radial (timelike) coordinate, which has been shown to be useful for studying some of the most important aspects of mass inflation [190,34,33]. This allows to write two spherically symmetric line elements as
ds 2 q = A(r)dt 2 + B(r)dr 2 + H 2 (r)dΩ 2 (4.52) ds 2 g = g tt dt 2 + g rr dr 2 + r 2 dΩ 2 (4.53)
where A(r), B(r), H(r), g tt (r) and g rr (r) are functions of the radial coordinate r alone. The total energy-momentum tensor is split into two pieces
T µ ν = e T µ ν + f T µ ν (4.54)
where e T µ ν and f T µ ν are the electromagnetic and fluid contributions, respectively. The components of such an energy-momentum tensor can be written as
T r r = −ρ = −ρ e − ρ f ; T t t = p = −ρ e + w ρ f ; T θ θ = T φ φ = p ⊥ = ρ e + w ⊥ ρ f (4.55)
where ρ e = Q 2 8πr 4 is the electromagnetic energy density, ρ f the fluid energy density and the factors {w , w ⊥ } are the fluid equations of state for the radial and tangential pressures, respectively. Since the electromagnetic and fluid contributions are assumed to be conserved independently, the conservation equation of the energy-momentum tensor of the latter can be explicitly integrated as ρ f,f = ρ f,i g tt,i gtt (1+w )/2 r i r 2(1+w ⊥ ) , where the subscripts {i, f } mean that physical quantities are evaluated at some initial and final radius, respectively. The above setup describes a charged EiBI black hole that accretes mass, the latter being described by a fluid, from an initial state which is the Reissner-Nordström solution of GR. Now, using Eqs.(2.57) the following relations between the metric functions in the line elements (4.52) and (4.53) are obtained
A = g tt (1 +ǭρ) 1/2 (1 −ǭp ⊥ ) (1 −ǭp ) 1/2 , (4.56) B = g rr (1 −ǭp ) 1/2 (1 −ǭp ⊥ ) (1 +ǭρ) 1/2 , (4.57) H = r(1 +ǭρ) 1/4 (1 +ǭp ) 1/4 ,(4.58)
whereǭ ≡ 8πǫ. When the fluid energy density vanishes, ρ f = 0, the solution reduces to the Reissner-Nordström one of GR.
To obtain analytical solutions one must introduce additional constraints. In particular, Avelino [31] studies the mass inflation regime in which w ∼ 1 and |ǭ|ρ ≪ 1, which simplifies the relations between metrics as A = g tt , B = g rr and H = r. In addition, it is assumed that mass inflation takes place near the inner horizon, r ∼ r − , and since during this regime the energy density becomes much larger than that of the electromagnetic field (so ρ ∼ ρ f ) one can approximate H ∼ r(1 +ǭρ) 1/4 (1 −ǭρ) 1/4 ∼ r − [1 − (ǭρ/2) 2 ]. Under these conditions, the tt and rr components of the field equations read (for the Reissner-Nordström solution of GR). Now, since mass inflation starts when the energy density of the fluid begins to dominate over the electromagnetic contribution, for the sake of finding analytical solutions one can assume ρ f = αρ e where α is some constant of order unity. The combination of the above equations implies that the ratio between metric components during mass inflation satisfies
− H ′ H B ′ B − B H 2 − H ′ H 2 + 2 H ′′ H = 8πBT t t ,(4.g rr g tt [MI] ∼ − α 2 Q 4 64π 2 ρ 2 f,i g 2 tt,i r 4(1−w ⊥ ) − r 4(1+w ⊥ ) i . (4.62)
Finally, assuming that mass inflation ends at r −ǭ 2 2 ρ|ρ ′ | = β, where β is another constant of order unity, one gets the maximum energy density attained at the end of mass inflation:
ρ [end] ∼ β 1/2 2π 1/2 α g 1/2 tt,i r 2−w ⊥ − r 1+w ⊥ i Q 2 ρ 1/2 f,i |ǫ| ,(4.63)
which implies the presence of a threshold of energy density for mass inflation not to be triggered in EiBI gravity, i.e.,
ρ 1/2 f,i |ǫ| < α 2 4π 1/2 β 1/2 Q 4 g 1/2 tt,i r 6−w ⊥ − r 1+w ⊥ i .
(4.64)
This threshold depends on the solution's mass and charge, the accretion rate, and the EiBI parameter ǫ. Note that mass inflation can always occur if the accretion rate is large enough, independently of the value of ǫ. To see the effect of this threshold in the behaviour of the local mass inside a sphere of radius r in the innermost region of these solutions, one considers the Misner-Sharp mass (MS), defined as M MS = r 2 1 + Q 2 r 2 − 1 grr [252], whose maximum is attained at the end of mass inflation, g rr [end] . The calculation of this mass in the present case yields the result
M MS[end] ∼ − r − 2g rr[end] ∼ 16π 3/2 β 1/2 α 3 g 3/2 tt,i r 9−3w ⊥ − r 3+3w ⊥ i Q 6 ρ 3/2 f,i |ǫ| . (4.65)
These analytical calculations complement and are in agreement with the numerical analysis presented also by Avelino in [30]. As depicted in Fig.13, for small values of ρ f,i the slope of the contours indicates that the Misner-Sharp mass is a function of ρ f,i /ǫ 3/2 and that no significant mass inflation occurs below a threshold on the fluid energy density, which is fully consistent with the analytic result obtained in Eq.(4.65). The conclusion of this analysis is that, under the restricted conditions considered in these works, in EiBI gravity there is a minimum accretion rate below which no mass inflation occurs, no matter how close the theory is to GR (which is obtained in the limit ǫ → 0). The underlying physical reason for this result still remains to be clarified.
Wormholes
Lorentzian wormholes are geometric structures representing a shortcut or tunnel between two asymptotically flat regions of spacetime. Such a geometry, for a static spherically symmetric and traversable (i.e. without horizons) solutions can be written as [354] ds 2 = −e 2Φ(r) dt 2 + 1
1 − b(r) r dr 2 + r 2 dΩ 2 (4.66)
where the (gravitational) redshift function Φ(r) and the (wormhole) shape function b(r) characterise the geometry. In order to describe a wormhole, two charts for the two asymptotically flat regions are needed, r ∈ [r 0 , +∞), where r 0 is the radius of the minimum area surface at which the two regions are joined. This defines the throat of the wormhole, for which b(r 0 ) = r 0 is fulfilled. In addition, from embedding calculations of the wormhole geometry, it follows that for the throat to be a minimum the flare-out condition
b(r) − b ′ (r)r b 2 (r) > 0 ,(4.67)
must be satisfied there by any wormhole geometry [256]. In GR, the flare-out condition at the wormhole throat (4.67) implies the violation of the null convergence condition via Raychaudhuri equation which, for a congruence of light rays with vanishing shear and rotation, is given by (for further details see [359], chapter 9)
dθ du + 1 2θ 2 + R αβû αûβ = 0,
whereû µ is the four-velocity of a light ray andθ the expansion of the congruence. In turn, via the Einstein equations, the Raychaudhuri equation entails the violation of the null energy condition [354], implying that in the context of GR wormholes are unavoidable sustained by exotic matter. However, such a restriction does not necessarily apply to extensions of GR and thus one could, in principle, obtain wormhole geometries without violations of the energy conditions. To investigate this issue in the context of EiBI gravity it is useful to write the field equations as
G µ ν = R µ ν − 1 2 δ µ ν R = κ 2 S µ ν (4.68)
where R µ ν ≡ R µ ν (q) and R = R µ µ , and the effective energy-momentum tensor S µ ν is given by
S µ ν = τ T µ ν − 1 − τ κ 2 ǫ + τ 2 T δ µ ν (4.69)
with τ ≡ g/q = |δ µ ν −κ 2 ǫT µ ν | −1/2 and T = g µν T µν is the trace of the energy-momentum tensor. This representation of the field equations makes clear that the effective energymomentum tensor S µ ν , assumed to be exotic, could be able to sustain wormhole geometries without violations of the null energy condition on the physical energy-momentum tensor T µ ν . In this section we shall consider the construction of such wormhole geometries in EiBI gravity. Consider a static spherically symmetric geometry, described by the line elements of the physical and auxiliary metrics as ds 2 g = −e ν(r) dt 2 + e ξ(r) dr 2 + f (r)dΩ 2 (4.70) ds 2 q = −e β(r) dt 2 + e α(r) dr 2 + r 2 dΩ 2 (4.71)
where {ν(r), ξ(r), f (r), β(r), α(r)} are some functions of the radial coordinate r. Observe that the gauge freedom has been imposed in this setup upon the line element for q µν in order to obtain two free functions there, which contrast with the Bañados-Ferreira geometry, where this restriction is made instead upon g µν (see Eqs.(4.23) and (4.24) in section 4.2.1). As a matter source, let us consider an anisotropic fluid given by the energymomentum tensor
T µν = (ρ + p t )u µ u ν + p t g µν + (p r − p t )χ µ χ ν (4.72)
where u µ is the four velocity in the metric g µν , normalized as u µ u ν g µν = −1, χ µ is the unit vector in the radial direction, i.e. χ µ = e ξ/2 δ µ r , while {ρ(r), p t (r), p r (r)} are the energy density, tangential pressure (measured in the direction of χ µ ) and radial pressure (measured in the orthogonal direction to χ µ ) of the fluid, respectively. With the line element (4.71), and assuming asymptotic flatness, λ = 1, the gravitational field equations for the auxiliary metric q µν read [194] 1
r 2 − e −α r 2 + α ′ e −α r = 1 2ǫ a hc 2 − h ac 2 − 2 ah + 2 , (4.73) − 1 r 2 + e −α r 2 + β ′ e −α r = 1 2ǫ a hc 2 − h ac 2 + 2 ah − 2 (4.74) e −α r 2β ′′ r − α ′ − β ′ 2 + β ′ r = 2 ǫ a hc 2 + h ac 2 − 2 ,(4.75)
with the functions a = 1 + κ 2 ǫρ, h = 1 − κ 2 ǫp r , and c = 1 − κ 2 ǫp t , respectively, while we have τ = (ahc 2 ) −1/2 . Like in the Bañados-Ferreira solutions, two of the metric functions can be removed using the relations (2.57), which imply e β = hc 2 a e ν , e α = ac 2 h e ξ , and f = r 2 ah . In addition, from the assumption of minimal coupling of the matter to the spacetime metric, the energy-momentum tensor of the fluid satisfies the conservation equation ∇ µ T µν = 0, computed with the covariant derivative constructed with the spacetime metric g µν . This equation reads explicitly
dν dr = 4 r p t − p r ρ + p r − 2 p r + ρ dp r dr = 4 r h 2 − c 2 a 2 − h 2 + 4d a 2 − h 2 dh dr . (4.76)
Now, the flare-out condition (4.67), which can be written in this case as ξ ′ e −ξ < 0, together with the field equations (4.73) and (4.74), and the relations above between q µν and g µν , imply that, for the energy conditions to be satisfied in these geometries, the inequality
κ 2 ǫ(ρ + p r ) < ǫh 2 r (c 2 ) ′ c 2 1 − b r (4.77)
(where we have redefined e −ξ(r) = 1−b(r)/r to convert (4.70) into the standard form of the wormhole geometry (4.66)) must be satisfied. Evaluation of this condition at the throat b(r) = r 0 , implies that if the factor (c 2 ) ′ /c 2 is finite, then (4.77) is violated, which means that exotic matter is needed in order to thread these geometries, like in GR. However, if (c 2 ) ′ /c 2 diverges or, alternatively, (c 2 ) ′ /c 2 e −ξ → K (with K some constant) as r → r 0 , then the condition (4.77) is satisfied if 0 < ρ + p r < K.
It is important to note that the set of field equations and relations provided so far do not constitute a closed system, since there are more independent functions than equations. Thus some restrictions have to be made. Harko et al [194] provide a particular wormhole geometry in this framework by introducing the equation of state p r (r) = ρ(r) 1 + κ 2 ǫρ(r)
, (4.78) which is equivalent to choosing the restriction a(r)h(r) = 1 on the matter components, and in turn implies f (r) = r 2 via the transformations between the metric q µν and g µν above. To close the system of solutions one introduces the additional constraint β = 0, and upon solving of the field equations one obtains the result
ds 2 = −dt 2 + 1 + 2ǫr 2 0 /r 4 1 − r 2 0 /r 2 dr 2 + r 2 dΩ 2 (4.79)
where ǫ > 0 has been assumed. This geometry describes two asymptotically flat spacetimes connected with a wormhole throat located at r 0 , so that r 0 < r < +∞. Alternatively one can describe both sides of the wormhole using the radial coordinate l defined as r 2 = l 2 +r 2 0 , so now −∞ < l < +∞ and the throat is located at l = 0. The wormhole geometry (4.79) reduces, in the GR limit ǫ → 0, to the Ellis and Bronnikov (EB) wormhole sustained by an exotic (phantom) scalar field [88]. In the present case, the energy density, ρ(r) = 1 κ 2 ǫ 1 1+2ǫr 2 0 /r 4 − 1 is negative throughout all space, so the NEC is violated everywhere no matter the value of the EiBI constant ǫ, which is an outrageous result. On other hand, the flare-out condition at the throat, ξ ′ e −ξ = − 2r 0 2ǫ+r 2 0 < 0, is satisfied. It should be stressed that if ǫ < 0 then the flare-out condition can only be satisfied if r 2 0 > 2|ǫ|, which suggests a lower bound of r 0 = 2|ǫ| for the wormhole throat in this case.
On each side of the wormhole throat l = 0 the masses seen by an observer can be computed as [355]: M ± = ±4π ±∞ 0 ρr 2 dr dl dl, which in the case under consideration yields the result [344]: M ± = ± r 0 2 ± 2κ 20r 0 ∓ 5κ 2 36r 2 0 + . . .. Thus, despite the fact that on each side of the throat an observer orbiting the wormhole would measure a mass M ± , the total mass M = M + + M − adds exactly to zero, which is a manifestation of the mass-withoutmass mechanism proposed by Wheeler [363] long ago (see section 4.4 for a more complete discussion of this issue). Regarding the effects on physical observers crossing the wormhole throat, Tamang et al. [344] analyse the effect of ǫ on the tidal forces experienced by a free falling observer by considering the relative tidal acceleration, ∆a j , between two nearby parts of the observer falling into the wormhole. In an orthonormal basis {e0, e1, e2, e3} of the observer radially moving towards the wormhole, this acceleration is given by [256] ∆a j = −R0ĵ0pξ p (4.80)
where ξ p is the deviation vector between these two parts and Rîĵkl are the components of the Riemann tensor. For the wormhole geometry (4.79) one has [344] R0ĵ0p = γ/(2ǫ + r 2 0 ) (where γ = (1 − v 2 /c 2 ) −1/2 and v = ± |g rr /g tt |dr/dt), which is finite for any nonvanishing ǫ, and thus the presence of a throat at r 0 may avoid the infinitely large tidal forces found in the EB black hole. Shaikh [327] also uses an anisotropic fluid (4.72) to investigate wormhole structures within EiBI gravity, taking the equations of state p r = −ρ and p t = αρ (where 0 ≤ α ≤ 1 in order for the energy conditions to be satisfied). This approach differs from the one of Harko et al. [194] in the gauge used for the line elements:
ds 2 g = −ψ 2 (r)f (r)dt 2 + dr 2 f (r) + r 2 (dθ 2 + sin 2 θdφ 2 ) (4.81) ds 2 q = −G 2 (r)F (r)dt 2 + dr 2 F (r) + H 2 (r)(dθ 2 + sin 2 θdφ 2 ). (4.82)
Integration of the conservation equation ∇ µ T µν = 0 yields the result ρ = C 0 r 2(α+1) , where C 0 is a constant (of dimension 2(1 − α)) whose explicit form will be determined from the asymptotic behaviour of the metric. Following the same strategy as in the previous spherically symmetric spacetimes considered in this section, the field equations (with λ = 1) provide the relations between the metric functions in the line elements (4.81) and (4.82) as
f (r) = F (r)(1 −ǭαρ), ; ψ(r) = G(r)(1 −ǭαρ) −1 ; H(r) = r 1 +ǭρ ,(4.83)
whereǭ ≡ κ 2 ǫ. With these relations, Eq.(2.57) can be explicitly written for this case, giving a set of three independent differential equations. Together with the fluid conservation equation, one can obtain the following solutions for the components of the line element (for ǫ < 0) as [327] ψ(r) = 1 + r 2(α+1) 0 .
r 2(α+1) − 1 2 (4.84) f (r) = 1 − r 2(α+1) 0 r 2(α+1) 1 + α r 2(α+1) 0 r 2(α+1) 1 − r 2(α+1) 0 3|ǫ|r 2α − 2M r 1 − r 2(α+1) 0 r 2(α+1) − 2(α + 1)r 2(α+1) 0 3|ǫ|r 1 − r 2(α+1) 0 r 2(α+1) I(r) ,
(4.86)
The interpretation of the radius r 0 is that of the minimum value of the radial coordinate, at which ψ(r 0 ) → ∞. The reason to choose ǫ < 0 follows from the analysis of the Raychaudhuri equation (4.68) for a radial null ray travelling towards the wormhole in the equatorial plane θ = π/2. For the line element (4.81) one hasû t = 1/(ψ 2 f ) andû r = ±1/ψ and thus the different contributions to (4.68) read
θ = ± 2 r 1 +ǭ C 0 r 2(α+1) ; dθ dλ = − 2 r 2 1 + (α + 2)ǭC 0 r 2(α+1) ; R (αβ)û αûβ = 2(α + 1)ǭC 0 r 2(α+2) , (4.87)
with ± for ingoing (outgoing) rays. From these expressions it is clear that, since one needs to have C 0 > 0 and 0 ≤ α ≤ 1 to satisfy the NEC, a wormhole can only exist if ǫ < 0. On the other hand, comparing the line element (4.81) with the canonical form of a wormhole geometry (4.66) it follows that e 2Φ = ψ 2 f and (1 − b/r) = f , which translates the flare-out condition (4.67) into
f ′ 2(1−f ) 2 > 0.
Regarding the regularity of the spacetime one can compute curvature scalars, with the result that they generically diverge, except when the mass is tuned to the value
M = − (α + 1)r 2(α+1) 0 3|ǫ| I(r 0 ) = (α + 1)r 3 0 3(2α − 1)|ǫ| 2 F 1 1 2 , 2α − 1 2α + 2 , 4α + 1 2α + 2 ; 1 (4.88)
(where the second equality is valid provided that α = 1/2) for which all curvature scalars are finite. If this constraint on the mass is assumed, then the parameter x = r 2 0 /|ǫ| separates those states without a horizon, x < 1, corresponding to traversable wormholes, from black holes with horizons, x > 1, and for which the curvature divergence is avoided. However, the mass M in Eq.(4.88) can only be positive if α > 1/2. In Fig.14 the metric components for the case α = 3/4 are depicted, where this transition between traversable wormholes and regular black holes with a horizon is observed. The metric functions gtt = ψ 2 (r)f (r) and g −1 rr = f (r) for the case α = 3/4 taking an EiBI parameter ǫ = −4. In these plots, x = r 2 0 /|ǫ|, in such a way that x = 1 sets the appearance/dissappeareance of a horizon. When x > 1, no horizon is found and the minimum value of the radial coordinate corresponds to r0, where the wormhole throat is located. Figure taken from Ref. [327].
Following the analysis of Tamang et al. [344], Shaikh also discusses the tidal forces upon an observer travelling through the wormhole [327]. Using the tidal acceleration equation (4.80) one can compute the components of such equation in the present case as (the subindex 0 denotes evaluation of the quantities at the wormhole throat)
∆a1 r 0 = − αc 2 r 2 0 1 − 2 3 x ξ1 ; ∆a2 r 0 = 1 r 2 0 (1 − x) γ 2 0 v 2 0 ξ2 ; ∆a3 r 0 = 1 r 2 0 (1 − x) γ 2 0 v 2 0 ξ3
(4.89) with the definition x = r 2 0 /ǫ (in terms of this variable, the flare-out condition reads
f ′ 2(1−f ) 2 | r 0 = (1 − x)/r 0 > 0, which is satisfied if x < 1,
implying that the wormhole throat r 0 < |ǫ| 1/2 ). Restricting the acceleration felt by a traveller of typical size ξ ∼ 2m to be below a certain value g, i.e, ∆a1 ′ < g, one obtains that the minimum wormhole throat radius is
r 2 0 ≥ 2αc 2 g 1 − 2 3 x . (4.90)
As r 0 is directly related to EiBI constant, this is translated into a maximum bound for ǫ. Now, from solar physics (see section 3.2.1) one has the constraint [103] |ǫ|/κ 2 1.8 × 10 14 m 2 . Take now for instance a model with α 1/2 and x 1, which implies a bound on the acceleration |∆a1 ′ | min ≃ 0.17 × 10 3 sec −2 , or roughly 17 times Earth's gravity g E . However, such a wormhole would have a typical minimum size r 0 = c 2 /(3g) = c 2 /(51g E ) ≃ 1.34 × 10 7 m, which is roughly 2.1 Earth's radius. To reduce the wormhole size, one needs to consider smaller values of ǫ, which in turn implies stronger accelerations at the throat. Note that the angular components of the tidal acceleration in Eq.(4.89) impose limits upon the radial velocity v 0 at the wormhole throat r 0 . Let us emphasize that the general solution for ǫ > 0 gives a singular spacetime, while for ǫ < 0 solutions for which the curvature scalars are finite can be found. However, we have not discussed the implications of having solutions with finite or divergence curvature yet. As an exception to this statement, the case α = 1, for which the structure of the energy-momentum tensor (4.72) coincides with that of a standard (Maxwell) electromagnetic field, have been derived and studied in detail, which we review thoroughly next in sections 4.4 and 4.5.
It should be pointed out that there are several difficulties on the consistence and viability of this kind of approaches to construct wormhole geometries supported by exotic fields in the context of EiBI gravity, such as the potential instabilities at the quantum level [343], which would require to perform stability analysis in the context of this theory, something not available in the literature yet.
Electromagnetic black holes and geons
The solutions we are going to discuss now correspond to EiBI gravity (2.33) coupled to the Maxwell Lagrangian (4.21). However, they will differ from the Bañados-Ferreira solutions [45] in that i) only the case of ǫ < 0 is considered (as comes from the analysis of Shaikh discussed above for the flare-out condition to be satisfied), and ii) the gauge is imposed in such a way that two independent functions are assumed for the auxiliary line element, in contrast with the line elements (4.23) and (4.24), but runs parallel with the analysis of Harko et al. [194] reviewed in section 4.3. This scenario was first considered in [282]. To start, the matter field equations, ∇ µ F µν = 0, for a generic static, spherically symmetric line element of the form
ds 2 = g tt dt 2 + g xx dx 2 + r(x) 2 dΩ 2 (4.91)
and an electrostatic field, yield the only non-vanishing component of the field strength tensor F tx = Q r(x) 2 √ −gttgxx . Though this component depends explicitly on the metric components g tt and g xx , the energy-momentum tensor (4.22) does not, which implies
T µ ν = X 8π −Î 2×202×2 0 2×2Î2×2 ⇒ |Ω| 1/2 (Ω −1 ) µ ν = (λ +X)Î 2×202×2 0 2×2 (λ −X)Î 2×2 (4.92)
(where we have combined Eqs.(2.52) and (2.57) for the second equality) and hats denote matrices. Here we have introduced by convenience a new length scale as ǫ → −2l 2 ǫ (to deal with ǫ < 0 solutions only) and introduced the objectX = − lǫκ 2 2π X. Given the structure of the right-hand-side of (4.92), one can introduce the ansatẑ
Ω = Ω +Î0 0 Ω −Î ⇒ Ω − = (λ + X) ; Ω + = (λ − X) (4.93)
for theΩ matrix, where the explicit expressions of Ω − and Ω + follow from solving Eq.(4.92). This way, the gravitational field equations, with the assumption of vanishing torsion and symmetric Ricci tensor [276] become
R µ ν (q) = −1 2l 2 ǫ (Ω − −1) Ω −Î0 0 (Ω + −1) Ω +Î ,(4.94)
where R µ ν (q) ≡ q αµ R (αν) . At this point it should be noted that the length-squared scale l 2 ǫ characterises the high-curvature corrections, as follows from the expansion of the EiBI action in series of l 2 ǫ ≪ 1:
S = 1 2κ 2 d 4 x √ −g R − 2Λ + l 2 ǫ − R 2 2 + R (µν) R (µν) + O(l 4 ǫ ) + S M (g µν , ψ m ) (4.95)
where Λ = 1−λ 2l 2 ǫ plays the role of the effective cosmological constant. Remarkably, the field equations for the action (4.95) up to order l 2 ǫ , turn out to be exactly the same as those of EiBI gravity in Eq.(4.94), as can be explicitly verified from Ref. [275]. The underlying reason for this result lies on the algebraic properties of the EiBI action and goes as follows: given the linear relation between T µ ν and |Ω| 1/2 , the diagonal character of T µ ν will make the matrix P α ν ≡ g αµ R (µν) (Γ) to be diagonal as well. Now, if P has two double eigenvalues, like happens in this case,P = diag(p 1 , p 1 , p 2 , p 2 ), then the fourth-order polynomial |Ω| 1/2 (Ω −1 ) µ ν | = |Î + ǫP | turns into the second-order polynomial appearing in Eq.(4.95) when the square root is evaluated. Moreover, this quadratic polynomial exactly coincides with the series expansion of the EiBI action. As a result, all the higherorder corrections beyond l 2 ǫ order cancel out, which means that the electrostatic spherically symmetric solutions of EiBI gravity exactly coincides with those obtained for the quadratic Lagrangian at order l 2 ǫ appearing in Eq.(4.95). Indeed, electrovacuum solutions in the context of such a quadratic gravity models (in Palatini approach) were previously found in Refs. [273,274,275]. Now, to solve the field equations (4.94) we introduce the static, spherically symmetric line element for the geometry q µν as
ds 2 q = −A(x)e 2ψ(x) dt 2 + 1 A(x) dx 2 +r(x)dΩ 2 . (4.96)
The three functions in this line element can be reduced to a single one by noting that the combination R t t − R x x = 0 of the field equations, which follows from the symmetry T t t = T x x of the energy-momentum tensor (4.92), implies thatr xx = ψ xrx . This allows to redefine the metric function and the radial coordinate as A → A/r 2 x andr 2 x dx 2 → dx 2 , respectively, so the line element can be written in the Schwarzschild-like form
ds 2 q = −B(x)dt 2 + 1 B(x) dx 2 + x 2 dΩ 2 . (4.97)
This leaves a single independent function to be determined from the R θ θ component of the field equations, which after introducing a standard mass ansatz,
B(x) = 1−2M (x)/x, reads −4l 2 ǫ M x = x 2 (Ω − −1)/Ω − .
Resolving this equation requires comparison with the spacetime metric Eq.(4.91) using the transformations (2.57), which can be split into two 2 × 2 blocks as q ab = g ab Ω + and q mn = g mn Ω − . The latter implies the relation of coordinates in the two line elements
x 2 = r 2 Ω − → dx dr = ± Ω + Ω 1/2 − (4.98)
which will play an important role later. This way, the equation to be solved reads −4l 2 ǫ M r = r 2 (Ω − − 1)Ω + /Ω 1/2 − , whose integration can be formally written as M (z) = M 0 (1 + δ 1 G(z)), where M 0 is an integration constant associated to the Schwarzschild mass, G(z) contains the electromagnetic contribution, and δ 1 isolates all the relevant constants out of this integration. A full solution for the spacetime line element (4.91), in the asymptotically flat case, λ = 1, can now be found explicitly as
ds 2 g = −A(x)dt 2 + dx 2 A(x)Ω 2 + + r 2 (x)dΩ 2 (4.99)
with the expressions
A(x) = 1 Ω + 1 − r S r(x) (1 + δ 1 G(r(x))) Ω 1/2 − (4.100) δ 1 = 1 2r S r 3 Q l ǫ (4.101) Ω ± = 1 ± r 4 c r 4 (x) (4.102) r 2 (x) = x 2 + x 4 + 4r 4 c 2 (4.103)
where r c = r Q l ǫ , with r 2 Q = κ 2 Q 2 /(4π) a length scale associated to the electric charge which, together with the Schwarzschild radius, r S = 2M 0 , and the EiBI length scale, l 2 ǫ , characterises the solution via the constant δ 1 in Eq.(4.101). Note that the relation (4.103) follows from explicitly solving Eq.(4.98). The function G(z) in Eq.(4.100), with the dimensionless variable z = r/r c , follows directly from the field equations as dG/dz = −Ω + /(z 2 Ω 1/2 − ), and can be explicitly written as
G(z) = − 1 δ c + 1 2 z 4 − 1 2 F 1 1 2 , 3 4 , 3 2 , 1 − z 4 + 2 F 1 1 2 , 3 7 , 3 2 , 1 − z 4 ,(4.104)
where 2 F 1 [a, b, c; t] is a hypergeometric function and δ c ≈ 0.572069 is a constant needed to recover the GR solution in the asymptotic regime, z ≫ 1. In this limit one has G(z) ≃ −1/z, Ω − ≃ 1 (so z 2 (x) ≃ x 2 ), and the metric function reduces to
A(x) ≈ 1 − r S r + r 2 Q 2r 2 ,(4.105)
which is the standard Reissner-Nordström solution of GR. This is confirmed by considering the behaviour of the curvature scalars for z ≫ 1 as where R(g) = g µν R (µν) , Q(g) = R (µν) R (µν) and K(g) = R α βµν R α βµν are the curvature scalar, the Ricci-squared and the Kretchsman, respectively. These expressions smoothly converge to their GR counterparts with higher-order corrections in l 2 ǫ .
R(g) ≈ −
Geometry and properties
It should be noted that the line element (4.99) can be written in a standard Schwarzschildlike form by absorbing the Ω + factor via a redefinition of the radial coordinate as dx 2 = dx 2 /Ω 2 + . This must be done with care since the radial coordinate x ∈] − ∞, +∞[, while r ≥ r c , as depicted in Fig.15, where one observes that the area of the two-spheres S = 4πr 2 (x) reaches a minimum of size S c = 4πr 2 c at x = 0, where it bounces off and re-expands again. As already discussed in section 4.3, the presence of a minimum value for the radial coordinate allows to infer the presence of a wormhole structure (here the flare-out condition (4.67) is automatically satisfied). Indeed, from the relation (4.98) between coordinates, it follows that it is ill-defined at r = r c , because dr/dx = 0 at this point, and thus the use of r as a radial coordinate is limited to those regions where r(x) is a monotonic function. Thus, in agreement with wormhole physics lore, one needs two charts of r to cover the entire manifold, but a single chart in terms of x. The line element (4.96) can be alternatively written in Eddington-Filkenstein coordinates using a local redefinition of the time coordinate, dt = dv − dx/(AΩ + ), which brings the metric into the form
ds 2 = −A(x)dv 2 + 2 Ω + dvdx + r 2 (x)dΩ 2 ,(4.107)
For null and time-like radial geodesics, ds 2 ≤ 0, which implies −Adv 2 + 2 σ + dvdx ≤ 0. Inside the event horizon A < 0, which means that all such geodesics move in the decreasing direction of x as the advanced time coordinate v moves forward. Now, since the radial function r(x) has a minimum at x = 0, the relation (4.98) becomes dx/dr = Ω + /Ω 1/2 − in the region x > 0 and dx/dr = −Ω + /Ω 1/2 − for x < 0. This way, ingoing geodesics, which always move in the direction of decreasing x, propagate in the direction of decreasing area of the radial function r(x) if x > 0, but in the growing direction if x < 0, i.e., they approach the wormhole throat if x > 0, but move away from it if x < 0 (a similar effect is found for outgoing geodesics). Later in section 4.5 we will discuss the potential troubles of the transit of physical observers across the wormhole throat x = 0.
As already stated, the geometry described by (4.99) reduces to the Reissner-Nordström of the Einstein-Maxwell field equations, Eq.(4.105), for z ≫ 1, but important departures are found as r ≈ r c . From the expansion of the G(z) function (4.104) in that region, G(z) ≈ −1/δ c + 2(z − 1) 1/2 − (11/6)(z − 1) 3/2 + . . ., one finds that the expansion of the metric function A(z(x)) there yields the result (4.108) where N q = Q/e (e is the electron charge) is the number of charges and N c = (2/α em ) 1/2 ≃ 16.55 (with α em the fine structure constant). From the expression (4.108) it is clear that there exists a transition on the behaviour of A(x) for δ 1 = δ c , yielding three different situations. This is consistent with the analysis of the horizons of these configurations, as given by the zeroes of A(x). A detailed analysis of such zeroes reveals the following structure for the horizons [282]:
A(x) ≈ N q 4N c (δ 1 − δ c ) δ 1 δ c r c r − r c + N c − N q 2N c + O √ r − r c ,
• If δ 1 < δ c a single horizon is located on each side of the wormhole throat r = r c , resembling the structure of the Schwarzschild spacetime.
• If δ 1 > δ c there may be two, one (degenerate) or none horizons, depending on the number of charges N q . These are Reissner-Nordström-like configurations.
• If δ 1 = δ c a single horizon is found for N q > N c and none otherwise. The spacetime metric g µν is finite at the throat r = r c , and the geometry there is Minkowski-like.
It is worth pointing out the resemblance of the three classes (in terms of horizons) of configurations above with those solutions resulting from the coupling of Born-Infeld electrodynamics to GR [321,139,132,162,142], described in section 4.1.1. Nonetheless, the presence of a finite-size wormhole throat introduces new features as compared to that case. In this sense, as the region z → 1 is approached, the resulting expressions strongly deviate from the GR case:
r 2 c R(g) = − 1 2δ 2 1 − δ c δ 1 1 (z − 1) 3/2 + a 1 √ z − 1 + −4 + 16δ c 3δ 2 + O (z − 1) (4.109) r 4 c Q(g) = 1 − δ c δ 1 6δ 2 − 5δ 1 3δ 2 2 (z − 1) 3/2 + a 2 √ z − 1 + 10 + 86δ 2 1 9δ 2 2 − 52δ 1 3δ 2 + O (z − 1) (4.110) r 4 c K(g) = 1 − δ c δ 1 2 (2δ 1 − 3δ 2 ) 3δ 2 2 (z − 1) 3/2 + a 3 √ z − 1 + 16 + 88δ 2 1 9δ 2 2 − 64δ 1 3δ 2 + O (z − 1) (4.111)
where {a 1 , a 2 , a 3 } are some constants. These expansion reveal the divergence of all curvature scalars at the wormhole throat, z = 1, but for the particular choice δ 1 = δ c they all become finite. The latter condition sets a particular charge-to-mass ratio, but says nothing on the particular amounts of them. Remember that, from the discussion on the structure of horizons above, the case δ 1 = δ c corresponds to the transition between the Reissner-Nordström-like and Schwarzschild-like case, where the geometry at z = 1 becomes Minkowskian. As already said, this is somewhat analogous to the case of Born-Infeld electrodynamics coupled to GR, though in such a case curvature divergences at r = 0 are always present. It should be noted that in the EiBI case, the presence of curvature divergences or not has no influence on the existence of a wormhole structure, so one could wonder about the physical meaning of such divergences (see section 4.5.2).
In the context of GR one could wonder what is the location of the sources generating the mass M and charge Q of the geometry of the Reissner-Nordström solution. It turns out that it is not possible to have a well defined point-like source generating both mass and charge of the Reissner-Nordström geometry and, at the same time, being a solution of the Einstein equations everywhere [287]. It is equally natural to wonder about the nature of both mass and charge that generate the wormhole geometries discussed here and in section 4.3. As first shown by Misner and Wheeler [253], the non-trivial topology of the wormhole allows to define by itself a charge without the need of considering sources for the electric field, an effect coined charge-without-charge in that paper. Indeed, an electric flux flowing through a 2-dimensional, spherical S 2 surface enclosing one of the sides of the wormhole mouths defines a charge as
Φ = 1 4π S 2 * F = ±Q (4.112)
where * F is the two-form dual to Faradays's tensor and the two signs ± come from the different orientation of the normal on each side of the wormhole throat. Note that this result holds true regardless of the particular details of the configurations as long as a wormhole throat exists and the topology does not change. In particular, it is not affected by the presence of curvature singularities. For the solutions considered in this section, the density of lines of flux crossing the spherical wormhole throat can be computed as
Φ 4πr 2 c = Q r 2 c = 2c 7 ( G) 2 (4.113)
which is independent of the particular amount of charge and mass, i.e, independent on the presence or not of curvature divergences.
In a similar fashion one could wonder about the origin of the mass generating the geometry (4.99). Following also the mass-without-mass mechanism introduced by Misner and Wheeler [253], and in analogy with the energy of an electric field in a Minkowski spacetime, S M = dt × E e one can estimate the total mass of the spacetime by evaluating the gravitational + matter action for these configurations, i.e., S = dt × (E G + E e ), which can be performed in terms of the variable dx 2 = dz 2 /Ω − , with the result [282]
S = 2M c 2 δ 1 δ c dt (4.114)
where the factor 2 comes from the need of integrating on both sides of the wormhole throat. Like the electric flux above, this result is finite and independent of the existence or not of curvature divergences. The explicit implementation of both charge-withoutcharge and mass-without-mass mechanisms make these objects be explicit realizations of Wheeler's geon [363], understood as self-gravitating electromagnetic entities without sources. It remains to be seen whether the case with ǫ > 0 (4.32), and the wormhole solutions constructed out of anisotropic fluids and described in section 4.3 admit a similar characterisation, though such an analysis is not available in the literature yet.
We have seen that in the solutions described in this section the presence of curvature divergences at the wormhole throat in the cases δ 1 = δ c seems to have no influence on the physical properties of the solutions such as total charge, mass and density of lines of electric field, which are as well defined as in the case δ 1 = δ c , where no curvature divergences arise. This is somewhat similar to the thin-shell approach to construct wormhole solutions, by which two spacetimes are joined together at a given hypersurface, where the throat is located [257,141,173]. The resulting manifold is geodesically complete by construction, but curvature divergences arise at the wormhole throat [304], which is interpreted as a surface layer with an energy-momentum tensor on it. To get an intuitive idea of the similarities and differences between the smooth, δ 1 = δ c , and divergent, δ 1 = δ c solutions described in this section, one can construct Euclidean embeddings of the spatial equatorial, θ = π/2, and t =constant section of the line element (4.99), expressed in terms of the coordinates dx 2 = Ω 2 + dr 2 /Ω − , which reads dl 2 = 1 Aσ − dr 2 + r 2 dϕ 2 , to embed it into a three-dimensional space with cylindrical symmetry as
dl 2 = dξ 2 + dr 2 + r 2 dϕ 2 , (4.115)
where the function ξ must be chosen so as to match the equatorial t =constant line element above. Around the wormhole throat r c one can make use of the expansions of the metric functions there, together with Ω − ≃ 4(r − r c )/r c , which yields so that one has
dl 2 = (Nc−Nq) 8Nc rc (r−rc) dr 2 + r 2 dϕ 2 if δ 1 = δ c Nc Nq δ 1 δc (δ 1 −δc) rc r−rc dr 2 + r 2 dϕ 2 if δ 1 = δ cξ(r) = ± (Nc−Nq) 4Nc √ r c √ r − r c if δ 1 = δ c ± 4Nc 3Nq δ 1 δc (δ 1 −δc) r c r−rc rc 3/4 if δ 1 > δ c (4.118)
In Fig.16 we have depicted these Euclidean embeddings for δ 1 = δ c (top figures) and δ 1 > δ c (bottom figures), in those cases where no horizons are present (recall the discussion of section 4.4.1). In both cases, the presence of a wormhole structure is manifest. The two-dimensional curvature, however, as given by the expression of the Kretchsman:
K 2D = 64(Nc−Nq) 2 N 2 c 1 r 2 c r 2 if δ 1 = δ c N 2 q N 2 c (δ 1 −δc) 2 4δ 2 1 δ 2 c 1 rc(r−rc)r 2 if δ 1 > δ c (4.119)
is finite for the former, but divergent for the latter. This highlights the fact that two similar wormhole structures can show very different properties regarding the behaviour of curvature invariants. This can also be observed in the case of wormholes supported by anisotropic fluids found in [327] and discussed in section 4.3, whose Euclidean embedding for the model with α = 3/4 in Eqs.(4.84) and (4.85) is also depicted in Fig.16. Remember that in such a case the tidal forces at the throat are finite, regardless of the behaviour of the curvature there. We shall review the issue of non-singular solutions in EiBI gravity in section 4.5.
Coupling to Born-Infeld electrodynamics
The coupling of EiBI gravity to Born-Infeld electrodynamics (4.12) has been considered by Jana and Kar in [218]. The strategy followed to obtain electrovacuum solutions to the field equations is pretty much the same than the one employed for the Maxwell field above, and therefore we shall omit the details. The resulting line element, with the redefinition b 2 = α/(4ǫ), becomes
ds 2 = −U (x)e 2ψ(x) dt 2 + V (x) U (x)
e −2ψ(x) dr 2 + r 2 dθ 2 + sin 2 θdφ 2 (4.120)
U (x) = 2 − α 2(1 − α) − α 2(1 − α) 1 1 + 4ǫQ 2 (1−α) αx 4 V (x) = 2 − α 2(1 − α) − α 2(1 − α) 1 + 4ǫQ 2 (1 − α) αx 4 ,
where the metric function ψ(x) splits into two subcases and takes the form
∞ > α > 1 : e 2ψ(x) = 1 − 2M x + αx 2 6ǫ(α − 1) 1 − 4ǫQ 2 (α − 1) αx 4 − 1 (4.121) + α 1/4 (4Q 2 ) 3/4 3ǫ 1/4 (α − 1) 1/4 x F arcsin 4ǫQ 2 (α − 1) 1/4 α 1/4 x , −1 −∞ < α < 1 : e 2ψ(x) = 1 − 2M x − αx 2 6ǫ(1 − α) 1 + 4ǫQ 2 (1 − α) αx 4 − 1 (4.122) + 4Q 2 3x 2 2 F 1 1 4 , 1 2 ; 5 4 ; − 4ǫQ 2 (1 − α) αx 4 ,
and the radial coordinates are related as , and thus in this case the charge is distributed over a 2-sphere of radius r 0 . This is in agreement with the analysis carried out in [277] where wormhole solutions with geonic properties were obtained in an extension of GR including quadratic corrections in the curvature and coupled to Born-Infeld electrodynamics. Now, for α > 2 there is also a minimum, which now occurs at r c = (κQ 2 α/(1 − α)) 1/4 , while in the range 0 < α ≤ 2, no minimum is found and a point-like charge arises (note that α = 0 corresponds to the GR case). This implies that, in particular, for α > 0, depending on the interplay between EiBI gravity and Born-Infeld electrodynamics wormhole solutions might be found, but this is not explicitly investigated by Jana and Kar. Nonetheless, they investigate in detail the case α = 1. By variation of the EiBI constant ǫ, solutions with one or two horizons may be found in that case (which is thus similar to what is found in Born-Infeld electrodynamics in GR, see section 4.1.1, and to geonic solutions supported by a Maxwell field, see section 4.4.1). Curvature divergences are always present either at the location of the throat r = r c (when a wormhole is present) or at r = 0, although the energy density of the electromagnetic field remains finite everywhere.
r 2 = V (x)x 2 .
To investigate the geodesic behaviour in these geometries, one first notices that in nonlinear theories of electrodynamics photons do not propagate along null geodesics of the background metric, but instead on null geodesics of an effective geometry [265] given by
g µν ef f = 1 + 1 b 2 F 2 g µν + 1 b 2 F µ σ F σν ,
where F µν is the background electromagnetic field. For any value of α = 1, the effective metric for the photon propagating in the EiBI background reads [218] ds
2 ef f = −U (x)e 2ψ(x) dt 2 + U (x)e −2ψ(x) dx 2 + αV 2 (x)x 4 + 4κQ 2 αV (x)x 2 (dθ 2 + sin 2 θdφ 2 )
(4.124) from which the expression for the deflection angle (see section 4.2.3 for the basic definitions) of the photon moving on this effective metric is obtained as
∆(ϕ) = 2 ∞ xtp U (x) r 2 (x) U (x tp )e 2ψ(xtp) r 2 (x tp ) − U (x)e 2ψ(x) r 2 (x) −1/2 dx − π
Non-singular solutions
Let us now consider an aspect of utmost importance regarding the internal structure of black holes resulting from gravitational collapse, namely, the presence of a singularity at their center. This is an unavoidable consequence of the singularity theorems provided that i) a trapped surface exists, ii) the null congruence condition holds and iii) global hiperbolicity is fulfilled [299,300,197,177] (see [326,125] for more pedagogical discussions of this issue). These theorems are formally based on the notion of geodesic (in)completeness, namely, on the impossibility of extending null and time-like geodesics to arbitrarily large values of their affine parameters. As null geodesics are associated to the transmission of information and time-like geodesics to the free-falling paths of physical observers, geodesic (in)completeness has become the most widely accepted criterium to detect the presence of spacetime singularities 39 . However, as geodesics are geometrical structures that represent idealized point-like observers without internal structure, it is unclear what a quantum theory of gravity should say about them. Indeed, from an intuitive point of view, since gravity is a matter of curvature, the blow up of curvature scalars could be seen as an indication of the presence of large tidal forces that would potentially rip apart a physical (extended) observer, which has shaped numerous approaches to get rid of spacetime singularities through bounded curvature scalars [35,21,20,259,254,241,65]. Indeed, the standard lore of the field states that, as the curvature grows to reach Planckian values, an improved theory of gravity properly incorporating quantum effects should avoid the formation of singularities during the last stages of the gravitational collapse [212,41,373,320,49,247]. In this section we will discuss the regular/singular character of the geonic configurations discussed in section 4.4 making use of these concepts.
Geodesic completeness
Our aim in this section is to determine whether the spacetimes considered in section 4.4 are geodesically complete, i.e., whether any time-like or null geodesic can be extended beyond the wormhole throat, since the latter can be reached in finite affine time. We will use the notations and conventions described in section 4.2.2. We shall focus on asymptotically flat spacetimes, λ = 1. In terms of the line element (4.99) the two conserved quantities of motion read E = A dt du and L = r 2 dϕ du or, alternatively, in terms of Eddington-Filkenstein coordinates (4.107), E = A dv du − 1 Ω + dx du . This way, the line element (4.99) can be used to write the modulus of the tangent vector u µ = dx µ /du, which satifies u µ u µ = −k, with k = 0(1) for null (time-like) geodesics, as
−k = −A dt du 2 + 1 AΩ 2 + dx du 2 + r 2 (x) dϕ du 2 .
(4.126)
In terms of the conserved quantities above, Eq.(4.126) reads
1 Ω 2 + dx du 2 = E 2 − V (x) ; V (x) = A k + L 2 r 2 (x)
, (4.127) which is just the equation of motion of a particle in a one-dimensional effective potential V (x). For radial (L = 0) null (k = 0) geodesics, Eq.(4.127) simplifies to
1 Ω − dr du 2 = E 2 , (4.128)
which admits an analytical integration of the form
±E · u(x) = 2 F 1 [− 1 4 , 1 2 , 3 4 ; r 4 c r 4 ]r if x ≥ 0 2x 0 − 2 F 1 [− 1 4 , 1 2 , 3 4 ; r 4 c r 4 ]r if x ≤ 0 , (4.129) where x 0 = 2 F 1 [− 1 4 , 1 2 , 3 4 ; 1] = √ πΓ[3/4] Γ[1/4]
≈ 0.59907 and the sign ± corresponds to ingoing/outgoing geodesics. For x → ∞, series expansion of the solution (4.129) yields Eu(x) ≈ r ≈ x and the GR behaviour is naturally recovered. In the GR case one has (dr/du) 2 = E 2 everywhere, whose integration is r(u) = ±Eu. Since in that case the function r(u) is strictly positive, then the affine parameter u(x) is only defined on the positive/negative (ongoing/ingoing) axis and thus geodesics cannot be extended beyond x = 0, hence such spacetime is geodesically incomplete. In the present case, however, the presence of a wormhole throat introduces significant deviations from the GR solution, and from (4.129) one finds that at r = r c (x = 0) the affine parameter behaves as Eu(x) ≈ ±x + √ r − r c ≈ x ± x/2, with the sign + (−) corresponding to the region with
x > 0 (< 0). As depicted in Fig.18, the affine parameter u(x) can be smoothly extended beyond x = 0 and thus radial null geodesics are complete regardless of the value of δ 1 . This is a relevant result since in the cases δ 1 = δ c curvature divergences arise at the wormhole throat, but they do not have any impact on the behaviour of the affine parameter, which is the same in all cases, being free of curvature divergences or not (see section 4.5.2 for a discussion on the impact of such divergences). For null geodesics with L = 0 and time-like geodesics, the effective potential in (4.127) can be approximated near the wormhole throat x = 0 as [280] V
(x) ≈ − a |x| − a ; a = k + L 2 r 2 c (δ c − δ 1 ) 2δ c δ 2 ; b = k + L 2 r 2 c (δ 1 − δ 2 ) 2δ 2 . (4.130)
From this expression it is clear that if δ 1 > δ c , corresponding to Reissner-Nordström-like configurations (see section 4.4.1), an infinite potential barrier prevents any such geodesics to reach the wormhole throat x = 0, which is the same behaviour found in the GR case. But if δ 1 < δ c , corresponding to Schwarzschild-like solutions, these geodesics see an infinite attractive potential as x → 0 and are unavoidably dragged to the wormhole throat. In the GR case, radial time-like geodesics behave near r = 0 as λ(r) ≈ ± 2 3 r(r/r S ) 1/2 and, likewise in the case of radial null geodesics above, the fact that r > 0 makes ingoing/outgoing geodesics to end/start at x = 0, with no possibility of further extension, and therefore geodesics in this case are incomplete. In the geonic wormhole case, however, the geodesic equation (4.127) can be integrated as [279] du dx ≈ ± 1 2
x a
L 2 = 1 2 (1 − Nq Nc ) + O(x)
(remember that in this case an event horizon is present if N q > N c ) for both null geodesics with L = 0 and time-like geodesics, which means that the potential is regular at x = 0. This way, all geodesics with energy E greater than the maximum of the potential V max will be able to go through the wormhole, while bounded orbits can exist for 0 < V max < E. The comparison of the behaviour of the effective potential for the three classes of configurations and different values of the number of charges is depicted in Fig.19, corresponding to time-like geodesics with L = 0.
From the description above, it follows that the presence of a spherically symmetric wormhole structure replacing the point-like singularity of GR allows for geodesically complete spacetimes, which is in agreement with the standard lore of wormhole physics [354]. Nonetheless, the physical meaning of curvature divergences at the wormhole throat requires a separate analysis.
The physical implications of curvature divergences
We have already discussed in section 4.3, following Shaikh [327], that for wormhole solutions supported by anisotropic fluids, tidal forces at the wormhole throat can be finite. In this section we shall follow a different approach to determine the impact of curvature divergences on physical (extended) observers and review the results of [281]. This approach is based on the concept of strong singularities, originally introduced by Ellis and Schmidt [152]. Such singularities are identified by the property that all objects approaching them are crushed to zero volume, no matter what their internal constituents or forces holding them might be. This is opposed to weak singularities, for which a body could retain its Figure 19: Effective potential V (x) for time-like geodesics with L = 2. Plots A, B and C depict the Reissner-Nordström-like (δ1 > δc), Minkowski-like (δ1 = δc) and Schwarzschild-like (δ1 < δc) cases, respectively, for three curves corresponding to Nq = 1, Nc, 8Nc (solid, dashed, dotted, respectively). Plots D, E and F depict three values of charge Nq = 1, Nc, 4Nc, respectively, for three curves corresponding to δ1 = δc, 1.5δc, 0.3δc, (solid green, dashed red, dotted blue, respectively). Figure taken from [279].
identity while crossing the divergent region. Built on the precise mathematical framework introduced by Tipler [346,347], Clarke and Krolak [117] and others [263,285], the idea is to idealize a physical (extended) observer as a set of points following their own geodesic path (i.e. a congruence), and to determine the relative separation between nearby geodesics as the divergent region is crossed. The congruence is characterised as x µ = x µ (u, ξ), where u corresponds to the affine parameter along a given geodesic and ξ labels the different geodesics on such a congruence. The separation between nearby geodesics (for fixed u) is measured by the Jacobi field Z µ ≡ ∂x µ /∂ξ, which satisfies the geodesic deviation equation
D 2 Z α du 2 + R α βµν u β Z µ u ν = 0 . (4.132)
Given the second-order character of this equation, it follows that there are six independent Jacobi fields along a given geodesic, which are obtained as
Z a (u) = A a b (u)Z b (u i ), where Z b (u i )u = u i , one can instead write Z a (u) = A a b (u) DZ b du u=u i ,
where A a is a 3 × 3 matrix that vanishes at u = u i . This way, three linearly independent solutions of (4.132) allow to define a volume element:
V (u) = det |A(u)|V (u i ) (4.133) (or as det |A(u)| if Z a (u i ) = 0)
. Thus a strong singularity is met if lim u→0 V (u) = 0 [117], where the singularity is approached if u → 0.
Following the analysis of Dolan [263] for spherically symmetric spacetimes, the Jacobi fields can be written as {Z (1) = B(u)(u x /A, Au t , 0, 0), Z (2) = (0, 0, P (u), 0), Z (3) = (0, 0, 0, Q(u)/ sin(θ))}, which are orthogonal to the time-like radial geodesic vector u µ = (u t , u x , 0, 0), whose components are defined via u t ≡ dt/du = E/A and u x ≡ dx/du. The geodesic deviation equation (4.132) allows to obtain the functions B(u), P (u) and Q(u) via the equations P (u) = P 0 + C du r 2 (u) , Q(u) = Q 0 + C ′ du r 2 (u) and B uu + Ayy 2 B(u) = 0. Close to the wormhole throat, the behaviour of these functions can be computed and the result compared to their GR counterparts as
B GR (u) ≈ C 1 1 |u| 1/3 − |u| 4/3 |u i | 5/3 → B EiBI (u) ≈ C ′ 1 1 |u| 1/3 − |u| 4/3 |u i | 5/3 (4.134) P GR (u) ≈ C 2 1 |u i | 1/3 − 1 |u| 1/3 → P EiBI (u) ≈ C ′ 2 (u − u i ) (4.135) Q GR (u) ≈ C 3 1 |u i | 1/3 − 1 |u| 1/3 → Q EiBI (u) ≈ C ′ 3 (u − u i ) , (4.136) where {C 1 , C ′ 1 , C 2 , C ′ 2 , C 3 , C ′ 3
} are arbitrary constants. Now, from [263] the resulting volume from these spacetimes can be written as
V (u) = |B(u)P (u)Q(u)|r 2 (u), (4.137)
Now, since in GR one has r GR ≈ (9r S /4) 1/3 u 2/3 and in EiBI geons r 2 (u) ≈ r 2 c + x 2 /2, then one finds that the volume in the former is V GR ≈ u 1/3 , while in the latter V EiBI ≈ 1/u 1/3 . Thus, in the GR case the volume vanishes as u → 0 and, according to Tipler's criterium [347,346], the divergence of curvature scalars is associated to a strong singularity. In the EiBI case, however, the finite radius of the wormhole prevents the convergence of geodesics of the GR case, and the volume element diverges instead as u → 0, a scenario that has been independently discussed by Nolan [264] and Ori [285].
To investigate in more detail the effect of such a divergent volume on physical observers let us rewrite the line element (4.99) in free-falling coordinates as [280,281]
ds 2 g = −du 2 + (u y ) 2 dξ 2 + r 2 (u, ξ)dΩ 2 , (4.138)
where ξ measures the radial separation between nearby geodesics and u y ≡ dy/du, where dy = dx/(1 + r 4 c /r 4 (x)). For the Scharwarzschild-like configurations, δ 1 < δ c , which is the only case in which time-like observers can go through the wormhole (recall the discussion of section 4.5.1), the vector (u y ) 2 can be approximated near the wormhole throat as (u y ) 2 ≃ a/|x| ≃ ( 3 a |u − Eξ|) − 2 3 . This turns (4.138) into
ds 2 g ≈ −du 2 + 3 a |u − Eξ| −2/3 dξ 2 . (4.139)
This expression states that, as the wormhole throat is approached, the distance between two infinitesimal nearby geodesics diverges, dl P hys = 3 a |u − Eξ| −1/3 dξ. However, for finite comoving separation between nearby geodesics, l ξ ≡ ξ 1 − ξ 0 , the physical separation l P hys ≡ |u y |dξ can be computed as
l P hys ≈ a 3 1/3 1 E |u − Eξ 0 | 2/3 − |u − Eξ 1 | 2/3 , (4.140)
which is finite. Due to the divergent volume carried by a physical observer, the meaning of this result is that, as the wormhole throat is approached, infinitesimally nearby geodesics are infinitely stretched in the radial direction, followed by an identical contraction as the wormhole is left behind, in a sort of spaghettisation process. The danger lies on the possibility that the constituents that make up and keep cohesioned the body could lose causal contact due to the spatial stretching affecting their infinitesimal elements, which would result in the unavoidable destruction of the body. To check this one can consider the propagation of radial null rays, ds 2 = 0, in the background (4.139), so the photon path satisfies
dξ du = ± 3 a (u − Eξ) 1/3 . (4.141)
Using a numerical integration, in Fig.(16) two main results are observed: i) a fiducial observer at ξ = 0 never loses causal contact with its nearby geodesics (left figure) and ii) the proper time taken in a round trip by a light ray from ξ = 0 to a nearby geodesic is always finite and casual as the wormhole throat is crossed (right figure), with just an additional delay in the travelling time. Thus, in these geometries, physical observers near the wormhole throat can remain in causal contact despite the spaghettisation process experienced as u → Eξ, and can apparently cross this region with curvature divergences, without experiencing absolutely destructive deformations.
Tests with scalar waves
As a third test to determine whether the presence of curvature divergences endangers the well posedness of the physical laws in these geonic geometries one can study the propagation of scalar waves near the wormhole throat, following the description of [280]. This analysis considers the case of Reissner-Nordström-like configurations, δ 1 > δ c , where the presence of a time-like Killing vector allows for a separation of variables. The field equation for a massive scalar field, ( − m 2 )φ = 0, can be decomposed in modes of the form φ ω,lm = e −iωt Y lm (θ, ϕ)f ω,l (x)/r(x), where Y lm (θ, ϕ) are spherical harmonics and the functions f ω,l (x)/r(x) are governed, in the radial coordinate dy/dx = 1/A(1 + r 4 c /r 4 ), by the Schrödinger-like equation
−f yy + V ef f f = ω 2 f ; V ef f = r yy r + A(r) m 2 + l(l + 1) r 2 , (4.142)
where the effective potential V ef f converges to the GR result for r ≫ r c , but behaves near the wormhole throat r = r c as
V ef f ≈ k |y| 1/2 ; k ≡ (δ 1 − δ c )N q δ 1 δ c N c (N c [m 2 r 2 c + 1 + l(l + 1)] − N q ) N c (8r 3 c ) 1/2 . (4.143)
While low-energy modes cannot overcome the potential barrier and are almost entirely reflected, much like in the GR case, high-energy models may overcome such a barrier and end hitting the wormhole throat. Considering an incoming wave packed travelling from null infinity (when no horizon is present, or from the event horizon otherwise), the wave equation (4.142) reduces to
f y ′ y ′ + α 2 ± 1 √ y ′ y = 0 , (4.144)
where the parameter α = |k| − 2 3 ω encodes all the relevant information for this problem. The sign ± determines an infinite well or potential, the former leading to a transmission coefficient that tends to one as α grows, while the latter has a typical sigmoid profile of barrier experiments, where a threshold around α = α th ∼ 1.5 from almost complete reflection to almost complete transition is found (see left panel of Fig.21). For constant ω there is another threshold, l = l max , such that the cross section, σ, can be roughly estimated by considering that the transmission factor is one for l > l max (almost entire transmission) and zero for l < l max (almost entire reflection) as
σ = π ω 2 lmax l=0 (2l + 1)1 = π ω 2 (1 + l max ) 2 (4.145)
which is depicted in the right panel of Fig.21, where for ω → ∞ one has σ ∝ ω −1/2 . As a summary of this section, the well posedness of the wave scattering problem, together with the geodesic completeness for null and time-like geodesics and all spectrum and mass and charge, and the fact that the constituents making up physical (extended) observers can remain in causal contact as the wormhole throat is crossed, imply the existence of classical non-singular black hole geometries in EiBI gravity. It should be pointed out that similar electromagnetic solutions as those analyzed here and in section 4.4 can be found in functional extensions of the form f (X ) = X n , where X ≡ det(ĝ −1q ) and the parameter n labels different models (n = 1/2 for EiBI gravity). It turns out that for any 1/4 < n ≤ 1/2 the corresponding electrically charged solutions, studied in Ref. [43], share a similar wormhole structure as those of EiBI gravity, yielding also geodesically complete structures, while for n > 1/2 no wormhole solutions were found in that reference.
Higher and lower dimensional models and solutions
Electromagnetic fields in higher dimensions
The setup derived and discussed in Sec. (4.4) can be extended to their higher-dimensional, D > 4, counterparts. Most of the corresponding expressions are easily obtained following a similar approach, see Bazeia et al [55]. The field equations in the q µν geometry read now
R µ ν (q) = ǫ 2 |Υ| 1 D−2 [L BI δ µ ν + T µ ν ] ; L BI = |Υ| 1 D−2 − λ ǫκ 2 , (4.146)
with the definitionΥ
≡ |Ω| 1/2 (Ω −1 ) µ ν = λδ µ ν − ǫκ 2 T µ ν (4.147) (remember thatΩ −1 ≡q −1ĝ , while the definition (4.147) implies |Ω| 1/2 = |Υ| 1 D−2 )
. The relation between the auxiliary q µν and the physical metric g µν is now given by
q µν = |Υ| 1 D−2 (Υ −1 ) µ α g αν ; q µν = 1 |Υ| 1 D−2 g µα Υ α ν . (4.148)
It is easy to see that the system of equations (4.146), with the definitions (4.147) and the transformation (4.148), satisfies the same second-order field equations and ghost-free character of their four-dimensional partners, besides the recovery of the D-dimensional Minkowski spacetime in vacuum. Electrovacuum solutions of the field equations (4.146) are easily derived following the same steps as in section 4.4, using now the set of transformations (4.147). One thus set two static, spherically symmetric line elements of the form
ds 2 g = g tt dt 2 + g xx dx 2 + r(x) 2 dΩ 2 (D−2) (4.149) ds 2 q = −A(x)e 2ψ(x) dt 2 + 1 A(x) dx 2 + x 2 dΩ 2 (D−2) , (4.150) (where dΩ 2 (D−2)
is the angular sector in the maximally symmetric subspace) for the metric g µν and q µν , respectively, so that the electromagnetic field satisfies
F tx = Q r(x) D−2 √ −gttgxx .
The ansatz forΩ compatible with the symmetry of the electromagnetic field becomeŝ
Ω = Ω +Î2×20(D−2)×2 0 2×(D−2) Ω −Î(D−2)×(D−2) ⇒ Ω − = (λ +X) 2 D−2 ; Ω + = (λ −X) (λ +X) D−4 D−2 (4.151)
where we have used Eq. (4.147). Like in the four dimensional case, the combination R t t − R x x = 0 allows to rewrite the line element for q µν in Eq.(4.149) in standard Schwarzschildlike form, while the introduction of a mass ansatz, A = 1 − 2M (x) (D−3)x D−3 , allows to solve the field equations for M (x), and transforming that solution back to g µν using that {q ab = g ab Ω + ; q mn = g mn Ω − }, where (a, b) contains the 2 × 2 block and (m, n) the maximally symmetric sector, one obtains the final solution for g µν as
ds 2 g = − A Ω + dt 2 + Ω + A dx Ω + 2 + z 2 (x)dΩ 2 (4.152) A(z) = 1 − 1 + δ 1 G(z) δ 2 Ω D−3 2 − z D−3 ; G z = −z D−2 Ω − − 1 Ω 1/2 − λ + 1 z 2(D−2) (4.153) δ 1 ≡ (D − 3)r D−1 c 2M 0 l 2 ǫ ; δ 2 ≡ (D − 3)r D−3 c 2M 0 , (4.154) with the definition z ≡ r/r c , where r 2(D−2) c ≡ 2l 2 ǫ r 2(D−3) Q with ǫ ≡ −2l 2 ǫ and r 2(D−3) Q ≡ κ 2 Q 2 /(4π)
, while M 0 is Schwarzschild mass. Again, to detect the presence of wormhole structures, we just need to inspect the relation between radial coordinates in the two line elements (4.149), obtained as
x 2 = r 2 Ω − ⇒ |x| r c (D−2) = 1 z D−2 z 2(D−2) − 1 , (4.155)
which is just a standard quadratic equation for z d−2 , which can consequently be solved as
r d−2 = |x| D−2 + |x| 2(D−2) + 4r 2(D−2) c 2 (4.156)
where the modulus in x comes from the fact that a square root has been extracted to obtain (4.156). The behaviour of the radial function r(x) is depicted in Fig.22 (left), where we observe the typical bouncing behaviour of a wormhole for any dimension D, with the throat located at x = 0 (z = 1). As follows from the analysis of Bazeia et al [55], expansions of the metric functions and the curvature scalars at the throat reveal that, as opposed to the four dimensional case, they always diverge there (in four dimensions, in the case δ 1 = δ c they become finite, see section 4.4.1). However, a similar analysis of the geodesic structure near the wormhole throat as in section 4.5.1 reveals the completeness of null and time-like geodesics for all the spectrum of mass and charge of the solutions. The case of radial null geodesics is depicted in Fig.22 (right), where we observe that they can be naturally extended beyond the wormhole throat x = 0. However, the impact of such curvature divergences on physical observers crossing the wormhole throat has not been analyzed in the literature yet. The geonic properties of such solutions have been also analyzed in [55], with similar qualitative results as those found in section 4.4.1.
Kaluza-Klein solutions
The original idea of extra dimensions was implemented by Kaluza and Klein by assuming that the four dimensional energy-momentum tensor of the electromagnetic field is originated from a part of a five dimensional metric tensor. This idea was reemployed in the EiBI scenario in Ref. [159], where they assume a five-dimensional metric given bŷ
g AB = g µν + αA µ A ν αA µ αA µ 1 (4.157)
where α is a parameter, latin indexes run from A = 0, . . . , 4 and greek from µ = 0, . . . , 3, while a hat denotes five dimensional objects. Now, tuning the compactification radius from five to four dimensions, denoted asR, to be given by 2πR/Ĝ 5 = 1/G 4 = 1 (wherê G 5 and G 4 are the five and four dimensional Newton's constant, respectively), and taking by convenience α = 4, one obtains that the five dimensional EiBI action reduces to [159] S = 1
8πĜ 5 ǫ d 5 x (ĝ AB + ǫR (AB) ) − λ |ĝ| ⇒ 1 8πG 4 ǫ d 4 x 1 + ǫF 2 (4.158) × g µν + ǫ(R (µν) + 2F µβ F β ν ) + (∇ δ F δ µ ∇ β F β ν ) ∞ n=0 (−1) n+1 ǫ n+2 F 2n − λ |g|
where it is clearly seen that it transforms into a four dimensional, Class-III gravitational action containing a number of curvature-matter couplings, and where F µν arises as the field strength tensor associated to the vector potential A µ . The field equations corresponding to the action (4.159) are highly involved, even to lowest order in ǫ (see Eqs.(4.4)-(4.7) of Ref. [159]). Nonetheless, in the spherically symmetric case, (electrostatic) solutions to first order in ǫ can be obtained under the form
ds 2 g = −f (r)dt 2 + f (r) −1 dr 2 + r 2 dΩ 2 (4.159) f (r) = 1 − 2M r − Λr 2 3 + Q 2 r 2 + ǫ 3Q 4 10r 6 − ΛQ 2 r 2 + O(ǫ 2 ) (4.160)
which corresponds to a modification of the Reissner-Nordström-Anti-de Sitter solution of GR (ǫ → 0). Computation of the curvature scalar for these solutions, R = 4Λ + 6ǫQ 4 /r 8 + O(ǫ 2 ), yields a curvature singularity at r = 0, but no further properties of these solutions (such as horizons, geodesic structure, etc) are investigated in that work.
Thick branes
Braneworld scenarios represent an interesting development of the Kaluza-Klein idea, boosted by the proposals introduced by Randall-Sundrum [313,314] and Arkani-Hamed-Dimopoulos-Dvali [24,22] models. They assume that the four-dimensional world to which standard model particles are attached (the brane), is embedded in a higher-dimensional spacetime (the bulk) with a warped geometry, in such a way that gravitons can propagate along the extra dimension (see e.g. [315] for a review). Though in the original proposals the brane is infinitely thin, in this section we shall consider instead a thick brane, namely, a five-dimensional bulk with a scalar field propagating in the extra dimension, and whose energy density is assumed to be localized around the point (say) y = 0 of the extra dimension. The analysis of this scenario can be carried out to a large degree of generality, by considering a Born-Infeld inspired modification of gravity (in Palatini formalism) defined by an arbitrary function F of the object P µ ν ≡ g µλ R (λν) and coupled to a scalar field as
S = 1 2κ 2 d D x √ −gF (P ) + d D x √ −gL(X, φ) (4.161)
where D = d + 1 is the number of spacetime dimensions. The Lagrangian density L(X, φ) contains, in general, a non-canonical contribution from the scalar field kinetic term X ≡ g αβ ∂ α φ∂ β φ (see [25] for the inception of these theories in Cosmology). The field equations for this system are derived in the usual way, i.e., by performing independent variations of the action (4.161) with respect to the metric and the connection, which can be handled also by introducing a new metric q µν as [53] q µν = 1
|FP | 1 D−2 g µλ (FP ) λ ν ; q µν = |FP | 1 D−2 (F −1 P ) µ λ g λν (4.162)
where (FP ) λ ν ≡ ∂F ∂P λ ν and |FP | represents its determinant. The resulting field equations are quite similar to those of EiBI gravity given by (4.146), which are written here by convenience as
R ν α (q) = κ 2 |FP | 1 D−2 L G δ ν α + T (φ)ν α ,(4.163)
where L G corresponds to the particular Lagrangian density considered. To implement the thick brane scenario one sets the line element for the physical metric g µν :
ds 2 g = a 2 (y)η ab dx a dx b + dy 2 (4.164)
where a(y) is the warp factor, which is assumed to depend only on the extra dimension y, and η ab is the metric on a d-dimensional spacetime brane of constant curvature K. The corresponding field equations (4.163) in this case can be conveniently written with the help of a similar ansatz for the auxiliary metric q µν :
ds 2 q =ã 2 (ỹ)η ab dx a dx b + dỹ 2 . (4.165)
Using (once more) the relation (2.57) it follows that in this case
Ω µ λ ≡ |FP | 1 D−2 (F −1 P ) µ λ = Ω + I d×d0 0 Ω − ,(4.166)
from where one obtains thatã 2 (ỹ) = Ω + a 2 (y) and dỹ 2 = Ω − dy 2 . The gravitational field equations follow now immediately as (see Bazeia et al [53] for details)
d(d − 1)[K − H 2 ] = κ 2 |Ω| 1/2 (d − 1)L G + d · T + − T − (4.167) (d − 1)[K + Hỹ] = κ 2 |Ω| 1/2 (T + − T − ) (4.168)
where H ≡ãỹ/ã, while T + = −L(φ, X)/2 and T − = L X φ 2 y + T + are the components of the scalar energy-momentum tensor, and Ω ± are model-dependent functions of X and φ. For the sake of the search for solutions below, L G in this equation represents EiBI gravity Lagrangian (2.33).
For the case of standard canonical kinetic term with a potential, L = X −V (φ), specific solutions were obtained by Liu et al. [242], using a fully equivalent approach to the one depicted above though written directly in terms of the functions {a,ã}, see Eqs.(17a), (17b) and (17c) of that paper. Specifically, they look for a kink solution interpolating between different vacua at asymptotic infinity y = ±∞. This can be achieved by introducing an additional constraint φ ′ (y) = Ka 2 (y), where K is a constant conveniently defined as
K = ± 7 3 3 4 1 2 √ ǫκ 2 .
This way, the scalar field equation
4 a ′ a φ ′ + φ ′′ = ∂V (φ) ∂φ (4.169)
can be integrated with the result V (y) = 3 2 K 2 a(y) 4 +V 0 , where V 0 is an integration constant that can be interpreted as the scalar field vacuum energy, fixed here as V (φ) = −λ/(κǫ). Inserting this result into the gravitational field equations (4.167) and (4.168) one finds an analytic solution for the warp factor and scalar field profile in closed form as
a(y) = sech 3 4 2 √ 21ǫ y , (4.170) φ(y) = ± 7 5/4 2 × 3 1/4 κ iE iy √ 21ǫ , 2 + sech 1 2 2y √ 21ǫ × sinh 2y √ 21ǫ , (4.171)
where E is an elliptic integral of second kind. This way the potential can be expressed as V (y) = 7 √ 21 24ǫκ 2 sech 3 ( 2y √ 21ǫ ) − λ ǫκ 2 , which allows to compute the energy density associated to these solutions as
ρ(y) = 7 √ 21 18ǫκ 2 sech 3 2y √ 21ǫ . (4.172)
These scalar field and energy density profiles naturally implement the defining properties of a kink, namely, its interpolating character between different vacua at y = ±∞, as well as the localized nature of the energy density around the center of the kink, y = 0. Regarding the curvature of the solutions, a simple calculation yields the result R = g M N R (M N ) = 1 ǫ 2 − 7 tanh 2 ( 2 √ 21ǫ )y , which asymptotically approaches the value R → −5/ǫ < 0, corresponding to an Anti-de Sitter space.
An important aspect of configurations on the brane is to determine its stability against tensorial perturbations there. As shown by Bazeia et al. [53] this can be also done in full generality for a theory F (P ) 40 . The idea is to write two perturbed line elements in Gaussian normal coordinates as
ds 2 g = a 2 (y) (η ab + h ab ) dx a dx b + dy 2 (4.173) ds 2 q =ã 2 (ỹ) (η ab + h ab ) dx a dx b + dỹ 2 (4.174)
where the scalar and vector modes are decoupled by imposing the conditions δg ab = a 2 (y)h ab and δg ay = 0 = δg yy . From (4.173) and (4.174), the perturbation of the field equations (4.163) in the q µν geometry reads simply δR µ ν (q) = 0 → δR (µν) (q) = R µ β t βν , where t ab =ã 2 h ab is the only non-vanishing component of t βν . Now, using standard covariant perturbation methods, and after some algebra, the tensorial modes, assumed to be written as h a b = X(z)ǫ a b (t, x) (where we have introduced a new coordinate z as dz 2 = dỹ 2 /ã 2 ), satisfy two sets of equations, namely
(η) ǫ a b − 2Kǫ a b − p 2 ǫ a b = 0 (4.175) −Y zz + V ef f (z)Y = p 2 Y (4.176)
where p 2 is a constant. Y , and the effective potential V ef f (z) is given by
V ef f = (d − 1) 2 H z + (d − 1) 2 4 H 2 ,(4.177)
where H ≡ã z /ã. The operator on the left-hand side of Eq.(4.176) can be factorized as
d dz − (d − 1) 2 H d dz − (d − 1) 2 H (4.178)
which is a non-negative operator, guaranteeing in this way that p 2 > 0, which implies the tachyonic-free and stable character of this class of theories of gravity under tensor perturbations. For the particular case of EiBI gravity, Liu et al. [242] computed the zero mode, p = 0, as Ψ 0 (z) = N 0 a 7/2 (z), where the normalization condition Ψ 2 (z)dz = 1 fixes the constant N 2 0 ≈ 0.35/ √ ǫ. This gravity zero mode is localized at the center of the kink, y = 0, while vanishes at y = ±∞. The effective potential (4.177) has a (asymptotically vanishing) volcano-like profile with a well at the center of the kink, with the result that a continuous set of massive Kaluza-Klein modes (not localized on the brane) arises for p > 0. The above study was further generalized by Fu et al. [172], where they considered a class of solutions defined by the ansatz φ ′ (y) = Ka(y) 2n , so that Liu et al. case [242] corresponds just to n = 1. The corresponding solutions for the warp factor and the scalar field can also be obtained in closed analytical form as a(y) = sech 3 4n (ky) ; φ(y) = 2K k iE(iky/2, 2)sech 1/2 (ky) sinh(ky) (4.179) where the constants K = ± (1+4n/3) 3/4 (n+1) n ǫκ 2 and k = 2n √ 3ǫ(4n+3)
, while the energy density can be computed simply as ρ = n+1 n K 2 sech 3 (ky). These configurations show similar features as those of n = 1 below, namely, interpolation of the kink between two asymptotic vacua at y = ±∞ and localized character around y = 0 with a maximum of the energy density there. The impact of increasing the value of n is just to decrease the width of the kink and to lower the maximum of the energy density. One could go on further in the standard strategy in the field, by investigating additional models which allow to modify the physical properties of the kink at will, but we shall stop here. Let us simply emphasize that the zero mode for any n not localized in the brane, while more complex models like φ ′ (y) = K 1 a(y) 2 (1 − K 2 a(y) 2 ) allow to find quasi-localized states on the brane for massive KK gravity modes.
Three dimensions
Electrovacuum solutions of EiBI gravity in D = 3 dimensions requires a separate analysis from that of section 4.6.1, due to the peculiarities of the integration of the metric on such a case. In this sense, the expressions (4.146), (4.147), (4.148), (4.149), (4.151) are still valid, but the integration of the metric (with a cosmological constant term, λ = 1) yields now the result
ds 2 g = − A(r) Ω + dt 2 + 1 A(r) dx Ω 1/2 + 2 + r 2 (x)dθ 2 (4.180) A(x) = −λ 2 M − λ 2 − 1 2s|ǫ| r 2 − Q 2 sr 2 c 2r 2 + 1 λ ln r 2 + sr 2 c /λ r 2 0 , (4.181)
where s in ǫ = s|ǫ| is the sign of ǫ, and r 0 is an integration constant. The line element (4.180) represents a natural generalization of Bañados, Teitelboim and Zanelli (BTZ) solution [48], which is recovered both in the limit ǫ → 0, and asymptotically, r ≫ 1. The BTZ solution raised a great deal of interest due to the fact that the states with M = −1 does not contain an event horizon but there is no curvature singularity to hide, either. In the EiBI scenario, the function r(x) in Eq.(4.180) can be explicitly written as
|r(x)| = |x| ± |x| 2 − 4sλr 2 c 2λ , (4.182)
which attains a minimum at r = r c /λ 1/2 both for s = ±1. When s = −1 one obtains a wormhole structure similar to that of the higher-dimensional case (compare with Eq.(4.156)), while for s = +1 a similar construction as the Einstein-Rosen bridge [149] can be obtained. In both cases, null and time-like geodesics can be indefinitely extended despite the presence of curvature divergences at the wormhole throat. However, this is done via two different mechanisms: when s = −1 the wormhole lies on the future (or past) boundary of the spacetime, as radial null geodesics take an infinite time to reach the wormhole throat 41 (see Fig.23, left), while when s = +1, the wormhole is reached on a finite affine time but, like their four and higher dimensional counterparts (see sections 4.5.1 and 4.6.1, respectively), it can be extended beyond this point to arbitrarily large values of the affine parameter (see Fig.23, right). This way, all the electrically charged solutions of EiBI gravity with a wormhole structure are geodesically complete in D ≥ 3 spacetime dimensions. Three dimensional, asymptotically flat, circularly symmetric charged solutions within the context of Born-Infeld inspired gravity formulated in Weitzenböck spacetime (Class-II) have been found by Ferraro and Fiorini [166]. This is a formulation of classical gravity in terms of a spacetime possessing absolute paralelism (or teleparallel gravity, see Ref. [202]). The action considered in this work is defined as
S BIT = 1 2κ 2 ǫ(A + B) d 3 x |g µν + 2ǫF µν | − λ |g µν | ,(4.183)
where F µν = AS µλρ T λρ ν + BS λµρ T λρ ν (with A and B some constants) is quadratic in the Weitzenböck torsion T ρ µν = e ρ a (∂ µ e a ν − ∂ ν e a µ ) build out of the set of 3-forms {e a (x)}, with the definitions (2.144) and (2.145). In the limit ǫ → 0 the teleparallel version of GR (TEGR) is obtained, which are equivalent to each other since the curvature scalar of the Levi-Civita connection, R, can be written as R = S µν ρ T ρ µν (+ total derivative terms). Upon resolution of the corresponding field equations for this theory one obtains the line element [166]
ds 2 = J 2 4r 2 + M 2 dt 2 − Y (r) 2 J 2 /(4 r 2 ) + M 2 dr 2 − r 2 − J 2 r 2 dt + dθ 2 (4.184)
described by a mass M and an angular momentum J, while the function Y is determined via the cubic equation Y 2 − Y 3 = ǫJ 2 /(4r 4 ) = ∆, and out of the three solutions of this system, imposing recovery of the GR limit, Y = 1 for ∆ → 0, one gets the result
3 Y = 1 + 1 − 27∆ 2 − 3 2 3∆ (27∆ − 4) −1/3 + 1 − 27∆ 2 − 3 2 3∆ (27∆ − 4) 1/3 .
(4.185) This geometry can be written, using a suitable change of coordinates given by {t, r} → {T = M t + Jθ/(2M ), ρ = M −2 (J 2 /4 + M 2 r 2 ) 1/2 }, as
ds 2 = dT 2 − Y (ρ) 2 dρ 2 − M 2 ρ 2 dθ 2 (4.186)
so that the TEGR limit, ǫ → 0, is naturally recovered. To further understand the geometry (4.186) one can consider the behaviour of the curvature scalars (in the case ǫ < 0) . The physical interpretation of this geometry is that of a spacetime with a deficit angle ranging between 2π(1 − M ) at spatial infinity and 2π at r = 0, corresponding to the circle of minimum radius ρ 0 = J/(2M 2 ) that can be attained in this geometry. Nonetheless, as radial null geodesics satisfy dT = Y dρ, this means that T diverges as a light ray approaches the minimal circle of radius ρ 0 , so they take an infinite affine time to reach it and the same applies for time-like geodesics. Therefore, this approach succeeds in removing the conical singularity of GR (and, as the same time, it removes the possibility of existence of closed time-like curves) in much the same way than electrically charged black holes in the s = +1 case of EiBI discussed above in this section, i.e., by setting the location of the wormhole throat at the future (or past) boundary of spacetime.
R = 2Y (ρ) ′ ρY (ρ) 3 = 2Y (r) ′ rY (r) 3 ; R (µν) R (µν) = 1 2 R 2 ; R α βγδ R βγδ α = R 2 .
Further analysis in three-dimensional scenarios, involving a Born-Infeld extension of New Massive Gravity [185] with a Chern-Simons term (Class-III), and defined by the action
S = 2m 2 κ 2 d 3 x − det(g µν − m −2 G µν + aF µν ) − 1 + Λ 2m 2 − det(g µν ) + µ 2 d 3 xε µνρ A µ ∂ ν A ρ (4.188)
where m is a mass scale, Λ represents a cosmological constant and a, µ are some constants, has been considered in [13]. In that work only Anti-de Sitter spacetimes are studied, while black holes were investigated instead in [179] and subsequently in [178] where, by expanding the New Massive Gravity action (4.188) to four and six derivative terms, the authors develop a method to find evidence of uncharged and charged black holes, but little is said about the deviations of such solutions with respect to the structure of the GR counterparts relevant for this review.
Magnetically charged solutions with cylindrical symmetry
Cylindrically symmetric solutions have only been considered in EiBI theories in the context of magnetically charged configurations (i.e. Melvin-type [250]) by Bambi et al [42]. The two line elements compatible with such a symmetry can be conveniently written as
ds 2 g = f (ρ)(−dt 2 + dz 2 ) + g(ρ)dρ 2 + h(ρ)ρ 2 dϕ 2 (4.189) ds 2 q =f (ρ)(−dt 2 + dz 2 ) +g(ρ)dρ 2 +h(ρ)ρ 2 dφ 2 . (4.190)
From the line element (4.189) the only non-vanishing component of Maxwell field equations, ∇ µ F µν = 0, reads F ρϕ = β/(ρf √ gh), where β is an integration constants related to the intensity of the magnetic field. The energy-momentum tensor (4.22) for these solutions allows to find the matrix Ω in Eq.(4.147) as 191) where X = −β 2 /f 2 , f c = l ǫ /l β and l 2 β = 4π/(κ 2 β 2 ). With this matrix at hand, by the transformation (2.57) one finds the relations {f = Ω + f,g = Ω − g,h = Ω − h} between the metric functions in Eqs. (4.189) and (4.190). The first of these relations can be written
T µ ν = X 8π diag(1, 1, −1, −1) ⇒Ω = Ω +Î0 0 Ω −Î ; Ω ± = 1 ± f 2 c f 2 ,(4.as f =f + √f 2 −4f 2 c 2
and implies thatf ≥ 2f c , the equality corresponding to f = f c and Ω − = 0. Now, since EiBI Lagrangian density reads now
L G = √ detΩ−1 −2κ 2 l 2 ǫ = β 2 f 2 c 8πf 4 , the field equations for q µν become R µ ν (q) = − ǫ 2 β 2 2f 2 1 Ω +Î0 0 − 1 Ω −Î . (4.192)
Computing the components of the Ricci tensor corresponding to the line element (4.190), and by taking appropriate combinations of the field equations (4.192) one obtains two independent equations
h ρ h + 2 ρ + 2f ρ f = 2f ρρ f ρ ;f ρρ − 3 4f 2 ρ f = κ 2 β 2 8πf f (4.193)
The first equation (4.193) can be directly integrated ashρ 2 = α(f ρ /f ) 2 , where α is an integration constant. To solve the second one in (4.193), in [42] the definitionsf = 2f c φ(x), ρ 2 = 8πfc κ 2 β 2 x 2 are introduced together with the new function Ω = φ 2 x (so that dΩ/dφ = 2φ xx ), in terms of which one finds the solution
Ω = Cφ 3 2 + 4φ 2 3 φ − φ 2 − 1 − 8φ 3 2 F 1 1 4 , 1 2 ; 5 4 ; 1 φ 2 (4.194)
where tuning the integration constant C = −4/3 + 2 √ πΓ 5 4 /Γ 7 4 ≈ 2.16274 guarantees the real character of Ω when expanded around φ ≈ 1. Unfortunately, it is not possible to integrate Ω to obtain φ(x) in analytic form, though one can resort to analytical expansions in the relevant regions. For φ(x) ≫ 1 one obtains the solution φ(x) = 4(1+(Cx) 2 /32) 2 /C 2 , which is nothing but the Melvin solution of GR [250], such that the line element reads
ds 2 f c ≈ 2 2 C 2 1 + C 2 x 2 32 2 −dt 2 + dz 2 + dρ 2 + C 2 2 ρ 2 1 + C 2 x 2 32 2 dϕ 2 , (4.195)
which, via a constant rescaling of (t, z, ρ) → (λt, λz, λρ) with λ 2 = 64f c /C 2 , becomes the GR solution. In the other limit, φ(x) → 1, the corresponding field equation φ 2 x ≈ 2(φ − 1), with a rescaling of the form dx 2 /x = dy 2 , yields the line element 196) up to first order in y 2 . This is just another Minkowski spacetime near the axis as follows from the definitions r = ρ 0 y/2 and α ≡ 2f c ρ 4 0 (the constant factor f c can be reabsorbed via another global rescaling of units). This kind of Melvin-type spacetimes are of great interest in the context of the generation of pairs of entangled black holes in high-intensity magnetic fields via instantons [175,174,144,153]. Indeed, very recently O(4) instantons have been studied in the context of the EiBI theory [26], with the result that both the physical metric and curvature scalars are finite. However, curvature divergences arise on the auxiliary metric, which in turn may induce the formation of singularities, as discussed in detail in section 2.6, and be problematic at the quantum level. In view of this, it would be convenient to investigate further and clarify the physical role played by the auxiliary metric.
ds 2 f c ≈ −dt 2 + dz 2 + ρ 2 0 4 dy 2 + α 8f c ρ 2 0 y 2 dϕ 2 ,(4.
Final remarks
In this section we have reviewed the developments on black hole physics in Born-Infeld inspired modifications of gravity described in section 2. Due to the fact that the Schwarzschild black hole is a vacuum solution of such theories, the literature on the topic has searched for scenarios going beyond it. In this sense, though astrophysically realistic black holes are not expected to have a significant amount of charge, the investigation of charged black holes is relevant in order to find theoretical insights on the modifications to their innermost structure, as well as observational deviations from the predictions of the Kerr black hole. In the influential paper of Bañados and Ferreira [45], where a coupling to Maxwell field was considered, a static, spherically symmetric geometry is obtained (for ǫ > 0), whose properties were further extended and complemented by several other authors [360,336]. The case of similar electrically charged black holes for ǫ < 0 was also considered [282], for which non-singular configurations can be found [280], results partially extended to the coupling to Born-Infeld electrodynamics [218]. On the other hand, wormhole solutions have been found using anisotropic fluid as the matter source [194], though they violate the energy conditions. Finally, higher and lower dimensional models have been the subject of different investigations, but their contributions to fundamental issues has been meager.
There are many open challenges regarding the understanding of black holes in these theories. In particular, rotating solutions in these theories have not been found yet 42 . For ǫ > 0, EiBI gravity black holes still require further analysis regarding its innermost structure and the possibility of finding a wormhole core there, and the physics of mass inflation requires further refinement beyond the approximations employed in the analysis of [31,30]. On the other hand, though the physics at the photon sphere has been explored and understood to some detail [360,337], much research is still needed in order to obtain observational signatures for gravitational waves out of the merging of two such black holes, as well as the potential existence of gravitational echoes in this context [99,4,97,50]. For ǫ < 0 the existence of non-singular solutions in EiBI gravity has been studied with great detail regarding geodesic completeness [279], but the physical meaning of curvature divergences still calls for an understanding [281]. The seemingly absence of pathologies in such curvature-divergent cases raises questions about what are the geometric degrees of freedom that quantum gravity should quantise, and what infinities should renormalise, if any. Two other interesting issues would be to investigate the existence of hairy black holes and superradiance, found in GR [206], in these theories 43 , as well as to extend the thermodynamic laws studied in other Palatini theories of gravity such as f (R) [38] to the Born-Infeld scenario. To conclude, though many appealing results have been found in the context of Born-Infeld inspired modifications of gravity, there is plenty of room for further research in many different directions. 42 In this sense we point out that the applicability of the Janis-Newman method (which allows to obtain a rotating solution from a seed static metric, see Erbin for a review [154]) in the context of Born-Infeld inspired modifications of gravity is still to be understood. 43 Indeed, very recently it was found evidence on the existence of wormhole configurations above a certain mass threshold when a free static and spherically symmetric scalar field is let to gravitate under the Born-Infeld dynamics [8].
Cosmology
The high precision of the cosmological observations made cosmology an ideal place to test fundamental theories of gravity [301,317,339,6,150]. On the assumption of General Relativity being the underlying theory of gravitational interactions together with the homogeneity and isotropy, cosmologists were able to construct the standard model of Big Bang cosmology. Even if this model is simple and stood up to intense scrutiny, it still lacks a fully satisfactory theoretical foundation. One of the challenges is the cosmological constant problem, posing a naturalness problem due to the giant mismatch between its observed value and the radiative contributions from known massive particles to the vacuum energy [361,292,249]. On the other hand, the observation of the accelerated expansion of the universe introduced the necessity of dark energy independently of the cosmological constant problem [25,298,124,120,16,89,224,17]. Furthermore, another problem that one has to face within the realm of General Relativity is the necessity of yet an additional dark component, dubbed dark matter, in order to correctly account for the formation of large scale structures, the anisotropies of the CMB, weak lensing measurements or observations of rotation curves of galaxies. Albeit great efforts [68,342,70], the true nature of dark matter still remains unknown.
The aforementioned challenges concern the late time evolution of the universe and thus, they motivated the consideration of infrared modifications of gravity. Remarkably, the tremendous progress made in observational cosmology also enabled us to probe the underlying physics of the early universe, which in fact shares a similar burden. In order to explain the observations the standard cosmological model is supplemented with the inflationary paradigm requiring an initial phase of accelerated expansion of the universe, that is commonly ascribed to yet another ingredient: the inflaton. It is believed that the primordial quantum fluctuations during inflation eventually become the seeds in the density field responsible for the cosmic large-scale structure via gravitational instability. Inflation is the most prominent model for a successful implementation of an extremely rapid exponential expansion, in which the perturbations of the inflationary field successively translate into the fluctuations of the gravitational potential. Since gravity is coupled to all other fields, these fluctuations are then imprinted onto all existing cosmic fluids. These density fluctuations leave imprints in the cosmic microwave background as temperature anisotropies and also in the matter distribution, that then can be probed by gravitational lensing and formation of galaxies. The inflationary scenario is realised in many different models based on different fields, and observations seem to favour models with a nearly scale invariant red power spectrum, a small value for the scalar to tensor ratio and a small non-Gaussianity. While the late time cosmology triggered searches for infrared modifications of gravity, the need for a primordial inflationary phase motivates modifications of gravity in the opposite regime. Furthermore, within the standard picture one is also prone to encounter a primordial classical singularity which calls for new physics beyond General Relativity at these scales. Moreover, the breakdown of unitarity at the Planck scale requires modifications of gravity in the ultraviolet regime to describe gravitational effects beyond M Pl . These additional challenges motivate to modify gravity at high energies.
It was precisely the cosmological Bing Bang singularities one of the motivations behind the inception of Born-Infeld inspired gravity theories in cosmology. The original construc-tion by Deser and Gibbons [140] formulated in the metric language was an early attempt in this direction. Unfortunately, this approach leads to the presence of ghostly degrees of freedom due to the presence of higher order field equations (see section 2.2 for more details), so any regular cosmological solution will not be reliable. In spite of the mentioned ghost instabilities of the metric formulation, a first quest of the cosmological implications of similar theories was pursued in [123]. There, although different realisations of (quasi) de Sitter solutions were shown to exist for appropriate choices of the parameters, due to the unavoidable ghost nature of the higher order derivative interactions, these solutions are unviable. More promising cosmological solutions without pathologies were found by considering Born-Infeld inspired gravity theoriesà la Palatini, where the connection is left arbitrary. In fact, Bañados and Ferreira showed the existence of non-singular solutions in [45] in the EiBI model, which have since then been extensively studied, and also found in other Born-Infeld theories of gravity. Although the avoidance of the singularities was the initial motivation, they provide very rich cosmological phenomenology, for instance these theories can support quasi de Sitter solutions with more standard forms of matter, like dust or radiation, as a consequence of modifying the high curvature regime of gravity. This behaviour permits to develop inflationary scenarios different from the more traditional models based on some scalar (or more general) degree of freedom. As we will see, in most of the modificationsà la Born-Infeld, the different cosmological evolution can be traced to a highly non-trivial dependence of the Hubble expansion rate on the density and pressure of the matter fields in a modified Friedman equation. In other words, the effects of the modifications in the gravity sector translates into a non-linear contribution from the matter fields density to the expansion rate. A remarkable property of these theories is that, while in most modified gravity theories the background expansion is determined by the equation of state parameter, Born-Infeld theories introduce a dependence on the sound speed already at the background level and not only for the perturbations. This is the cosmological analogue of the modified Poisson equation (3.5) with gradients of the density sourcing the equation for the gravitational potential, with its general case being discussed in section 2.5.1.
The goal of this section will be to review all these cosmological applications and show the novel and interesting phenomenology derived from Born-Infeld inspired gravity theories. However, before starting with that, let us take a moment to fix the notation that we will use throughout this section. Cosmological observations seem to indicate a homogenous and isotropic universe. Compatible with these symmetries, we will assume the metric tensor to be of the Friedman-Lemaitre-Robertson-Walker (FLRW) form, so the line element will read
ds 2 = −N (t) 2 dt 2 + a 2 (t)d x 2 ,(5.1)
where N represents the lapse, a the scale factor and t the cosmic time. Sometimes, it will be useful to work in conformal time η defined as adη = dt. We will also extensively refer to the Hubble function H =ȧ/a or, in conformal time, H = a ′ /a, where a dot and a prime denote derivatives with respect to cosmic and conformal time, respectively. It will be sometimes convenient to keep the lapse explicitly because of the presence of two metrics in the Born-Infeld theories, as we extensively discussed in section 2.
Eddington-inspired Born-Infeld gravity
We will start our survey on the cosmological applications of Born-Infeld inspired gravity theories by considering the most extensively studied case of EiBI, whose action we rewrite here for convenience as
S BI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R (µν) (Γ) − λ √ −g + S matter ,(5.2)
where S matter stands for the action of the standard matter fields, that we assume minimally coupled to the metric g µν . As shown in section 2.5.1, varying the above action with respect to the metric yields the modified field equations
det g µν + 1 M 2 BI R (µν) (Γ) √ −g ĝ + 1 M 2 BIR −1 µν − λg µν = − 1 M 2 BI M 2 Pl T µν . (5.3)
Similarly, we can vary the action with respect to the independent connection. Since the connection Γ does not carry any dynamics, its algebraic equation can be used to solve it in terms of g µν and R (µν) . The resulting solution is such that the connection can be written as the corresponding Christoffel symbols of the effective metric
q µν = g µν + 1 M 2 BI R (µν) . (5.4)
On the other hand, we can use the metric field equations to express R (µν) in terms of g µν and the matter fields. This amounts to writing the equations as in General Relativity but with a modified non-linear matter coupling. See section 2.5.1 for more details on that. This feature becomes more apparent when we write the resulting modified Friedman equation of a homogeneous and isotropic background (5.1). Compatible with the symmetries of the background metric, we assume the following Ansatz for the stress energy tensor T µ ν = diag(−ρ(t), p(t), p(t), p(t)), where ρ and p represents the energy density and pressure of the matter fields, respectively. The Friedman equation modifies into the general form (see section 5.2 for more details)
H 2 = f (ρ, p, c s ) ,(5.5)
with a non-trivial function f , that depends non-linearly on ρ, p and c s . In the case of the EiBI model, one can compute this function exactly [45]. In terms of the auxiliary metric we have
q 00 = − (1 −p T ) 3 (1 +ρ T ) and q ij = a 2 (1 +ρ T )(1 −p T )δ ij , (5.6) whereρ T ≡ ρ T M 2 BI M 2 Pl andp T ≡ p T M 2 BI M 2
Pl with the total energy density ρ T = ρ+(λ−1)M 2 BI M 2 Pl and total pressure p T = ρ − (λ − 1)M 2 BI M 2 Pl and the lapse set to N = 1. In terms of these quantities, the function f (ρ, p) corresponds to [45] f (ρ, p) = 1 3
G F 2 ,(5.7)
with the short-cut notations standing for 8) and the equation of state parameter w = p/ρ. Note, that the dependence on c s drops in f because we have so far w = const. In general, the dependenceẇ will appear as well, as we will see in section 5.1.2 and also in section 5.2 for the more general case. We can study this background equation for two different epochs. At late times for a dust filled universe (w = 0) together with a cosmological constant, one recovers the standard Friedman equation in General Relativity
F = 1 − 3(ρ T +p T )(1 − w −ρ T −p T ) 4(1 +ρ T )(1 −p T ) G = M 2 BI 2 1 − 2q 00 − 3 (1 −p T ) (1 +ρ T ) ,(5.3H 2 ∼ ρ + Λ + ρ 2 Λ − (ρ + Λ) κΛ + O(κΛ) 2 with Λ = (λ − 1)/κ ,(5.9)
where κ = 1 M 2 BI and M 2 Pl = 1 with the notation used in [45]. On the other hand, at early times, when the universe is dominated by radiation, we have w = 1/3 and the modified Friedman equation becomes in this case
3H 2 = 1 κ ρ − 1 + 1 3 √ 3 (1 +ρ)(3 −ρ) 3 (1 +ρ)(3 −ρ) 2 (3 +ρ 2 ) 2 . (5.10)
As one can see from the above expression, forρ = 3 (with κ > 0) one obtains H 2 = 0. The same is true forρ = −1 (with κ < 0). These stationary points correspond to a maximum density. This is shown in figure 24. The evolution of the scale factor in terms of the maximum density is given in figure 25. The maximum energy density would translate into a minimum value of the scale factor of the order a B = 10 −32 κ 1/4 a 0 , with a 0 representing the scale factor today. Depending on the sign of κ, the scale factor can evolve in two different ways. If κ < 0 one obtains H 2 ∝ a − a B ∝ |t − t B | 2 , which corresponds to a universe undergoing a bounce. On the other hand, if κ > 0 one has H 2 ∝ (a − a B ) 2 , so that ln(a/a B − 1) = 8/(3κ)(t − t B ). In this scenario there is no bounce, and the universe loiters for a long time. These two behaviours can be visualised nicely by plotting the scale factor normalised by the scale a B as a function of time, which can be seen in figure 25. A more detailed analysis of these cosmological solutions was further investigated in [323,111,79,80,78].
For positive values of κ, the primordial nucleosynthesis constraints were used in [28] in order to impose stringent restrictions on the allowed region in the parameter space. The agreement between the observed light element abundances and the predictions of the primordial nucleosynthesis is only ensured if the dynamics of the universe deviates from General Relativity only at a few percentage level at the initial epoch of nucleosynthesis. This, on the other hand, imposes the stringent constraint on the energy density at the start of nucleosynthesis to be of the order ρ nuc ∼ 3H 2 nuc /(8πG) < 3/κ, which translates into κ < 6 × 10 8 m 5 kg −1 s −2 . [45] and illustrates the dependence of the Hubble rate in terms of the energy density for a radiation dominated universe in the EiBI model. In [45] the notation ρB stands for the maximum energy density where H 2 = 0. Furthermore, κ = 1 M 2 BI and M 2 Pl = 1 in terms of our notation. The Born-Infeld inspired gravity model was also applied to a three dimensional spacetime cosmology by S. Jana and S. Kar in [217]. There the authors compute some explicit analytical and numerical solutions for the scale factor in a curved and flat FLRW background with two different scenarios for the matter fields, namely a pressureless dust field with p = 0 and a field with p = ρ/2. They show that also in three dimensions the branch of solutions with M −2 BI > 0 is singular, with an exception of specifically conditioned open universes. For the other case with M −2 BI < 0, they also find non-singular solutions in the same spirit as the four dimensional Born-Infeld gravity model.
Cosmological tensor instabilities
In the previous subsection, we have seen that the original EiBI theory yields interesting homogeneous and isotropic solutions, where the cosmological singularities might be avoided by a bounce. We have seen that a bouncing solution with H 2 = 0 at a B is achievable in the presence of a radiation fluid with w = 1/3. We have also seen the presence of loitering solutions, where the scale factor approaches a B for t → −∞. As next, we shall see whether the perturbations on top of these possible cosmological solutions are stable in order for them to be viable. This was investigated in detail in [155,237]. We shall summarise their results here. For this purpose let us start with the tensor perturbations and describe the tensor modes of the spacetime and auxiliary metric in conformal time dη = dt/a in the following form
g 00 = −a 2 , g ij = a 2 (δ ij + h ij )
and
q 00 = −Ñ 2 , q ij =Ã 2 (δ ij + f ij ) . (5.11)
The tensor perturbations h ij and f ij are transverse and traceless, respectively. Since we are interested in the dynamics of the perturbations in the early universe epoch, we will again assume a relativistic perfect fluid for the matter fields and hence the background evolution will be as in section 5.1. First of all, using the field equations
NÃ 3 a 4Ã2 f ij + λ a 2 h ij = h ij ÑÃ 3 a 4Ã2 + λ a 2 (5.12)
we immediately observe that the two perturbations are identical even if the background scale factors were different, namely
h ij = f ij . (5.13)
This is a remarkable property of the EiBI model. In fact, only in the presence of anisotropic stresses, the two tensor perturbations will be different from each other. This proportionality of the tensor perturbations turns out to be a generic feature of Born-Infeld inspired gravity theories beyond the standard formulation. We will see that for a general function of the metric and the Ricci tensor in section 5.2. See also [60] for more details. The tensor perturbations of the dynamical metric follow the evolution equation [155] h ′′ ij + 3Ã
′ A −Ñ ′ N h ′ ij + Ñ A 2 k 2 h ij = 0 ,(5.14)
where we made use of the background equations of motion. In the regime of low energy densities, one recovers the standard evolution equation of the tensor modes as in General
Relativity. On the other hand, in the Born-Infeld regime at high energy densities, the modifications in the evolution equation due to the scale factor of the auxiliary metric become appreciable. For the stability of the tensor perturbations, it will be crucial that both scale factors are well-behaved. It is not enough to impose this condition solely on the background variables of the spacetime metric. Similarly, one has to guarantee that the auxiliary metric does not vanish. In fact, as we have seen in the previous section, the evolution of the scale factor for κ > 0 goes as ln(a/a B − 1) = 8/(3κ)(t − t B ), hence the lapse and the scale factor of the auxiliary metric evolve as
A = 2 1/4 a exp 8/(3κ)(t − t B ) , N = 1 √ 2Ã 3 a 2 . (5.15)
As it becomes clear from these expressions, the scale factor of the auxiliary metric becomes singular for t → −∞. This non-singular behaviour has a crucial impact on the tensor perturbations, since their evolution equation scales with the quantities of the auxiliary metric. In the far asymptotic past, the pre-factors of the last two terms in equation (5.14) are suppressed and the evolution equation simply becomes h ′′ ij ∼ 0. The solution for the metric perturbations is hence of the form h ij ≈ Aη + B. This represents an unstable growth and therefore, the loitering solution in the case κ > 0 suffers from an instability. This instability is a mild one and can be easily cured by slightly modifying the set-up.
The presence of tensor instabilities in the loitering solution is unfortunately also shared by the bouncing solution and it is even more virulent. For the case κ < 0, we have seen in previous section that a bouncing solution is obtained since H 2 ∼ a − a B ∼ |t − t B | 2 . In terms of the conformal time, the scale factor evolves as a = a B 1 + tan 2 (βη) , We can Taylor expand these expressions around the bounce η = 0. By doing so, the evolution equation of the tensor perturbations close to the bounce becomes
h ′′ ij + 2 η h ′ ij + k 2 3β 2 η h ij = 0 . (5.18)
The solution scales this time as h ij ≈ η n with n = − 1 2 ± 1
Thus, in the EiBI model in the presence of a radiation fluid the interesting loitering and bouncing solutions suffer from tensor instabilities. This unsatisfactory result might change if one considers a more general fluid with varying equation of state parameter or if one extends the EiBI model to a more general Born-Infeld inspired gravity model.
Varying equation of state parameter
In the previous subsections we have seen that the EiBI theory admits interesting bouncing and loitering solutions for early universe cosmology in the presence of a radiation fluid. However, as we have seen, these solutions are plagued by tensor instabilities if the matter field is assumed to be a perfect fluid with the equation of state parameter w = 1/3. It is possible to find more general solutions if we abandon this restriction and this might alleviate the found tensor instabilities. In fact, this was precisely considered in [32]. It could be that additional dynamical fields are present in the early universe, giving rise to matter fields withẇ = 0. In this case, the modified Friedman equation (5.5) generalizes to [32]
H 2 = a √ g 1 +ẇg 3 g 2 2 , (5.19)
with the functions g i given by the energy density and pressure of the matter fields
g 1 = 2M −2 BI 1 + 2ρ M 2 BI 1 − 2ρw M 2 BI 2 −2 + 2 M 2 BI
(1 + 3w) + 2D , [32] (remember that κ = M −2 BI in our units). In that work it is shown that the possibility with time varying equation of state parameter can ameliorate the tensor instabilities found for w = const. As an example, a scalar field with a general kinetic and potential term is considered. In the presence of this scalar field, with the Lagrangian L(X, φ) where X = − 1 2 ∂ µ φ∂ µ φ, the equation of state parameter is given by
g 2 = 4 + 2 M 2 BI ρ 1 − 2w 2 − 2 M 2 BI ρ + 3w 2 1 + 2 2 M 2 BI ρ , g 3 = −3ρ 1 + 2 M 2 BI ρ ,(5.w = L 2XL ,X − L ,(5.21)
with the pressure p = L and energy density ρ = L ,X − L accordingly. It turns out, that for κ = −1, the instability of the tensor perturbations cannot be avoided. This is the reason why the authors in [32] consider the case κ = 1. Sinceẇ = 0, one achieves a bouncing solution with H 2 = 0 andḢ = 0, which differs from the case studied in [155], where ρ → w −1 as η → −∞. Remember that the authors in [32] use the units |κ| = 1.
For an initial density of ρ i = 10 −4 and w i = 0, this behaviour is illustrated in figure 26 for a scalar field with L = X − 1 2 m 2 φ 2 . The tensor perturbations on top of this background are given by
h ′′ ij + g 4 h ′ ij + g 5 k 2 h ij = 0 ,(5.22)
with the two functions g 4 = 2H + κρ 1 + κρ ,
g 5 = 1 − κρw 1 + κρ . (5.23)
For the case κ = 1, the pre-factor in the friction term near the bounce vanishes g 4 ∼ 0 and therefore, the tensor instabilities reported in [155] are avoided. This simple example for time varying equation of state parameter was achieved with a standard scalar field with a mass term. As we have seen, this simple set-up already helps with the encountered tensor instabilities. In the next subsection, we will discuss in more detail the presence of a scalar matter field in EiBI model and summarise the works done in this context.
Born-Infeld with a scalar matter field
We have seen above that the reported tensor instabilities of the interesting cosmological solutions might be avoided by considering matter fields with varying equation of state parameter. As a specific model, one can consider the presence of a scalar field as matter field. This was for instance done in the works [155,237,112,371]. In this way, the underlying physics of the early universe will be determined by both the Born-Infeld modification and the presence of the scalar field. As a simple realisation one can consider a scalar field with a quadratic potential. In standard General Relativity an inflationary scenario with sufficiently long duration based on such a simple scalar field might require very large field values. This is due to the fact that the time derivative of the scalar field increases rapidly as going back in time with the scalar field itself climbing up the potential giving rise to an increasing energy density until the Planck scale is reached quickly. The hope to use this same scalar field in the Born-Infeld inspired gravity theory is to alleviate this requirement. The crucial point with this respect is that the pressure in EiBI gravity is bounded from above due to the square root structure. Hence, there is an upper bound for the value of the field velocity as it was shown in [112]. This guarantees a real value for the Hubble parameter. Due to this upper limit, one does not run into the same problem as in the standard inflationary model. Let us consider the following action [237]
S BI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R (µν) (Γ) − λ √ −g + d 4 x √ −g − 1 2 g µν ∂ µ ϕ∂ ν ϕ − m 2 2 ϕ 2 . (5.24)
In this model, the curvature scale remains finite thanks to the square root structure of EiBI gravity and the early universe undergoes a pre-inflationary accelerated expansion in order then to end in an ordinary chaotic inflationary epoch. Since the scalar Lagrangian is the same as in General Relativity, it follows the same evolution equation. For a homogeneous and isotropic background metric and correspondingly only time dependent scalar field, the equation of the scalar field is simplyφ + 3Hφ + m 2 ϕ = 0. The maximum value for the field velocity is achieved whenφ 2 = m 2 ϕ 2 + 2λM 2 BI . So we can define this moment of maximum velocity byφ = m 2 ϕ 2 + 2λM 2 BI with the Hubble parameter taking the form H = − 2 3 m 2 ϕ/ m 2 ϕ 2 + 2λM 2 BI at this point. These equations can be integrated to have the evolution of the scalar field and the scale factor giving rise to solutions that respect the maximal pressure condition. By doing so, the explicit analytic solutions with this bound are given by
ϕ = 2λM 2 BI m sinh [m(t − t 0 )] and a = a 0 (2λM 2 BI ) 1/3 cosh −2/3 [m(t − t 0 )] . (5.25)
These solutions describe a universe that expands until the bouncing stage is achieved at t = t 0 and then starts contracting whereas the scalar field tracks the symmetric potential. At early times t → −∞, there is no singularity and the universe expands exponentially with a ∼ a 0 (2/λM 2 BI ) 1/3 e 2 3 m(t−t 0 ) and ϕ ∼ − λM 2 BI /(2m 2 )e m(t−t 0 ) . During this period, the Hubble parameter is nearly constant and purely determined by the scalar field's mass H ≈ 2m/3 and in this limit m 2 ϕ 2 ≫ 2λM 2 BI , i.e. the potential of the scalar field is larger than λM 2 BI . Thus, the upper limit in the pressure guarantees that the curvature scale remains finite.
In figure 27 an example of the phase map is plotted for ϕ andφ, where the Hubble function is denoted by the colour. We borrowed this figure from [112], where one can nicely see the evolution of the Hubble parameter and the scalar field and the realisation of the different phases. One can see that the universe starts off close to the region of the upper bound of the pressure or field velocity respectively and decreases as time passes. Sufficiently away from this region, the universe undergoes the first slow-roll where the field In the left panel one can see the behaviour of the Hubble parameter denoted by different colours. The red region corresponds to H > 1, whereas the blue colour shows the regions with small Hubble parameter. The region encoded in white is the physically forbidden region. In the right panel the behaviour ofφ is represented. The blue region corresponds to the field space where the upper bound limit is violated. The different trajectories correspond to different initial conditions for the scalar field. The grey region is the high-curvature regime and the solid and dashed curves represent trajectories that start from the left top and right bottom, respectively. velocity drops rapidly. As time evolves, the universe passes through the second phase of slow roll following the attractor solution represented by the standard chaotic inflationary expansion. This stage ends when the scalar field starts oscillating around the minimum of the potential going over to a possible reheating epoch. As it can be clearly seen in figure 27, all the trajectories start either from right bottom or from left top, where the forbidden region is avoided and converge to the attractor solution. Note also that, since the scalar field has a non-constant equation of state parameter, the tensor instabilities on top of these cosmological backgrounds can be eluded [32]. The tensor and scalar perturbations of this model were investigated in detail in one of the pioneering works [237], where the authors constructed the general algorithm in terms of the bimetric interpretation of the model. They were able to show that the theory admits indeed the expected two tensor modes and one scalar mode, corresponding to the matter field. The authors further found scale-invariant power spectra for the tensor and scalar perturbations. However, they also reported a too large tensor-to-scalar ratio in contradiction with current observations. This is in the case of a scalar matter field that couples minimally to the Born-Infeld gravity. The tensor perturbations within this model were further studied in the work [110], where it was shown that the same properties of the standard chaotic inflation are maintained for very short wavelength modes, whereas the model gives rise to a distinctive feature in form of a peculiar rise in the power spectrum for long wavelength modes. This peculiarity could be then tested with the CMB observations.
The preliminary findings of tensor and scalar perturbations of [237] were further investigated in great detail in [110,113,115,114,109], where the authors study the scalar and tensor spectral indices and show that the contributions are second order in the slow roll approximation for the scalar perturbations and first order in the tensor perturbations. In the framework of EiBI gravity the tensor-to-scalar ratio r can be suppressed significantly in difference to the standard chaotic inflation in General Relativity. For the analysis of the scalar perturbations of the model, let us adapt to the useful approach of considering parallel variables for the g metric and the auxiliary metric, as we did above for the tensor perturbations. Let us consider the following scalar perturbations [237]
ds 2 q =ã 2 − (1 + 2φ q ) Z dη 2 + 2 B 1,i √ Z dηdx i + [(1 − 2ψ 1 )δ ij + 2E 1,ij ] dx i dx j ds 2 g = a 2 −(1 + 2φ g )dη 2 + 2B 2,i dηdx i + ((1 − 2ψ 2 )δ ij + 2E 2,ij ) dx i dx j . (5.26)
Similarly, we shall perturb the scalar field as ϕ = ϕ 0 + δϕ. Note that the auxiliary metric carries the additional background quantity Z =
factorã = (1 + ρ 0 /M 2 BI ) 1/4 (1 − p 0 /M 2 BI ) 1/4 a with ρ 0 = ϕ ′ 0 /(2a 2 ) + m 2 ϕ 2 /2 and p 0 = ϕ ′ 0 /(2a 2 ) − m 2 ϕ 2 /2.
One can use the gauge freedom in order to eliminate some of the perturbations. One could for instance choose ψ 1 = 0 and E 1 = 0. However, not all of the remaining quantities are dynamical. In fact, except for the scalar field, all the remaining perturbations of the metrics can be integrated out using their algebraic equations. This is to be expected, since the q µν metric is related algebraically with the g µν metric and the scalar perturbations in the space-time metric are not dynamical (see sections 2.5.
+ k 2 − 2 (τ − τ 0 ) 2 χ ≈ 0 ,(5.27)
where τ 0 denotes the end of inflation and f 1 = 3Z 2 −2Z+3 (Z+1)(3Z−1) a 2 here. In the works [113,115,109] it was shown that the perturbations start from an initial point where the maximal pressure condition holds and evolve towards an intermediate stage, where the WKB approximation can be applied to then end at the attractor stage. Finally, the solutions of these three stages are matched together. Furthermore, they compute the comoving curvature perturbation Rψ 2 + Hδϕ/ϕ 0 and from that the scalar power spectrum P R = k 3 |R| 2 /(2π 2 ). Last but not least, from this they were able to evaluate the spectral index n R − 1 = d log P R /d log k. They observe that the spectral index is of second order in the slow-roll approximation and a suppression of the tensor-to-scalar ratio. The exact form of the scalar power spectrum and the spectral index can be extracted from [109,114] and we refer the reader to these works for more details.
We have seen that in the presence of a scalar field one can realise different epochs in the early universe. One can have a preinflationary scenario followed by a standard chaotic inflationary expansion. Due to the squared root structure of the gravitational interactions, there is an upper limit for the pressure and hence the field velocity. So far we have considered the case where the scalar field is minimally coupled to the gravity and has standard kinetic and mass terms. In the following subsection we will pay attention to the case where the scalar sector obeys the Born-Infeld structure as well.
Born-Infeld in gravity and matter sector
In the following we would like to discuss the EiBI gravity theory in the presence of a scalar Born-Infeld matter field. The Born-Infeld structure in both the gravity and matter sector with their corresponding scales might have interesting implications. This idea was pursued by S. Jana and S. Kar in [219], where they provide interesting analytical cosmological solutions for a particular choice of the time derivative of the Born-Infeld scalar. For a positive constant M −2 BI > 0, they were able to realise solutions with two separate de Sitter expansions with an intermediate sandwiched phase of deceleration. The action of this model is given by [219]
S BI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R (µν) (Γ) − λ √ −g + α 2 T d 4 x √ −gV (φ) 1 + α −2 T g µν ∂ µ φ∂ ν φ ,(5.28)
with the scales M BI and α T representing the Born-Infeld scales in the gravity and matter sector, respectively, and V (φ) denoting a potential for the scalar field. The scalar sector is a Dirac-Born-infeld like action. The equation of motion of the scalar field yields
∂ ν V √ −gg µν ∂ µ φ 1 + α −2 T g µν ∂ µ φ∂ ν φ = α 2 T V ′ √ −g 1 + α −2 T g µν ∂ µ φ∂ ν φ . (5.29)
Similarly, the corresponding stress energy tensor reads
T µν = V (φ) (g µα g νβ − g µν g αβ )∂ α φ∂ β φ − g µν α −2 T 1 + α −2 T g µν ∂ µ φ∂ ν φ . (5.30)
We will be interested in the possible cosmological solutions that one can construct in this particular model. For this purpose, we will again consider a FLRW metric for the background metric g µν with lapse N and scale factor a. For this specific simple background, the scalar field equation becomes
φ α 2 T N −φ 2 + 3φH α 2 T N + V ′ V −φṄ 2N (α 2 T N −φ 2 ) = 0 . (5.31)
The equation of motion of the scalar field can also be written asρ φ /ρ φ = −3Hφ 2 /(N α 2 T ), where the corresponding energy density of the field is given as
ρ φ = α 2 T V 1 −φ 2 N −1 α −2 T . (5.32)
Similarly, we can compute the pressure of the scalar field, which for the considered DBI action yields p φ = −α 2 T V 1 −φ 2 N −1 α −2 T , which we can use to define the corresponding equation of state parameter of the scalar field. In contrast to the standard single field inflation where one assumes a specific form of the potential, here one can choose a specific form forφ. In [219] the following solution forφ is contructed:
φ 2 = N α 2 T 1 + C 1 a n ,(5.33)
with positive constant variables C 1 and n. For n = 3 for instance one can obtain a constant negative pressure p φ = −α 2 T C 2 with the integration constant C 2 . Using the metric field equations
√ −qq µν − λ √ −gg µν = √ −gT µν M 2 Pl M 2 BI ,(5.34)
we can relate the lapseÑ and shiftã of the auxiliary metricq with the scale factor of thê g metric and the energy density of the scalar field. By doing so, one obtains with the new constant parameterα T = 1 + α 2 T C 2 /(M 2 BI M 2 Pl ). Similarly, the equations of motion for the connection yield the following relationṡ
a =ã Ñ /Ñ α T , ρ φ = M 2 BI M 2 Pl α 3 T Ñ N 2 − 1 ,(5.a 2 a 2 = M 2 BI 6 2Ñ + N − 3Ñ 2 C 2 0 N , d dt ȧ a −ȧṄ 2ãÑ = M 2 BI 2 −N +Ñ 2 C 2 0 N . (5.36)
In [219] this system of equations is analysed for a particular solution of the scale factor of the auxiliary metric, namely,ã =ã 0 eH b t with two constantsã 0 andH b . Introducing the quantity Y φ = C 0 (1 − 3H 2 bÑ −1 M −2 BI ), the last equation of (5.36) translates intȯ
Y φ 2Y φ − Y 2 φ + 3 = −2H b . (5.37)
In terms of this new variable, the deceleration parameter d can be expressed as 44
d = − aa ′′ a ′2 = −1 + Y 2 φ + 3 − 2Y φ Y 2 φ + 3 − Y φ 1 + Y φ Y 2 φ + 3 + Y 2 φ + 3 + Y φ C 0 2(C 0 − Y φ ) ,(5.38)
where prime denotes the derivative with respect to cosmological time τ = √ N dt. One immediate observation is that for a → ∞ (Y φ → 1) one has d = −1 and similarly for a → 0 (Y φ → −∞) one also has d = −1. Hence, one obtains two de Sitter phases, one at early and one at late time universe. In figure 28 taken from [219] one can see the dependence of the deceleration parameter from X for different values of C 0 . Furthermore, one can choose the value of the constant C 0 such that one can realise an initial loitering phase with an acceleration and subsequently a decelerated and again an accelerated expansion phase afterwards. This can be seen in figure 29, where the scale factor is plotted as a function of cosmological time. During the loitering phase, the scale factor grows approximately as a loit ∼ a 0 e 2 √ 2M BI τ / BI < 0. Of course, due to the Born-Infeld term in the gravity sector, one can construct bouncing solutions as we have seen before. The novelty of the scalar Born-Infeld term results in an additional accelerating phase at late time universe. Nevertheless, this solution with M −2 BI < 0 yields a bounce at an unacceptable low redshift as it is shown in [219], therefore we do not report more on this solution here. More details can be taken from [219]. There the authors show also the comparison of the obtained solutions with the supernovae Ia Union2.1 data and find that the agreement with the data is as good as in ΛCDM model. Summarizing, the combined Born-Infeld model in the gravity and scalar sector delivers an interesting framework to study effects both on early and late time universe cosmology. The ongoing physics at early times is dictated by the standard Born-Infeld scale M BI , where one can either realise loitering, accelerating or bouncing solutions, whereas the physics at late times is governed by the scalar Born-Infeld scale α T , which gives rise to an accelerated expansion.
Anisotropic cosmological solutions
So far we have studied the cosmological applications of the EiBI gravity theory for homogeneous and isotropic backgrounds. We have seen the appearance of different interesting cosmological solutions for early universe, including loitering, quasi de Sitter and bouncing solutions and discuss their stability. Another interesting question along this line is the evolution of cosmological backgrounds with anisotropies. Some of the anomalies observed in the cosmic microwave background might be due to the presence of small anisotropies. In this context, Bianchi type models could be a natural and simple extension of the standard FLRW with small anisotropies, which could be for instance at the origin of the power suppression at large scales of the cosmic microwave background. This was exactly pursued in [229,191]. Let us consider the following Bianchi type I background for the dynamical metric g µν
ds 2 g = g µν dx µ dx ν = −dt 2 + g 1 (t)dx 2 + g 2 (t)dy 2 + g 3 (t)dz 2 ,(5.ds 2 q = q µν dx µ dx ν = −dt 2 + q 1 (t)dx 2 + q 2 (t)dy 2 + q 3 (t)dz 2 ,(5.40)
with q i standing for the quantities q i = g i A/B i . In terms of these variables, the field equations of the Bianchi Type I geometry of the EiBI theory can be calculated easily.
Plugging the two Ansaetze into the covariant field equations yields
1 − A B 1 B 2 B 3 = M 2 BI (q 1 q 2 q 3 + q 1q2 q 3 + q 1 q 2q3 ) q 1 q 2 q 3 , (5.41) 1 − B 1 AB 2 B 3 = M 2 BI (q 1q2 q 3 +q 1 q 2q3 +q 1 q 2 q 3 ) q 1 q 2 q 3 , (5.42) 1 − B 2 AB 1 B 3 = M 2 BI (q 1q2 q 3 + q 1q2q3 + q 1q2 q 3 ) q 1 q 2 q 3 , (5.43) 1 − B 3 AB 1 B 2 = M 2 BI (q 1 q 2q3 + q 1q2q3 + q 1 q 2q3 ) q 1 q 2 q 3 . (5.44)
First, we can consider the simple case with isotropic pressure where p i = p and therefore B i = B. For clarity of the notation, we can further introduce the Hubble functions in the different spatial directions as H i =q i /q i and ∆H i = H − H i where H = 1/3 3 i=1 H i is the mean Hubble expansion rate. We can define the degree of anisotropy as the shear
σ = 1 3 3 i=1 ∆H i H . (5.45)
In the field equations the multiplication of the scale factors appears very often. For this reason, we can define a new variable here as Q = q 1 q 2 q 3 and express the field equations in terms of H and Q, which read
3Ḣ + 3 i=1 H 2 i = M −2 BI 1 − A B 3 , (5.46) 1 Q d dt (QH i ) = M −2 BI 1 − 1 AB . (5.47)
After simple manipulations of the field equations, they can be combined into d dt [Q(H i − H)] = 0, which can be simply integrated to give H i = H + C i /Q with integration constants C i . Further integration gives for the scale factors
q i = q i0 Q 1/3 exp C i 1 Q dt dQ dQ .
For consistency, the integration constants have to satisfy C 1 + C 2 + C 3 = 0. Furthermore, the product of the scale factors follows the second order differential equation Q = 3M −2 BI (1 − 1/(AB))Q, which can be also integrated easily. From these solutions, we can also determine the quantities of theĝ metric. For instance, the Hubble functions of theĝ metric in the different directions can be obtained from H g i = H i +Ḃ/B −Ȧ/A and similarly the mean Hubble parameter as well. The ordinary matter fields couple to the standardĝ metric, therefore their conservation equation is dictated by the mean Hubble parameter of theĝ metric. Thus, they follow aṡ
ρ + 3 H +Ḃ B −Ȧ A (ρ + p) = 0 . (5.48)
In terms of the energy density and pressure of the matter fluid, we can also compute the degree of anisotropy in theĝ sector, which takes the following from
σ g = 3C 2 (ρ + p) 2 Q 2ρ2 ,(5.49)
with C = C 1 +C 2 +C 3 . In [191] the quantity Q is used as a parameter in order to obtain the general solution in a parametric form. Furthermore, they provide the general solutions for the Hubble functions and the anisotropy parameter in theĝ sector for three different fluid types: for stiff, radiation and dust fluid. Let us for example consider the dust component with p = 0 and hence B = 0. After making the following change of variables θ = tM BI and ρ = rM 2 BI , the factor Q = q 1 q 2 q 3 becomes Q = M −2 BI ρ 0 /(r(1 + r) 3/2 ) with the initial density ρ 0 . From the differential equationQ = 3M −2 BI (1 − 1/(AB))Q one obtains in this case 2(r + 1)(5r + 2)rr ′′ − (7r(5r + 4) + 9)r ′2 − 12 √ r + 1r 2 (r − (1 + r) 3/2 + 1) = 0 (5.50) where prime denotes here the derivative with respect to θ. The volume element in theĝ metric sector, so in other words G = g 1 g 2 g 3 , is given by G = M −2 BI ρ 0 /r. The evolution of the rescaled energy density r and the volume element is plotted in figure 30 for different choices of the initial energy density, which we took from [191]. In the presence of a dust fluid, the mean anisotropy parameter is given by (M −2 BI ρ 0 ) 2 /(3K 2 )σ g = r 4 (1 + r) 3 /r ′2 and is plotted in figure 31. As it can be clearly seen in this figure, the mean anisotropy parameter evolves BI ρ (which is v g in the notation used in [191] and furthermore M Pl = 1) as a function of the rescaled time θ = tMBI for three different initial conditions: r(0) = 0.9 (solid line), r(0) = 0.85 (dotted line), r0 = 0.8 (short dashed line) and r(0) = 0.75 (dashed line) in a universe filled with dust. Figures were taken from [191]. [191] the evolution of the physical Hubble function in a dust filled universe in a Bianchi type I space-time together with the mean anisotropy parameter σ g (in terms of the notation used in [191] this corresponds to a to zero after some time elapses and the universe ends up in an isotropic phase in a Bianchi type I space-time in the EiBI gravity theory. This seems to be complementary to the standard picture in General Relativity where shear decays in the presence of a cosmological constant [51]. Within the framework of EiBI gravity this property is maintained in the presence of a dust component as well. However, this property does not seem to be general. For instance, instead of a dust fluid, if one considers a radiation fluid with w = 1/3 or a stiff fluid, then the universe does not isotropise at late times as it was happening for dust. In fact, for a stiff fluid, the mean anisotropy parameter in theĝ metric sector takes rather the form σ g = r 3 (1 + r) 3 /((1 − r) 3 r ′2 ). Its evolution together with the evolution of the Hubble function can be extracted from figure 32, which we borrowed from [191] as well. As one can see in figure 32, the mean anisotropy parameter increases with time. Starting from an initial state with a vanishing anisotropy, the degree of the anisotropy increases until it reaches a maximum constant value. [191] illustrates the evolution of the Hubble function and the mean anisotropy parameter σ g (which corresponds to a (g) p in the notation used in [191]) as a function of the rescaled time θ = tMBI in a universe filled with stiff fluid with different initial values.
Late-time cosmology
As stressed several times throughout this review, the general motivation for Born-Infeld inspired theories of gravity is to modify the gravitational interactions in the high curvatures regime. This means that deviations with respect to GR will typically arise when the curvatures become of the order of the Born-Infeld scale M 2 BI . Since the source of gravity is weighted by the Planck scale, an equivalent formulation of this statement is that one only expects deviations from standard gravity when the densities are of the order of 45 ρ ∼ M 2 BI M 2 Pl . For this reason, the natural place where these theories manifest themselves in cosmological scenarios is the early universe, being ideal candidates for inflationary models or bouncing solutions as we have reviewed above. Applications of these theories for models of the late-time universe are instead dissonant as a consequence of their very own defining properties. Since the Born-Infeld effects will become negligible whenever the cosmological energy density drops below M 2 BI M 2 Pl , from that moment on we will have the usual cosmological evolution with GR governing the gravitational interaction. If we want to have non-negligible effects on cosmological scales today (or somewhere between decoupling and today) that would mean that the whole cosmological evolution would have taken place in the Born-Infeld regime. For this reason, late-time cosmology constitutes an inefficient way to constrain EiBI theories and dark matter and/or dark energy models based on this type of theories are likely to fail in their goal. Models for the dark components of the universe find a better suited arena within the framework of infrared modifications of gravity so that they become relevant in the late-time evolution of the universe. If we want to be on the safe side, we can impose the Born-Infeld corrections be important only before the onset of Big Bang Nucleosynthesis (BBN). Since BBN takes place when the temperature of the universe is roughly 1 − 10 MeV, we obtain the conservative constraint 45 More precisely, one expects modifications whenever any component of the energy-momentum tensor becomes comparable to M 2 BI M 2 Pl . In the most standard cosmological backgrounds the energy density is the relevant quantity. but in more general scenarios other components of the energy-momentum tensor could play an important role as well. This is the case for instance of anisotropic cosmological solutions where anisotropic stresses can be present.
M BI H BBN ≃ T 2 BBN /M Pl ≃ 10 −13 eV. Notice that this bound is less stringent that the one discussed in section 2.6 where we obtained M BI 10 −1 eV from the absence of anomalous interactions in collider experiments.
Despite the general arguments given in the precedent paragraph, there are some works in the literature attempting to explain the dark matter problem in galaxies as an effect of modifying gravity as in EiBI theory [322]. In [193], the authors study spherical dark matter haloes and conclude that the value of the Born-Infeld parameter that allows to realistically reproduce the dark matter haloes is M −2 BI ≃ 10 44 cm 2 which translates into M BI ≃ 10 −27 eV. Even if this value allows to reproduce the haloes, we must remember that the Born-Infeld coupling is universal and this value is in contradiction with the constraints discussed in the previous paragraph so that it is excluded. Analogous studies like e.g. those in [309,216] find similar results and are, thus, subjected to the same limitations. Similarly, the bounds obtained on M BI from other cosmological and astrophysical probes explain the results found in [145] where the authors compute the matter power spectrum for the EiBI theory. They find that the deviations of the power spectrum with respect to that of ΛCDM is completely negligible for realistic values of M BI .
Let us also notice that, still within the class of Born-Infeld inspired gravity theories, in order to avoid the aforementioned triviality for late-time cosmological applications, there has been some attempts in the literature to include additional corrections to the EiBI action that could give some effects at late times. We should note however, that this goes against the Born-Infeld spirit and it is very likely that the EiBI sector will not play any role and the whole effect will come from the new terms. As an example of this approach, some works introduced an Einstein-Hilbert term supplementing the Born-Infeld sector, but this class of modifications seriously compromise the stability of the theory. In fact, such variations of the Born-Infeld actions belong to the Class 0 described in the section 2.7.1 and which are precisely characterised by the presence of pathologies. Thus, even if one can achieve non-negligible effects in the late-time cosmology, this would come at the expense of possibly losing the ghost freedom of the theory. Explicit examples of this type of modifications will be summarised in section 5.6, but it is worthwhile to stress here that this road of tracing late-time cosmological solutions are doomed to fail due to the mentioned instabilities. As discussed in 2.7.1, if one really wants to add an additional Einstein-Hilbert term in the Lagrangian, then the Born-Infeld interactions have to be modified so that the ghost-free massive (bi-) gravity potential interactions in its formulation in terms of the auxiliary metric are recovered.
From the above discussion it is clear that Born-Infeld inspired theories of gravity cannot play a relevant role for the cosmological evolution from roughly BBN (where we need to have standard gravity) until today. There is however a more natural place to study potential effects of EiBI in the late-time cosmology residing within the framework of future cosmological singularities. The properties of dark energy are crucial for the future evolution of the universe and its eventual fate. At this respect, some models predict the existence of future singularities that can be broadly classified according to the divergence of some cosmological quantity (see for instance [52,262,126,93,160,81,62,11] for some related literature). It is indeed common to perform such a classification attending to which derivative of the scale factor diverges first [161]. In some scenarios with future singularities, the Hubble expansion rate or its derivative show divergences so that the Born-Infeld corrections will eventually be relevant again and one could wonder if such future singularities could be tamed. This was studied in [79,78] and it was found that generally future cosmological singularities can remain, although in some cases the divergences can be somewhat smoothed. In [77] the authors argued that the classical big rip singularity might be avoided by applying the quantisation based on the Wheeler-DeWitt equation to the EiBI model.
General cosmological framework for Born-Infeld inspired gravity theories
After reviewing the cosmological applications of the EiBI model, we will discuss the cosmological studies performed for other Born-Infeld inspired theories of gravity. Most of them share the same underlying features and mechanisms, although leading to different cosmologies depending on the specific model under consideration. Thus, instead of studying the individual extensions one by one, we will develop here a general framework to study cosmological solutions within these theories. In fact, without increasing the level of difficulty, we can consider the general class of theories already analysed in 2.7.1 and harvest the results of that section to obtain the relevant equations to study the cosmology of these theories. To avoid the reader to thumb through the review, we will rewrite the main equations here for convenience. The starting action is given by 46
S = 1 2 M 2 Pl M 2 BI d 4 x √ −gF P ,(5.51)
with P µ ν = M −2 BI g µα R αν (Γ). The analysis of the general field equations, even in the presence of torsion, was discussed in great detail in section 2.7.1. For our purposes here, the important equations will be those relating the auxiliary metric q µν , which determines the connection as its Christoffel symbols, with the spacetime metric and the matter content. The two metrics are related byq =ĝΩ, whereΩ is the deformation matrix defined aŝ
Ω −1 = 1 detFP ∂F ∂P T . (5.52)
Notice that this definition relatesΩ andP so that all the equations below will admit equivalent formulations in terms ofΩ orP alone. By using the definition of the deformation matrix, the metric field equations can be expressed aŝ
Ω −1P = 1 M 2 BI M 2 Pl detΩ L G ½ +Tĝ (5.53)
where L G = 1 2 M 2 Pl M 2 BI F is the Lagrangian. These equations give the deformation matrix Ω (or the fundamental objectP ) in terms of the matter content and the spacetime metric. The resolution of the problem will be completed with the differential equations for the auxiliary metric (see 2.7.1)
R µ ν (q) = 1 M 2 Pl detΩ L G δ µ ν + T µ ν . (5.54)
After briefly reviewing the relevant equations, we can proceed to the study of cosmological scenarios. As usual, we will consider a homogeneous and isotropic background metric described by the FLRW line element
ds 2 g = −N 2 (t)dt 2 + a 2 (t)d x 2 (5.55)
and a perfect fluid with isotropic pressures as matter sector
T µ ν = −ρ 0 0 pδ i j . (5.56)
As an additional condition, we will assume that all relevant quantities inherit this form so that we will haveΩ
= Ω 0 0 0 Ω 1 δ i j andP = P 0 0 0 P 1 δ i j .
(5.57)
As we have explained several times above, the recovery of GR at low curvatures imposeŝ Ω ≃ ½ for P 0 , P 1 ≪ 1. The form of the deformation matrix ensures that the auxiliary metric will also have a FLRW line element
ds 2 q = −Ñ 2 (t)dt 2 +ã 2 (t)d x 2 ,(5.58)
withÑ 2 = N 2 Ω 0 andã 2 = a 2 Ω 1 . We keep the explicit dependence on the lapse function N (t) for later convenience. Once we have specified the assumptions for our homogeneous and isotropic Ansätze, we can now proceed to write the background metric field equations (5.53), which read
P 0 Ω 0 = 1 M 2 BI M 2 Pl Ω 0 Ω 3 1 L G + T 0 0 , P 1 Ω 1 = 1 M 2 BI M 2 Pl Ω 0 Ω 3 1 L G + 1 3 T i i . (5.59)
As anticipated above, for a given function F (P ), these equations will allow to obtain the components ofΩ (or those ofP ) in terms of the energy density ρ = −T 0 0 and the pressure p = 1 3 T i i of the matter fields. An important point to keep in mind is that these equations are non-linear so that, in general, we will find several branches. Out of those branches, the condition (2.73) on the function F will guarantee the existence of one particular branch that will be continuously connected with GR at low curvatures. This will be the interesting branch for most applications, although other branches might also offer interesting cosmological scenarios.
As we have seen in the previous section for the EiBI model, the crucial step to study the cosmological evolution is to extract the dependence of the Hubble expansion rate in terms of the energy density and pressure of the matter fields. For this purpose, we will make use of the Einstein tensor of the auxiliary metric and express its 00 component in two different ways. We will start from the definition of the Einstein tensor of q µν given bŷ
G(q) =R − 1 2q Tr q −1R . (5.60)
First, we will compute its 00 component in terms of the auxiliary metric, which will simply give the corresponding Hubble expansion rate:
G 00 (q) = 3H 2 = 3 d lnã dt 2 = 3 H 2 + 1 2 d ln Ω 1 dt 2 .
(5.61)
Since Ω 1 = Ω 1 (ρ, p) as obtained from (5.59), we can express the time derivative of Ω 1 in terms of derivatives with respect to ρ and p. Thus, if we use that matter fields are assumed to be minimally coupled so that they satisfy the usual conservation equatioṅ
ρ + 3H(ρ + p) = 0, (5.62)
we can finally arrive at
G 00 (q) = 3H 2 1 − 3 2 ρ + p ∂ ρ ln Ω 1 + c 2 s ∂ p ln Ω 1 2 (5.63)
where we have introduced the sound speed c 2 s ≡ṗ/ρ. If we further assume a barotropic equation of state p = p(ρ) the sound speed can also be written as c 2 s = dp/dρ. This completes the first part of our computation of the Hubble expansion rate. The second part consists in writing the Einstein tensor of the auxiliary metric by using the definition P = M −2 BIĝ −1R and the relation between the two metrics through the deformation matrix q =ĝΩ so that we obtain
G(q) =R − 1 2q Tr q −1R = M 2 BIĝ P − 1 2Ω Tr Ω −1P . (5.64)
We can again extract the expression for G 00 , this time in terms of the components ofP andΩ, as follows:
G 00 (q) = 1 2 M 2 BI g 00 P 0 − 3 Ω 0 Ω 1 P 1 . (5.65)
If now we equal the right hand sides of (5.63) and (5.65) and solve for H 2 we finally obtain
3H 2 M 2 BI N 2 = 3Ω 0 P 1 − P 0 Ω 1 2Ω 1 1 − 3 2 (ρ + p) ∂ ρ ln Ω 1 + c 2 s ∂ p ln Ω 1 2 ,(5.66)
where we have used that g 00 = −N 2 . This is the master equation providing the modified Friedman equation for the theories under consideration, i.e., it gives the dependence of the Hubble function in terms of the matter field variables. Let us remember that the components ofP andΩ are functions of ρ and p as obtained from the resolution of (5.59) and, hence, the right hand side of the above equation only depends on the matter sector variables. A very distinctive feature of these theories is the appearance of c 2 s in this modified Friedman equation. This means that, unlike the case of GR and many other modified gravity theories, the sound speed not only affects the evolution of the perturbations, but it also affects the background evolution. In particular, this includes one additional parameter for the homogeneous cosmologies of these theories. While in the most extensively studied modified gravity theories the equation of state fully determines the background evolution, in the theories under consideration here (among which many Born-Infeld inspired theories are included) there is a further dependence encoded in c 2 s . Moreover, some matter sources can actually have a non-constant sound speed (like in the case of several interacting fluids) and it could even depend on H(t) so that (5.66) will be an implicit equation for the Hubble expansion rate. We have already encountered a particular case of this result in the EiBI theory rephrased in terms of a time-dependent equation of state parameter and we saw that the background cosmology depends not only on w(t) but also onẇ.
There is a number of interesting general features that can be directly inferred from (5.66). The first thing to notice is that now it is very easy to understand the mechanism by which these theories can give rise to bouncing solutions without violating the NEC. For that, let us rewrite the modified Friedman equation in the more familiar form
H 2 = 8πG eff (ρ, p, c 2 s ) 3 ρ (5.67)
that is closer to its usual form and we have encoded all the modified effects into the effective Newton's constant
8πG eff (ρ, p, c 2 s ) = M 2 BI 3Ω 0 P 1 − P 0 Ω 1 2Ω 1 1 − 3 2 (ρ + p) ∂ ρ ln Ω 1 + c 2 s ∂ p ln Ω 1 2 (5.68)
where we have momentarily set the lapse to N = 1. If we take the time derivative of (5.67) and use the conservation equation (5.62) we finḋ
H = −4πG eff ρ + p + 4πĠ eff 3H ρ. (5.69)
In GR with minimally coupled fields, the existence of bouncing solutions (omitting the possible presence of spatial curvature for the sake of simplicity) characterised by an evolution whereḢ is initially negative (contracting phase) and becomes positive (expanding phase) is subjected to a regime where the NEC is violated in the initial regime and it holds in the final stage. For the theories under consideration here 47 , the presence of the time derivative of the effective Newton's constant makes it possible to have bouncing solutions where the NEC holds throughout the entire evolution. Notice that, at the bounce, the Hubble expansion rate must vanish at a finite but non-vanishing density, so the bouncing will generally occur when
G eff ρ b , p b , c 2 s,b = 0 (5.70)
where the subscript b stands for their values at the bounce. By looking at (5.68) we can see that the bounce can generally happen in two ways, namely:
• i) The numerator vanishes so that 3Ω 0 P 1 − P 0 Ω 1 = 0.
• ii) The denominator diverges so that
Ω 1 1 − 3 2 (ρ + p) ∂ ρ ln Ω 1 + c 2 s ∂ p ln Ω 1 2 → ∞.
Let us stress that these two possibilities are the most straightforward (and perhaps smooth) ways to realise the bouncing solution, but they are not exhaustive. For instance, one could envisage situations where both the numerator and the denominator diverge (or vanish) while the quotient is a well-behaved function with some roots at ρ = 0. Leaving this possibility aside, the bounce realised by means of ii) will generally rely on the existence of a divergence either in Ω 1 (or one of its derivatives) or in c 2 s . Since both of this quantities have a physical relevance, Ω 1 relates the two metrics and c 2 s typically gives the adiabatic sound speed, a divergence in them can potentially give rise to divergent physical effects. On the other hand, the bounce characterised by i) takes place when 3Ω 0 P 1 − P 0 Ω 1 = 0. From (5.59), we can obtain that
3Ω 0 P 1 − P 0 Ω 1 = 1 M 2 BI M 2 Pl Ω 0 Ω 1 2L G + T (5.71)
with T = T µ µ the trace of the energy momentum tensor. Thus, a bouncing solution where both metrics are regular (i.e., finite and non-vanishing Ω 0 and Ω 1 ) will be characterised by the equation 2L G + T = 0. Interestingly, for a radiation dominated universe the energy-momentum tensor is traceless and the condition reduces to F (P ) = 0. In a similar way as we studied the tensor perturbations for the EiBI model in the precedent sections, we can extend the analysis to the general class of theories considered here. We will closely follow the analysis in [60] where tensor perturbations are analyse in detail for an even larger class of theories formulated in the affine formalism. We will recognise that most of the properties we discussed for the EiBI theory are actually generic features for the theories described by (5.51). Let us then consider tensor perturbations on top of the homogeneous and isotropic background defined as
δg µν = 0 0 0 a 2 h ij δq µν = 0 0 0 a 2 Ω 1 f ij and δT µ ν = 0 0 0 Π i j (5.72)
theories involving non-minimally coupled fields. In general, the argument presented here will be valid for all theories giving rise to a non-constant effective Newton's constant for the cosmological evolution. Once again, the distinctive features of the theories considered here arise from the dependence on c 2 s that is not present in other classes of modified gravity theories, and this is what can introduce novel features.
with Π i j representing the anisotropic stress 48 . An important property that considerably simplifies the computations with tensor perturbations is that they only live in the spatial 3dimensional space and all the background quantities are diagonal in that box. This means that, at first order in tensor perturbations, any possible pair of matrices appearing in the equations will commute. Furthermore, the tensor perturbation of any scalar quantity vanishes identically, for instance we will have δ detΩ = 0 and so on. The equation (5.53) at first order in tensor perturbations reads
δΩ −1P +Ω −1 δP = 1 M 2 BI M 2 Pl Ω 0 Ω 3 1Π . (5.73)
Again,Ω andP are related by means of the definition ofΩ so the above equation can be seen as an equation for δΩ, whose solution will have the general form
δΩ i j = ω(ρ, p) Π i j (5.74)
with ω(ρ, P ) some function obtained from solving (5.73) which only depends on background quantities. This expression for the perturbation ofΩ allows to express the perturbation of the auxiliary metric as δq = δĝΩ +ĝ δΩ ⇒ δq ij = Ω 1 δg ij + a 2 δΩ ij = Ω 1 δg ij + a 2 ω Π ij (5.75) where the spatial indices have been lowered with the Kronecker delta. In terms of h ij and f ij , we then have
f ij = h ij + ω Ω 1 Π ij . (5.76)
This result generalises the one already found for the EiBI theory in section 5.1.1. An important property is that, in the absence of any anisotropic stresses, the tensor perturbations of the two metrics are identical and we can simply talk about metric tensor perturbations without referring to any specific metric. In other words, there is only one class of gravitational waves.This roots in the conformal relation for the two background metrics in the spatial 3-hypersurfaces. The field equations for these gravitational waves can be easily computed from (5.54), whose tensor perturbation yields
δR i j (q) = 1 M 2 Pl Ω 0 Ω 3 1 Π i j (5.77)
As it becomes clear, this evolution equation for gravitational waves is exactly the same as the one found in GR barring the replacement
M 2 Pl → M 2 Pl Ω 0 Ω 3 1 . (5.78)
In terms of the metric perturbations the equation (5.77) can be equivalently written in the familiar formf
ij + 3H(t) −Ṅ (t) N (t) ḟ ij −Ñ (t) 2 a(t) 2 ∇ 2 f ij = 16πG gw Π ij ,(5.
79) 48 We are dropping here the perfect fluid assumption in the perturbed sector for generality.
where G gw = G N / Ω 0 Ω 3 1 . In the regime of small energy densities, Ω 0 ≃ 1 and Ω 1 ≃ 1 so G gw ≃ G N . Once the solution for f ij is computed from the above equation, the evolution of the perturbation h ij is determined by the relation (5.76). It should not come as a surprise by now that the gravitational waves f ij satisfy the usual equation for cosmological tensor perturbations but with respect to the auxiliary background metric and a modified coupling to the source. This is simply the cosmological application of the discussion presented in 2.7.1 where it was shown that, in the Einstein frame, the auxiliary metric acquires the standard Einstein-Hilbert kinetic term, but it is coupled in a non-standard way to the matter fields. Since we are working at first order in tensor perturbations, the modified coupling to matter fields was expected to appear as a modified Newton's constant. From (5.79) we can also understand the rising of tensor perturbations discussed in section 5.1.1 for the bouncing and loitering solutions of EiBI as a consequence of having a non-regular auxiliary metric.
Elementary symmetric polynomials extension: Minimal model
We will now consider the Class-I theory introduced in [59] that we already discussed in section 2.7.3. This family of theories consists in extending the EiBI theory to include all the elementary symmetric polynomials and is described by the actions (2.122). The cosmology of the general case including all the elementary polynomials has not been performed yet in the literature. The fourth polynomial coincides with EiBI so that its cosmology is the one extensively discussed above. The other polynomial whose cosmology has also been investigated is the first one, that was called Minimal model. The corresponding action is given by
S min = M 2 BI M 2 Pl d 4 x √ −gTr ½ + M −2 BIĝ −1R − ½ ,(5.80)
where the constants have been chosen as to match GR without a cosmological constant in the low curvature regime. A possible cosmological constant term will be considered as part of the matter sector. As shown in the corresponding part of section 2.7.3, it is convenient to introduce the fundamental matrix of the model given bŷ
M ≡ ½ + M −2 BIĝ −1R .(5.M −1 −M − Tr M − ½ ½ = 1 M 2 BI M 2 PlTĝ . (5.82)
Before studying the more general cosmological solutions within this model, we shall first consider Einstein space solutions characterised by R (µν) (Γ) = R E g µν , with R E some constant curvature. In terms of the fundamental matrix, this means M µ ν = m 2 δ µ ν where m 2 = 1 + R E M −2 BI and the field equations (5.82) simplify to
4 − 3m 2 − 1 m 2 g µν = 1 M 2 Pl M 2 BI T µν . (5.83)
The conservation of the energy-momentum tensor implies that m 2 = const. Since we have that R E = (m 4 − 1)M 2 BI , the curvature R E must indeed be a constant and cannot be promoted to some arbitrary function. In other words, only a fluid corresponding to a cosmological constant can support Einstein space solutions, as one would have expected. For T µν = −ρ Λ g µν , the above equations give
4 − 3m 2 − 1 m 2 +ρ Λ = 0 ,(5.84)whereρ Λ = ρ Λ /(M 2 Pl M 2 BI )
. The solution of this equation is
m 2 = 4 +ρ Λ ± 4 +ρ Λ (8 +ρ Λ ) 6 . (5.85)
For these Einstein space solutions the deformation matrix is simplyΩ = m −2 ½ so that both metrics are conformally related as q µν = m −2 g µν . In the absence of the cosmological constant ρ Λ = 0, i.e., the vacuum solutions, the two branches give m 2 = 1 and m 2 = 1/3 respectively. The former corresponds to a Ricci-flat space with R (µν) = 0 and represents the branch continuously connected with GR, whereas the latter gives R (µν) = (−8M 2 BI /9)g µν and represents a de Sitter (anti-de Sitter) space for negative (positive) M 2 BI without the need for a cosmological constant. The two branches of solutions for m 2 in terms of ρ Λ are illustrated in figure 33. We can clearly see how the physical condition m 2 > 0 gives the boundρ Λ ≥ 2( √ 3 − 2). After briefly going through the simplest case of Einstein and vacuum space solutions, we shall consider the general cosmological solutions. Our homogeneous and isotropic Ansatz make the fundamental matrix take the form M µ ν = diag[M 0 (t), M 1 (t), M 1 (t), M 1 (t)] and the equations (2.138) read 1 M 0 + 3M 1 = 4 +ρ, These equations are of course (5.53) adapted to the present case. We can now obtain from them the quantities M 0 and M 1 algebraically in terms ofρ andp, i.e., we will have M 0 (ρ,p) and M 1 (ρ,p). If we solve for M 1 from the first equation and plug it in the second one we obtain (4 +ρ)M 3 0 + P (4 +ρ) + It is possible to solve this equation analytically, but the explicit expression is not very illuminating, so we will omit it here (the interested reader can find it in [61]) and instead we will plot the solutions in figure 34 for some interesting matter sources. As usual, one finds several branches of solutions, 3 in this case owed to above equation being cubic. Out of those 3, one is always unphysical because either M 0 or M 1 is negative and we do not consider it. The remaining two branches satisfy the positivity requirement, but only one is continuously connected with GR in the low densities regime (see figure 34). Even for these physical branches, the positivity of M 0 and M 1 impose constraints on the allowed values ofρ andp as shown in figure 35. From that figure we can see that the Born-Infeld corrections impose the bounds p M 2 BI M 2 Pl and ρ −4M 2 BI M 2 Pl . In particular, these constraints make the allowed region for a radiation fluid be compact, i.e., there is a maximum allowed value for its energy density. As can be easily understood from the left panel in figure 35, this will be the case for fluids with constant and strictly positive equation of state parameter. However, for dust or a cosmological constant, the energy density can grow arbitrarily large.
M 0 + 2M 1 + 1 M 1 = 4 −p,(5.
The definition of the fundamental matrixM also allows to obtain the corresponding curvature as [59]. In the left panels we show the solutions for M0 (green) and M1 (blue) for the Branch I (solid) and Branch II (dashed). In the right panels the corresponding solutions for the scalar curvature R are illustrated. Three types of fluids are considered from top to bottom: radiation (p = ρ/3), dust (p = 0) and a fluid with p = −0.8ρ. The dotted-purple lines represent the corresponding solutions in GR. We can see that the solutions for radiation are bounded for both ρ and R, for dust the density can grow to infinity but the curvature is bounded by ∼ M 2 BI and, finally, for the fluid with w = −0.8 neither the density nor the curvature are bounded.
R(Γ) = g µν R (µν) (Γ) = M 2 BI M 2 0 + 3M 2 1 − 4 .
parameter space (ρ, p, dp/dρ), while having a positive definiteM only gives constraints on its subspace (ρ, p).
In the right panel of figure 35 it is shown the Hubble expansion rate as a function of the density for radiation, dust and a cosmological constant. These fluids are important representatives of the following typical behaviour depending on the equation of state parameter w = p/ρ:
• Fluids with the equation of state parameter in the range 0 < w < 1 give rise to a maximum value for the energy density ρ M 2 BI M 2 p . We had already observed this type of behaviour for the standard EiBI theory in the previous section. Thus, it is quite typical to find an upper bound for the allowed energy densities in theoriesà la Born-Infeld.
• For fluids with −2/3 < w ≤ 0 one does not observe any upper bound for ρ. Interestingly, the Hubble function can become constant at high energy densities.
• Finally, for fluids with −1 ≤ w < −2/3 the Hubble function evolves as H 2 ∝ ρ 2 which is even worse than in GR in terms of singularities at high energy densities. For this type of fluids the realisation of the Born-Infeld mechanism fails.
One distinctive and crucial feature of this minimal Born-Infeld extension is the saturation of the Hubble function to a constant value at high energy densities appearing for −2/3 < w ≤ 0, which could offer an interesting alternative to realise a de Sitter inflationary epoch in the presence of a dust fluid. This idea was developed in [61], where, in order to achieve an inflationary scenario eventually evolving to a radiation dominated phase, it was considered a cascade of decaying dust fluids at the end of which there is a radiation component. This system is thus described by the following system of equations:
ρ i + 3Hρ i = Γ i−1 ρ i−1 − Γ i ρ i i = 1, ..., n (5.94) ρ r + 4Hρ r = Γ n ρ n (5.95)
where Γ 0 = ρ 0 = 0, ρ r is the energy density of radiation representing the final state and Γ i is the decay rate of the ith particle. In order to ensure the stability of the dust components during inflation we need to impose Γ i < H dS with H dS the (nearly constant) Hubble expansion rate during the inflationary phase and that will be H dS ≃ M BI . The idea is then that the quasi de Sitter phase is supported by the dust components as long as ρ dust ≫ M 2 BI M 2 Pl . Since ρ dust ∝ a −3 , the energy density of the dust components will eventually drop below M 2 BI M 2 Pl and the Born-Infeld regime will be abandoned. This will determine the end of the inflationary regime and the beginning of the reheating phase. In this phase, the Hubble expansion rate will evolve as H 2 ≃ ρ dust /(3M 2 Pl ) so that the different decay rates will become larger than the expansion rate and, therefore, the dust will start decaying. At the same time, the radiation component will be populated and, after all the dust components have decayed, we will be left with a radiation dominated universe.
This inflationary model has some interesting features that we will summarise here without entering into too many details and refer to [61] for a more rigorous treatment. The first important property is that there is a maximum value for the allowed energy density in the inflationary regime. This can be traced back to the very presence of the cascade that will lead to a non-trivial and time-dependent sound speed c 2 s (t) and a non-vanishing pressure. As we discussed above, this will give rise to a bounded range of values for the energy density in the physical space. This bound on the energy density will depend on the decay rates and, in turn, it will lead to a maximum number of e-folds for the inflationary phase. Thus, imposing that the inflationary phase lasts for at least 60 e-folds will give bounds on the parameters of the model, namely M BI and Γ i . Another constraint can be obtained from the fact that the reheating phase should end before the onset of BBN. Since the end of the reheating period is determined by the last decaying dust component, this will give a direct constraint on the smallest decay rate. These constraints are summarised in figure 36. Finally, a remarkable property of this inflationary scenario is that the first slow parameter ǫ 1 ≡ −d log H/d log a is negative so that we actually have a super-inflationary phase. Nevertheless, one cannot directly infer anything on the perturbations from here since the gravity sector is highly modified with respect to the standard inflationary models. This can be illustrated by looking at the tensor perturbations. Since they propagate on the auxiliary metric we need to compute the effective expansion seen by them. It turns out that they see an effective equation of state w = 1 and, thus, they are oblivious to the inflationary background and no primordial gravitational waves are generated within this model. Therefore, this inflationary model would come with the distinctive feature of the absence of primordial gravitational waves so that the detection of B-modes in the CMB generated by primordial gravitational waves would immediately rule it out. [61]. The left panel shows the bounds on MBI to have at least 60 e-folds of inflation as a function of the number of dust species n and assuming the decay rates so that reheating ends right before BBN. The black line denotes the Planck scale and the dotted line gives the lower bound on MBI so that the dust components are stable during inflation. It becomes clear that for n = 1 the allowed region is above Planck scale and hence no realistic inflationary scenario can be constructed and for more than 20 species BBN constraints and stability during inflation cannot be realized at the same time. In the right panel the bounds on (Γi, MBI) are shown together by assuming that all the decay rates are of the same order. The bound on Γi coming from the condition that reheating should end before BBN is encoded by the orange region. The blue region represents the stable region of dust components during inflation. Furthermore, the green curves indicate the bounds for MBI in order to have 60 e-folds of inflation for different number of components.
Functional extensions of Born-Infeld gravity
In the previous section we have studied the cosmological implications of the minimal extension of Born-Infeld inspired gravity theory, which was based on the trace of the square root structure rather than the determinant. We have seen how one can construct interesting quasi de Sitter solutions with a dust component in this model. In this section, we shall draw our attention to another interesting extension of the original EiBI gravity and discuss its potential impact to the early universe cosmology. Instead of a square root one could consider an arbitrary function of the determinant. This modification would still share the same properties as in EiBI gravity in the sense that General Relativity would be recovered in low energy density regimes with the modifications becoming appreciable only at very high energy density regime. Exactly this idea was pursued in detail in [268] and we shall summarise the main results of this study here. For this purpose, let us adapt to the notation used in [268] with Ω α β = g αµ q µβ where again q µν = g µν + M −2 BI R (µν) (Γ). In terms of Ω the Born-Infeld inspired gravity theory can be simply expressed as
S = M 2 Pl M 2 BI d 4 x √ −g |Ω| − λ + S matter . (5.96)
A natural extension arises by promoting the square root to an arbitrary function as it was proposed in [268]. In this case, the action generalises to
S = M 2 Pl M 2 BI d 4 x √ −g f (|Ω|) − λ + S matter . (5.97)
The gravitational Lagrangian density, L G , in (5.97), can be conveniently expressed in terms of an auxiliary scalar field A via the general function f (A) with the Lagrange multiplier
(|Ω| − A)f A ) as L G = M 2 Pl M 2 BI (Φ|Ω| − V (Φ) − λ) . (5.98)
where Φ = df /dA and V (Φ) = Af A − f (A). Written in this language, the connection field equations are simply given by ∇ µ Φ|Ω| 1/2 √ −qq µν = 0. One can use the same trick as in the previous sections to express the Riemann tensor in terms of the quantities of the matter field. By doing so, one obtains [268]
R α β = L G δ α β + T α β 2M 2 Pl Φ 2 |Ω| 3/2 . (5.99)
For the cosmological application, we will consider a perfect fluid as a representative of the matter fields. Using exactly the same procedure as above, we can compute the evolution of the Hubble function in terms of the matter field quantities. The resulting Hubble function in this particular extension takes the following form
H 2 = M 2 BI 6(1 +∆/(2H∆)) 2 (ρ + 3p)/(M 2 Pl M 2 BI ) + 2(Φ|Ω| − V − λ) Φ|Ω| + V + λ + ρ/(M 2 Pl M 2 BI ) (5.100)
where the short-cut notation is introduced ∆ = 2Φ 2 |Ω| 3/2 /(Φ|Ω| + V + λ − p/(M 2 Pl M 2 BI )). In [268] a family of power law functions f (|Ω|) = |Ω| n was investigated in detail. For
Extensions including a Ricci scalar
There exist extensions of the original EiBI gravity theory that relies on the presence of an additional Ricci scalar, which we will review in this section. These modifications are constructed either by including a Ricci scalar R(Γ) directly into the determinantal structure, or by including an additional function as a separate sector into the theory (belonging to the Class II theories in section 2.7.4).
Born-Infeld-f (R) gravity
One of the early predominant extensions of the EiBI gravity theory is the Born-Infeldf (R) gravity, where the original EiBI gravity theory is combined with an additional func-tion, that depends on the Ricci scalar [245]. In order to avoid any ghost instabilities, the theory is constructed in the Palatini formalism in both sectors. The Riemann tensor and the Ricci scalar both depend only on the connection and not on the metric. This idea of combining Born-Infeld and f (R) gravity was further investigated in [246,244,151]. Due to the presence of the Ricci scalar, the model exhibits more freedom for simultaneous applications to early and late time universe cosmology. In [245] it was shown, that for f (R) = aR 2 , the model does not alter much the physical properties of bouncing solutions found in the original EiBI model, but it does have crucial impact on the loitering solutions. The Lagrangian proposed in [245] has the following explicit structure:
S BI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI R (µν) (Γ) − λ √ −g + αM 2 BI 2 d 4 x √ −gf (R) + S matter ,(5.101)
where the first term is the standard EiBI Lagrangian and the second term is the novelty in form on an additional function of R = g µν R (µν) (Γ). The matter fields in S matter couple in a standard manner to the metric. The variation of this action with respect to the metric yields the modified metric field equations with respect to the standard EiBI model
√ −q √ −g q µν − g µν λ − α 2M 2 BI f + αf R M 2 BI g µβ g νγ R (βγ) = 0 ,(5.102)
where again we can define theq metric as q µν = g µν + 1 M 2 BI R (µν) (Γ) and f R is the derivative with respect to R. Similarly, the variation with respect to the connection can be written as
∇ σ √ −qq µν + αf R √ −gg µν = 0 . (5.103)
The connection equation can be written in the for us more useful form ∇ σ √ −qq µν = 0
whereq plays now the role of the auxiliary metric and is defined asq = |Σ| 1/2Σ−1ĝ and its inverse asq −1 = |Σ| −1/2ĝ−1Σ with Σ representing Σ µ ν = |Ω| 1/2 (Ω −1 ) µ ν + αf R δ µ ν with the standard notationΩ =ĝ −1q andM = Ω in the previous sections. For the cosmological application of the model, we are interested in homogeneous and isotropic backgrounds. We consider again the metric to be FLRW with N (t) = 1 and similarly we make an homogeneous and isotropic Ansatz forq orΣ directlyΣ = diag(σ 1 , σ 2 δ i j ). This on the other hand determines the form of the metricq to beq 00 = − −σ 3 2 /σ 1 and q ij = √ σ 1 σ 2 a 2 δ ij . For the matter fields, we again assume a perfect fluid withT = (ρ, pδ j i ). We can use the same procedure as in standard EiBI gravity model in order to obtain the evolution equation of the Hubble function in terms of the energy density and pressure of the matter fields (see section 5.2 for the general cosmological framework). For that we can use the field equations and the definition of the Einstein tensor and equal them. By doing so, one obtains [245]
H 2 = M 2 BI σ 1 − 3σ 2 − 2|Ω| 1/2 (σ 1 w 1 − 3σ 2 w 2 ) 2σ 1 1 − 3(1+w)ρ∆ρ 2∆ 2 ,(5.104)
where ∆ = √ σ 1 σ 2 and ∆ ρ = ∂∆/∂ρ. In this way, we again have a parametric representation of the Hubble function of the energy density and pressure of the matter fluid. For a particular choice of the function f (R) and the equation of state parameter of the matter fluid w, one can estimate the evolution of the Hubble function and examine whether different bouncing and loitering solutions exist in this extension of the EiBI model. In [245] a simple example was studied assuming a quadratic dependence in the form f (R) = aR 2 , since this allows to computeΩ and H 2 analytically in terms of the variables of the matter field. For this simple model, the presence of bouncing solutions is assured for M −2 BI < 0. Pl M 2 BI ) for three different cases: the original EiBI theory a = 0 (solid blue) and the Born-Infeld-f (R) theory with f (R) = aR 2 , with two different values a = 1/2 (dashed orange) and a = 1 (dashed red), and different equations of state (w = −1/5, 1/20, 1/10, and 1/3) respectively. One immediate observation is that the zero of ǫH 2 in the case ω = 1/3 is unstable under the changes of the parameter a. Furthermore, when the equation of state saturates to ω → 0, the Hubble function H 2 /M 2 BI might become again zero for sufficient high densities. However, the corresponding derivative of the functionḢ/M 2 BI would vanish, thus representing rather a minimum of H 2 /M 2 BI . This does not correspond to a bounce but rather signals an instability representing a state of minimum volume.
due to the presence of a in the function f (R). On the contrary, for different equation of state parameters one encounters novel loitering solutions, which were not present in the EiBI theory. These properties are shown in figure 39 taken from [245]. One additional interesting property is observable for the case w = 1/10. After reaching a local maximum, H 2 evolves towards a non-zero minimum to then diverge at a finite value of large energy densities. The non-zero value of the minimum depends on the parameter a. Since H 2 does not reach the solution H 2 = 0 in this case, one does not have a bounce. Nevertheless, they could offer an interesting alternative for a quasi de Sitter inflation due to the long plateau between the local minimum and maximum. In this way, one could achieve an inflationary scenario in the presence of radiation. In the minimal extension of the EiBI theory in section 5.3 we saw that one could realise a quasi de Sitter evolution in the presence of dust with w = 0. In this modification with f (R) this is achievable with radiation.
It is worth mentioning that this same model of Born-Infeld-f (R) theories was also used in [151] in order to construct singular inflationary cosmologies. For this purpose, they borrow ideas from singular f (R) inflation [267]. A requirement is that the scale factor evolves in the following form a(t) = e −(c 0 (−t+ts) 1+c 1 )/(1+c 1 ) , (5.105) with the constant variables c 0 , c 1 and t s . The Hubble parameter in this case is H = c 0 (−t + t s ) c 1 . With this Ansatz of the scale factor, one can establish the required relation between the dynamical and auxiliary metric. For large values of the constant c 1 > 1, singular inflation with a graceful exist can be realised [151].
Ricci scalar in the determinant
Other modifications based on the Ricci scalar have been constructed in the literature, where the Ricci scalar enters directly the determinantal structure of Born-Infeld [106]. The inclusion of this pure trace term in the determinant might offer interesting and promising cosmological implications. The proposed model has the following action [106]
S BI = M 2 BI M 2 Pl d 4 x − det g µν + 1 M 2 BI (αR (µν) (Γ) + βg µν R(Γ)) − λ √ −g +S matter .
(5.106) In order to recover General Relativity in the low energy density limit, the parameters of the theory have to satisfy α + 4β = 1. Furthermore, in the corresponding limits, one recovers Palatini R 2 theories or the original EiBI theory. In the following we will follow the notation of [106], where M Pl = 1. The variation of the action yields the modified field equations
√ −q √ −g 1 + βR M 2 BI q µν − β M 2 BI q αβ g αβ g µρ g νσ R (ρσ) − λg µν = − T µν M 2 BI ,(5.107)
where q µν = g µν + 1 M 2 BI (αR (µν) (Γ) + βg µν R(Γ)) in this particular modification of the EiBI theory. The variation with respect to the connection, on the other hand, results in ∇ ν √ −q(αq µν + βq αβ g αβ g µν ) = 0 . (5.108)
As in the previous sections, we can manipulate the equations on top of a homogeneous and isotropic background such that the Hubble expansion rate can be expressed in terms of the energy density and pressure of the matter fluid. For the homogeneous and isotropic evolution we can make a diagonal Ansatz forΩ =ĝ −1 q as Ω 0 0 = Ω 1 and Ω j i = Ω 2 δ j i . In terms of a dimensionless parameter x these components can be also written as Ω 1 = x 3 |Ω| 1/4 and Ω 2 = |Ω| 1/4 /x. After the adequate manipulations, the dependence of the Hubble expansion rate in terms of the energy density can be expressed as follows [106]
H 2 = 2M 2 BI α + |Ω| 1 4 (4βz − x 3 ) + 3 σ 2 σ 1 (|Ω| 1 4 (x 3 − 4βzx 4 ) − αx 4 ) 3α 2 − 3 q 2 dq 2 dρ ρ(1 + w) 2 ,(5.109)
with the short-cut notations
σ 1 = α + β(1 + 3x 4 ), σ 2 = α + β(x −4 + 3) andq 2 = √ σ 1 σ 2 Ω 2 .
Furthermore, the variable z satisfies x 3 + 3x −1 = 4z. After having brought the expression of the Hubble function H 2 in the desired form, we can study its evolution for different equation of state parameter of the matter fluid as we did in the previous sections. This will enable us to directly compare the type of bouncing, loitering and quasi de-Sitter solutions [106] and shows the dependence of the Hubble expansion function in terms of the energy density for a radiation fluid with w = 1/3 and for β > 0 but very small values close to zero. The evolution for β = 0 (solid blue), β = 10 −3 (solid red) and for β = 10 −2 (dashed blue) are plotted respectively. In the notation used in [106]
κ = M −2 BI .
within this class of modifications with respect to the standard EiBI gravity theory. In [106] this analysis was performed for radiation with w = 1/3 for different values of β. It was observed, that on the contrary to the previous modification in form of an additional f (R), the inclusion of the Ricci scalar into the determinant alters the robustness of the bouncing solutions for M −2 BI < 0. These solutions seem to be very sensitive to the presence of the parameter β for even very small values. This behaviour can be seen in figure 40. The Hubble function scales as H 2 ∼ ρ at large energy densities in this case. Another difference in the model rises for the loitering solutions of the standard EiBI model when M −2 BI > 0. In this modification the loitering solutions become a bounce with H 2 ∼ ρ − ρ max , where ρ max represents the maximum energy density at the bounce. The evolution of the Hubble function in the case 0 < β ≤ 1/4 together with β > 1/4 are shown in figure 41. For the increasing value of β getting closer to 1/4, the cosmological singular solutions resemble more those obtained in R + R 2 theories with H 2 ∼ ρ/3 for M −2 BI < 0. For the opposite case with M −2 BI > 0, for instance for β = 1/10 and β = 3/25, the loitering solutions of the standard EiBI theory become again a bounce in the past with the Hubble function saturating to H 2 ∼ ρ − ρ max , which has a quasi-sudden singularity in the past. For other values of β, for example β = 1/5 and β = 21/100, the asymptotic behaviour of the Hubble expansion rate becomes on the other hand H 2 ∼ (ρ−ρ max ) −2 corresponding to a big freeze singularity in the past.
Other extensions
Gravity coupled to Born-Infeld
Born-Infeld inspired gravity theories were mainly applied to early universe cosmology, since the effects of the modifications become appreciable at high energies. We have seen that interesting alternatives to the standard inflationary paradigm can be constructed within this framework and promising roads to avoid cosmological singularities can be successfully realized. Since the modificationsà la Born-Infeld are dominant at early times, for a possible application to dark energy and dark matter a change of the framework is needed. This was pursued by Bañados in the work [44], where the standard Einstein-Hilbert Lagrangian of GR is coupled to a "Born-Infeld" field in the hope to reproduce interesting phenomenology for late-time universe. Even if it was proposed as a modification of the original EiBI gravity theory, we would like to emphasise once again that these models do not comply the original Born-Infeld spirit of not modifying the field content. The action considered in [44] can be expressed as where α is a dimensionless parameter and R is the Ricci scalar associated to the metric g and R (µν) is the Ricci curvature of the independent connection Γ. This model constitutes General Relativity with the Einstein-Hilbert term coupled to the Born-Infeld connection Γ. In fact the model can be analogously written as a bimetric theory, where the potential interactions of the two metricsĝ andq do not satisfy the potential structure of massive gravity. Therefore, the theory probably might contain dangerous ghostly degrees of freedom. Independently of these ghost issues, the model was studied in [44], where it was found that the model admits de Sitter solutions at late times. In fact, it is argued that the parameters can be chosen such that the Born-Infeld field contributes ∼ 73% of the total energy density in form of vacuum energy and 23% in form of dark matter with the equation of state parameter varying between w = −1 and w = 0 respectively. The constructed cosmological solution is such that the scale factor evolves as a ∼ e Ht at late times and a ∼ t 2/3 at early times. The field equations of the model are given by G µν = −M 2 BI q/gg µρ q ρβ g βν + T µν 2M 2 Pl , R µν = M 2 BI (g µν + αq µν ) , (5.111) where q µν is the metric associated to the connection Γ. As it can be seen from the field equations, the structure of the interactions betweenq andĝ in −M 2 BI q/gg µρ q ρβ g βν does not correspond to the ghost-free massive gravity interactions, signalling the presence of ghostly degrees of freedom. For Einstein space solutions R µν = Λg µν , the two metrics have to be proportional to each other q µν = C 2 g µν , where the constant C is determined by the field equations to be C 2 = 1/(1−α). The modification of the Einstein equations is encoded in the term −M 2 BI q/gg µρ q ρβ g βν in equation (5.111) and acts as a cosmological constant for the Einstein space Ansatz, where its corresponding value can be expressed as Λ = C 2 M 2 BI = M 2 BI /(1−α). As mentioned above, even if these interactions allow for a constant contribution in form of a cosmological constant, they correspond to ghostly interactions, which will render the cosmological solutions unviable. For general cosmological solutions beyond Einstein space solutions, the following homogeneous and isotropic Ansatz for the two metrics were considered in [44]:
ds 2 g = −N (t) 2 dt 2 + a(t) 2 d x 2 , ds 2 q = −Ñ (t) 2 dt 2 +ã(t) 2 d x 2 . (5.112)
The background field equations (5.111) for these metrics become 113) where N = 1 and ρ c = 3H 2 0 /2M 2 Pl with H =ȧ/a, H q =ȧ/ã and H 0 denoting the Hubble parameter today. The Born-Infeld field contributes to the field equations in form of a fluid with the following effective energy density and pressure: Hence, the scale factor evolves between a ∼ t 2/3 at early times and a ∼ e Ht at late times.
H 2 = M 2 BI 3H 2 0ã 3 N a 3 + ρ ρ c , a 3 = 3Ñ 2ã a 2 H , H 2 q =Ñ 2 M 2 BI 3H 2 0 − 1 2Ñ 2 + α 3 2 a 2 a 2 ,(5.ρ BI = M 2 Pl M 2 BI
In [44] Bañados provides also the numerical solutions to confirm the approximate solutions of these two regimes. In figure 42, we see the numerical solution for the scale factor. In order to achieve the standard evolution as in ΛCDM model, the parameter α should be very close to 1, whereas the exact value α = 1 is singular. As next, we can compute the effective equation of state parameter of the Born-Infeld field. In terms of its energy density and pressure, it can be simply expressed as
w BI = p BI ρ BI = − aÑ a 2 .
(5.117)
As it can be seen in the right panel of figure 42, at early times the pressure is p BI = 0 behaving as matter and at late times p BI = −ρ BI behaving as dark energy.
As mentioned above, even if this model provides an interesting phenomenology for dark energy and dark matter, the ghostly interactions between theĝ andq metrics cast serious doubts on the physical viability of these cosmological solutions.
Teleparallel inspired Born-Infeld
A Born-Infeld approach to the Teleparallel equivalent of General Relativity was also pursued in the literature in the hope that Born-Infeld teleparallelism might cure the cos-mological singularities. For this purpose, Fiorini and Ferraro have considered an extension Figure 43: In this figure we illustrate the evolution of the scale factor of the modified teleparallel model in the presence of a radiation fluid w = 1/3 for different choices of α = Hmax/H0, which we extracted from [165].
In the case, where the energy density saturates to ρ → ∞, one has d → −1 corresponding to an exponential expansion. For a fluid with a 3(1+w) ρ = const = a 12 . The evolution of the scale factor is depicted in figure 43. Even if one can achieve interesting phenomenology for the early universe cosmology, this model barely represents a modificationà la Born-Infeld but should be rather considered as a f (T ) model. A model more close to the spirit of Born-Infeld will be discussed in the following.
Born-Infeld in Weitzenböck space-time
Still within the same framework of the previous section, there has been also the attempt in the literature to consider more general setups in the Weitzenböck space-time and study the consequences for early universe cosmology [167]. The main ingredients are again the super-potential S defined in equation (5.119) and torsion T µν ρ . Let us consider the following (Class-III) action S = M 2 Pl M 2 BI d 4 x det η ab e a µ e b ν + 2M −2 BI F µν − λ det(e a µ ) , (5.126) where the tensor F µν has the following general form F µν = α 1 S µ ρσ T νρσ + α 2 S ρµ σ T ρ νσ + α 3 η ab e a µ e b ν T . (5.127) For the interesting cosmological applications we shall consider a homogeneous and isotropic background for the vielbein e a = diag(1 + a(t), a(t), a(t)) and a barotropic matter field with p = wρ. By varying the action with respect to the vielbein, one obtains the following modified Friedman equation
√ 1 + A 2 H 2 √ 1 − A 1 H 2 (1 + 2A 2 H 2 − 3A 1 A 2 H 4 ) − 1 = ρ M 2 Pl M 2 BI ,(5.128)
with the short-cut notations Since the matter fluids are assumed to couple minimally to e, they follow the standard conservation equationρ+3(ρ+p)H = 0, which imposes the evolution of the energy density in the form ρ = ρ 0 a 0 a 3(1+w) with the subscript "0" denoting again the present day value.
In order to obtain General Relativity in the low energy density regime, we have to fulfil the condition α 1 + α 2 + 4α 3 = 1. For simplicity, let us first concentrate on the case with A 2 = 0, in other words, 2α 1 + α 2 + 6α 3 = 0. This leaves A 1 = 12M −2 BI . In this particular case, the Friedman equation simplifies to This specific case with A 2 = 0 recovers the modified Friedman equation of the previous subsection that we had categorised as f (T ) theories. Therefore, in this case one obtains the same non-singular cosmological solutions for radiation and dust matter as the ones reported in the previous subsection 5.6.2. For a more general case of the background with spatial curvature, the Ansatz for the vielbein is a little bit more involved e 0 = dt, e 1 = a(t)ẽ 1 , e 2 = a(t)ẽ 2 and e 3 = a(t)ẽ 3 ,
where theẽ i components are given bỹ e 1 = K(−K cos θdψ + sin(Kψ) sin θ cos(Kψ)dθ − sin 2 (Kψ) sin 2 θdφ) , e 2 = K(K sin θ cos φdψ − sin 2 (Kψ)(sin φ − cot(Kψ) cos θ cos φ)dθ − sin 2 (Kψ) sin θ(cot(Kψ) sin φ + cos θ cos φ)dφ) , e 2 = K(−K sin θ sin φdψ − sin 2 (Kψ)(cos φ + cot(Kψ) cos θ sin φ)dθ − sin 2 (Kψ) sin θ(cot(Kψ) cos φ − cos θ cos φ)dφ) , (5.132) with K denoting the spatial curvature. In this case, the modified Friedman equation (5.130) becomes
(1 ± M −2 BI a −2 ) 3/2 1 − 12H 2 M −2 BI − 1 = ρ M 2 Pl M 2 BI ,(5.133)
where ± represents the closed (K = 1) and open (K = −1) universe, respectively. In the high energy density regime, the solution to the modified Friedman equation gives the following evolution for the scale factor in the presence of a radiation fluid a(t) ≈ exp( M 2 BI /12t) as a/a 0 → 0 . (5.134) This is again the same type of non-singular solution as we found in the previous subsection, which cures the initial singularity and seems to be insensitive to the presence of spatial curvature. Another interesting case arises when one chooses the parameters as α 2 = 0 and α 1 − 12α 3 = 0. For this particular case, the Friedman equation modifies to
1 ± M −2 BI a −2 1 ± M −2 BI a −2 − 12H 2 M −2 BI − 1 = ρ M 2 Pl M 2 BI . (5.135)
We can again abstract the evolution of the scale factor for the high energy density regime. However, now the solution depends highly on the sign of curvature. For the closed universe scenario with the + sign, the scale factor evolves as a(t) ≈ t in the high energy regime, which therefore corresponds to a singular solution with the singularity appearing at t = 0. Maybe a more interesting scenario appears for the case of open universe, where the scale factor evolves as a(t) ≈ a min + M BI 24 t 2 with a min = M −1 BI , constituting a bounce at t = 0. The accelerated expansion period takes over when the energy density has its maximum value ρ max ∼ a −4 min = M 4 BI and the volume its minimum value a 3 min = M −3 BI . This model with the three components in (5.127) was further investigated in [168], where the realisation of a primordial brusque bounce was studied in detail. The author investigates the unexplored case with A 1 = A 2 and finds yet other type of interesting cosmological solutions. We shall summarise his findings for this case in the following. In the case with A 1 = A 2 , the modified Friedman equation becomes
6H 2 1 − 9H 2 2M 2 BI = ρ M 2 Pl ,(5.136)
where α 1 = α 2 and α 3 was reabsorbed into M BI . We can solve this equation for H 2 , which results in H 2 = M 2 BI (1 ± √ 1 − 3ρ)/9, whereρ stands for the dimensionless energy densitȳ ρ = ρ/(M 2 Pl M 2 BI ). The conservation equation for the energy density has the standard formρ = −3(1 + w)Hρ. Due to the different signs in the expression for H 2 , we have disconnected two different branches. The branch with the positive sign corresponds to a solution completely disconnected from the GR limit and therefore we can discard this branch. Concerning the negative branch, because of the presence of M BI one will have different solutions depending on the sign of this parameter. The type of solutions with M 2 BI < 0 are not interesting either, since they do not admit any regularisation process with the standard diverging behaviour as in GR. On the other hand, the branch with M 2 BI > 0 gives rise to the wanted regularisation effect with the maximum values H 2 max = M 2 BI /9 and ρ max = 3M 2 Pl M 2 BI . The conservation equation together with the equation (5.136) can be solved exactly. These exact solutions can be found in [168]. We shall only report on the behaviour of these exact solutions in the interesting limits. For a universe filled with radiation (w = 1/3) and for the branch with M 2 BI > 0, the scale factor evolves approximately as
a(t) a 0 4 = ρ 0 3M 2 Pl M 2 BI (1 ± 4M BI t) + O(M 2 BI t 2 )
. with H 2 (t = 0) = M 2 BI . This represents a brusque bounce. Even if the Ricci scalar suffers from indefiniteness at this point, the solution does not represent a singularity. The author shows explicitly that the geodesics are well behaved at the bounce in the sense that a point particle traveling along causal geodesics does not experience any singular behaviour. Furthermore, the author extends this analysis to a finite size extended object and confirms the same behaviour.
Final remarks
As we have seen in detail in this section, Eddington-inspired Born-Infeld gravity and its known extensions have received much attention in the literature. Since the modifications a la Born-Infeld become appreciable at large energy densities or in high curvature regimes, the direct cosmological applications can be only for early universe physics. The main goal of most of the studies was to construct cosmological solutions curing the standard Big Bang singularities. The inflationary scenario with a standard single field suffers from cosmological singularities. The idea behind using the Eddington-inspired Born-Infeld theory or its extensions was to deliver an alternative scenario for early universe, for instance in form of a bounce or loitering solutions. We have also seen in this section that interesting bouncing and loitering solutions can be constructed within these theories, that are based on a radiation or dust, depending on the explicit model. In the standard inflationary scenario the matter fields couple minimally to the gravity sector. As we have seen in various occasions in this section, Born-Infeld type theories can be seen as nothing else but General Relativity with a non-trivial and non-linear matter coupling. Specially, concerning the cosmological solutions, the essence of the modifications can be encapsulated in the Friedman equation as H 2 = f (ρ, p) with a non-linear function. The resulting cosmological evolution correspond to either quasi de-Sitter or bouncing or loitering solutions. We have seen that in the simplest realisation of the bouncing and loitering solutions in the EiBI model, the tensor perturbations were becoming unstable in the presence of matter fields with constant equation of state parameter. This of course renders these simplest realisations unviable. More general scenarios with non-constant equation of state parameters however can alleviate these issues. In this respect, we have seen concrete examples of an additional scalar field as matter field with varying equation of state parameter, which avoids the tensor instabilities in the bouncing and loitering solutions.
After reviewing the works of the standard EiBI model, we then systematically summarised similar cosmological studies of other extensions and modifications of Born-Infeld inspired gravity theories. Since most of these extensions were sharing the same mechanisms and features, before studying the individual cases, we have first developed the general framework of cosmological solutions for a general class of theories constructed out of the Ricci tensor and the inverse metric. Within this general framework, we have derived the master equation that determines the Hubble function in terms of the matter field variables and discussed the general mechanism that provides bouncing solutions. As next, for concrete models we considered the family of Born-Infeld inspired gravity theories based on all of the elementary polynomials and discussed in detail the cosmological solutions of the first polynomial as an example. We considered again fluids with different equation of state parameters and saw that interesting quasi de Sitter solutions can be constructed in a universe with dust component. We summarised briefly the arising inflationary scenario with a cascade of dust components in the early universe. Another interesting extension of the original Born-Infeld gravity is the functional extension in the sense that the square root of the EiBI model is replaced by an arbitrary function of the determinant. The resulting evolution of the Hubble function is such that the bouncing solutions are robust to the functional enlargement, whereas the loitering solutions do undergo notable changes. We have also discussed extensions of the original theory by means of an additional Ricci scalar, appearing either directly in the determinantal structure or as an additional separate function and explored the features of new cosmological solutions beyond the ones present in the EiBI model. Finally, we have also reported on other extensions along the line of teleparallel formulations of Born-Infeld theories and discussed the presence of interesting brusque bouncing solutions.
Concluding remarks, open questions and prospects
This review has been devoted to provide a comprehensive survey on theories of modified gravity that take inspiration from the Born and Infeld approach to nonlinear electrodynamics. The underlying logic is that a modification of the high curvature/density regime of the gravitational interaction could effectively introduce upper bounds that cannot be surpassed. As we have seen in this review, the richness of the theory transcends the mere bound of certain invariants, leading to physically sound results even in the presence of curvature divergences in black hole scenarios.
We started our journey on Born-Infeld gravity from the most reasonable place, namely, by reviewing the original construction of Born and Infeld for electromagnetism and the different routes leading to a transcription of its remarkable properties into gravity, specially its determinantal structure. The unsuccessful first attempt of the work of Deser and Gibbons to construct gravity theoriesà la Born-Infeld was rooted in the use of the metric formalism, which inevitably leads to the appearance of ghosts. So far the only ghostless theories in the metric approach are the so-called f (R) theories, but these can hardly be considered proper Born-Infeld gravity theories. The scrutiny presented in section 2 suggests that any theory of gravity realizing the Born-Infeld construction and formulated in the metric formalism will either be pathological or reduce to other known theories of gravity. A challenging problem is to find counter-examples to this general no-go result. The story becomes more interesting when resorting to a metric-affine framework, as Vollick did. When the connection is regarded as an independent field, the aforementioned pathologies arising in the metric formalism are avoided. A further refinement introduced by Bañados and Ferreira, making the matter-gravity coupling more standard, resulted in the most extensively explored Born-Infeld inspired gravity theory so far, dubbed EiBI. In this theory, the connection is generated by an auxiliary metric q µν that is non-trivially related to the metric g µν . Although q µν made its debut as an auxiliary object helping to solve the field equations, it soon showed its real significance and allowed to establish the existence of two frames for these theories, similar to what happens in scalar-tensor theories. In the original Born-Infeld frame, matter fields are minimally coupled to the spacetime metric g µν , which satisfies second-order dynamical equations. In the Einstein frame, the metric q µν behaves as in standard gravity, with an Einstein-Hilbert term governing its dynamics, but it couples in a non-standard way to the matter fields. Elucidating the existence of these two frames allowed to discern that, while matter fields follow geodesics of g µν , the geodesic motion of gravitons is determined by q µν .
Most of the existing developments in the literature make two important assumptions (though not always explicitly said) that we also adopted here. The first one is related to the class of theories considered where only the symmetric part of the Ricci tensor is included. This condition is useful to simplify the field equations and express the solutions for the connection solely in terms of the auxiliary metric. Although it could seem to be rather ad hoc, imposing a projective symmetry naturally results in this type of theories. However, it remains to be explored more general frameworks without the projective invariance and clarify to what extent such a symmetry should be considered as a fundamental ingredient. The second condition that is usually made has to do with the class of solutions that are considered, where the torsion is set to zero. Very little can be found at this respect in the literature of Born-Infeld inspired gravity theories and it is not rare to find works where this issue is simply omitted. Certainly, in most applications assuming vanishing torsion is a consistent Ansatz, but studying the stability of such solutions should also consistently incorporate fluctuations of the torsion. As with the projective invariance, the actual role played by the torsion is to be clarified within the context of these theories. In fact, it would not be too surprising to find links between the projective symmetry and the absence (or irrelevance) of torsion in the solutions. Let us remember that for the Einstein-Hilbert action in the Palatini formalism, the full solution for the connection only contains the trace of the torsion and it precisely enters as a projective mode, thus being pure gauge.
Our first contact with Born-Infeld inspired theories of gravity concluded with a glance at the different approaches adopted in the literature to incorporate the Born-Infeld ideas into gravity and a classification of the existing proposals. We first presented a general formalism showing that most of the features are actually shared by a wide variety of theories. We then decided to perform a classification based on the proximity to the original Born-Infeld construction, and taking the most studied case of EiBI as baseline. We could appreciate the richness of the field where the imagination of the community led to numerous searches along several directions. This was in high contrast with the case of Born-Infeld electrodynamics where the Lagrangian does not admit immediate alterations. This is so because such a Lagrangian was singled out by precise conditions, among which a symmetry guiding principle was paramount. In the case of gravity, an analogous guiding principle permitting to isolate some unique Lagrangian has not been found yet. The projective invariance could be invoked, but we have seen its incapability to sufficiently reduce the number of possible actions. Finding a better suited principle would considerably reduce the different possibilities and would give improved guidance to pursue the exploration of Born-Infeld gravity in closer relation with its electromagnetic ancestor, that turned out to exhibit a number of remarkable features. Until then, a prolific family of different Born-Infeld gravity theories is expected to remain. So far, most of them are based on the EiBI model and extensions along different paths. An interesting alternative was introduced taking TEGR as starting point. This allowed to study a different branch of theories which are formulated in a Weitzenböck space. Since TEGR gives an alternative description of GR as a gauge theory of the translational group, this route could lead to Born-Infeld theories of gravity closer in their structure to the original construction for electromagnetism. This gauge character could be appropriately exploited and it could serve as the desired symmetry principle so it deserves a further exploration.
Once the general landscape of Born-Infeld inspired theories was overviewed, we moved on to the different territories where these theories find applications. The first pertinent place to test the modifications introduced by Born-Infeld inspired theories of gravity was inhabited by the illustrious family of astrophysical objects. Since the Born-Infeld corrections are designed to only appear at high curvatures or densities, compact objects exhibit excellent prospects to put these theories on trial. The first explicit applications, however, already showed some subtleties in the weak-field limit associated with the energy-density dependence of the modified dynamics proper of metric-affine theories. In diluted systems, the fluid approximation may lead to unphysical effects depending on the weight functions used in the transition from the discrete to the continuum description. Potential patholo-gies associated to this can be found in Newtonian pressureless fluids and in compact star models based on polytropic equations of state, for instance. As discussed in detail for white dwarfs and neutron stars, polytropic equations of state are very useful to model their structural properties, but the transition to the external (idealized) Schwarzschild solution must be improved in order to construct realistic models able to account for certain observational features (like electromagnetic spectra and radiation fluxes), which at the same time may avoid artificial effects associated with the peculiarities of certain equations of state and/or the fluid approximation. After clarifying the importance of correctly modeling the outermost regions of stars, a number of results related with the structural and dynamical properties of compact objects and the Sun were reported. We can highlight the ability of solar observations to constrain the EiBI model via neutrino emission and seismic waves, the possibility of accommodating higher masses with soft equations of state in neutron stars, the stability of these objects against radial perturbations, and the possibility of using the low-mass spectrum of neutron stars to discriminate EiBI from GR. On the other hand, it really came as a surprise the existence of universality relations between quantities constructed using the moment of inertia, the quadrupolar moment, and the Love numbers (I-Love-Q relations), which turn out to coincide with those of GR. Dipolar magnetic fields also converge to the GR prediction at the crust and surface of neutron stars. These results imply a degeneracy which poses obstacles to the observational discrimination between GR and the EIBI theory.
After spending some time with the best known members of the family of compact objects, we continued our trip and arrived at the place where some of their more exotic acquaintances dwell, namely, black holes and their closest relatives. Obviously, as the black hole terrain has occupied the efforts and imagination of countless theoreticians and astrophysicists alike for decades, is not surprising that a few years of research in the context of Born-Infeld inspired theories of gravity has only allowed to touch a few of the relevant physical aspects regarding these objects. In this sense, our trip quickly went over some dubious proposals for these theories, either because they are formulated in metric approach (and thus plagued by ghostly-like instabilities) or because matter is included in an unconventional form. Nonetheless, this allowed us to naturally introduce the well known black hole solutions for Born-Infeld electromagnetism within GR, of which the familiar Reissner-Nördstrom solution is a particular (limiting) case. We thus introduced some of the trademark features of such black holes that bear a close resemblance with those obtained in Born-Infeld inspired theories of gravity, such as the appearance of different number and types of horizons, depending on characteristic parameters of the matter and gravity models. This way we naturally entered into the terrain of black hole solutions within EiBI gravity, where most research in this context has been carried out in the literature. First we reviewed and enlarged the description provided in the paper by Bañados and Ferreira and other works in the field, were we paid special attention to the deviations regarding geodesic motion, strong gravitational lensing and mass inflation. But we also described a different family of solutions, whose study revealed the presence of all types of exotic objects, like geons or wormholes. Geons are self-sustained electromagnetic objects without charges. On the other hand, wormholes represent the promised behaviour of Born-Infeld theories so that the center of black holes in GR is replaced by a regular object of finite size. We saw that the construction of wormhole solutions without any pathologies (i.e. violation of energy conditions) is a hard task, but EiBI gravity managed to surpass our expectations and provided, in analytical form, both wormholes and geons.
Our analysis of the geodesic structure over the innermost region of these objects revealed that, although these places look inhospitable at first, they actually are less perilous than expected and, in fact, geodesics can smoothly pass through, while the impact of curvature divergences on physical observers did not seem to pose any absolutely destructive threat. These results were confirmed by the well posedness of the problem of scattering of scalar waves off the wormhole. Further developments on this field involve higher and lower dimensional models, though with much less impressive results. It should be pointed out that the counterparts of the rotating Kerr solution of GR (and Kerr-Newman when charge is included) in Born-Infeld inspired theories of gravity are not available in the literature and, without such solutions, realistic black hole scenarios for astrophysical purposes cannot be put to a test. This is a very relevant point, since the present (and future) observations from gravitational wave astronomy offer a great opportunity for testing deviations with respect to GR solutions. We cannot but to encourage researchers working in the context of these theories to look for such rotating black hole solutions.
The last stage of our pilgrimage throughout the Born-Infeld land took us to a completely different scenery shaped by cosmological applications. It should not come as a surprise at this point that the natural home for such applications is the early universe because Born-Infeld theories are designed to affect the regime of high densities. In that context, it has been extensively shown that both EiBI and other Born-Inspired theories can provide singular-free solutions of two types, namely: bouncing solutions, where the universe transits from a contracting phase to an expanding one without hitting a singularity, and loitering solutions, where the universe asymptotically approaches a Minkowski universe as the energy density goes to infinity. Both of these solutions were shown to present some tensor instabilities for the original EiBI theory and in the simplest case of one single perfect fluid, although it was later shown that such instabilities could be avoided in more contrived scenarios. These solutions have recurrently been found in other formulations of Born-Infeld inspired gravity. Furthermore, other singular-free solutions have also been found like, e.g. the brusque bounce solution where the Hubble expansion rate is not defined at the bounce, but all relevant geometrical quantities are smooth. In most treatments of these solutions, the analysis is limited to studying the isotropic background evolution and, at most, the tensor perturbations. In some works, homogeneous and anisotropic solutions have also been studied, what is closely related to the analysis of tensor perturbations. However, the full viability of the bouncing solutions can only be claimed once all potential sources of instabilities have been shown to be under control. This is a paramount issue that needs to be properly addressed.
Providing singularity-free cosmological evolutions was precisely the job the Born-Infeld theories were designed for. However, it did not take long to find other jobs for which these theories could serve just as well. In fact, since they are constructed to modify the regime of high densities for gravity, they are also compelling frameworks to have models of inflation. This is achieved for theories that exhibit a nearly constant Hubble expansion rate in the Born-Infeld regime. Such a behavior has been found in several of the proposed models, some even for a radiation dominated universe. In particular, a specific model of inflation was developed in detail where inflation is supported by a dust component that decays into radiation, giving a model of the early universe similar to the usual inflationary scenarios with a re-heating phase, but from a completely different perspective. This in turn led to different predictions, in particular, a super-inflationary phase is achieved where primordial gravitational waves are not produced. One important feature of inflationary models based on these theories is that we have a naturally graceful exit of inflation. This is due to the fact that the density will typically decrease during the inflationary phase so it will eventually become smaller than the transition scale given by M 2 BI M 2 Pl and the GR regime will be restored, thus matching the standard cosmological evolutions. In general, and as with most of the cosmological analysis within Born-Infeld gravity theories, a proper treatment of the scalar and vector perturbations is still to be performed. This is crucial for the viability of these inflationary scenarios since it is of paramount importance to show that a red and nearly scale invariant spectrum of primordial scalar perturbations is generated, in order to be compatible with CMB measurements. However, given the highly non-standard gravitational sector of theses theories, a general and rigorous analysis of the subject will likely be an arduous task. At this respect, simplified models and, perhaps, the use of the Einstein frame could permit advancing in this direction.
Since the expansion of the universe makes the total energy density be diluted during the standard radiation and matter dominated epochs, the Born-Infeld corrections are expected to be negligible for the late-time evolution of the universe. In fact, a safe assumption is to impose that the transition to the GR regime is achieved before BBN. An important consequence of this is that Born-Infeld theories are not fruitful frameworks for dark matter and/or dark energy models and the use of cosmological observables to constrain them by studying their late-time evolution is futile. These theories can only affect the late-time cosmological observables by modifying the initial conditions in the early universe, perhaps set during a Born-Infeld inflationary scenario. There is however a family of cosmologies where Born-Infeld theories can become relevant at late times, namely models with future singularities. If the dark energy component happens to have some exotic features like a phantom behaviour, the asymptotic evolution of the universe in GR will end up with some type of singularity. In the presence of Born-Infeld gravity theories, these models will lead to scenarios where the Born-Infeld regime is reached again when the growing density trespasses the transition density M 2 BI M 2 Pl . Some works have studied the effects of the Born-Infeld corrections on these future singularities and found that, in general, there is not a universal regularisation of such singularities. This might not be too surprising since the existence of future singularities in GR is tightly linked to exotic properties of dark energy and, thus, the very cosmological model containing those singularities could already present pathologies. At this respect, we find fair to say that Born-Infeld inspired theories are entailed to regularise the Big Bang singularity with standard forms of matter, that is radiation and/or matter. The failure in regularising more general types of singularities should not be regarded as a flaw, but rather their eventual success in this task would be an additional gift granted by these theories.
As we have extensively discussed, the most outstanding feature and the raison d'être of Born-Infeld inspired theories of gravity is the possibility of regularising the singularities of GR without resorting to quantum gravity effects that should appear at the Planck scale M Pl . At this respect, we should say that the actual problem in GR is not the existence of singularities per se, but rather that the classical solutions near those divergences go beyond the regime of validity of GR as an effective field theory (presumably near the Planck scale) and, thus, we cannot trust those solutions anymore. The main idea behind Born-Infeld inspired theories is to introduce a new scale M BI at which the gravitational interaction is modified so that curvature divergences are classically regularised before reaching the quantum regime. However, a proper treatment of the validity of Born-Infeld theories as actual effective field theories is still missing. In particular, an issue that should be clarified is the existence of some regime above M BI , that one could naively identify with the strong coupling scale of the theory, where quantum corrections remain under control and, thus, the resulting classical solutions without singularities can be trusted. As with other open questions, perhaps the best starting point to address this issue would be the Einstein frame where all the effects are moved to the matter sector. It is not hopeless to expect a nice quantum behaviour, at least for some matter fields. For instance, if we start from a massless scalar field in the Born-Infeld frame, in the Einstein frame we would have a K−essence type of theory whose Lagrangian would be of the form K(X), for which the quantum stability of the classical action has already been discussed in detail in [137,85]. A crucial point to notice here is that for the singular free-solutions provided by Born-Infeld theories, NEC violations are not required and, therefore, the usual arguments for the instability and breaking of unitarity of these solutions do not directly apply. In general, the question would be as to what extent the Born-Infeld scale determines the strong coupling scale or the cutoff of the theory and the radiative stability for known types of matter.
The voyage undertaken throughout this review has permitted us to encounter an interesting family of gravitational theories that revealed fascinating novel effects in astrophysics, black hole physics and cosmology. They offer excellent opportunities for the exploration of the gravitational interaction and the open questions exposed above should serve, even if not exhaustively, as a guidance for future research within the field. We hope the accompanying traveller profited and enjoyed reading this work as much as we did in its elaboration.
and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Born-Infeld theories 15 2.1 Born-Infeld electromagnetism in a nutshell . . . . . . . . . . . . . . . . . . . 17 2.2 The Deser-Gibbons proposal: The ghost problem of the metricformalism . 21 2.3 Other proposals in the metric formalism . . . . . . . . . . . . . . . . . . . . 23 2.4 Eddington-Born-Infeld gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Simplified case: Vanishing torsion and projectively invariant case . . 31 2.6 The two frames of Born-Infeld gravity and the physical relevance of the auxiliary metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7 Classes of Born-Infeld inspired gravity. . . . . . . . . . . . . . . . . . . . . . 40 2.7.1 General mathematical framework . . . . . . . . . . . . . . . . . . . . 41 2.7.2 Class 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.7.3 Class I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.7.4 Class II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7.5 Class III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.7.6 Class IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 Astrophysics 62 3.1 Newtonian limit and fluid approximation . . . . . . . . . . . . . . . . . . . . 63 3.1.1 The modified Poisson equation . . . . . . . . . . . . . . . . . . . . . 63 3.1.2 Non-relativistic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.3 The issue with the matter profiles at a star surface . . . . . . . . . . 65 3.1.4 Limitations and improvements of the polytropic description . . . . . 66 3.2 Non-relativistic stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.1 Solar physics constraints . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Relativistic stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Stellar structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.3 Observational discriminations from GR . . . . . . . . . . . . . . . . 75 3.3.4 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.5 Universality relations: f -mode and I-Love-Q. . . . . . . . . . . . . . 77 3.3.6 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.1 Cosmological tensor instabilities . . . . . . . . . . . . . . . . . . . . 144 5.1.2 Varying equation of state parameter . . . . . . . . . . . . . . . . . . 146 5.1.3 Born-Infeld with a scalar matter field . . . . . . . . . . . . . . . . . 147 5.1.4 Born-Infeld in gravity and matter sector . . . . . . . . . . . . . . . . 151 5.1.5 Anisotropic cosmological solutions . . . . . . . . . . . . . . . . . . . 155 5.1.6 Late-time cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.2 General cosmological framework for Born-Infeld inspired gravity theories . . 160 5.3 Elementary symmetric polynomials extension: Minimal model . . . . . . . . 166 5.4 Functional extensions of Born-Infeld gravity . . . . . . . . . . . . . . . . . . 174 5.5 Extensions including a Ricci scalar . . . . . . . . . . . . . . . . . . . . . . . 175 5.5.1 Born-Infeld-f (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . 175 5.5.2 Ricci scalar in the determinant . . . . . . . . . . . . . . . . . . . . . 179 5.6 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.6.1 Gravity coupled to Born-Infeld . . . . . . . . . . . . . . . . . . . . . 180 5.6.2 Teleparallel inspired Born-Infeld . . . . . . . . . . . . . . . . . . . . 183 5.6.3 Born-Infeld in Weitzenböck space-time . . . . . . . . . . . . . . . . . 185 5.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 1. Preamble
is the n × n identity and e i the elementary symmetric polynomials which, for the case of interest here of n = 4, read: 4 − 6[M ] 2 [M 2 ] + 8[M ][M 3 ] + 3[M 2 ] 2 − 6[M 4 ] .(1.26)
Figure 1 :
1In this plot we show the regularisation occurring in Born-Infeld electromagnetism (solid lines) as compared to the case of Maxwell's theory (dotted lines).
n some dimensionless constants and e n (M ) the elementary symmetric polynomials defined as e 0 (M ) = 1, e 1 (M ) = [M ],
F
(
Figure 2 :
2Density profile (3.9) of the pressureless configurations. Note that the density is not positive definite beyond kJ r > π.
Figure 3 :
3Zero-temperature relativistic white dwarfs in EiBI theory (in this plot, ǫ → κ) in units of the typical density ρW D = 10 9 kg/m 3 . The horizonal line denotes the Chandrasekhar limit, M = 1.4M⊙. Note that an upper limit on the mass of these stars arises but can be much larger than in GR. Figure taken from Ref.[295].
Figure 4 :
4Analytic fit of the BBB2 and FPS models, taken from Ref.[329]. The crosses and pluses represent the data points in the EOS tables and the lines are the analytic fit functions.
Figure 5 :
5Gravitational mass M and fundamental mode frequency square ω 2 plotted against the central density ρc (denoted as εc in the plots) for two EOS models: (a) APR and (b) BBB2.
Figure 6 :
6Radii of neutron stars with 0.5M⊙, R05 in Eq.(3
FitFigure 7 :
7Universality relations between the moment of inertia, the spin-induced quadrupole moment, and the tidal Love number as shown in[330]. The curves are insensitive to both the EOS and the Born-Infield parameter ǫ.
Figure 8 :
8Born-Infeld black holes in GR for b = Q = 1/2 (Ũ ≃ 3.882). From top to bottom we find: naked singularities (M =Ũ − 3.5), extreme black holes (M ≃Ũ − 2.1152), two-horizon black holes (M =Ũ − 0.5), finite-metric solutions with a single horizon (M =Ũ ), and black holes with a single horizon (M =Ũ + 1). Note the transition between Reissner-Nordström-like configurations (M <Ũ , blue curves) to Schwarzschild-like black holes (M >Ũ, red) via the critical case M =Ũ (brown). Solutions with M =Ũ and no horizons are also possible. All solutions are asymptotically flat (horizontal dashed green line).
Figure 9 :
9Behaviour of the function f (r) in Eq.(4.31) for EiBI gravity with ǫ > 0 (solid red curves), compared to the Reissner-Nordström black holes of GR (dashed blue curves). In these plots we fix ǫ = Q = 1 and vary the mass. Left plot: a Reissner-Nordström black hole with two horizons may transform either into a naked singularity (M = 1.07) or in a Schwarzschild-like black hole with a single horizon (M = 1.15) in EiBI gravity. Right plot: a naked singularity in Reissner-Nordström black hole (M = 0.95) always remains a naked singularity in EiBI gravity. Note that EiBI solutions are only defined beyond a certain radius r > rc with r 2 c = √ ǫQ. All solutions are asymptotically flat (horizontal dashed green line).
= r 0 Figure 10 :
010Effective potential V (r) for time-like geodesics, k = +1 (left) and null geodesics, k = 0 (right) in Eq.(4.40) for the choice Q/M = 0.5 and ǫ/M 2 = 6 (in the notation of this plot, ǫ → κ). Left figure: four values of the impact parameter b ≡ L/M = 3.3, 3.5, 3.7, 4 (in the notation of this plot, L → l) have been depicted. On each of such curves the open circle corresponds to the radius of the innermost circular orbit (ISCO). Right figure: three values of the impact parameter b = 2M, bc, 7M . On the b = 7M curve the photon is scattered by the black hole at r = r0. Figures taken from Ref.[336] and[337], respectively.
Figure 11 :
11Deflection angle ∆ϕ in Eq.4.45 for fixed charge Q/M = 0.5 (left) and Q/M = 1.0 (right) for the EiBI black hole as a function of r0/M for different values of ǫ/M 2 (dashed lines) as compared to the GR case (solid), corresponding to ǫ = 0. Figures taken from Ref.[337].
Figure 12 :
12First relativistic image in Eq.(4.49) as function of the EiBI parameter ǫ/M 2 (in this figure, ǫ → κ) for different values of the electric charge (set of curves), as compared to GR (solid thick line).Figure taken from Ref.[337].
Figure 13 :
13The maximum value of Misner-Sharp mass MMS as a function of the logarithm of the final density ρ f,i and the EiBI parameter ǫ, assuming ω ⊥ = 1 (left), resulting from a numerical simulation (taking ri ∼ 0.95r− and a Reissner-Nordström solution of GR as initial conditions) of the field equations to obtain the metric component g rr[end] . Figure taken from Ref.[30].
≡ |ǭ|C 0
0, the constant −2M arises from imposing the asymptotically flat behavior of the metric component f (r), and the function I(r)
Figure 14 :
14Figure 14: The metric functions gtt = ψ 2 (r)f (r) and g −1 rr = f (r) for the case α = 3/4 taking an EiBI parameter ǫ = −4. In these plots, x = r 2 0 /|ǫ|, in such a way that x = 1 sets the appearance/dissappeareance of a horizon. When x > 1, no horizon is found and the minimum value of the radial coordinate corresponds to r0, where the wormhole throat is located. Figure taken from Ref.[327].
+ . . . ; K(g) ≈ K
Figure 15 :
15Representation of the radial function r(x) in Eq.(4.103) as a function of the coordinate x and measured in units of rc. In this plot the wormhole throat is located at x = 0 (r = rc). The dotted line represents |x| (the two asymptotic spaces).
Figure 16 :
16Left plot: Euclidean embedding of the equatorial θ = π/2 and t =constant section of the cases with curvature divergences for electromagnetic geons. The vertical axis represents the function ξ(r).Figure extracted from Ref.[279]. Right plot: Euclidean embedding of the wormhole described by Eqs.(4.84) and (4.85) with α = 3/4, ǫ = 4 and x = 1/2. Figure extracted from [327].
analysis of these solutions it follows that for α < 0 there is a minimum value of the radial coordinate r = r 0 in Eq.(4.123), where r 0 = (α−2)(4ǫQ 2 (1−1/α))
Figure 17 :
17expressions appearing in the line element in Eq.(4.120) and with the definitions of Eqs.(4.121) and (4.122). Numerical integration of (4.125) yields the plots of Figs.17, where the light deflection angle ∆Φ is depicted against the turning point radius r tp (for which dr/dφ r th = 0) for different values of α, both positive and negative, and compared to Maxwell case. This way, like in the case of EiBI gravity coupled to a Maxwell field (see section 4.2.3), gravitational lensing could be used in order to put experimental constraints on the size of ǫ and on possible nonlinear corrections to Maxwell theory. Deflection angle ∆Φ for EiBI gravity coupled to Born-Infeld electrodynamics as a function of the turning radial point rtp (defined as dr/dφr th = 0), for different values of b 2 = α/(4ǫ), both positive (left plot) and negative (right plot). The dashed curve represents the coupling of EiBI gravity to Maxwell electrodynamics. Figures taken from[218].
Figure 18 :
18Affine parameter u(x) (in this plot u → λ) as a function of the radial coordinate x for null radial geodesics in Eq.(4.129), compared to the GR behaviour (dashed green curve). In this plot E = 1, and the horizon axis is measured in units of rc. Figure taken from[279].
, the fact that x ∈] − ∞, +∞[ allows to smoothly extend the affine parameter u(x) across x = 0 to the whole real axis, which contrasts with the geodesics ending at x = 0 of the GR case. Finally, if δ 1 = δ c (finite curvature cases), the leading order term of the expansion of the effective potential in Eq.(4.130) vanishes, and the new expression becomes V (x)
corresponds to the value of the Jacobi fields at some initial instant u i and A a b (u) is a 3 × 3 matrix (the identity if u = u i ). If all Jacobi fields vanish at
Figure 20 :
20Left plot: trajectories of light rays (in this plot u → λ) emitted by a time-like observer from ξ = 0 at different times before reaching the wormhole throat (oblique line u − Eξ = 0) in a Schwarzschildlike configuration, δ1 < δc (in this plot, E = 1, a = 3). Right plot: proper time ∆u taken in a round trip by a geodesic at ξ = 0 to another separated by a distance ξ = {0.01rc, 0.005rc, 0.001rc}, as a function of the proper time u at which the light ray was sent. At u = 0 the divergent region is reached, but the (finite) travelling time tends to zero as the comoving distance tends to zero too. Figures taken from Ref.[281].
Figure 21 :
21Left plot: transmission coefficient for the potential well −|y ′ | −1/2 (blue dots) and the barrier +|y ′ | −1/2 (red crosses). Right plot: transmission cross section in Eq.(4.145) as a function of ω for δ1 = 1 and Nq = 10 (configurations without horizons), from numerical calculation of V ef f (dots) in Eq.(4.142) and compared to the approximation σ ∝ ω −1/2 (continuous line). Figures extracted from Ref.[280].
Figure 22 :
22Left plot: representation of r(x) in Eq.(4.156) for D = 4 (solid), D = 6 (dashed) and D = 10 (dotted), with both axes measured in units of rc. The wormhole throat is located at x = 0. Right plot: representation of the affine (null radial geodesics) parameter Eu(x) (in this plot u → τ ) as a function of the radial coordinate x in D = 4 (solid), D = 5 (dashed) and D = 10 (dotted). Figures taken from Ref.[55].
( 4 .
4176) is a Klein-Gordon-type equation for the massless, p = 0, and massive, p = 0, gravitons, while in the Schrödinger-like equation (4.176) for the Kaluza-Klein modes we have redefined X =ã − (d−1) 2
Figure 23 :
23Left plot: affine parameter u(x) (in this plot u → σ) for ingoing and outgoing radial null geodesics in the case s = −1, as compared to the GR case (dashed lines). In this case, the wormhole lies on the future (or past) boundary of the spacetime. Right plot: Affine parameter u(r(y)) (where y is a new suitable radial coordinate) for radial null geodesics in the case s = +1, where the wormhole is reached in finite time but can be indefinitely extended. Figures extracted from Ref.[54].
Fiorini analyse the structure of this spacetime in the two regions of interest. At asymptotic infinity, ρ → ∞, where Y → 1, all these scalars vanish, and the geometry (4.186) describes a BTZ-type spacetime with a conical singularity. On the central region, r → 0, the scalars vanish as well. In particular, the curvature scalar behaves as R
Figure 24 :
24This figure is taken from
Figure 25 :
25This figure is taken from[45] and shows the evolution of the scale factor in the EiBI model in the presence of a radiation fluid. The scale factor is normalised by the minimum length scale aB. For κ < 0, the universe undergoes a bounce (in the upper panel), whereas for κ > 0 the universe loiters, where the scale factor approaches the constant value aB for t → −∞ (in the lower panel).
N
≡ a B 2/(3|κ|). This, on the other hand, means that the lapse and the scale factor of the auxiliary metric evolve this time as =
20) with D = (1 + ρ/M 2 BI )(1 − p/M 2 BI ) 3 and the choice of units M Pl = 1 and |κ| = 1 used in
Figure 26 :
26This figure is taken from[32] and shows the evolution of the scale factor in the presence of a scalar field with varying equation of state parameter in the EiBI model. The universe undergoes a bounce at t = 0. The initial values are chosen to be ρi = 10 −4 and wi = 0 and the mass of the scalar field is assumed to be m = 100. Note, that they use the unfortunate choice of units M Pl = 1 and |κ| = 1.
Figure 27 :
27This figure illustrates an example of the phase map taken from [112] with the parameters chosen as m = 1/4, M −2 BI = 1/4 and λ = 1, where the authors use the units M Pl = 1.
W 2 /f 1 dη, the equation of the dynamical scalar field perturbation takes the simple form at the attractor stageχ
Figure 28 :
28This figure is extracted from[219] and illustrates the deceleration parameter (which in their notation is denoted by q) as a function of X in the case M −2 BI > 0. One can see the transition of the deceleration parameter from negative to positive values and then afterwards to become negative again. The outcome for different values of the constant C0 is shown by the different curves.
Figure 29 :
29This figure is borrowed from [219], where one can see the evolution of the scale factor in the different regimes. In [219] the following values have been chosen for this plot: M Pl = 1, M −2 BI = 0.5, αT = 5, C2 = 0.001 and finally C0 = 1.0025 with the initial value a0 = 0.06 at τ0 = 0.06. In the left panel, one can see the evolution of the scale factor during the loitering, deceleration and acceleration phases. The right panel is just a zoom of the first two phases.
√ 3 .
3During the period of inflation the universe grows by 60 efolds in 10 −32 seconds. This on the other hand puts the bound M −2 BI 0.67 × 10 −50 m 2 . The second phase of accelerated expansion at late times has the scale factor growing as a DE ∼ e √ C 2 /3α T τ /M Pl . As it becomes clear from the expressions of the scale factor in these two regimes, the evolution at early times depends on the Born-Infeld scale M BI of the gravity sector whereas at late times on the scalar Born-Infeld parameter α T . The intermediate phase depends on both as α 2 T M −2 BI . Similarly, one can study the cosmological solutions of the background equations in the other case when M −2
Figure 30 :
30This figure represents the evolution of the rescaled energy density r = ρM −2 BI and the rescaled volume element GM −2
Figure 31 :
31In this figure taken from
are shown as a function of the rescaled time θ = tMBI.
Figure 32 :
32This figure borrowed from
Figure 33 :
33In this plot (adapted from[59]) we show the two branches of solutions for m 2 as a function ofρΛ given in (5.85) (left panel) and the characteristic scale RE normalized to M 2 BI of the corresponding Einstein space (right panel). The blue-solid solution is continuously connected with GR in vacuum, while the red-dashed solution represents the branch giving rise to dS/AdS in vacuum. Interestingly, the dS/AdS branch is almost insensitive to the presence ofρΛ. In this branch, the value of RE quickly saturates to −M 2 BI and remains constant irrespectively of the cosmological constant. Finally, we can see how the positivity of m 2 selects the physical solutions and imposes some bounds on the values of theρΛ that can be accommodated.
Figure 34 :
34the physical branches of solutions for M 0 and M 1 we can then obtain the dependence of the curvature on the density. In the low energy density limit theFigure adapted from
Figure 35 :
35Figures adapted from [61]. Left panel: physical region determined by imposing the positivity of the fundamental matrixM . It is also shown some important equation of state parameters to illustrate the bounds on ρ and p imposed by the Born-Infeld corrections. Right panel: Evolution of the Hubble function in terms of the energy density for different equation of state parameters. At low energy densities the standard evolution of GR is recovered (depicted by the dotted line), whereas at high energy densities the translated modifications in the matter source from Born-Infeld become dominant. A crucial property of this model is that the Hubble function becomes constant in the Born-Infeld regime for a dust component with w = 0 giving rise to a de Sitter phase.
Figure 36 :
36Figure adapted from
Figure 37 :
37These figures are taken from[268] where the evolution of the Hubble function is plotted in terms of the energy density of the fluid ρ/(M 2 Pl M 2 BI ) for different values of the index n. The notation used there corresponds to ǫ → M −2 BI and κ 2 → M −2 Pl in our notation. In the left panel we see the evolution for a radiation fluid with w = 1/3. The solutions represented by the dashed lines correspond to M 2 M BI < 0 (plotted in positive quadrant for graphical convenience) and represent the bouncing solutions whereas the solid lines (M 2 BI > 0) represent unstable solutions with H 2 = 0 and H,ρ = 0 at high energy densities. One can further see that these non-singular solutions have very similar behaviour for small deviations in n. In the right panel the evolution is shown for a fluid with equation of state parameter w = −1/5. In this case one striking observation is that the solid solutions start resembling bouncing solutions for sufficiently large values for the index n.values of the parameter close to n = 1/2, of course the features are very close to the original Born-Infeld gravity. In general, one has again two types of branches of solutions, the branch with M −2 BI > 0 and the branch with M −2 BI < 0. It turns out that the first type of solutions are more sensitive to the changes in the index n. The second type of solutions representing a bounce are more robust. In figure 37 extracted from [268] we can see the evolution of the Hubble function for different values of the index n for a fluid with a positive equation of state parameter in the left panel and with a negative equation of state parameter in the right panel, respectively. The bouncing solutions are depicted by the dashed lines and the solid lines represent the unstable solutions with H 2 = 0 and H ,ρ = 0 for sufficiently high energy densities. The qualitative behaviour of these two branches of solutions remains the same for small deviations in the index parameter. On the other hand, for fluids with negative equation of state parameter the solid line solutions start hitting the horizontal line converting more and more into a rather bouncing solutions for large values of n.
Figure 38 :Figure 39 :
3839One obtains H 2 = 0 at |ρ/(M 2 Pl M 2 BI )| = 1 independently of the sign of the equation of state parameter. On the other hand, for M −2 BI > 0, the Hubble function strongly depends on the sign of w and shows a divergent behaviour for w ≤ 0. The parameter a in the function f (R) does not effect significantly the type of bouncing solutions for M −2 BI < 0 within this model as one can see in figure 38. The novelty of this modification coming This figure from [245] represents the evolution of the dimensionless Hubble function −H 2 /M 2 BI in terms of the dimensionless energy density −ρ/(M 2 Pl M 2 BI ) for both the original EiBI theory (solid blue) and the modification with the function of the form f (R) = aR 2 , with the value a = 1/2 (dashed orange) and a = 1 (dashed red), in the presence of a matter fluid with two different equations of state (w = −1/5, 0, and 1/3) respectively. The presence of bouncing solution does not alter with the difference in a and hence is a robust property of the M −2 BI < 0 branch. The notation in [245] translated into ours as 1/κ → M 2 Pl and 1/ǫ → M 2 BI . from f (R) becomes apparent in the other branch of solutions when M −2 BI > 0 and is very sensitive to the sign of the equation of state parameter. For instance, the standard loitering solutions of EiBI gravity theory for w = 1/3 disappear as one moves away In this figure we show the evolution of the dimensionless Hubble function H 2 /M 2 BI as a function of the dimensionless energy density ρ/(M 2
Figure 40 :
40This figure is taken from
Figure 41 :
41This figure is taken from[106] for the evolution of the Hubble function in the case 0 < β ≤ 1/4 (left panel) and β > 1/4 (right panel) with κ = M −2 BI .
R
(µν) (Γ) + L matter ,(5.110)
Figure 42 :
42now study the behaviour of the equations at late and early times. For large values of the scale factor we can neglect the contribution of the ordinary matter fields. In this case, the scale factors evolve as a = a 0 e This figure is borrowed from [44], where one can see the evolution of the scale factor in the Born-Infeld gravity model in the presence of the standard Einstein-Hilbert action versus the standard Friedman universe in the left panel. Both evolutions are almost indistinguishable if one chooses α = 0.99. In the right panel on the other hand the evolution of the effective equation of state parameter wBI is illustrated. with the constant variable c 1 = 3(1 − α)M BI H 0 . This corresponds to the de Sitter solution with Ω Λ = 1/c 2 1 . For small values of the scale factor, on the other hand, the solutions can be approximated as a = a 0 t 2/3 (1 + O(t 4/3 )) andÑ =Ñ 3 0 (1 + O(t)),ã =Ñ 0 (1 + O(t)) . (5.116)
subscript "0" denoting the values today and β 0 = (1 − 12H 2 0 /M 2 BI ) −1/2 − 1. The second term in equation (5.125) is always negative and approaches zero if w > −2/3 for a → ∞. In the contrary case, if w < −2/3 then this term decreases. In the opposite limit, with a → 0 for w > −1, one has a ∼ e √ M 2 BI /12t and hence H max = M 2 BI
the dynamics of Hubble function are recast byH(t) = ∓M BI /(1 ± 4M BI t) + O(M 2 BI t 2 ). From these expressions one immediately observes that there is a minimum value for the scale factor at t
ρ|ρ ′ | ≪ 1, where one obtains the additional simplifications H ′ ∼ 1, A ∼ g tt , and B ∼ g rr . This way, Eqs.(4.59) and (4.60) become approximately Combining the last two equations and integrating the results one gets grr gtt | [MI] ∼ constant where MI stands for quantities evaluated during mass inflation. This equation means that grr gtt | [start] ∼ grr gtt | [end] and we recall that g tt[start] ∼ −g −1 rr[start]59)
−
H ′
H
A ′
A
+
B
H 2 −
H ′
H
2
= 8πBT r
r .
(4.60)
Mass inflation takes place for r −ǭ
2 g ′
rr
g rr
∼ −8πr − ρg rr ;
g ′
tt
g tt
∼ −8πr − ρg rr .
(4.61)
81 )
81This fundamental matrix must be positive definite on physically acceptable solutions and is related to the deformation matrix byΩ =M / detM , as shown in equation (2.137), thus guaranteeing that both the auxiliary and the spacetime metrics have the same signature. Furthermore,M satisfies the equation (2.138), that we write here again
86 )
86with the dimensionless density and pressurēρ ≡
ρ
M 2
BI M 2
Pl
,p ≡
p
M 2
BI M 2
Pl
.
(5.87)
Concluding remarks, open questions and prospects 190
Those models avoiding these shortcomings and, at the same time, being able to provide a consistent cosmological expansion which is coherent with the GR limit are usually termed as viable, see e.g.[15,121,131].
This is true for true tensorial densities. For pseudo-tensorial densities the transformation also picks up a sign for parity odd transformations.
Let us remember that a form is nothing but a completely antisymmetric tensor.
Since the graviton propagator trivialises in 3 dimensions, the problem of the potential ghosts discussed above are less virulent.11 Deser and Gibbons already made reference to this approach in[140], but they did not consider it any further in favour of a metric formalism.
Here we prefer to restore all the dimensionful constants as opposed to[45], where the authors set 8πG = 1. Furthermore, we correct a typo in form of a factor of 2 appearing there, which has propagated in the literature.
In the literature of Born-Infeld theories it is customary to denote the inverse of the matrix qµν simply as q µν , in accordance with the usual convention of denoting the inverse of a metric with upper indices. Since we will have two metrics, we prefer to explicitly keep the inverse for the moment in order to avoid any confusion to the unfamiliar reader in these first steps into the formalism of Born-Infeld theories, since q µν could very well be confused with g µα g µβ q αβ . We will eventually drop the explicit mention for the inverse ofq to alleviate the notation and whenever there is no risk of confusion.
Again, remember that we are considering minimally coupled fields, so the energy-momentum tensor does not depend on the connection, but only on the matter fields and, perhaps, the spacetime metric gµν .
Here we consider the λ−term as part of the matter sector, where it will contribute as a cosmological constant.
Of course, we need to integrate out the connection. Since for the Einstein-Hilbert term at hand we know that the connection is given by the Levit-Civita connection of the metric qµν , we omit this step here and assume that this operation has already been carried out.23 By consistent one usually means unitary and Lorentz invariant, although locality is a frequent implicit condition. See for instance[71] for consistent theories including non-localities. See also[207,208] for other constructions based on higher but finite derivatives.
A treatment of more general theories can be found for instance in[60].
In order to illustrate this point, let us remember the case of a Proca field coupled to conserved currents whose equations read ∂νF µν + m 2 A µ = J µ . The gauge invariance of the charged sector implies the conservation of the current ∂µJ µ = 0, while the mass term for the vector field breaks the gauge invariance in the vector field sector. However, this does not introduce any inconsistency in the equations as, by taking their divergence one obtains the constraint ∂µA µ = 0 which, not only it does not represent an inconsistency, but it plays in fact a crucial role to remove additional polarizations for the massive vector.
The same will however apply if we consider more general metric sectors like, e.g., f (R) terms.
We adapt the notation of that reference to be consistent with the notation of this review, so we reservê q andΩ for the auxiliary metric and the deformation matrix respectively.
Actually, in[122], the author considers a family of theories that would generically belong to the Class 0, but a particular model is eventually selected that would belong to the Class IV and is the one we refer to here.
For the purpose of this section, we shall consider just the case of both symmetric connection and Ricci tensor for this theory, a case discussed in detail in section 2.5.1.
In the determination of the relation between pressure and energy density in nuclear matter equations of state, the so-called symmetry energy quantifies the change in nuclear energy associated with modifying the neutron-proton asymmetry. Accurate determination of the thickness of the neutron skin of neutron rich heavy nuclei would provide crucial experimental constraints on the symmetry energy and, as a consequence, on the structural properties of neutron stars.
Indeed, the existence of gravitational waves was already indirectly hinted by the observations of the Hulse-Taylor binary pulsar[213] and others.
See Berti et.al.[67] for an overview on experimental constraints on the many gravitational modifications of GR proposed in the literature.36 It should be stressed that, though in astrophysically realistic situations the amount of net electric charge is negligible, its consideration for black holes may yield relevant lessons regarding gravitational physics beyond GR, in particular, on the spacetime singularities issue.
Note in passing by that finiteness of the self-energy can be achieved in other non-linear theories of electrodynamics, which indeed share most of the features regarding the structure of horizons and behaviour of curvature scalars when coupled to GR[142].
Note, however, that a given spacetime can be geodesically complete and still be pathological since it can contain finite paths for observers with bounded acceleration, see Geroch[177].
A general treatment of tensorial perturbations in EiBI gravity can be found in[371].
A similar result has also been found in other theories of gravity formulated in the Palatini approach, like f (R)[278,40,56].
1 − (4k 2 /(3β 2 )), representing an unstable growth. Hence, the bouncing solution suffers also from an instability in the same way as the loitering solution, even though in the latter case it was much milder.
Usually, q is used for the deceleration parameter in the literature but we use d here in order to avoid confusion with the auxiliary metric.
We are not assuming any projective symmetry a priori on the Ricci tensor so, in principle, we could consider both the symmetric and the antisymmetric parts of the Ricci tensor. The background cosmological evolution where all the relevant objects are assumed to be diagonal will be, in general, oblivious to the presence of the projective symmetry. However, it is crucial when studying the perturbations. For an analysis of the cosmological scenarios in an extended class of theories see[60].
This discussion is not specific of these theories, but it also applies to other modified gravity theories or
Acknowledgments 196AcknowledgmentsWe are grateful to many colleagues for useful discussions and comments regarding the many topics and results reported in this work. In particular, we thank Pedro Avelino, Alejandro Cárdenas-Avendaño, Joaquín Díaz-Alonso, Rafaelwhere the Branch I refers to the solution that connects with GR at low energy densities whereas Branch II stands for the branch that connects with the dS/AdS solutions in vacuum. Infigure 34we show the full solutions. Three different types of fluids are considered: dust with p = 0, radiation with p = ρ/3 and a fluid with the equation of state parameter w = −0.8. As we commented above, a radiation fluid naturally shows an upper bound on its possible energy density, which further translates into an upper bound for the scalar curvature ∼ M 2 BI . This actually happens for equation of state parameters 0 < w < 1. This will have important consequences for early universe cosmology where the relativistic degrees of freedom are supposed to dominate. Concerning dust fluids, the allowed range for ρ is no longer compact and, in fact,ρ is not bounded from above. However, it is interesting to notice that, even ifρ can grow arbitrarily large, the scalar curvature saturates beyond ρ ≃ 1 and this makes it be bounded by M 2 BI , thus avoiding curvature singularities. Finally, fluids with equation of state close to that of a cosmological constant do not have a compact allowed region and the curvature divergences at high energy densities are even more severe than those in GR in the branch I. While GR gives R ∝ ρ, in the Minimal model we have R I ∝ ρ 2 . Finally, for the cases with a non-compact allowed region, the branch II shows the same behaviour found for the Einstein space solutions above, i.e., the curvature is insensitive to the value ofρ.Once we have the fundamental matrix in terms of ρ and p we can easily compute the auxiliary metric by using thatΩ =M / detM so thatThen, we can use the general expression for the Hubble expansion rate (5.66) adapted to the Minimal model to obtainwhere we have used the equations (5.86) to express M 1 in terms of M 0 , which can then be solved for from(5.88). This expression for the Hubble expansion rate shows once again the distinctive property of depending on ρ, p and c 2 s . Let us notice that the allowed region discussed above arising from the positivity of the fundamental matrixM constrained the possible values for ρ and p. Here we have one additional constraint for the cosmological solutions given by the condition H 2 ≥ 0. This condition will in fact be more restrictive since it will depend on c 2 s . In other words, if we consider a barotropic fluid with p = p(ρ) (not necessarily linear) so that c 2 s = dp/dρ, the constraint H 2 ≥ 0 will restrict the of a teleparallel modelà la "Born-Infeld" in[165,169]with the following action:where e a µ represents the four one-forms, T µν ρ the torsion and S the super-potentialAs before, this model can be barely categorised as a Born-Infeld inspired gravity theory according to our criterium, but rather it should be better considered as belonging to the class of f (T ) theories (Class-IV). For a cosmological background e a µ =diag(1, a, a, a), the field equations read , regularizing the divergences of standard General Relativity. Let us consider the above action (5.118) without the cosmological constant Λ = 0. We can combine the two field equations (5.120) into a single equation[165]1 + d = 3 2
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| {'fraction_non_alphanumeric': 0.06253274833496418, 'fraction_numerical': 0.045410498405984656, 'mean_word_length': 4.054169392020165, 'pattern_counts': {'":': 0, '<': 112, '<?xml version=': 0, '>': 90, 'https://': 10, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 504, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'General Relativity has shown an outstanding observational success in the scales where it has been directly tested. However, modifications have been intensively explored in the regimes where it seems either incomplete or signals its own limit of validity. In particular, the breakdown of unitarity near the Planck scale strongly suggests that General Relativity needs to be modified at high energies and quantum gravity effects are expected to be important. This is related to the existence of spacetime singularities when the solutions of General Relativity are extrapolated to regimes where curvatures are large. In this sense, Born-Infeld inspired modifications of gravity have shown an extraordinary ability to regularise the gravitational dynamics, leading to non-singular cosmologies and regular black hole spacetimes in a very robust manner and without resorting to quantum gravity effects. This has boosted the interest in these theories in applications to stellar structure, compact objects, inflationary scenarios, cosmological singularities, and black hole and wormhole physics, among others. We review the motivations, various formulations, and main results achieved within these theories, including their observational viability, and provide an overview of current open problems and future research opportunities.', 'arxivid': '1704.03351', 'author': ['Jose Beltrán Jiménez ', 'Lavinia Heisenberg [email protected] ', 'Gonzalo J Olmo [email protected] ', 'Diego Rubiera-Garcia ', 'Jose Beltrán Jiménez ', '\nInstitute for Theoretical Studies\nAix Marseille Univ\nUniversité de Toulon\nCNRS\nMarseilleCPTFrance\n', '\nDepto. de Física Teórica and IFIC\nDepartamento de Física\nETH Zurich\nClausiusstrasse 47, Centro Mixto Universidad de Valencia-CSIC, Burjassot-461008092Zurich, ValenciaSwitzerland., Spain\n', '\nInstituto de Astrofísica e Ciencias do Espaço\nUniversidade Federal da Paraíba\n58051-900João Pessoa, ParaíbaBrazil\n', '\nFaculdade de Ciencias\nUniversidade de Lisboa\nCampo GrandePT1749-016LisboaPortugal\n'], 'authoraffiliation': ['Institute for Theoretical Studies\nAix Marseille Univ\nUniversité de Toulon\nCNRS\nMarseilleCPTFrance', 'Depto. de Física Teórica and IFIC\nDepartamento de Física\nETH Zurich\nClausiusstrasse 47, Centro Mixto Universidad de Valencia-CSIC, Burjassot-461008092Zurich, ValenciaSwitzerland., Spain', 'Instituto de Astrofísica e Ciencias do Espaço\nUniversidade Federal da Paraíba\n58051-900João Pessoa, ParaíbaBrazil', 'Faculdade de Ciencias\nUniversidade de Lisboa\nCampo GrandePT1749-016LisboaPortugal'], 'corpusid': 119492881, 'doi': '10.1016/j.physrep.2017.11.001', 'github_urls': [], 'n_tokens_mistral': 205260, 'n_tokens_neox': 172004, 'n_words': 104995, 'pdfsha': '5385eff504171e521bdb9256d7d32e946b72a941', 'pdfurls': ['https://arxiv.org/pdf/1704.03351v2.pdf'], 'title': ['Born-Infeld inspired modifications of gravity', 'Born-Infeld inspired modifications of gravity'], 'venue': []} |
arxiv |
Partial wave effects in the heavy quarkonium radiative electromagnetic decays
6 Dec 2022
Su-Yan Pei
Department of Physics and Technology
Hebei University
071002BaodingChina
Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province
BaodingChina
Wei Li
Department of Physics and Technology
Hebei University
071002BaodingChina
Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province
BaodingChina
Ting-Ting Liu
Department of Physics and Technology
Hebei University
071002BaodingChina
Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province
BaodingChina
Meng Han
Department of Physics and Technology
Hebei University
071002BaodingChina
Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province
BaodingChina
Guo-Li Wang
Department of Physics and Technology
Hebei University
071002BaodingChina
Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province
BaodingChina
Tianhong Wang
School of Physics
Harbin Institute of Technology
150001HarbinChina
Partial wave effects in the heavy quarkonium radiative electromagnetic decays
6 Dec 2022
In a previous paper[1], it was pointed out that the wave functions of all particles are not pure waves, besides the main partial waves, they all contain other partial waves. It is very interesting to know what role these different partial waves play in particle transitions. Therefore, by using the Bethe-Salpeter equation method, we study the radiative electromagnetic decays ψ → γχ cJ and Υ → γχ bJ (J = 0, 1, 2). Find that for the S and P wave dominated states, like the ψ(2S), Υ(2S), χ cJ (1P ) and χ bJ (1P ), etc, the dominant S and P waves provide main and non-relativistic contrition to the decays, other partial waves mainly contribute to the relativistic correction. For the states, like the ψ(1D), Υ(2D), χ c2 (1F ) and χ b2 (1F ), etc, they are the S − P − D mixing states dominated by the D wave or the P − D − F mixing states dominated by the F wave. The contribution of P wave in the former and D wave in the later are small. So in a rough calculation, their contribution can be ignored, then the mixing states degenerate into the usual S − D and P − F mixtures. And large mixing angles around |θ| ∼ 30 • for them are obtained. Large decay widths are found in the transitions ψ(2D) → χ c2 (1F ), Υ(1D) → χ bJ (1P ) and Υ(2D) → χ bJ (2P ), etc, which may be helpful to study the missing states χ c2 (1F ), Υ(1D) and Υ(2D). *
I. INTRODUCTION
In a previous paper [1], we pointed out that the wave functions of all particles are not pure waves. For example, the wave function of a 0 − state is S wave dominant, but it also contains a small amount of P partial wave. For the 1 − case, there are two kinds of states.
One is dominated by the S wave, and contains small amounts of P and D waves. The other is dominated by the D wave, while contains considerable S and P wave components.
In order to know the behavior of each component of wave function in particle transition, we study the radiative electromagnetic decays of heavy quarkonia, namely the transition 1 −− → 0 ++ , 1 ++ , 2 ++ , which includes the decays of ψ(nS) → γχ cJ (mP ), Υ(nS) → γχ bJ (mP ) (n = 2, 3, m = 1, 2), ψ(n ′ D) → γχ cJ (m ′ P ), Υ(n ′ D) → γχ bJ (m ′ P ) (n ′ = 1, 2, m ′ = 1, 2), ψ(2D) → γχ c2 (1F ) and Υ(2D) → γχ b2 (1F ), etc, where J = 0, 1, 2.
We will first give the contents of different partial waves in the wave functions of the heavy quarkonia involved, and then show the details of the contributions of each partial wave in the decay process.
Heavy quarkonia have attracted great interest both in theory and experiment since they were discovered [2][3][4][5]. So far, great progress has been made in experiment [6][7][8][9][10][11][12][13] and theory [14][15][16][17][18][19][20][21][22][23][24]. However, there are still some radiative decays that have not been detected experimentally, for example, ψ(4040) → χ cJ (2P )γ and ψ(4160) → χ cJ (2P )γ (J = 0, 1, 2), etc.
There are even some particles that have not been discovered such as the Υ(1D), Υ(2D), Υ(1F ) and ψ(1F ), etc. Therefore, the study of the radiative transitions will help to detect the missing channels and discover the missing particles experimentally.
In this paper, we choose the Bethe-Salpeter (BS) equation method. BS equation [25] is a relativistic dynamic equation dealing with bound states in quantum field theory. The Salpeter equation [26] is its instantaneous approximation, which is suitable for heavy quarkonia. Since we solve the complete Salpeter equation without other approximations, we can obtain the meson wave function which contains components of multiple partial waves. This method has been proved to have good results in many aspects [27][28][29][30][31][32][33].
This paper is organized as follows. In Sec. II, we will show the method to calculate the transition matrix element of the EM decay. The emphasis is given on the partial waves of the wave functions. In Sec. III, we give the results of the EM decay and discuss the contribution of different partial waves to the decay width.
II. THEORETICAL METHOD
A. Transition matrix element
The transition matrix for the single-photon radiative decay of a 1 −− quarkonium can be written as:
< P f ; k, ǫ 0 |S|P, ǫ >= (2π) 4 ee q δ 4 (P − P f − k)ǫ µ 0 M µ ,(1)
where P , P f and k are the momenta of the initial meson, the final meson and the final photon, respectively; ee q are the charge of the quark, e q = 2/3 and 1/3 for the charm and bottom quarks, respectively; ǫ and ǫ 0 are the polarization vectors of the initial quarkonium and the final photon, respectively.
In the BS equation method, the transition amplitude M µ can be expressed as the overlapping integral over the initial and final state wave functions M µ = d 4 qd 4 q 1 (2π) 4 δ 4 (p 2 − p f 2 )T r χ P f (q 1 )γ µ χ P (q)S −1 2 (−p 2 ) − d 4 qd 4 q 2 (2π) 4 δ 4 (p 1 − p f 1 )T r χ P f (q 2 )S −1 1 (p 1 )χ P (q)γ µ ,
where the subscript f means the final state; χ P and χ P f are respectively the BS wave functions of the initial and final mesons with χ = γ 0 χ † γ 0 . S 1 (p 1 ) and S 2 (−p 2 ) are the propagators of quark 1 with momentum p 1 and antiquark 2 with momentum p 2 , respectively; q, q 1 and q 2 are the relative momenta between quark and antiquark in the mesons, for example, q = p 1 − 1 2 P = 1 2 P − p 2 . The final state relative momentum q 1 or q 2 is related to the initial relative momentum q by the δ-function for the spectator, so we have the relations q 1 = q + 1 2 (P f − P ) and q 2 = q + 1 2 (P − P f ). Since both the charmonium and bottomonium are double heavy mesons, the instantaneous approximation is a good choice to avoid solving the complicated full BS equation.
With instantaneous approximation, the BS equation becomes the Salpeter equation, and the BS wave function becomes the Salpeter wave function. Then the transition amplitude can be written as [33],
M µ = d 3 q ⊥ (2π) 3 T r / P M ϕ ++ P f (q 1⊥ )γ µ ϕ ++ P (q ⊥ ) − d 3 q ⊥ (2π) 3 T r ϕ ++ P f (q 2⊥ ) / P M ϕ ++ P (q ⊥ )γ µ ,(3)
where q ⊥ = q− q·P M 2 P , M is the mass of initial meson. In the center of mass frame of the initial state, we have q ⊥ = (0, q). Note that in Eq. The wave function will be given in the center of mass system of the corresponding meson.
q ⊥ is the relative momentum between quark and anti-quark. P and M are the momentum and mass of the meson, respectively.
1. 1 −− quarkonium
The positive energy wave function of the initial 1 −− heavy quarkonium can be expressed as [34],
ϕ ++ 1 −− (q ⊥ ) = (ǫ · q ⊥ ) A 1 + / P M A 2 + / q ⊥ M A 3 + / P / q ⊥ M 2 A 4 +M/ ǫ A 5 + / P M A 6 + / P / q ⊥ M 2 A 7 ,(4)
where ǫ µ is the polarization vector of the 1 −− quarkonium; A i (i = 1, 2, ...7) is related to the four independent radial wave functions a 3 , a 4 , a 5 and a 6 which are functions of −q 2 ⊥ and their numerical values are solutions of full Salpeter equation for 1 −− state,
f = 1 2 a 3 + m w a 4 , A 1 = q 2 ⊥ Mm f + M 2m a 5 − m w a 6 , A 5 = 1 2 a 5 − w m a 6 ,A 2 = − M w A 5 , A 3 = f − M 2 2mw a 6 , A 4 = w m f − M 2 2mw a 5 , A 6 = − m w A 5 , A 7 = A 2 ,
where m and w = m 2 − q 2 ⊥ are the mass and energy of the quark (anti-quark), respectively.
We have pointed out that the wave function of the 1 − B * c state is not a pure S wave, it includes P and D partial waves [1]. The same conclusion is applicable to the 1 −− quarkonium. In the wave function of Eq. (4), the terms including A 5 and A 6 are S waves, A 1 , A 2 and A 7 terms are P waves, while A 3 and A 4 terms are D waves. Following the method of Ref. [1], the normalization condition shows that d 3 q ⊥ (S + P + D) 2 → 1. Under this condition, we can calculate the ratios between different partial waves, and the results are shown in Table I. We can see that in the solutions of Salepter equation [34], the first, second and fourth results which corresponding to the mesons of J/ψ, ψ(3686) and ψ(4040), are dominated by the S partial waves, so they are marked as ψ(1S), ψ(2S) and ψ(3S) in Table I. The third and fifth results which are corresponding to ψ(3770) and ψ(4160) indicate that the components of D waves are the dominant ones in their wave functions, while the components of S and P partial waves are sizable. So different from the widely known S − D mixing states in literature, we mark them the S − P − D mixing states in Table I. We obtain similar conclusions for the bottomonia.
1 S 2 S 1 S − P − D 3 S 2 S − P − D
In Table I, taking the 1S states as examples, we can see that the content of the P or D partial wave in the wave function of J/ψ is much larger than those of Υ(1S), which means the relativistic correction of J/ψ is much larger than that of Υ(1S). Since in the non-relativistic limit, the small components of P and D partial waves are all disappear, they are relativistic corrections. The same conclusion is applicable to the small partial wave in other states, and we will not mention again.
0 ++ quarkonium
The positive energy wave function of the 0 ++ state is expressed as [35]
ϕ ++ 0 ++ (q ⊥ ) = / q ⊥ B 1 + / P / q ⊥ M B 2 + B 3 ,(5)
with
B 1 = 1 2 (b 1 + m w b 2 ), B 2 = w m B 1 , B 3 = q 2 ⊥ m B 1 ,
where b 1 and b 2 are function of −q 2 ⊥ , and they are two independent radial parts of the 0 ++ wave function which will be obtained numerically by solving the Salpeter equation for a 0 ++ state [35]. In Eq.(5), the B 1 and B 2 terms are P waves, while the B 3 term is S wave. We show the partial wave ratios of P : S for χ c0 and χ b0 in Table II. We can see that, they are all P wave dominant states. The positive energy wave function of 1 ++ state is expressed as [35] ϕ
++ 1 ++ (q ⊥ ) = iε µναβ P ν q α ⊥ ǫ β γ µ D 1 M + D 2 / P + D 3 / P / q ⊥ /M 2 ,(6)
with
D 1 = 1 2 (d 1 + w m d 2 ), D 2 = − m w D 1 , D 3 = − 1 w D 1 ,
where ε µναβ is the Levi-Civita symbol, ǫ β is the polarization vector of the 1 ++ state. The two radial wave functions d 1 and d 2 are solutions of the corresponding Salpeter equation [35].
In Eq. (6), the terms including D 1 and D 2 are P waves, while the D 3 term is D wave. We obtain the ratios of P : D for χ c1 and χ b1 and show them in Table II. The results indicate that they are all P wave dominant states.
2 ++ quarkonium
The positive energy wave function of 2 ++ state is expressed as [36] ϕ
++ 2 ++ (q ⊥ ) = ǫ µν q ν ⊥ q µ ⊥ F 1 + / P M F 2 + / q ⊥ M F 3 + / P / q ⊥ M 2 F 4 +Mǫ µν γ µ q ν ⊥ F 5 + / P M F 6 + / P / q ⊥ M 2 F 7 ,(7)
where ǫ µν is the polarization tensor of the 2 ++ state.
g = 1 2 f 3 + m w f 4 , F 1 = q 2 ⊥ Mm g + M 2m f 5 − m w f 6 , F 5 = 1 2 f 5 − w m f 6 , F 2 = − M w F 5 , F 3 = g − M 2 2mw f 6 , F 4 = w m g − M 2 2mw f 5 , F 6 = − m w F 5 , F 7 = F 2 .
In the wave function of Eq.(7), F 5 and F 6 terms are P waves, F 1 , F 2 and F 7 terms are D waves, F 3 and F 4 terms are F waves. The ratios P : D : F of partial waves for some 2 ++ heavy quarkonia are shown in Table III. Similar to the case of 1 −− , the first, second and fourth solutions correspond with the P wave dominant states, and they are marked as 1P , 2P and 3P states, respectively. In the third and fifth results, the largest components are F waves, and the components of P and D partial waves are also sizable. So different from the literature, they are not P − F mixing states, but typical P − D − F mixing states. of 1 −− → 0 ++ , 1 ++ , 2 ++ , the transition amplitudes can be written as
1 P 2 P 1 P − D − F 3 P 2 P − D − F χ c2 P : D :M µ 1 −− →0 ++ = P µ (ǫ · P f )t 1 + ǫ µ t 2 , M µ 1 −− →1 ++ = P µ ǫ ǫǫ f P P f x 1 + (ǫ · P f )ǫ µǫ f P P f x 2 + ǫ µǫǫ f P f x 3 , M µ 1 −− →2 ++ = P µ (ǫ · P f )ǫ P P f y 1 + ǫ µ ǫ P P f y 2 + P µ ǫ ǫP f y 3 + (ǫ · P f )ǫ µP f y 4 + ǫ µǫ f y 5 ,(8)
where t 1 , t 2 , x i (i = 1, 2, 3) and y j (j = 1, 2, ...5) are the form factors; P f is the momentum of the final state; ǫ µ f or ǫ µν f is the polarization vector or tensor of the final quarkonium.
In the upper formula, we have used the following shorthand notations ǫ P P
f ≡ ǫ µν f P µ P ν and ǫ ǫǫ f P P f = ǫ ρσαβ ǫ ρ ǫ f σ P α P f β , etc.
In the transition amplitude, because of the relation k · ǫ 0 = 0, where k = P − P f and ǫ 0 are the momentum and polarization vector of the photon, some of the expressions do not exist at the same time. For example, in the transition 1 −− → 0 ++ , there is the term of P µ (ǫ · P f ), but no P µ f (ǫ · P f ) term. For the same reason, there are no P µ f terms in the transitions of 1 −− → 1 ++ and 1 −− → 2 ++ . Since the transition amplitude must meet the gauge invariance, we also have the following relations
t 2 = M(M − E f )t 1 , x 3 = −M(M − E f )x 1 ,y 2 = y 4 + M(M − E f )y 1 , y 5 = −M(M − E f )y 3 .(9)
III. RESULTS AND DISCUSSIONS
A. Charmonium's radiative decays
The masses of most low excited charmonia have experimental measurement, including ψ(1D) and ψ(2D), so we use the experimental values in our calculation. However, there are still some particles that have not been detected by experiments, such as ψ(1F ). For such particles, our theoretical predictions are used, for example, the mass of ψ(1F ) is predicted as 4037 MeV [37].
ψ(4160) → γχ c2 (1F ) 78.0 17
The theoretical results as well as data from Particle Data Group [43] about the charmonia radiative decays 1 −− → 0 ++ , 1 ++ , 2 ++ are shown in Table IV. ψ(4040) is the state ψ(3S); ψ(3770) and ψ(4160) are the ψ(1D) and ψ(2D); χ c0 (3860), χ c1 (3872) and χ c2 (3930) are the χ c0 (2P ), χ c1 (2P ) and χ c2 (2P ), respectively. Although the masses may be different between models, especially the χ cJ (2P ) states, we will not show detail masses of different models because usually the radiative decay are not very sensitive to the masses except for some special channels, for example, ψ(4040) → χ cJ (1P )γ and ψ(4160) → χ cJ (1P )γ which we will discuss later.
At present, only the decays of ψ(2S) and ψ(3770) have experimental results. Our results of them are comparable to experimental data and other theoretical values. For channels ψ(4040) → χ cJ (2P )γ and ψ(4160) → χ cJ (2P )γ, the values between different theoretical models are also comparable. Compared with the case of bottomonium, where the predictions of different models are in good agreement, the results of charmonium differ slightly between different models. We believe that this is mainly due to the large relativistic correction in the charmonium system.
For other processes, such as ψ(4040) → χ cJ (1P )γ and ψ(4160) → χ cJ (1P )γ, there are huge differences between the results predicted by different models. We find that the huge difference comes from the uncertainty of theory. We take the decay ψ(4040) → χ cJ (1P )γ as an example to illustrate this. The radial wave function of initial state has two nodes. The contributions of wave functions on both sides of the node to the amplitude are cancelled. The decay width of this channel is much smaller than those of other processes, which indicates that the cancellation is very strong, and lead to the strong dependence of the results on model parameters. Therefore, the theoretical error of this process is huge and this is the reason that the predictions of different models differ greatly.
B. Contributions of different partial waves in Chamonia decays
1. ψ(2S) → χ c0 (1P )γ
In Sec.II, we show the ψ(2S) is a 2S dominant state with small admixtures of P and D partial waves. The ratio of its partial waves is S : P : D = 1 : 0.147 : 0.0793. For the χ c0 (1P ), its wave function is 1P dominant but mixing with a small amount of S wave since we have P ′ : S ′ = 1 : 0.127 (here and later, the superscript 'prime' is used to denote the partial wave in the final state). In Table V, we show the detailed contributions of the different partial waves to the decay width of ψ(2S) → χ c0 (1P )γ. Where the 'whole' means the result is obtained using the complete wave function, while the 'S wave' in column or row means the corresponding result is obtained only using the S partial wave and ignoring others, etc.
(keV) of ψ(2S) → χ c0 (1P )γ. 1 −− 0 ++ whole (P ′ + S ′ ) P ′ wave (B 1 ,B 2 ) S ′ wave (B 3 ) whole (S + P + D) 39.9 33.6 0.269 S wave (A 5 ,A 6 ) 34.4 34.4 0 P wave (A 1 ,A 2 ,A 7 ) 0.215 4.2 × 10 −3 0.279 D wave (A 3 ,A 4 ) 1.8 × 10 −4 1.4 × 10 −5 9.0 × 10 −5
From Table V, we can see that, the dominant S partial wave in ψ(2S) state and P ′ wave in χ c0 (1P ) provide the overwhelming contribution. The small P wave in ψ(2S) and S ′ wave in χ c0 (1P ) give small contribution, while the D partial wave in ψ(2S) has tiny contribution, which can be ignored safely.
In non-relativistic limit, only S partial wave in ψ(2S) and P ′ wave in χ c0 (1P ) survive.
Their contribution S × P ′ to the decay width is 34.4 keV, which is closer to the data than the relativistic case of 39.9 keV. Using the relativistic decay width Γ rel and non-relativistic Γ non−rel , we predict the relativistic effect
Γ rel − Γ non−rel Γ rel = 14%,
which is not as large as we expected. The reason may be due to that, see Table V, the main contribution of the relativistic correction does not come from the interaction S × S ′ between the dominant S wave in ψ(2S) with the small S ′ wave in χ c0 (1P ), or those of the small P wave in ψ(2S) with dominant P ′ wave in χ c0 (1P ), P × P ′ , but from the interaction between the two small terms P × S ′ . We also note that there is no interaction (zero in Table V) between the S partial wave in ψ(2S) with the S ′ wave in χ c0 (1P ), since S × S ′ = 0.
2. ψ(2S) → χ c1 (1P )γ(keV) of ψ(2S) → χ c1 (1P )γ. 1 −− 1 ++ whole (P ′ + D ′ ) P ′ wave (D 1 ,D 2 ) D ′ wave (D 3 ) whole (S + P + D) 35.6 27.8 0.484 S wave (A 5 ,A 6 ) 31.1 26.9 0.155 P wave (A 1 ,A 2 ,A 7 ) 0.137 4.8 × 10 −4 0.122 D wave (A 3 ,A 4 ) 5.9 × 10 −4 5.0 × 10 −3 2.1 × 10 −3
The ratio of the partial waves in χ c1 (1P ) is P ′ : D ′ = 1 : 0.15, where the small term is not a S ′ wave like in χ c0 (1P ), but a D ′ partial wave. In Table VI, we show the details of decay
Ψ(2S) → χ c1 γ.
Where the largest contribution comes from the interaction S × P ′ between the dominant partial waves, the S wave in Ψ(2S) and the P ′ wave in χ c1 (1P ). The small P wave in ψ(2S) and D ′ wave in χ c1 (1P ) have small contribution. Similar to the previous case, the smallest D partial wave in ψ(2S) has tiny contribution, and can be safely ignored.
When we take the non-relativistic limit, only the dominant S partial wave in ψ(2S) and P ′ wave in χ c1 (1P ) have contribution, the obtained decay width is 26.9 keV. The relativistic effect is calculated as
Γ rel − Γ non−rel Γ rel = 24%,
which is about 1.4 times larger than the those of χ c0 (1P ) case. In this decay, the relativistic corrections come from the interaction S × D ′ between the dominant S wave in ψ(2S) and the small D ′ wave in χ c1 (1P ), also from the interaction P × D ′ between the small P wave in ψ(2S) and the D ′ wave in χ c1 (1P ). Their contributions are comparable, S × D ′ ∼ P × D ′ , which indicates that the interaction between the later (P × D ′ ) is much stronger than those of the former (S × D ′ ) since the component of P partial wave is one order smaller than those of S wave in ψ(2S). Similar to the case of Ψ(2S) → χ c0 γ, the interaction between P waves, P × P ′ , is very small in the Ψ(2S) → χ c1 γ.
The wave function of χ c2 is more complicated than those of χ c0 (1P ) and χ c1 (1P ). Besides the dominant P ′ partial wave, it also contains small amounts of D ′ and F ′ partial waves, and their ratios are P ′ : D ′ : F ′ = 1 : 0.145 : 0.070. In Table VII, we show the contributions of different partial waves to the decay ψ(2S) → χ c2 (1P )γ. We can see, the dominant S and P ′ partial waves from initial ψ(2S) and final χ c2 (1P ) states give main contributions to the decay width, the P partial wave in ψ(2S) and D ′ wave in χ c2 (1P ) give small contributions, while the contributions from the D wave in ψ(2S) and F ′ wave in χ c2 (1P ) are tiny, which can be ignored safely. indeed a D wave dominant state, but its wave function also contains sizable and comparable S and P partial waves. It is a S − P − D mixing state in our method.
1 −− 2 ++ whole (P ′ + D ′ + F ′ ) P ′ wave (F 5 ,F 6 ) D ′ wave (F 1 ,F 2 ,F 7 ) F ′ wave (F 3 ,
In Table VIII Table VIII should have sizable contributions, for example, (S × P ′ ) × (D × P ′ ) and (P × S ′ ) × (D × P ′ ). We find the decay width will change from 290 keV to 257 keV if we ignore the contribution of P partial wave. So in a rough estimate, P partial wave can be ignored, but in a precise calculation, its contribution should be taken into account. provides the dominant contribution, S wave also give sizable contribution. Although the content of P wave is almost equal to that of S wave in the wave function, the contribution of P wave is obviously smaller than that of S wave. But the decay width changes from 90.8 keV to 69.0 keV if ignoring the P partial wave. So we draw the same conclusion, in a precise calculation, the P partial wave should be considered since its contribution may not be very small after considering the cross interaction. The column of 'whole' in Table X shows that for ψ(3770), the main contribution is not from the largest component of its wave function, that is, the D partial wave, but from the S wave.
1 −− 1 ++ whole (P ′ + D ′ ) P ′ wave (D 1 ,D 2 ) D ′ wave (D
And the contribution of P wave alone in ψ(3770) is small, but this term should not be ignored in a precise calculation. Considering the wave function of χ c2 (1P ), its dominant component, P ′ partial wave, provides the largest contribution, D ′ wave has small contribution, while F ′ wave in χ c2 (1P ) can be ignored safely.
We also note that, the maximum interaction occurs between the S wave in ψ(3770) and P ′ in χ c2 (1P ), S × P ′ , which is obviously larger than that between the two dominant partial waves, D × P ′ . The weak interaction of D × P ′ is responsible for the small decay width of In literature, ψ(4160) is primarily the state ψ(2D) with admixtures of ψ(3S), that is, it is a 3S − 2D mixing state. In our method, ψ(4160) is the 2D dominant state, but mixed with sizable S and P partial waves, so it is a S − P − D mixing state. Its main radiative decays are the channels of ψ(4160) → χ cJ (2P )γ (J = 0, 1, 2). We will not show the details of these decays since they are similar to those of ψ(3770) → χ cJ (1P )γ.
ψ(3770) → χ c2 (1P )γ.χ c2 (1P )γ. 1 −− 2 ++ whole P ′ wave (F 5 ,F 6 ) D ′ wave (F 1 ,F 2 ,F 7 ) F ′ wave (F 3 ,F 4 )
As the 2D dominant state, its mass is heavier than the 1F dominant state χ c2 (1F ) in our theoretical prediction, so the decay process ψ(4160) → χ c2 (1F )γ exists. This channel is another typical process not encountered before in this article, since the final meson χ c2 (1F )
is also a typical mixed state. In literature, χ c2 (1F ) is the 2P −1F mixing state, 1F dominant but mixed with sizable 2P component. But in our method, it is 1F dominant state mixed with sizable P and D components, namely a P −D−F mixing state. χ c2 (1F ) is not available in experiment, so for its mass, we use our theoretical prediction, 4038 MeV [37].
Some details of the contributions of different partial waves to the decay channel ψ(4160) → χ c2 (1F )γ are listed in Table XI. First, we note that the main components of the initial and final states, namely, the D wave in ψ(4160) and the F ′ wave in χ c2 (1F ),
give the maximum contribution. Second, both the S wave of the initial state and the P ′ wave of the final state play an important role. For example, the interactions S × F ′ , S × P ′ and D × P ′ are all strong, but their contribution to the overall result is cancelled due to the existence of cancellation. Third, the contributions of P wave and the D ′ wave are small. Their wave functions have similar partial wave content, that is, dominant S partial wave, small amount of P and D partial waves. Although the component of D partial wave is not tiny in the wave function, especially compared with that of P wave, the contribution of D wave in the radiative decay can be ignored safely. So we point out that the result of radiative decay seems not support the 2S − 1D mixing mode for ψ(2S) and ψ(3770), and 3S − 2D mixing for ψ(3S) and ψ(4160), since the D partial wave can be ignored safely.
1 −− 2 ++ whole F ′ wave(F 3 ,F 4 ) P ′ wave(F 5 ,F 6 ) D ′ wave(F 1 ,F 2 ,F 7 )
ψ(3770) and ψ(4160)
In our method, the wave functions of these two particles are D partial wave dominant, mixed with sizable S and P waves, so they are S − P − D mixing states. In radiative decays, usually D partial wave provides the dominant contribution, and the contribution of S wave is also sizable, while those of P wave is the smallest. But in some special channel, for example ψ(3770) → χ c2 (1P )γ, the contribution of S wave is larger than that of D wave, and the P wave contribution is still minimal. So in both cases, D and S waves play an important role, while P wave can be ignored when high precision calculation is not required. In our method, χ c0 (1P ) and χ c0 (2P ) are P ′ partial wave dominant states with small amount of S ′ wave. In radiative decay, the P ′ partial wave provides the main contribution which is also the non-relativistic result, while the S ′ wave mainly gives the relativistic correction of the decay. The situation of χ c1 (1P ) and χ c1 (2P ) is similar to that of χ c0 (1P ) and χ c0 (2P ). The difference is that the small component term in their wave functions is not S ′ partial wave, but D ′ wave.
For the states of χ c2 (1P ) and χ c2 (2P ), we draw the same conclusion, they are P ′ wave dominant states, but their wave functions include small amount of D ′ and F ′ wave. In radiative decays, P ′ provides the main contribution which is the non-relativistic result, the relativistic correction mainly comes from the contribution of D ′ wave, while the contribution of F ′ can be ignored safely. So the tiny contribution of F ′ wave does not support the 2P −1F mixing mode for χ c2 (2P ) and χ c2 (1F ).
χ c2 (1F )
Although we have only studied one channel containing χ c2 (1F ) at present, we can still draw a conclusion due to the similarity between the 1 −− state and the 2 ++ state [1]. That is, in the transition process including χ c2 (1F ), the F ′ and P ′ partial waves in χ c2 (1F ) play an important role, and we can ignore its D ′ wave in a rough calculation. If we delete the D ′ wave, the mixing changes from P ′ − D ′ − F ′ to P ′ − F ′ , then using the relation χ c2 (1F ) = |F ′ cosθ + |P ′ sinθ, we obtain the mixing angle |θ| = 32.2 • for χ c2 (1F ).
Similarly, in our method, the P ′ − F ′ mixing is not the 2P ′ − 1F ′ mixing in literature, and our result of χ c2 (2P ) does not support the mixing mode of 2P ′ − 1F ′ . If we ignore the D ′ wave, χ c2 (2P ) becomes to the P ′ − F ′ mixing state. With the definition χ c2 (2P ) = −|F ′ sinθ +|P ′ cosθ, the mixing angle is calculated as |θ| = 4.89 • for χ c2 (2P ), which is much different from the large mixing angle |θ| = 32.2 • of χ c2 (1F ).
D. Bottomonium's radiative decays
At present, the Υ(1D), Υ(2D) and Υ(1F ) have not been detected by the experiment.
So in our calculation, their masses are taken from our previous study [37], Table XII, for comparison. Theoretical results from other models and data from PDG are also shown in the same table.
Since the mass of the bottomonium is very heavy, the relativistic correction is small.
Then from Table XII, we can see that, except the channels Υ(3S) → χ bJ (1P )γ which have large uncertainties in theory, the results by most of the theoretical models are in agreement, at least comparable with each other, and also consist with data from PDG. For example, Our results of Υ(2S) → χ bJ (1P )γ and Υ(3S) → χ bJ (2P )γ consist very well with experimental data and the theoretical predictions in Refs. [39,48,49].
For the mixing states Υ(1D) and Υ(2D), we must point out that in other models, these particles are treated as pure D waves, while in our model, they are all mixed particles, including the final state Υ(1F ). Therefore, our results about the Υ(1D) and Υ(2D) decays are comparable with other theoretical predictions, but there are some small differences. For example, our results are close to those in Refs. [39,48,49], but not as good as the processes Υ(2S) → χ bJ (1P )γ and Υ(3S) → χ bJ (2P )γ.
Similar to the case of charmonium, because the contributions of wave functions on the two sides of the nodes strongly cancel each other out, the theoretical errors of processes Υ(3S) → χ bJ (1P )γ and Υ(2D) → χ bJ (1P )γ, especially the former, are large, resulting in large differences between the results of different models.
E. Contributions of different partial waves in bottomonia decays
For the decays of bottomonia which have the same quantum numbers with charmonia, the calculations are similar, and we will not repeat them one by one, but focus on some different contents. The bottomonium is much heavier than charmonium, so the relativistic correction of bottomonium is much smaller than that of charmonium. This leads to the fact that the content of the small partial wave in bottomonium is much smaller than the corresponding charmonium case, except for the mixing state (we will discuss this when considering Υ(1D)), see Tables I,II,III for details.
1. Υ(2S) → χ b0 (1P )γ
As expected, we note that in Table XIII, is very small.
2. Υ(2S) → χ b1 (1P )γ
Some details of the channel Υ(2S) → χ b1 (1P )γ are listed in Table 2. We can see that, similar to the process Υ(2S) → χ b0 (1P )γ, the non-relativistic result plays a major role in this process. The relativistic effect of this process is
Γ rel − Γ non−rel Γ rel = 7.2%.
The relativistic correction mainly comes from the contributions of P wave in the initial state and D ′ wave in the final state, while the contribution of D wave in the initial state can be safely ignored.
3. Υ(2S) → χ b2 (1P )γ
In the process Υ(2S) → χ b2 (1P )γ, see Table XV Compared with the case of ψ(3770), whose ratio is S : P : D = 0.573 : 0.509 : 1, we find that the content of S and P waves in the bottomonium dose not decrease significantly. It shows that the contributions of S and P waves can not be simply attributed to the relativistic corrections, as we did in the Υ(2S) case. Therefore, unlike Υ(nS) (n = 1, 2, 3), where P and D wave terms are both relativistic corrections, the mixed P wave and D wave in Υ(1D) still exist in the non-relativistic limit, so S − P − D mixing needs to be considered even in the non-relativistic model. For the charmonium ψ(3770), we have the same conclusion. Table XVI shows some details of the decay Υ(1D) → χ b0 (1P )γ. We can see that, in the wave function of Υ(1D), the D wave and S wave provide the main and sizable contributions, respectively, while the contribution of P wave is the smallest. We get the same conclusion as in case of ψ(3770), in a high-precision calculation, the contribution of P wave needs to be calculated, while in a rough estimation, the P wave contribution can be ignored.
P wave (A 1 ,A 2 ,A 7 ) 3.9 × 10 −3 8.3 × 10 −5 2.9 × 10 −3 5. Υ(1D) → χ b1 (1P )γ
In Table XVII, we show some details about the contributions to the decay width of Υ(1D) → χ b1 (1P )γ from different partial waves. It can be seen that, although there are some differences, we can get the same conclusion as in process Υ(1D) → χ b0 (1P )γ, which will not be repeated here.
6. Υ(1D) → χ b2 (1P )γ
This process is similar to the decay ψ(3770) → χ c0 (2P )γ, and we can draw similar conclusion. The D wave is the dominant one in the wave function of Υ(1D), but the main contribution to the radiative decay Υ(1D) → χ b2 (1P )γ is from S wave (see Table XVIII for details), which results in the decay width of this process is much smaller than that with χ b0 (1P ) or χ b1 (1P ) as the final state. And the P wave in Υ(1D) provides the smallest contribution. χ b2 (1F ) has not been detected by experiment, and we predict its mass to be about 10374
1 −− 2 ++ whole P ′ wave (F 5 ,F 6 ) D ′ wave (F 1 ,F 2 ,F 7 ) F ′ wave (F 3 ,F 4 )
MeV [37].
From Table XIX, it is worth noting that, the large components, D wave in Υ(2D) and F ′ wave in χ b2 (1F ), give the maximum contribution of decay width, and the interaction D × F ′ is the maximum. S wave in Υ(2D) and the P ′ wave in χ b2 (1F ) also play an important role.
For example, there are sizable interactions S × F ′ , S × P ′ and D × P ′ . The contribution of P wave and the D ′ wave is very small, which can be ignored. In this case, the S − P − D mixing for Υ(2D) and P ′ − D ′ − F ′ mixing for χ b2 (1F ) become the S − D mixing and P ′ − F ′ mixing, respectively. The wave functions of them are both S wave dominant with small amount of P and D waves. In radiative decays, P wave provides the relativistic correction, which is small and indicates that the non-relativistic approximation is good for bottomonium. The contribution of D wave is tiny, which can be ignored safely. So this result also does not support the mode of 2S − 1D mixing.
Υ(1D) and Υ(2D)
These two states are typical S − P − D mixing states. In addition to the largest D wave component, their wave functions also contain sizable P and D wave contents. When studying their radiative decay, it is found that both D wave and S wave play an important role, and their contributions need to be calculated even in the non-relativistic case. Although the content of P wave in wave function is not small, its contribution to radiative decay is relatively small, which can be ignored in rough calculation. In this case, the S − P − D mixing becomes S − D mixing. We calculated the mixing angles of Υ(1D) and Υ(2D), both of which are |θ| = 30.0 • .
3. χ bJ (1P ) and χ bJ (2P ) (J = 0, 1, 2)
For the states χ bJ (1P ) and χ bJ (2P ) (J = 0, 1, 2), the dominant P ′ wave plays a major role, and only it contributes under non-relativistic conditions. Other partial wave contributions are all relativistic corrections. The F ′ wave in state χ b2 (1P ) or χ b2 (2P ) can be safely ignored.
Therefore, this result does not support the 2P − 1F mixing mode. 4. χ b2 (1F ) χ b2 (1F ) is also a typical P − D − F mixing state. Its wave function is mainly 1F ′ wave, and contains sizable P ′ and D ′ waves. However, in the radiation decay, only F ′ and P ′ waves give an important contribution, while D ′ wave has a small contribution, which can be ignored in the rough calculation. Therefore mixing mode becomes the P − F mixing. The mixing angle is calculated as |θ| = 33.6 • .
IV. SUMMARY
We study the partial waves of heavy quarkonium wave functions and their contributions to radiative electromagnetic decays. The results show that for the S and P wave dominated states, for example, ψ(nS), Υ(nS) (n = 2, 3), χ cJ (mP ) and χ bJ (mP ) (m = 1, 2; J = 0, 1, 2), the dominant S and P waves provide main and non-relativistic contribution, while the partial waves of the small components mainly contribute to the relativistic correction.
The mixed states, for example ψ(nD), Υ(nD) (n = 1, 2), χ c2 (1F ) and χ b2 (1F ) which dominated by D wave and F wave, their wave functions are mixtures of S − P − D or P − D − F waves. But in the radiative decays, P partial wave in the S − P − D mixture and D wave in the P − D − F mixture have small contribution and can be ignored, then the mixture degenerates into the common S − D or P − F mixture. In this case the mixing angle can be calculated, our result support a large mixing angle around |θ| ∼ 30 • for these mixing states. But the mixing in our method is not the 2S − 1D or 2P − 1F mixing in literature.
Our results of charmonium electromagnetic decay are comparable with the experimental data, and the results of bottomonium are in good agreement with the existing data. We calculate the radiative decays of the mixed states and find that the ψ(2D) → χ cJ (2P ), Υ(1D) → χ bJ (1P ) and Υ(2D) → χ bJ (2P ) (J = 0, 1) transitions have large partial decay widths, which may be helpful to find these undiscovered particles.
(3), we only keep the dominant contribution of positive energy wave function ϕ ++ , but ignore other tiny contributions from negative wave functions, etc. B. The positive wave functions and their partial waves
F i (i = 1, 2, ..., 7) is a function of the four independent radial wave functions f 3 , f 4 , f 5 and f 6 which are solutions of the Salpeter equation for 2 ++ state [36],
the positive energy wave functions, the transition amplitude in Eq. (3) are calculated straightly, and they are expressed as functions of the form factors. For the transitions
In the non-relativistic limit, only S partial wave in ψ(2S) and P ′ wave in χ c2 (1P ) have contribution, their interaction S × P ′ contribute 20.4 keV, which is close to the relativistic result. The relativistic corrections mainly come from the interactions of S × D ′ , P × D ′ and P × P ′ . And the relativistic effect in this decay isΓ rel − Γ non−rel Γ rel = 17%.4. ψ(3770) → χ c0 (1P )γ ψ(3770) is usually treated as 1D dominant state with a component of 2S partial wave, so it is famous as the 1D − 2S mixing state. But in our solution of the corresponding Salpeter equation for the 1 −− charmonium, we find besides the D and S wave, there is also the P partial wave in its wave function, their ratios are S : P : D = 0.573 : 0.509 : 1. So it is
, we show the contributions of different partial waves to the radiative decay ψ(3770) → χ c0 (1P )γ. In the column of 'whole', where the wave function of χ c0 (1P ) is complete, we show the contributions of S, P and D partial waves of ψ(3770) separately. We can see that, the D partial wave provides the dominant contribution, S wave give sizable result, while the contribution from P wave is small. But based on this, we cannot simply draw a conclusion that the contribution of P wave is ignorable. Because the whole decay width, 290 keV, is much larger than the summed contributions from lines of D, S and P , which are 180 keV, 6.69 keV and 0.944 keV, separately. This indicates that the cross interaction terms which are not listed in the
ψ(4160) → χ c2 (1F )γ
ψ(2S) and ψ(3S)
If we ignore the P wave, there is only the S − D mixing for ψ(3770) and ψ(4160), then using the definition, ψ(3770) = |D cosθ + |S sinθ, we obtain the mixing angle |θ| = 29.8 • for ψ(3770), and similarly we obtain |θ| = 29.6 • for ψ(4160). Our result support the large mixing angle in literature, −(27 ± 2) • in Ref.[44], 26 • in Ref.[45], or |θ| ≈ 40 • in Ref.[46], not the small angle (12 ± 2) •[44] or −13 •[45].We use the symbol S − D (or the complete S − P − D) for mixing, but in our method, it is not the 2S − 1D mixing, because our result of ψ(2S) does not support the 2S − 1D mixing mode. For example, if we artificially ignore the P wave, then ψ(2S) is also S − D mixing state. Using the relation ψ(2S) = −|D sinθ + |S cosθ, we obtain the mixing angle |θ| = 4.53 • for ψ(2S), which does not match the large angle for ψ(3770).
3. χ cJ (1P ) and χ cJ (2P ) (J = 0, 1, 2)
M Υ(1D) = 10130 MeV, M Υ(2D) = 10435 MeV and M Υ(1F ) = 10374 MeV. Our results of bottomonium radiative decays are shown in
the small component terms, namely, P wave and D wave of Υ(2S), and S ′ wave of χ b0 (1P ) make very little contribution to the decay Υ(2S) → χ b0 (1P )γ. And the D partial wave in Υ(2S) can be ignored safely. The relativistic effect Γ rel − Γ non−rel Γ rel = 4.4%,
for details, the contributions of the initial D wave and the final F ′ wave can be ignored. The main contribution of the decay width is still non-relativistic and comes from the main partial waves of the initial and final states, that is, from the interaction of S ×P ′ . The contributions of the initial P wave and the final D ′ wave are mainly relativistic corrections. And the relativistic effect in this process is 4.9%. 4. Υ(1D) → χ b0 (1P )γIn our method, Υ(1D) is a D wave dominant state, but its wave function includes sizable S and P waves, so it is a S −P −D mixing state. And the ratio is S : P : D = 0.577 : 0.425 : 1.
P
wave (A 1 ,A 2 ,A 7 ) 3.0 × 10 −4 8.3 × 10 −4 2.9 × 10 −4 4.4 × 10 −6 7. Υ(2D) → χ b2 (1F )γ Υ(2D) is a state dominated by 2D wave and mixed with a sizable amount of S and P partial waves. Its radiative decay to the final state χ bJ (1P ) or χ bJ (2P ) (J = 0, 1, 2) has many similarities with Υ(1D) → χ bJ (1P )γ, and they belong to the same type of process. So we will not give details about them, only show the details of the decay Υ(2D) → χ b2 (1F )γ in
TABLE I :
IRatios of the partial waves in the 1 −− wave functions for heavy quarkonia.
TABLE II :
IIRatios of the partial waves in the 0 ++ and 1 ++ wave functions for heavy quarkonia.1 P
2 P
3 P
χ c0
P : S
1 : 0.127
1 : 0.142
1 : 0.157
χ b0
P : S
1 : 0.0361
1 : 0.0399
1 : 0.0444
χ c1
P : D
1 : 0.137
1 : 0.150
1 : 0.165
χ b1
P : D
1 : 0.0358
1 : 0.040
1 : 0.0463
3. 1 ++ quarkonium
TABLE III :
IIIRatios of the partial waves in the 2 ++ wave functions for heavy quarkonia.
TABLE IV :
IVThe decay widths (keV) of ψ → χ cJ γ. Where χ c0 (3860), χ c1 (3872) and χ c2 (3930) are treated as the 2P states; χ c2 (1F ) is the 1F dominant state with sizable P and D partial waves, our theoretical prediction about its mass is 4037 MeV.Process
Ours [18] [38] [39] [40] [41] [42]a [42]b PDG[43]
ψ(2S) → γχ c0 (1P ) 39.9 50 26.3 47.0 25.2 26
22
22 28.8 ± 1.4
ψ(2S) → γχ c1 (1P ) 35.6 45 22.9 42.8 29.1 29
42
45 28.7 ± 1.5
ψ(2S) → γχ c2 (1P ) 24.5 29 18.2 30.1 25.2 24
38
46 28.0 ± 1.4
ψ(3770) → γχ c0 (1P ) 290
355 299 243.9 213 272 261 188 ± 23
ψ(3770) → γχ c1 (1P ) 90.8
135 99.0 104.9 77 138 135 67.7 ± 8.7
ψ(3770) → γχ c2 (1P ) 3.13
6.9 3.88 1.9 3.3 7.1 8.1
< 17.4
ψ(4040) → γχ c0 (1P ) 0.29
2.1 12.7 5.9 6.7
ψ(4040) → γχ c1 (1P ) 1.42
0.3 0.85 4.0 6.7
< 272
ψ(4040) → γχ c2 (1P ) 2.71
2.4 0.63 0.25 2.5
< 400
ψ(4040) → γχ c0 (3860) 31.0
30.1 22
19
27
ψ(4040) → γχ c1 (3872) 74.4
45.0 43
55
67
ψ(4040) → γχ c2 (3930) 46.7
36.0 48
67
82
ψ(4160) → γχ c0 (1P ) 8.42
23.3 35 150 189
ψ(4160) → γχ c1 (1P ) 6.66
0.02 3.4
37
63
ψ(4160) → γχ c2 (1P ) 2.32
0.23 0.027 17
20
ψ(4160) → γχ c0 (3860) 462
191 332 360
ψ(4160) → γχ c1 (3872) 281
114 309 347
ψ(4160) → γχ c2 (3930) 14.7
6.3
24
29
TABLE V :
VContributions of different partial waves to the decay width
TABLE VI :
VIContributions of different partial waves to the decay width
TABLE VII :
VIIContributions of different partial waves to the decay width (keV) of ψ(2S) →χ c2 (1P )γ.
TABLE VIII :
VIIIContributions of different partial waves to the decay width (keV) of ψ(3770) →χ c0 (1P )γ.
1 −−
0 ++
whole (P ′ + S ′ ) P ′ wave (B 1 ,B 2 ) S ′ wave (B 3 ,B 4 )
whole (S + P + D)
290
257
0.925
D wave (A 3 ,A 4 )
180
177
0.0206
S wave (A 5 ,A 6 )
6.69
6.69
0
P wave (A 1 ,A 2 ,A 7 )
0.944
0.0235
0.670
5. ψ(3770) → χ c1 (1P )γ
Table IX
IXshows the detail contributions of different partial waves to the decay ψ(3770) →χ c1 (1P )γ. Similar to the case of ψ(3770) → χ c0 (1P )γ, here the D partial wave of ψ(3770)
TABLE IX :
IXContributions of different partial waves to the decay width (keV) of ψ(3770) → χ c1 (1P )γ.
It is very interesting to see the details of what happens to this decay. Some details of our calculations are shown in Table X, where the contributions to the decay width from different partial waves are listed. It can be seen that it is indeed different.3 )
whole (S + P + D)
90.8
91.8
2.5 × 10 −3
D wave (A 3 ,A 4 )
106
126
0.866
S wave (A 5 ,A 6 )
4.04
8.18
0.725
P wave (A 1 ,A 2 ,A 7 )
1.53
1.45
1.3 × 10 −3
6. ψ(3770) → χ c2 (1P )γ
From Table IV, we can see that, all the theoretical results including ours show that the
decay width of ψ(3770) → χ c2 (1P )γ is much smaller than those of ψ(3770) → χ c0 (1P )γ and
ψ(3770) → χ c1 (1P )γ.
TABLE X :
XContributions of different partial waves to the decay width (keV) of ψ(3770) →
TABLE XI :
XIContributions of different partial waves to the decay width (keV) of ψ(4160) →
χ c2 (1F )γ, where ψ(4160) and χ c2 (1F ) are 2D and 1F dominant states, respectively.
TABLE XII :
XIIThe decay widths (keV) of Υ → χ bJ γ.Process
Ours [38] [39] [40] [47] [48] [49] [50]
PDG[43]
Υ(2S) → γχ b0 (1P ) 1.13 1.62 1.29 0.74 1.19 0.91 1.09 1.09 1.22 ± 0.23
Υ(2S) → γχ b1 (1P ) 1.80 2.45 2.00 1.40 2.28 1.63 1.84 2.17 2.21 ± 0.23
Υ(2S) → γχ b2 (1P ) 1.85 2.46 2.04 1.67 2.58 1.88 2.08 2.62 2.29 ± 0.30
Υ(1D) → γχ b0 (1P ) 15.5 23.4 20.1 12.5
16.5 20.98 19.8
Υ(1D) → γχ b1 (1P ) 7.94 12.7 10.7 7.59
9.7 12.29 13.3
Υ(1D) → γχ b2 (1P ) 0.389 0.69 0.564 0.44
0.56 0.65 1.02
Υ(3S) → γχ b0 (1P ) 0.009 0.027 0.001 0.03 0.12 0.01 0.15 0.097 0.055 ± 0.012
Υ(3S) → γχ b1 (1P ) 0.071 0.067 0.008 0.003 0.0 0.05 0.16 0.0005 0.018 ± 0.012
Υ(3S) → γχ b2 (1P ) 0.075 0.097 0.015 0.11 0.20 0.45 0.0827 0.14 0.203 ± 0.039
Υ(3S) → γχ b0 (2P ) 1.19 1.49 1.35 1.07 1.31 1.03 1.21 3.330 1.20 ± 0.23
Υ(3S) → γχ b1 (2P ) 2.18 2.41 2.20 2.05 2.66 1.91 2.13 2.61 2.56 ± 0.48
Υ(3S) → γχ b2 (2P ) 2.52 2.67 2.40 2.51 3.18 2.30 2.56 3.16 2.66 ± 0.57
Υ(2D) → γχ b0 (1P ) 2.16
3.60
2.9 3.52 5.56
Υ(2D) → γχ b1 (1P ) 1.70
0.9 1.58 2.17
Υ(2D) → γχ b2 (1P ) 0.097
0.02 0.0608 0.44
Υ(2D) → γχ b0 (2P ) 12.5
13.1
10.6 8.35 9.58
Υ(2D) → γχ b1 (2P ) 6.14
6.5 4.84 6.74
Υ(2D) → γχ b2 (2P ) 0.37
0.4 0.24 0.47
Υ(2D) → γχ b2 (1F ) 0.43
0.833
1.6 2.05
TABLE XIII :
XIIIContributions of different partial waves to the decay width (keV) of Υ(2S) →χ b0 (1P )γ.
1 −−
0 ++
whole P ′ wave (B 1 ,B 2 ) S ′ wave (B 3 )
whole
1.13
1.08
6.9 × 10 −4
S wave (A 5 ,A 6 )
1.08
1.08
0
P wave (A 1 ,A 2 ,A 7 ) 5.8 × 10 −4
5.2 × 10 −6
7.0 × 10 −4
D wave (A 3 ,A 4 ) 4.0 × 10 −8
1.4 × 10 −8
5.7 × 10 −9
TABLE XIV :
XIVContributions of different partial waves to the decay width (keV) of Υ(2S) →
χ b1 (1P )γ
1 −−
1 ++
whole
P ′ wave (D 1 ,D 2 ) D ′ wave (D 3 )
whole
1.80
1.67
2.2 × 10 −3
S wave (A 5 ,A 6 )
1.73
1.67
6.0 × 10 −4
P wave (A 1 ,A 2 ,A 7 ) 4.9 × 10 −4
1.2 × 10 −6
5.3 × 10 −4
D wave (A 3 ,A 4 )
3.2 × 10 −7
2.2 × 10 −6
8.5 × 10 −7
TABLE XV :
XVContributions of different partial waves to the decay width (keV) of Υ(2S) → χ b2 (1P )γ
1 −−
2 ++
whole
P ′ wave (F 5 ,F 6 ) D ′ wave (F 1 ,F 2 ,F 7 ) F ′ wave (F 3 ,F 4 )
whole
1.85
1.81
2.5 × 10 −4
1.8 × 10 −7
S wave (A 5 ,A 6 )
1.79
1.76
1.2 × 10 −4
2.8 × 10 −8
P wave (A 1 ,A 2 ,A 7 ) 5.7 × 10 −4
3.7 × 10 −4
2.1 × 10 −5
7.6 × 10 −9
D wave (A 3 ,A 4 )
6.9 × 10 −8
7.7 × 10 −8
6.9 × 10 −7
8.6 × 10 −8
TABLE XVI :
XVIContributions of different partial waves to the decay width (keV) of Υ(1D) →χ b0 (1P )γ.
1 −−
0 ++
whole
P ′ wave (B 1 ,B 2 ) S ′ wave (B 3 )
whole
15.5
15.0
4.1 × 10 −3
D wave (A 3 ,A 4 )
9.94
9.87
1.2 × 10 −4
S wave (A 5 ,A 6 )
0.517
0.517
0
TABLE XVII :
XVIIContributions of different partial waves to the decay width (keV) of Υ(1D) →χ b1 (1P )γ.
1 −−
1 ++
whole P ′ wave (D 1 ,D 2 ) D ′ wave (D 3 )
whole
7.94
7.99
8.5 × 10 −5
D wave (A 3 ,A 4 )
13.5
14.1
7.4 × 10 −3
S wave (A 5 ,A 6 )
0.929
1.08
5.9 × 10 −3
P wave (A 1 ,A 2 ,A 7 ) 0.0132
0.0131
2.9 × 10 −7
TABLE XVIII :
XVIIIContributions of different partial waves to the decay width (keV) of Υ(1D) → χ b2 (1P )γ.
Table XIX ,
XIXwhere χ b2 (1F ) is the 1F ′ dominant state mixed with sizable P ′ and D ′ waves.
TABLE XIX :
XIXContributions of different partial waves to the decay width (keV) of Υ(2D)→ ++ whole F ′ wave(F 3 ,F 4 ) P ′ wave(F 5 ,F 6 ) D ′ wave(F 1 ,F 2 ,F 7 ) P wave (A 1 ,A 2 ,A 7 ) 1.02 × 10 −4 2.7 × 10 −6 1.54 × 10 −6 3.96 × 10 −5F. Discussions about the bottomonia 1. Υ(2S) and Υ(3S)χ b2 (1F )γ.
1 −−
2 whole
0.43
0.54
8.90 × 10 −3
1.82 × 10 −4
D wave (A 3 ,A 4 )
0.50
1.10
0.12
4.24 × 10 −5
S wave (A 5 ,A 6 ) 3.27 × 10 −3
0.10
0.067
3.4 × 10 −6
. ψ(2S) → χ c2 (1P )γ
Acknowledgments This work was supported in part by the National Natural Science
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| {'fraction_non_alphanumeric': 0.09173173101140575, 'fraction_numerical': 0.07273414169035369, 'mean_word_length': 3.012632642748863, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 87, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In a previous paper[1], it was pointed out that the wave functions of all particles are not pure waves, besides the main partial waves, they all contain other partial waves. It is very interesting to know what role these different partial waves play in particle transitions. Therefore, by using the Bethe-Salpeter equation method, we study the radiative electromagnetic decays ψ → γχ cJ and Υ → γχ bJ (J = 0, 1, 2). Find that for the S and P wave dominated states, like the ψ(2S), Υ(2S), χ cJ (1P ) and χ bJ (1P ), etc, the dominant S and P waves provide main and non-relativistic contrition to the decays, other partial waves mainly contribute to the relativistic correction. For the states, like the ψ(1D), Υ(2D), χ c2 (1F ) and χ b2 (1F ), etc, they are the S − P − D mixing states dominated by the D wave or the P − D − F mixing states dominated by the F wave. The contribution of P wave in the former and D wave in the later are small. So in a rough calculation, their contribution can be ignored, then the mixing states degenerate into the usual S − D and P − F mixtures. And large mixing angles around |θ| ∼ 30 • for them are obtained. Large decay widths are found in the transitions ψ(2D) → χ c2 (1F ), Υ(1D) → χ bJ (1P ) and Υ(2D) → χ bJ (2P ), etc, which may be helpful to study the missing states χ c2 (1F ), Υ(1D) and Υ(2D). *', 'arxivid': '2212.02838', 'author': ['Su-Yan Pei \nDepartment of Physics and Technology\nHebei University\n071002BaodingChina\n\nKey Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina\n', 'Wei Li \nDepartment of Physics and Technology\nHebei University\n071002BaodingChina\n\nKey Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina\n', 'Ting-Ting Liu \nDepartment of Physics and Technology\nHebei University\n071002BaodingChina\n\nKey Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina\n', 'Meng Han \nDepartment of Physics and Technology\nHebei University\n071002BaodingChina\n\nKey Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina\n', 'Guo-Li Wang \nDepartment of Physics and Technology\nHebei University\n071002BaodingChina\n\nKey Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina\n', 'Tianhong Wang \nSchool of Physics\nHarbin Institute of Technology\n150001HarbinChina\n'], 'authoraffiliation': ['Department of Physics and Technology\nHebei University\n071002BaodingChina', 'Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina', 'Department of Physics and Technology\nHebei University\n071002BaodingChina', 'Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina', 'Department of Physics and Technology\nHebei University\n071002BaodingChina', 'Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina', 'Department of Physics and Technology\nHebei University\n071002BaodingChina', 'Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina', 'Department of Physics and Technology\nHebei University\n071002BaodingChina', 'Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province\nBaodingChina', 'School of Physics\nHarbin Institute of Technology\n150001HarbinChina'], 'corpusid': 254275177, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 22485, 'n_tokens_neox': 19221, 'n_words': 11110, 'pdfsha': 'e654865bc0d42473ffb8cf01f032390892e8419a', 'pdfurls': ['https://export.arxiv.org/pdf/2212.02838v1.pdf'], 'title': ['Partial wave effects in the heavy quarkonium radiative electromagnetic decays', 'Partial wave effects in the heavy quarkonium radiative electromagnetic decays'], 'venue': []} |
arxiv |
MultiCAM: A multivariable framework for connecting the mass accretion history of haloes with their properties
2023
Ismael Mendoza
Department of Physics
The University of Michigan
48109Ann ArborMIUSA
Philip Mansfield
Kavli Institute of Particle Astrophysics and Cosmology
Stanford University
94305StanfordCAUSA
SLAC National Accelerator Laboratory
94025Menlo ParkCAUSA
Kuan Wang
Department of Physics
The University of Michigan
48109Ann ArborMIUSA
Leinweber Center for Theoretical Physics
University of Michigan
450 Church St48109Ann ArborMIUSA
Camille Avestruz
Department of Physics
The University of Michigan
48109Ann ArborMIUSA
Leinweber Center for Theoretical Physics
University of Michigan
450 Church St48109Ann ArborMIUSA
MultiCAM: A multivariable framework for connecting the mass accretion history of haloes with their properties
MNRAS
0002023Accepted XXX. Received YYY; in original form ZZZPreprint 6 February 2023 Compiled using MNRAS L A T E X style file v3.0cosmology: galaxy clusters -dark matter -galaxies: haloes -galaxies: evolution -methods: numerical
Models that connect galaxy and halo properties often summarize a halo's mass accretion history (MAH) with a single value, and use this value as the basis for predictions. However, a single-value summary fails to capture the complexity of MAHs and information can be lost in the process. We present MultiCAM, a generalization of traditional abundance matching frameworks, which can simultaneously connect the full MAH of a halo with multiple halo and/or galaxy properties. As a first case study, we apply MultiCAM to the problem of connecting dark matter halo properties to their MAHs in the context of a dark matter-only simulation. While some halo properties, such as concentration, are more strongly correlated to the early-time mass growth of a halo, others, like the virial ratio, have stronger correlations with late-time mass growth. This highlights the necessity of considering the impact of the entire MAH on halo properties. For most of the halo properties we consider, we find that MultiCAM models that use the full MAH achieve higher accuracy than conditional abundance matching models which use a single epoch. We also demonstrate an extension of MultiCAM that captures the covariance between predicted halo properties. This extension provides a baseline model for applications where the covariance between predicted properties is important.
INTRODUCTION
Characterizing the properties and growth of dark matter haloes has been an important goal of cosmological N-body simulations (Diemand & Moore 2011 ;Frenk & White 2012). Dark matter haloes are groups of dark matter particles that have gravitationally collapsed into bound structures. In the ΛCDM cosmological model, every galaxy forms within the potential well provided by a dark matter halo (White & Rees 1978;Blumenthal et al. 1984). Thus, galaxies and their dark matter haloes are closely connected, meaning that models which attempt to predict the properties of galaxies must account for the behaviour and properties of their dark matter haloes (e.g. Hearin & Watson 2013;Hearin et al. 2016;Wechsler & Tinker 2018) Previous work has established a deep connection between a halo's present-day ( = 0) properties and its mass accretion history (MAH), i.e. its mass growth as a function of time. Properties such as concentration, virial ratio, centre of mass offset, spin, and axis ratio have been studied in relation to MAH. Early-forming haloes tend to have a higher concentration on average than late-forming haloes (e.g. Wechsler et al. 2002), and merger events induce lasting changes in halo structure which are encoded as a universal signatures in the halo's concentration (e.g. Wang et al. 2020). Other properties like the centre of mass offset and virial ratio have strong positive correlations with the halo's recent mass growth history and merging activity (e.g. ★ E-mail: [email protected] Power et al. 2012). This joint dependence leads to substantial covariance between halo parameters (e.g. Lau et al. 2021). Much of this dependence comes from long-term growth trends: it has been found that a significant percentage of the variance in the concentration, axis ratio, and spin of a dark matter halo can be explained by the first principal component of the mass assembly history (e.g. Chen et al. 2020).
The mass accretion history of a halo directly impacts the dynamical state of a halo, which in turn determines the reliability of structural measurements of its properties. Previous studies have established that haloes that have recently experienced one or more major mergers are more likely out of dynamical equilibrium (Tormen et al. 1997;Hetznecker & Burkert 2006). These major merger events can cause temporary deviations from a halo's equilibrium state during which its structural properties change rapidly and might not be well-defined (Ludlow et al. 2016). Thus, it is critical that we characterize the dynamical state of haloes so that their structural measurements can be robustly propagated to downstream analysis. Previous work measuring the distribution of halo properties in simulations attempted to address this by selecting a sub-sample of relaxed haloes, i.e., those haloes considered to be close to dynamical equilibrium (e.g. Neto et al. 2007;Klypin et al. 2011Klypin et al. , 2016. A closely related line of work seeks to identify relaxed galaxy clusters to avoid similar biases in the corresponding measurements (e.g. Cui et al. 2017;Zhang et al. 2022). However, there is a significant ambiguity on how to exactly define this relaxed sample for both cases, which usually rely on hard-cuts. Klypin et al. (2011) which is based on the WMAP5 cosmology (Dunkley et al. 2009).
This further highlights the need for increasing our understanding of the relationships between a galaxy's or halo's properties, MAH, and dynamical state. A common way to connect galaxy or halo properties to their MAH is to use a single parameter summary of the MAH, such as the halfmass scale (e.g. Gao et al. 2005;Hearin & Watson 2013) or the value returned by a single-parameter fit (e.g. Wechsler et al. 2002). This framework leads to a one-to-one parameter correlation analysis called abundance matching, which corresponds to a prediction model that assumes perfect correlation between the two parameters (e.g. half-mass scale and halo concentration) (Kravtsov et al. 2004). Abundance matching and its hierarchical extension, conditional abundance matching (CAM, Hearin et al. 2014, see subsection 3.3.1 for a description of these methods), have been effective models for a range of applications. For example, CAM can predict low-redshift galaxy statistics like two-point correlation functions in SDSS to reasonable accuracy (Hearin et al. 2014). However, the MAH of a dark matter halo is a complex multi-dimensional quantity that contains richer predictive information than single parameter summaries.
MAHs are typically made up of a smooth accretion component consisting of an early-fast accretion phase and a late-slow accretion phase, which was successfully captured with a three parameter model in . The MAH also includes a non-smooth accretion component in the form of an arbitrary number of discrete major merger events that can significantly change halo properties on a short time-scale (e.g. Hetznecker & Burkert 2006;Power et al. 2012;Wang et al. 2020). Separately, it has been shown that different present-day halo properties correlate more or less strongly with different parts of the MAH (e.g. Wong & Taylor 2012). Thus, summarizing the MAH with a single quantity leads to discarding a significant amount of useful information. Another significant drawback of one-to-one parameter models is that they are unable to capture the covariance between predictions. If the same single parameter MAH summary is chosen, CAM-like models necessarily output a perfect correlation between any pair of predicted halo properties. Thus, if one is in-terested in emulating multiple halo properties from a given MAH, one-to-one models are insufficient.
To address the aforementioned limitations we propose a new method for connecting galaxy or halo properties with their formation history: MultiCAM. MultiCAM is a generalization of the traditional abundance matching framework that consistently incorporates the full formation history into a prediction of single-epoch properties while preserving the key benefits of CAM. MultiCAM utilizes the full covariance between features and targets in its predictions. Moreover, MultiCAM can predict multiple properties simultaneously and correctly capture the correlations between them. As a first demonstration of our new method, we apply it to connecting dark matter halo properties with their MAH. In the future, our main focus will be in applying this method to predict baryonic properties.
This paper is organized as follows. Section 2 describes the simulation suite and halo sample used in our studies. Section 3 presents the parameterizations of MAH we consider in this work, gives an overview of CAM, and a detailed description of MultiCAM. In Section 4 we characterize the covariance of MAH and halo present-day properties, and evaluate MultiCAM on our halo sample. Section 5 discusses future applications of MultiCAM and how it compares to other methods. Finally, in Section 6 we present our conclusions.
DATASET
Simulation Suite
For our dataset we use the Bolshoi dark matter-only cosmological simulation (Klypin et al. 2011) which was performed with the Adaptive-Refinement-Tree (ART) code described in (Kravtsov et al. 1997). The simulation has outputs at 180 snapshots starting at 179 = 0.07835 and ending at 0 = 1.00035 ≈ 1. The spacing between early snapshots is Δ = 0.006 between 179 = 0.07835 and 77 = 0.80835, and Δ = 0.003 between late snapshots 77 = 0.80835 and 0 = 1.00035. The cosmological parameters and other simulation details are shown in Table 1.
The halo catalogues were generated by the R halo finder (Behroozi et al. 2013a), as run by Rodríguez-Puebla et al. (2016). This catalogue uses both position and velocity information to identify each halo in the simulation. Halo finder comparison projects have found this algorithm to perform well at halo finding tasks, including detecting substructure and tracing mergers (e.g., Knebe et al. 2011).
We use catalogues generated by - (Behroozi et al. 2013b) to construct the merger history that we use for our analysis (Rodríguez-Puebla et al. 2016). Given a merger event, we define the main progenitor halo as the one that contains the most particles that end up in the resulting halo after the merger. Given a presentday ( = 0) halo, a merger tree can be constructed by following its evolution at each snapshot in the simulation going backwards in time.
The main progenitor branch of a given present-day halo is the branch in the merger tree resulting from following the main progenitor halo backwards in time at each snapshot.
Defining the Halo Sample
Throughout this work we use the same dataset of a random sample of 10 4 haloes from the Bolshoi simulation in the mass bin of vir ∈ [10 12 , 10 12.2 ]ℎ −1 which we denote as M12. Here, vir is the bound mass within a radius enclosing an average density corresponding to the overdensity threshold defined in Bryan & Norman (1998). We take this radius to be the virial radius vir .
For each of the haloes in this sample, we use the R catalogue at each snapshot and -to extract the corresponding main progenitor branch and the virial masses of progenitors at each snapshot in this branch. We do not use all of the 180 snapshots in the Bolshoi simulation, rather we impose a cutoff based on the mass resolution of our simulation. We pick our first snapshot to be the earliest snapshot out of the 180 where at most 5% of haloes have a virial mass lower than 50 times the particle mass. This ensures that we never attempt to analyze snapshots where a substantial portion of our sample is unresolved. For our M12 sample, we consider a total of snap = 165 scales ranging from 164 = 0.18635 up to 0 = 1. For the small percentage ≤ 1% of haloes in our sample that do not have a corresponding main line progenitor at 164 (or in any subsequent snapshots), we assign them a virial mass at those missing snapshots to be the mass of a single particle of the simulation. This is so that there are no missing values in the mass accretion history for all haloes in our M12 sample.
Halo properties and their convergence
In this study we mainly consider halo concentration, vir , defined as the ratio of the virial radius to the NFW scale radius; the normalized maximum value of the halo's rotation curve max / vir ; the offset between the halo's center of mass and its most bound particle off ; the virial ratio, /| |; its dimensionless spin parameter, bullock ; and its second minor-to-major axis ratio, / . See Mansfield & Avestruz (2021) for the exact definitions of these properties as computed by R . Mansfield & Avestruz (2021) measured the minimum converged masses for each of these properties in Bolshoi at different levels of acceptable numerical bias. No detectable bias is measured in max / vir at vir > 10 11.8 ℎ −1 at vir > 10 11.6 ℎ −1 , /| | at vir > 10 11.1 ℎ −1 , bullock at vir > 10 10.2 ℎ −1 , and / at vir > 10 10.9 ℎ −1 . Mansfield & Avestruz (2021) do not report a vir convergence limit for Bolshoi, but do report a vir convergence limit for Erebos_CBol_L125 (Diemer & Kravtsov 2015) at vir > 10 11.6 ℎ −1 , which has an identical cosmology, identical particle mass, coarser force softening, and coarser timesteps than Bolshoi. Therefore, all the considered properties are converged within our mass window of [10 12 , 10 12.2 ]ℎ −1 . We also briefly consider several other, more minor halo properties. For example, the average of the first minor-to-major axis ratio / and the second minor-to-major ratio / , which we denote with :
= 1 2 + .
Because this property is derived from / and / , it is converged at about 10 10.9 ℎ −1 . For all other halo properties, their definitions can be found in Mansfield & Avestruz (2021) and they are also converged within our mass window.
METHODS
Parameterizations of MAHs
First, we introduce the notation that we use to parameterize the mass accretion history and its properties. We measure time through the cosmological scale factor:
( ) = 1 1 + (1)
We track mass growth through the normalized peak mass,
( ) = peak ( ) peak ( = 1) ,(2)
where we take the ratio of peak values to force monotonicity,
peak ( ) = max 0≤ ≤ [ vir ( )].(3)
The difference between peak ( ) and vir ( ) is significant for subhalos due to the large amount of mass loss they experience (e.g.
Wechsler & Tinker 2018), but the difference is less important for the central haloes in M12, since their masses will typically increase over time. The main impact on our sample of host haloes is that it allows ( ) to be inverted. To that end, we define
( ) = ( ) −1 .(4)
Since ( ) is monotonic, but not strictly increasing, we take ( ) to be the first scale factor at which the halo reaches a given mass. When inverting ( ) we use piecewise power-law interpolation between adjacent snapshots of a halo's MAH.
In addition, per convention, we sometimes use the notation 1/ where is an integer to mean:
1/ = ( = 1/ )(5)
This notation, usually with = 2, is often used in the literature as a tracer of formation (e.g. Gao et al. 2005).
We define a halo's dynamical time dyn as the time it takes for a test particle to travel a distance of virial radius vir at a speed of vir , the orbital speed of a particle on a circular orbit at vir . Since all haloes have the same enclosed density within vir , dyn is only a function of redshift and cosmology:
dyn = vir √︁ vir / vir = 1 ( )2 (
For convenience, Eq. 7 is normalized to the = 0 virial density in the Bolshoi simulation. Following from this definition, dyn is the mass fraction at a time dyn before the present day. dyn is a commonly used measure of late-time accretion rates and its unnormalized equivalent is tracked by -catalogues by default. We also analyze the best-fitting exponential scale factor of each mass accretion history :
( )/ ( = 0) = − .(8)
DiffMAH Model of Smooth Mass Accretion History
We also consider the best-fitting parameters of the DiffMAH model of smooth mass accretion histories presented in Hearin et al. (2021). This model consists of the following fitting function:
peak ( )/ peak ( = 0 ) = ( / 0 ) ( )(9)
where is age of the universe, and 0 is the present-day age of the universe. Finally, ( ) is a sigmoid function defined as:
( ; , , early , late ) ≡ early + late − late 1 + exp(− ( − ))(10)
and has parameters early , late , , and with an explicit physical meaning. First, early , late determine the asymptotic value of the power-law index at early and late times respectively; controls the transition time between the early-and late-time indices; and determines the speed of transition between the two phases. As in Hearin et al. (2021), we fix = 3.5. In this diagram we illustrate the novel method presented in this work to connect mass accretion history information to present-day halo properties: 'MultiCAM'. Each step of our algorithm is marked with a green circle. Each box represents the 1D distribution of one of the features or targets. The curve of the 1D distribution is delineated so that the blue and red curve intersect at the median. The rhombuses represent algorithms, either a quantile transformer to marginally map variables to Gaussian distributions, or a linear regression prediction model. The algorithm and each of the steps are described in detail in subsection 3.3.2.
Statistical Algorithms
In this section we introduce the statistical algorithms we use for predictions connecting MAH and present-day halo properties.
Conditional Abundance Matching
One of the methods we use is an adapted Conditional-Abundance Matching (CAM). The CAM algorithm is a method which was originally developed to study and model the connection between halo ages -traced through properties like 1/2 -to observable galaxy properties -like galaxy color or star formation rate ( (Kravtsov et al. 2004), which assigns stellar masses or luminosities to simulated dark matter haloes. Traditional abundance matching evaluates the function −1 ★ ( dm ( vir )), where ★ and dm are some observed cumulative stellar mass function and theoretical cumulative mass function, respectively. Similarly, CAM assigns galaxy properties via −1 gal ( halo ( mah | ★ )| ★ ), where halo and gal are the conditional CDFs at a fixed stellar mass ★ for some theoretical tracer of halo age, mah , and the CDF for the target observable galaxy property, respectively.
The primary application of CAM is generating empirical models of observable properties. But more generally, CAM is a method that optimally implements a specific assumption for the connection between halo growth and halo/galaxy properties: a given halo property halo is entirely and monotonically determined by a given feature of a halo's MAH mah . If this assumption is correct, CAM predictions will be the exact values of the given halo property, and failures in this assumption propagate into inaccuracies in CAM predictions. Therefore, throughout this paper, we use the CAM prediction strength as a measure of how well a given halo property halo can be understood to be determined by a given proxy of halo growth mah . Moreover, multi-parameter models which have improved predictive power over CAM are evidence that the halo property in question is influenced by multiple features in a halo's MAH.
In this work, the CAM algorithm is used to abundance match a given MAH feature mah to a given present-day halo property halo , at fixed present-day halo mass vir . Specifically with the equation:
halo = −1 halo ( mah ( mah | vir )| vir )(11)
where halo and mah are the conditional CDF of the present-day halo property and the MAH feature respectively. Throughout, we condition at a fixed mass bin equal to the one used for constructing the M12 dataset. We pick the MAH property mah for abundance matching to be the scale ( opt ) (see Eq. 4) at a fixed mass bin opt that optimally correlates with halo across all . For example, when halo = vir , we find opt ≈ 0.5 in our M12 dataset, so that mah = ( opt ) = (0.5) = 1/2 . The optimal mass bin opt satisfies the equation:
max ( sp ( ( ), halo )) = sp ( ( opt ), halo ).(12)
Similarly, we could have chosen mah = ( opt ) where the optimal scale opt satisfies:
max ( sp ( ( ), halo )) = sp ( ( opt ), halo ),(13)
but we find that ( opt ) has overall higher correlations across all halo properties than ( opt ). We refer to the algorithm that uses ( opt ) to abundance match between MAH and halo properties at a given halo mass as 'CAM a( opt )'. We use CAM a( opt ) to predict a given halo property halo from the MAH of a halo in subsection 4.3.
CAM is a simple, yet powerful empirical non-parametric approach to matching any pair of strongly correlated variables. It however has some important limitations: (1) It is unable to match multiple variables to another set of multiple variables. (2) It does not incorporate the scatter between prediction and target when matching. We address these limitations of CAM in the algorithms described next.
MultiCAM
We propose the new algorithm MultiCAM to address these limitations of CAM. MultiCAM generalizes CAM to match multiple MAH properties to multiple present-day halo properties simultaneously. To accomplish this, MultiCAM first introduces a multi-variable linear regression between the multiple features and target variables. Then, MultiCAM marginally matches the distribution of outputs to the true distribution of targets. In our context, different halo properties correlate more or less strongly at different time scales of a halo's growth history (e.g. Wang et al. 2020). This means that matching multiple variables consistently is essential for exploring the connections in this work.
MultiCAM also includes a pre-processing step where all features and target variables are marginally transformed to Gaussian distributions. At the end of the procedure, all variables are transformed to their original space. This pre-processing step is beneficial in the context of linear regression since it allows for a version of MultiCAM that introduces scatter between the features and targets, as discussed in detail in subsection 3.3.3 and subsection 3.3.4.
The MultiCAM algorithm is illustrated in Fig. 1 and in detail consists of the following:
(i) Collect all desired features for prediction, , and targets, , from a given dataset. For example, can be set to the full MAH of all haloes in the dataset:
mah = { ( )} =1 , where { } =1
are some pre-defined mass bins with = 1. Similarly, can be set to all the halo present-day properties we consider in this work halo = { vir , /| |, off , bullock , / }, which are described in subsection 2.3.
(ii) We marginally transform each individual feature from its empirical distribution to a normal distribution (top left of figure) to a Gaussian distribution. We do this via the inverse transform method (e.g. Devroye 1986), which can map any 1D dataset of variables to have any other desired empirical distribution without changing the rank-ordering of its points.
(iii) We then take the subset of marginalized Gaussian features train and targets˜t rain in the training set, and train a linear regression model for prediction in this Gaussianized space for these features and targets.
(iv) We then use the marginalized Gaussian features in the testing set˜t est and apply linear regression to obtain the corresponding set of predictions˜p red .
(v) The predictions from the linear regression model˜p red are not guaranteed to follow the empirical distribution of target features (they tend to be narrower) so we apply one more quantile transformer to˜p red and make its distribution (marginally) Gaussian, which then matches the distribution of the Gaussianized training targets˜t rain . This is illustrated in the bottom-left corner of Fig. 1. (vi) Finally, we transform the Gaussianized predictions˜p red back into original target space by applying the inverse of the original quantile transformer used to map training target variables to the Gaussianized space. The result is the final MultiCAM prediction pred .
This approach incorporates the multi-variable prediction accuracy from linear regression while preserving the properties of the marginal predictor distributions. Due to the quantile transformations illustrated in Fig. 1, our procedure automatically outputs predictions whose marginal distributions match the marginal distributions of the training data. This means that the outputs from MultiCAM have a correlation strength with the true targets that is at least as high as CAM (see subsection 4.2). In fact, MultiCAM exactly reduces to CAM in the case of 1D features and targets. In summary, MultiCAM also has the added advantage of (1) predicting multiple properties from multiple input properties and (2) taking advantage of the increased accuracy from linear regression.
This version of MultiCAM that uses linear regression still faces one key limitation in that it doesn't account for the scatter between target features and predictions and thus will not reproduce the correct correlations between output properties. To address this, we first discuss the relationship between linear regression and sampling from a conditional Gaussian. Second, we discuss a method that maintains the correlation between sampled properties that is based on using conditional Gaussian sampling within MultiCAM instead of linear regression.
Linear Regression and Conditional Gaussian Sampling
We start by discussing the theoretical framework of conditional Gaussian prediction, and then connect it with linear regression and Multi-CAM. Assume that you have some multi-dimensional features and multi-dimensional targets that are jointly distributed as a multivariate Gaussian , . Given a new feature test point ★ , we consider the conditional distribution | ★ in order to choose our new prediction based on ★ . The conditional distribution | ★ is also Gaussian with mean¯( ★ ) and covariance matrixΣ. The equations to derive the conditional parameters¯( ★ ) andΣ from empirical estimates of the joint distribution , parameters can be found in Appendix A. Given this framework, there are two different goals we could choose to pursue: (1) Minimize (squared) residuals of the prediction pred ( ) relative to the target or (2) Sample points such that their distribution matches the true target distribution ( ), including in its correlations between different target variables.
The first goal is achieved by using the mode of the conditional distribution directly as the prediction:
pred ( ★ ) ≡¯( ★ ).(14)
Based on the expression for¯( ★ ) in Eq. A5, we can see how this prediction would not take into account the intrinsic scatter of the target distribution, as there is no term with Σ -the covariance matrix between target variables. The second goal can be achieved by sampling the conditional distribution | after sampling ( ). Concretely, given a test point ★ ∼ ( ), we choose as our prediction samples directly from the conditional normal distribution | ★ :
pred ( ★ ) ∼ N (¯( ★ ),Σ).(15)
This second approach does incorporate the intrinsic scatter in the target distribution asΣ depends on Σ , as can be seen in Eq. A6 in Appendix A. We denote this approach conditional Gaussian sampling.
In Appendix B, we prove that the mode of the conditional distribution | (Eq. 14) is equivalent to the linear regression output if , are jointly normal distributed. Additionally, MultiCAM already includes a pre-processing step (step 2 of the algorithm in subsection 3.3.2) where we try to bring features and targets close to a joint Gaussian. These two facts combined imply that the conditional Gaussian sampling approach (Eq. 15) is a natural replacement for the linear regression prediction algorithm within MultiCAM that could allow us to account for the scatter between targets.
Finally, note that we restrict analysis in this paper to simulation data, where we can train the entirety of Σ and account for the explicit covariance between all features and predicted quantities. However, conditional Gaussian sampling provides an avenue to use Multi-CAM as an interpretable empirical model. In the simplest case, if we consider traditional CAM as such an empirical model, the "fit" procedure would consist of Σ containing one row for mah , one row for the target galaxy observable, and off-diagonal terms artificially fixed to assume perfect correlation. In the more general case using MultiCAM with conditional Gaussian sampling, we would perform an analogous "fit" procedure by taking any subset of the elements in Σ as free parameters.
MultiCAM with scatter
As mentioned previously, the MultiCAM algorithm presented in subsection 3.3.2 cannot correctly capture the correlation between targets. Table 2. Correlations between halo properties predicted from each model. We show the Spearman correlation between each pair of predicted target = 0 halo properties given their MAH using three different methods. The training and 'true' sample is equivalent to the one used for Fig. 2 in subsection 3.3.4.
As seen in subsection 3.3.3 this is because the prediction model connecting features and targets, linear regression, does not account for the scatter in the target distribution. However, given that the MultiCAM presented in subsection 3.3.2 already includes a normalizing pre-processing step (step 2), we can replace the prediction model from linear regression (step 3 and 4) to conditional Gaussian sampling (Eq. 15) to solve this problem. As explained in subsection 3.3.3, the pre-processing step allows us to interpret this replacement as using the same joint normal distribution to solve a different goal, that of directly sampling ( ). This can be achieved by using the conditional Gaussian sampling approach within MultiCAM, since we will be explicitly incorporating the scatter between targets in our predictions. Therefore, for the rest of this subsection, we denote this new version of MultiCAM as MultiCAM (with scatter) to distinguish it from the method in subsection 3.3.2 which we will denote as MultiCAM (no scatter). Unless otherwise stated, in the rest of the paper 'MultiCAM' refers to MultiCAM (no scatter).
Importantly, the MultiCAM (with scatter) approach explicitly models scatter between features and targets, i.e. a given test data point of features can be used to sample multiple predictions from the conditional normal distribution. This means that the point estimate accuracy of MultiCAM (with scatter) will be lower compared to MultiCAM (no scatter), since we are introducing noise into the prediction. However, we will show how this simple extension allows for capturing the lion's share of the covariance between variables while still matching the marginal distributions exactly.
To demonstrate this, we first train each of the models presented so far -CAM a( opt ), MultiCAM (no scatter), and MultiCAM (with scatter) -on 7000 random haloes from the M12 dataset using the full MAH { ( )} =1 of each halo as features and three present-day halo properties as targets: / , bullock , and off . The three models are then tested on full MAH of remaining 3000 haloes from the M12 dataset and the 2D, 1-sigma, 2-sigma, and 3-sigma contours between each pair of target predicted variables are plotted as shown in Fig. 2. The true contours are shown in orange and the predicted contours by each model in green.
In Fig. 2 we see that CAM a( opt ) and MultiCAM (no scatter) fail to match the 2D distributions of halo properties. For CAM a( opt ), the width of the green contours in each panel directly corresponds to the covariance between the ( opt ) of each property, since CAM does a one-to-one matching between these. For example, off and bullock are the target variables with the largest difference in their corresponding opt , as shown in Table 3. Fig. 3 demonstrates that a larger difference in mass bins between a pair of scales ( ) implies a lower covariance between them. Thus, we expect a weaker correlation between the CAM a( opt )-predicted off and bullock than for the other pairs of variables. This is exactly what we see in the leftmost subplot in Fig. 2. MultiCAM (no scatter) has the narrowest contours out of the three methods. This is because the predicted variables use the same sets of MAHs and there is substantial overlap in the relative importance of different epochs (see subsection 4.2). However, MultiCAM (with scatter) has contours that seem to match the true contours more closely.
Additionally, Table 2 shows the correlation between each pair of = 0 halo properties for each of the three models. We can quantitatively reach the same conclusions suggested by Fig. 2: the correlations between target properties outputted by MultiCAM (with scatter) agree closely with the true correlations, but this is not the case for CAM ( opt ) and MultiCAM (no scatter).
The full triangle plot applying MultiCAM (with scatter) to all the present-day properties considered in this work is shown in Fig. C1 of Appendix C, which shows good agreement in both 1D marginals and 2D contours. As explained in subsection 3.3.3, MultiCAM (with scatter) can successfully capture the covariance between target variables since the sampling scatter depends directly on this covariance (Eq. 15).
In summary, Table 2, Fig. 2, and Fig. C1 demonstrate that Multi-CAM (with scatter) can be used to successfully emulate present-day halo properties given the full MAH of a dark matter halo.
RESULTS
In this section, we focus on understanding the statistical properties of our M12 dataset through correlations and evaluate the MultiCAM approach. We choose to focus on the following = 0 halo properties for our analysis: concentration, vir , virial ratio, /| |, center of mass displacement, off , spin Bullock, bullock , and second minor-axis to major-axis ratio, / .
We analyze the M12 dataset as defined in Section 2. We divide the M12 halo sample into a training set of 7000 haloes and a test set of 3000 haloes (unless otherwise stated). The performance metrics of trained models are evaluated only on the test set. The error bars reported in all our results are standard errors estimated from jackknife resampling over 8 equal volume sub-cubes of the simulation. Figure 3 shows two-dimensional histograms where we color code each pixel (bin) by Spearman correlation strength. The top plot shows the Spearman correlation, sp ( ( ), ( )), between mass fraction at a given pair of formation times. The bottom plot shows the Spearman correlation, sp ( ( ), ( )), between the formation time at a given pair of mass fractions in our M12 dataset.
Autocorrelation of Halo Mass Accretion History
In the top panel, we see that ( ) values are strongly correlated with one another for small (Δ ≈ 0.1) changes in . Similarly, in the bottom plot we see that ( ) values are strongly correlated with one another for small (Δ ≈ 0.1) changes in . This suggests that we can achieve a similar prediction accuracy with a sparser subset of the MAH information. For example, if we wanted to retain information at a level of sp ∼ 0.9 between adjacent bins, we could choose data at approximately a spacing of Δ = 0.05 which would result in approximately ten times less data.
Another takeaway from the top plot is that adjacent snapshots at both early and late times are strongly correlated (see subsection 2.1). The distinct output cadence of Bolshoi should therefore have minimal impact in the following analysis.
The takeaways for the bottom plot are similar to those from the top plot.
Correlations of MAH and present-day halo properties
In Fig. 4 we show the Spearman correlation coefficient between several present-day halo properties and the halo accretion history, parameterized as ( ) (left) and ( ) (right). We compute the correlation using the full 10 4 halo sample M12. The colored bands correspond to the uncertainty on each curve as estimated by jackknife resampling. In this figure, solid lines are used to represent positive correlation values and dotted lines represent negative values.
Both figures illustrate that present-day halo properties contain information about the growth of haloes back to very early times, ≈ 4, and at times when haloes were ≈10% to 20% of their current mass. As expected, formation times correlate positively with vir , max / vir (this follows directly from the vir correlation with growth), / (Allgood et al. 2006;Chen et al. 2019), and negatively with /| |, off (Maccio et al. 2007), and bullock (Vitvitska et al. 2002).
Inner halo structure, tracked by vir and max / vir , most strongly correlates with early times, ≈ 3.4 dyn in the past, when haloes were roughly half their current mass. This is consistent with models of halo structure in which the inner profile is primarily set by long term growth trends (e.g. Dalal et al. 2010;Ludlow et al. 2013). More recently, Wang et al. (2020) systematically examined the correlation between the present-day concentration and different stages of halo mass assembly. They found that there are extended periods in the assembly history that correlate strongly with the present-day halo structure, which justifies the use of various definitions of halo formation time with which to predict present-day concentrations. These findings are qualitatively consistent with our results.
The other properties that we track, off , /| |, bullock , and / have relatively larger predictive power at late times compared with properties that more closely describe the halo inner structure, such as vir . All four are expected to be tracers of dynamically unrelaxed haloes that have recently experienced major mergers or rapid, anisotropic smooth accretion from nearby filaments. More relaxed haloes will be more spherical, more centered on its most bound point, and will have a virial ratio closer to 0.5 (Mo et al. 2010). Any deviations would be caused by recent external influences, which are typically mergers for non-subhalos (although mass loss due to tidal stripping can also influence halo properties, e.g., Tucci et al. 2021). The correlation with spin is generally understood to arise because a slowly accreting halo will generally accrete isotropically, reducing its normalized angular momentum over time, while a rapidly accreting halo will experience larger mergers which will inject large amounts of angular momentum into the system (e.g. Vitvitska et al. 2002). However, halo spin also plays a large role in the early collapse of dark matter perturbations prior to forming haloes (e.g. Sheth et al. 2001), meaning that it should not be thought of as a purely late-time phenomenon. Table 3 contains the values of optimal correlations between halo properties and MAH which correspond to the peaks of the curves in Fig. 4. As an example, we include a dashed vertical line in the left panel of Fig. 4 which intersects the peak of the correlation curve for the max / vir property. In other words, the -value of the vertical orange line is opt when = max / vir , which corresponds to the second row of Table 3.
We also measured correlations with other measures of triaxiality, and semi-minor axis ratio / . The opt of and the ellipticity ratio / are the same, but has a slightly higher peak absolute Spearman correlation with MAH of | sp | = 0.533 compared to | sp | = 0.510 for / . The correlation between / and MAH is comparable with that between / and MAH. Analogously, we compared the results for Peebles , the Peebles spin parameter. This measurement was comparable with Bullock , but Bullock has a higher peak correlation of sp = 0.473 compared to Peebles which has a peak correlation of sp = 0.384, likely due to the fact that measurements of internal energy for Peebles is less stable leading to weaker signals. We use Bullock in all subsequent analyses considering the spin of the haloes.
Finally, in comparing max / vir to vir , the peak correlation occurs slightly earlier in the former quantity with comparable correlation strength.
Predictions of present-day properties based on MAH
In Fig. 5, we show the Spearman correlation between several predicted halo properties and their true value for four different models described in subsection 3.3.
In blue circles, we show results for our canonical MultiCAM model, using the full mass accretion history of each halo. Under this metric, MultiCAM either outperforms or performs comparably well to the other tested models.
With orange squares, we show results of applying MultiCAM to the best-fitting DiffMAH curve for each MAH (see subsection 3.2 for more information). This model is next in predictive power for the target halo properties shown. We highlight the similar performance between this model and MultiCAM trained on the full nonparametrized MAH (blue circles) for most halo properties. The consistency of performance implies that our method leans heavily on information contained within the smooth accretion history.
Next, we show the performance of a model that applies Multi-CAM to the three best-fitting parameters from DiffMAH in green diamonds. We note that the DiffMAH parameters alone have systematically lower prediction power than the full MAH curve that the DiffMAH parameters describe. This may be due to the non-linear mapping of DiffMAH parameters onto the mass accretion histories that cannot be captured by the linear modeling we employ in Multi-CAM. Further investigation might include testing non-linear models to map DiffMAH parameters to halo properties.
Relatedly, the decrease in prediction power for MultiCAM on DiffMAH parameters suggests a degeneracy between DiffMAH parameters and present-day halo properties. In that case, the exact parametrization of the DiffMAH curve matters. Indeed, one can show from Eq. 9 and 10 that we can pick a parametrization where we replace early with 1/2 and still get a complete set of DiffMAH parameters that uniquely characterizes a MAH curve. We find an increase of ≥ 0.05 in the correlation with vir , bullock , and / with this alternative parametrization. This indicates that the DiffMAH parametrization chosen impacts the predictive power of DiffMAH parameters, which is also further evidence of the aforementioned degeneracy.
Finally, in the purple pluses, we show model predictions for CAM evaluated at opt , which only uses the scale at a single mass fraction of a halo that best correlates with that halo property (see Table 3). We see that MultiCAM on the full MAH significantly outperforms CAM a( opt ) for prediction of most halo properties including: vir , /| |, and off . For the other two halo properties, bullock and / , Multi-CAM and CAM a( opt ) have (statistically) the same performance. Moreover, CAM performs significantly better than MultiCAM on DiffMAH parameters, which might be related to the fact that CAM opt is using (by construction) the best single feature in predicting MAH.
Comparing the individual models within different types of halo properties, we notice a few trends. First, the full curve from the DiffMAH fit performs at least as well as the model trained with CAM opt on all halo properties. The MultiCAM on DiffMAH fit information provides better predictions on properties that are most strongly correlated with overall MAH, e.g. vir and /| |. For properties whose predicted values are more weakly correlated with truth (i.e. Bullock and / ), all models, except for the one using the DiffMAH parameters only, perform similarly.
The halo property predictions where CAM applied to opt performs comparably well tend to be in the "worst" cases of target predictions (e.g. off , Bullock , and / ). We surmise that the comparable performance is due to the fact that these halo properties are largely dependent on the most recent MAH of the haloes and that Table 3. Optimal correlations between present-day halo properties and single-epoch measurements of the MAH. In this table we show the optimal scale factors, opt , and mass fractions, opt , at which the present-day halo properties of our halo sample achieve their maximum absolute Spearman correlation with ( opt ) or ( opt ) respectively. These values corresponds to the maxima of the curves in Fig. 4. The precise definition of opt and opt can be found in subsection 3.1.
these properties are even more sensitive to the non-smooth component of the MAH, comprised of moderate and major mergers, which our model does not yet account for. We additionally investigated whether using the gradient of the MAH could successfully capture the missing major merger information. Specifically, we computed the first-order derivative of MAHs using a Savitzky-Golay Filter (Savitzky & Golay 1964) and used these derivatives as additional features for MultiCAM. However, we found no significant difference between MultiCAM trained on the full MAH and its gradients compared to our canonical MultiCAM model trained only on the full MAH.
Predictions of MAH summaries based on present-day properties
In Fig. 6 we use MultiCAM to perform the inverse of the test shown in Fig. 5: predicting summary statistics of a halo's MAH from its = 0 halo properties. We attempt to predict 1/2 , the half-mass scale (Eq. 5), , the characteristic time in an exponential MAH fit (Eq. 8), ( dyn ), the accretion rate over a dynamical time (Eq. 2), and the three DiffMAH parameters, , late , and early (Eq. 9 and Eq. 10). We use MultiCAM to predict these values with different combinations of vir , /| |, off , bullock , and / .
Using MultiCAM on the full suite of halo properties (purple plus signs) results in strictly more accurate predictions than using a single halo property, as expected. As expected from Fig. 4, vir (blue circles) does a better job predicting tracers of early accretion history like 1/2 , , and early than off (orange squares) and /| | (green diamonds). The opposite is true for tracers of late accretion history, like ( dyn ) and late .
Simpler parameterizations, like for an exponential growth history, are well-predicted, but for more complicated non-linear parameters like in the DiffMAH fits, predictions are quite poor. This is most likely due to the degeneracy between DiffMAH parameters and halo properties as discussed in subsection 4.3.
Despite being comparatively poorly predicted, the same trends can been seen in the DiffMAH parameters that are seen in the singleepoch MAH tracers. late encodes behavior at late times and is better predicted by off and /| | than by vir . is sensitive to earlier times and better predicted by vir than off and /| |. early probes even earlier times and shows no statistically significant trends. This may either be due to early having a relatively small impact on the overall MAH or it corresponding to such an early time period that no present-day properties do a good job at tracing it.
In Fig. 7 we use MultiCAM with the same models as in Fig. 6 to predict the full MAH either with the ( ) parametrization (left) or the ( ) parametrization (right). We use the same scale and mass bins for prediction as in Fig. 4 which consist of the Bolshoi simulation cadences for ( ) (left) and uniformly spaced log bins on between 0 and 1 for ( ) (right). For each parametrization, we show the Spearman correlation coefficient between predicted MAH and true MAH of our test set from the M12 dataset.
MultiCAM with all properties (purple diamonds) produces strictly more accurate predictions for the mass-accretion histories of haloes than any individual property. It leverages properties like vir to maintain high accuracy ( spearman ≈ 0.75) at early times and switches to later-time properties like /| | and off to maintain that accuracy after vir ceases to be a good tracer of growth.
Overall, the curves follow the same trends as in Fig. 4. For example, off and /| | are better are predicting late history than vir . /| | has higher correlation than off throughout. This is expected since higher Spearman correlation corresponds to higher prediction power.
MultiCAM (all models) drops in predictive power at very early times and very late times. The former drop is because the halo properties across the board are no longer correlated with those early times, and the latter drop is because, as shown in Fig. 4, at very late times masses are still strongly correlated with the present-day mass, meaning that there isn't much room to gain predictive power, unless we were able to predict mergers in detail, which we leave for future work.
Finally, as discussed in subsection 3.3.2, MultiCAM reduces to CAM in the case of connecting a single feature with a single target variable. This means that the MultiCAM predicted correlations for the models using a single halo property as a feature in Fig. 6 and Fig. 7 are equivalent to the CAM predictions for the corresponding MAH summary.
DISCUSSION
In this work, we have studied the correlations between a halo's present-day properties and multiple intermediate epochs of their MAH. In particular, we investigated the time and mass scales at which different halo properties correlate most strongly with the MAH (see Fig. 4 and Table 3). We find a significant non-zero correlation between all the halo properties we studied and its formation history for most time and mass scales, with most halo properties, including concentration, achieving their strongest correlation with the MAH at intermediate time and mass scales. This is in disagreement with the findings in Wong & Taylor (2012), where the authors find that correlation between concentration and the MAH was strongest when the halo had accumulated only 20% of its mass for a relaxed halo sample. However, we see a high level of agreement both quantitatively and qualitatively for the correlation between concentration and MAH with Wang et al. (2020), who use the same halo finder (R ) as our work. We thus hypothesize that the disagreement with Wong & Taylor (2012) is due to differences in halo finder and halo sample, but leave confirmation of this for future work.
We also studied the autocorrelations between different epochs of mass growth. Fig. 3 shows that a sparser representation of the MAH can provide a similar amount of predictive information to model galaxy or halo properties. This conclusion is similar to the one reached in Wong & Taylor (2012), where their principal component analysis of MAHs suggested that only a few principal components explained the majority of the scatter in the MAHs. Physically, this indicates that longer timescales of mass accretion likely set halo properties.
Our model and subsequent analysis adds to a growing body of literature that models the connections between galaxies, dark matter haloes, and their mass accretion histories (Wechsler & Tinker 2018). Such models have ranged from one-to-one mappings of properties in the form of abundance matching (Kravtsov et al. 2004), to complex machine learning approaches (e.g. Hausen et al. 2022;Horowitz et al. 2022;Stiskalek et al. 2022;de Andres et al. 2023). We provide a generalization of CAM and quantify its ability to connect halo properties with their full MAH. Other recent models enable connections between more details of a halo's full MAH with corresponding halo or galaxy properties. For example, Jespersen et al. (2022) builds a graph neural network that directly uses the full dark matter merger tree of a halo to accurately emulate galaxy properties and their scatter. They find that using the full formation history always outperform predictions compared to only using the = 0 halo properties and a traditional abundance matching approach, which is consistent with our conclusions in Fig. 5 and Fig. 6. As another example, Lucie-Smith et al. (2022) uses gradient-boosted-tree algorithms to predict the final mass profiles of cluster-sized haloes based on the initial density field and the MAH. Their model is able to identify time-scales in the MAHs that are most predictive of the final mass profiles.
As demonstrated in Wang et al. (2020), one major source of scatter in the concentration mass relation comes from mergers, and the scatter depends on fine grained details of these mergers. However, Fig. 4 shows that the last dynamical time of the halo is not providing much predictive information. This suggests that MultiCAM is not able to successfully extract the relevant merger and non-smooth information from the MAH features given. In addition, we attempted to capture merger information by incorporating gradient features of MAH in MultiCAM's prediction. We found that the prediction performance of MultiCAM remained the same when adding these additional features across all halo properties. We therefore plan to explicitly incorporate major merger information from merger trees in future development and studies with MultiCAM.
Additional future applications of our method include (1) applying MultiCAM to connecting DM halo accretion histories to baryonic properties in the context of hydrodynamical simulations such as the Table 3 for the specific values of optimal correlations between halo properties and MAH (peaks in these plots). The annotated orange dashed vertical line in the left plot illustrates one such optimal correlation opt for the max / vir property (whose exact value is the second row in Table 3).
T T H project (Haggar et al. 2021), (2) using Multi-CAM to build fast emulators that paste small-scale properties into cheaply generated ensembles of accurate mock halo catalogues (e.g. Tassev et al. 2013;Feng et al. 2016), or parametric models of MAHs (e.g. Hearin et al. 2021), (3) exploring other extensions of MultiCAM that incorporate more advanced non-linear methods, such as neural networks, that could provide higher predictive accuracy, and (4) applying MultiCAM as an empirical method where we can constrain the internal covariance matrix of the model with observational data.
Finally, previous work indicates that the mass accretion history closely connects to proxies for the dynamical state of galaxies, galaxy clusters, and their host haloes (e.g. Hetznecker & Burkert 2006;Gouin et al. 2021). An improved understanding of this connection can better inform the interpretation of measurements of galaxy and galaxy cluster properties (e.g. Ludlow et al. 2012;Mantz et al. 2015;Ludlow et al. 2016). The flexibility of MultiCAM provides a simple and interpretable framework to explore various measures of the dynamical state of DM haloes, galaxies, or galaxy clusters and to see how their dynamical state connects with their structural properties and accretion history. Specifically, MultiCAM provides a framework to study the predictive power of any combination of galaxy or halo properties on the MAH. Such studies could enable optimal combinations of properties that strongly correlate with merger information or other indicators of dynamical state. Thus, MultiCAM complements approaches to classifying the dynamical state of haloes or galaxies similar to the ones proposed in works such as De Luca et al. (2021) and Vallés-Pérez et al. (2023), which attempt to construct tracers of halo relaxedness from multiple halo properties.
CONCLUSION
In this study, we present MultiCAM, a generalization of traditional abundance matching algorithms. MultiCAM connects halo and galaxy properties with their mass accretion histories (MAH). As a case study, we apply MultiCAM to connect the present-day properties of dark matter haloes with their full mass accretion histories using the Bolshoi dark matter-only cosmological simulation.
Our key result is that we can use the entire MAH with MultiCAM to significantly outperform CAM in such connections. Our MultiCAM models are particularly successful in connecting the entire MAH with halo properties often used to trace MAH, such as vir , /| |, and off . For other halo properties considered (e.g. bullock ), MultiCAM performs at least as well as CAM. See Fig. 5 and Fig. 6 for relevant figures.
Our other main results are the following:
(i) There is a significant auto-correlation in dark matter haloes' mass accretion history. We find that values of normalized peak masses ( ) are strongly correlated with one another for small changes in . This indicates that a subset, or a sparser representation of MAH, might be sufficient to model some galaxy or halo properties with comparable information content. For more details, see Fig. 3.
(ii) The entire formation history of a halo leaves imprints on present-day properties. We find that all the properties in our subset of present-day halo properties have significant non-zero correlations with their MAH between ≈ 4 and = 0. See Fig. 4.
(iii) We find that MultiCAM applied to the DiffMAH smooth parametrization of MAH performs comparably with MultiCAM applied on the full MAH for halo properties known to be strongly correlated with late-time merger events such as off and bullock . This suggests that MultiCAM is not able to fully capture merger information in detail, which we leave for future work. See Fig. 5 for more details.
(iv) We show how a simple extension of MultiCAM based on conditional Gaussian sampling is able to simultaneously sample multiple halo properties based on the MAH and capture the true correlation between properties. See Fig. 2 and Fig. C1.
(v) Finally, we apply MultiCAM to the inverse problem of predicting the MAH of a halo from its present-day properties. We show that vir is better at predicting the early formation history of a halo, and /| | and off are better at predicting the late time formation history. MultiCAM enables simultaneous use of all halo properties for MAH prediction, which outperforms predictions from any individual property. See Fig. 6 and Fig. 7. Here we show the Spearman correlation between parameters characterizing the MAH of haloes in our testing set, and their predictions using the MultiCAM algorithm trained on subsets of the = 0 halo properties. The definitions of these MAH properties can be found in subsection 3.1 and subsection 3.2. The last model (purple cross) corresponds to MultiCAM trained on the following = 0 halo properties: vir , max / vir , off , / | |, bullock , and / . The correlation from the first three models (blue circle, orange square, green diamond) is equivalent to the CAM predicted correlation. See subsection 4.4 for additional discussion on this figure. eration with the Spanish MultiDark Consolider Project CSD2009-00064. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de). The Bolshoi simulations have been performed within the Bolshoi project of the University of California High-Performance AstroComputing Center (UC-HiPACC) and were run at the NASA Ames Research Center.
We acknowledge the use of the scikit-learn software for linear regression models and quantile transformers (Pedregosa et al. 2011). We also acknowledge the use of numpy (Harris et al. 2020), scipy (Virtanen et al. 2020), colossus (Diemer 2015), astropy (Astropy Collaboration et al. 2013, matplotlib (Hunter 2007), corner (Foreman-Mackey 2016), and lmfit (Newville et al. 2016).
We thank Andrew Hearin and Daisuke Nagai for feedback on early results of our model and analysis.
DATA AVAILABILITY
The code to reproduce all results in this work is publicly available in the following github repository: https://github.com/ ismael-mendoza/nbody-relaxed.
The dark matter halo catalogue data is publicly available in https: //www.cosmosim.org. Here we use the same models as in Fig. 6 to predict full MAHs, parameterized by ( ) (top) and ( ) (bottom). The left curves are the Spearman correlations between the predicted ( ) and the true ( ) at each scale of the Bolshoi simulation starting at = 0.2 for our test set from the M12 dataset. Similarly, the right curves are the Spearman correlations between the predicted ( ) and the true ( ) on uniformly spaced log bins on between 0 and 1. The bands correspond to 68% confidence intervals, estimated by jackknife resampling. See Section subsection 4.4 for additional discussion on this figure.
For simplicity, we assume that our dataset has been mean-centered so that = 0 and = 0 (Eq. A2). From standard statistical literature (see, eg., Freedman 2009), we know that given a new data point 0 ∈ R the linear regression prediction 0 is:
0 = ( ) −1 0 .(B1)
Let us rewrite the matrix . Given that = we have from Eq. A3 that:
( ) = ∑︁ = ( Σ ) ( − 1),(B2)
where the last equality holds since we assumed = 0. Similarly = Σ ( − 1), so that combining this with Eq. B1 we get:
0 = Σ ( − 1) Σ −1 ( − 1) −1 0 = Σ Σ −1 0 ,(B3)
which is the same as Eq. A5 in the mean-centered case. This proves that linear regression is the same as multivariate Gaussian sampling without scatter.
APPENDIX C: MULTICAM CAPTURES COVARIANCES OF PRESENT-DAY HALO PROPERTIES
In Fig. C1 we plot 1D marginals and 2D histograms with 1-, 2-, and 3-sigma contours of our main present-day properties of 3000 haloes from our M12 dataset. The orange contours come from the true values of these halo properties and the green contours are samples from MultiCAM (with scatter) applied on the full MAH of each of these 3000 haloes. We overall see good agreement between truth (green) and samples from our model (orange) in both the 1D marginal distributions and the 2D scatter contour plots. This paper has been typeset from a T E X/L A T E X file prepared by the author. The orange contours in each subplot are the same and correspond to a 3000 random sample of vir , / | |, off , bullock , and / from our M12 dataset. The green contours of each subplot were produced by applying MultiCAM (with scatter) on the full MAH of each of the 3000 haloes. See subsection 3.3.4 for more discussion on MultiCAM (with scatter). See Appendix C for additional discussion of this figure.
Figure 1 .
1Schematic illustrating the MultiCAM method.
Figure 2 .
22D Scatter with contours of samples of = 0 halo properties comparing different models. We show plots of 1-, 2-, and 3-sigma contours for the 2D histograms of 3000 samples of = 0 halo properties given their MAH using three different methods. Each method is applied to bullock , / , and off within our M12 dataset. The orange contours in each subplot show the true distributions of these properties. The green contours of each subplot were produced by applying three different prediction methods to these halo properties: CAM a( opt ) (left), MultiCAM with no scatter (middle), MultiCAM with scatter (right). These models were trained on the remainder of the M12 dataset. For more details on the different methods used see subsection 3.3 and for more discussion on the figure see subsection 3.3.4.
Figure 3 .
3Internal MAH Spearman correlation between formation times as a function of mass fraction. The color in each 2D bin (pixel) of these plots corresponds to the Spearman correlation sp ( ( ) , ( )) between the mass fraction at a given pair of formation times, ( , ) (top), and the Spearman correlation sp ( ( ), ( )) between the formation time at a given pair of mass fractions, ( , ) (bottom), for all the 10 4 haloes in our M12 dataset. See subsection 4.2 for additional discussion.
Figure 4 .
4Correlation of accretion history with present-day properties. We show the Spearman correlation coefficient between different present-day halo properties, , and accretion history, parameterized as ( ) or ( ). The correlation is calculated based on our complete 10 4 halo sample M12. The coloured bands around each curve show the error estimated by jackknife resampling. In both figures, solid lines indicate positive correlation value and dotted lines a negative correlation value. See subsection 4.2 for additional discussion on these plots. See
Figure 5 .
5Correlation between predictions of = 0 properties based on MAH and true halo properties. We show the Spearman correlation between several true = 0 halo properties and predicted = 0 halo properties using four trained models on the M12 dataset. The first three models are based on MultiCAM trained on full MAH (blue circle), on DiffMAH curve fits to the MAH curves evaluated at the same scale factors as the full MAH (orange square), and on the parameters of the DiffMAH fit (green diamond). The last model (purple plus) is the prediction of the CAM algorithm using the corresponding opt (defined in Eq. 13) for each halo property. See subsection 4.3 for additional discussion on this figure.
Figure 6 .
6Correlation between MultiCAM predictions of mass accretion history summaries from = 0 halo properties.
Figure 7 .
7MultiCAM predictions of full MAHs based on = 0 halo properties.
Figure C1 .
C11D Marginals and 2D Scatter with contours of samples of = 0 halo properties from MultiCAM (with scatter) We show plots of 1D marginals and of 1-, 2-, and 3-sigma contours for the 2D histograms of 3000 samples of = 0 halo properties given their full MAH using MultiCAM (with scatter).
Table 1. Simulation and Cosmological parameters of the Bolshoi dark matteronly cosmological ΛCDM simulation presented inParameter
Value
Box size
250 Mpc/ℎ
Number of particles
2048 3
Particle mass
1.35 × 10 8 M ℎ −1
Force resolution
1.0 kpc/ℎ
Initial redshift
80
Number of snapshots
180
Hubble parameter ℎ
0.7
Ω Λ
0.73
Ω
0.27
Ω
0.0469
Tilt
0.95
8
0.82
Hearin & Watson 2013; Hearin et al. 2014; Watson et al. 2015). It is similar to the traditional abundance matching algorithm
MNRAS 000, 1-15(2023)
ACKNOWLEDGEMENTSIM and CA acknowledge support from DOE grant DE-SC009193. IM, KW, and CA acknowledge support from the Leinweber foundation at the University of Michigan. IM acknowledges the support of the Special Interest Group on High Performance Computing (SIGHPC) Computational and Data Science Fellowship. IM acknowledges support from the Michigan Institute for Computational Discovery and Engineering (MICDE) Graduate Fellowship.This research was supported in part through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor. The CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark database was developed in coop-APPENDIX A: CONDITIONAL MULTIVARIATE GAUSSIAN SAMPLING ALGORITHMLet ∈ R be a vector of features and ∈ R ℓ be a vector of targets. We also assume we have access to a training dataset of pairs:The conditional multivariate Gaussian sampling approach consists of the following steps:• Assume that , are jointly Gaussian distributed so that:where we separated the mean and covariance matrix into corresponding blocks for each variable.• Empirically compute estimates for the mean and the covariance matrix using the training set {( , )} =1 . For example:• Given a feature vector in the testing set 0 we are interested in obtaining a prediction 0 for the corresponding true target 0 . In order to do this we need to determine the conditional distribution ( | 0 ). From statistics we know that the conditional distribution of two or more jointly normal distributed variables is also normal, in particular:Note that to calculate this quantities we would replace and with their corresponding empirical estimates , . Importantly¯depends on a particular test point 0 but¯does not.• Finally, from Eq. A4 we see that there are two natural options for our predictor 0 . We could choose to simply make our predictor the mean of the posterior distribution, i.e., setting 0 =¯( 0 ). This is reasonable since¯( 0 ) is the most likely value of 0 given 0 (assuming our statistical model is correct). In fact, in Appendix B we show that this approach is equivalent to linear regression.Another option is to sample from the distribution in Eq. A4 and in this way account for scatter. This is the approach used in subsection 3.3.4, and it allows us to accurately capture correlations between target variables. This latter approach is what we refer to as conditional Gaussian sampling.APPENDIX B: LINEAR REGRESSION: SCATTER AGNOSTIC PREDICTIONS FROM MULTIVARIATE GAUSSIANIn this short appendix, we will show that the predictions from linear regression and the conditional multivariate Gaussian sampling algorithm in subsection 3.3 are equivalent up to the scatter coming from (Eq. A6) in the Gaussian approach. In other words, linear regression outputs the mean of the posterior ( | ) (Eq. A4) outputted by the multi-Gaussian approach.Let ∈ R × be a training dataset of features and ∈ R ×ℓ a training set of predictors. The matrix contains training feature vectors each with dimensionality , and contains predictors each of dimensionality ℓ. Explicitly, the matrices are:
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| {'fraction_non_alphanumeric': 0.048892944038929444, 'fraction_numerical': 0.03291970802919708, 'mean_word_length': 4.309113220952012, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 6, 'https://': 1, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Models that connect galaxy and halo properties often summarize a halo's mass accretion history (MAH) with a single value, and use this value as the basis for predictions. However, a single-value summary fails to capture the complexity of MAHs and information can be lost in the process. We present MultiCAM, a generalization of traditional abundance matching frameworks, which can simultaneously connect the full MAH of a halo with multiple halo and/or galaxy properties. As a first case study, we apply MultiCAM to the problem of connecting dark matter halo properties to their MAHs in the context of a dark matter-only simulation. While some halo properties, such as concentration, are more strongly correlated to the early-time mass growth of a halo, others, like the virial ratio, have stronger correlations with late-time mass growth. This highlights the necessity of considering the impact of the entire MAH on halo properties. For most of the halo properties we consider, we find that MultiCAM models that use the full MAH achieve higher accuracy than conditional abundance matching models which use a single epoch. We also demonstrate an extension of MultiCAM that captures the covariance between predicted halo properties. This extension provides a baseline model for applications where the covariance between predicted properties is important.", 'arxivid': '2302.01346', 'author': ['Ismael Mendoza \nDepartment of Physics\nThe University of Michigan\n48109Ann ArborMIUSA\n', 'Philip Mansfield \nKavli Institute of Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA\n\nSLAC National Accelerator Laboratory\n94025Menlo ParkCAUSA\n', 'Kuan Wang \nDepartment of Physics\nThe University of Michigan\n48109Ann ArborMIUSA\n\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church St48109Ann ArborMIUSA\n', 'Camille Avestruz \nDepartment of Physics\nThe University of Michigan\n48109Ann ArborMIUSA\n\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church St48109Ann ArborMIUSA\n'], 'authoraffiliation': ['Department of Physics\nThe University of Michigan\n48109Ann ArborMIUSA', 'Kavli Institute of Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA', 'SLAC National Accelerator Laboratory\n94025Menlo ParkCAUSA', 'Department of Physics\nThe University of Michigan\n48109Ann ArborMIUSA', 'Leinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church St48109Ann ArborMIUSA', 'Department of Physics\nThe University of Michigan\n48109Ann ArborMIUSA', 'Leinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church St48109Ann ArborMIUSA'], 'corpusid': 256598208, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23564, 'n_tokens_neox': 20006, 'n_words': 12871, 'pdfsha': 'ee2145733b28dadfa80c266ecbef14a000b13ade', 'pdfurls': ['https://export.arxiv.org/pdf/2302.01346v1.pdf'], 'title': ['MultiCAM: A multivariable framework for connecting the mass accretion history of haloes with their properties', 'MultiCAM: A multivariable framework for connecting the mass accretion history of haloes with their properties'], 'venue': ['MNRAS']} |
arxiv |
ELEMENTARY PROOFS OF INFINITE FAMILIES OF CONGRUENCES FOR MERCA'S CUBIC PARTITIONS
24 Aug 2022
Robson Da
Silva
James A Sellers
ELEMENTARY PROOFS OF INFINITE FAMILIES OF CONGRUENCES FOR MERCA'S CUBIC PARTITIONS
24 Aug 2022
Recently, using modular forms and Smoot's Mathematica implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all n ≥ 0,Here A(n) is closely related to the function which counts the number of cubic partitions, partitions wherein the even parts are allowed to appear in two different colors. Indeed, A(n) is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts.In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non-nested Ramanujan-like congruences modulo 3 satisfied by A(n) wherein Merca's original two congruences serve as the initial members of each family.
Introduction
In a recent work, Merca [4] extensively studied the function which he called A(n) which is defined to be the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts. (Cubic partitions were introduced by Chan [2,3] to be integer partitions in which even parts are allowed to appear in two different colors. Chen introduced these in connection with Ramanujan's cubic continued fraction.) Merca notes that the generating function for A(n) is given by
∞ n=0 A(n)q n = (q; q 2 ) ∞ (q 2 ; q 4 ) ∞ = f 1 f 4
where the q-Pochhammer symbol (a; q) ∞ is defined by
(a; q) ∞ = (1 − a)(1 − aq)(1 − aq 2 )(1 − aq 3 ) . . . and f b a is defined by f b a = (q a ; q a ) b ∞
for positive integers a and any integer b.
In [4], Merca proved the following two Ramanujan-like congruences satisfied by A(n) :
A(9n + 5) ≡ 0 (mod 3),(1)A(27n + 26) ≡ 0 (mod 3).(2)
Merca's proof of these two congruences relies solely on Smoot's Mathematica implementation of Radu's algorithm for proving partition congruences (which relies heavily on the machinery of modular forms). Indeed, Merca's proof of (1) involves finding a generating function for A(9n + 5) which is produced by Smoot's Mathematica package. In this case, the generating function in question turns out to be a non-trivial linear combination of four ratios of eta products (in essence, ratios of products whose terms involve q-Pochhammer symbols as defined above), while the generating function for A(27n + 26) contains a dozen such terms.
Our goal in the work below is to provide elementary proofs of Merca's two congruences via classical generating function manipulations and dissections. We then significantly extend Merca's work on such divisibility properties for A(n) by proving two infinite families of non-nested Ramanujan-like congruences modulo 3 satisfied by A(n) wherein Merca's two congruences above serve as the initial members of each family. Indeed, we will prove the following:
For all j ≥ 0 and all n ≥ 0,
A 9 j+1 n + 39 · 9 j + 1 8 ≡ 0 (mod 3)
and A 3 · 9 j+1 n + 23 · 9 j+1 + 1 8 ≡ 0 (mod 3).
In order to accomplish the above goals, we require a few classical tools. First, we recall Ramanujan's theta functions
f (a, b) := ∞ n=−∞ a n(n+1) 2 b n(n−1) 2 , for |ab| < 1, φ(q) := f (q, q) = ∞ n=−∞ q n 2 = (q 2 ; q 2 ) 5 ∞ (q; q) 2 ∞ (q 4 ; q 4 ) 2 ∞ , ψ(q) := f (q, q 3 ) = ∞ n=0 q n(n+1)/2 = (q 2 ; q 2 ) 2 ∞ (q; q) ∞ .
These functions satisfy many interesting properties (see Entries 18, 19, and 22 in [1]), including:
φ(−q) = (q; q) 2 ∞ (q 2 ; q 2 ) ∞ , ψ(−q) = (q; q) ∞ (q 4 ; q 4 ) ∞ (q 2 ; q 2 ) ∞ .
As noted above, the generating function for A(n) is given by
∞ n=0 A(n)q n = (q; q 2 ) ∞ (q 2 ; q 4 ) ∞ = f 1 f 4 = f 2 1 f 2 f 2 f 1 f 4 = φ(−q) ψ(−q) .(3)
In order to prove the congruences in question, we require a few well-known q-series dissections.
Lemma 1.2. We have φ(−q) = f 2 9 f 18 − 2q f 3 f 2 18 f 6 f 9 .
Proof. A proof of this identity can be seen in [5, Eq. 14.3.4]. These are all of the tools that we need in order to prove Merca's congruences in an elementary fashion. We now transition to providing these proofs. Dividing by q 2 and replacing q 3 by q yields
Lemma 1.3. We have 1 ψ(−q) = f 9 18 f 2 3 f 3 9 f 2 12 f 3 36 + q f 2 6 f 3 18 f 3 3 f 3 12 + q 2 f 4 6 f 3∞ n=0 A(3n + 2)q n = f 4 2 f 5 3 f 3 12 f 4 1 f 4 4 f 4 6 − 2 f 2 f 5 6 f 2 1 f 3 f 3 4 ≡ f 2 f 1 f 4 f 4 3 f 2 12 f 3 6 − 2 f 2 f 2 1 f 5 6 f 3 f 12 (mod 3).(4)
Using Lemmas 1.3 and 1.4 we extract the terms of the form q 3n+1 from both sides of the last congruence to obtain
∞ n=0 A(9n + 5)q 3n+1 ≡ q f 3 f 3 18 f 6 f 12 − 4q f 8 6 f 3 9 f 8 3 f 12 (mod 3) ≡ q f 3 f 3 18 f 6 f 12 − 4q f 3 f 6 f 12 f 9 6 f 3 9 f 9 3 (mod 3) ≡ −3q f 3 f 3 18 f 6 f 12 (mod 3) ≡ 0 (mod 3),
which proves (1). Using Lemmas 1.3 and 1.4 we extract the terms of the form q 3n+2 from both sides of (4):
∞ n=0 A(9n + 8)q 3n+2 ≡ q 2 f 6 f 3 9 f 3 36 f 2 12 f 3 18 − 8q 2 f 7 6 f 3 18 f 7 3 f 12 (mod 3).
Dividing by q 2 and replacing q 3 by q, we obtain
∞ n=0 A(9n + 8)q n ≡ f 2 f 3 3 f 3 12 f 2 4 f 3 6 − 8 f 7 2 f 3 6 f 7 1 f 4 (mod 3) ≡ f 2 f 2 4 f 3 3 f 3 12 f 3 6 − 8 f 2 f 1 f 4 f 5 6 f 2 3 (mod 3).(5)
Now we employ Lemmas 1.3 and 1.5 to extract the terms of the form q 3n+2 from the congruence above. The resulting congruence after division by q 2 and replacing q 3 by q is given by
∞ n=0 A(27n + 26)q n ≡ − f 3 1 f 3 12 f 4 4 − 8 f 9 2 f 3 3 f 3 12 f 6 1 f 4 4 f 3 6 (mod 3) ≡ − f 3 1 f 3 12 f 4 4 − 8 f 3 1 f 3 12 f 4 4 (mod 3) ≡ 0 (mod 3),
which proves (2).
Infinite families of congruences modulo 3
While the elementary proofs of Merca's original congruences that we have provided above are very satisfying, it turns out that much more is true about A(n) modulo 3, and our elementary approach to proving these congruences yields the insights needed in order to see this. In this light, we now proceed to proving two infinite families of non-nested Ramanujan-like congruences modulo 3 satisfied by A(n).
We begin by proving the following internal congruence satisfied by A(n) which will serve as an important component in our remaining proofs. for all n ≥ 0 as well. Note that 9 j+2 n + 39 · 9 j+1 + 1 8 = 27 3 · 9 j n + 13 · 9 j 8 − 7 24 + 8.
Thanks to (6), we know that, for all n ≥ 0,
A(27n + 8) ≡ A(3n + 1) (mod 3).
Thus,
A 9 j+2 n + 39 · 9 j+1 + 1 8 = A 27 3 · 9 j n + 13 · 9 j 8 − 7 24 + 8 ≡ A 3 3 · 9 j n + 13 · 9 j 8 − 7 24 + 1 (mod 3)
= A 9 j+1 n + 39 · 9 j + 1 8 ≡ 0 (mod 3)
by the induction hypothesis. The result follows. Theorem 3.3. For all j ≥ 0 and all n ≥ 0, A 3 · 9 j+1 n + 23 · 9 j+1 + 1 8 ≡ 0 (mod 3).
Proof. As above, this theorem follows from a straightforward proof by induction on j. Note that the basis case, j = 0, is simply the statement that, for all n ≥ 0,
A(27n + 26) ≡ 0 (mod 3).
This is Theorem 1.1 equation (2) above, the second of Merca's original congruences. Next, we assume that the statement is true for some j ≥ 0. We then wish to prove that A 3 · 9 j+2 n + 23 · 9 j+2 + 1 8 ≡ 0 (mod 3) for all n ≥ 0 as well. Note that 3 · 9 j+2 n + 23 · 9 j+2 + 1 8 = 27 9 j+1 n + 3(23) · 9 j 8 − 7 24 + 8.
Thanks to (6), we know that, for all n ≥ 0, A(27n + 8) ≡ A(3n + 1) (mod 3).
Thus,
A 3 · 9 j+2 n + 23 · 9 j+2 + 1 8 = A 27 9 j+1 n + 3(23) · 9 j 8 − 7 24 + 8 ≡ A 3 9 j+1 n + 3(23) · 9 j 8 − 7 24 + 1 (mod 3)
= A 3 · 9 j+1 n + 23 · 9 j+1 + 1 8 ≡ 0 (mod 3) by the induction hypothesis. The result follows.
. 6 ,.
6Proof. A proof of this result can be seen in [TheoremProof. A proof of this result can be seen in [7, Lemma 2.2].
2 .
2Elementary proof of Theorem 1.1 Initially, we use Lemmas 1.2 and 1.3 to extract the terms involving q 3n+2 in (3): ∞ n=0 A(3n + 2)q 3n+2 = q 2 f
Theorem 1.1 ([4, Theorem 1.10]). For all n ≥ 0,
Proof. We prove this result by showing that both A(27n + 8) and A(3n + 1) have the same generating function modulo 3. Using Lemmas 1.3 and 1.5 we extract the terms of the form q 3n from(5):which after replacement of q 3 by q yieldsThe generating function for A(3n + 1) is obtained in the same way using (3) and Lemmas 1.2 and 1.3:Dividing this expression by q and replacing q 3 by q, we are left withwhich coincides with (7) and the proof is complete.We can now prove the following two theorems.Theorem 3.2. For all j ≥ 0 and all n ≥ 0,A 9 j+1 n + 39 · 9 j + 1 8 ≡ 0 (mod 3).Proof. This theorem follows from a straightforward proof by induction on j. First, we note that the basis case, j = 0, is simply the statement that, for all n ≥ 0,This is Theorem 1.1 equation (1) above, the first of Merca's original congruences. Next, we assume that the statement is true for some j ≥ 0. We then wish to prove that A 9 j+2 n + 39 · 9 j+1 + 1 8 ≡ 0 (mod 3)
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Arithmetic properties of partitions with odd parts distinct. M D Hirschhorn, J A Sellers, Ramanujan J. 22M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct. Ramanujan J. 22 (2010), 273-284.
Ramanujan type identities and congruences for partition pairs. P C Toh, Discrete Math. 312P. C. Toh, Ramanujan type identities and congruences for partition pairs. Discrete Math. 312 (2012), 1244-1250.
| {'fraction_non_alphanumeric': 0.08303317535545024, 'fraction_numerical': 0.07345971563981042, 'mean_word_length': 3.1392703020792467, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 36, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Recently, using modular forms and Smoot's Mathematica implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all n ≥ 0,Here A(n) is closely related to the function which counts the number of cubic partitions, partitions wherein the even parts are allowed to appear in two different colors. Indeed, A(n) is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts.In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non-nested Ramanujan-like congruences modulo 3 satisfied by A(n) wherein Merca's original two congruences serve as the initial members of each family.", 'arxivid': '2208.11249', 'author': ['Robson Da ', 'Silva ', 'James A Sellers '], 'authoraffiliation': [], 'corpusid': 251765342, 'doi': '10.1007/s11139-022-00660-7', 'github_urls': [], 'n_tokens_mistral': 4304, 'n_tokens_neox': 3494, 'n_words': 2201, 'pdfsha': '12f07e360dca00c00182dade84a1eb08099f2b00', 'pdfurls': ['https://export.arxiv.org/pdf/2208.11249v1.pdf'], 'title': ["ELEMENTARY PROOFS OF INFINITE FAMILIES OF CONGRUENCES FOR MERCA'S CUBIC PARTITIONS", "ELEMENTARY PROOFS OF INFINITE FAMILIES OF CONGRUENCES FOR MERCA'S CUBIC PARTITIONS"], 'venue': []} |
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