Holonomy of the Ising model form factors

Boukraa, S.; Hassani, S.; Maillard, J.‐M.; McCoy, Barry M.; Orrick, William P.; Zenine, N.

doi:10.1088/1751-8113/40/1/005

Cited by 33 publications

(224 citation statements)

References 49 publications

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“…This deep relation between elliptic curves and Painlevé VI explains the occurrence of Painlevé VI on the Ising model, and on other lattice Yang-Baxter integrable models which are canonically parametrized in term of elliptic functions (like the eight-vertex Baxter model, the RSOS models, see for instance [15]). One can see in Section 6 of [25], other examples of this deep connection between the transcendent solutions of Painlevé VI and the theory of elliptic functions, modular curves and quasi-modular functions. Along this line one should note that other linear differential operators, not straightforwardly linked to L E but more generally to the theory of elliptic functions and modular forms (quasimodular forms .…”

Section: The Elliptic Representation Of Painlevé VImentioning

confidence: 97%

“…The expressions (or forms) of the linear differential operators L 5 (N ), L 7 (N ) and L 9 (N ) are given in [25].…”

Section: Fuchsian Linear Dif Ferential Equations Formentioning

confidence: 99%

“…where L 2 (N ) is the linear differential operator for f (1) N,N and where the fourth order operator M 4 (N ) is displayed in [25] for successive values of N . One remarks on these successive expressions that the degree of each polynomial occurring in these linear differential operators M 4 (N ) grows linearly with N .…”

Section: Direct Sum Structurementioning

confidence: 99%

“…Other examples are given in [25]. [121] and for a sketch of how the algebro-differential structures generalize in that anisotropic case [26]).…”

Section: Equivalence Of Variousmentioning

confidence: 99%

“…Let us first recall the form factors [25] of the lattice Ising model. Our starting point will be the expansions of the diagonal correlations in an exponential form [123], both for T < T c where E h and E v are the horizontal and vertical interaction energies of the Ising model.…”

Section: Introductionmentioning

confidence: 99%

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From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

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Abstract. We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a "Russian-doll" nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral E). The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure, the "scaled" linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the n-particle contributions χ (n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for n = 1, 2, 3, 4 and, only modulo a prime, for n = 5 and 6, thus providing a large set of (possible) new singularities of the χ (n) . We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w 2 = 0, that occurs in the linear differential equation of χ (3) , actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.

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Section: The Elliptic Representation Of Painlevé VImentioning

confidence: 97%

“…The expressions (or forms) of the linear differential operators L 5 (N ), L 7 (N ) and L 9 (N ) are given in [25].…”

Section: Fuchsian Linear Dif Ferential Equations Formentioning

confidence: 99%

Section: Direct Sum Structurementioning

confidence: 99%

“…Other examples are given in [25]. [121] and for a sketch of how the algebro-differential structures generalize in that anisotropic case [26]).…”

Section: Equivalence Of Variousmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

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The Romance of the Ising Model

McCoy

2013

Springer Proceedings in Mathematics &Amp; Statistics

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The essence of romance is mystery. In this talk, given in honor of the 60th birthday of Michio Jimbo, I will explore the meaning of this for the Ising model beginning in 1946 with Bruria Kaufman and Willis Lamb, continuing with the wedding by Jimbo and Miwa in 1980 of the Ising model with the Painlevé VI equation which had been first discovered by Picard in 1889. I will conclude with the current fascination of the magnetic susceptibility and explore some of the mysteries still outstanding.

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A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

Forrester

Rains

2011

Commun. Math. Phys.

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We characterize averages of N l=1 |x − t l | α−1 with respect to the Selberg density, further contrained so that t l ∈ [0, x] (l = 1, . . . , q) and t l ∈ [x, 1] (l = q + 1, . . . , N ), in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the t j (j = 1, . . . , N ). In the case q = 0 and α < 1 we compute the explicit leading order term in the x → 0 asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case q = 0 and α ∈ Z + , and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.

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Holonomy of the Ising model form factors

Cited by 33 publications

References 49 publications

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

The Romance of the Ising Model

A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

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