On the singularity structure of the 2D Ising model susceptibility

Nickel, B. G.

doi:10.1088/0305-4470/32/21/303

Cited by 88 publications

(496 citation statements)

References 17 publications

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“…2 This conjecture is verified by the free energy and by the susceptibility at h = 0 [10] and by the free energy F (0, h) [2]. Note that this is expected to be true only in the thermodynamic limit.…”

Section: Extrapolation To the Thermodynamic Limitmentioning

confidence: 82%

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High-precision estimate ofg₄in the 2D Ising model

Caselle

Hasenbusch

Pelissetto

et al. 2000

J. Phys. A: Math. Gen.

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Section: Extrapolation To the Thermodynamic Limitmentioning

confidence: 82%

“…We thank Alan Sokal for useful discussions and Bernie Nickel for sending us his unpublished addendum, Ref. [10]. This work was partially supported by the European Commission TMR programme ERBFMRX-CT96-0045.…”

mentioning

confidence: 98%

High-precision estimate ofg₄in the 2D Ising model

Caselle

Hasenbusch

Pelissetto

et al. 2000

J. Phys. A: Math. Gen.

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“…The motivation of introducing an isomonodromic parameter λ, in the framework of isomonodromy deformations, is, at first sight, quite different from the "coupling constant" motivation at the origin of the form factor λ-extensions (9) and (10). In sec.…”

Section: Introductionmentioning

confidence: 99%

Holonomy of the Ising model form factors

Boukraa¹,

Hassani²,

Maillard³

et al. 2006

J. Phys. A: Math. Theor.

217

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Abstract. We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f N,N is expressed polynomially in terms of the complete elliptic integrals E and K. The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure. The previous λ-extensions, C(N, N ; λ) are, for singledout values λ = cos(πm/n) (m, n integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

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“…The form factor expressions for the two-point correlation functions C(M, N ) of [87,88,94,93,123,124] are obtained by expanding the exponentials in (1.1), and (1.2), in the form given in [123] as multiple integrals, and integrating over half the variables. The form of the result depends on whether the even, or odd, variables of [123] are integrated out.…”

Section: Nn 'Smentioning

confidence: 99%

“…B. Nickel found [87,88] that each of these χ (n) 's is actually singular on a set of points located on the unit circle |s| = | sinh(2K)| = 1, where K = J/kT is the usual Ising model temperature variable. These singularities are located at solution points of the following equations…”

Section: Other N-fold Integrals Linked To the Susceptibility Of The Imentioning

confidence: 99%

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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On the singularity structure of the 2D Ising model susceptibility

Cited by 88 publications

References 17 publications

High-precision estimate ofg₄in the 2D Ising model

High-precision estimate ofg₄in the 2D Ising model

Holonomy of the Ising model form factors

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

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On the singularity structure of the 2D Ising model susceptibility

Cited by 88 publications

References 17 publications

High-precision estimate ofg4in the 2D Ising model

High-precision estimate ofg4in the 2D Ising model

Holonomy of the Ising model form factors

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

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High-precision estimate ofg₄in the 2D Ising model

High-precision estimate ofg₄in the 2D Ising model