Two-dimensional Ising correlations: Convergence of the scaling limit

Palmer, John; Tracy, Craig A.

doi:10.1016/0196-8858(81)90010-5

Cited by 68 publications

(382 citation statements)

References 57 publications

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“…The form factor expressions for C(M, N ) of [9,10,11,26,27,14] are obtained by expanding the exponentials in (3), and (5), in the form given in [9] as multiple integrals and integrating over half the variables. The form of the result depends on whether the even, or odd, variables of [9] are integrated out.…”

Section: New Integral Representations For Thementioning

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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Section: New Integral Representations For Thementioning

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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“…The n-particle contribution to the magnetic susceptibility of the Ising model have integral representations. They are given by (n − 1)-dimensional integrals [2, 17,18] that read (omitting the prefactor)…”

Section: Introductionmentioning

confidence: 99%

Landau singularities and singularities of holonomic integrals of the Ising class

Boukraa¹,

Hassani²,

Maillard³

et al. 2007

J. Phys. A: Math. Theor.

159

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Abstract.We consider families of multiple and simple integrals of the "Ising class" and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable w = s/2/(1 + s 2 ), with s = sinh(2K), where K = J/kT is the usual Ising model coupling constant. Switching to the variable s, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A 38 (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model χ (3) at the points 1+3w +4w 2 = 0 and show that χ (3) (s) is not singular at the corresponding points inside the unit circle |s| = 1, while its analytical continuation in the variable s is actually singular at the corresponding points 2 + s + s 2 = 0 oustside the unit circle (|s| > 1).

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“…It clearly satisfies translation invariance constraint (2.14). In the thermodynamic limit ξ T → 1, ν θ → 0, the spectra of quasimomenta become continuous and (6.6) reproduces infinite-lattice form factors found in [27,29]. In particular, considering the N → ∞ limit of (5.47), it is straightforward to recover Yang's formula σ = √ ξ for the spontaneous magnetization [33].…”

Section: Multiparticle Form Factorsmentioning

confidence: 87%

“…The problem can be circumvented on the infinite lattice (N → ∞), where the distinction between two types of fermions effectively disappears. In that case, two-particle Ising spin form factors have been found in [29], Theorems 5.2.1-5.2.3, and multiparticle ones in [27], Theorem 5.0.…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

Ising Correlations and Elliptic Determinants

Iorgov¹,

Lisovyy²

2011

J Stat Phys

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Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors -matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.

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Two-dimensional Ising correlations: Convergence of the scaling limit

Cited by 68 publications

References 57 publications

Holonomy of the Ising model form factors

Holonomy of the Ising model form factors

Landau singularities and singularities of holonomic integrals of the Ising class

Ising Correlations and Elliptic Determinants

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