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Ising Correlations and Elliptic Determinants

N. Z. Iorgov, Nikolai Iorgov

Published Jan 1, 2011• 6 citations
OpenAlex

Abstract

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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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.
We consider the general -symmetric free-fermion model on the finite periodic lattice, which includes as special cases the Ising model on the square and triangular lattices and the -symmetric BBS τ (2) -model with n = 2. Translating Kaufman's fermionic approach to diagonalization of Ising-like transfer matrices into the language of Grassmann integrals, we determine the transfer matrix eigenvectors and observe that they coincide with the eigenvectors of a square lattice Ising transfer matrix. This allows us to find exact finite-lattice form factors of spin operators for the statistical model and the associated finite-length quantum chains, of which the most general is equivalent to the XY chain in a transverse field.
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length N∈2Z≥2N\in 2Z_{\ge 2}N∈2Z≥2​. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to 2\sqrt{2}2​. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient G(12)G(32)G(\frac{1}{2})G(\frac{3}{2})G(21​)G(23​) appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.
Explicit factorized formulas for the matrix elements (form factors) of the spin operators σ x and σ y between the eigenvectors of the Hamiltonian of the finite quantum periodic XY -chain in a transverse field were derived. The derivation is based on the relations between three models: the model of quantum XY -chain, Ising model on 2D lattice and N = 2 Baxter–Bazhanov–Stroganov τ (2) -model. Due to these relations we transfer the formulas for the form factors of the latter model recently obtained by the use of separation of variables method to the model of quantum XY -chain. Hopefully, the formulas for the form factors will help in analysis of multipoint dynamic correlation functions at a finite temperature. As an example, we re-derive the asymptotics of the two-point correlation function in the disordered phase without the use of the Toeplitz determinants and the Wiener–Hopf factorization method.
We study U (1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions reduce to form factors of the exponential fields of the sine-Gordon model at the free-fermion point.