Ising correlations and elliptic determinants

TL;DR

This paper presents a novel approach to calculating correlation functions in the 2D Ising model using elliptic Cauchy matrices and theta functions, providing a new proof of known form factor formulas.

Contribution

It introduces an explicit solution to the inversion of elliptic Cauchy matrices arising in Ising model form factors, offering a new proof of factorized formulas.

Findings

Explicit inversion of elliptic Cauchy matrices is achieved.

New proof of the factorized form factor formulas for the Ising model.

Connection between form factors and elliptic determinants is established.

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.