Fredholm Determinant evaluations of the Ising Model diagonal correlations and their \lambda - generalisation

TL;DR

This paper presents a novel Fredholm determinant approach to evaluate the diagonal correlations in the 2D Ising model, revealing new integral and summation representations and their -generalizations.

Contribution

It introduces two equivalent Fredholm determinant formulations for Ising correlations, extending classical methods with scattering theory and bi-orthogonal polynomial systems.

Findings

Demonstrates equivalence of two Fredholm determinant representations

Provides -parameter generalizations of form factors

Employs advanced scattering theory techniques

Abstract

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants - one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural \lambda - parameter generalisation and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case, applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi-orthogonal situation.