Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model

TL;DR

This paper connects isomonodromic deformation theory with the computation of next-to-diagonal correlations in the anisotropic square lattice Ising model, extending previous work on diagonal correlations through bi-orthogonal polynomial methods.

Contribution

It demonstrates that next-to-diagonal correlations can be expressed via isomonodromic systems and bi-orthogonal polynomials, providing a new approach to evaluate these correlations.

Findings

Next-to-diagonal correlations are given as elements of an isomonodromic system.

Correlations are represented as Cauchy-Hilbert transforms of bi-orthogonal polynomials.

The approach extends the isomonodromy framework to off-diagonal correlations.

Abstract

In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $\tau$-function of the sixth Painlev\'e system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal polynomials.