The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

TL;DR

This paper expresses the diagonal two-point correlations of the Ising model on an anisotropic triangular lattice using a Garnier system, linking statistical mechanics with advanced integrable systems and deriving nonlinear recurrence relations.

Contribution

It establishes a novel connection between Ising model correlations and Garnier systems, extending the theory of Painlevé equations to a three-variable setting with explicit recurrence relations.

Findings

Correlations characterized by a Garnier system.

Derivation of nonlinear recurrence relations for correlations.

Extension of discrete Painlevé V to a Garnier framework.

Abstract

The diagonal spin-spin correlations $ \langle \sigma_{0,0}\sigma_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlev\'e system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathbb{Z}_{\geq 0} $. These later recurrence relations are an example of the discrete Garnier equations which, in turn, are extensions to the "discrete Painlev\'e V" system.