The anisotropic Ising correlations as elliptic integrals: duality and differential equations

TL;DR

This paper expresses anisotropic Ising model correlations as elliptic integrals, extends duality to anisotropic cases, and derives simple factorized differential equations governing these correlations.

Contribution

It provides a novel representation of anisotropic Ising correlations using elliptic integrals and extends duality and differential equations to the anisotropic case.

Findings

Correlation functions are homogeneous polynomials of elliptic integrals.

Exact dual transformations for anisotropic correlations are established.

Linear differential operators are factorized into simple components.

Abstract

We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorised in a very simple way, in operators of decreasing orders.