Scaling functions in the square Ising model

TL;DR

This paper derives linear differential operators for integrals in the two-point correlation function of the Ising model, revealing solutions related to Painlevé VI and analyzing their algebraic properties.

Contribution

It introduces explicit differential operators for the integrals in the Ising model's correlation functions and explores their solutions and Galois groups.

Findings

The integrals satisfy specific linear differential equations of order q.

The solution r^{1/4} exp(r^2/8) relates to Painlevé VI in the scaling limit.

The differential Galois groups of the operators are characterized.

Abstract

We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $ F_{\pm}(r)= \lim_{scaling} {\cal M}_{\pm}^{-2} < \sigma_{0,0} \, \sigma_{M,N}> = \sum_{n} I_{n}(r)$. The integrals $ I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $ r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlev\'e VI equation in the scaling limit. Combinations of the (analytic at $ r= 0$) solutions of $ {\cal L}^{scal}_q$ sum to $ \exp(r^2/8)$. We show that the expression $ r^{1/4} \exp(r^2/8)$ is the scaling limit of the correlation function $ C(N, N)$ and $ C(N, N+1)$. The differential Galois groups of the factors occurring in the operators $ {\cal L}^{scal}_q$ are given.