From Form Factors to Correlation Functions: The Ising Model

TL;DR

This paper presents a new proof that the correlation functions in the Ising model satisfy Painlevé III equations and connects their generating functions to soliton solutions of the sinh-Gordon hierarchy.

Contribution

It introduces a simplified proof linking Ising model correlations to Painlevé equations and relates generating functions to soliton tau-functions, expanding understanding of integrable structures.

Findings

Correlation functions obey Painlevé III equations

Generating functions are N-soliton tau-functions of sinh-Gordon hierarchy

Connection to isomonodromy deformation problems

Abstract

Using exact expressions for the Ising form factors, we give a new very simple proof that the spin-spin and disorder-disorder correlation functions are governed by the Painlev\'e III non linear differential equation. We also show that the generating function of the correlation functions of the descendents of the spin and disorder operators is a $N$-soliton, $N\to\infty$, $\tau$-function of the sinh-Gordon hierarchy. We discuss a relation of our approach to isomonodromy deformation problems, as well as further possible generalizations.