Differential Equations for Sine-Gordon Correlation Functions at the Free Fermion Point

TL;DR

This paper shows that at the free fermion point, sine-Gordon correlation functions can be described by a sinh-Gordon-like equation, using form factors and Fredholm determinants, providing new insights into integrable models.

Contribution

It introduces a novel method to derive differential equations for sine-Gordon correlation functions at the free fermion point via Fredholm determinants and a $ ext{Z}_2$ graded multiplication law.

Findings

Correlation functions are parameterized by solutions to a sinh-Gordon-like equation.

A new proof of differential equations for Ising model correlators is provided.

The approach uses form factors and integral operator properties to establish the results.

Abstract

We demonstrate that for the sine-Gordon theory at the free fermion point, the 2-point correlation functions of the fields $\exp (i\al \Phi )$ for $0< \al < 1$ can be parameterized in terms of a solution to a sinh-Gordon-like equation. This result is derived by summing over intermediate multiparticle states and using the form factors to express this as a Fredholm determinant. The proof of the differential equations relies on a $\Zmath_2$ graded multiplication law satisfied by the integral operators of the Fredholm determinant. Using this methodology, we give a new proof of the differential equations which govern the spin and disorder field correlators in the Ising model.