Near-critical Ising, sine-Gordon at the free fermion point, and bosonization

TL;DR

This paper establishes a rigorous connection between the near-critical 2D Ising model and the sine-Gordon model at the free fermion point through bosonization, using complex analysis and perturbative techniques.

Contribution

It provides a mathematical proof linking Ising correlations to sine-Gordon at the free fermion point, extending bosonization to near-critical regimes.

Findings

Correlation functions match between models in perturbative regime

Analyticity in mass parameter enables rigorous comparison

Construction of massive holomorphic functions and Mayer expansion techniques

Abstract

In this article, we study the continuous correlations of the near-critical Ising model in two dimensions with plus boundary conditions, and prove that doubled correlation functions of primary fields (spin, disorder, fermions, energy) in the Ising model are given by correlation functions of the sine-Gordon model at the free fermion point. This is an instance of bosonization. The main ideas involve analyticity of correlation functions in a mass parameter in finite volume and proving that in a perturbative regime, the Taylor coefficients of the correlation functions match due to known bosonization results for the critical Ising model in terms of the Gaussian free field. The main techniques on the Ising side involve construction and precise estimates of certain massive holomorphic functions while on the sine-Gordon side, we control an iterated Mayer expansion with techniques going back to Brydges and Kennedy.