Integrable field theory and critical phenomena. The Ising model in a magnetic field

TL;DR

This paper reviews the application of integrable quantum field theory to the two-dimensional Ising model in a magnetic field, highlighting exact solutions, correlation functions, and universal ratios, and emphasizing the generality of these methods for critical phenomena.

Contribution

It provides a comprehensive review of the integrable quantum field theory approach to the Ising model in a magnetic field, including scattering theory and form factors, with insights applicable to other universality classes.

Findings

Exact scattering solutions for the Ising model in a magnetic field

Calculation of correlation functions and universal amplitude ratios

Comparison of field theory results with lattice numerical estimates

Abstract

The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the forties, exact results for the magnetic case have been missing until the late eighties, when A.Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this article we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.