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

TL;DR

This article surveys how integrable quantum field theory methods illuminate the Ising universality class in two dimensions, focusing on the magnetic-field case solved by Zamolodchikov in the scaling limit.It explains how the IQFT framework—rooted in conserved currents, elastic, factorised scattering, and the form-factor bootstrap—yields exact results for correlation functions, mass spectra, and universal amplitude ratios, and it demonstrates consistency with lattice results where available.The discussion covers the two integrable directions (zero field and Tc with nonzero field), the identification of scaling operators σ and ε, and the construction of the eight-particle E8 spectrum, culminating in a detailed map between continuum predictions and lattice observables.Beyond integrability, the work outlines perturbative approaches to non-integrable deformations, including mass and energy shifts and confinement phenomena, thereby highlighting the broader applicability of the integrable framework to scaling regions of two-dimensional critical systems.

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.

Figures (9)

• Figure 1: Coupling space of the Ising field theory (\ref{['A']}). The oriented lines indicate some renormalisation group trajectories flowing out of the critical point at the origin. The integrable directions coincide with the principal axes. In our conventions $\tau>0$ corresponds to $T>T_c$.
• Figure 2: Space-time diagrams for a three-particle scattering process. A genuine three-body collision (a) or a sequence of widely separated two-body collisions (b) can be obtained by a suitable choice of initial conditions on the wave packets.
• Figure 3: Space-time diagram associated to the scattering amplitude $S_{ab}(\theta_1-\theta_2)$; time runs upwards.
• Figure 4: Simple pole diagram associated to equation (\ref{['pole']}).
• Figure 5: Pictorial representation of the bootstrap equation (\ref{['bootstrap']}).