The importance of the Ising model

TL;DR

This paper reviews the significance of the two-dimensional Ising model in a magnetic field, highlighting its role in studying the relationship between integrable and non-integrable models, and discusses recent findings on magnetic susceptibility and potential limitations of scaling theory.

Contribution

It provides a comprehensive review of recent advances in understanding the Ising model's magnetic susceptibility and its implications for the completeness of scaling theory.

Findings

Revealed an unexpected natural boundary in magnetic susceptibility

Linked Fermionic representations with conformal characters

Suggested potential incompleteness of scaling theory for H ≠ 0

Abstract

Understanding the relationship which integrable (solvable) models, all of which possess very special symmetry properties, have with the generic non-integrable models that are used to describe real experiments, which do not have the symmetry properties, is one of the most fundamental open questions in both statistical mechanics and quantum field theory. The importance of the two-dimensional Ising model in a magnetic field is that it is the simplest system where this relationship may be concretely studied. We here review the advances made in this study, and concentrate on the magnetic susceptibility which has revealed an unexpected natural boundary phenomenon. When this is combined with the Fermionic representations of conformal characters, it is suggested that the scaling theory, which smoothly connects the lattice with the correlation length scale, may be incomplete for $H \neq 0$.