Random Matrix Theory and Classical Statistical Mechanics: Spin Models

TL;DR

This paper uses spectral analysis of transfer matrices in classical lattice spin models to identify integrability and criticality, providing a numerical criterion and exploring complex spectra in these systems.

Contribution

It introduces a spectral statistical method to distinguish integrable from non-integrable models and applies it to classical spin models, including the Ising and Potts models.

Findings

Spectral properties serve as criteria for integrability.

Method distinguishes integrability from criticality.

Eigenvalue independence persists in complex spectra.

Abstract

We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper. We show that the statistical properties of these spectra can serve as a criterion of integrability. It provides also an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.