Random Matrix Theory and Classical Statistical Mechanics. I. Vertex Models

TL;DR

This paper explores the spectral properties of the asymmetric eight-vertex model using random matrix theory, revealing distinct statistical behaviors linked to integrability and identifying algebraic varieties related to free parafermions.

Contribution

It establishes a connection between integrability and spectral statistics, and identifies algebraic varieties where free parafermions may occur in the model.

Findings

Poissonian level statistics in integrable cases

Wigner-like level repulsion in non-integrable models

Identification of algebraic varieties related to free parafermions

Abstract

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.