Spectral Properties of Statistical Mechanics Models

TL;DR

This paper investigates the spectral properties of the eight-vertex model's transfer matrices, revealing different eigenvalue behaviors in integrable and non-integrable regimes, and suggesting parallels with quantum spin systems.

Contribution

It provides the first numerical analysis of eigenvalue statistics in classical statistical mechanics models, confirming conjectures about integrability and spectral behavior.

Findings

Eigenvalue repulsion in non-integrable regimes

Poissonian eigenvalue distribution in integrable regimes

Level clustering observed at some points

Abstract

The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analyzed. In non-integrable regimes we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in random matrix theory. By contrast, in integrable regimes we have found eigenvalue independence leading to a Poissonian behavior, and, for some points, level clustering. These first examples from classical statistical mechanics suggest that the conjecture of integrability successfully applied to quantum spin systems also holds for classical systems.