Level density distribution for one-dimensional vertex models related to Haldane-Shastry like spin chains

TL;DR

This paper demonstrates that the energy level density distributions of a broad class of one-dimensional vertex models, including those related to Haldane-Shastry spin chains, asymptotically follow a Gaussian distribution as the number of vertices increases.

Contribution

It extends previous work by analyzing vertex models with polynomial energy functions of arbitrary degree, establishing their asymptotic Gaussian level density distribution.

Findings

Level density distributions follow a Gaussian pattern for large vertex numbers.

Analytical estimates of mean and variance support the Gaussian behavior.

Numerical evidence confirms the theoretical predictions.

Abstract

The energy level density distributions of some Haldane-Shastry like spin chains associated with the $A_{N-1}$ root system have been computed recently by Enciso et al., exploiting the connection of these spin systems with inhomogeneous one-dimensional vertex models whose energy functions depend on the vertices through specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions depend on the vertices through arbitrary polynomials of any possible degree. We estimate the order of mean and variance for such energy functions and show that the level density distribution of all vertex models belonging to this class asymptotically follow the Gaussian pattern for large number of vertices. We also present some numerical evidence in support of this analytical result.