Spin chains of Haldane-Shastry type and a generalized central limit theorem

TL;DR

This paper demonstrates that the energy level density of certain quantum spin chains approaches a Gaussian distribution as the system size grows, using a modified central limit theorem tailored for models with known partition functions.

Contribution

It introduces a novel version of the central limit theorem applicable to specific quantum models and explains the asymptotic Gaussian behavior of Haldane-Shastry spin chains.

Findings

Energy level density tends to Gaussian in large systems

First theoretical explanation for Haldane-Shastry spin chains' level density

Modified central limit theorem applicable to models with explicit partition functions

Abstract

We show that the density of energy levels of a wide class of finite-dimensional quantum systems tends to a Gaussian distribution as the number of degrees of freedom increases. Our result is based on a nontrivial modification of the classical central limit theorem, and is especially suited to models whose partition function is explicitly known. In particular, we provide the first theoretical explanation of the fact that the level density of several spin chains of Haldane-Shastry type is asymptotically Gaussian when the number of sites tends to infinity.