The Polychronakos-Frahm spin chain of BC_N type and Berry-Tabor's conjecture

TL;DR

This paper analyzes the spectral properties of the BC_N type Polychronakos-Frahm spin chain, revealing unique spacing distributions that challenge the Berry-Tabor conjecture and showing Gaussian level density for large systems.

Contribution

It provides an exact partition function calculation and derives a simple analytic expression for the spacings distribution, highlighting the exceptional integrability of Haldane-Shastry type spin chains.

Findings

Level density follows a Gaussian distribution for large N.

Spacing distribution is neither Poisson nor Wigner, similar to Haldane-Shastry.

Analytic expression matches numerical data and applies to related chains.

Abstract

We compute the partition function of the su(m) Polychronakos-Frahm spin chain of BC_N type by means of the freezing trick. We use this partition function to study several statistical properties of the spectrum, which turn out to be analogous to those of other spin chains of Haldane-Shastry type. In particular, we find that when the number of particles is sufficiently large the level density follows a Gaussian distribution with great accuracy. We also show that the distribution of (normalized) spacings between consecutive levels is of neither Poisson nor Wigner type, but is qualitatively similar to that of the original Haldane-Shastry spin chain. This suggests that spin chains of Haldane-Shastry type are exceptional integrable models, since they do not satisfy a well-known conjecture of Berry and Tabor according to which the spacings distribution of a generic integrable system should be Poissonian. We derive a simple analytic expression for the cumulative spacings distribution of the BC_N-type Polychronakos-Frahm chain using only a few essential properties of its spectrum, like the Gaussian character of the level density and the fact the energy levels are equally spaced. This expression is in excellent agreement with the numerical data and, moreover, there is strong evidence that it can also be applied to the Haldane-Shastry and the Polychronakos-Frahm spin chains.