The Berry-Tabor conjecture for spin chains of Haldane-Shastry type

TL;DR

This paper investigates the level spacings distribution of Haldane-Shastry type spin chains, proposing it follows a specific 'square root of a logarithm' law, contrasting with the typical Poisson or Wigner distributions.

Contribution

It demonstrates that the spacings distribution of Haldane-Shastry type chains follows a unique 'square root of a logarithm' law, extending previous conjectures and analyzing the rational case.

Findings

The spacings distribution follows the 'square root of a logarithm' law.

The law is validated for the rational Haldane-Shastry chain.

Contrasts with Poisson and Wigner distributions for integrable and chaotic systems.

Abstract

According to a long-standing conjecture of Berry and Tabor, the distribution of the spacings between consecutive levels of a "generic'' integrable model should follow Poisson's law. In contrast, the spacings distribution of chaotic systems typically follows Wigner's law. An important exception to the Berry-Tabor conjecture is the integrable spin chain with long-range interactions introduced by Haldane and Shastry in 1988, whose spacings distribution is neither Poissonian nor of Wigner's type. In this letter we argue that the cumulative spacings distribution of this chain should follow the "square root of a logarithm'' law recently proposed by us as a characteristic feature of all spin chains of Haldane-Shastry type. We also show in detail that the latter law is valid for the rational counterpart of the Haldane-Shastry chain introduced by Polychronakos.