A new perspective on the integrability of Inozemtsev's elliptic spin chain

TL;DR

This paper investigates the integrability of Inozemtsev's elliptic spin chain using spectral statistics, confirming its integrability and similarities to the Haldane-Shastry chain, with analytical and numerical evidence on its spectral properties.

Contribution

It provides a novel spectral analysis approach to confirm the integrability of Inozemtsev's elliptic spin chain and compares its properties to related models.

Findings

Spectral statistics support the integrability of Inozemtsev's chain.

The level density is asymptotically Gaussian as the number of spins increases.

Analytical calculations show the spectrum's mean and standard deviation match those of the Haldane-Shastry chain.

Abstract

The aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some well-established conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane-Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev's chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane-Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane-Shastry chain.