Rational quantum integrable systems of D_N type with polarized spin reversal operators

TL;DR

This paper analyzes the spectrum and partition function of a D_N-type spin Calogero model with polarized spin reversal operators and its related Haldane-Shastry type spin chain, revealing invariance under Yangian symmetry and connections to vertex models.

Contribution

It provides a closed-form spectrum and partition function for the D_N-type model, introduces a recursive method for high-spin chain evaluation, and uncovers Yangian invariance and spectral equivalences.

Findings

Exact spectrum and partition function derived

Recursive evaluation method for high N implemented

Evidence of Yangian symmetry in the spectrum

Abstract

We study the spin Calogero model of D_N type with polarized spin reversal operators, as well as its associated spin chain of Haldane-Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos-Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the D_N-type spin chain for relatively high values of the number of spins N. We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.