Resurrecting the partially isotropic Haldane-Shastry model

TL;DR

This paper introduces a simplified, physically meaningful Hamiltonian for the partially isotropic Haldane-Shastry model, clarifying its properties, symmetries, and spectrum, including extensions to higher rank spins.

Contribution

It provides a new, more efficient expression for the Hamiltonian of the partially isotropic Haldane-Shastry model, incorporating multi-spin interactions and extending the model to higher rank spins.

Findings

New Hamiltonian expression simplifies computations.

Exact spectrum matches previous results with anisotropy effects.

Extended model to $SU(n)$ spins with spectrum characterized by $rak{sl}_n$-motifs.

Abstract

We present a new and simpler expression for the Hamiltonian of the partially isotropic (XXZ-like) version of the Haldane-Shastry model, which was derived by D. Uglov over two decades ago in an apparently little-known preprint. While resembling the pairwise long-range form of the Haldane-Shastry model our formula accounts for the multi-spin interactions obtained by Uglov. Our expression is physically meaningful, makes hermiticity manifest, and is computationally more efficient. We discuss the model's properties, including its limits and (ordinary and quantum-affine) symmetries. In particular we introduce the appropriate notions of translational invariance and momentum. We review the model's exact spectrum found by Uglov for finite spin-chain length, which parallels the isotropic case up to level splitting due to the anisotropy. We also extend the partially isotropic model to higher rank, with $SU(n)$ 'spins', for which the spectrum is determined by $\mathfrak{sl}_n$-motifs.