Resurrecting the partially isotropic Haldane-Shastry model

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.

arXiv comment5 pages, 1 figure, 1 table; v2: various minor changes and additions; v3: 6 pages, 1 figure, 1 table; added proof of equality of formulae, minor further changes; v4: minor changes; v5: minor change

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