Modular families of elliptic long-range spin chains from freezing

TL;DR

This paper develops a method to construct elliptic long-range spin chains with quantum integrability by freezing elliptic spin-Ruijsenaars systems at classical equilibria, unifying various known models.

Contribution

It introduces a modular group action on elliptic Ruijsenaars-Schneider systems and uses deformation quantisation to generate a broad class of integrable long-range spin chains.

Findings

Constructed a family of elliptic long-range spin chains with real spectra.

Unified several known integrable spin chains within a common framework.

Showed the preservation of quantum integrability during the freezing process.

Abstract

We consider the construction of quantum-integrable spin chains with q-deformed long-range interactions by `freezing' integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars-Schneider system. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbour to long-range interacting spins. We focus on the elliptic case. We first define an action of the modular group on the spinless elliptic Ruijsenaars-Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantisation to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium whilst preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane-Shastry chains along with their xxz-like q-deformations (face-type), or the antiperiodic Haldane-Shastry chain of Fukui-Kawakami, its elliptic generalisation of Sechin-Zotov, and their completely anisotropic q-deformations due to Matushko-Zotov (vertex type). Finally, we show how freezing fits in the setting of 'hybrid' integrable systems.