Modular families of elliptic long-range spin chains from freezing

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 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 spin- less 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 pre- serving quantum integrability. As we showed in previous work, the results include the Heisen- berg, 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 generalisa- tion of Sechin–Zotov, and their completely anisotropic q-deformations due to Matushko–Zotov (vertex type).