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 (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.

Figures (2)

• Figure 1: The generator $T$ of $\mathrm{SL}(2,\mathbb{Z})$ simply shifts the modular parameter of the lattice $\Lambda_\tau \subset \mathbb{C}$ as $\tau \longmapsto \tau+1$. The generator $S$ acts by a signed inversion, $\tau \longmapsto -1/\tau$, which requires a simultaneous coordinate rescaling $x\longmapsto -\tau \, x$ in order to fix the lattice, as indicated
• Figure 2: Examples of equilibrium positions $x_j^\star{}^{\mspace{2mu}(B)}$ for simple $B \in \mathrm{PSL}(2,\mathbb{Z})$, where we take $\tau \in \mathrm{i} \, \mathbb{R}_{>0}$. For freezing, the choices $B = 1$ ($x_j^\star{}^{\mspace{2mu}(1)} = j/N$ real) and especially $B = S$ ($x_j^\star{}^{\mspace{2mu}(S)} = \tau \, j/N$ imaginary) are particularly important. In general, equilibrium positions lie equispaced on any line $[0,t]$ (for $t\in\Lambda_\tau \setminus \{0\}$) that does not intersect any other point in $\Lambda_\tau$

Theorems & Definitions (78)

• Theorem : Theorem \ref{['thm:RS_modular']}
• Theorem : Theorem \ref{['thm:equilibria_modular']}
• Theorem : Theorem \ref{['thm:HnB commute']}
• Theorem : Theorem \ref{['thm:comm diagram']}
• Definition
• Lemma 2.1
• proof : Proof.
• Theorem : Ruijsenaars ruijsenaarsCompleteIntegrabilityRelativistic1987
• Lemma 2.2
• proof : Proof.