Modular families of elliptic long-range spin chains from freezing
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| Format: | Preprint |
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2025
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| _version_ | 1866914381837631488 |
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| author | Klabbers, Rob Lamers, Jules |
| author_facet | Klabbers, Rob Lamers, Jules |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular families of elliptic long-range spin chains from freezing Klabbers, Rob Lamers, Jules Mathematical Physics High Energy Physics - Theory Quantum Algebra Exactly Solvable and Integrable Systems 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. |
| title | Modular families of elliptic long-range spin chains from freezing |
| topic | Mathematical Physics High Energy Physics - Theory Quantum Algebra Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2507.13104 |