Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain

Abstract

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars-Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin ${\mathrm{GL}}_M$ $R$-matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars-Schneider model. Trigonometric degenerations are studied as well. For example, in $M=2$ case our construction provides q-deformation for anisotropic XXZ Haldane-Shastry model. The standard Haldane-Shastry model and its Uglov's q-deformation based on ${\mathrm{U}}_q({\widehat{\mathrm{gl}}}_M)$ XXZ $R$-matrix are included into consideration by separate verification.

arXiv comment36 pages, minor corrections

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Cited by in corpus (6)

• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• From fermionic spin-Calogero-Sutherland models to the Haldane-Shastry chain by freezing
• The deformed Inozemtsev spin chain
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• Quantum group deformation of the Kittel--Shore model
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type