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 ${\rm GL}_M$ $R$-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 ${\rm U}_q({\widehat {\rm gl}_M})$ XXZ $R$-matrix are included into consideration by separate verification.