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

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 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 U-q(<(gl) over the cap >(M)) XXZ R-matrix are included into consideration by separate verification.

Automated summary

We describe an integrable elliptic q-deformed anisotropic long-range spin chain. The Polychronakos freezing trick is applied to derive a set of commuting Hamiltonians for this spin chain, which is constructed using the elliptic Baxter-Belavin GL(M)R-matrix. The freezing trick is reduced to a set of elliptic function identities, serving as equilibrium conditions in the classical spinless Ruijsenaars-Schneider model.