Elliptic generalisation of integrable q-deformed anisotropic Haldane–Shastry long-range spin chain
M. G. Matushko, A. Zotov
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mi>M</mml:mi> </mml:msub> <mml:mi>R</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mover> <mml:mrow> <mml:mi mathvariant="normal">g</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mi>M</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> XXZ R -matrix are included into consideration by separate verification.