Spin-Ruijsenaars, q-Deformed Haldane–Shastry and Macdonald Polynomials

Abstract

We study the -analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of D. Uglov, via ‘freezing’ from the affine Hecke algebra. To this end we first obtain explicit expressions for the spin-Macdonald operators of the (trigonometric) spin-Ruijsenaars model. Through freezing these give rise to the higher Hamiltonians of the spin chain, including another Hamiltonian of the opposite ‘chirality’. The sum of the two chiral Hamiltonians has a real spectrum also when $$|\mathsf {q}|=1$$ | q | = 1 , so in particular when is a root of unity. For generic $$\mathsf {q}$$ q the eigenspaces are known to be labelled by ‘motifs’. We clarify the relation between these patterns and the corresponding degeneracies (multiplicities) in the crystal limit $$\textsf {q}\rightarrow \infty $$ q → ∞ . For each motif we obtain an explicit expression for the exact eigenvector, valid for generic , that has (‘pseudo’ or ‘l-’) highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of A and D and annihilated by C. It has a simple component featuring the ‘symmetric square’ of the -Vandermonde polynomial times a Macdonald polynomial—or more precisely its quantum spherical zonal special case. All other components of the eigenvector are obtained from this through the action of the Hecke algebra, followed by ‘evaluation’ of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the exact spectrum is complete. The entire model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the Y-operators from the affine Hecke algebra. From a more mathematical perspective the key step in our diagonalisation is as follows. We show that on a subspace of suitable polynomials the first M ‘classical’ (i.e. no difference part) Y-operators in N variables reduce, upon evaluation as above, to Y-operators in M variables with parameters at the quantum zonal spherical point.

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

“…These are equilibrium positions in the classical trigonometric Ruijsenaars-Schneider model [41], see also [12,38]. In this way the described above recipe for the freezing trick is performed as was recently shown by Lamers et al [29].…”

“…The first example of anisotropic long-range spin chain was introduced by Uglov [47] (see also [28,29] and [20,46]), and it is also a q-deformed model, which we discuss below. But before proceeding to q-deformed models let us describe a natural extension of the previously discussed long-range chains to anisotropic case.…”

“…The first q-deformed long-range spin chain was proposed in [47] using a different approach motivated by studies of Hecke algebras and results of [20,46]. The Uglov's construction was revisited and clarified recently in [28,29]. Main idea is as follows.…”

“…Uglov suggested a q-deformed version of the construction from [20] (see also [29]). The result is that the Macdonald operators (1.10) admit anisotropic spin generalisation of the form:…”

“…Here following ideas of [29,35,47] we deduce the long-range spin chain Hamiltonians and find conditions for their commutativity. These conditions are unified to a set of identities, which are proved in the next section.…”