Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials

Abstract

We study the $q$-analogue of the Haldane-Shastry model, a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of Uglov, via 'freezing' from the affine Hecke algebra. We obtain explicit expressions for the spin-analogue of Macdonald operators. Through freezing these yield the higher Hamiltonians of the spin chain, including a Hamiltonian of the opposite chirality. The sum of the two chiral Hamiltonians has a real spectrum also for $q$ a root of unity. We clarify the relation between patterns labelling the eigenspaces, known as 'motifs', and the corresponding degeneracies in the crystal limit $q\to \infty$. For each motif we obtain an explicit expression for the exact eigenvector, valid for generic $q$, 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 $q$-Vandermonde times a Macdonald polynomial - or more precisely its quantum spherical zonal special case. Its other components are obtained 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 spectrum is complete. The model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the $Y$-operators of the affine Hecke algebra. The key step in our diagonalisation is 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.

v1: 79 pages, 5 figures, 6 tables. v2: improved structure for easier navigation, incl glossary of notation; other small changes; 73 pages, 5 figures, 6 tables. v3: further improved structure; other small corrections; 76 pages, 5 figures, 6 tables

References in corpus (3)

• Fractional Quantum Hall States and Jack Polynomials
• On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2

Cited by in corpus (10)

• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• 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
• A solvable non-unitary fermionic long-range model with extended symmetry
• Quantum group deformation of the Kittel--Shore model
• Bethe ansatz inside Calogero-Sutherland models
• On the Q operator and the spectrum of the XXZ model at root of unity
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type
• The q-deformed Haldane-Shastry chain at q=i with even length