The XXZ spin chain at $\Delta =-1/2$: Bethe roots, symmetric functions and determinants

Abstract

A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at $\Delta =-\frac{1}{2}$ and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter's $Q$ function for the XXZ chain for periodic, twisted and open boundary conditions. These include expressions for the elementary symmetric functions evaluated at the groundstate solution of the Bethe roots. In this approach Schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the Bethe roots. When evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices.

11 pages, revtex

References in corpus (6)

• Spin chains and combinatorics
• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions
• The Importance of being Odd
• Combinatorial nature of ground state vector of O(1) loop model
• Spin chains and combinatorics: twisted boundary conditions
• Temperley-Lieb Stochastic Processes

Cited by in corpus (27)

• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Lattice fermion models with supersymmetry
• Exact expressions for correlations in the ground state of the dense O(1) loop model
• Exact Ground State and Finite Size Scaling in a Supersymmetric Lattice Model
• Spin chains with dynamical lattice supersymmetry
• New Developments in the Eight Vertex Model II. Chains of odd length
• The eight-vertex model and lattice supersymmetry
• Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
• Ground-state properties of a supersymmetric fermion chain
• Open spin chains with dynamic lattice supersymmetry
• Bethe roots and refined enumeration of alternating-sign matrices
• Exact results for the sigma^z two-point function of the XXZ chain at Delta=1/2
• The TQ equation of the 8 vertex model for complex elliptic roots of unity
• The Finite Size SU(3) Perk-Schultz Model with Deformation Parameter q=exp(i 2 pi/3)
• Exact Large Deviations of the Current in the Asymmetric Simple Exclusion Process with Open Boundaries
• The Wave Functions for the Free-Fermion Part of the Spectrum of the $S{U}_q(N)$ Quantum Spin Models
• Root of unity symmetries in the 8 and 6 vertex models
• A staggered fermion chain with supersymmetry on open intervals
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder
• On Baxter's Q operator of the higher spin XXZ chain at the Razumov-Stroganov point
• On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta =-1/2$
• Fully packed loop models on finite geometries
• Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
• Laws of large numbers in the Raise and Peel model
• The open XXZ chain at $\Delta =-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice