The open XXZ chain at $\Delta=-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations

TL;DR

This paper explicitly computes the ground-state eigenvalues and eigenvectors of an open XXZ spin chain at a specific anisotropy, linking solutions of boundary quantum KZ equations to combinatorial enumeration problems.

Contribution

It provides explicit polynomial characterizations of ground states and constructs contour-integral solutions to boundary quantum KZ equations for the six-vertex model.

Findings

Explicit finite-size ground-state eigenvalues for real x>0

Polynomial characterization of ground-state components with integer coefficients

Connection between boundary quantum KZ solutions and alternating sign matrices

Abstract

The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$ and diagonal boundary magnetic fields that depend on a parameter $x$ is studied. For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in $x$ with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal $K$-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.