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

TL;DR

This paper explores the integrability of the XXZ spin chain at .5, deriving new determinant formulas and symmetric function expressions that connect Bethe roots to combinatorial objects like alternating sign matrices.

Contribution

It introduces novel determinant and symmetric function formulas for Bethe roots of the XXZ chain at .5, linking integrability to combinatorial enumeration.

Findings

Derived explicit expressions for elementary symmetric functions at Bethe roots.

Established determinant formulas involving Schur functions and Bethe roots.

Connected Bethe root structures to enumeration of symmetry classes of alternating sign matrices.

Abstract

A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at $\Delta = - \frac12$ 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.