Completeness of Bethe's states for generalized $XXZ$ model

TL;DR

This paper proves the combinatorial completeness of Bethe's states for a generalized $XXZ$ model, assuming the string conjecture, and derives an explicit inverse matrix form related to vacancy numbers.

Contribution

It introduces an integer version of vacancy numbers and proves completeness, extending the understanding of Bethe ansatz solutions for the generalized $XXZ$ model.

Findings

Proves combinatorial completeness of Bethe's states.

Derives an explicit inverse matrix with a tridiagonal form.

Generalizes the Cartan matrix of type A.

Abstract

We study the Bethe ansatz equations for a generalized $XXZ$ model on a one-dimensional lattice. Assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of Bethe's states for a generalized $XXZ$ model. We find an exact form for inverse matrix related with vacancy numbers and compute its determinant. This inverse matrix has a tridiagonal form, generalizing the Cartan matrix of type $A$.