XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity

TL;DR

This paper demonstrates that regular Bethe eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the $sl_2$ loop algebra, explicitly computes their weights, and discusses their representation properties.

Contribution

It establishes the highest weight property of Bethe eigenvectors at roots of unity and explicitly evaluates their weights in terms of Bethe roots, including reducibility aspects.

Findings

Regular Bethe states generate highest weight vectors.

Explicit highest weight evaluation in terms of Bethe roots.

Existence of reducible Weyl modules from Bethe states.

Abstract

We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the $sl_2$ loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator $S^Z$, and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities {\it regular Bethe roots}. We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.