Completing Bethe's equations at roots of unity

TL;DR

This paper extends Bethe's equations to fully specify eigenvectors of the XXZ model at roots of unity, especially in degenerate cases, and explores the role of $sl_2$ loop algebra symmetry in this context.

Contribution

It derives additional equations to complete Bethe's equations at roots of unity and demonstrates the significance of $sl_2$ loop algebra symmetry in determining eigenvector properties.

Findings

Bethe's equations are incomplete at roots of unity for degenerate states.

Additional equations are identified to fully specify eigenvectors.

$sl_2$ loop algebra symmetry determines highest weights of irreducible representations.

Abstract

In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the $sl_2$ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.