On the Degenerate Multiplicity of the $sl_2$ Loop Algebra for the 6V Transfer Matrix at Roots of Unity

TL;DR

This paper reviews the $sl_2$ loop algebra symmetry in the XXZ spin chain and six-vertex model at roots of unity, analyzing the dimensions of degenerate eigenspaces generated by Bethe states.

Contribution

It provides a rigorous analysis of the dimensions of degenerate eigenspaces and the highest weight structure of Bethe eigenvectors at roots of unity.

Findings

Regular Bethe eigenvectors are highest weight vectors

Derived highest weight and evaluation parameters

Determined dimensions of highest weight representations

Abstract

We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight ${\bar d}_k^{\pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.