Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

TL;DR

This paper bridges Baxter's auxiliary matrices and the algebraic Bethe ansatz for the six-vertex model, deriving Bethe eigenvalues of Q-operators and analyzing their implications for the XXZ spin-chain at various quantum group parameters.

Contribution

It introduces a general framework connecting auxiliary matrices with the algebraic Bethe ansatz and derives explicit Bethe eigenvalues for Q-operators, including proofs for states with up to three Bethe roots.

Findings

Derived a formula for Bethe eigenvalues of Q-operators.

Proved results for states with up to three Bethe roots.

Related findings to the six-vertex fusion hierarchy.

Abstract

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q-operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyze the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.