Functional relations and Bethe Ansatz for the XXZ chain

TL;DR

This paper demonstrates that solving functional relations from the fusion hierarchy effectively recovers Bethe Ansatz solutions for the XXZ chain, including cases without a pseudovacuum, expanding solution methods for integrable models.

Contribution

It applies the functional relation approach to solve the XXZ chain, including nonstandard boundary conditions, without relying on the pseudovacuum assumption.

Findings

Recovered Bethe Ansatz solutions for closed and open XXZ chains.

Extended method to solve cases with nondiagonal boundary terms.

Validated the effectiveness of functional relations in integrable models.

Abstract

There is an approach due to Bazhanov and Reshetikhin for solving integrable RSOS models which consists of solving the functional relations which result from the truncation of the fusion hierarchy. We demonstrate that this is also an effective means of solving integrable vertex models. Indeed, we use this method to recover the known Bethe Ansatz solutions of both the closed and open XXZ quantum spin chains with U(1) symmetry. Moreover, since this method does not rely on the existence of a pseudovacuum state, we also use this method to solve a special case of the open XXZ chain with nondiagonal boundary terms.