Bethe Ansatz and boundary energy of the open spin-1/2 XXZ chain

TL;DR

This paper reviews Bethe Ansatz solutions for the open XXZ spin-1/2 chain, focusing on boundary energy calculations at roots of unity and general boundary conditions, highlighting new functional relations and generalized Baxter equations.

Contribution

It introduces a generalized Bethe Ansatz solution involving multiple Q-functions and computes boundary energy for specific boundary parameter cases.

Findings

Derived a generalized Baxter T-Q equation with multiple Q-functions.

Computed boundary energy in the thermodynamic limit for certain boundary conditions.

Reviewed results for fully general boundary parameters.

Abstract

We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{$\alpha_-$,$\alpha_+$,$\beta_-$,$\beta_+$}} are nonzero. A generalization of the Baxter $T-Q$ equation that involves more than one independent $Q$ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.