Bethe Ansatz derived from the functional relations of the open XXZ chain for new special cases

TL;DR

This paper derives the Bethe Ansatz solution for the open XXZ quantum spin chain at roots of unity using functional relations, focusing on special boundary parameter cases and extending to more general boundary conditions.

Contribution

It provides a novel Bethe Ansatz solution for specific boundary parameter configurations of the open XXZ chain at roots of unity, expanding understanding of its integrable structure.

Findings

Bethe Ansatz solutions for special boundary cases at roots of unity.

Extension to cases with arbitrary boundary parameters and specific conditions.

Enhanced understanding of the functional relations in integrable quantum spin chains.

Abstract

The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe Ansatz solution for the transfer matrix eigenvalues for the special cases that all but one of the boundary parameters are zero, and the bulk anisotropy parameter is \eta = i\pi/3, i\pi/5 ,... In an Addendum, these results are extended to the cases that any two of the boundary parameters {\alpha_-, \alpha_+,\beta_-, \beta_+} are arbitrary and the remaining boundary parameters are either \eta or i \pi/2.