Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity

TL;DR

This paper develops an exact functional relation for the open XXZ spin chain with nondiagonal boundary terms at roots of unity, enabling Bethe-Ansatz-like solutions for eigenvalues.

Contribution

It introduces a novel (p+1)-order functional relation that simplifies solving the model with nondiagonal boundaries at roots of unity.

Findings

Derivation of a (p+1)-order functional relation for the transfer matrix.

Expression of higher-spin transfer matrices in terms of lower-spin matrices.

Truncation of the fusion hierarchy at roots of unity.

Abstract

We consider the open XXZ quantum spin chain with nondiagonal boundary terms. For bulk anisotropy value \eta = i \pi/(p+1), p= 1, 2, ..., we propose an exact (p+1)-order functional relation for the transfer matrix, which implies Bethe-Ansatz-like equations for the corresponding eigenvalues. The key observation is that the fused spin-(p+1)/2 transfer matrix can be expressed in terms of a lower-spin transfer matrix, resulting in the truncation of the fusion hierarchy.