Exact solution for the spin-$s$ XXZ quantum chain with non-diagonal twists

TL;DR

This paper derives an exact eigenvalue expression for the spin-$s$ XXZ quantum chain with non-diagonal twists, extending functional relation methods to models lacking trivial reference states.

Contribution

It generalizes the functional relation method to spin-$s$ XXZ chains with non-diagonal boundary conditions, providing exact solutions where Bethe ansatz was previously challenging.

Findings

Eigenvalue expression for twisted spin-$s$ XXZ chain obtained

Bethe ansatz equations reduce to Hofstadter problem equations for $N=1$

Method applicable to models without trivial reference states

Abstract

We study integrable vertex models and quantum spin chains with toroidal boundary conditions. An interesting class of such boundaries is associated with non-diagonal twist matrices. For such models there are no trivial reference states upon which a Bethe ansatz calculation can be constructed, in contrast to the well-known case of periodic boundary conditions. In this paper we show how the transfer matrix eigenvalue expression for the spin-$s$ XXZ chain twisted by the charge-conjugation matrix can in fact be obtained. The technique used is the generalization to spin-$s$ of the functional relation method based on ``pair-propagation through a vertex''. The Bethe ansatz-type equations obtained reduce, in the case of lattice size $N=1$, to those recently found for the Hofstadter problem of Bloch electrons on a square lattice in a magnetic field.