Structure of the two-boundary XXZ model with non-diagonal boundary terms

TL;DR

This paper analyzes the structure of the XXZ quantum spin chain with non-diagonal boundary conditions, identifying invariant subspaces at critical points where Bethe Ansatz solutions are valid, and connecting algebraic and spectral properties.

Contribution

It introduces a basis that diagonalizes a conserved charge, computes the action of boundary generators, and characterizes invariant subspaces at critical points for the two-boundary XXZ model.

Findings

Invariant subspaces are identified at critical points.

The dimension of invariant subspaces matches Bethe Ansatz eigenvalue splitting.

The algebraic structure relates to the solutions of Bethe Ansatz equations.

Abstract

We study the integrable XXZ model with general non-diagonal boundary terms at both ends. The Hamiltonian is considered in terms of a two boundary extension of the Temperley-Lieb algebra. We use a basis that diagonalizes a conserved charge in the one-boundary case. The action of the second boundary generator on this space is computed. For the L-site chain and generic values of the parameters we have an irreducible space of dimension 2^L. However at certain critical points there exists a smaller irreducible subspace that is invariant under the action of all the bulk and boundary generators. These are precisely the points at which Bethe Ansatz equations have been formulated. We compute the dimension of the invariant subspace at each critical point and show that it agrees with the splitting of eigenvalues, found numerically, between the two Bethe Ansatz equations.