On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint

TL;DR

This paper extends the analysis of the open XXZ spin-1/2 chain with boundary fields by characterizing the spectrum and eigenstates using the Separation of Variables framework, enabling computation of matrix elements under more general boundary conditions.

Contribution

It introduces a method to describe the spectrum with non-longitudinal boundary fields constrained, using the homogeneous TQ-equation within the SoV framework, and computes matrix elements for these states.

Findings

Derived multiple integral representations for matrix elements in the half-infinite chain limit.

Extended previous results to more general boundary conditions with a constraint.

Connected the spectrum description to solutions of the homogeneous TQ-equation.

Abstract

This paper is a continuation of [1], in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in [1].