The open XXZ spin chain in the SoV framework: scalar product of separate states

TL;DR

This paper extends the scalar product determinant representations within the quantum separation of variables framework to the XXZ open spin-1/2 chain with general boundary conditions, facilitating calculations of form factors and correlations.

Contribution

It generalizes previous rational case results to the trigonometric XXZ case and derives new determinant formulas for scalar products with arbitrary boundary terms.

Findings

Derived determinant representations for scalar products in the XXZ open chain.

Showed how to take the homogeneous limit of these determinants.

Provided simplified formulas for form factors and correlation functions.

Abstract

In our previous paper [1] we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. To solve this model by means of SoV we use an algebraic Vertex-IRF gauge transformation reducing one of the boundary K-matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to - although more cumbersome than - the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.