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

Abstract

In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor. 50 224001) 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. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce 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.