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

TL;DR

This paper computes scalar products of states in the open XXX spin-1/2 chain with non-diagonal boundaries using the quantum separation of variables, providing determinant formulas that generalize known results for periodic chains.

Contribution

It introduces determinant representations for scalar products of separate states in the open XXX chain with general boundary conditions using the SoV approach, extending known Bethe ansatz results.

Findings

Determinant formulas for scalar products of separate states.

Homogeneous limit can be taken easily with new representations.

Special cases recover known Bethe ansatz scalar product formulas.

Abstract

We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T-Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.