On the gauge invariant boundary condition in open XXZ chains: SoV bases, elementary blocks and overlaps

Abstract

We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $α$ and $β$, $β$ being the so-called dynamical parameter and $α$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $α$ and $β$ through the difference $α-β$ only. Considering the ungauged limit in which $α-β$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.