SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra

The analysis of the transfer matrices associated with the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method, which generalizes to these integrable quantum models the method first introduced by Sklyanin. For representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras, for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which particularly apply to transfer matrix eigenstates. A comparison of our analysis of the 8-vertex reflection algebra with that of (Niccoli G 2012 J. Stat. Mech. P10025, Faldella S et al 2014 J. Stat. Mech. P01011) for the 6-vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products, which exists for these two different realizations of the reflection algebra once they are described by the SOV method. As will be shown in a future publication, this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8-vertex and 6-vertex type.