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

The analysis of the transfer matrices associated to 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. More in detail, for the 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 apply in particular to transfer matrix eigenstates. The comparison of our analysis for the 8-vertex reflection algebra with that of [1, 2] for the 6-vertex one leads to the interesting remark that a profound similarity in both the characterization of the spectral problems and of the scalar products exists for these two different realizations of the reflection algebra once they are described by SOV method. As it will be shown in a future publication, this remarkable similarity will be at the basis of the simultaneous determination of form factors of local operators of integrable quantum models associated to general reflection algebra representations of both 8-vertex and 6-vertex type.

Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms

We use the quantum separation of variable (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates in the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.