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

TL;DR

This paper develops a quantum separation of variables method for analyzing the spectral problem of transfer matrices in integrable 8-vertex reflection algebra models on spin-1/2 chains, enabling complete characterization of eigenvalues and eigenstates.

Contribution

It introduces a generalized SOV approach for the 8-vertex reflection algebra, extending Sklyanin's method to more general representations and boundary conditions.

Findings

Explicit construction of SOV representations for the 8-vertex reflection algebra

Complete spectral characterization including eigenvalues and eigenstates

Determinant formulae for matrix elements on separated states

Abstract

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.