Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators

The integrable quantum models, associated to the transfer matrices of the 6-vertex reflection algebra for spin 1/2 representations, are studied in this paper. In the framework of Sklyanin's quantum separation of variables (SOV), we provide the complete characterization of the eigenvalues and eigenstates of the transfer matrix and the proof of the simplicity of the transfer matrix spectrum. Moreover, we use these integrable quantum models as further key examples for which to develop a method in the SOV framework to compute matrix elements of local operators. This method has been introduced first in [1] and then used also in [2], it is based on the resolution of the quantum inverse problem (i.e. the reconstruction of all local operators in terms of the quantum separate variables) plus the computation of the action of separate covectors on separate vectors. In particular, for these integrable quantum models, which in the homogeneous limit reproduce the open spin-1/2 XXZ quantum chains with non-diagonal boundary conditions, we have obtained the SOV-reconstructions for a class of quasi-local operators and determinant formulae for the covector-vector actions. As consequence of these findings we provide one determinant formulae for the matrix elements of this class of reconstructed quasi-local operators on transfer matrix eigenstates.Comment: 40 pages. Minor modifications in the text and some notations and some more reference adde

Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model

The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states by determinant formulae. The matrices entering in these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of separate states. This SOV analysis is not reduced to the case of the elliptic roots of unit and the results here derived define the required setup to extend to the dynamical 6-vertex model the approach recently developed in [1]-[5] to compute the form factors of the local operators in the SOV framework, these results will be presented in a future publication. For the periodic 8-vertex transfer matrix, we prove that its eigenvalues have to satisfy a fixed system of equations. In the case of a chain with an odd number of sites, this system of equations is the same entering in the SOV characterization of the antiperiodic dynamical 6-vertex transfer matrix spectrum. This implies that the set of the periodic 8-vertex eigenvalues is contained in the set of the antiperiodic dynamical 6-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate to any of such eigenvalues one nonzero eigenstate of the periodic 8-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy of the periodic 8-vertex spectrum is also presented.Comment: 36 pages, main modifications in section 3 and one appendix added, no result modified for the dynamical 6-vertex transfer matrix spectrum and the matrix elements of identity on separate states for chains with an odd number of site

The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method

The most general cyclic representations of the quantum integrable tau_2-model are analyzed. The complete characterization of the tau_2-spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin's Separation of Variables (SOV) method by extending and adapting the ideas first introduced in [1, 2]: i) The determination of the tau_2-spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials. ii) The determination of the tau_2-eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proven to be polynomials for a quite general class of tau_2-self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: i) Simplicity of the spectrum, for general representations. ii) Complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of tau_2-model on the chiral Potts algebraic curves.Comment: 40 pages. Minor modifications in the text and some notation

On the form factors of local operators in the lattice sine-Gordon model

We develop a method for computing form factors of local operators in the framework of Sklyanin's separation of variables (SOV) approach to quantum integrable systems. For that purpose, we consider the sine-Gordon model on a finite lattice and in finite dimensional cyclic representations as our main example. We first build our two central tools for computing matrix elements of local operators, namely, a generic determinant formula for the scalar products of states in the SOV framework and the reconstruction of local fields in terms of the separate variables. The general form factors are then obtained as sums of determinants of finite dimensional matrices, their matrix elements being given as weighted sums running over the separate variables and involving the Baxter Q-operator eigenvalues.Comment: 43 page