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

TL;DR

This paper fully characterizes the spectrum and matrix elements of certain quasi-local operators in non-diagonal open spin-1/2 XXZ quantum chains using Sklyanin's separation of variables, advancing the analysis of integrable models.

Contribution

It develops a method within the SOV framework to compute matrix elements of local operators for non-diagonal boundary conditions in open XXZ chains, including determinant formulas.

Findings

Complete eigenvalues and eigenstates characterization.

Determinant formulas for matrix elements of quasi-local operators.

Proof of the simplicity of the transfer matrix spectrum.

Abstract

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.