The integrable quantum models, associated with 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 is based on the resolution of the quantum inverse problem (i.e. the reconstruction of 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-reconstruction for a class of quasi-local operators and determinant formulae for the covector–vector actions. As a consequence of these findings we provide one determinant formula for the matrix elements of this class of reconstructed quasi-local operator on transfer matrix eigenstates.

https://iopscience.iop.org/article/10.1088/1742-5468/2012/10/P10025/pdf

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

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

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 variables (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates for 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.

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

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.

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

We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q-function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.

https://iopscience.iop.org/article/10.1088/1742-5468/2014/05/P05015/pdf

Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables

We study integrable lattice regularizations of the sine–Gordon model with the help of the separation of variables method of Sklyanin and the Baxter -operators. This leads us to the complete characterization of the spectrum (eigenvalues and eigenstates), in terms of the solutions to the Bethe ansatz equations. The completeness of the set of states that can be constructed from the solutions to the Bethe ansatz equations is proven by our approach.

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The sine–Gordon model revisited: I

We study integrable lattice regularizations of the sine-Gordon model with the help of the separation of variables method of Sklyanin and the Baxter Q-operators. This leads us to the complete characterization of the spectrum (eigenvalues and eigenstates), in terms of the solutions to the Bethe ansatz equations. The completeness of the set of states that can be constructed from the solutions to the Bethe ansatz equations is proven by our approach.

J. Teschner

Papers

The sine–Gordon model revisited: I

The Sine-Gordon model revisited I