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Journal of Physics Conference Series

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra by quantum separation of variables

B Pezelier

Published Feb 1, 2018• 0 citations
Publisher

Abstract

In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin's Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter's like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.

DOI
10.1088/1742-6596/965/1/012030

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In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin's Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter's like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.
We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxter’s gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.
This article is a direct continuation of where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the K K -matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary K K -matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary K K -matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter’s T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.
Abstract We implement our new separation of variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the Y ( gl n ) reflection algebra associated to general integrable boundary conditions. Moreover, we give the conditions on the boundary K -matrices allowing for the transfer matrix to be diagonalizable with simple spectrum. These SoV basis are then used to completely characterize the transfer matrix spectrum for the rank one and two reflection algebras. The rank one case is developed for both the rational and trigonometric fundamental representations of the 6-vertex reflection algebra. On the one hand, we extend the complete spectrum characterization to representations previously lying outside the SoV approach, e.g. those for which the standard algebraic Bethe Ansatz applies. On the other hand, we show that our new SoV construction can be reduced to the generalized Sklyanin’s one whenever it is applicable. The rank two case is developed explicitly for the fundamental representations of the Y ( gl 3 ) reflection algebra associated to general integrable boundary conditions. For both rank one and two our SoV approach leads to a complete characterization of the transfer matrix spectrum in terms of a set of polynomial solutions to the corresponding quantum spectral curve equation. Those are finite difference functional equations of order equal to the rank plus one, i.e. here two and three respectively for the Y ( gl 2 ) and Y ( gl 3 ) reflection algebras.
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.
The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.