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International Journal of Modern Physics A

CYCLIC MONODROMY MATRICES FOR THE R-MATRIX OF THE SIX-VERTEX MODEL AND THE CHIRAL POTTS MODEL WITH FIXED SPIN BOUNDARY CONDITIONS

V. Tarasov

Published Apr 1, 1992• 57 citations
Publisher

Abstract

Irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model are described. As a consequence, the direct computation of spectra for transfer-matrices of the chiral Potts model with special fixed-spin boundary conditions is done. The generalization of simple Baxter's Hamiltonian is proposed.

DOI
10.1142/s0217751x92004129

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Irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model are described. As a consequence, the direct computation of spectra for transfer-matrices of the chiral Potts model with special fixed-spin boundary conditions is done. The generalization of simple Baxter's Hamiltonian is proposed.
Abstract Monodromy matrices of the model are known to satisfy a Yang–Baxter equation with a six-vertex R -matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this R -matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
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.
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.