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arXiv:hep-th/9505007 · Communications in Theoretical Physics

Three-Dimensional Vertex Model Related BCC Model in Statistical Mechanics

Hu Zhanning

Published Mar 15, 1998• 0 citations
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Abstract

In this paper, a three-dimensional vertex model is obtained. It is a duality of the three-dimensional integrable lattice model with N states proposed by Boos, Mangazeev, Sergeev and Stroganov. The Boltzmann weight of the model is dependent on four spin variables, which are the linear combinations of the spins on the corner sites of the cube, and obeys the modified vertex-type tetrahedron equation. This vertex model can be regarded as a deformation of the one related to the three-dimensional Baxter–Bazhanov model. The constrained conditions of the spectrums are discussed in detail and the symmetry properties of weight functions of the vertex model are presented.

DOI
10.1088/0253-6102/29/2/217

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•Jan 1, 2008•1 citation
In this paper, a three-dimensional vertex model is obtained. It is a duality of the three-dimensional integrable lattice model with N states proposed by Boos, Mangazeev, Sergeev and Stroganov. The Boltzmann weight of the model is dependent on four spin variables, which are the linear combinations of the spins on the corner sites of the cube, and obeys the modified vertex-type tetrahedron equation. This vertex model can be regarded as a deformation of the one related to the three-dimensional Baxter–Bazhanov model. The constrained conditions of the spectrums are discussed in detail and the symmetry properties of weight functions of the vertex model are presented.
We apply a 3-dimensional approach to describe a new parametrization of the L-operators for the 2-dimensional Bazhanov-Stroganov (BS) integrable spin model related to the chiral Potts model. This parametrization is based on the solution of the associated classical discrete integrable system. Using a 3-dimensional vertex satisfying a modified tetrahedron equation, we construct an operator which generalizes the BS quantum intertwining matrix S. This operator describes the isospectral deformations of the integrable BS model.
We report progress in constructing Boltzmann weights for integrable 3-dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation can be conveniently parameterized in terms of N-th roots of theta-functions on the Jacobian of a compact algebraic curve. Fay’s identity guarantees the Fermat relations and the classical equations of motion for the parameters determining the Boltzmann weights. Our parameterization allows to write a simple formula for fused Boltzmann weights R which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation. Imposing periodic boundary conditions we observe that the R satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model and the Chessboard model as special cases.