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arXiv:solv-int/9909019 · Progress of Theoretical Physics

The Structure of the Bazhanov-Baxter Model and a New Solution of the Tetrahedron Equation

M. Horibe, Kazuyasu Shigemoto

Published Aug 1, 1999• 1 citation
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Abstract

We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state integrable model. There are two essential points, i) the cubic symmetries, and ii) the spherical trigonometry parametrization, to understand the structure of this model. We propose two approaches to find a candidate as a solution of the tetrahedron equation, and we find a new solution.

DOI
10.1143/ptp.102.221

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We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state integrable model. There are two essential points, i) the cubic symmetries, and ii) the spherical trigonometry parametrization, to understand the structure of this model. We propose two approaches to find a candidate as a solution of the tetrahedron equation, and we find a new solution.
The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra Uq(sl^(n))U_q(\hat{sl}(n))Uq​(sl^(n)), where the rank nnn coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)sl(n)sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.
The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra Uq(hatsl(n))U_q(\\hat{sl}(n))Uq​(hatsl(n)), where the rank nnn coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)sl(n)sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.
A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator [Formula: see text] in the space of a triple Weyl algebra. [Formula: see text] is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for [Formula: see text] follows without further calculation. If the Weyl parameter is taken to be a root of unity, [Formula: see text] decomposes into a matrix conjugation operator R1,2,3 and a c-number functional mapping [Formula: see text]. The operator R1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.