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arXiv:hep-th/9407146 · International Journal of Modern Physics A

MODIFIED TETRAHEDRON EQUATIONS AND RELATED 3D INTEGRABLE MODELS, I

Herman E. Boos, Vladimir V. Mangazeev, С. М. Сергеев

Published Nov 10, 1995• 18 citations
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Abstract

Using a modified version of the tetrahedron equations we construct a new family of N- state three-dimensional integrable models with commuting two-layer transfer matrices. We investigate a particular class of solutions to these equations and parametrize them in terms of elliptic functions. The corresponding models contain one free parameter, k—an elliptic modulus.

DOI
10.1142/s0217751x9500187x

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Using a modified version of the tetrahedron equations we construct a new family of N- state three-dimensional integrable models with commuting two-layer transfer matrices. We investigate a particular class of solutions to these equations and parametrize them in terms of elliptic functions. The corresponding models contain one free parameter, k—an elliptic modulus.
In this paper we present a new series of three-dimensional integrable lattice models with N colors. The case N=2 generalizes the elliptic model of Ref. 8. The weight functions of the models satisfy modified tetrahedron equations with N states and give a commuting family of two-layer transfer matrices. The dependence on the spectral parameters corresponds to the static limit of the modified tetrahedron equations, and weights are parametrized in terms of elliptic functions. The models contain two free parameters: elliptic modulus and additional parameter η. Also, we briefly discuss symmetry properties of weight functions of the models.
In this paper we present a new series of three-dimensional integrable lattice models with N colors. The case N=2 generalizes the elliptic model of Ref. 8. The weight functions of the models satisfy modified tetrahedron equations with N states and give a commuting family of two-layer transfer matrices. The dependence on the spectral parameters corresponds to the static limit of the modified tetrahedron equations, and weights are parametrized in terms of elliptic functions. The models contain two free parameters: elliptic modulus and additional parameter η. Also, we briefly discuss symmetry properties of weight functions of the models.
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.
Abstract We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q -oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n , they ultimately yield a new proof of the steady state formula for the n -species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.