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arXiv:1308.4773 · Journal of Physics A Mathematical and Theoretical

An integrable 3D lattice model with positive Boltzmann weights

Vladimir V. Mangazeev, Vladimir V. Bazhanov, С. М. Сергеев

Published Nov 6, 2013• 21 citations
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Abstract

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0

DOI
10.1088/1751-8113/46/46/465206

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In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0
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.
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.
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.
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 obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on