Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics A

SPATIAL SYMMETRY, LOCAL INTEGRABILITY AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

Rinat Kashaev, Vladimir V. Mangazeev, Yu. G. Stroganov

Published Jan 30, 1993• 32 citations
Publisher

Abstract

We show that the Baxter-Bazhanov model is invariant under the action of the cube’s symmetry group. The three-dimensional star-star relations, proposed by Baxter and Bazhanov as local integrability conditions, correspond to a particular transformation from this group. Invariant Boltzmann weights, parametrized in terms of the Zamolodchikov angle variables, apparently satisfy the tetrahedron equations.

DOI
10.1142/s0217751x93000230

Related papers

STAR-SQUARE AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

Rinat Kashaev, Vladimir V. Mangazeev, Yu. G. Stroganov

Spatially symmetrical Baxter-Bazhanov model’s Boltzmann weights are proved to satisfy the tetrahedron equation. The latter is a consequence of the “inversion” and “star-square” relations for the Boltzmann weight elementary building blocks.

International Journal of Modern Physics A•Mar 30, 1993•60 citations

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

M. Horibe, Kazuyasu Shigemoto

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.

arXiv:solv-int/9909019•Progress of Theoretical Physics•Aug 1, 1999•1 citation

Star-Triangle Relation for a Three Dimensional Model

Bazhanov, V V, R. J. Baxter

The solvable $sl(n)$-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.

•Jan 1, 1993•64 citations

Three-Dimensional Vertex Model Related BCC Model in Statistical Mechanics

Hu Zhanning

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.

arXiv:hep-th/9505007•Communications in Theoretical Physics•Mar 15, 1998•0 citations

On the Star-Triangle Relation and the Star-Square Relation in the Baxter-Bazhanov Model

Hu Zhan-Ning

The connection is found between the star-triangle relation and the star-square relation in the Baxter-Bazhanov model which is an Interaction-Round-a-Cube model. This gives a proof of the guess suggested by Bazhanov and Baxter.

Chinese Physics Letters•Jun 1, 1995•1 citation

Remarks on the Three-dimensional Lattice Model

Zhan-Ning Hu

We find the connection between the weight functions of the three-dimensional (3D) Bazhanov-Baxter model and the 3D vertex model proposed by Mangazeev et al. And the weight function of the 3D integrable model with the modified tetrahedron can be formulated as the vertex form.

arXiv:hep-th/9501062•Chinese Physics Letters•Jun 1, 1996•4 citations
We show that the Baxter-Bazhanov model is invariant under the action of the cube’s symmetry group. The three-dimensional star-star relations, proposed by Baxter and Bazhanov as local integrability conditions, correspond to a particular transformation from this group. Invariant Boltzmann weights, parametrized in terms of the Zamolodchikov angle variables, apparently satisfy the tetrahedron equations.
The solvable sl(n)sl(n)sl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.
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.