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

Theta function parameterization and fusion for 3-D integrable Boltzmann weights

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

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.

Related papers

Theta-function parametrization and fusion for 3D integrable Boltzmann weights

G von Gehlen, Stanislav Zdislavovich Pakuliak, С. М. Сергеев

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.

arXiv:nlin/0305031•Journal of Physics A Mathematical and General•Jan 9, 2004•5 citations

THE MODIFIED TETRAHEDRON EQUATION AND ITS SOLUTIONS

G. VON GEHLEN, S. PAKULIAK, S. SERGEEV

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.

arXiv:nlin/0303043•International Journal of Modern Physics A•May 1, 2004•5 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

SU(2) parameterisation of Baxter's model

Omar E. Foda

A parametrisation of the Boltzmann weights of Baxter's symmetric eight-vertex model (1982) is derived, using Cherednik's representation (1979) of the Zamolodchikov algebra, in terms of theta functions based on the weight lattice of SU(2). The Yang-Baxter equations are satisfied as a consequence of the associativity of the Zamolodchikov algebra. The derivation makes the connection of Baxter's model to the spin-1/2 representation of SU(2) explicit.

Journal of Physics A Mathematical and General•Jul 21, 1990•0 citations

An integrable 3D lattice model with positive Boltzmann weights

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

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

arXiv:1308.4773•Journal of Physics A Mathematical and Theoretical•Nov 6, 2013•21 citations

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 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.
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.
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.
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.