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

Physical Review Letters

Partition Function of the Three-Dimensional Zamolodchikov Model

R. J. Baxter

Published Nov 5, 1984• 21 citations
Publisher

Abstract

It is shown that the Zamolodchikov model on the cubic lattice of $N$ sites is equivalent to a spin model on a lattice of $2N$ sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function $Z$ (for $N$ large) factors into a product of single-variable functions. These functions, and hence $Z$, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.

DOI
10.1103/physrevlett.53.1795

Related papers

Free-fermion, checkerboard and Z -invariant lattice models in statistical mechanics

R. J. Baxter

Abstract It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘Z-invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 8, 1986•103 citations

The free energy of the three-dimensional Zamolodchikov model

M. L. Glasser

Details are given for the reduction of Baxter’s expression for the exact partition function per site of the Zamolodchikov model to explicit form.

Journal of Mathematical Physics•Nov 1, 1986•0 citations

On the partition function for the three-dimensional Ising model

Yogendra P. Valani

Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.

•Aug 1, 2011•2 citations

Transfer matrix approach to the three-dimensional Ising model

Philippe Audit

The (111) plane to (111) plane transfer matrix of the Ising model on a simple cubic lattice is obtained in a simple product form by using a purpose-built matrix product previously defined by this author. As a first application of this theoretical result, the exact analytic expression of the partition function of an Ising model on a 2*2* infinity lattice is derived.

Journal of Physics A Mathematical and General•Jun 1, 1987•3 citations

Diagrammatical Techniques for Two-Dimensional Ising Models. III. Ising Model to Vertex Model

Tohru Morita, Kazuyuki Tanaka

It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.

Journal of the Physical Society of Japan•Mar 15, 1993•3 citations

Symmetry relations in exactly soluble models

Marc-Thierry Jaekel, J.-M. Maillard

Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.

Journal of Physics A Mathematical and General•Apr 1, 1982•42 citations
It is shown that the Zamolodchikov model on the cubic lattice of NNN sites is equivalent to a spin model on a lattice of 2N2N2N sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function ZZZ (for NNN large) factors into a product of single-variable functions. These functions, and hence ZZZ, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.
Abstract It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘Z-invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.
Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.