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

Journal of Physics A Mathematical and General

Towards an exhaustive classification of the star-triangle relation. I

J.-M. Maillard, Thomas Garel

Published Apr 21, 1984• 8 citations
Publisher

Abstract

The authors study the star-triangle relation for two-component spin models on a square lattice, in order to classify exhaustively such a relation. For this purpose, they obtain necessary conditions for the star-triangle relation to be satisfied, by considering the commutation of transfer matrices of arbitrary size N, for small N. The number of such conditions is compared with the number of relevant parameters of the model, and we are led to distinguish two very different cases. Only the first one is dealt with in this paper.

DOI
10.1088/0305-4470/17/6/021

Related papers

Towards an exhaustive classification of the star-triangle relation. II

J.-M. Maillard, Thomas Garel

For pt.I see ibid, vol.17, p.1251 (1984). The authors continue their study of the star-triangle relation for two-component spin models, with the help of the inversion relation. The conclusion is that the most general interaction-round-a-face models (16 non-zero parameters) that satisfy the star-triangle relations are mainly Baxter (1972) and Fan and Wu (1970) models. In this way they recover Krichever's result (1981) for the classification of the star-triangle relations for vertex models. They sketch a generalisation of this approach to models with excluded configurations, to staggered models, and to three-dimensional models (tetrahedron relations).

Journal of Physics A Mathematical and General•Apr 21, 1984•5 citations

Necessary versus sufficient conditions for exact solubility of statistical models on lattices

Pierre Lochak, J.-M. Maillard

It is shown that under rather mild conditions the triangle relation represents a necessary condition for the existence of commuting transfer matrices of arbitrary size. The cases of spin models and vertex models are treated separately.

Journal of Mathematical Physics•Feb 1, 1986•10 citations

The star-triangle (Yang-Baxter) equation

Barry M. McCoy

Abstract This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.

•Dec 1, 2009•0 citations

From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices

Jeffrey Robert Schmidt

Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).

Axioms•Aug 27, 2012•3 citations

Star-Triangle and Star-Star Relations in Statistical Mechanics

R. J. Baxter

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".

International Journal of Modern Physics B•Jan 20, 1997•27 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
The authors study the star-triangle relation for two-component spin models on a square lattice, in order to classify exhaustively such a relation. For this purpose, they obtain necessary conditions for the star-triangle relation to be satisfied, by considering the commutation of transfer matrices of arbitrary size N, for small N. The number of such conditions is compared with the number of relevant parameters of the model, and we are led to distinguish two very different cases. Only the first one is dealt with in this paper.
For pt.I see ibid, vol.17, p.1251 (1984). The authors continue their study of the star-triangle relation for two-component spin models, with the help of the inversion relation. The conclusion is that the most general interaction-round-a-face models (16 non-zero parameters) that satisfy the star-triangle relations are mainly Baxter (1972) and Fan and Wu (1970) models. In this way they recover Krichever's result (1981) for the classification of the star-triangle relations for vertex models. They sketch a generalisation of this approach to models with excluded configurations, to staggered models, and to three-dimensional models (tetrahedron relations).
Abstract This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.
Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).
The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".
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.