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 B

Star-Triangle and Star-Star Relations in Statistical Mechanics

R. J. Baxter

Published Jan 20, 1997• 27 citations
Publisher

Abstract

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".

DOI
10.1142/s0217979297000058

Related papers

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

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

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

J.-M. Maillard, Thomas Garel

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.

Journal of Physics A Mathematical and General•Apr 21, 1984•8 citations

Exact solution of Z(N) invariant spin models on a triangular lattice

Roberto Nicolau Onody, V. Kurak

Based on Baxter's ideas the authors present a method to solve spin models on triangular lattices. If the model has only nearest neighbour interaction and it satisfies the star-triangle relation then the exact free energy can be obtained. They apply the method to reobtain the critical triangular Potts model and to extend the self-dual Z(N) solutions of Fateev and Zamolodchikov to the triangular lattice.

Journal of Physics A Mathematical and General•Aug 1, 1984•4 citations

Integrable Quantum Circuits from the Star-Triangle Relation

Yuan Miao, Eric Vernier

The star-triangle relation plays an important role in the realm of exactly solvable models, offering exact results for classical two-dimensional statistical mechanical models. In this article, we construct integrable quantum circuits using the star-triangle relation. Our construction relies on families of mutually commuting two-parameter transfer matrices for statistical mechanical models solved by the star-triangle relation, and differs from previously known constructions based on Yang-Baxter integrable vertex models. At special value of the spectral parameter, the transfer matrices are mapped into integrable quantum circuits, for which infinite families of local conserved charges can be derived. We demonstrate the construction by giving two examples of circuits acting on a chain of Q− state qudits: Q -state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and ZQ circuits, which are novel to our knowledge. In the first example, we present for Q=3 a connection to the Zamolodchikov-Fateev 19-vertex model.

arXiv:2302.12675•Quantum•Nov 3, 2023•17 citations

The Yang-Baxter Equation for the Chiral Potts Model and Integrable Parafermions

Zhao Zhang

A new type of Yang-Baxter equation (YBE) for $R$-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the $Z_N$ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for $N>2$, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated $R$-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter $R$-matrix---realized as an intertwiner of cyclic representations of $U_q(\mathfrak{sl}_2)$ at a root of unity---and Shastry's two-parameter $R$-operator obtained from the decorated YBE.

math-phCross-listedarXiv:2606.11146•Jun 10, 2026•0 citations
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".
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.
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.
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.
The star-triangle relation plays an important role in the realm of exactly solvable models, offering exact results for classical two-dimensional statistical mechanical models. In this article, we construct integrable quantum circuits using the star-triangle relation. Our construction relies on families of mutually commuting two-parameter transfer matrices for statistical mechanical models solved by the star-triangle relation, and differs from previously known constructions based on Yang-Baxter integrable vertex models. At special value of the spectral parameter, the transfer matrices are mapped into integrable quantum circuits, for which infinite families of local conserved charges can be derived. We demonstrate the construction by giving two examples of circuits acting on a chain of Q− state qudits: Q -state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and ZQ circuits, which are novel to our knowledge. In the first example, we present for Q=3 a connection to the Zamolodchikov-Fateev 19-vertex model.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.