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

Staggered lattice models

David Hansel, Th. Jolicœur, J.-M. Maillard

Published Oct 1, 1987• 3 citations
Publisher

Abstract

Some staggered models are studied along three different lines: exact solvability (Yang-Baxter equations); the existence of other exact solutions called disorder solutions; their group of symmetries. The analysis, based on the commutation of row to row transfer matrices of arbitrary size N, suggests that the exactly solvable models are either of non-staggered type or of free-fermion staggered type. In contrast, disorder solutions can easily be exhibited for staggered models: the example of the Ashkin-Teller model is analysed. Finally, the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models.

DOI
10.1088/0305-4470/20/14/030

Related papers

NEW LOCAL RELATIONS FOR LATTICE MODELS

Jean Avan, J.-M. Maillard, Michel Talon +1 more

We describe new local relations leading to non-trivial (non-homogeneous) equations for the row-to-row transfer matrices of arbitrary size for two dimensional I.R.F. and vertex models. We sketch the connection between this relation and the Yang-Baxter equations, and we describe the example of the hard hexagon model.

International Journal of Modern Physics B•Sep 1, 1990•3 citations

FUNCTIONAL RELATIONS IN SOLVABLE LATTICE MODELS I: FUNCTIONAL RELATIONS AND REPRESENTATION THEORY

ATSUO KUNIBA, TOMOKI NAKANISHI, JUNJI SUZUKI

We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.

arXiv:hep-th/9309137•International Journal of Modern Physics A•Dec 10, 1994•272 citations

Functional relations in solvable lattice models II: Applications

Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki

Abstract. We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r + 1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.

•Jan 31, 2013•72 citations

The Six-Vertex Yang-Baxter Groupoid

Daniel Bump, Slava Naprienko

A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$, $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.

arXiv:2503.05960•Mar 7, 2025•0 citations

Yang-Baxter Relations for Spin Models and Fermion Models

Eugenio Olmedilla, Miki Wadati, Yasuhiro Akutsu

A systematic method to derive the Yang-Baxter relations for the fermion models which are equivalent to the solvable spin models is presented. The method is applied to prove the Yang-Baxter relation and the commutability of the transfer matrices for the one-dimensional Hubbard model.

Journal of the Physical Society of Japan•Jul 15, 1987•88 citations

GENERALIZED YANG-BAXTER EQUATION

Rinat Kashaev, Yu. G. Stroganov

A generalization of the Yang-Baxter equation is proposed. It enables us to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Boltzmann weights of the sl (3) chiral Potts models.

arXiv:hep-th/9305144•Modern Physics Letters A•Aug 10, 1993•8 citations
Some staggered models are studied along three different lines: exact solvability (Yang-Baxter equations); the existence of other exact solutions called disorder solutions; their group of symmetries. The analysis, based on the commutation of row to row transfer matrices of arbitrary size N, suggests that the exactly solvable models are either of non-staggered type or of free-fermion staggered type. In contrast, disorder solutions can easily be exhibited for staggered models: the example of the Ashkin-Teller model is analysed. Finally, the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models.
We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.
Abstract. We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r + 1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid G\mathfrak{G}G together with a map π:G→End⁡(V⊗V)π:\mathfrak{G}\to\operatorname{End}(V\otimes V)π:G→End(V⊗V) for some vector space VVV such that the Yang-Baxter commutator [[π(u),π(w),π(v)]]=0[[ π(u),π(w),π(v)]]=0[[π(u),π(w),π(v)]]=0 if u,v∈Gu,v\in\mathfrak{G}u,v∈G are such that the groupoid composition w=u⋆vw=u\star vw=u⋆v is defined. An important role is played by an object map Δ:G→MΔ:\mathfrak{G}\to MΔ:G→M for some set MMM such that Δ(u)=Δ(v′)Δ(u)=Δ(v')Δ(u)=Δ(v′), Δ(w)=Δ(v)Δ(w)=Δ(v)Δ(w)=Δ(v) and Δ(w′)=Δ(u′)Δ(w')=Δ(u')Δ(w′)=Δ(u′), where v↦v′v\mapsto v'v↦v′ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group GL⁡(2)×GL⁡(1)\operatorname{GL}(2)\times\operatorname{GL}(1)GL(2)×GL(1). For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.