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 de Physique I

Solvable weak-graph duals of partially frozen vertex models

M. Kolesík, Ladislav Šamaj, P. Markoš

Published Jul 1, 1992• 0 citations
Publisher

Abstract

A family of q-state vertex models on a square lattice is solved exactly using the generalized weak-graph transformation. Concrete physical symmetries of vertex weights turn out to be closely related to abelian groups. The possibility of occurrence of the first-order phase transition is discussed.
DOI
10.1051/jp1:1992212

Related papers

Exact solution of a ten-vertex model in two dimensions

Roberto Nicolau Onody, M. Karowski

A ten-vertex model, which is a multi-state generalisation of the six-vertex model, is formulated on a square lattice and solved exactly in the thermodynamic limit using the inverse scattering method. It is found that this system undergoes a first-order phase transition into a 'frozen' ferroelectric state.

Journal of Physics A Mathematical and General•Jan 11, 1983•3 citations

The 16-vertex model and its even and odd 8-vertex subcases on the square lattice

Michael Assis

Abstract We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the S L ( 2 ) × S L ( 2 ) symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field H = i π / ( 2 β ) , valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.

arXiv:1702.02110•Journal of Physics A Mathematical and Theoretical•Aug 4, 2017•9 citations

Full nonuniversality of the symmetric 16-vertex model on the square lattice

Eva Pospíšilová, Roman Krčmár, Andrej Gendiar +1 more

We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically with the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter eight-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.

arXiv:2002.02757•Physical review. E•Jul 13, 2020•0 citations

General Square Lattice Vertex Models

Hans-Peter Eckle

Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.

•Jul 25, 2019•0 citations

New Factorized S-Matrix and Its Application to Exactly Solvable q-State Model. II

Kiyoshi Sogo, Yuko Akutsu, T. Abe

The partition function of the q-state vertex model is derived by the use of the inversion method. The connection of the vertex model to the solid on solid model is established by the generalized Wu-Kadanoff-Wegner transformation. The critical behavior of the model is discussed in relation with the roughening transition.

Progress of Theoretical Physics•Sep 1, 1983•106 citations

Integrable Vertex Models with General Twists(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Marcio Jose Martins

We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions.Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method.Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the $B_{n},$ $D_{n}$ and $C_{n}$ Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

数理解析研究所講究録•Apr 1, 2006•0 citations
Abstract We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the S L ( 2 ) × S L ( 2 ) symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field H = i π / ( 2 β ) , valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.
We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically with the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter eight-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions.Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method.Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the Bn,B_{n},Bn​, DnD_{n}Dn​ and CnC_{n}Cn​ Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.