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. B, Solid state

Staggered eight-vertex model

C.S. Hsue, Ken‐Yu Lin, Fa Yueh Wu

Published Jul 1, 1975• 45 citations
Publisher

Abstract

An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent $\ensuremath{\alpha}=\frac{1}{2}$ and the system is frozen below a unique transition point. Conditions for these special cases are given.

DOI
10.1103/physrevb.12.429

Related papers

Staggered eight-vertex model on the Kagome lattice

Ken‐Yu Lin

An eight-vertex model on the Kagome lattice with staggered (site-dependent) vertex weights is considered. The soluble case of a free-fermion model is solved by the Pfaffian method. The staggered free-fermion model may exhibit up to five phase transitions. In general the specific heat has logarithmic singularities, except in some special cases where the system exhibits first- or second-order phase transition(s).

Journal of Physics A Mathematical and General•Apr 1, 1976•6 citations

Staggered eight-vertex model with four sublattices

Ken‐Yu Lin, I P Wang

The staggered eight-vertex model on a square lattice which allows different vertex weights for the four sublattices of the square lattice has been studied. The soluble case of a free-fermion model is solved by the Pfaffian method. It is shown that the model may exhibit up to five phase transitions. In general the specific heat has logarithmic singularities, except in some special cases where the system exhibits first- or second-order phase transitions.

Journal of Physics A Mathematical and General•May 1, 1977•9 citations

Eight-vertex model on a ruby lattice

Ken‐Yu Lin

An eight-vertex model on a ruby lattice with staggered (site-dependent) vertex weights is considered. The special case of a free-fermion model is solved by the Pfaffian method. In general the special heat has logarithmic singularity at the critical temperature, except in some special cases where the system exhibits a first- or second-order phase transition.

Journal of Physics A Mathematical and General•Nov 11, 1984•3 citations

Ice-Rule Vertex Model on a 34-6 Lattice

Ken‐Yu Lin

An ice-rule vertex model on a 3-4-6 lattice with staggered (site-dependent) vertex weights is considered. The special case of a free-fermion model is solved exactly by the Pfaffian method. The system exhibits second-order phase transition and the specific heat diverges with an exponent 1/2.

Chinese Journal of Physics•Jul 1, 1985•0 citations

The odd eight-vertex model

Fa Yueh Wu, H. Kunz

We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the freefermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions. Key words: Odd eight-vertex model, free-fermion model, exact solution. 1 2 1

•Jan 1, 2008•2 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
An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent \ensuremathα=12\ensuremath{\alpha}=\frac{1}{2}\ensuremathα=21​ and the system is frozen below a unique transition point. Conditions for these special cases are given.
We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the freefermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions. Key words: Odd eight-vertex model, free-fermion model, exact solution. 1 2 1
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.