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Physical Review Letters

Eight-Vertex Model in Lattice Statistics

R. J. Baxter

Published Apr 5, 1971• 627 citations
Publisher

Abstract

The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, $F$, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.

DOI
10.1103/physrevlett.26.832

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Partition Function of the Eight-Vertex Lattice Model

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The partition function of the zero-field “Eight-Vertex” model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase transition, with an irrational exponent.

Advanced series in mathematical physics•Mar 1, 1990•1 citation

Solution of eight-vertex model excites critical-point theorists

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A difficult problem in two-dimensional statistical mechanics has been solved by R. J. Baxter of the Australian National University in Canberra (Phys. Rev. Lett. 26, 832, 1971). The so-called “eight-vertex” model that he solved contains as special cases the square lattice Ising, dimer, ice, F and KDP (potassium dihydrogen phosphate) models. Baxter's elegant and beautiful solution has challenged one of the widely held precepts of critical-point theory, namely that the character of the critical behavior is independent of the substance provided the symmetry properties of the Hamiltonian are the same.

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Critical properties of the eight-vertex model in a field

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The general eight-vertex model on a square lattice is studied numerically by using the Corner Transfer Matrix Renormalization Group method. The method is tested on the symmetric (zero-field) version of the model, the obtained dependence of critical exponents on model's parameters is in agreement with Baxter's exact solution and weak universality is verified with a high accuracy. It was suggested long time ago that the symmetric eight-vertex model is a special exceptional case and in the presence of external fields the eight-vertex model falls into the Ising universality class. We confirm numerically this conjecture in a subspace of vertex weights, except for two specific combinations of vertical and horizontal fields for which the system still exhibits weak universality.

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SOLVABLE EIGHT-VERTEX MODEL ON AN ARBITRARY PLANAR LATTICE

R. J. Baxter

Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.

Advanced series in mathematical physics•Mar 1, 1990•0 citations

Eight-Vertex Model in Lattice Statistics and One-Dimensional Anisotropic Heisenberg Chain: II. Equivalence to a Generalized Ice-type Lattice Model

R. J. Baxter

Abstract We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.

Advanced series in mathematical physics•Mar 1, 1990•0 citations
The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, FFF, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.
A difficult problem in two-dimensional statistical mechanics has been solved by R. J. Baxter of the Australian National University in Canberra (Phys. Rev. Lett. 26, 832, 1971). The so-called “eight-vertex” model that he solved contains as special cases the square lattice Ising, dimer, ice, F and KDP (potassium dihydrogen phosphate) models. Baxter's elegant and beautiful solution has challenged one of the widely held precepts of critical-point theory, namely that the character of the critical behavior is independent of the substance provided the symmetry properties of the Hamiltonian are the same.
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.
The general eight-vertex model on a square lattice is studied numerically by using the Corner Transfer Matrix Renormalization Group method. The method is tested on the symmetric (zero-field) version of the model, the obtained dependence of critical exponents on model's parameters is in agreement with Baxter's exact solution and weak universality is verified with a high accuracy. It was suggested long time ago that the symmetric eight-vertex model is a special exceptional case and in the presence of external fields the eight-vertex model falls into the Ising universality class. We confirm numerically this conjecture in a subspace of vertex weights, except for two specific combinations of vertical and horizontal fields for which the system still exhibits weak universality.
Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.