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Physics Today

Solution of eight-vertex model excites critical-point theorists

Gloria B. Lubkin

Published Sep 1, 1971• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.3022922

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Solvable eight-vertex model on an arbitrary planar lattice

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Abstract 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 Kagomé 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.

Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences•May 2, 1978•301 citations
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.
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.
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.
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.
The ice model Introduction The six-vertex model , which is the object of this chapter, is a special case of the eight-vertex model on a two-dimensional square lattice introduced by Fan and Wu (1970) in order to summarize a class of exactly solvable models in classical statistical mechanics. The thermodynamics of the six-vertex model is by now known in its full generality (Yang and Yang, 1966a–d; Lieb, 1967a) which is not the case for the eight-vertex model, but only the self-conjugate one (Baxter, 1971a). The general model can be considered as a two-dimensional idealization of a crystalline system in which pairs of adjacent atoms or radicals on the network are linked by ‘hydrogen bonds’. Of ionic type, this link between two neighbouring electronegative atoms is realized by a proton H + which is located closer to one of the atoms than to the other. On each link of the network there thus exist two equilibrium positions for H + (Pauling, 1960). If the coordination number at each site equals 4, as is the case for the oxygen atom in hexagonal ice, we indeed have eight possible proton configurations on a given site, which give rise to the eight vertices. In contrast, for the physical systems for which this model could be viewed as a valid idealization – ice H 2 O, the ferroelectric PO 4 H 2 K, the antiferroelectric PO 4 H 2 NH 4 – there exist at most two H + next to each site.
Abstract 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 Kagomé 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.