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Advanced series in mathematical physics

SOLVABLE EIGHT-VERTEX MODEL ON AN ARBITRARY PLANAR LATTICE

R. J. Baxter

Published Mar 1, 1990• 0 citations
Publisher

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 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.

DOI
10.1142/9789812798336_0004

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Advanced series in mathematical physics•Mar 1, 1990•1 citation
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.
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.
This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.
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.
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.
Simple procedures are first used to obtain exact solutions of highly-localized odd-number Ising correlations on the kagom\'e, square, and honeycomb lattices. To extend these results, a systematic and unifying method is then developed and demonstrated for finding exact solutions of n-site (with n an odd integer) Ising correlations on various planar lattices. The method combines five transformation or mapping theorems and linear-algebraic correlation identities of the triangular Ising model supplemented by a foreknowledge of its spontaneous magnetization and three select triplet correlations. In particular, considering a select seven-site cluster of the triangular Ising model, the knowledge of all its eleven odd-number correlations defined upon this cluster is shown sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising odd-number correlations upon their correspondingly select 10-, 19-, and 9-site clusters, respectively. The direct applicability of the catenated mapping theorems and relative ease of the calculational procedures are highlighted by the resulting large numbers of multisite correlation solutions (e.g., approximately 80 and 50 for the honeycomb and kagom\'e Ising models, respectively), the large nmax{\mathit{n}}_{\mathrm{max}}nmax​ values (nmax{\mathit{n}}_{\mathrm{max}}nmax​=9, 9, and 19, respectively, for the honeycomb, kagom\'e, and decorated-honeycomb Ising models), and convenient prescriptions for extracting critical amplitudes. The results also offer examples of correlation degeneracies and other linear-algebraic correlation identities that do not depend explicitly upon the interaction parameters.