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Advanced series in mathematical physics

Partition Function of the Eight-Vertex Lattice Model

R. J. Baxter

Published Mar 1, 1990• 1 citation
Publisher

Abstract

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.

DOI
10.1142/9789812798336_0003

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Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully nonuniversal

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arXiv:1708.06115•Physical review. E•Jan 9, 2018•3 citations
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.
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.
Partition function zeros are powerful tools in understanding critical behavior. In this paper, we present results of the Fisher zeros of two-dimensional Ising models, in the framework of free-fermion eight-vertex model. First, we succeed in finding special boundary conditions for the free-fermion model, under which the partition function of a finite lattice can be expressed in a double product form. Using appropriate mappings, these boundary conditions are transformed into the corresponding versions of the square, triangular, and honeycomb lattice Ising models. Each Ising model is studied in the cases of a zero field and of an imaginary field i ( π / 2 ) k B T . For the square lattice model, we rediscover the famous Brascamp-Kunz (B-K) boundary conditions. For the triangular and honeycomb lattice models, we obtain the B-K-type boundary conditions, and the Fisher zeros are conveniently solved from the product form of partition function. The advantage of B-K-type boundary conditions is that the Fisher zeros of any finite lattice exactly lie on certain loci, and the accumulation points of zeros can be easily determined in the thermodynamic limit. Our finding and method would be very helpful in studying the partition function zeros of vertex and Ising models.
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.
The partition function of the symmetric (zero electric field) eight-vertex model on a square lattice can be formulated either in the original "electric" vertex format or in an equivalent "magnetic" Ising-spin format. In this paper, both electric and magnetic versions of the model are studied numerically by using the corner transfer matrix renormalization-group method which provides reliable data. The emphasis is put on the calculation of four specific critical exponents, related by two scaling relations, and of the central charge. The numerical method is first tested in the magnetic format, the obtained dependencies of critical exponents on the model's parameters agree with Baxter's exact solution, and weak universality is confirmed within the accuracy of the method due to the finite size of the system. In particular, the critical exponents η and δ are constant as required by weak universality. On the other hand, in the electric format, analytic formulas based on the scaling relations are derived for the critical exponents η_{e} and δ_{e} which agree with our numerical data. These exponents depend on the model's parameters which is evidence for the full nonuniversality of the symmetric eight-vertex model in the original electric formulation.