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Physical review. B, Solid state

Modified universality hypothesis for the eight-vertex model

J. D. Gunton, Th. Niemeijer

Published Jan 1, 1975• 17 citations
Publisher

Abstract

This paper is a comment on recent papers by Kadanoff and Wegner, and Barber and Baxter, concerning the critical exponents of the eight-vertex model. It is shown that the critical exponents of this model as derived by these authors can be obtained from the square Ising lattice critical exponents by a renormalization of the temperature only. A new "universality" hypothesis is proposed which is consistent with these results.

DOI
10.1103/physrevb.11.567

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Generalized codes and their application to Ising models with four-spin interactions including the eight-vertex model

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Generalized codes are defined for the simple quadratic lattice and it is shown how they may be used to derive high magnetic field or low temperature expansions for two Ising-type models with four-spin interactions. One of these models, namely Baxter's eight-vertex model, has critical exponents which are known to depend on x, the ratio of the strengths of the four-spin to two-spin interactions. The series so derived are analysed to yield estimates of the critical exponent delta for the magnetization as a function of magnetic field at T c . The results for both models are consistent with a constant value delta =15 independent of x as predicted by the scaling laws.

Journal of Physics A Mathematical Nuclear and General•Sep 1, 1974•6 citations
This paper is a comment on recent papers by Kadanoff and Wegner, and Barber and Baxter, concerning the critical exponents of the eight-vertex model. It is shown that the critical exponents of this model as derived by these authors can be obtained from the square Ising lattice critical exponents by a renormalization of the temperature only. A new "universality" hypothesis is proposed which is consistent with these results.
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.
Results on the critical temperature of the eight-vertex model in any dimension are combined with the Griffiths inequalities for correlations in Ising ferromagnets to yield relations between critical exponents of eight-vertex models of differing dimensionality. Then Baxter's exact results for the two-dimensional, eight-vertex model are able to yield information on the three-dimensional exponents of these models. As in the planar eight-vertex models, the exponents in all dimensions probably vary continuously with the energy parameters.
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.
Generalized codes are defined for the simple quadratic lattice and it is shown how they may be used to derive high magnetic field or low temperature expansions for two Ising-type models with four-spin interactions. One of these models, namely Baxter's eight-vertex model, has critical exponents which are known to depend on x, the ratio of the strengths of the four-spin to two-spin interactions. The series so derived are analysed to yield estimates of the critical exponent delta for the magnetization as a function of magnetic field at T c . The results for both models are consistent with a constant value delta =15 independent of x as predicted by the scaling laws.