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Journal of Physics C Solid State Physics

Note on universality and the eight-vertex model

K Jungling, Gustav M. Obermair

Published Oct 21, 1974• 16 citations
Publisher

Abstract

It is shown that the critical behaviour of the general eight-vertex model is equal to that of an Ising model with only two-spin interactions between first, second and third neighbours. The critical exponents of the latter must therefore, in general, depend on the ratios of coupling constants as those of Baxter's eight-vertex model have been shown to do, in contrast to universality so far assumed to hold for all short-range two-spin interactions.

DOI
10.1088/0022-3719/7/20/001

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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.

Physical Review Letters•Feb 18, 1974•4 citations
It is shown that the critical behaviour of the general eight-vertex model is equal to that of an Ising model with only two-spin interactions between first, second and third neighbours. The critical exponents of the latter must therefore, in general, depend on the ratios of coupling constants as those of Baxter's eight-vertex model have been shown to do, in contrast to universality so far assumed to hold for all short-range two-spin interactions.
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.
The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, the critical indices obey $\frac{\ensuremath{\gamma}}{{\ensuremath{\gamma}}_{0}}=\frac{\ensuremath{\beta}}{{\ensuremath{\beta}}_{0}}=\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}}=1\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}$, $\frac{\ensuremath{\delta}}{{\ensuremath{\delta}}_{0}}=\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}=1$, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model.
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.