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Journal of Physics C Solid State Physics

On the spontaneous order of the eight-vertex model

Michael N. Barber, R. J. Baxter

Published Oct 16, 1973• 71 citations
Publisher

Abstract

A functional form for the spontaneous order M 0 of the eight-vertex model is conjectured from an investigation of the structure of a low-temperature expansion of M 0 . The precise expression then follows from the known result for the Ising case. The exponent beta is then obtained and three-exponent scaling used to predict the remaining exponents of the eight-vertex model from the known exponents.

DOI
10.1088/0022-3719/6/20/004

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Physical review. B, Solid state•Feb 1, 1970•113 citations
A functional form for the spontaneous order M 0 of the eight-vertex model is conjectured from an investigation of the structure of a low-temperature expansion of M 0 . The precise expression then follows from the known result for the Ising case. The exponent beta is then obtained and three-exponent scaling used to predict the remaining exponents of the eight-vertex model from the known exponents.
Below its critical temperature, the triangular Ising model with pure three-spin interactions has a spontaneous magnetization M. The first 13 terms of a series expansion for M are known. Here these are re-arranged into a form which can be extrapolated to all terms in a natural way. Thus a conjecture is obtained for M. It fits the scaling prediction beta =1/12. Another order parameter is similarly discussed, together with the difficulties encountered in applying the technique to the two-spin Ising model susceptibility.
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.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.