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Physics Letters A

The ϵ-expansion and parametric models for the ising equation of state in the critical region

D. J. Wallace, Royce K. P. Zia

Published Dec 1, 1973• 6 citations
Publisher
DOI
10.1016/0375-9601(73)90212-0

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The authors report a microscopic derivation of the parametric representation of the equation of state for an Ising-like system by means of the skeleton expansion technique. Such a representation explicitly describes the crossover behaviour from the small correlation length region to the critical region. The Gaussian-Ising crossover is obtained at criticality. The derivation is based on differential relations that connect natural thermodynamical variables with the parameters defining the representation. Differentiability properties are checked in the small epsilon limit.
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.
We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.