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Journal of Statistical Physics

A self-consistent Ornstein-Zernike approximation for the site-diluted Ising model

Édouard Kierlik, Martin Luc Rosinberg, Gilles Tarjus

Published Oct 1, 1997• 19 citations
Publisher
DOI
10.1007/bf02770762

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The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Hückel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.
The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Huckel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.
We use a self-consistent Ornstein-Zernike approximation to study the Blume-Capel ferromagnet on three-dimensional lattices. The correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations. The theory provides a comprehensive and accurate description of the phase diagram in all regions, including the wing boundaries in a nonzero magnetic field. In particular, the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion. Numerical and analytical analysis strongly suggest that the theory predicts a universal Ising-like critical behavior along the lambda line and the wing critical lines, and a tricritical behavior governed by mean-field exponents.