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Physica A Statistical Mechanics and its Applications

A general theory of spin-one Ising models in the correlated effective-field approximation

T. Kaneyoshi

Published Mar 1, 1992• 13 citations
Publisher
DOI
10.1016/0378-4371(92)90353-r

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The bimodal random crystal field(?) effects are investigated on the phase diagrams of spin-3/2 Ising model by using the effective-field theory with correlations based on two approximations:the general van der Waerden identity and the approximated van der Waerden identity.In our approach,the crystal field is either turned on or turned off randomly for a given probability p or q = 1- p,respectively.Then the phase diagrams are constructed on the(?,kT /J) and(p,kT /J)planes for given p and ?,respectively,when the coordination number is z = 3.Furthermore,the effect of randomization of the crystal field is illustrated on the(?,kT /J) plane for p = 0.5 when z = 3,4,and 6.All these are carried out for both approximations and then the results are compared to point out the differences.In addition to the lines of second-order phase transitions,the model also exhibits first-order phase transitions and the lines of which terminate at the isolated critical points for high p values.
The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inverse Ising problem without running the Susceptibility Propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in presence of an external field.