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Mean-field theories

Julia M. Yeomans

Published May 7, 1992• 3 citations
Publisher

Abstract

Abstract Very few statistical mechanical models have been solved exactly. In three dimensions not even the nearest neighbour spin-1/2 Ising model is tractable. Therefore it is necessary to resort to approximation methods. One of the most widely used of these is mean-field theory. A systematic way of deriving the mean-field theory for a given microscopic Hamiltonian is to start from the Bogoliubov inequality This gives the best possible approximation to the true free energy for a given choice of Ho because the inequality (4.1) insists that the meanfield free energy cannot fall below the true free energy. (This is analogous to the variational principle in quantum mechanics.) The usual choice for Ho is a free Hamiltonian -one with no interactions-as this enables explicit calculation of the right-hand side of (4.1).

DOI
10.1093/oso/9780198517290.003.0004

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Abstract Very few statistical mechanical models have been solved exactly. In three dimensions not even the nearest neighbour spin-1/2 Ising model is tractable. Therefore it is necessary to resort to approximation methods. One of the most widely used of these is mean-field theory. A systematic way of deriving the mean-field theory for a given microscopic Hamiltonian is to start from the Bogoliubov inequality This gives the best possible approximation to the true free energy for a given choice of Ho because the inequality (4.1) insists that the meanfield free energy cannot fall below the true free energy. (This is analogous to the variational principle in quantum mechanics.) The usual choice for Ho is a free Hamiltonian -one with no interactions-as this enables explicit calculation of the right-hand side of (4.1).
The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.
Before embarking on an exploration of the modern theories of critical behaviour, it is wise to consider first the more traditional approaches, generally lumped together under the heading of mean field theory . Despite the fact that such methods generally fail sufficiently close to the critical point, there are nonetheless several good reasons for their study. Mean field theory is relatively simple to apply in most cases, and often gives a qualitatively correct picture of the phase diagram of a given model. In some cases, where fluctuation effects are suppressed for some physical reason, its predictions are also quantitatively accurate. In a sufficiently large number of spatial dimensions, it gives exact values for the various critical exponents. Moreover, it often serves as an important adjunct to the renormalization group, since the latter by itself may give direct information about the existence and location of phase transitions, but not about the nature of the phases which they separate. For this, further input is necessary, and this often may be provided by applying mean field theory in a region of the phase diagram far from the critical region, where it is applicable. The mean field free energy There are as many derivations of the basic equations of mean field theory as there are books written on the subject, all of varying degrees of rigour and completeness.
Abstract A procedure is developed for obtaining approximate thermodynamic properties of statistical models, based on a well known variational principle for the free energy. The procedure involves the choice of a suitable parametrized trial Hamiltonian which contains some elementary interactions. The free trial Hamiltonian leads to the usual mean‐field approximation. When the trial Hamiltonian includes a certain number of pair interactions only, the results are regained of the constant‐coupling approximation for the Ising model if the number of interacting pairs is conveniently chosen by a comparison with the first few terms of the exact high‐temperature expansion. This procedure is simpler than existing approximations and can be easily generalized. Besides treating the Ising model, some applications are presented to the spin‐1/2 antiferromagnetic Heisenberg model, and to the spin‐1 Heisenberg model with uniaxial anisotropy.
In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.