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Journal of the Physical Society of Japan

Generalization of Mean-Field Approximations by the Feynman Inequality and Application to Long-Range Ising Chain

Minoru Takahashi

Published Jun 1, 1981• 19 citations
Publisher

Abstract

Mean-field approximations such as molecular field approximation for spin system, Hartree-Fock approximation and BCS approximation for electron system are universally formulated by the Feynman inequality: -β -1 ln Tr exp (-β H ) ≤-β -1 ln Tr exp (-β H 0 )+Tr exp (-β H 0 )( H - H 0 )/Tr exp (-β H 0 ). The method of generalization of mean-field approximation is given. This is applied to the problem of one-dimensional long-range Ising model.

DOI
10.1143/jpsj.50.1854

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Mean-field approximations such as molecular field approximation for spin system, Hartree-Fock approximation and BCS approximation for electron system are universally formulated by the Feynman inequality: -β -1 ln Tr exp (-β H ) ≤-β -1 ln Tr exp (-β H 0 )+Tr exp (-β H 0 )( H - H 0 )/Tr exp (-β H 0 ). The method of generalization of mean-field approximation is given. This is applied to the problem of one-dimensional long-range Ising model.
The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.
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.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
The excitation of a many-body system by a one-body perturbation is considered. The stationary phase approximation to a functional-integral representation of the many-body evolution operator shows that the optimal mean field for describing the final expectation values of one- and two-body operators is that given by the time-dependent Hartree-Fock method. The forced Lipkin model is considered as a test of the mean-field approximation.NUCLEAR REACTIONS Mean-field approximation, inclusive observables. Time-dependent Hartree-Fock.
We develop a generalization of the two-dimensional (2D) Ising model in which a 'spin variable' lies in the 2D plane and can have four internal degrees of freedom (four possible pointing directions: up, down, right and left). By considering various forms of near-neighbour interactions, we are able to solve the model with a generalization of the Bragg–William approximation. Within this mean-field theory we examined the behaviour of heat capacity and an 'order parameter' which captures the short-range correlations between the spins. Below a transition temperature these short-range correlations are enhanced and lead to a long-range order. This is examined (for the first time) through the breakdown of statistical independence.
The mean-field approximation is used for the evaluation of the Niemeijer---Van Leeuwen real-space renormalization. Although this new method is very simple, it gives reasonable results for the critical properties of two-, three-, and four-dimensional Ising models, for the transition temperature Tc{T}_{c}Tc​ of an Ising model with clusters and for the two-dimensional Ising antiferromagnet in an external magnetic field. Using this new mean-field renormalization in connection with the real-space renormalization of the kinetic Ising model, which recently has been suggested by the author, we obtain the long-time relaxation of the magnetization for all temperatures above Tc{T}_{c}Tc​. In the last application the critical properties of nnn-dimensional spins in two and three space dimensions are calculated for arbitrary nnn. Here the mean-field approximation and Monte Carlo methods are used to evaluate the renormalization step.