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Journal of Physics Condensed Matter

Fluctuations in the random-field Ising model

Terufumi Yokota

Published Jun 15, 1998• 0 citations
Publisher

Abstract

Fluctuations can break down mean-field critical behaviour. For the random-field Ising model, fluctuations caused by the randomness are important. The Ginzburg criteria based on two kinds of mean-field theory are discussed. One is the mean-field theory using the replica method and the other is the site-dependent mean-field theory. An argument that justifies the use of the site-dependent mean-field theory to obtain critical properties for the random-field Ising model is given.

DOI
10.1088/0953-8984/10/23/018

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Fluctuations can break down mean-field critical behaviour. For the random-field Ising model, fluctuations caused by the randomness are important. The Ginzburg criteria based on two kinds of mean-field theory are discussed. One is the mean-field theory using the replica method and the other is the site-dependent mean-field theory. An argument that justifies the use of the site-dependent mean-field theory to obtain critical properties for the random-field Ising model is given.
The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d≤2d\le 2d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.