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arXiv:cond-mat/9706042 · Series on directions in condensed matter physics

EXPERIMENTS ON THE RANDOM FIELD ISING MODEL

David Peter Belanger

Published Dec 1, 1997• 25 citations
arXivHTMLPDF

Abstract

New advances in experiments on the random-field Ising model, as realized in dilute antiferromagnets, have brought us much closer to a full characterization of the static and dynamic critical behavior of the unusual phase transition in three dimensions (d=3). The most important experiments that have laid the ground work for our present understanding are reviewed. Comparisons of the data with Monte Carlo simulations of the d=3 critical behavior are made. We review the current experimental understanding of the destroyed d=2 transition and the experiments exploring the d=2 metastability at low T. Connections to theories most relevant to the interpretations of all the experiments are discussed.

DOI
10.1142/9789812819437_0008

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New advances in experiments on the random-field Ising model, as realized in dilute antiferromagnets, have brought us much closer to a full characterization of the static and dynamic critical behavior of the unusual phase transition in three dimensions (d=3). The most important experiments that have laid the ground work for our present understanding are reviewed. Comparisons of the data with Monte Carlo simulations of the d=3 critical behavior are made. We review the current experimental understanding of the destroyed d=2 transition and the experiments exploring the d=2 metastability at low T. Connections to theories most relevant to the interpretations of all the experiments 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.
We report results of large Monte Carlo simulations of an experimentally realizable random-field system: the three-dimensional dilute Ising antiferromagnetic in a field. We find that the correlation time diverges dramatically as $T\ensuremath{\rightarrow}{T}_{c}$; the results are consistent with a proposed new type of activated dynamic scaling. The transition appears to be continuous, with effective critical exponents $\ensuremath{\eta}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.3$ for strong fields, away from the weak-field regime where crossover effects distort exponent estimates. These strong-field exponents satisfy recently derived inequalities.
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.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.