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Journal of Physics A Mathematical and General

Dipolar interaction in random-field systems

Thomas Nattermann

Published Jun 21, 1988• 20 citations
Publisher

Abstract

It is shown that the dipolar interaction breaks the universality of the low-temperature and critical behaviour between the random-field Ising model (RFIM) and the diluted antiferromagnet in an external field (DAFF). For the RFIM, the dipolar interaction changes the roughness and the critical exponents; in particular, the upper critical dimensionalities are diminished by one. The dipolar interaction exerts no influence on these exponents in the DAFF, nor on the lower critical dimensionality d l =2 of both systems.

DOI
10.1088/0305-4470/21/12/005

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It is shown that the dipolar interaction breaks the universality of the low-temperature and critical behaviour between the random-field Ising model (RFIM) and the diluted antiferromagnet in an external field (DAFF). For the RFIM, the dipolar interaction changes the roughness and the critical exponents; in particular, the upper critical dimensionalities are diminished by one. The dipolar interaction exerts no influence on these exponents in the DAFF, nor on the lower critical dimensionality d l =2 of both systems.
The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).
We have studied the equilibrium properties of the random-field (H) ferromagnetic Ising model (RFIM) and the dilute antiferromagnetic Ising model (DAF) in a uniform field. For RFIM we have used the transfer-matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S versus H. Our results are consistent with S\ensuremath{\sim}exp(c/H2{H}^{2}H2) as predicted by theories which yield a lower critical dimension dc{d}_{c}dc​=2, whereas the expected behavior for dc{d}_{c}dc​=3, S\ensuremath{\sim}${H}^{\mathrm{\ensuremath{-}}4}$ is inconsistent with our results. We have carried out analogous calculations for DAF. Our results for DAF suffer seriously from finite-size effects. To circumvent this difficulty, we have also performed calculations on interface models. Together, these results indicate that DAF crosses over to RFIM behavior at H\ensuremath{\simeq}J.
A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.
We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
The random-field Ising model (RFIM) is one of the simplest statistical-mechanical models that captures the anomalous irreversible collective response seen in a wide range of physical, biological, or socioeconomic situations in the presence of interactions and intrinsic heterogeneity or disorder. When slowly driven at zero temperature, it can display an out-of-equilibrium phase transition associated with critical scaling (``crackling noise''), while it undergoes at equilibrium, under either temperature or disorder-strength changes, a thermodynamic phase transition. We show that the out-of-equilibrium and equilibrium critical behaviors are in the same universality class: they are controlled, in the renormalization-group (RG) sense, by the same zero-temperature fixed point. We do so by combining a field-theoretical formalism that accounts for the multiple metastable states and the exact (functional) RG. As a spin-off, we also demonstrate that critical fluids in disordered porous media are in the same universality class as the RFIM, thereby unifying a broad spectrum of equilibrium and out-of-equilibrium phenomena.