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arXiv:cond-mat/0207415 · Physical Review Letters

Scale Invariance in Disordered Systems: The Example of the Random-Field Ising Model

Giorgio Parisi A, Nicolas Sourlas

Published Dec 4, 2002• 73 citations
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Abstract

We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.

DOI
10.1103/physrevlett.89.257204

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Scale invariance and self-averaging in disordered systems

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In a previous paper (Parisi G. and Sourlas N., Phys. Rev. Lett. 89 (2002) 257204) we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.

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Abstract We analyze the nonequilibrium athermal random field Ising model (RFIM) at equilateral cubic lattices of finite size L and show that the entire range of disorder consists of three distinct domains in which the model manifests different scaling behaviour. The first domain contains the values of disorder R that are below the critical disorder R c where the spanning avalanches almost surely appear when the system is driven by the external magnetic field. The spanning avalanches become unlikely for disorders above the size-dependent effective disorder R c eff ( L ) > R c , and the system response is size-independent. Between the foregoing two lies the domain of transitional disorders R c < R < R c eff ( L ) vanishing in the thermodynamic limit. In this domain, not recognized in the literature so far, all types of spanning avalanches exist, whereas for R < R c only the avalanches spanning all three dimensions are present. Like for R < R c the data collapsing is possible only for distributions having the same value of (1 − R c / R ) L 1/ ν , however with different universal scaling functions than in R < R c case; ν is the correlation length exponent. The foregoing findings follow from the extensive simulations of L ⩽ 2048 systems enabling us to propose modified values of some of the RFIM critical exponents and nonuniversal critical parameters as well as the analytic forms of universal scaling functions and different definitions of reduced magnetization and reduced magnetic field.

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We present new numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 ± 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self-averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.

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Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.

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We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.
In a previous paper (Parisi G. and Sourlas N., Phys. Rev. Lett. 89 (2002) 257204) we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.
Abstract We analyze the nonequilibrium athermal random field Ising model (RFIM) at equilateral cubic lattices of finite size L and show that the entire range of disorder consists of three distinct domains in which the model manifests different scaling behaviour. The first domain contains the values of disorder R that are below the critical disorder R c where the spanning avalanches almost surely appear when the system is driven by the external magnetic field. The spanning avalanches become unlikely for disorders above the size-dependent effective disorder R c eff ( L ) > R c , and the system response is size-independent. Between the foregoing two lies the domain of transitional disorders R c < R < R c eff ( L ) vanishing in the thermodynamic limit. In this domain, not recognized in the literature so far, all types of spanning avalanches exist, whereas for R < R c only the avalanches spanning all three dimensions are present. Like for R < R c the data collapsing is possible only for distributions having the same value of (1 − R c / R ) L 1/ ν , however with different universal scaling functions than in R < R c case; ν is the correlation length exponent. The foregoing findings follow from the extensive simulations of L ⩽ 2048 systems enabling us to propose modified values of some of the RFIM critical exponents and nonuniversal critical parameters as well as the analytic forms of universal scaling functions and different definitions of reduced magnetization and reduced magnetic field.
We present new numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 ± 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self-averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.
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 investigate the critical properties of the d=3d=3d=3 random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength hhh and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength hc{h}_{\mathrm{c}}hc​ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.
Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.