Universality in the off-equilibrium critical dynamics of the three-dimensional diluted Ising model

Abstract

We study the off-equilibrium critical dynamics of the three dimensional diluted Ising model. We compute the dynamical critical exponent z and we show that it is independent of the dilution only when we take into account the scaling-corrections to the dynamics. Finally we will compare our results with the experimental data.PACS numbers: 05.50.+q, 75.10.Nr, 75.40.Mg The issue of Universality in disordered systems is a controversial and interesting subject.Very often in the past it has been argued that critical exponents change with the strength of the disorder [1]. While, on a deeper analysis, it has turned out that those exponents were "effective" ones, i.e. they are affected by strong scaling corrections. So, when one studies the critical behavior of a disordered system it is mandatory to control the leading correction-to-scaling in order to avoid these effects that could modify the dilution-independent values of the critical exponents. For instance, in Ref.[2] the equilibrium critical behavior of the three dimensional diluted Ising model was studied. The authors showed that taking into account the corrections-to-scaling it was possible to show that the static critical exponents (e.g. ν and η) and cumulants were dilution-independent. These numerical facts supports the (static) perturbative renormalization group picture: all the points of the critical line (with p < 1) belong to the same Universality class (with critical exponents given by the random fixed point) [3]. Their final values of the exponents [2] were in very good agreement with the experimental figures (see below).We will show that an analogous effect also happens in the off-equilibrium dynamics of the diluted ferromagnetic model and we will take it into account in our data analysis, in order to get the best estimate of the critical dynamical exponent.The critical dynamics of the diluted Ising model has been studied experimentally in Ref.[4] using neutron spin-echo inelastic scattering on samples of Fe 0.46 Zn 0.54 F 2 (antiferromagnetic diluted model) and has been compared with the results obtained in pure samples (FeF 2 ) [4]. For the pure model a dynamical critical exponent z = 2.1(1) was found (in good agreement with the theoretical predictions based on one-loop perturbative renormalization group (PRG) [5]) whereas in the diluted case the exponent z = 1.7(2) was computed (three standard deviations away of the analytical prediction based on (one-loop) PRG that provides z ≃ 2.34 [6]). Furthermore, the dynamical exponent was computed in the framework of the PRG up to two loops and it was obtained z = 2.237 [7] and z = 2.180 [8] (the experimental value is at 2.5 standard deviation of the two loops analytical result).In the experiment [4] were measured critical amplitudes 100 times smaller than those computed in the pure case. It is clear that a more precise experiment on this issue will be welcome. We should point out that the critical dynamics of a diluted antiferromagnet is the same as of a diluted ferromagnet.A numerical study of the on-equilibr...

Finite-time Scaling and its Applications to Continuous Phase Transitions

“…Second, they help to unify the exponents. For example, our dynamic critical exponent of z = 2.114(51) supports a lower value found by renormalization-group analyzes, experiments, and some Monte Carlo simulations rather than the larger values of z ≈ 2.6 (Parisi et al, 1999;Schehr & Paul, 2005) and z ≈ 2.35 (Calabrese et al, 2008;Hasenbusch et al, 2007). Third, they corroborate the universality of the random fixed point with respect to the form of disorders.…”

Nonsteady relaxation and critical exponents at the depinning transition

“…This suggests two simple pictures: (i) In the first picture we simply introduce phenomenological corrections to the dynamical exponent as (t) ∼ t 1/z (1 + |c 3 |t − ) with c 3 and parameters of the correction. This type of correction has been introduced in studies of the off-equilibrium critical dynamics of the three-dimensional diluted Ising model, for instance, and attributed to the biggest irrelevant eigenvalue of the renormalization group in the dynamics [62]. If we assume that v(t) ∼ (t) −β/ν holds, it is easy to see that such a correction would lead to a corrected velocity v(t) ∼ t −β/zν (1 + |c 2 |t −σ ) with parameters c 2 and σ .…”

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“…28 In principle w and ω have no fundamental reason to be exactly equal but can be expected to be similar. Finally, if the measurement is continued long enough, the long-time susceptibility will cross over to a saturation value which is just the equilibrium susceptibility χ(L, T ) for the measurement temperature T and the sample size L used.…”