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Philosophical Magazine B

Non-exponential relaxation near a transition in the presence of disorder: The case of spin glasses

Jean Souletie

Published Jun 1, 1992• 3 citations
Publisher

Abstract

Within the framework of a model which associates Kadanoff's renormalization with a hierarchical condition, we justify dynamic scaling (τ/τ ≈ (ξ/ξ0)z) and non-exponential relaxations (m(t) = t −1/zv at T = T C). In the presence of some disorder the same model predicts β → α transition from a short-time regime where we observe activated T In t scaling for short-range effects, to a regime where we have stretched exponential behaviour and dynamic scaling (τ ≈ τ0(1 – T C/T)−zv) for long-range effects.

DOI
10.1080/13642819208215097

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The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).

Cambridge University Press eBooks•May 2, 1991•0 citations
Within the framework of a model which associates Kadanoff's renormalization with a hierarchical condition, we justify dynamic scaling (τ/τ ≈ (ξ/ξ0)z) and non-exponential relaxations (m(t) = t −1/zv at T = T C). In the presence of some disorder the same model predicts β → α transition from a short-time regime where we observe activated T In t scaling for short-range effects, to a regime where we have stretched exponential behaviour and dynamic scaling (τ ≈ τ0(1 – T C/T)−zv) for long-range effects.
Assuming that hierarchical procedure governs the evolution from one to the next step of Kadanoff's renormalization in the presence of some disorder one finds that the dynamic response is described by a log-normal distribution of the relaxation times. There are two natural limits to the model which restitute either the dynamic scaling or the activated dynamic hypotheses. This provides the basis for a simple justification to the success of the usual fits by a stretched exponential or by a Cole-Cole law which are commonly made to the non-exponential responses observed in the time or in the frequency domains in spin glasses. We provide expressions for the temperature dependences of the parameters which control these laws. We also propose a model for freezing which is based on the idea that ∂\partial∂ ln τ/∂(1/T)\tau/\partial(1/T)τ/∂(1/T) should be continuous at the freezing point. We discuss available data in spin glasses but our conclusions should be very general.
A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’
We extend an earlier fractal cluster model of spin glasses to study (1) the magnetization relaxation M(t) induced by a magnetic field, (2) magnetic noise, and (3) dynamics near a critical line Tg{T}_{g}Tg​(H). Above the zero-field transition temperature Tg{T}_{g}Tg​, M(t) follows a stretched exponential form exp(-${t}^{1\mathrm{\ensuremath{-}}n}$) with n=\ensuremath{\nu}z/(\ensuremath{\varphi}+\ensuremath{\nu}z), where \ensuremath{\varphi},\ensuremath{\nu}, and z are standard static and dynamical critical exponents. A dynamic scaling relation is derived for the entire region Tg{T}_{g}Tg​(H)Tg{T}_{g}Tg​ in agreement with experiments. The equations associated with lines of constant relaxation time are obtained and it is shown how nonuniversality of exponents along the critical line can be tested.
Following a brief introduction to spin-glass systems, the current status of experimental evidence on the occurrence of a phase transition is reviewed. Recent analyses of dynamical properties of spin glasses are discussed in terms of several model predictions, including the conclusion of a divergent relaxation time τ at a finite temperature, although with large, non-mean-field indices z and v. The role of regular terms (leading to corrections to scaling) in the nonlinear response are reviewed, stressing their influence on the evaluation of the susceptibility and order-parameter exponents γ′ and β′.
The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).