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Journal de Physique I

Dynamic scaling and non exponential relaxations in the presence of disorder. Application to spin glasses

B. Castaing, Jean Souletie

Published Mar 1, 1991• 30 citations
Publisher

Abstract

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 $\tau/\partial(1/T)$ should be continuous at the freezing point. We discuss available data in spin glasses but our conclusions should be very general.

DOI
10.1051/jp1:1991142

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Physical review. B, Condensed matter•Mar 1, 1986•44 citations
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.
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.
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.’’
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
A model for thermally activated dynamics in disordered systems shows that the linear and nonlinear susceptibility follows a generic exponential form with a ``critical rounding,'' ${\ensuremath{\chi}}_{1}\ensuremath{\propto}{\ensuremath{\chi}}_{3}\ensuremath{\propto}[T\mathrm{ln}(t/{\ensuremath{\tau}}_{0}^{\ensuremath{'}})/K{]}^{\ensuremath{\gamma}/b\ensuremath{\phi}}\mathrm{exp}\ensuremath{-}[Tt_{g}{}^{\ensuremath{\phi}b}\mathrm{ln}(t/{\ensuremath{\tau}}_{0}^{\ensuremath{'}})/K){]}^{\ensuremath{\nu}/b}$ (T=temperatureT=\mathrm{\text{temperature}}T=temperature, t=timet=\mathrm{\text{time}}t=time, K=barrier constantK=\mathrm{\text{barrier constant}}K=barrier constant, ${t}_{g}=1\ensuremath{-}{T}_{\mathrm{SG}}/T$, and TSG=transition temperature{T}_{\mathrm{SG}}=\mathrm{\text{transition temperature}}TSG​=transition temperature; $\ensuremath{\gamma}>0$ for ${\ensuremath{\chi}}_{3}$ and <0<0<0 for ${\ensuremath{\chi}}_{1}$). This model, also valid in the presence of resonant tunneling states at energies ${K}_{0}\ensuremath{\ll}K$ [provided that KKK is replaced by ${K}_{0}+2T\mathrm{ln}$ ($1/{\ensuremath{\Gamma}}_{0}$), where $\ensuremath{\Gamma}_{0}{}^{2}\ensuremath{\propto}$ tunnel splitting of a spin S=1S=1S=1], is potentially applicable to a wide variety of systems opening the way for the study of thermally activated quantum phase transitions. The famous spin-glass system ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}$ seems to follow this model.
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.