Aging In Spin Glasses Research Articles

Mesoscopic real-space structures in spin-glass aging: The Edwards-Anderson model

Isothermal simulational data for the 3D Edwards-Anderson spin glass are collected at several temperatures below $T_{\rm c}$ and, in analogy with a recent model of dense colloidal suspensions,interpreted in terms of clusters of contiguous spins overturned by quakes, non-equilibrium events linked to record sized energy fluctuations. We show numerically that, to a good approximation, these quakes are statistically independent and constitute a Poisson process whose average grows logarithmically in time. The overturned clusters are local projections on one of the two ground states of the model, and grow likewise logarithmically in time. Data collected at different temperatures $T$ can be collapsed by scaling them with $T^{1.75}$, a hitherto unnoticed feature of the E-A model, which we relate on the one hand to the geometry of configuration space and on the other to experimental memory and rejuvenation effects. The rate at which a cluster flips is shown to decrease exponentially with the size of the cluster, as recently assumed in a coarse grained model of dense colloidal dynamics. The evolving structure of clusters in real space is finally sssociated to the decay of the thermo-remanent magnetization. Our analysis provides an unconventional coarse-grained description of spin glass aging as statistically subordinated to a Poisson quaking process and highlights record dynamics as a viable common theoretical framework for aging in different systems.

Effect of the thermal quench on aging in spin glasses

Weprobetheeffectsofdifferentinitialconditionsontheprocessofaginginspinglassesarisingfromalternative temperature quenching protocols for the canonical and representative Cu:Mn (6 at.%) dilute magnetic alloy. The effects of these changes are observed through the decay of the thermoremanent magnetization, MTRM(t). We find significant changes in the initial state as a consequence of the different cooling processes, reflected through an effective waiting time, t eff . The effective waiting time shortens as the time spent above the measuring temperature in the cooling protocol is reduced, reaching a minimum when the temperature is not allowed to rise above the final measurement temperature after the initial quench. This behavior is consistent with the picture of a rapid temperature dependence of the barrier heights separating metastable states.

Effect of cooling rate on aging in spin glasses

Aging is a well known property of spin glass materials and has been investigated extensively in recent years. This aging effect is commonly observed by thermal remnant magnetization (TRM) experiments in which the relaxation of the magnetization is found to be dependent on the time, tw, spent at constant temperature before a field cut. The TRM curves scale with tw^\mu, where \mu is less than 1, which is known as a "subaging" effect. The question of whether this subaging effect is intrinsic, or due to experimental artifacts, remains as yet unanswered. One possible experimental origin of subaging arises from the cooling of the sample to the measuring temperature and it has been proposed that with fast enough cooling \mu would go to 1 . Here we investigate this possibility by studying the effect of cooling protocol on aging for 3 well characterized spin glasses, CdCr1.7In0.3S4, Au:Fe8% and Fe0.5Mn0.5TiO3. We find no strong influence of the cooling rate on \mu and no evidence that \mu would go to 1 for very short cooling times. We propose additionally an argument which shows that small (+/-150 mK) variations in the temperature of the sample during the first tens of seconds of the TRM can significantly influence the behavior of the relaxation of a spin glass which in turn may result in a misleading interpretation of \mu values.

Aging, rejuvenation and memory phenomena in spin glasses

In this paper, we review several important features of the out-of-equilibrium dynamics of spin glasses. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low-temperature phase. We report in particular new results on the sub-aging behaviour of spin glasses. We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, from the point of view of both energy landscape pictures and real-space pictures. We highlight the fact that both approaches point out the necessity of hierarchical processes involved in aging. Finally, we report an investigation of the effect of small temperature variations on aging in spin glass samples with various anisotropies which indicates that this hierarchy depends on the spin anisotropy.

Nonequilibrium relaxation method — an alternative simulation strategy

One well-established simulation strategy to study the thermal phases and transitions of a given microscopic model system is the so-called equilibrium method, in which one first realizes the equilibrium ensemble of a finite system and then extrapolates the results to infinite system. This equilibrium method traces over the standard theory of the thermal statistical mechanics, and over the idea of the thermodynamic limit. Recently, an alternative simulation strategy has been developed, which analyzes the nonequilibrium relaxation (NER) process. It is called theNER method. NER method has some advantages over the equilibrium method. The NER method provides a simpler analyzing procedure. This implies less systematic error which is inevitable in the simulation and provides efficient resource usage. The NER method easily treats not only the thermodynamic limit but also other limits, for example, non-Gibbsian nonequilibrium steady states. So the NER method is also relevant for new fields of the statistical physics. Application of the NER method have been expanding to various problems: from basic first- and second-order transitions to advanced and exotic phases like chiral, KT spin-glass and quantum phases. These studies have provided, not only better estimations of transition point and exponents, but also qualitative developments. For example, the universality class of a random system, the nature of the two-dimensional melting and the scaling behavior of spin-glass aging phenomena have been clarified.

Aging and scaling laws inβ-hydroquinone-clathrate

The dielectric permittivity of the orientational glass $\mathrm{methanol}(x=0.73)\ensuremath{-}\ensuremath{\beta}$-hydroquinone-clathrate has been studied as function of temperature and waiting time using different temperature-time protocols. We study aging, rejuvenation, and memory effects in the glassy phase and discuss similarities and differences to aging in spin glasses. We argue that the diluted methanol-clathrate, although conceptually close to its magnetic pendants, takes an intermediate character between a true spin glass and a pure random-field system.

Spin anisotropy and slow dynamics in spin glasses.

We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.

Exploring valleys of aging systems: the spin glass case

We present a statistical method for complex energy landscape exploration which provides information on the metastable states--or valleys--actually explored by an unperturbed aging process following a quench. Energy fluctuations of record size are identified as the events which move the system from one valley to the next. This allows for a semi-analytical description in terms of log-Poisson statistics, whose main features are briefly explained. The bulk of the paper is devoted to thorough investigations of Ising spin glasses with Gaussian interactions of both short and long range, a well established paradigm for glassy dynamics. Simple scaling expressions with universal exponents for (a) barrier energies, (b) energy minima, and (c) the Hamming distance as a function of the valley index are found. The distribution of residence time inside valleys entered at age $t_w$ is investigated, along with the distribution of time at which the global minimum inside a valley is hit. Finally, the correlations between the minima of the landscape are presented. The results fit well into the framework of available knowledge about spin glass aging. At the same time they support a novel interpretation of thermal relaxation in complex landscapes with multiple metastable states. The marginal stability of the attractors selected is emphasized and explained in terms of geometrical properties of the landscape.

Full aging in spin glasses.

The discovery of dynamic memory effects in the magnetization decays of spin glasses in 1983 marked a turning point in the study of the highly disordered spin glass state. Detailed studies of the memory effects have led to much progress in understanding the qualitative features of the phase space. Even so, the exact nature of the magnetization decay functions has remained elusive, causing confusion. In this Letter, we report strong evidence that the thermoremanent magnetization decays scale with the waiting time t(w). By employing a series of cooling protocols, we demonstrate that the rate at which the sample is cooled to the measuring temperature plays a major role in the determination of scaling. As the effective cooling time t(eff)(c) decreases, t/t(w) scaling improves and for t(eff)(c)<20 s we find almost perfect t/t(w) scaling, i.e., full aging.

Role of initial conditions in spin-glass aging experiments

Effect of initial conditions on aging properties of the spin-glass state is studied for a single crystal Cu:Mn 1.5 at %. It is shown that memory of the initial state, created by the cooling process, remains strong on all experimental time scales. Scaling properties of two relaxation functions, the TRM and the IRM (with $t_{w1}=0$ and $t_{w2}=t_{w}$), are compared in detail. The TRM decay exhibits the well-known subaging behavior, with $\mu<1$ and $t_{w}^{eff}>t_{w}$. The IRM relaxation demonstrates the superaging behavior, with $\mu>1$ and $t_{w}^{eff}<t_{w}$. It is shown that an average over different initial conditions leads to systematic improvement in $t/t_{w}$ scaling. An effective barrier, describing influence of the initial state on spin-glass relaxation, is found to be almost independent of temperature. These results suggest that departures from full $t/t_{w}$ scaling, observed in aging experiments, are largely due to cooling effects.

Publisher’s Note: Temperature dependence of aging in spin glasses [Phys. Rev. B60, 10 164 (1999)

Received 29 January 2003DOI:https://doi.org/10.1103/PhysRevB.67.059901©2003 American Physical Society

Heterogeneous aging in spin glasses.

We introduce a set of theoretical ideas that form the basis for an analytical framework capable of describing nonequilibrium dynamics in glassy systems. We test the resulting scenario by comparing its predictions with numerical simulations of short-range spin glasses. Local fluctuations and responses are shown to be connected by a generalized local out-of-equilibrium fluctuation-dissipation relation. Scaling relationships are uncovered for the slow evolution of heterogeneities at all time scales.

A microscopic mechanism for rejuvenation and memory effects in spin glasses

Aging in spin glasses (and in some other systems) reveals astonishing effects of `rejuvenation and memory' upon temperature changes. In this paper, we propose microscopic mechanisms (at the scale of spin-spin interactions) which can be at the origin of such phenomena. Firstly, we recall that, in a frustrated system, the effective average interaction between two spins may take different values (possibly with opposite signs) at different temperatures. We give simple examples of such situations, which we compute exactly. Such mechanisms can explain why new ordering processes (rejuvenation) seem to take place in spin glasses when the temperature is lowered. Secondly, we emphasize the fact that inhomogeneous interactions do naturally lead to a wide distribution of relaxation times for thermally activated flips. `Memory spots' spontaneously appear, in the sense that the flipping time of some spin clusters becomes extremely long when the temperature is decreased. Such memory spots are capable of keeping the memory of previous ordering at a higher temperature while new ordering processes occur at a lower temperature. After a qualitative discussion of these mechanisms, we show in the numerical simulation of a simplified example that this may indeed work. Our conclusion is that certain chaos-like phenomena may show up spontaneously in any frustrated and inhomogeneous magnetic system, without impeding the occurrence of memory effects.

Multiple aging mechanisms in relaxor ferroelectrics

In three relaxor ferroelectrics strong low-field aging effects are found for the dielectric susceptibility. Different regimes, marked by different time dependence and by different stability under subsequent thermal cycles and field changes, are found in different materials and at different temperatures. One regime strongly resembles aging in spin glasses, but others do not.

Temperature dependence of aging in spin glasses

Experimental data for ${\mathrm{Cu}}_{0.9}{\mathrm{Mn}}_{0.1}$ are presented for the change with temperature of: the dependence of the initial magnetization on waiting time before applying a magnetic field to a zero-field-cooled sample, the maximum value of $S=\ensuremath{\partial}M/\ensuremath{\partial}\mathrm{ln}t,$ and the time at which the maximum occurs. The results as well as published data [Granberg et al., J. Magn. Magn. Mater. $92,$ 228 (1990)] on the dependence of S on $\mathrm{ln}t$ and waiting time are found to be in good agreement with a domain model.

Domain models for aging in spin glasses

A simple domain growth model is shown to account for aging in spin glasses. It accounts quantitatively for the change in glass temperature with frequency. It is shown that percolation clusters present in disordered systems are responsible for the peak in ∂M/∂ ln t, where M is the magnetization, and t is the time.

Relaxation and aging in spin glasses and other complex systems

We describe how a hierachical model of spin glass relaxation can display aging behaviour, similarly to what is found in spin glasses and other complex systems out of thermodynamic equilibrium. Since we deal with a nonequilibrium situation, the usualF(luctuation)D(issipation)T(heory) does not apply. We therefore derive a general relation between the linear response function and the non equilibrium propagator of the unperturbed system. The relation is shown to be very similar to the equilibrium FDT under certain conditions, which one can reasonably assume for spin glass systems. Having thus related the linear response of the system to a small external field to the autocorrelation function of the magnetization, we calculate the latter quantity by a master equation on a set of states which have the topology of a tree. The model can reproduce the main qualitative features of theZ(ero)F(ield)C(ooled) spin glass experiments, i.e. the maximum in the logarithmic time derivative of the magnetization, with only two free parameters.