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Nonequilibrium Dynamics and Aging in the

arXiv:hep-lat/9303013v1 26 Mar 1993Nonequilibrium Dynamics and Aging in theThree–Dimensional Ising Spin Glass ModelHeiko Rieger*Physics Department, University of California, Santa Cruz, CA 95064, USAHLRZ c/o KFA J¨ulich, Postfach 1913, 5170 J¨ulich, GermanyThe low temperature dynamics of the three dimensional Ising spin glass in zerofield with a discrete bond distribution is investigated via MC simulations. The ther-moremanent magnetization is found to decay algebraically and the temperature de-pendent exponents agree very well with the experimentally determined values.

Thenonequilibrium autocorrelation function C(t, tw) shows a crossover at the waiting (oraging) time tw from algebraic quasi-equilibrium decay for times t≪tw to another,faster algebraic decay for t≫tw with an exponent similar to one for the remanentmagnetization.PACS numbers: 75.10N, 75.50L, 75.40G.∗Present address: Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, 5000 K¨oln 41, Germany.1

The measurement of dynamical nonequilibrium quantities in real spin glasses [1] has along history. The typical experiment that has been conducted many times [2,3] is the follow-ing: Within a magnetic field the spin glass (for instance Cu(Mn), Au(Fe), Fe0.5Mn0.5TiO3,(FexNi(1−x))75P16B6Al3, CdxMn(1−x)Te, etc.

)is cooled down to temperatures below thefreezing temperature Tg and either immediately or after a certain waiting time tw the fieldis switched off. Then the so called (thermo)remanent magnetization Mrem(t) is measuredas a function of time t. The asymptotic time dependence of this quantity is found to bealgebraic well below Tg (T/Tg ≤0.98) in the short range Ising spin glass Fe0.5Mn0.5TiO3 [4,5]and in an amorphous metallic spin glass (FexNi(1−x))75P16B6Al3 [6].

Furthermore the time–dependence of the remanent magnetization depends on the waiting time tw, a phenomenoncalled aging [2].Several attempts have been made to explain this behavior theoretically [7,8,9,10,11] anda wide variety of functional forms for the time dependence of the remanent magnetizationis found. The problem lays in the fact that starting from a microscopic model or model–Hamiltonian one encounters insurmountable difficulties in trying to solve the nonequilibriumdynamics.Therefore additional assumptions have to be made and the final outcome —stretched exponential [7], algebraic [10,11] or logarithmic [9] decay — depends on them.Even within the mean field approximation it is hard to obtain any analytical [12,13] orsemianalytical [14] results.Once a microscopic model for a spin glass has been formulated, one can in principle tryto extract its macroscopic behavior via Monte Carlo (MC) simulations.

In contrast to ananalytical treatment, where the calculation of dynamical nonequilibrium quantities withinthe spin glass phase (instead of those characterizing equilibrium, see [15]) is even more com-plicated, MC simulation can be done with less effort for certain nonequilibrium situations,since equilibration times reach astronomical values in case of spin glasses [16,17,18]. Theremanent magnetization with zero waiting time has been investigated numerically for themean field version of a spin glass model [9,19] and for the three dimensional EA (Edwards–2

Anderson) model only right at the critical temperature [20]. Quite recently attempts havebeen made to investigate the whole temperature range below Tg numerically [21,22].

How-ever, a systematic MC study of nonequilibrium correlations and aging phenomena withinthe frozen phase (T < Tc) of the three–dimensional EA spin glass model has not been madeup to now. The results such an investigation, its theoretical implications and comparisonwith experiments will be reported in this letter.The system under consideration is the three–dimensional Ising spin–glass with nearestneighbor interactions and a discrete bond distribution.

Its Hamiltonian isH = −X⟨ij⟩Jijσiσj ,(1)where the spins σi = ±1 occupy the sites of a L × L × L simple cubic lattice with periodicboundary conditions and the random nearest neighbor interactions Jij take on the values+1 or −1 with probability 1/2. We consider single spin flip dynamics and used a special,very fast implementation of the Metropolis algorithm on a Cray YM-P (see ref.

[23] fordetails). The simulations were done in the frozen phase, that means at temperatures belowTc = 1.175 ± 0.025 (see ref.

[16,17,18]). All measured quantities are averaged over at least128 samples (smaller system sizes were averaged over up to 1280 samples).

The system sizewas increased until no further size dependence of the results has been observed within thesimulation time (t ≤106), which is measured in MC sweeps through the whole lattice. Itturns out that L = 32 is large enough for this time range (cf.

[18]).The system was prepared in a fully magnetized initial configuration and then the sim-ulation was run for a time tw (=waiting time) and then the spin configuration σ(tw) wasstored. From now on after each MC step (data were then averaged over appropriate timeintervals, cf.

[18]) the following correlation function was measured:C(t, tw) = 1NXi⟨σi(t + tw)σi(tw)⟩,(2)where ⟨· · ·⟩means a thermal average (i.e. an average over different realizations of the ther-mal noise, but the same initial configuration) and the bar means an average over different3

realizations of the bond–disorder. The quantity C(t, 0) corresponds to the remaining mag-netization of the system after a time t.Mrem(t) = C(t, 0) .

(3)This quantity is directly related to the experimentally determined thermoremanent magneti-zation with zero waiting time (i.e. without aging) and nearly saturated initial magnetization.The result for the remanent magnetization Mrem(t) is shown in fig.

1 within a log–log plot. Itsdecay clearly obeys a power law for large times and temperatures in the range 1.1 ≥T ≥0.5.The exponent λ(T) for the fitMrem(t) ∝t−λ(T) ,(t ≥102)(4)is plotted in fig.

2, upper curve. It starts at λ = 0.36 ± 0.01 for T = 1.1 (and can beextrapolated via the fit indicated in figure 2 to 0.39±0.01 for T=Tc, which was alreadyfound in [20]) and decreases monotonically with temperature.

For the short range Isingspin glass Fe0.5Mn0.5TiO3 and for certain amorphous metallic spin glasses not only the samealgebraic decay of the remanent magnetization has been observed, but also the shape of thefunctional temperature dependence of the exponent λ(T) and even its numerical values arein excellent agreement: from figure 4b in ref. [6] one may for instance read offλ(Tg) ≈0.38and λ(0.5 Tg) ≈0.12, concurring within the errorbars to the corresponding data plotted infig.

2.It was argued [9] that the decay of Mrem(t) should be logarithmic (i.e. Mrem(t) ∝(ln t)−λ/ψ) below Tc, but a fit of the data in fig.

1 for T ≥0.5 does not yield accept-able results over the range of the observation time. We want to focus some attention to theT = 0.4 curve: It bends upward in the log–log plot for t > 104, which could indicate theonset of a slowlier than algebraic decay for temperatures smaller than 0.5, logarithmic forinstance.Another indication that something new happens at lower temperatures can be obtainedby looking at the short time behavior of Mrem(t): At T ≈0.5 a plateau begins to develop4

for t < 102, which can clearly be seen for T = 0.4 and becomes even more pronounced andwider for even smaller temperatures. It can be excactly reproduced in shape and locationfor smaller and larger sizes and number of samples, which means that it is a physical effectand not only a fluctuation.

A possible interpretation might be that the system gets trappedin metastable states, whose lifetime grows with decreasing temperatures.Next we turn our attention to the correlation function C(t, tw) with tw = 10a (a =1, . .

. , 5).

In contrast to Mrem(t) the correlation function C(t, tw) for tw ̸= 0 is not directly re-lated to the thermoremanent magnetization M(t, tw) at time t+tw in a temperature–quenchexperiment, where the field H is switched offat time tw after the quench (see e.g. [24]).

Inequilibrium (tw →∞) C(t, ∞) is related to the relaxation function R(t, ∞) = M(t, ∞)/Hvia the fluctuation dissipation theorem (FDT) R(t, ∞) = C(t, ∞)/kBT.However, in anonequilibrium situation, like the one considered here, slight differences between them exist[21,25] (e.g. in the location of the maximum relaxation rate).

The magnetization that isinduced by a small external field (H ≪1) for model (1) is rather small, therefore the func-tional form of M(t, tw) is harder to determine accurately via MC–simulations. This is thereason why in this letter the focus is on C(t, tw).A typical set of data for a particular temperature (T = 0.8) is shown in fig.

3 in a log–logplot. One observes a crossover from a slow algebraic decay for t ≪tw to a faster algebraicdecay for t ≫tw.

The crossover time is simply defined as the intersection of the two straightline fits for short– and long–time behavior in the log–log plot. For the long–time behaviorthe fit toC(t, tw) ∝t−λ(T,tw) ,t ≫tw(5)yields a set of exponents that is depicted in fig.

2. For increasing tw the exponent λ(T, tw)decreases only slightly and the waiting time dependence becomes weaker for lower temper-atures.

By looking at fig. 3 one observes that it is difficult to extract λ(T, tw) for tw = 104and 105 since there are only 2 or 1 decades left to fit the exponent — therefore they are notshown in fig.

2. The exponent describing the short time (t ≪tw) behavior of C(t, tw),5

C(t, tw) ∝t−x(T) ,t ≪tw ,(6)which is depicted in fig. 4, is independent of the waiting time tw.

Since the system was ableto equilibrate over a time tw, all processes occuring on timescales smaller than tw have thecharcteristics of equilibrium dynamics and therefore the exponent x(T) is identical to thatdescribing the decay of the equilibrium autocorrelation function q(t) = limtw→∞C(t, tw).The latter was investigated in [18] and the exponents that are reported there for T ≥0.7Tcagree with the values shown in fig. 4.

They also agree with those determined experimentally[5] in the short range Ising spin glass Fe0.5Mn0.5TiO3 via the above mentioned relaxationfunction R(t, tw) = M(t, tw)/H for t ≪tw (note that in this quasiequilibrium–regime C(t, tw)and R(t, tw) are related via the FDT [21,25], yielding the same exponents for both): closeto Tg (T/Tg = 1.029) they obtain x = 0.07. Furthermore there seems to be a temperatureat about 0.3, where x(T) becomes zero, which could be another indication for the abovementioned onset of a logarithmic decay of the correlation functions [26].Although the decay of the nonequilibrium correlations in the temperature range 0.5 ≤T ≤1.1 is algebraic rather than logarithmic as predicted by the droplet picture proposed in[9], this picture might not be inappropriate: Let us assume the following scaling law for thedependence of the free energy barriers B on a length scale L of the regions to be relaxed:B ∝Λ lnL instead of B ∝Lψ as in [9].

Then one ends up with an algebraic decay of e.g.the remanent magnetization by observing (see [9]) that the typical length scale of domainsRt now grows with time according to Λ lnRt ∼T lnt, which means Rt ∝tT/Λ(T), leading toequations (3–6).In the context of the phenomenological model for the dynamics and aging in disorderedsystems developped in ref. [11], the algebraic decay of correlations found so far implies thatthe probability distribution of free energy barriers is exponential in the temperature rangeof 0.5 ≤T ≤1.1 for the system under consideration.

Furthermore we would like to mentionthat a fit to the functional form for the short time behavior (t ≪tw) C(t, tw) ∼1 −a(t/tw)yproposed in [11] works also quite well for our data, although not as convincingly as equation6

(6).Guided by equations (4) and (5) we tried to put our results into the following scalingform:C(t, tw) = cT t−x(T) ΦT(t/tw) ,(7)where ΦT (y) = 1 for y = 0 and ΦT(y) ∝yx(T)−˜λ(T) for y →∞. The form (7) has re-cently been used [27] successfully to extract the critical dynamical exponent z from thenonequilibrium correlation function (2) via finite size scaling, where the waiting time twhas been replaced by the relaxation time τ ∝Lz in the critical region.

For temperaturesbelow T = 0.8 equation (7) yields an acceptable fit (which can already be deduced from theneglegible waiting time dependence of the exponents λ(T, tw) for T ≤0.7, see fig. 2).Concluding we have reported new results of numerical nonequilibrium simulationsthat show an excellent concurrence with experiments on the short range Ising spin glassFe0.5Mn0.5TiO3 and on amorphous metallic spin glasses: not only a single exponent but awhole continuum of (temperature dependent) exponents for the remanent magnetization arefound to agree within the numerical errors.

Although the values for the exponents extractedfrom experiments might vary somewhat depending on the microscopic details (range of inter-actions, spin–type) the main features of the relaxation and the dynamics of many differentthree–dimensional spin glasses are very similar and the functional forms of the remanentmagnetization decay should be the same for different systems [28].Furthermore we have shown that aging phenomena in the spin glass model under con-sideration can be observed via the measurement of a particular correlation function andthat its nonequilibrium dynamics is indeed gouverned by its equilibrium characteristics fortime scales smaller than the imposed waiting (or aging) time. This gives an interesting newperspective (see also [27,29]) to extract equilibrium quantities, which are hard to obtain viaMC simulations within the spin glass phase.

Finally, by observing plateaus in the shorttime behavior and slowing down of the algebraic decay of the remanent magnetization, werevealed a dynamical scenario at very low temperatures that is not yet fully understood.7

The author would like to thank A. P. Young for many extremely valuable discussions.He is grateful to J. O. Andersson, D. Belanger, J. P. Bouchaud and P. Nordblad for variouscomments, hints, suggestions and explanations.The simulations were performed on theCRAY Y-MP at the supercomputer center in J¨ulich and took about 100 hours of CPU–time.Financial support from the DFG (Deutsche Forschungsgemeinschaft) is also acknowledged.8

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FIGURESFIG. 1.

The remanent Magnetization Mrem(t) versus the time t in a log–log plot for varyingtemperatures. From top to bottom we have T=0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 and 1.1.

The sytemsize is L=32 and the data are averaged over 128 samples. The errorbars are of the size of thesymbols for Mrem ≤0.01 and much smaller for larger Mrem.FIG.

2. Upper curve: The exponent λ(T) for the remanent magnetization (3) versus tem-perature.

The points represented by diamonds (⋄, plus errorbars) are those extracted from figure1 and the full curve is a least square fit to a quadratic polynomial as a guideline to the eye.Lower Points:The nonequilibrium exponent λ(T, tw) (see equation (5)) extracted from thelong-time behavior (t ≫tw) of the nonequilibrium correlation function C(t, tw) (see figure 3)for fixed values of tw versus the temperature T. From top to bottom we have: (△) tw=10, (✷)tw=100 and (◦) tw=1000. The errorbars are indicated.FIG.

3. The averaged nonequilibrium spin autocorrelation function C(t, tw) of equation (2) forfixed values of tw versus time t on a double logarithmic time–scale.

The temperature is fixed to beT=0.8 and from top to bottom we have tw=105, 104, 103, 102 and 10. The system size is L=32and the data are averaged over 128 samples.

The size of the errorbars is only a fraction of the sizeof the symbols.FIG. 4.

The equilibrium exponent x(T) for the equilibrium autocorrelation funcion q(t) ex-tracted from the short–time behavior (t ≪tw) of the nonequilibrium correlation function C(t, tw)(see equation (6)) versus the temperature T. The errorbars are smaller than the circles, as in-dicated. For T ≤0.8 the data are fitted to a straight line, which shows that at approximatelyT = 0.3 the exponent x(T) vanishes.11

0.010.11101001000100001000001e+06Mrem(t)t❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝Fig. 112

00.050.10.150.250.30.350.40.20.40.60.811.2λ(T,tw)T✸✸✸✸✸✸✸✸△△△△△△△△✷✷✷✷✷✷✷✷❝❝❝❝❝❝❝❝Fig. 213

0.111101001000100001000001e+06C(t,tw)t❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝Fig. 314

00.010.020.030.040.050.060.20.40.60.811.2x(T)T✸✸✸✸✸✸✸✸Fig. 415

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