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arXiv:cond-mat/0205276v1 [cond-mat.dis-nn] 14 May 2002

Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. III. Cumulative Memory and ‘Chaos’ E?ects in the Temperature-Shift Protocol

Hajime Takayama? and Koji Hukushima??

Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581 (Received February 1, 2008)

The temperature (T )-shift protcol of aging in the 3 dimensional (3D) Edwards-Anderson (EA) spin-glass (SG) model is studied through the out-of-phase component of the ac susceptibility simulated by the Monte Carlo method. For processes with a small magnitude of the T -shift, ?T , the memory imprinted before the T -shift is preserved under the T -change and the SG short-range order continuously grows after the T -shift, which we call the cumulative memory scenario. For a negative T -shift process with a large ?T the deviation from the cumulative memory scenario has been observed for the ?rst time in the numerical simulation. We attribute the phenomenon to the ‘chaos e?ect’ which, we argue, is qualitatively di?erent from the so-called rejuvenation e?ect observed just after the T -shift.

KEYWORDS: spin glass, slow dynamics, aging, droplet theory

§1.

Introduction

Recently, in studies on slow dynamics in spin glasses,1, 2, 3) the apparently contradictory phenomena, i.e., rejuvenation (or chaos) and memory e?ects in aging dynamics, have been intensively investigated.4, 5) In fact the phenomena were already observed in the early stage of study on aging in spin glasses by the socalled temperature-cycling protocol.6) In the protocol we quench a spin glass to a temperature, say T1 , below the spin-glass (SG) transition temperature Tc from above it and let the system equilibrate (or age) for a period of tw1 . Subsequently, we change the temperature to T2 (< T1 ), age the system for a period of tw2 , and then the temperature is turned back to T1 . For a certain range of the parameters T1 , T2 , tw1 and tw2 some quantities such as the out-of-phase component of ac susceptibility, χ′′ (ω; t), exhibit the following behavior. Just after the ?rst T -shift χ′′ (ω; t) behaves as if the system were quenched to T2 directly from above Tc , or it looks having forgotten the aging at T1 before the T -shift. This is called the rejuvenation (or chaos) e?ect. However, after the temperature is turned back to T1 , χ′′ (ω; t) observed is the one we expect as a simple extension of χ′′ (ω; t) already aged by tw1 at T1 . Thus the system de?nitely preserves the memory of the previous aging at T1 , while it apparently exhibits the rejuvenation behavior, in the aging process at T2 . The proper understanding of such a peculiar phenomenon is believed to shed light not only on the mechanism behind the aging dynamics but also the nature of the SG phase itself. Furthermore it will provide us with powerful concepts to understand the glassy dynamics in various related systems such as orientational

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??

E-mail: takayama@issp.u-tokyo.ac.jp E-mail: hukusima@issp.u-tokyo.ac.jp

glasses,7) polymers,8) and interacting nanoparticles systems.9) By the real-space interpretation, or by the droplet picture,10, 11, 12) which we have been adopting in our recent studies,13, 14, 15, 16) the SG order is considered to grow up slowly in aging processes. In particular, we have demonstrated that the SG coherence length, which we regards as the mean size of SG domains developed in aging, continuously grows even under the T -shift process.15) This we call the cumulative memory e?ect. We have further extended this characteristics to a scenario that the SG short-range order ever grows continuously with growth rates sensitively dependent on the temperature so long as the system is in the SG phase.16) Let us call this the cumulative memory scenario, and denote the mean size of SG domains as RT [t] (t), where T [t] symbolically represents the temperature changes that the system has experienced up to time t from the ?rst quench to the SG phase at t = 0. It has been demonstrated that the time evolution of zero-?eld-cooled (ZFC) magnetizations observed in various schedules of temperature changes but with a common initial quench condition17) are well described by a unique function of RT [t] (t).16) The purpose of the present work is to numerically explore to what extent the simulated data of χ′′ (ω; t) in the T -shift protocol of the 3D Ising EA SG model are compatible or incompatible with the cumulative memory scenario, and with the rejuvenation (chaos) and memory e?ects mentioned above. For this purpose we certainly need a few more length and time scales than RT [t] (t). One is RT [t] (t) at t = tw1 , i.e., the mean domain size grown in the isothermal aging after quench to the SG phase, which is denoted as RT1 (tw1 ). After the temperature is shifted to T2 spin con?gurations within each domains, which were in local equilibrium of T1 just before the T -shift, gradually become in local equilibrium of T2 .

2

Hajime Takayama and Koji Hukushima

By the word ‘gradually’ we mean that the change associates with slowly growing droplets (or subdomains) of a mean size LT2 (τ ) (< RT1 (tw1 )) with τ = t ? tw1 . Here we call this the droplets-in-domain scenario (previously called the quasi-domains-within-domains picture15) ) . A key quantity of the present work is the time scale required for LT2 (τ ) to catch up RT1 (tw1 ).15) At time scales larger than this one, denoted as te? , behavior of physiw1 cal quantities such as χ′′ (ω; t) cannot be distinguished, within the accuracy of measurement, from the corresponding behavior in the isothermal aging at T2 . In other words, the system merges to a T2 -isothermal aging state at the time scale of te? after the T -shift. The latter is w1 regarded as the e?ective waiting time of the tw1 -aging at T1 reread as an T2 -isothermal aging. If the cumulative memory scenario holds, te? is the time required for the w1 SG coherence to grow in the T2 -isothermal aging up to RT1 (tw1 ), i.e., LT2 (te? ) = RT2 (te? ) = RT1 (tw1 ). w1 w1 (1.1)

waiting time te? from the obtained data of χ′′ (ω; t), and w1 present the results of te? , or RT2 (te? ) for T -shift prow1 w1 cesses with various sets of T1 and T2 . In the ?nal section we discuss our results, emphasizing on the length and time scales involved as well as on the relation to the experimental results. §2. Method of Analysis

Numerical simulation on a well-de?ned microscopic SG model, such as the EA model investigated in the present work, is of quite importance in studying aging phenomena. It enables us to faithfully realize any T -shift process and observe any quantity in principle. For example, the SG coherence length ξT [t] (t), which we regard as the mean SG domain size RT [t] (t), has been calculated from the replica overlap function.15) For the isothermal process of the 3D Ising Gaussian EA model which we study in the present work, it is well described by the power law15, 22, 23) written as RT (t)/l0 = bT (t/t0 )1/z(T ) , (2.1)

is expected to hold. Here the ?rst equality simply indicates that LT (t) has the same functional form as that of RT (t) since the both growth processes are governed by common thermally-activated dynamics. In the present work we have extensively examined T -shift processes in the 3D Gaussian EA model with Tc ? 0.95J 18) through the ac susceptibility χ′′ (ω; t) simulated by the standard Monte Carlo (MC) simulation. Here J is the variance of the interactions. The temperature range investigated is T ? [0.4, 0.7] in unit of J, and the time range is up to 105 MC steps. One of the results we have found is that in negative (positive) T -shift protocols with T1 → (←) T2 Eq.(1.1) (the same equation but with the su?x 1, 2 interchanged) holds well when ?T = T1 ? T2 = 0.1. This con?rms the cumulative nature of aging in both negative and positive T -shift protocols with a small ?T . A more interesting result is that, for the negative T -process with T1 = 0.7, T2 = 0.4 signi?cant violation of Eq.(1.1) has been observed, while in the corresponding positive T -shift protocol Eq.(1.1) is satis?ed within accuracy of the present numerical analysis. The former non-cumulative memory e?ect has been, for the ?rst time to our knowledge, observed in simulations on the 3D EA models. Although the phenomena appear asymmetrically with respect to the direction of temperature changes, the deviation from the cumulative memory scenario in the negative T -shift appears qualitatively similarly to the one observed recently in experiments on the AgMn spin glass.19) We tentatively regard them as an e?ect associated with the temperature-chaos e?ect in the equilibrium SG phase,10, 11) and call them the ‘chaos e?ect’. In contrary, as in the previous work,20, 21) such rejuvenation e?ects observed experimentally just after the T -shift have not been detected even with ?T = 0.3 in the present work. The organization of the paper is as follows. In the next section we describe our strategy of the simulation and the method to evaluate χ′′ (ω; t) from the spin autocorrelation function making use of the ?uctuation-dissipation theorem. In §3 we explain how to specify the e?ective

where l0 and t0 are microscopic length and time scales (l0 = 1 lattice distance and t0 = 1 MC step for simulated results), bT is a weakly T -dependent constant, and the exponent 1/z(T ) linearly depends on T except for the region near Tc .13) The growth law of RT (t) in isothermal aging di?erent from Eq.(2.1) was proposed in the droplet theory due to Fisher and Huse.12) It is written as24) ? RT (t)/L0 (T ) = L(t/τ0 (T )), ? with the scaling function L(x) given by ? L(x) ? x1/z log1/ψ (x) (x ? 1), (x ? 1). (2.3) (2.2)

Here L0 (T ) (? l0 ??ν ) is the crossover length, τ0 (T ) (? t0 ??zν ) is the attempt time for thermal activation process of droplets, where ? = (Tc ? T )/Tc, and z and ν are the critical exponents associated with the criticality at Tc . The exponent ψ in Eq.(2.3) is, on the other hand, intrinsic to the activation dynamics of droplets. We have recently con?rmed the above growth law, including the crossover from the critical dynamics (x ? 1) to the activated dynamics (x ? 1), in the EA SG model but in 4 dimension24) (see also21) ). However, the simulated data of RT (t) in the 3D EA model are compatible with both the power law of Eq.(2.1) and the logarithmic law in Eq.(2.3).21, 22) Our strategy in the present work is as follows. Because of the circumstances of the 3D EA model mentioned just above as well as those of the recent experiment,25) we do not go into the question which growth law is a correct one for RT (t) observed in the time-window < ( ? 105 MCS) of the simulations, and simply use our results, i.e., Eq.(2.1), in relating a time scale of observation to a length scale of the SG short-range order. Then we examine whether the cumulative memory scenario is su?cient or not to interpret the obtained results of the length scales. We also restrict ourselves to the temperature range of T /J = [0.4, 0.7] as already noted in §1. In

Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. III.

3

this temperature range 1/z(T ) in Eq.(2.1) is well proportional to T and the dynamics is considered to be dominated by the activated process. But the prefactor bT still exhibits weak dependence on T even in this range which will turn out not to be neglected in our present analysis. Lastly we have examined T -shift processes with various values of the temperature di?erences ?T , in particular, a larger one than that studied in our previous work.15) In the aging study at T through the ac susceptibility at frequency ω we need to introduce another length scale, LT (tω ), with tω = 2π/ω. It is a mean size of spin clusters (or droplets) which can respond to the ac ?eld at T . In the droplet picture the aging (or t-dependent) part of χ′′ (ω; t) in an isothermal aging is described by a function of LT (tω )/RT (t).12, 14) As will be discussed in §4 below, χ′′ (ω; t) in some T -shift process to T at t = tw is given in terms of LT (tω )/LT (τ ) where τ = t ? tw . Thus it provides us information of RT (t) or LT (τ ) in the aging process since LT (tω ) is independent of t or τ . By the experimental condition of measuring χ′′ (ω; t), t or τ is necessarily larger than tω in general. This time regime is called the quasi-equilibrium one, where the ?uctuation-dissipation theorem (FDT) is expected to hold well, though approximately.26, 27) Therefore, in the present work, the out-of phase component of ac susceptibility, χ′′ (ω; t), is evaluated from the spin autocorrelation function C(τ ; t) = Si (τ + t)Si (t) , via the FDT as14) χ′′ (ω; t) ? ? π ? C(τ ; t) 2T ?lnτ .

τ =tω

shown below indicate the variance in the results of the numerical derivative on these several sets of C(τ ; t). §3. Results

3.1 Isothermal aging Before going into discussions on the T -shift protocol, let us here present the results of χ′′ (ω; t) obtained in the isothermal aging. In Fig. 1 we show χ′′ (ω; t) with tω = 64 in the isothermal aging at various temperatures. They play an important role in the following arguments on the T -shift protocol, and we call them the reference curve at each temperature.

0.18 0.16 0.14

Isothermal aging tω=64

T=0.7 T=0.6 T=0.5 T=0.4

χ"(t;ω)

0.12 0.1 0.08 0.06 0.04 0.02 100 1000

10000

t

Fig. 1. χ′′ (ω; t) with tω = 64 in the isothermal aging at various temperatures.

(2.4)

(2.5)

In Eq.(2.4), Si (t) is the sign of the Ising spin at site i at time t which is measured in unit of one MC step. The over-line denotes the averages over sites and over di?erent realizations of interactions (samples), and the bracket the average over thermal noises (or di?erent MC runs). In this evaluation of χ′′ (ω; t) we are completely free from any nonlinear e?ect of the ac-?eld amplitude.20) In our previous work15) we studied the susceptibility de?ned by χ(ω; t) = ? 1 [1 ? C(τ ; t)] T .

τ =tω

(2.6)

It is just the ZFC susceptibility: the induced magnetization (divided by h) at an elapsed time of τ under the ?eld h which is switched on after the system has aged under h = 0 by a period of t. For slow processes of our present interest, χ(ω; t) is essentially regarded as the in? phase component of the ac susceptibility, χ′ (ω; t). As in the experiments, simulated χ′′ (ω; t) exhibits larger effects of aging relatively to its own absolute magnitude than χ′ (ω; t) or χ(ω; t) does. However we have to nu? merically evaluate the logarithmic derivative in Eq.(2.5) to estimate χ′′ (ω; t). In the present work we have calculated several C(τ ; t) in Eq.(2.4), each of which is the average over one MC run for each sample but typically over 1600 samples. The linear system size is ?xed to L = 24. The error bars on χ′′ (ω; t) drawn in the ?gures

In Fig 2 χ′′ (ω; t) at T = 0.6 for tω = 16 ? 256 are shown. The ?lled symbols are raw data plotted directly against t, while the open symbols are the same χ′′ (ω; t) plotted against ωt with no vertical shifts of the data sets. All the sets of data thus plotted nicely collapse to a universal curve. This recon?rms that the ωt-scaling of χ′′ (ω; t) holds also in the present model spin glass20) as observed experimentally.1) As pointed out in §2, the time evolution of χ′′ (ω; t) is considered to be a function of LT (tω )/RT (t) in the droplet picture. The ωt-scaling then comes out from the ?rst equality of Eqs.(1.1) and (2.1).14) The response in equilibrium, χ′′ (ω) = limt→∞ χ′′ (ω; t), is hardly extracted from our eq present data. We could not detect even its relative difference with ω, which should be re?ected as the vertical shifts of the data sets in the above scaling analysis.

3.2 T -shift protocol In Fig. 3 we show χ′′ (ω; t) with tω = 64 numerically observed in the negative (positive) T -shift protocol. The temperature is changed from T1 = 0.7 (T2 = 0.5) to T2 = 0.5 (T1 = 0.7) at di?erent waiting times tw1 (tw2 ). The observation starts from t = twi + tω after each T shift. Similarly to χ(ω; t) previously investigated,15) we ? see the following characteristic features. a) Each χ′′ (ω; t) rapidly undershoots (overshoots) the T2 (T1 )-reference curve.

4

0.11 0.1 0.09

Hajime Takayama and Koji Hukushima

Isothermal aging: T=0.6

tω=16 tω=32 tω=64 tω=128 tω=256

χ"(t;ω)

0.08 0.07 0.06 0.05 0.04 1000 t 10000

100000

Fig. 2. χ′′ (ω; t) in the isothermal aging at T = 0.6. The ?lled symbols are the data plotted directly against t, while the open symbols are the same data plotted against 64t/tω .

b) χ′′ (ω; t) merges to the T2 (T1 )-reference curve from below (above). For the negative T -shift, in particular, the value of χ′′ (ω; t) just after the negative T -shift is the larger relatively to the T2 -reference curve, the larger is tw1 . However, χ′′ (ω; t) just after the shift does not exhibit overshooting of the T1 -reference curve, a phenomenon which we call the strong rejuvenation e?ect in the present paper. This is also the case even if we dare to measure χ′′ (ω; t) at τ = t ? tw1 smaller than tω .21)

0.09 0.08 0.07

value of tsh1 (? 3300 ? 512 in the ?gure) it merges to the reference curve and lies on it afterwards. We regard time τ required for the shifted branch to merge to the reference curve in this situation as the e?ective waiting time, te? , introduced in §1. If tsh1 is further increased w1 the branch merges to the reference curve from above and again at larger t than te? . Thus the chosen tsh1 which w1 corresponds to te? yields the shortest time for the shifted w1 branch to merge to the reference curve. An interesting observation in the above analysis is that the time at which the properly shifted branch merges to the T2 -reference curve is nearly equal to 2te? ; w1 tw1 + tsh1 + te? ? 2te? and so te? ? tw1 + tsh1 . This w1 w1 w1 aspect, which is by no means trivial, has been commonly observed in most of the T -shifts examined in the present work. Using this observation, we estimate errors of te? w1 as follows. We judge by eyes the largest tsh1 for which the shifted branch of χ′′ (ω; t) certainly crosses with but not merges to the reference curve, and this value of tsh1 gives a smallest estimate of te? . Similarly the smallest w1 tsh1 for which the shifted branch merges to the refer> w1 ence curve at t ? 3te? yields a largest estimate of te? . w1 The three shifted branches shown in Fig. 4 correspond to these smallest, mean, and largest estimates for te? . w1

0.048 0.046 0.044

T1=0.7→T2=0.5 ω?1=64,tw1=512

tsh1=0 tsh1=2600?512 tsh1=3300?512 tsh1=4000?512

χ"(t;ω)

tw1=512 tw1=2048 tw2=4096 tw2=16384

0.042 0.04 0.038 0.036 0.034 0 5000 10000 t

T-shift protocols T1=0.7,T2=0.5;ω?1=64

χ"(t;ω)

0.06 15000 20000 0.05 0.04 0.03 1000 t 10000

Fig. 4. χ′′ (ω; t) in the negative T -shift protocol from T1 = 0.7 to T2 = 0.5. Each symbol represents the branch of χ′′ (ω; t) at t > tw1 shifted by an amount of tsh1 indicated in the ?gure. the line with the smaller symbols is the T2 -reference curve.

Fig. 3. χ′′ (ω; t) in aging with negative and positive T -shifts between T1 = 0.7 and T2 = 0.5 at t = twi indicated in the ?gure. The upper and lower curves with the smaller symbols represent the T1 - and T2 -reference (isothermal) curves, respectively.

3.3 E?ective waiting time Feature b) above is examined in more details in Figs. 4 and 5. We note that the t-axis in these ?gures is linear in t. Within the time window of Fig. 4 the merging of bare χ′′ (ω; t) (denoted by tsh1 = 0) to the T2 -reference curve is not seen. If, however, the branch of χ′′ (ω; t) at t > tw1 is shifted to the right by an amount denoted by tsh1 , it crosses the reference curve and merges to it at a smaller t than that with tsh1 = 0. At a certain

The above analysis for the negative T -shift protocol also works for the positive T -shift protocol. A typical example from T2 = 0.5 to T1 = 0.7 with tw2 = 16384 is shown in Fig. 5, for which we have to reread the su?x 1 by 2 and vice versa in the above argument. Also in this case the branch of χ′′ (ω; t) at t > tw2 is shifted to the left by tsh2 . A too large tsh2 makes the shifted branch to overshoot the T1 -reference curve, while a too small tsh2 signi?cantly delays the merging. With a proper chosen tsh2 we obtain te? (? 1600 from the ?gure) which satis?es w2 te? = tw2 ? tsh2 . Its error bar is similarly evaluated from w2 the two other tsh2 ’s indicated in the ?gure.

Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. III.

0.059 0.058 0.057 0.056

5

T2=0.5→T1=0.7

ω?1=64,tw2=16384

tsh2=16384?1300 tsh2=16384?1600 tsh2=16384?1900

χ"(t;ω)

0.055 0.054 0.053 0.052 0.051 0.05 t 2000 3000 4000

?T = 0.1 the e?ect of the T -dependence of bT is negligibly small. The e?ect is, however, signi?cant for ?T ≥ 0.2. Thus the T -dependence of bT has to be properly taken into account to judge the cumulative nature of memory observed even in the temperature range examined in the present simulation.

0.4 → 0.7 0.7 → 0.4 0.5 → 0.7 0.7 → 0.5 0.6 → 0.7 0.7 → 0.6

100000

teff ,tw2 w1

5000

6000

7000

8000

10000

Fig. 5. χ′′ (ω; t) with tω = 64 in the positive T -shift protocol from T2 = 0.5 to T1 = 0.7 with tw2 = 16384. The data in a large time scale are shown in Fig. 3. The three sets of symbols represent the branches of χ′′ (ω; t) at t > tw2 shifted to the left by the amount tsh2 indicated in the ?gure. The line is the T1 -reference curve.

1000 100 1000 10000

tw1,teff w2

Fig. 6. te? vs twi for the negative (i = 1) and positive (i = 2) wi T -shift protocols for three sets of T1 and T2 indicated in the ?gure. The lines represent the expected behavior from Eqs.(1.1) and (2.1) as explained in the text.

3.4 Cumulative memory and ‘chaos’ e?ects In Fig. 6 we plot te? (tw2 ) as a function of tw1 (te? ) w1 w2 obtained in the previous subsection in the negative (positive) T -shift protocol for three sets of (T1 , T2 ). Here we have followed the idea of ‘twin-experiments’ in the recent work.19) Before the explanation of the lines drawn in the ?gure, we note that the data points of both negative and positive T -shifts with T1 = 0.7, T2 = 0.6 are seen to lie on a certain common curve, while this is not the case for those with T1 = 0.7 and T2 = 0.4. The former is expected from the cumulative memory scenario. But the latter data points clearly indicate a violation to the scenario irrespectively of the growth law of the SG domains. Now let us explain the lines in Fig. 6. The solid ones represent the relation between the te? (tw2 ) and tw1 (te? ) w2 w1 when the cumulative memory scenario represented by Eq.(1.1) (the one whose su?x 1, 2 interchanged) combined with the growth law of Eq.(2.1) holds. For the latter we have explicitly used the following sets of the parameter values (T ; z(T ), bT ) we previously obtained:13) (0.7; 8.71, 0.779), (0.6; 9.84, 0.782), (0.5; 11.76, 0.800) and (0.4; 14.80, 0.818). As seen in the ?gure, for a small ?T (= 0.1), both te? and te? lie on the solid line. The rew1 w2 sults con?rm the cumulative memory scenario described in §1. With ?T = 0.2, te? still lie on the solid line w2 but te? tends to deviate, though a little, from it. For w1 ?T = 0.3, te? signi?cantly deviate from the solid line, w1 which is incompatible with the cumulative memory scenario. It should be emphasized, however, that the corresponding χ′′ (ω; t) (not shown) in this process exhibit features a), b) mentioned in §3.2, i.e., no strong rejuvenation. If the weak T -dependence of bT is discarded the condition of Eq.(1.1) is reduced to te? w1 τ0 = tw1 τ0

T1 /T2

2.2 2

1.8 1.6 1.4 1.2

0.7→0.6 0.6→0.7 0.6→0.5 0.5→0.6 0.5→0.4 0.4→0.5 0.7→0.5 0.5→0.7 0.6→0.4 0.7→0.4 0.4→0.7

Reff,R2 1

1.2

1.4

1.6

1.8

2

2.2

R1,Reff 2

Fig. 7. Relations te? vs. twi drawn by means of the correspondwi ing domain sizes evaluated by Eq.(1.1) with Eq.(2.1). Here we omit the error bars. For all the sets of T1 , T2 their magnitudes are comparable with those shown in Fig. 6.

,

(3.1)

which is also shown by the dotted lines in Fig. 6. For

In Fig. 7 we replot our data in Fig. 6 as well as those of other sets of (T1 , T2 ) in terms of the lengths, where e? R1 ≡ RT1 (tw1 ) and R1 ≡ RT2 (te? ) which are evaluated w1 by Eq.(2.1) using tw1 and te? extracted in the negative w1 e? T -shift processes (R2 and R2 in the positive T -shift process are similarly evaluated). The line in the ?gure is what is expected from the cumulative memory scenario, e? i.e., Ri = Ri for both i = 1, 2. We see clearly that this is the case for both negative and positive T -shift processes with ?T = 0.1 within the time window of the present simulation. For ?T = 0.2 the data of the pose? itive T -shift satisfy the condition R2 = R2 , but those e? of the negative T -shift exhibit the tendency R1 < R1 .

6

Hajime Takayama and Koji Hukushima

Behavior of the T -shift with ?T = 0.3, i.e., T1 = 0.7 and T2 = 0.4 is as already described above and is interpreted below to be due to the ‘chaos e?ect.’ According to the theory for the temperature-chaos in spin glasses,11, 10) the SG equilibrium con?gurations at di?erent temperatures, T1 and T2 , are completely uncorrelated with each other in the length scale larger than l?T ∝ ?T ?1/ζ , where l?T is called the overlap length and ζ (> 0) the chaos exponent. An important problem here is how the existence of l?T a?ects the nonequilibrium aging dynamics. Let us consider a negative T -shift process with ?T , for which l?T is supposed to be su?ciently smaller than RT1 (tw1 ), and introduce the time scale tov1 by LT2 (tov1 ) = l?T . At a time range after the T -shift speci?ed as te? ? τ ? tov1 , a longer part w1 of the memory imprinted before the T -shift is still preserved, but such a memory is expected to be irrelevant to the equilibration process to the SG ordered state at T = T2 . Thus the system looks as if it is already in the isothermal aging state at T2 . Then, if our analysis to determine te? is applied to this T -shift process, the w1 expected result is te? ? tov1 irrespectively of tw1 . The w1 circumstances are the same for the positive T -shift protoe? col. Consequently, Ri in Fig. 7 is expected to saturate to l?T at large Ri both for i = 1, 2, and the data for the negative and positive T -shifts come out symmetrically e? with respect to the line of Ri = Ri . We tentatively interpret our data of the negative T shift with ?T = 0.3 as an early stage of the saturation described above. Unfortunately, the data are so limited that we cannot ?gure out a value of l?T . Also the corresponding positive T -shift data nearly coincide with the e? line Ri = Ri , i.e., the two sets of data are by no means symmetric with respect to the line. One of the reason of this asymmetric behavior may be due to our method to specify the e?ective aging time combined with the time scales in our simulation. Although l?T is common to the negative and positive T -shifts, the separation of time scales tw2 , tov2 and te? in the positive T -shift is much w2 smaller than that of tw1 , tov1 and te? in the negative T w1 shift. This is due to a large di?erence in the growth rates at the two temperatures. It is then rather hard to detect a possible small di?erence between tov2 and te? within w2 our present analysis. With these reservations, we interpret our results of the T -shift process with ?T = 0.3 as a dynamic process which re?ects the temperature-chaos predicted for the equilibrium SG phase. §4. Discussions

In the T -shift protocol examined in the present study, the cumulative memory scenario has been con?rmed for T -shift processes with a small magnitude of the shift, i.e., ?T = T1 ? T2 = 0.1. This has been done by close comparisons of χ′′ (ω; t) after the T -shift with that in the isothermal aging at T2 (reference curve). However, there have been little experiments which directly measure te? similarly to our analysis.28, 29) An example is w1 the one by Mamiya et al,28) who analyzed the aging dynamics in the SG-like phase of a ferromagnetic ?ne particles system. In the T -shift process with T1 = 49K and T2 = 47K (with Tg ? 70K) they observed te? ? 3 × tw1 w1

for tw1 = 2.0, ..., 15.0ks. If we suppose τ0 = 10?6 s for a magnetic moment carried on by each ?ne particle,30) we obtain te? ? (2.4 ? 2.7) × tw1 from Eq.(3.1). The result w1 is rather satisfactory and implies that the cumulative memory scenario works as well for the T -shift process with a small ?T in this SG material. Our results on the negative T -shift protocol with ?T = 0.3 have turned out to be incompatible with the cumulative memory scenario. The period te? necessary w1 for the system to become in a T2 -isothermal aging state after the T -shift is signi?cantly smaller than the value of te? estimated from Eq.(1.1) combined with Eq.(2.1). w1 In this negative T -shift process which violates the cumulative memory scenario, however, the strong rejuvenation phenomenon just after the T -shift, which is described in §3.2, has not been detected. We have therefore attributed the non-cumulative memory e?ect we have found to the ‘chaos e?ect’. Quite recently J¨nsson et al. (JYN) have reported o the chaos e?ect which symmetrically appears in the positive and negative T -shifts in a Heisenberg-like spin glass AgMn.19) They have measured the ZFC magnetization with schedules of temperature change corresponding to the T -shift protocol discussed in the present work but within a very small range of ?T (≤ 0.012Tc). The logarithmic-time derivative of the ZFC magnetization, S(t; tw1 ), exhibits a peak, whose position is considered to be at τ = t ? tw1 ? te? , the time required for the w1 merging to an isothermal state at the new temperature just investigated in the present work (see the discussion e? below). In fact, our R1 -vs-R1 plot in Fig. 7 of the negative T -shift with ?T = 0.3 is in qualitative agreement with their Le? -vs-LTi (tw ) plot, where their Le? (LT1 (tw )) e? just corresponds to our R1 (R1 ). In contrast to our numerical observation, however, their data for the positive T -shift plotted in our way appear symmetrically to the e? negative one with respect to the line Ri = Ri . The overlap length l?T estimated by scaling analysis has turned out to be larger than Ri (= LTi (tw )), or before the sate? uration of Ri mentioned in §3.4. One of the reasons of this discrepancy between their experiment and our simulation may be the Heisenberg-spin nature in their material AgMn; Heisenberg spin glasses are more chaotic than Ising spin glasses.19, 31) This point is very interesting and to be further pursued. Next let us make a few comments on aging at a time < w1 range τ = t ? tw1 ? te? after the T -shift, which we have called the transient regime of the T -shift process.15) A main idea for this regime is the droplets-in-domain scenario which involves at least two characteristic length scales as mentioned in § 1. One is the mean domain size at τ = 0, i.e., RT1 (tw1 ) and the other is LT2 (τ ), the mean size of droplets (or subdomains) which are already in local equilibrium of the shifted temperature T2 at time τ after the T -shift. Associated with the growth of LT2 (τ ) some peculiar features have been observed. An example is a non-monotonic time evolution of the energy density in a positive T -shift process (see Fig. 4 in15) ). It is recently named as the Kovacs e?ect in21) since the qualitatively similar phenomenon was ?rst observed in polymer glasses.32)

Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. III.

7

In the droplets-in-domain scenario here we implicitly assume that droplets (or subdomains) in local equilibrium of T2 distinguish themselves from those in local equilibrium of T1 . On the other hand, the Kovacs effect has been observed in negative T -shift processes with ?T = 0.2, for which the ‘chaos e?ect’ is not clearly detected in Fig. 7, or l?T > RT1 (tw1 ). This strongly suggests that in nonequilibrium aging dynamics the spin con?gurations which we have so far supposed to be in local equilibrium at two di?erent temperatures di?er from each other even in length scales shorter than the equilibrium l?T of the corresponding temperatures. This viewpoint is in contrast to the argument of the temperaturechaos in equilibrium, and is worthy to be investigated. From the droplets-in-domain scenario mentioned just above, the strong rejuvenation experimentally observed in the ac susceptibility measurement discussed in §3.2 can be regarded as one of such peculiar phenomena in the transient regime. As mentioned in §2, the ac response associates another short length scale LT2 (tω ). In the time range tω ? τ ? te? so that LT2 (tω ) < LT2 (τ ) < w1 RT1 (tw1 ) holds, the t-dependent part of χ′′ (ω; t) is governed dominantly by the ratio LT2 (tω )/LT2 (τ ) since droplets responding to the ac ?eld less feel the existence of larger domains of RT1 (tw1 ) than that of smaller subdomains of LT2 (τ ). The strong rejuvenation alone is therefore not necessarily incompatible with the cumulative memory scenario. In neither the previous15, 20, 21) nor the present simulations, however, the strong rejuvenation has been detected. This may be again attributed to the smallness of the time window; the separation of the time scales, tω ? τ ? te? , is not enough in the w1 simulations. Quite recently Yoshino and the present authors have argued based on the numerical results on the 4D Ising EA model that ?uctuations of droplets, whose size becomes comparable to that of the preexisting domains, become anomalously large, and that they are responsible to the occurrence of a peak in S(t; tw ) of the isothermal ZFC magnetization at τ ? tw where τ = t ? tw is the time elapsed after the measuring ?eld is applied.24) At the end of the transient regime of the T -shift process, or at the merging to a T2 -isothermal aging state, similar large ?uctuations and so a peak in S(t; tw1 ) at τ ? te? are w1 expected to appear so long as ?T is relatively small. In fact this has been experimentally observed19) as already mentioned above. Combining the arguments based on our numerical results, in particular, the droplets-in-domain scenario and the cumulative memory one, with a possible existence of the ‘chaos e?ect’, we can think of the following behavior that a spin glass exhibits in the negative T -shift protocol of aging depending on the magnitude of ?T (and similar behavior also for the positive T -shift protocol). For a su?ciently small ?T , the merging to the T2 -isothermal aging state is observed at τ ? te? , where te? is given by w1 w1 Eq.(1.1), i.e., by the cumulative memory scenario. When ?T becomes large, both the strong rejuvenation just after the T -shift and the merging to the T2 -isothermal aging state at τ ? te? are expected to be observed. The latw1 ter, however, becomes to be hardly detected by such an

ac susceptibility analysis done in the present work since LT2 (tω ) is much smaller than LT2 (te? ). Instead, it is obw1 served through a peak in S(t; tw1 ) of the ZFC magnetization.33) The extracted value of te? in this case either satw1 is?es the cumulative memory scenario (tov1 ? te? ) or is w1 already strongly a?ected by the chaos e?ect (tov1 ? te? ). w1 What is the expected behavior for T -shift processes with a su?ciently large ?T for which tov1 ? te? or even w1 tov1 ? tobs holds? Here tobs is the shortest time that min min the temperature T2 is experimentally stabilized after the T -shift. The recent experimental results are claimed to reach this regime,19, 33) and a theory for the chaos e?ect on T -shift and T -cycling processes in this regime has been proposed by Yoshino et al.34) Unfortunately this regime has not been realized in the numerical simulations on the 3D EA model. Probably it needs a su?ciently large tw1 , larger than the time-window of our simulation < ( ? 105 t0 ), to realize the condition RT1 (tw1 ) > l?T . In order to further explore aging dynamics in the T -shift protocol, one has to systematically choose values of the parameters ?T and twi even in experiments, since their time-window is similarly small (? 5 decades) to that of the numerical simulation though its absolute magnitude is large (1s ? 1012 t0 ). To conclude, we have numerically studied the T -shift protocol of aging in the 3D EA spin-glass model through the measurement of the ac susceptibility. For processes with a small magnitude of the T -shift, ?T , the memory imprinted in the ?rst stage of isothermal aging is preserved under the T -shift and the SG short-range order continuously grows with a rate intrinsic to the temperature changed (cumulative memory scenario). For T -shift processes with a large ?T the deviation from the cumulative memory scenario has been observed for the ?rst time in the numerical simulation. We attribute the phenomenon to the ‘chaos e?ect’ which, we argue, is qualitatively di?erent from the so-called the rejuvenation e?ect observed just after the T -shift. Acknowledgements We thank H. Yoshino for many fruitful discussions, and P.E. J¨nsson, P. Nordblad, V. Dupuis, E. Vincent o and H. Mamiya for discussions on their experimental results. This work is supported by a Grant-in-Aid for Scienti?c Research Program (# 12640369), and that for the Encouragement of Young Scientists(# 13740233) from the Ministry of Education, Science, Sports, Culture and Technology of Japan. The present simulations have been performed using the facilities at the Supercomputer Center, Institute for Solid State Physics, the University of Tokyo.

[1] E. Vincent, J. Hammann, M. Ocio, J.-P. Bouchaud and L.F. Cugliandolo: in Proceeding of the Sitges Conference on Glassy Systems, Ed.: E. Rubi (Springer, Berlin, 1996) [2] J. P. Bouchau, L. F. Cugliandolo, J. Kurchan and M. M?zard: e in Spin glasses and random ?elds, edited by A. P. Young, (World Scienti?c, Singapore, 1997). [3] P. Nordblad and P. Svendlidh: in the same book as Ref. 2. [4] K. Jonason, E. Vincent, J. Hammann, J.P. Bouchaud, and P. Nordblad: Phys. Rev. Lett. 81 (1998) 3243.

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Hajime Takayama and Koji Hukushima

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