VOLUME 85, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 OCTOBER 2000

Hole-Burning Experiments within Glassy Modelswith Infinite Range Interactions

Leticia F. Cugliandolo1 and José Luis Iguain2

1Laboratoire de Physique Théorique de l’École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, Franceand Laboratoire de Physique Théorique et Hautes Energies, Jussieu, 5ème étage, Tour 24, 4 Place Jussieu, 75005 Paris, France

2Departamento de Física, Universidad Nacional de Mar del Plata, Deán Funes 3350, 7600, Mar del Plata, Argentina(Received 8 February 2000)

We reproduce the results of nonresonant spectral hole-burning experiments with glassy models withinfinite-range interactions that generalize the mode-coupling approach to nonequilibrium situations. Weshow that an ac field modifies the integrated linear response and the correlation in a way that dependson the amplitude and frequency of the pumping field. We study the effect of the waiting and recoverytimes and the number of oscillations applied. This calculation will help discriminate which results canand which cannot be attributed to spatial heterogeneities in real systems.

PACS numbers: 64.70.Pf, 75.10.Nr

One of the most interesting questions in glassy physicsis whether spatially localized heterogeneities are generatedin supercooled liquids and glasses [1].

In most supercooled liquids, the linear response to smallexternal perturbations is nonexponential in the time differ-ence t. Within the “heterogeneous scenario,” the stretch-ing is due to the existence of dynamically distinguishableentities in the sample, presumably localized in space, eachof them relaxing exponentially with its own characteristictime. A different interpretation is that the macroscopic re-sponse is intrinsically nonexponential. In the glass phase,the relaxation is nonstationary and the dependence in t isalso much slower than exponential.

The heterogeneous regions, if they exist, are expectedto be nanoscopic. The development of experimental tech-niques capable of giving evidence for the existence of suchdistinguishable spatial regions has been a challenge forexperimentalists.

With nonresonant spectral hole-burning (NSHB) tech-niques one expects to probe, selectively, the microscopicresponses [2]. The method is based on a wait, pump, re-covery, and probe scheme depicted in Fig. 1. The ampli-tude of the ac perturbation is sufficiently large to pumpenergy in the sample, modifying the response as a linearfunction of the absorbed energy. The perturbation e is veryweak and serves as a probe to measure the integrated lin-ear response of the full system. The large ac and smalldc fields can be magnetic, electric, etc. The comparison ofthe modified (perturbed by the oscillation) and unmodified(unperturbed) integrated responses is expected to yield in-formation about the microscopic structure of the sample.On the one hand, a spatially homogeneous sample absorbsenergy uniformly and its modified integrated response isassumed to be a simple translation towards shorter timedifferences t of the unmodified one. On the other hand,in a heterogeneous sample, the degrees of freedom that re-spond near the pump frequency V are expected to absorban important amount of energy and a maximum difference

3448 0031-9007�00�85(16)�3448(4)$15.00

in the relaxation (equivalently, a spectral hole) is expectedto generate around t � 1�V. The NSHB technique hasbeen first applied to the study of supercooled liquids. Thepolarization response of dielectric samples, glycerol andpropylene carbonate, was measured after being modifiedby an ac electric field [2]. More recently, ion-conductingglasses like CKN [3], relaxor ferroelectrics (90PMN-10PTceramics) [4], and spin glasses (5% Au:Fe) [5] were stud-ied with similar methods. The results have been interpretedas evidence for the existence of spatial heterogeneities.

Our aim is to prove that the main features of NSHBexperiments can be reproduced by a glassy model with nospatial structure. We show that a frequency and amplitudedependent distortion in the integrated response is generatedby the ac perturbation.

We use one model, out of a family, that captures manyof the experimentally observed features of supercooled liq-uids and glasses as, for instance, a two-step equilibriumrelaxation close and above Tc [6], aging effects below Tc

[7], etc. The model is the p spherical spin glass [8] that isintimately related to the Fp21 mode-coupling model [9].It can be interpreted as a system of N fully connected con-tinuous spins or as a model of a particle in an infinite di-mensional random environment [10]. In both cases, noidentification of spatially distinguishable regions can bemade. We expect the same qualitative behavior in other

FIG. 1. Wait, pump, recovery, and probe scheme.

© 2000 The American Physical Society

VOLUME 85, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 OCTOBER 2000

glassy models like the Potts glass, the random energymodel, etc. In the presence of a field, the model is

HJ�s� �X

i1#···#ip

Ji1···ip si1 · · · sip 1

NXi�1

hisi . (1)

The interactions Ji1···ip are quenched independent Gauss-ian variables with zero mean and variance �J2

i1···ip� J �

J̃2p!��2Np21�. p is a parameter and we take p � 3.Hereafter � � J represents an average over P�J� and J̃ �1. The continuous variables si are constrained sphericallyPN

i�1 s2i � N . A stochastic evolution is given to s, �si�t� �

2dHJ�s��dsi�t� 1 ji�t� with ji a white noise with�ji� � 0 and �ji�t�ji�t0�� � 2kBTd�t 2 t0�. When N !`, standard techniques lead to a set of equations for theautocorrelation NC�t, t0�

PNi�1��si�t�sj�t0��� J and the

linear response NR�t, t0� PN

i�1 d��si�t��� J�dei�t0�je�0,with ei�t0� an infinitesimal perturbation modifying theenergy at time t0 according to H ! H 2

Pi eisi . The

dynamic equations read [11]

≠tC�t, t0� � 2z�t�C�t, t0� 1p2

Z t0

0dt00 Cp21�t, t00�R�t0, t00�

1p�p 2 1�

2

Z t

0dt00 Cp22�t, t00�R�t, t00�C�t00, t0�

1 2TR�t0, t� 1 h�t�Z t0

0dt00 h�t00�R�t0, t00� ,

(2)

≠tR�t, t0� � 2z�t�R�t, t0� 1p�p 2 1�

2

3Z t

t0dt00 Cp22�t, t00�R�t, t00�R�t00, t0� . (3)

The Lagrange multiplier z�t� enforces the spherical con-straint and it is determined by an integral equation follow-ing from Eq. (2) and C�t, t� � 1. In deriving Eqs. (2) and(3), a random initial condition at t0 � 0 has been used. Itcorresponds to an infinitely fast quench from equilibriumat T � ` to the working temperature T . The evolutioncontinues in isothermal conditions.

In the absence of energy pumping, these models have adynamic phase transition at a ( p-dependent) critical tem-perature, Tc� p � 3� � 0.61. When an external ac field isapplied, it drives the system out of equilibrium and station-arity and fluctuation-dissipation theorem do not necessarilyhold at any temperature. We do not address the question asto whether the clear-cut dynamic transition survives underan oscillatory field [12]. We simply study the dynamicsclose to the critical temperature in the absence of the fieldby constructing a numerical solution to Eqs. (2) and (3)with a constant grid algorithm of spacing a. We presentdata for small spacings, typically a � 0.02, to minimizethe numerical errors. Because of the fact that Eqs. (2) and(3) include integrals ranging from t0 � 0 to present timet, the algorithm is limited to a maximum number of it-

erations of the order of 8000 that imposes a lower limitV � 2p�8000a � 0.1 to the frequencies we use.

A word of caution concerning the scheme in Fig. 1 andthe times involved is in order. For the purpose of collectingdata for each reference unmodified integrated response,the sample is prepared at the working temperature T att0 � 0 and let freely evolve during a total waiting timetw 1 t1 1 tr . Depending on T , this interval may or maynot be enough to equilibrate the sample. (t1 � 2pnc�V

with V the angular velocity of the field that will be usedto record the modified curve.) An infinitesimal probe e isapplied after tw 1 t1 1 tr to measure

F�t� Z t

0dt0 R�tw 1 t1 1 tr 1 t, tw 1 t1 1 tr 1 t0� .

As an abuse of notation we eliminate the possible tw 1

t1 1 tr dependence from F�t�. The modified integratedresponse F� is measured after waiting tw , applying nc

oscillations of duration t1 � 2pnc�V, further waiting tr ,and only then applying the probe e. The effect of the acperturbation is quantified by studying

DF F� 2 F . (4)

We have examined DF at T � 0.8 . Tc and T �0.59 , Tc. We pump one oscillation with amplitude hF �0.1 that implies a linear modification in the absorbed en-ergy (see Fig. 4). For simplicity, we choose tw � tr � 0.In Fig. 2 we show DF against logt for different Vs atT � 0.8. All the curves vanish at short t by definitionand they slowly approach zero at long t due to the

FIG. 2. The distortion DF vs logt for a single oscillation.T � 0.8 . Tc, hF � 0.1, and tw � tr � 0. (a) Both DFm andtm decrease with increasing V if V . Vc � 1. The dotted,solid, and dashed curves correspond to V � 1, 2, and 5, respec-tively. (b) If V , Vc � 1, DFm increases with V while tmis almost unchanged. The dotted and solid curves correspond toV � 0.1 and 0.2, respectively.

3449

VOLUME 85, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 OCTOBER 2000

FIG. 3. As in Fig. 2 for T � 0.59 , Tc. The peak movestowards shorter times and its height decreases for increasingfrequencies. The dashed, solid, and dotted curves correspond toV � 0.1, 0.5, and 1, respectively.

property of “weak long-term memory” that ensures thatany perturbation of finite duration will eventually be for-gotten [7]. Thus, the perturbations generate bell-shapedcurves with a maximum DFm max�DF� at tm. In2a, the Vs are larger than a threshold value Vc � 1.The height of the peak DFm decreases with increasingfrequency reaching the limit DFm � 0 for V ! `. Inaddition, the location of the peak tm moves towardslonger times when V decreases. In 2b, V , Vc andthe behavior of DFm is the opposite, it decreases whenV decreases and its position is either independent ofV or it very smoothly moves towards shorter times forincreasing V. The nonmonotonic behavior of DFm

with V is a consequence of the interplay between thestructural relaxation time ta and the period of the os-cillation 2p�V. Indeed, the response has a slow decaycharacterized by ta , defined by R�ta 1 tw , tw� � 1�2, fortw fixed. The term

Rmin�t0,t1�0 dt00 h�t00�R�t0, t00� in Eq. (2)

controls the effect of the field and vanishes in the limitsV ! ` and V ! 0. Heuristically, one expects its effectto be maximum if V � 2p�ta , leading to the criticalfrequency Vc � 2p�ta . The dependence of tm with V

does not follow from a simple argument but from the fullsolution of Eqs. (2) and (3). These results qualitativelycoincide with the measurements of the electric relaxation

FIG. 4. Check of DFm ~ h2F . nc � 1, V � 1. 1 corresponds

to T � 0.59 and 3 to T � 0.8. The line has a slope equal to2 and is a guide to the eye.

3450

FIG. 5. The normalized maximum distortion for several recov-ery times tr . In panel (a) T � 0.59 and crosses, diamonds, andplusses correspond to V � 1, 2, and 3, respectively. In panel(b) T � 0.8 and crosses, plusses, and diamonds correspond toV � 1, 5, and 10, respectively.

in CKN at T , Tg in Figs. 1a and 1b of Ref. [3]. In Fig. 3we show DF against logt for different Vs at T � 0.59.For all V we reproduced the situation of panel a in Fig. 2,as if V . Vc. We have not found a threshold Vc that hasgone below the minimum V reachable with the algorithm.

The maximum modification of the relaxation DFm in-creases quadratically with the square of the amplitude ofthe pumping field hF , and hence linearly in the absorbedenergy, as long as hF # 1. In Fig. 4 we display the rela-tion DFm ~ h2

F in a log-log scale for the two temperaturesexplored. hF � 0.1, used in Figs. 2 and 3, is in the linearregime.

The effect of the pump diminishes with increasingrecovery time tr . A convenient way of displaying this

FIG. 6. Effect of several cycles at T � 0.8 for V � 10. Thedashed, solid, and dotted curves correspond to nc � 10, 2, and1, respectively.

VOLUME 85, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 OCTOBER 2000

FIG. 7. Distortion for tw � 8 with dots compared to the onefor tw � 0 with solid line at T � 0.59.

result is to plot the normalized maximum deviationDFm�tr��DFm�0� vs Vtr . Using several frequenciesand recovery times, we verified that this scaling holdsfor T � 0.59 but does not hold for T � 0.8, as shownin Fig. 5. This simple scaling holds very nicely in therelaxor ferroelectric [4] and in the spin glass [5] but it isvery different from the V independence of the propylenecarbonate [2].

Up to now, the effect of a single cycle of differentfrequencies has been studied. Another procedure can beenvisaged. Since t1 � 2pnc�V, we can change t1 by ap-plying different numbers of cycles nc while keeping V

fixed. In Fig. 6 we show the distortion due to nc � 10, 2,and 1 cycles with V � 10 at T � 0.8. The qualitativedependence on nc is indeed the same as the dependenceon 1�V: the peaks are displaced towards longer timeswith increasing nc (longer t1). This behavior is similarto the results obtained for propylene carbonate in Fig. 11of Ref. [2(b)] though we do not reach the expected satu-ration within our accessible time window.

Below Tc the nonperturbed model never equilibrates andthe relaxation depends on tw . Indeed, ta is an approxi-mately linear function of tw [7,10] and the distortion mightdepend on tw . We compare DF vs logt for two tw’s inFig. 7.

Finally, we checked that the effect of one or many pumposcillations on the difference DC C��tw 1 t1 1 tr 1

t, tw 1 t1 1 tr� 2 C�tw 1 t1 1 tr 1 t, tw 1 t1 1 tr �is very similar to the one observed in DF. Figure 8shows the modification observed at T � 0.8 and V . Vc

(cf. Fig. 2). This observation is interesting since correla-tions are easier to compute via numerical simulations. Ina future publication we shall show that the distortion inthe correlations of the mean-field Sherrington-Kirkpatrickmodel for spin glasses is much more pronounced than inglasses, as also observed experimentally [5].

We conclude by stressing that we do not claim that spa-tial heterogeneities are absent in real glassy systems. Wejust wish to stress that the ambiguities in the interpretation

FIG. 8. Change in the autocorrelation at T � 0.8. V � 1(dashed line), V � 2 (solid line), and V � 5 (dotted line).

of experimental results have to be eliminated in order tohave unequivocal evidence for them. The comparison ofexperimental results to the behavior of glassy systems withdroplet structures or, as in the case studied in this Letter,fully connected mean-field models will certainly help usrefine the experimental techniques. Numerical simulationscan play an important role in this respect.

L. F. C. and J. L. I. thank the Department of Physics(UNMDP) and LPTHE (Jussieu) for hospitality, andECOS-Sud, CONICET, and UNMDP for financial sup-port. We thank R. Böhmer, H. Cummins, G. Diezemann,M. Ediger, J. Kurchan, and G. McKenna for very usefuldiscussions, and T. Grigera, N. Israeloff, and E. Vidal-Russel for introducing us to the hole-burning experiments.

[1] H. Sillescu, J. Non-Cryst. Solids 243, 81 (1999); M. T.Cicerone and M. D. Ediger, J. Chem. Phys. 103, 5684(1995); R. Böhmer et al., J. Non-Cryst. Solids 235–237,I-9 (1998); W. Kob et al., Phys. Rev. Lett. 79, 2827 (1997).

[2] (a) B. Schiener et al., Science 274, 752 (1996); (b) B.Schiener et al., J. Chem. Phys. 107, 7746 (1997).

[3] R. Richert and R. Böhmer, Phys. Rev. Lett. 83, 4337(1999).

[4] O. Kircher, B. Schiener, and R. Böhmer, Phys. Rev. Lett.81, 4520 (1998).

[5] R. V. Chamberlin, Phys. Rev. Lett. 83, 5134 (1999).[6] W. Götze, J. Phys. C 11, A1 (1999).[7] L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. 71, 173

(1993).[8] A. Crisanti and H.-J. Sommers, Z. Phys. B 87, 341 (1992).[9] T. R. Kirkpatrick and D. Thirumalai, Phys. Rev. B 36, 5388

(1987).[10] J.-P. Bouchaud et al., in Spin Glasses and Random Fields,

edited by A. P. Young (World Scientific, Singapore, 1998).[11] L. F. Cugliandolo and J. Kurchan, Phys. Rev. B 60, 922

(1999).[12] L. Berthier, L. F. Cugliandolo, and J. L. Iguain (unpub-

lished).

3451