Glass and polycrystal states in a lattice spin model

Glass and polycrystal states in a lattice spin modelAndrea Cavagna, Irene Giardina, and Tomás S. Grigera Citation: The Journal of Chemical Physics 118, 6974 (2003); doi: 10.1063/1.1560937 View online: http://dx.doi.org/10.1063/1.1560937 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/118/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in “Ideal glassformers” vs “ideal glasses”: Studies of crystal-free routes to the glassy state by “potential tuning”molecular dynamics, and laboratory calorimetry J. Chem. Phys. 138, 12A549 (2013); 10.1063/1.4794787 Transmission electron microscopy and in situ Raman studies of glassy sanbornite: An insight into nucleationtrend and its relation to structural variation J. Appl. Phys. 108, 063507 (2010); 10.1063/1.3487473 Molecular simulation of the crystallization of aluminum from the supercooled liquid J. Chem. Phys. 127, 144509 (2007); 10.1063/1.2784120 Characterization of polyethylene crystallization from an oriented melt by molecular dynamics simulation J. Chem. Phys. 121, 2823 (2004); 10.1063/1.1768515 Influence of the liquid states on the crystallization process of nanocrystal-forming Zr–Cu–Pd–Al metallic glasses Appl. Phys. Lett. 75, 3644 (1999); 10.1063/1.125415

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Glass and polycrystal states in a lattice spin modelAndrea Cavagna, Irene Giardina, and Tomas S. Grigeraa)

Center for Statistical Mechanics and Complexity, INFM Roma ‘‘La Sapienza’’and Dipartimento di Fisica, Universita` di Roma ‘‘La Sapienza,’’ 00185 Roma, Italy

~Received 5 November 2002; accepted 23 January 2003!

We numerically study a nondisordered lattice spin system with a first order liquid–crystal transition,as a model for supercooled liquids and glasses. Below the melting temperature the system can bekept in the metastable liquid phase, and it displays a dynamic phenomenology analogous to fragilesupercooled liquids, with stretched exponential relaxation, power law increase of the relaxationtime, and high fragility index. At an effective spinodal temperatureTsp the relaxation time exceedsthe crystal nucleation time, and the supercooled liquid loses stability. BelowTsp liquid propertiescannot be extrapolated, in line with Kauzmann’s scenario of a lower metastability limit ofsupercooled liquids as a solution of Kauzmann’s paradox. The off-equilibrium dynamics belowTsp

corresponds to fast nucleation of small, but stable, crystal droplets, followed by extremely slowgrowth, due to the presence of pinning energy barriers. In the early time region, which is longer thelower the temperature, this crystal-growth phase is indistinguishable from an off-equilibrium glass,both from a structural and a dynamical point of view: crystal growth has not advanced enough to bestructurally detectable, and a violation of the fluctuation–dissipation theorem~FDT! typical ofstructural glasses is observed. On the other hand, for longer times crystallization reaches a thresholdbeyond which crystal domains are easily identified, and FDT violation becomes compatible withordinary domain growth. ©2003 American Institute of Physics.@DOI: 10.1063/1.1560937#

I. INTRODUCTION

When a liquid is cooled fast enough below its meltingpoint, crystallization is avoided and the system enters thesupercooled phase. Relaxation time increases rapidly in thistemperature regime, and when it becomes comparable to thelargest experimentally accessible time the system falls out ofequilibrium, remaining stuck in a disordered phase calledstructural glass.1

The configurational disorder of a structural glass is notcaused by the presence of intrinsic disorder in the Hamil-tonian, since this is just the sum of deterministic interactionelements. This notwithstanding, a lot of attention has beendevoted to the phenomenological analogies between struc-tural glasses and mean-field spin-glasses, which are systemsdirectly containing quenched disorder in the Hamiltonian inthe form of random couplings among the spins.2 In particu-lar, models of spin-glasses withp-body interactions andp>3, seem indeed to have many features in common withstructural glasses,3 with the obvious~but crucial! differencethat in mean-field systems there is a purely dynamical tran-sition which is absent in real finite-dimensional glasses. De-spite the close phenomenology, however, it is natural to askwhy we use models with quenched disorder to understandthe behavior of physical systems with deterministic Hamil-tonians.

Disorder, one of the characteristics of the glassy state,cannot be expected to arise spontaneously from a determin-istic mean-field Hamiltonian, because in this case the me-

chanically stable configurations are ordered: the local field isthe same for all spins, and thus all tend to point in the samedirection. Thus if we wish to stick to mean-field and theassociated analytical advantages, we must resort to quencheddisorder to produce glassy behavior. The argument does notapply to finite range systems, however, and we can askwhether finite dimensional spin systems with no quencheddisorder andp>3 display a glassy phenomenology similarto structural glasses~we of course know thatp52, i.e., theIsing model, is not glassy under many respects!. This is arelevant question, first because it is important to understandwhat the key ingredients responsible for glassy dynamics are,and second because lattice spin systems are normally easierto study than off-lattice liquid models.

The answer to this question seems to be affirmative. Anumber of spin models without quenched disorder now existthat reproduce a phenomenology similar to supercooled liq-uids and glasses. Generally, one can say that in all thesesystems the nontrivial dynamic behavior is related to thepresence of frustration, due to complex even if non-disordered interaction terms in the Hamiltonian. In the SSmodel of Ref. 4, this is achieved by mixing nearest- andnext-nearest-neighbor interactions, which gives an interest-ing phenomenology, although not precisely glassy. In theplaquette~PQ! model of Refs. 5, 6, 14, on the other hand,glassiness is achieved by ap-body interaction term withp54, in close analogy withp-spin models of mean-field spin-glasses. In the lattice glass model of Ref. 9 frustration arisesfrom a constraint on the numbers of neighboring particles onthe lattice. The PQ model has been heavily studied in recentyears, and the results seem to suggest that its phenomenol-ogy has at least some points in common with structural

a!Also at: Centro Studi e Ricerche ‘‘Enrico Fermi,’’ Via Panisperna, 89/A,00184 Roma, Italy.

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glasses. In particular, a sufficiently rapid cooling of the PQmodel forms an off-equilibrium disordered phase whichclosely resembles real glasses. Therefore, it would seem thatp-body interactions in finite dimensional spin models withoutquenched disorder are sufficient to reproduce a phenomenol-ogy similar to supercooled liquids and glasses.

The aim of this paper is to study a model in the sameclass as those just described, namely the deterministic ver-sion of the CTLS model introduced in Ref. 7~a concisedescription of some of our results has recently appeared8!. Asdiscussed below, this model has various advantages with re-spect to its predecessors and will enable us to check moredeeply to what extent lattice spin systems can mimic struc-tural glasses. Moreover, important questions on the low tem-perature behavior of glasses, such as the stability of the su-percooled phase and the role of the crystal, can be addressed,whereas in previous models such an analysis was very lim-ited. Let us now explain why.

Liquids in nature have a crystalline ground state, usuallywith low degeneracy, and display a first order thermody-namic transition at the melting temperature. In the PQ modelin three dimensions a first order transition takes place, butthe crystalline ground state has an unusual property for liq-uids, in that it is strongly degenerate: for linear sizeL, thePQ model has 23L different ground states with the same en-ergy. The problem is not the degeneracy in itself~the groundstate entropy density is zero!, but rather the consequencesthat this degeneracy has on the possibility to measure thedegree of order in the system. More precisely, in the PQmodel, given a ground state configuration~for example, theferromagnetic one!, we can obtain another ground state byflipping any plane of spins. Clearly, by iterating this proce-dure we can produce ground state configurations which lookhighly disordered. The problem then is how to distinguish atruly disordered, or glassy, configuration, from a configura-tion made up of many droplets of different ground states. Theenergy of both configurations will be high, in the first casebecause it is essentially a liquid, in the second case becauseof the interfaces among different droplets. In other words,energy is not a sufficiently precise measure of order in thePQ model, and due to the strong ground state degeneracy noconfigurational means can be devised to quantify order anddomain growth.

The system we introduce here does not suffer from thisproblem: it has a doubly degenerate ground state, such that itis possible to measure the amount of crystalline order and tomonitor domain growth. A likely objection is that glassyphysics does not have anything to do with the crystal, andthat the possibility of measuring the formation and growth ofcrystal domains is far from essential to study the physics ofglasses. We reply to this objection with two observations.First, that simple glass formers usuallydo havea crystallinephase, whose possible connection with the properties of theglass cannot be completely ruled out in principle. In fact, ourmodel explicitly shows such a connection. Second, if on theother hand, spin models with a crystal phase fail somehow toreproduce some basic property of glasses, then we shouldunderstand why, and make sure that all spin models are freefrom this problem. Indeed, the precise nature of what is nor-

mally called the glassy phase in systems like PQ5 and inlattice glass models9 is the main focus of our work.

Our model shows, in this respect, an interesting phenom-enology. As we will show, it has a first order liquid–solidtransition at a melting temperatureTm . Crystallization canbe avoided by fast cooling, and a supercooled liquid phase,metastable with respect to the crystal, is found. ‘‘Equilib-rium’’ measurements can be performed as long as the equili-bration time of the supercooled liquid is much shorter thanthe nucleation time of the crystal. In this phase the behaviorof our system is very similar to real supercooled liquids, andin particular to fragile systems. However, a temperatureTsp

exists, where the crystal nucleation time becomes of thesame order as the liquid equilibration time: at this point thesupercooled liquid loses stability and it no longer exists be-low Tsp. We callTsp theeffective spinodaltemperature. It isimportant to stress the following fact: we are able to detectthe loss of stability of the liquid only because the equilibra-tion time atTsp, tsp, is shorter than our longest experimentaltime. We can state this point in a different way. A systemfalls out of equilibrium when the relaxation time becomestoo large for the~real/numerical! experiments, and this is theoperational definition of the glass transition temperatureTg .Therefore, the reason why we can observe the loss of stabil-ity of the supercooled liquid in our model is thatTsp.Tg .The Kauzmann paradox~i.e., the fact that the extrapolatedliquid entropy becomes smaller than the crystal entropy at afinite temperature! is avoided by this loss of stability: noequilibrium supercooled liquid exists belowTsp and the en-tropy extrapolation is meaningless. In fact, this is the sce-nario proposed by Kauzmann himself10 as solution of theparadox, which is thus exactly reproduced in this system.Below Tsp the system enters a completely different phase,where crystal nucleation and crystalline domain growth arethe main dynamic processes. However, by quenching tolower temperatures the growth of crystalline order becomesso slow that it is impossible to distinguish this phase from atypical off-equilibrium realistic glass. Moreover, we shallshow that dynamical measurements of the fluctuation–dissipation~FD! ratio are compatible with the results ob-tained in ordinary structural glasses belowTg .

It is our impression that the scenario described above isalso shared by the PQ, and possibly other spin models. Inparticular, that what is normally called the glassy phase forthe PQ model is an off-equilibrium regime of very slow crys-tal growth. However, as already mentioned, without a directmeasure of crystalline order it is hard to assess the amount oforder in the PQ model, so the question whether there is infact a glassy phase qualitatively different from the crystalgrowth phase belowTsp is open. This is certainly not the casein the present model, though.

Throughout the paper we will use a notation common tosupercooled liquids and glasses. We callTm the melting tem-perature. In analogy with mode coupling theory,11 Tc will bethe temperature where a power law fit locates a divergence ofthe relaxation time, whileT0 will be the temperature where aVogel–Fulcher–Tamman fit puts the same divergence. Theso-called Kauzmann temperature, i.e., the temperature wherethe extrapolated liquid entropy becomes equal to the crystal

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entropy, is calledTs . The effective spinodal temperature,i.e., the temperature below which it becomes impossible toobserve the metastable liquid is calledTsp. This temperatureis calledTg in Ref. 5, but we reject this choice because, asalready mentioned, in the liquid literature the glass transitionis defined rather as the point where the equilibration time islonger than the experimental time.

The paper is structured as follows. In Sec. II we intro-duce our model and describe the properties of the crystallineground state. In Sec. III we show, by comparing the liquidand crystal free energies, that the model has a first orderphase transition at a melting temperatureTm . The dynamicsof the supercooled liquid belowTm and the metastabilitylimit Tsp are considered in Sec. IV, while aging dynamics andits relation to domain growth are studied in Sec. V. Finally,we draw our conclusions in Sec. VI.

II. THE MODEL

The model is described by the Hamiltonian

H5(i 51

N

f i~11si !, ~1!

where the variablessi561 are Ising spins belonging to atwo-dimensional square lattice of sideL, and the plaquettevariable f i is the product of the four nearest neighbors ofspin i:

f i5siWsi

SsiEsi

N ~2!

~W is for west,S for south, and so on!. The model is studiedby Metropolis Monte Carlo simulations with single-spin–flipdynamics for lattices of sizeL5100 andL5500.

A disordered version of this model was first introducedin Ref. 7, as a way to model two-level systems in a lattice,interacting with their nearest neighbors, whence its nameCTLS ~coupled two level systems!. In fact, one can take thespin si to be the variable describing the state of a two-levelsystem sitting at sitei. Then statesi521 has always zeroenergy, while the other state can be at 1 or21 depending onthe factorf i . Thus when one of the neighbors switches, therelative height of the two states ati is reversed. In Ref. 7,however, quenched random coupling constants were consid-ered. Here we shall see that the multibody nature of theinteraction is sufficient to give a nontrivial dynamic andstatic behavior.

Frustration in the CTLS comes from the fact that it is notpossible to satisfy simultaneously the four-spin and the five-spin interaction. After careful inspection of the Hamiltonianit is found that the ground state of the CTLS is obtained bycovering the lattice with the following nonoverlapping ele-ments:

si521, siW5si

S5siE5si

N511. ~3!

As Fig. 1 shows, the covering can be done in two ways,giving rise to two different ground states which we call dex-trocrystal and levocrystal. These two crystals are connectedby the symmetryx→2x, y→y, or x→x, y→2y, whilethey are both invariant underx→2x, y→2y. Therefore,the CTLS has a crystalline ground state, with energy density

EGS521.6. In Fig. 1 it can be seen that the crystal can beobtained by periodic repetition of a 535 unit cell. Apartfrom the two distinct dextro and levo forms, additional crys-tals can be formed by translating the unit cell up to fivelattice spacings in each direction. There are then differentways to cover the lattice, which are not locally compatiblewith each other. Upon a quench, the system will greedilyoptimize the energy locally in an uncoordinated way, creat-ing many different competing domains with boundaries thatturn out to be pinned, thus giving rise to slow dynamics.

III. THERMODYNAMICS

In the present section we will show that the CTLS has afirst order phase transition at a melting temperatureTm . Westart with the numerical observation of hysteretical behaviortypical of first order transitions: we find that, in a range oftemperatures, the system stays for very long times@106

Monte Carlo steps~MCS! or more forL5100] in the phase~liquid or solid! in which it was prepared. In Fig. 2 we plotthe equilibrium internal energy as a function of the tempera-ture for the crystal and the liquid phases.

FIG. 1. Dextrocrystal~left! and levocrystal~right!: empty circles correspondto positive spins, full circles to negative spins. In the left figure the squaremarks the five-element unit cell. In the right figure the small square showsthe elementary unit used in the definition of the crystal mass~see Sec. III!.

FIG. 2. Liquid and crystal energy per spin as a function of temperature. Thecontinuous lines correspond to the two fits reported in the text. The horizon-tal dotted line marks the energy of the crystal at zero temperatureEGS

521.6. The vertical dotted line marksTm51.30.L5100.

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To find Tm we need to compute the free energy in thetwo phases by integration of the internal energy. To this aim,we fit the liquid energy~LQ! as

ELQ~T!52a tanh~b/T!, a51.84, b51.10, ~4!

while for the crystal~CR! we find

ECR~T!5EGS1cTn, c55.531025, n513. ~5!

By using the thermodynamic formula,

bF~b!5b0F~b0!1Eb0

b

db8 E~b8!, b51/T, ~6!

we obtain the free energy of the liquid and crystal phases:

FLQ~T!52T log 22a

bT log cosh~b/T!, ~7!

FCR~T!5EGS2c

n21Tn, ~8!

where we have takenb050 for the liquid, andb05` for thecrystal. The melting temperature is fixed by requiringFLQ(Tm)5FCR(Tm), which gives

Tm51.30. ~9!

We have checked that our energy data are independent ofsystem size for 100<L<500.

In order to check that a first order liquid-crystal transi-tion actually takes place atTm we proceed as in Ref. 5: weconsider an initial condition where half of the system is inthe crystal phase and half in a random phase, and we run thisconfiguration atTm10.01 and atTm20.01. If our determi-nation of Tm is correct, after a transient time the systemshould relax either to the liquid or crystalline phase, since thestable phase will expand and take over the whole systemafter a sufficiently long time. The time evolution of the en-ergy is shown in Fig. 3, confirming our previous determina-tion of Tm .

Even though the energy of a configuration may be con-sidered as an index of the degree of crystallization of thesystem, it is certainly not a very sensitive one. Given that weknow the crystal in this system, and thanks to its low degen-eracy, we introduce some methods to directly quantify the

amount of crystalline order in our samples. This can be doneeither by measuring the typical crystal droplet sizej, or bymeasuring the total crystal massm. To measurej we observethat both crystals have a periodic structure with a unit cell ofsize 5. Thus, the Fourier transform of the spatial correlationfunction,

G~r !5^s~0!s~r !&2^s&2

12^s&2 , ~10!

has a peak atk052p/5. The widthl of this peak is propor-tional to the inverse of the typical crystalline domain size.We want to extract froml a measure of how large is thetypical crystal domain, a question which is better answeredby comparing it to the system size. So we actually definej asa density, i.e.,j51/(lL)P@0:1#. It is important to note thatthis definition of j has a limited power of resolution: in arandom configuration we find on average thatj050.05,which can therefore be considered as the effective zero forj.But a complementary measure of crystal order is needed: in aconfiguration entirely made up of tiny mismatched crystal-lites j would be very small, even though the total amount ofcrystal is large. The crystalline massm can be measured bycounting the fraction of crystallized spins. To this aim, wehave to define an elementary unit, which is large enough tohave a small probability to be formed randomly, but smallenough to be sensitive to small amounts of crystal. Wechoose the nine-spin unit shown in Fig. 1~right!: in the purecrystal each minus spin is surrounded by eight pluses, and inthe unit cell we have five of these elements~which clearlyoverlap!. To count all the spins in the cell we must multiplythe number of these units by 5. Thus, we define the crystalmass densitymP@0:1# as the number of these elementaryunits times 5 normalized byL2. As for j, we have a randomaverage value of the mass,m050.01, which is the effectivezero of m. Had we chosen the 535 unit cell, the accuracywould be higher, but the sensitivity would decrease as well.In order to test the definition ofj and m, we measure theirtime evolution in the experiment described above for thedetermination ofTm . The results are shown in Fig. 4. Notethat in the liquid phase atT5Tm10.01 we havem;0.15@m0 ~whereasj is compatible with zero!, meaning that also

FIG. 3. Energy per spin as a function of time below and aboveTm .L5100.

FIG. 4. Crystal massm as a function of time below and aboveTm . Sametemperatures and symbols as in Fig. 3.



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in the liquid aboveTm there is a certain amount of shortrange order. Both these quantities are thus excellent configu-rational indicators of the amount of crystalline order in thesystem.

We finish this section by noting that if we extrapolateELQ(T) and FLQ(T) to low temperatures, we can find thetemperatureTs where the entropy of the supercooled liquidbecomes equal to the entropy of the crystal. We find

Ts50.91. ~11!

Ts is the temperature where the Kauzmann paradox occurs:10

below Ts the supercooled liquid entropy would becomesmaller than that of the crystal. This temperature is usuallycalled Kauzmann temperature in the literature, a denomina-tion we find somewhat unsatisfactory given that Kauzmannhimself did not believe the extrapolation was really meaning-ful.

IV. DYNAMICS

The aim of this section is to study the dynamical behav-ior of the CTLS in the supercooled liquid phase~i.e., for T,Tm) and in particular to analyze how the relaxation timeincreases when decreasing the temperature. There is amarked difference in the behavior of the system if our obser-vational time window is longer or shorter than a given timetsp;104 MCS. For t,tsp the supercooled liquid is stable atall temperatures where equilibration may be achieved: wecan either study the equilibrium supercooled liquid or theoff-equilibrium glass, formed by a sufficiently fast cooling.On the other hand, if we are able to observe the dynamics ontime scalest.tsp we discover that the supercooled liquidloses stability at low enough temperatures, and the systementers a completely different regime. Therefore, it is useful todivide our dynamical investigation into two parts: first weshall assume that our maximum experimental timetexp isshorter thantsp and we will measure the equilibrium proper-ties of the supercooled liquid. Second, we will consider timeslarger thantsp, to analyze the loss of stability of the super-cooled liquid.

A. Moderate times dynamics: tËt sp

In this section we will assume that the longest experi-mental time available to us istexp<tsp;104 MCS. To probethe dynamics of the CTLS we measure the equilibrium re-laxation timet as a function of the temperatureT. This canbe obtained from the decay rate of the~normalized! equilib-rium spin–spin correlation function, defined as

C~ t,tw!5^s~ tw!s~ tw1t !&2^s&2

12^s&2 , ~12!

where^¯& indicate averages over spins and initial configu-rations. We quench the system from infinite temperaturedown to the target temperatureT and wait for it to equili-brate. At equilibrium the dependence ontw disappears andC5C(t). This is our first equilibration test. The correlationfunction can be adequately fit with a stretched exponential,which is the expected equilibrium behavior forC(t) in su-percooled liquids,

C~ t !5exp@2~ t/t!b#. ~13!

From this fit we extract the relaxation timet. To be sure thatthe system has equilibrated we check that at each tempera-ture the equilibration time that we have waited for is muchlarger thant. More precisely, sinceC(t) decays to zero inapproximately 20t we requireteq>20t.

The relaxation time as a function of the temperature isplotted in Fig. 5. We note that there is no dynamic signatureof the melting transitionTm . Below Tm the supercooled liq-uid is metastable with respect to the crystal, and there is asharp increase of the relaxation time. In the inset of Fig. 5 wecan see that the Kohlrauschb exponent of the stretched ex-ponential fit decreases with the temperature, as it happens forrealistic models of liquids~see, for example, the data of Ref.12!.

In the temperature regime of Fig. 5 the relaxation timecan be fitted very well by a power law,

t5A

~T2Tc!g , with Tc51.06, g52.29. ~14!

This fact suggests that the CTLS is a fragile system.13 Tosupport this conclusion, we compare the CTLS with anArrhenius/strong system, like the PQ model ind52.14 Togive a quantitative measure of fragility we need to fit therelaxation time with a Vogel–Fulchner–Tamman form,

t5t0 expS D

T2T0D . ~15!

The fit works pretty well for the CTLS and givesT050.76for [email protected]:1.7#, andT050.90 [email protected]:1.4#. The quan-tity KVFT5T0 /D is a measure of the degree of fragility ofthe system.13 For the two models, we find

2dPQ: KVFT50.017, ~16!

CTLS: KVFT50.095, ~17!

consistent with a high fragility of the CTLS. Incidentally, wenote that an Arrhenius fit of the 2d– PQ correlation time

FIG. 5. Relaxation time as a function of temperature. The full line is thepower law fit Eq.~14!. Inset, Kohlrauschb exponent of the stretched expo-nential fit vs temperature.L5500.



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gives a barrier sizeD55.8. This can be reconciled with theresultD53 of Ref. 14 noting that the spin coupling constantis taken as 1/2 in that work.

For T,1.25 we are no longer able to equilibrate thesystem within our experimental timetexp5104, therefore thisis our glass transition temperature within this time regime. If,however we give up the requirement to be at equilibrium, wecan explore the low temperature region. For quenches belowT51.25 the system should be considered, according to usualdefinitions, as an out of equilibrium glass. In fact, out-of-equilibrium glasses can be formed when cooling the liquid atlow temperatures at a rate fast enough to prevent equilibra-tion. In Fig. 8 we show the results of cooling experimentswith a linear cooling scheduleT5Tin2rt with cooling rater 5dT/dt and Tin51.67. We plot the system energy as afunction of T for different cooling rates. In the same graphwe report the equilibrium supercooled energyELQ(T), whichof course must be extrapolated below the last equilibriumtemperature. Considering for now only the three fastest rates,we find as expected that at high temperatures even for rela-tively fast coolings the system stays in equilibrium, whereasE(T) departs from the equilibrium curve at lower tempera-tures. The slower the cooling, the lower the temperaturewhere the energy freezes in the glassy phase: this is the glasstransitionTg(r ) corresponding to that particular cooling rate.Since we have set a maximum experimental timetexp, tobring the system through the interesting temperature regionof width [email protected]:1.3#, within a time texp we can afford atmost cooling rates of the order of 531025. Slower rates willbe considered in the next section.

The energy of the configurations reached atT50 uponcooling, as a function of the cooling rate, is a quantity whichis considered as an important indicator of glassy behavior.What we find forr P@531024,0.01# is a slow dependence ofE2EGS on r, compatible with a power law with low expo-nent or with logarithmic behavior~see Fig. 13 and Sec. V fora wider discussion of this point!, which is very similar towhat was observed in real glasses.

Summarizing the results of this section, for times smallerthan tsp;104 MCS we observe in the CTLS a phenomenol-ogy typical of fragile structural glasses: in the supercooledliquid the correlation function displays stretched exponentialrelaxation. The relaxation timet grows very sharply and iswell fitted by a power law, and the stretching exponentbdecreases with decreasing temperature. The fragility index ismuch larger than a typical Arrhenius/strong spin system. Fi-nally, on rapid cooling, the system goes out of equilibriumremaining stuck in the glassy phase, at a glass transitiontemperature which is lower the smaller the cooling rate.

B. Long times dynamics: tÌt sp and the metastabilitylimit Tsp

We analyze now the behavior of the system for timeslonger thantsp5104. In Fig. 6 we plot the energy as a func-tion of time for three different temperatures belowTm , withrandom initial condition.

At the higher temperature,T51.26, the system relaxesin the supercooled liquid and remains in this phase up to ouractual experimental time,texp523106. On the other hand,

for T51.23, after initial equilibration in the supercooled liq-uid phase, the system makes the transition to the crystal.Note that atT51.23 the supercooled liquid survives up to105 MCS when the transition to the crystal starts, and crys-tallization is completed in 106 MCS. At T51.18, on theother hand, we see that the supercooled liquid lasts roughly104 MCS, while the approach to the crystal takes more than106 MCS. In other words, the lower the temperature, thefaster is the departure from the liquid, but the slower is thecompletion of the crystal. If we further decrease the tempera-ture, the plateau corresponding to the supercooled liquid dis-appears, and the energy decreases steadily toward the crystalground state. The dynamics becomes ever slower in this tem-perature regime, and the system remains out of equilibriumup to very long times~see Sec. V!. These results can beinterpreted in terms of nucleation theory.15 Even in the su-percooled liquid phase the system nucleates droplets of crys-tal with finite probability. Most of these droplets are toosmall to be stable and soon disappear, however a critical sizeexists such that whenever a droplet of that size is created itwill not shrink back, but rather expand and bring the systeminto the stable crystal phase. This phenomenon, criticalnucleation, is characteristic of metastability and determinesthe average lifetime of the metastable phase. The timeneeded by the system to nucleate critical droplets, i.e., thenucleation timetnuc, depends on temperature and may bevery long. It is often not accessible experimentally, so thatthe loss of stability of the metastable phase cannot be ob-served. Once critical nucleation takes place, critical crystaldroplets start to grow until all the system has a crystallinestructure. However, the growth of the crystal may becomevery slow if kinetic constraints are present. What we observein the CTLS is that the lower the temperature, the faster iscrystal nucleation, but the slower is crystal growth.

We can formalize the loss of stability of the liquid byestimating the nucleation timetnuc. In two dimensions, thefree energy difference due to formation of a crystal droplet ofsizej in the liquid phase is given by15

D~j!5Asj2Bd f j2, ~18!

FIG. 6. Energy per spin close to the spinodal as a function of time. Thedotted line marks the energy of the crystal.L5100. Error bars are onlyshown if larger than symbols size.



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whereA andB are geometric factors depending on the shapeof the droplet, s(T) is the surface tension, andd f (T)5FLQ(T)2FCR(T) is the bulk free energy difference be-tween supercooled liquid and crystal belowTm . The func-tion D~j! has a maximum at a value of the droplet sizej!,with an associated valueD! of the excess free energy:

j!~T!5A

2B

s~T!

d f ~T!, D!~T!5

A2

4B

s~T!2

d f ~T!. ~19!

For j,j! the system still needs energy to enlarge the dropletwhich therefore naturally shrinks its size to zero; forj.j!

on the other hand, it becomes favorable for the droplet togrow. Therefore,j! is the critical droplet size andD! is thefree energy barrier to crystal nucleation. The Arrhenius esti-mate for the timetnuc of crystal nucleation is then

tnuc~T!5t0 expS A2

4B

s~T!2

Td f ~T! D . ~20!

A necessary condition for the existence of the super-cooled liquid phase at temperatureT is that the nucleationtime tnuc(T) is much longer than the time to make an equi-librium measurement, i.e., the equilibration timeteq(T). Inother words, only iftnuc(T).teq(T) can we speak of a meta-stable supercooled liquid. We want now to test this relationin the CTLS. To findtnuc we need the surface tensions,which we estimate at a reference temperatureTref and as-sume that depends weakly onT close toTref . To do this weinvert formula~20!,

s254BTrefd f ~Tref!log@tnuc~Tref!/t0#/A2, ~21!

and after settinga5Trefd f (Tref)/Td f (T), we can write

tnuc~T!'t0 expH a logFtnuc~Tref!

t0G J ~22!

5tnuc~Tref!at0

12a ~23!

'tnuc~Tref!a, ~24!

where the last approximation is valid whenT is nearTref ,since thena'1. To have a result as accurate as possibleclose to the loss of stability of the liquid, we takeTref

51.234, wheretnuc553104.Regarding the estimate ofteq(T), we have already no-

ticed that the correlation functionC(t) drops to zero afterroughly 20 relaxation times, and this we take as a reasonableestimate of the equilibration time. The relaxation time isgiven by the power law fit, Eq.~14!.

In Fig. 7 we plottnuc and teq versusT from the aboveestimates. We see that the nucleation time drops below theequilibration time at a temperatureTsp51.22, which there-fore marks the metastability limit of the supercooled liquidphase. BelowTsp the liquid can only be observed on timescales shorter than the equilibration time, meaning that whatwe are actually observing is an out-of-equilibrium glass.Therefore, the equation

tnuc~Tsp!5teq~Tsp! ~25!

can be considered as a definition of the effective spinodaltemperatureTsp for the metastable liquid phase. Note that the

two curves cross at a timetsp;104 MCS: this is the reasonwhy if our maximum experimental time is smaller thantsp

we do not detect this metastability limit. As we have alreadysaid, the glass transition temperatureTg is fixed by the ex-perimental timetexp via the relationteq(Tg)5texp. Therefore,if Tsp,Tg the metastability limit cannot be observed,whereas we detect it ifTg,Tsp.

The presence of a lower metastability limitTsp is clearlyvisible also in cooling experiments. If our experimental timeexceedstsp5104 MCS, we can afford smaller cooling ratesthan r sp;DT/tsp to explore the relevant [email protected]:1.3#. The results are reported in Fig. 8. Forr .r sp

;531025 the energy departs from the equilibrium line re-mainingaboveit, while for r ,r sp it dropsbelow it. We canunderstand qualitatively this behavior in terms of equilibra-tion and nucleation processes. Forr .r sp, given the fastcooling rate, the system has not the time to equilibrate at lowtemperatures where the equilibration time becomes verylarge and its energy therefore remains higher than the equi-librium one. On the contrary, forr ,r sp the cooling is so

FIG. 7. Nucleation and equilibration times vs temperature.

FIG. 8. Cooling experiments above and below the spinodal cooling rater sp.The cooling protocol is linear,T(t)5Tin2rt , from Tin51.67 down toT50. L5100.



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slow that at low enough temperatures the system has the timeto nucleate critical crystal droplets and its energy is driventoward the ground state one, thus becoming smaller than thesupercooled liquid energy. In both cases, however, the sys-tem eventually remains stuck in an off the equilibrium phase.

This experiment poses an interesting question about theactual nature of the out-of-equilibrium glassy phase. For fastcooling rates the system remains stuck at a high energy level,into a clearly disordered, or glassy-like, phase. On the otherhand, for the slowest cooling rates the system approaches theordered crystal phase, even though it remains stuck in anout-of-equilibrium polycrystalline configuration~this can beclearly seen inspecting configuration snapshots visually!.The decrease in energy, and the associated increase in crys-talline order that we observe as we slow down the cooling isa continuous process, making it very difficult to distinguishwhat is a glass and what is a highly disordered polycrystal.

This structural continuityfrom glass to polycrystal canbe appreciated quantitatively in Fig. 9, where we plot theenergy of the asymptotic configurations the system reaches atT50, as a function of the corresponding crystalline domainsizej and crystal massm, for various cooling ratesr. All thepoints on this plot correspond to configurations obtainedwith a well-defined cooling experiment. Moreover, all theseconfigurations are~local! minima of the energy. As we cansee, the data interpolate without gaps between crystalline~orpolycrystalline! configurations with lowE and highj or m,to glassy configurations with highE and lowj or m. In thissense we may say that in the present system there is noqualitative difference between the off-equilibrium disor-dered, i.e., glassy, regime, and the crystal growth regime. Onthe other hand, thanks to the presence of the pseudospinodaltransition, there is a clear distinction betweenequilibriumliquid and crystal phases.

C. Metastability limit and relaxation time divergence:Tsp versus Tc

It has been shown in Ref. 6 that in the PQ model thetemperatureTc where a power law fit of the relaxation time

locates a divergence, is very close to the metastability limitTsp. The interpretation of this fact given in Ref. 6 is veryclear: in mean-field the PQ model has a sharp spinodal tem-perature at the point where the liquid metastable state disap-pears, and this transition is accompanied by a divergence ofthe correlation times. Therefore, it is argued that in the finite-dimensional version of the PQ model the temperatureTc isactually a remnant of the mean-field spinodal. The sameview has been advanced for off-lattice models: in Ref. 16 thebehavior of a supercooled two-component Lennard-Jonessystem ind52 has been interpreted as due to the influenceof a pseudospinodal, while in Ref. 17 arguments are given toexplain why ind53 spinodal effects may be much harder todetect.

The situation is very similar in the CTLS, even thoughour definition of Tsp makes it indispensable to distinguishbetween the two temperatures. The temperatureTc where therelaxation time would diverge is not actually a true criticaltemperature: when the temperature is decreased the equili-bration time increases and finally becomes as large as thenucleation time atTsp, whenteq5tnuc is still finite. There-fore we have by definitionTsp.Tc . This is simply due to thefinite-dimensional nature of the CTLS. In mean-field systemsa metastable state has infinite lifetime and is separated fromthe stable phase by an infinite free energy barrier: the spin-odal temperature, which marks the disappearance of themetastable state, corresponds to the temperature where thebarrier goes to zero, and is naturally associated with a diver-gent time scale. In a finite-dimensional system like theCTLS, on the contrary, the metastable phase~the liquid! isseparated from the stable one~the crystal! by a finite freeenergy barrier—see Eq.~19!—and has always a finite life-time. An effective spinodal temperature can only be definedby comparing the different and finite time-scales present inthe system, as done in the preceding section. However, thephysical interpretation of Ref. 6 may remain valid, i.e., theincrease of the relaxation time close toTsp may be an effectof approaching incoming instability of the supercooled liq-uid.

We stress again that had our maximum experimentaltime been shorter thantsp5104 MCS, we would not havedetected the metastability limit atTsp, while we still wouldfit the equilibrium relaxation time as a power law with criti-cal temperatureTc . In that case, it would have been harderto realize the connection between divergence of the relax-ation time and loss of stability.

D. Kauzmann’s resolution of the Kauzmann paradox

In 1948, Kauzmann outlined in a famous paper10 theparadox now named after him. In the supercooled phase theviscosity of the system increases with decreasing tempera-ture, until at the glass transitionTg relaxation time becomestoo long and equilibration cannot be achieved within experi-mentally accessible times. However, Kauzmann noted that ifthe entropy of a supercooled liquid is extrapolated belowTg ,it becomes equal to the crystal entropy at a temperatureTs

.0, and even smaller than zero if extrapolated further. Thisentropy crisisis never actually observed, because the glasstransition intervenes before. However, Kauzmann found it

FIG. 9. Energy of the configurations reached atT50 during cooling experi-ments as a function of the mass and the domain size.r P@1026,1023#,L5100.



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paradoxical that it was just a kinetic phenomenon~the glasstransition! that saved the liquid from thermodynamic non-sense.

In the context of the Adams–Gibbs–DiMarzio theory,18

the entropy crisis has however an interpretation: the entropydifference between crystal and liquid is related to the con-figurational entropyS, that is the entropic contribution dueto the presence of an exponentially high number of differentglassy minima. The vanishing ofS at Ts signals a thermo-dynamic transition to a new phase, characterized by a subex-ponential number of glassy states, separated by infinite free-energy barriers. This picture is exact for some mean-fieldspin-glass systems,19 and it may be the correct resolution ofthe Kauzmann paradox even for real structural glasses. Ac-cording to Adam–Gibbs–DiMarzio, the entropy crisis is ac-companied by a divergence of the relaxation time atTs givenby t;exp(A/TS). Expanding linearly the configurational en-tropy close toTs a Vogel–Fulchner–Tamman behavior isobtained@see Eq.~15!# with T05Ts . T0 and Ts have beenobserved to be quite close in various systems,1 a fact that hasbeen advocated as an indirect argument in favor of this in-terpretation.

Despite recent analytic and numerical work supportingthe entropy crisis scenario,20,21 there is another way to avoidthe Kauzmann paradox, which, interestingly enough, wasproposed by Kauzmann himself.10 He rejected the idea of athermodynamic glassy phase, and instead hypothesized theexistence of a metastability limit of the supercooled liquidphase, below which crystal nucleation becomes faster thanliquid equilibration. More specifically, he defined an effec-tive spinodal temperatureTsp.Ts below which ‘‘the free en-ergy barrier to crystal nucleation becomes reduced to thesame height as the barrier to simpler motions.’’ Below Tsp

the supercooled liquid is operationally meaningless and thusthe paradox is avoided.

We have shown in this paper that the CTLS is a systemwith a supercooled phase and a fragile-glass phenomenologythat precisely reproduces the scenario hypothesized by Kauz-mann. If we look at the behavior of the extrapolated excessentropySLQ–SCR we find that in this modelTs and T0 areactually very similar~see Secs. III and IV A!. However, de-spite this fact, no entropy crisis does actually take place. Themetastable liquid ceases to exist belowTsp51.22 and thusthe paradox is avoided.

As we have already stressed, the metastability limit mayprove impossible to observe experimentally if the equilibra-tion time atTsp is much larger than the experimental time,that is if Tsp!Tg . One is then tempted to suggest that theKauzmann’s resolution of the Kauzmann paradox that wehave explicitly analyzed in the CTLS model possibly appliesto many other systems, but is actually out of experimentalrecognition. However, we must be careful in judging howgeneral this behavior can be.

The nucleation timetnuc of the CTLS has a minimum ata temperatureT50.98, below which it starts increasingagain. This is simply due to the balance between the freeenergy barrierD!(T) andT in the Arrhenius factor, and it iscertainly a very model-dependent property. In the CTLS allthis is of little relevance, becausetnuc and teq cross at a

much higher temperature, as shown in Fig. 7 and also in Fig.10 ~left!. Yet, in general the interplay betweens(T), d f (T),andT may be such that a minimum oftnuc(T) actually oc-cursand that at this minimumtnuc@teq. However, even inthis case it may still happen that the curves cross below theminimum of tnuc, as shown in Fig. 10~center!. In this casethe Kauzmann paradox would still be resolved by the meta-stability limit, exactly as in CTLS. Note that this particularscenario may actually prove to be quite deceiving: at tem-peratures smaller than the minimum of the nucleation curveone can think that the ‘‘dangerous’’ zone for crystal nucle-ation has been safely crossed and that crystallization is there-fore ruled out at lower temperatures. However, very carefulextrapolations of bothtnuc andteq would be needed to rec-ognize that a metastability limit actually exists atTsp!Tg .

But is it possible thattnuc@teq at all temperatures? Theanswer is yes for strong liquids, whereTs50, and thusteq

stays finite forT.0; but this case is of little interest for ussince then the Kauzmann paradox does not arise. However infragile liquids, whereteq is supposed to diverge atTs , thecondition for the existence of the liquid,tnuc@teq, requiresthat tnuc diverges at a temperatureTnuc>Ts . In our opinionit is possible thatTnuc5Ts , since at low temperatures thecontribution of t0 in Eq. ~20! could be important due tokinetic constraints.22 In nucleation theory, this prefactor istaken as proportional to the viscosity, in which casetnuc

would diverge when the viscosity does, that is atTs . But it ishard to imagine howtnuc could divergestrictly abovethethermodynamic transition temperatureTs . Therefore, itseems to us that the best one can have isTsp5Ts , as in Fig.10~right!.

Let us stress once again the fact that crystal nucleation isa different process from crystal growth: saying that nucle-ation is fast, is not the same as saying that the critical crystaldroplets will grow quickly, since the speed of growth willstrongly depend on kinetic considerations. From an experi-mental point of view, it is clear that having a very longnucleation time~longer than the experimental time! and fastgrowth, is not much different from having a fast nucleationof very small, undetectable, crystal droplets, which howevergrow exceedingly slowly. In order to avoid crystallization

FIG. 10. Pictorial view of three different scenarios: Left,tnuc andteq crossat a spinodal temperatureTsp, which is larger than the temperature wheretnuc has a minimum, as in the CTLS. Center,tnuc andteq cross at a spinodaltemperatureTsp, which is smaller than the temperature wheretnuc has aminimum. Right, tnuc and teq both diverge at the same temperatureTs

5Tsp. The circles indicate the spinodal temperatureTsp.



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and form the glass we have either to avoid nucleation, or toensure that crystal growth is kinetically blocked. To clarifythis point, we can ask how long is the timetx the systemtakes to develop a substantial amount of crystalline order,large enough to prove that crystallization has started. We cananswer this question by following the time evolution of thecrystal massm(t) at various temperatures, and definingtx

from the relationm(tx)>mth , where mth is an arbitrarythreshold. The functiontx(T) is normally called in the con-text of supercooled liquids time–temperature–transformation~TTT! curve, and it is the only experimentally availablemeans to check how likely crystal formation is at a giventemperature. Note that the value oftx(T) will strongly de-pend onmth , and it cannot therefore bequantitativelycom-pared toteq.

In Fig. 11 we plot the TTT curves of the CTLS for twodifferent values ofmth . As in realistic supercooled liquids,the TTT curves have a minimum at a temperatureTx : herecrystallization has the fastest rate, in the sense that atTx wehave to wait the minimum possible time to reach a levelmth

of crystallization. The origin of this minimum is obvious inCTLS: for high temperaturestx is large becausecrystalnucleation is very slow, i.e., becausetnuc is very large,whereas for small temperaturestx is large becausecrystalgrowth is very slow, which is essentially due to the highviscosity of the system~i.e., to the same reason why, if theliquid still existed,teq would be very large!. Thus, in CTLSthe two branches of the TTT curve correspond to two differ-ent phenomena, that is nucleation and growth. Not surpris-ingly we findTx;Tsp. On the other hand, in a system wherethe scenario depicted in Fig. 10~center! holds, also the leftbranch of the TTT curve forT,Tx would be due to slownucleation,23,24 until Tsp,Tx is reached. In this case theminimum of the TTT curve would be an effect of the mini-mum of tnuc(T) described above. BelowTsp, however, theTTT curve would again increase due to slow crystal growth.

It may be argued that there is little experimental differ-ence between the case where crystallization at lowT isblocked because of arrested nucleation or arrested growth.However, from a theoretical point of view these two situa-

tions are quite different: in the first case the equilibrium liq-uid phase exists, at least on certain time scales, whereas inthe second case it does not, the only liquid phase is theout-of-equilibrium glass and the Kauzmann paradox isavoided. We can characterize these two scenarios asequilib-rium arrested nucleationvs off-equilibrium arrested growth.

V. OFF-EQUILIBRIUM BEHAVIOR: DOMAIN GROWTHAND AGING

In this section we study the dynamical behavior of theCTLS after a quench toT,Tsp. We shall first focus on one-time quantities, and in particular on the behavior of energy,crystal domain size, and crystal mass. Then we will analyzetwo-time quantities to understand whether our system exhib-its aging, and of what kind. The key result of this section isthat there are two time regions for off-equilibrium dynamics.In the earlier time regime there is the formation of a largenumber of critical crystal droplets all over the system. As wehave seen, this mixture of crystallites cannot be structurallydistinguished from a glass or liquid configuration, as long asthe crystallites are very small. Their growth is controlled byactivation, and thus this regime will last longer the lower thetemperature. In this phase, which we callbubbling, off-equilibrium dynamics is similar to what was found in struc-tural glasses. The later time regime kicks in when the energybarriers to crystal growth have been crossed, so that crystaldomains are large enough to be structurally recognizable.This phase we callcoarsening, and off-equilibrium dynamicsin this regime is consistent with models where simple coars-ening occurs, as the Ising model. The minimum time wehave to wait to see the coarsening regime in our system~andthus realize that crystal domains are actually growing! is aslong as the time needed to directly see the loss of stability ofthe liquid, that istsp. In other words, the dynamical checksfor the growth of the crystal are as inefficient as the struc-tural ones we considered in the former section.

A. One time quantities

The rate of decay of the energy gives a useful character-ization of the off-equilibrium dynamics. In Fig. 12 we reportthe behavior of the excess energydE5E2EGS as a functionof time, together with power law fits of the formdE5At2n. For all temperaturesT,Tsp we find an initial timeregime wheren,0.2, which is compatible with that found insome model systems of structural glasses~see, e.g., Refs. 25and 26!. This is followed by a second late time regime wherethe energy decreases faster, withn;0.5. However, the datashow that the crossover to this second regime takes place ata time which increases on lowering the temperature. AtT50.67 we are unable to enter this late time region within ourmaximum experimental time, and we only observe the slowregime.

The existence of these two different regimes can be re-lated to the presence ofactivated coarsening, as is com-monly found in frustrated lattice spin models.4,5 Below thespinodal temperature the system rapidly nucleates smallstable crystal domains, which tend to grow. However, due tothe high viscosity, local spin rearrangements are difficult, andmoving a domain boundary may be rather costly. Thus, con-trary to the case of standard coarsening in the Ising model,

FIG. 11. TTT curves for two different values of the threshold valuemth .L5100.



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when frustration is present domain boundaries are pinned,and domain growth requires overcoming some energy barri-ers. The system therefore needs activation to grow crystaldomains, and this may lead to a nonstandard decay of theenergy as a function of time.

More precisely, according to Ref. 27 we must distinguishtwo classes of activated coarsening, depending on whether ornot the maximum barrierD to domain growth depends on thelinear domain sizeLj(t), and therefore ont itself. In the firstclass of systemsD does not depend onj(t). In this case thesystem evolves slowly within a time scaletD;exp(bD),while for logt.log tD barriers are no longer a problem, and astandard power law behavior is recovered, i.e.,E2EGS

;t21/2 ~we are only considering systems with nonconservedorder parameter!. On the contrary, if the barrier to domaingrowth is proportional toj(t), then it is larger the longer thetime. In this second class of activated coarsening one expectsa logarithmic asymptotic dependence of the energy on time.

The data regarding the energy in Fig. 12 show that, afteran early slow regime, the standardt21/2 decay of the energyis recovered. This fact suggests that our model belongs to thefirst class of activated coarsening, and that the early timeregime is due to activated domain growth with a constantenergy barrier.28 The same can be inferred from the behaviorof j(t), which has two regimes: a first in whichj fluctuatesaround its effective zero, and a second where it increaseswith an exponent close to 1/2.

The behavior of the system when we perform a coolingexperiment is consistent with the interpretation above. Theconfigurations reached atT50 for low cooling rates exhibitthe typical pattern of a coarsening system and are stable,indicating that coarsening is indeed activated. Besides, if weplot the excess energydE as a function of the cooling rater,once again we find two regimes~Fig. 13!: for fast coolings~short time! we have a slow energy decay, which can befitted either with a logarithm, or with a power lawdE;r n,with n;0.15. On the other hand, for slow coolings~longtime! we findn;1/2, as in standard coarsening for the Ising

model. This behavior ofdE(r ) is what we expect for acti-vated coarsening of the first class and can be qualitativelyexplained in the following way. As we have seen, belowTsp

the system exhibits a nontrivial dynamical behavior wherethe energy evolution is determined by the formation andgrowth of pinned crystal domains. In a cooling experiment itis then crucial how much time the system spends below thespinodal temperature before remaining trapped in theasymptotic state. If we callTf the temperature where thesystem reaches the asymptoticT50 energy value,29 then at agiven cooling rater the time elapsed fromTsp to Tf is givenby t(r )5(Tf2Tsp)/r . During this time the energy of thesystem evolves approximately as described previously in thissection, i.e., dE5At2n. Therefore we haveE(r )2EGS

;t(r )2n;r n and the two regimes observed in Fig. 13 arenothing else than a manifestation of the two regimes alreadydiscussed for Fig. 12. The exponentn corresponds to the firstregime value ift,tsp, that is if r .r sp, while for r ,r sp thestandard valuen51/2 is recovered.

B. The bubbling and coarsening regimes

The two regimes we have presented above can be explic-itly related to different patterns of domain dynamics. In orderto do this we compare the different time dependence of thedomain sizej and crystal massm.30 Up to now we used bothquantities as equivalent markers of crystal order, but in factin the growth phase their role is quite different.

After a quench toT!Tsp we expect that fast nucleationwill lead to an increase inm due to the formation of manycrystallites, but that the typical domain sizej will remainvery small, actually smaller than our resolution. This pictureis confirmed by Fig. 14, where we plotj andm as a functionof time for two different temperatures belowTsp. In bothcases we see that the initial increase ofm is faster than thatof j. For T51.0, which is quite close toTsp51.22, we cansee a rise ofj only for times longer than 105 MCS, while forT50.67 there is practically no increase ofj up to t5106

MCS, while a clearly detectable increase ofm can be seen.To further support this picture we plot in Fig. 15 the

massm as a function of domain sizej for many different

FIG. 12. Excess energy as a function of time after a quench atT,Tsp. Fulllines, power law fit of the early~slow! time region:n50.17 for T50.67;n50.19 for T50.83; n50.18 for T51.00. Dashed lines, power law fit ofthe late ~fast! time region: n50.40 for T50.83; n50.51 for T51.00.L5100.

FIG. 13. Energy of the configurations reached atT50 during cooling ex-periments~see Fig. 8! as a function of the cooling rate, together with powerlaw and logarithmic fits.L5100.



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temperatures belowTsp, and for different times. All thesedata collapse on a master curve, which shows that for shorttimes ~small j and m! the crystal mass grows much fasterthan the domain size, whereas for long timesm tends tosaturate, andj start increasing rapidly.

We can therefore define two time regimes for crystaldomain growth. For shorter times, in what we may call abubbling regime, many small crystallites form at the ex-penses of a liquid background, their linear size growing verylittle with time, whereas the total crystal mass grows quitesharply.31 In this phase domains are so small that they are noteven well defined, in the sense that their linear sizeLj is ofthe same order as the widtha of the interfaces~or domainwalls! among them. The excess energy is dominated by theregions not yet crystallized, and the interfacial and domaincontributions are of the same order. For longer times, thelargest part of the system is crystallized, so the massm satu-rates, and the small domains start growing one at the expenseof each other.

Therefore a propercoarsening regime starts with aclearly increasingj, andLj@a. Now domains are well de-fined and the excess energy is concentrated in the interfaces.These are now much thinner than the domain size and can beconsidered as lower dimensional manifolds. This must give,

E~ t !2EGS;j~ t !d21

j~ t !d 51

j~ t !, ~26!

as in standard coarsening.27,32 In Fig. 15 we plot the excessenergyE(t)2EGS versusj for a number of temperatures andtimes. Consistently with our conclusions, for shorter timesthere is a drop in energy with basically no increase inj. Onthe other hand, forj.0.2 the 1/j law fits reasonably well thedata, suggesting that this is the late-time coarsening regime.

C. Two time quantities

We now focus on the behavior of two-time quantities,for quenches belowTm . In equilibrium dynamics, correla-tion functionsC(t,tw)5^A(tw)A(t)& and their associated re-sponse functionsR(t,tw)5d^A(t)&/dh(tw) ~A is a genericvariable andh is the field conjugate to it!, depend only on thedifferencet2tw . This is not the case when the system is outof equilibrium. The aging regime is that in which one-timequantities ~such as energy or density! are stationary orchange very little, but two-time quantities depend explicitlyon both timestw , t.33

An important quantity which allows to distinguishamong different kinds of aging behavior33 is the fluctuation–dissipation~FD! ratio,

X5TR~ t,tw!

]C~ t,tw!/]tw. ~27!

In equilibrium, the fluctuation–dissipation theorem~FDT!states thatX51, while out of equilibrium the FDT is violatedand X depends ont, tw in a nontrivial way. It has beenconjectured34 that during agingX depends on time onlythrough the correlation function, conjecture that has provedvalid in all systems studied to date. If one considers thesusceptibility, or integrated response,

x~ t,tw!5Etw

t

R~ t,s!ds, ~28!

which is much easier to compute numerically, then assumingX5X@C(t,tw)# one finds from~27!,

x~ t,tw!5bEC~ t,tw!

C~ t,t !X~C!dC. ~29!

Therefore one can plotx vs C at fixedtw , and extractX fromthe slope of the curve. In equilibriumX51 and the curve isa straight linex(t2tw)5b@C(0)2C(t2tw)#, with slope21/kBT. For a system out of equilibrium one finds in gen-eral two regimes: a short time quasiequilibrium regime, cor-responding to equilibration of fast degrees of freedom, wherethe curve follows the equilibrium straight line and FDTholds; and the aging regime at larger times wherex(t,tw)departs from the equilibrium line. In particular, for structuralglass models studied so far via numerical simulations25,35 it

FIG. 14. Mass and domain size as a function of time for two differenttemperatures.L5100.

FIG. 15. Crystal mass as a function of the domain size parametrically int,after a quench at various temperatures. Inset, excess energy as a function ofthe domain size. Full line, 1/j fit. All runs are 23106 MCS long,L5100.



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has been found that after leaving the equilibrium line, thecurve follows another straight line, with smaller~in absolutevalue! slope.

To study the FD relations in CTLS we follow~as hasbeen done before in other spin systems!36 the evolution ofthe staggered magnetization (h i561 are quenched, inde-pendent random variables!,

ms~ t !51

N (i

h isi~ t !, ~30!

when adding a term2hNms to the Hamiltonian. If the fieldh is applied at time tw , the h-averaged susceptibilityx(t,tw)5ms(t)/h is related to the spin–spin correlationfunction:37

Cs~ t,tw!5^Nms~ t !ms~ tw!&51

N (i j

h ih j^si~ tw!sj~ t !&

51

N (i

^si~ tw!si~ t !&. ~31!

We have used fieldsh50.1kBT and h50.15kBT, andchecked that they are within the linear regime. The resultspresented here correspond to the highest field.

We look for aging behavior in the regionsTsp,T,Tm

and T,Tsp. We first quench the system toT51.25 ~abovethe spinodal!. A plot of x vs Cs for different waiting times~not shown! gives a straight line with slope21/kBT, inde-pendent oftw , i.e., the FDT is obeyed. No aging is observed,and the behavior is compatible with a system in equilibrium:this is themetastable liquid. On the other hand, a quench toT50.67!Tsp shows aging very clearly~Fig. 16!. The FDratio ~Fig. 17! also shows very clearly that the system is outof equilibrium. In fact, we know that the liquid is unstable atthis temperature, and that the crystal phase is growing. In asystem undergoing coarsening, like the Ising modelquenched belowTc , the susceptibility rapidly saturates at thevalue corresponding to the low-temperature phase, becausethe contribution of the interfaces is negligible. The correla-tion function, however, continues to drop because of themovement of the domain walls. Thus the FD curve is flat for

small values of the correlation function.36,38,39We would ex-pect this same behavior whenever the dynamics is given bygrowing domains with bulk equilibrium properties, separatedby sharp interfaces. However, the FD curve does not becomeflat in the CTLS atT50.67, even for our longer waitingtimes. Instead, the system shows FDT violations similar tothose commonly exhibited by structural glass models25,35andcertain mean field spin-glasses~solved by a one step replicabroken solution, one-step RSB!.34 There are two regimes: aquasiequilibrium one with slope2b and an off-equilibriumone with slope2beff52Xb. Note that the FD curves hardlydepend ontw , at least for the times the simulations canreach.

Taken at face value, this result would lead to interpretthis as an out-of-equilibrium glassy phase~i.e., a liquid withweakly broken ergodicity!. However, in the light of the dis-cussion of the preceding section we known that forT,Tsp

the crystal growth dynamics proceeds through two qualita-tively different time regimes. Due to the extremely slow ac-tivated dynamics the bubbling regime can last for long, andindeed at very low temperatures it is the only one accessiblein our simulation times. In this case, we do not expect to seein the response– correlation relation the FDT violations typi-cal of coarsening systems, since the coarsening regime hasnot yet been attained. This is exactly what happens forT50.67: we can check that the crystal mass for the longesttime explored in Fig. 17 ism;0.3 @Fig. 14~right!#, well be-low the valuem;0.5 that characterizes the crossover to thecoarsening regime. The 1-RSB-like FDT violation observedat T50.67 thus occurs in the bubbling regime, while if wecould wait longer times we would observe a crossover totypical coarsening behavior.

Since the duration of the bubbling regime is shorter forhigher temperatures, we expect to explicitly observe an FDTcrossover to the coarsening regime at higher temperatures.From the former sections we know that atT51.00 the cross-over from the bubbling to the coarsening regime takes placeat about 104– 105 MCS. In Fig. 18 we plot the FD curves forT51.00, and the data confirm our expectations. For short

FIG. 16. Correlation function as a function oft2tw , at different values ofthe waiting timetw . T50.67,L5500.

FIG. 17. Parametric plot ofx(t,tw) vs Cs(t,tw), at various values of thewaiting time tw for T50.67, well below the spinodal temperature.L5500. The full line represents the equilibrium relationx5b(12Cs).



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waiting times, the curves initially follow what seems to bethe path of a 1-RSB-like violation. This is the bubblingphase, pretending to be a glass. The curves depend stronglyon waiting time, and for highertw evolve toward the normaldomain-growth curve~two straight lines, with slope2b atshort times and 0 at long times!, which is nearly reached fortw5106 MCS.

VI. CONCLUSIONS

We have presented a lattice model, the deterministicCTLS, which exhibits a phenomenology typical of glassforming systems: it has a melting transition and a low tem-perature crystalline ground state, but it exists as a super-cooled liquid for a wide range of temperatures; in the meta-stable supercooled phase the relaxation time increases in away resemblant of typical fragile glasses and upon fast cool-ing crystallization can be easily avoided to bring the systemin disordered glassy-like states.

The feature that makes this model particularly interest-ing is that it displays a metastability limit which can be ex-plicitly observed within the time scales available for numeri-cal simulations. What we have shown is that the typicalphenomenology of fragile glass forming systems is entirelycompatible with the existence of a metastability limit of thesupercooled phase, even if it is plausible that, contrarily tothe CTLS, for many models this limit cannot be observed. Acrucial point in this regard concerns the length of experimen-tal times under which measurements are performed. By defi-nition the supercooled liquid loses stability atTsp, whencrystal nucleation becomes faster than liquid equilibration. Ifthe equilibration time atTsp is much larger than the experi-mental time, however, the metastability limit cannot be ob-served. Since the glass temperatureTg is the lowest tempera-ture at which the system is equilibrated within theexperimental time, this situation takes place whenTsp,Tg .

If the system has a metastability limit, below the pseu-dospinodalTsp the dynamical behavior basically consists ofvery slow crystal growth. This suggests some criteria to de-tect whether a system has such a metastability limit or not,

even whenTsp is experimentally inaccessible. For example,one woulda priori think that there is a qualitative differencebetween a disordered glassy configuration, obtained byquenching a liquid, and a crystalline configuration, howeverrich in defects this is, and however slow crystal growth maybe. Also, since off-equilibrium dynamics of simple domaingrowth has very peculiar characteristics one would hope todetect crystal growth by analyzing the dynamical behavior ofthe system, and in particular the pattern of FDT violations.What we have shown in the CTLS is that actually neither ofthese criteria is sharp enough, and if our experimental timewere not long enough toexplicitly observe the loss of stabil-ity of the liquid at Tsp, it would be impossible to makestatements about it, either by structural or dynamical means.On the other hand, on shorter time scales the system exhibitsan off-equilibrium behavior which is typical of structuralfragile glasses. Indeed, we have seen that there is a long timeregime where FDT violations are 1-RSB like, as in mostfragile glass models studied so far, while the proper coarsen-ing regime is attained only at much larger times. Also, thereis structural continuity between the low-temperature configu-rations reached via fast coolings and those approached withslower cooling rates, which means that the degree of order inthe system increases gradually: what appears as a disorderedglassy like configuration is just the beginning of a continuousprocess that eventually, on much longer time scales, leads tothe crystal. The reason for this behavior is that belowTsp

crystal nucleation is fast, but crystal growth becomes veryslow, with many crystal droplets trying to expand in a liquidbackground. In such a situation distinguishing between atruly disordered glass and a mixture of tiny mismatched crys-tallites becomes very hard, and FDT violations are nontrivial.

An important consequence of the metastability limit sce-nario in the CTLS is that the Kauzmann paradox is avoided:below Tsp the liquid does not exist in any reasonable sense,thus extrapolation of the excess entropy is meaningless. Thiswas the resolution of the paradox proposed by Kauzmannhimself in his paper. While we cannot claim that this is thecase for real glasses~and we have discussed qualitative al-ternative scenarios in terms of crystal nucleation andgrowth!, this system shows that Kauzmann’s way out of theKauzmann paradox should be considered seriously. In thestudy of glassy systems the crystal phase is often neglected,assuming that if crystallization is avoided atTm , then thecrystal does not play any role in the low temperature physics.What this model shows is that the stability of the super-cooled liquid should not be taken for granted.

Finally, even if stability were to be proved not to be aconcern in real systems, this remains a cautionary note whenformulating nondisordered glass models, especially on thelattice, which is so attractive. One can also wonder if theseconsiderations might be applicable to disordered systems, orsystems with complex~highly degenerate! low-temperaturephases, were the supercooled liquid could be unstable withrespect to a crystal in disguise.

ACKNOWLEDGMENTS

We thank V. Martin-Mayor and F. Ricci-Tersenghi formany important suggestions, and J. P. Garrahan for kind re-

FIG. 18. Parametric plot ofx(t,tw) vs Cs(t,tw), at various values of thewaiting time tw for T51.00, below the spinodal temperature.L5500. Thefull line represents the equilibrium relationx5b(12Cs).



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marks on the paper. We also acknowledge interesting discus-sions with P. Debenedetti, H. Horner, G. Parisi, F. Sciortino,D. Sherrington, and P. Verrocchio. A.C. thanks in particularM. A. Moore.

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Glass and polycrystal states in a lattice spin model

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