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Correlation Time Scales in the Sherrington-Kirkpatrick Model Alain Billoire CEA/Saclay, Service de Physique Théorique, 91191 Gif-sur-Yvette, France. 1 Enzo Marinari 0 Dipartimento di Fisica, INFN and INFM, Università di Roma La Sapienza, 0 P. A. Moro 2, 00185 Roma, Italy. 2 (February 1, 2008) n We investigate the dynamical behavior of the Sherrington-Kirkpatrick mean field model of spin a glasses by numerical simulation. All the time scales τ we have measured behave like ln(τ )∝Nǫ, J x x where N is the numberof spins and ǫ≃ 1. This is true whether the autocorrelation function used 2 3 to defineτ is sensitive to thefull reversal of thesystem or not. 1 x ] h c e Today many features of the Sherrington-Kirkpatrick subset with T = 0.4, 0.5 , ...) with a simple Metropolis m mean field model of spin glasses [1–3] have been clari- dynamics, and perform 4 ·106 Metropolis sweeps. We - fied. Probably most questions that still need investiga- have in all cases two replica and 512 realizations of the t a tions are related to the very interesting dynamics of the disorder. t s model (see for example [4]). Here, following Mackenzie For each of these samples we compute the flip times at. and Young [5], we examine the equilibrium dynamics of τ1, τ2 and τ3. We define τ1(J) as the time after which, on the model. Inthis classicpaper the authorsgavenumer- a given sample J, the time dependent self-overlap m ical evidences, from systems with up to 192 spins, for d- the existence ofa spectrumofrelaxationtimes1whichdi- q(0,t)≡ N1 Xσi(0)σi(t) (1) n vergewiththenumberofspinsN asln(τ)∝N4,andofa i o second,longer “ergodic”time scale τ whichis the time eg has become smaller than +Σ, with [c neededtoturnoverallthespins,with<ln(τeg)>∝N12. For doing that one looks both at processes that require Σ≡phq2iJ , (2) 1 a full reversal of all the spins and at processes that on v the contrary are not sensitive to this phenomenon. In wherehq2iJ,theusualsquareParisioverlap,iscomputed 7 this letter, we establish that indeed all dynamical scales during the second half of the thermalization run for the 7 1 have the same behavior, compatible with barrier heights givensample. The time t is measuredin units ofsweeps, 1 growing like Nǫ, where ǫ≃0.3 close to the N31 behavior with t=0 at the beginning of the Metropolis dynamics. 0 suggested in [6,8] (see also the numerical simulations in Wedefine analogouslyτ2 asthetime ittakestoq(0,t)to 1 [7]). decay from its initial value of 1 down to 0, and τ3 as the 0 Let us start by giving some details about our simu- time it takes to q(0,t) to decay down to −Σ. / at lation. We study systems with N = 64, 128, 256, 512, Weexpect1 τ1,τ2 andτ3 toobeythesamescalinglaw. m and 1024 spins, with ±1 couplings. We first thermalize In the following we will try to check if an exponential the systemusingthe parallel temperingoptimizedMonte scaling of the kind - d Carloprocedure[9]withasetof38T valuesintherange τ ≃A exp(α Nǫ) (3) n 0.4−1.325 (i.e. ∆T = 0.025). We perform 400000 it- 1,2,3 1,2,3 1,2,3 o erations (one iteration consists of one Metropolis sweep givesa goodfittothe data,andwewilltry todetermine c : plus one tempering update cycle), and store the final ǫ. v well equilibrated configurations. Next we start updat- We base our analysis on empirical medians for ln(τ), i X ing these equilibrium configurations (more precisely the i.e. wesortthe512valuesofln(τ)asln(τ(0))≤ln(τ(1))≤ r a 1Notice that while τ2 and τ3 are unambiguous signatures of thetransition to thereversed part of thephase space, τ1 can be ambiguous, since depending on T it can still characterize a transition in the short time regime or already an ergodic transition. The fact that the three τ turn out to be com- i patiblegivesfurthersupporttotheexistenceof asingle time scale exponent ǫ. 1 ... ≤ ln(τ(511)) (more precisely the 512 values of ln(τ) estimateatallǫ ). ForT from0.6upto0.8thebestfitis τ3 averaged over the two replica) and define the median as very stable with an exponent close to 0.25−0.30. When ln(τ(255)). For large N and small T, the probability dis- goingtooclosetothecriticalpointthebehaviorbecomes tribution of τ has a very long tail (for large values of τ), less clean. ǫ (where we wait for q becoming zero) can τ2 and in many cases we are not able to compute average be determined down to T = 0.5 ( τ is smaller than τ ). 2 3 values, since for some samples τ is larger than the num- Here fluctuations are slightly larger than in the former ber of sweeps performed. On the contrary, the median case, but again up to T =0.8 the exponent fluctuates in approach works, and allows a fair estimate. In all cases the range 0.2−0.3. In the ǫ case (where we only wait τ1 where we are also able to estimate the average value of for q decreasing from 1 down to +Σ) we succeed to get ln(τ), we find thatit is verysimilar to the medianvalue. a goodestimate down to T =0.4. Again here, for exam- Thanks to this approach we have been able to estimate ple, we estimate ǫ (T =0.4)≃0.25, and we get a quite τ1 τ on all our lattice sizes down to T = 0.4, τ down to stable fit in T. We remark that when T approaches T 1 2 c T = 0.5 and τ down to T = 0.6. Statistical errors have the estimates of ǫ have largeerrors: α becomes very 3 τ1,2,3 been computed using the usual bootstrap procedure. small (one expects α → 0 for T → T ) and the leading c The second decay time of interest is the time scale Nǫ behavior cannot be distinguished, with the present that governs the decay of, for example, the square range of system sizes, from sub-leading corrections. It is (time-dependent) overlap. We monitor the decay of alsoimportantto notice that our datafully confirmthat <q(0,t)> and of different ways to estimate the correlation times (the 1, J 2 and 3 τ’s) lead to the same scaling behavior, with a qc2(t)≡<q2(0,t)>J −hq2iJ , (4) scaling exponent close to ≃0.3. and we call τ and τ the time scales that characterize q q2 theshorttimedecayoftheseobjects(seelaterfordetails 0.6 L=64 about the exact definition). L=128 0.5 L=256 Let us start by the results for τ , τ and τ . In figure 1 2 3 L=512 (1) we plot one of our most successful fits of τ : here we L=1024 3 0.4 are at T =0.6, the fit is very good and we estimate 0.6 = 14 (2)q(t), Tc 00..23 13 12 0.1 11 0 u) 10 1 10 100 1000 10000 100000 1e+06 1e+07 a og(t 9 time l FIG.2. q2(t) versusln(t) at T =0.6. 8 c 7 Let us notice here (and this is the focal point of this 6 note,thatwewilldiscussbetterinthefollowing)thatthe T=0.6 5 result of equation (5) does not manifest, as opposite 0 100 200 300 400 500 600 700 800 900 10001100 1 to the findings of [5], a scale of the order of exp(cN2). N FIG.1. Pointswith errorsareforln(τ3)versusN, andthe Thescaleweobserveisgovernedbyanexponentcloseto continuousline for our best fit tothe form (3). 0.3. We discussnowthe measurementsofcorrelationtimes ǫ (T =0.6)=0.25±0.04 , (5) τ3 that do not involve the reversal of all the spins. As an example we plot in figure (2) q2(t) versus ln(t) at that can be compared to c T =0.6, and in figure (3) the same quantity at T =0.4. ǫ (T =0.6)=0.20±0.16 ,ǫ (T =0.6)=0.19±0.07 . The two figures exhibit two regimes separated by some τ1 τ2 crossovervaluet : asmalltimeregime,whereq2(t)de- (6) max c caysslowlywithln(t),andalargetregimewhereq2(t)is c Ourestimatesforǫ ,ǫ andǫ turnouttobeverysim- very small. This is very suggestive of the existence of a τ1 τ2 τ3 wholespectrumofrelaxationtimes,uptosomemaximal ilar. Thegeneralpatternthatemergesfromthesefitisof value ≈t . a very good consistency. Let us go in some more details. max Ftiivtesatondǫτe3q(uhaelreto,a−sΣw)easraeida,vwaielawbaleitofnolryqdboewcnomtoinTgn=eg0a.6- theWteimheavneededefiednefdorthqc2e(cto)rtroeldaetciorneatsiemferoτmq2 tbhyecvoamlupeu0ti.n2g5 (at lower T values τ is too large and we are not able to to a threshold value T that we vary (reference [5] was 3 2 looking directly to the moment in which q2(t) is close We have also measured q(t), that we plot in figures 5 c enough to zero)2. In the case of q2(t) we have used the and 6 for T = 0.6 and T = 0.4 respectively. At large c two threshold values T =0.125 and T =0.050. times q(t) goes to zero, on the contrary we expect the 1 2 The exponents we estimate by best fits to the form initial decay to be governed from the same process that (3) are again quite stable (even if in this case we have determines the decay of q2(t). It is also interesting to c not been able to produce reliable error estimates) and, note that we are observing the expected plateau at the let us note right ahead, if any they are larger than the Edwards-Anderson value of the self-overlap, q : with EA one estimated for the full reversal times τ : we can good approximation one estimates [2] q (T = 0.6) ≃ 1,2,3 EA be quite precise on the claim that the scenario where a 0.50 and q (T = 0.4) ≃ 0.74. These two values co- EA slower time scale governs the full spin reversal while a incide very well with the locations where on our larger faster time scale governs the valley to valley migration lattice we see a plateau: this is very clear at T = 0.4 in doesnotapply. Asanexampleweplotinfigure4theτ figure 6 and a bit less clean but also evident at T = 0.6 q2 time as a function of N, and our best fit to the form (3) in figure 5. The finite, large system, spends a long time atT =0.4andfora thresholdT =0.050: the estimated at q before having q(t) → 0 because of the ergodic 2 EA exponent is here 0.38±0.05. The exponent values are transition. very stable when changing the value of the lower threshold, that is a very good sign. In the T range 0.5−0.8 13 the estimated value of ǫ are in the range 0.28−0.38, i.e. 12 completelycompatiblewiththevalue 1 thatisreasonable 3 11 from a theoretical point of view (see for example [6,8]). The quality ofthe best fit degeneratesagainwhen T be- 10 comes too close to T . It is maybe worth to stress here u) 9 c a that the determination of the exponent ǫ is a very diffi- og(t 8 l cultproblem,exponentiallymoredifficultthanthe usual 7 determinationof criticalexponents, since here insteadof 6 a power behavior we are trying to fit an exponential to a power behavior: if τ is ranging over 5 order of magni- 5 T=0.400 tudes (that would be more than acceptable for a power 4 0 100 200 300 400 500 600 700 800 900 10001100 fit) its logarithmis ranging over half a decade only, that N givesapoorbasisforourfittotheexponentialofapower FIG. 4. Data points are for ln(τ ) (without error-bars) q2 law. versusN,andthecontinuouslineforourbestfittotheform (3). 0.4 L=64 We have checked that by fitting with the same proce- 0.35 L=128 L=256 dure used for q2(t), using this time the q interval going c L=512 0.3 from1downto0.63(weuseahigherlowthresholdtostay L=1024 4 0.25 farfromtheactualdecaytozero). Thingsworkwell,and 0. = we fit a scaling exponent for the correlationtimes statis- (2)q(t), Tc 00.1.25 stihcoawllyincofimgupraet7ibtlehewainthalothgeouosnoefofibgtuariene4d,wfohrerqec2(tth)e.bWeset fit gives ǫ = 0.34±0.02 (again, very well compatible 0.1 τq with the value 1). Consistent results (slightly lower, of 0.05 3 the order of 0.25) are obtained at higher T values. 0 Itisclearfromthefigureswehaveshownthatq(t)and 1 10 100 1000 10000 100000 1e+06 1e+07 q2(t) decay very slowly to zero, on a logarithmic scale. time c We try to be more quantitative in figure 8, where we FIG.3. q2(t) versusln(t) at T =0.4. c show that the q2(t) data are very linear when plotted, c 2Itisimportanttonotethattheergodiccorrelationtimesτ i andtheseτ ,τ aredefinedinverydifferentways,andnone q2 q of them as a simple, bona fide coefficient of an exponential decay e−t/τ. The fact that we find that they satisfy reason- able scaling laws shows that the definitions we use are well founded. 3 for example, as a function of ln(t)β, with β = 0.25: we FIG.7. Datapointsareforln(τq)(withouterror-bars)ver- susN,andthecontinuouslineforourbestfittotheform(3). do not consider that as a fair determination of β, since thereisalargerangeofvalueofβ thatmaketheplotlin- ear. What we can claimis that β is surely a smallvalue, In figure 9 we show, for our largest system, ln(τ ) as ofthe orderofmagnitude of0.25. At higherT values we q2 function of T. The data are very well explained by the have the same kind of behavior. fact that we expect an Arrhenius like behavior, exp(A), T withA≃(T −T)[6]: a coefficientproportionalto Tc−T 1 c T fits indeed the data very well. We can sketch a few conclusions. In the Sherrington- 0.8 Kirkpatrick mean field model of spin glasses one sin- gle time scaling dictates the behavior of the correlations 0.6 0.6 times related to the complete reversalof all spins and to = T the transitions through the different states that consti- q(t), 0.4 tutethephasespace: thespeculationsuggestingthatone L=64 could get two different scaling laws is not founded. It is 0.2 L=128 noteasytogetprecisevaluesfortheexponentthatchar- L=256 L=512 acterizesthisexponentialscaling,butallourfindingsare 0 L=1024 compatible with a ǫ = 31 scaling: this is consistent with 1 10 100 1000 10000 100000 1e+06 1e+07 1 barriers scaling like N3 [6,8]. We have also been able to time show that the connected squared overlap decays to zero FIG.5. q(t) versusln(t) at T =0.6. with a power of the logarithm of the order of 0.25 (and clearly not like a power law). 1 0.35 L=64 L=128 0.3 0.8 L=256 L=512 0.25 L=1024 4 0.4 0.6 =0. 0.2 = T q(t), T 0.4 (2)q(t), c 0.15 L=64 0.1 L=128 0.2 L=256 0.05 L=512 0 L=1024 0 1 10 100 1000 10000 100000 1e+06 1e+07 0.8 1 1.2 1.4 1.6 1.8 2 time log(t)**0.25 FIG.6. q(t) versusln(t) at T =0.4. FIG.8. q2(t) versusln(t)0.25 at T =0.4. c 16 14 15 N=1024 14 12 13 12 10 u) 11 g(ta 10 au) 8 lo 9 og(t 6 l 8 7 4 6 T=0.400 2 5 0 100 200 300 400 500 600 700 800 900 10001100 0 N 0.4 0.5 0.6 0.7 0.8 0.9 1 T 4 FIG.9. ln(τ )versusT forN =1024. Thebehaviorclose q2 [2] M.Mézard,G.ParisiandM.A.Virasoro,SpinGlassThe- toT =1isverylinear,whileatsmallerT valuestheincrease c ory and Beyond (World Scientific, Singapore 1987). of ln(τ ) becomes sharper. q2 [3] K. H. Fisher and J. A. 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