Analysis of landscape hierarchy during coarsening and aging in Ising spin glasses
AAnalysis of landscape hierarchy during coarsening and aging in Ising spin glasses
Stefan Boettcher and Mahajabin Rahman
Department of Physics, Emory University, Atlanta, GA 30322, USA
We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phe-nomenology known as "aging", as a diagnostic tool to find universal features that distinguishbetween the energy landscapes of Ising spin models and the ferromagnet. According to RD, anon-equilibrium system after a quench relies on fluctuations that randomly generate a sequence ofirreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higherbarriers of meta-stable states within a complex, hierarchical energy landscape. Once these recordevents allow the system to overcome such barriers, the system relaxes by tumbling into the followingmeta-stable state that is marginally more stable. Within this framework, a clear distinction can bedrawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass,which are often put in the same category. To that end, we interpolate between the spin glass andferromagnet by varying the admixture p of ferromagnetic over anti-ferromagnetic bonds from theglassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation ofrecord events grows logarithmic with time in the glassy regime, with a sharp transition at a specificadmixture into the ferromagnetic regime where such activations saturate quickly. We show this ef-fect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick(mean-field) spin glass. While this transition coincides with a previously observed zero-temperatureequilibrium transition in the former, that transition has not yet been described for the latter. I. INTRODUCTION
The morphology of complex energy landscapes [1, 2],and the parameters that control it, are of continuing in-terest in a large variety of scientific endeavors, from pro-tein folding and evolutionary landscapes in biology [3–7],the design of amorphous materials [8–18], to the hard-ness of combinatorial optimization problems [5, 19–23].The challenges encountered in describing the geometry ofthe extremely high-dimensional space of attainable con-figurations are enormous [1, 2, 12, 23–28]. The structureof such energy landscapes hugely impacts the dynamicsof statistical systems evolving through them. While re-laxation in simple, smooth landscapes is rapid, like theexponential cooling of a cup of coffee [29], relaxationin complex energy landscapes can possess a myriad ofmetastable states to temporarily or permanently trap anydynamic process. In turn, simple relaxation processescan serve as diagnostic tools to explore features of land-scapes [19, 24, 25, 27, 30–32]. It is particularly enticingwhen it is possible to discover universal aspects of suchlandscapes that allow to categorize those features and,ultimately, predict and control dynamic behavior.Variations in temperature can be used to take a fullmeasure of landscapes. At high temperature correspond-ingly higher echelons in energy get explored, while an-nealing or quenching is used to trace out a descentthrough the landscape towards configurations of lower en-ergy. A conceptually simple protocol consists of prepar-ing a system at a high temperature, where it equilibrateseasily, and then instantaneously quenching it down to afixed, low temperature, to explore how it relaxes towardsequilibrium thereafter. Such an “aging” protocol [33],when applied to systems in a complex energy landscape,elicits quite subtle relaxation behaviors which, unlike forthe coffee mentioned above, keeps the system far from a new equilibrium for very long times. Anomalously slowrelaxation and full aging in a complex landscape ensueswhen downward paths are obstructed by barriers, ener-getic or entropic, that trap the system in neighborhoodswith many local minima.The aging phenomenology is associated with memoryeffects by which the current activity is imprinted by adependence on the waiting time t w since the quench. Fora wide class of systems, generally considered to be glassy,it is found that correlations, instead of being time trans-lational invariant, G ( t, t w ) ∼ f ( t − t w ) , roughly dependon a ratio, G ∼ f ( t/t w ) . Although memory effects in out-of-equilibrium systems are generally of interest, that factalone is not sufficient to categorize its energy landscapeas complex or glassy. To emphasize this fact, and to pro-vide a deeper insight into the relation between landscapemorphology and aging dynamics, we investigate here theaging in families of models that interpolate between awell-known spin glass [34, 35] and the corresponding fer-romagnet. Albeit glass and ferromagnet exhibit similarscaling with age t , it stands to reason that the agingdynamics of a homogeneous ferromagnet differs signifi-cantly from that of a glass. In contrast to the hierar-chical, multimodal energy landscape of a glass [36–40],that of a ferromagnet is smooth. Yet, in much of theliterature [41–43], the prevailing mode of relaxation viacoarsening in a ferromagnet is taken as a model for glassyaging. Technically, one could argue that the fact that ineither extreme a growing length-scale emerges is indica-tive of coarsening domains. We posit that the process bywhich those length-scales grow with age, logarithmicallyin the glassy case and with a power-law for a ferromagnet,is fundamentally different.As discussed in Ref. [44], energy barriers that scalewith the size of a domain to be flipped imply that fur-ther growth in those domains is curtailed to be merely a r X i v : . [ c ond - m a t . d i s - nn ] S e p logarithmic in time. Such a feedback does not emergein the coarsening of a ferromagnetic Ising system, whereenergy barriers remain insensitive to the size of the do-main to be flipped. Accordingly, the landscape of a glassysystem has a hierarchical structure in that, the lower anenergy it has reached, the higher the barriers get, andthus, the harder it becomes to escape local minima [45].In a homogeneously coarsening system, energy barriersremain largely independent of the depth reached withinthe landscape, providing some roughness and metastabil-ity but of bounded scale beyond which the structure isrelatively smooth. Within the aging process, this differ-ence manifests itself dramatically in the manner that therelaxing system responds to fluctuations, as illustrated inFig. 2. In a ferromagnet, average fluctuations in energy,beyond some low, fixed threshold, suffice to cross typicalbarriers, often followed by disproportionately large ex-pulsions of heat. In contrast, to advance glassy systemswith diverging energy barriers, mere average fluctuationsbecome ineffective. To be able to relax, those fluctua-tions have to produce ever new records to overcome eversteeper barriers. Such record production, decorrelatedby a wide separation in time, is known to unfold only ona logarithmic scale [46, 47].Although irrelevant for the cases studied in Ref. [44](and here), it should be noted that entropic effects canbecome dominant and may entail diverging free-energybarriers with domain size, even in an otherwise homoge-nous system. One example is a 3-spin Ising ferromag-net [48]. Systems driven by entropic barriers, such as thefree volume in a hard-core colloidal system, are referredto as “structural” glasses. In those systems, a hierarchicalfree-energy landscape emerges dynamically [17].In the following, we define a simple coarse-graining pro-cedure, counting the number of “valleys” traversed in theenergy landscape, that effectively probes the impact offluctuations on the aging dynamics. It reveals the natureof the irreversible, intermittent events that allow the ex-pulsion of excess energy from the system. It shows a dy-namical transition between a glassy and a ferromagneticrelaxation regime based on this measure that reproducessimilar findings using two-time correlation functions [49],indicating that this dynamical transition is closely relatedwith a zero-temperature equilibrium transition betweena glass and a ferromagnet [50]. This transition highlightsthe fact that aging in a glassy system is a distinct processthan is found in homogeneous systems, characteristic ofa distinct, hierarchical landscape.Our paper is organized as follows: In the next Sec-tion II, we introduce the families of Ising spin models weemploy in our study. In Sec. III, we will discuss recorddynamics and the measures we will apply to detect recordfluctuations. In Sec. IV, we present the results of our in-vestigation, and we conclude in Sec. V. II. MODELS
Ising spin systems, consisting of spin variables σ i = ± , have been widely used, first of all as ferromagnets,to model spontaneous symmetry breaking and contin-uous phase transitions [51]. With the random admix-ture of anti-ferromagnetic bonds, they have also servedas models for disordered materials and glasses gener-ally [34, 35, 52, 53]. The relevance of such spin modelsreaches far beyond physics, into biological and sociologi-cal applications, for example [54]. Here, we are employ-ing families of such spin models that interpolate betweenthe randomly disordered spin glass on a cubic lattice,called the Edwards-Anderson model (EA) [34], as well asits mean-field version, the Sherrington-Kirkpatrick model(SK) [35], on one side and the respective homogeneousferromagnetic systems [51] on the other. Each systemconsists of a random mixture of ferromagnetic and anti-ferromagnetic bonds J between neighboring spins σ i and σ j that are drawn from a distribution P ( J ) we have cho-sen to be bi-modal, i.e., J ij = ± J , with energy unitssuch that J = 1 in 3D and J = 1 / √ N in the mean fieldcase. A fraction p of ferromagnetic bonds is balanced outwith a fraction − p of anti-ferromagnetic bonds such that P ( J ) = pδ ( J − J ) + (1 − p ) δ ( J + J ) . (1)For each, the Hamiltonian (without external field) reads H = − (cid:88) (cid:104) ij (cid:105) J ij σ i σ j , (2)where (cid:104) ij (cid:105) refers to all extant bonds between neighboringspins σ i and σ j , either on a cubic lattice for EA or allmutual pairs of spins for SK.In the family EA of models we study on the cubic lat-tice [49, 50], we change the admixture of bonds by varying p between ≤ p ≤ , from the pure glass with an equalmix of bonds ( p = ) to a pure ferromagnet when allbonds are ferromagnetic ( p = 1 ). The situation is morecomplicated for SK, where already a sub-extensive excessof ferromagnetic bonds, away from the pure glass, resultsin ferromagnetic behavior. Specifically, since all N spinsare mutually connected, there are N ( N − bonds,and it only takes an imbalance between either type ofbond, merely of order (cid:118) √ N , to achieve ferromagneticordering. Thus, we define a family of mean-field modelsparametrized by α with p = + α √ N , varying between ≤ α ≤ to explore the full range of behaviors [55]. III. AGING AND RECORD DYNAMICSA. Simulation of Quenches in Spin Glasses
The distinction between slow relaxation in glassy ver-sus homogeneous systems is succinctly analyzed in thesimplest conceivable protocol of a hard quench from an
Figure 1. Illustration of the definition of valleys. The tracethrough an energy landscape produces a time sequence of en-ergy records (E) and of barrier records (B), relative to themost recent “E” [30, 31]. Only the highest and lowest recordsof the “E and “B” are kept to give a strictly alternating se-quence “EBEBE...”. Then, any sequence “BEB” demarcates avalley (vertical lines). easily equilibrated high-temperature state into an or-dered phase, whether glassy or ferromagnetic, crossinga phase transition in the process. Such a pure ag-ing protocol has been studied extensively in the last 40years [33, 43, 46, 56, 57]. In this process, the systemis thrown far out of equilibrium, left with an enormousamount of excess heat to be released to the bath to beable to descent deeper into its energy landscape to reachstates with the appropriate energy.To facilitate such a quench for the family of Ising spinmodels considered in our study, for each instance at time t = 0 , we initiate with randomly assigned spins, either σ i = ± , which corresponds to T = ∞ , and run the sim-ulation for t > at a low, finite temperature. For ourfamily of models on the cubic lattice, the critical temper-ature for a transition into an ordered state varies from T c ≈ . J in EA [58], to about T c ≈ . J for the ferro-magnet [59]. In our Monte Carlo simulations, we quenchto T q = 0 . J for all p , similar to Ref. [49], and monitorthe aging process for about sweeps. For the fam-ily of mean field models, we only vary the admixture offerromagnetic bonds minutely, so that the transition tem-perature does not deviate much from that of SK, whichis known to be T c = J [35]. Here, we also quench to T q = 0 . J throughout. For each value of p in our study,we have averaged results over at least realizations. B. Valleys in an Energy Landscape
Key to our analysis of coarsening versus glassy relax-ation is the definition of a measure that can serve todistinguish the effect of fluctuations on the irreversible events by which a system relaxes. One such measure hasbeen provided by Dall and Sibani in Ref. [30]. There, theinternal energy of an entire system of finite size is mon-itored to observe its time-trace for the ensuing quench.Since the system is expelling energy into the bath to re-lax, on average, the energy gradually decreases, albeitvia localized, intermittent events [60], in line with experi-mental observations of glassy systems [42, 61–63]. In par-ticular, record-sized fluctuations are needed for a glassysystem to relax [45, 64].As illustrated in Fig. 1, Dall and Sibani defined the“valley” production as an observable. [30] in the fol-lowing way: Let E be the up-to-now lowest energyvalue encountered up to time t and let E ( t ) be theinstantaneous energy. In turn, let the “barrier” B bethe up-to-now highest energy attained, relative to themost recent E, i.e., B = E ( t ) − E . An energy tracethen maps into a random sequence of symbols, like,...EEEBBBBBEEEEEEBBEEE.... Note that the tracecan generate a sub-sequence of records in the lowest en-ergy, i.e., multiple E’s in a row, before it encounters itsnext barrier record B, and also a sub-sequence of such B’sbefore it meets the next E, and so on. Clearly, it is thelatest E or B in either type of sub-sequence that is signif-icant: Each prior one is merely transitory, while the lastone supersedes each prior one as record that reaches itsultimate significance only after a new record fluctuationin the opposite direction is attained. Thus, we squashthe entire sequence into a strict alternation between Eand B, as the stricken letters in Fig. 1 imply. Then, a“valley” is defined as the part of the trace between twoconsecutive record barrier-crossings, as indicated by ver-tical dashed lines there. If the ground state would bereached, the sequence would terminate, of course.To focus truly on locally correlated record barriercrossings, it would be useful to refine this definition ofvalley [65]. However, unless a system gets too large, withtoo many simultaneous but spatially distant quakes, byconsidering a small enough system these events becomesufficiently rare to dominate the fluctuations in the entiresystem trace, instead of being “washed out”. This pointillustrates also that, to understand a thermodynamic sys-tem out-of-equilibrium , it is often not helpful to take thethermodynamic limit.Examples of a valley sequence from our simulations isshown for single energy traces in Fig. 2 for the EA spinglass (top) and the corresponding ferromagnetic system(bottom) on a cubic lattice. These plots exemplify thestark difference in the effect of fluctuations on either typeof system that we discuss in the following. C. Dynamics Driven by Record-Sized Fluctuations
As alluded to in the introduction, glassy and other-wise homogeneous systems such as a ferromagnet distin-guish themselves in the manner fluctuations affect theirrelaxation dynamics. In the latter, barriers are compa- -1.78 E n e r gy D e n s it y E n e r gy D e n s it y Figure 2. Typical trajectories of an aging process through theenergy landscape of the spin glass model on a -lattice with L = 16 spins and a fraction p of ferromagnetic bonds and − p anti-ferromagnetic bonds, here with p = 0 . (top) and p = 0 . (bottom). Energy ( (cid:72) ) and barrier ( (cid:78) ) records, asdefined in Fig. 1, are marked along each trajectory, where thevertical dashed lines indicate the transition between consecu-tive valleys. While the energy decreases, on average, graduallyas a logarithm in time with an ongoing but random produc-tion of further records in the glassy case ( p = 0 . ), the moreferromagnetic system ( p = 0 . ) expels energy in a few largeevents which appear to be triggered by typical fluctuations,record-sized fluctuations are seemingly irrelevant. rably low and remain invariant independent of the depthwithin the landscape and, thus, of the age of the process.As Fig. 2 exemplifies, large releases of energy are pre-ceded by typical fluctuations at any stage of the process.Fewer events, like the evaporation of a domain in coars-ening, happen not because individual events become somuch harder but rather because so many fewer events canhappen when only few domains is left. Larger domainsmay take a little more time to evaporate, as meanderinginterfaces need to find each other and collide, but such anentropy barrier does not dominate the otherwise domain-size independent energetic barriers [44]. Yet, ordinaryfluctuations suffice to bring those interfaces together.In the glassy system, however, it is the barrier height growing with domain size that decelerates the event-rate.Although many domains remain available even after along aging time, few muster the chance fluctuation re-quired to break up. In a landscape with those barriers,ordinary fluctuations become ineffective to drive the re-laxation process. They merely “rattle” the system duringincreasingly longer quasi-equilibrium interludes. Onlyrare, extraordinary large, in fact, record-size fluctuationsmanage to scale such barriers to expel excess heat, ad-vance the relaxation, and grow domain size, minutely.These features, widely shared across many disorderedmaterials, inspire a phenomenological description knownas Record Dynamics (RD) [64]. In RD, the relaxationprocess of a non-equilibrium system after a hard quenchis determined by large, irreversible fluctuations whichmove the system from one meta-stable state to the next(usually only marginally more stable than the last one)within its complex energy landscape [45, 66]. This can bethought of as the system overcoming energy barriers ina hierarchical energy landscape [36–40]. The rate λ ( t ) ofsuch record events, also termed “quakes”, decelerates withtime as /t . Therefore, the expected number of events ina time interval [ t , t w ], is (cid:104) n ( t, t w ) (cid:105) (cid:29) t ˆ t w λ ( t (cid:48) ) dt (cid:48) (cid:29) ln (cid:18) tt w (cid:19) (3)implying that the dynamics of the system is self-similarin the logarithm of time. That time-homogeneity is acommon feature of many aging systems [43, 56, 67]. Inour studies here, we are more concerned with the rate ofevents λ ( t ) and the logarithmic growth of observables intime. The dependence on waiting time t w has been thefocus elsewhere [45, 60, 66]. IV. NUMERICAL RESULTSA. Edwards-Anderson Model
Applying the measure of a valley number defined inSec. III B to the cubic Ising spin model introduced inSec. II provides a notable distinction between glassy andhomogeneous coarsening behavior, as Fig. 3 shows. Forall p < p c ≈ . , the critical threshold found in Ref. [50],we find that the valley count progresses logarithmically intime (in fact, like the root of that logarithm [66]), consis-tent with Eq. (3). For larger values of p , the valley countslows ever more significantly to eventually plateau at afinite value, apparently. All the results shown here wereobtained for systems with N = 16 = 8096 spins, usingperiodic boundaries, since we found very little variationwith system size for larger N .The fact that the underlying ordered state is eitherglassy or ferromagnetic affords us to also measure theincrease in magnetization with time, as demonstrated inFig. 4. This measure actually exhibits a more dramatic Sweeps V a ll e y s p = 0.5p = 0.55p = 0.6p = 0.65p = 0.7p = 0.75p = 0.8p = 0.85p = 0.9p = 0.95p = 1.0 Figure 3. Average number of valleys, as defined in Fig. 1,that are traversed with time after a quench to T = 0 . J in a L d = 16 spin glass with a fraction p of ferromagnetic bondsand − p anti-ferromagnetic bonds. For p ≤ . , the gen-eration of valleys evolves essentially independent of p , whilefor a larger admixture of ferromagnetic bonds valley genera-tion progresses to cease ever more rapidly and the number ofvalleys reached plateaus. Sweeps M a gn e ti za ti on p = 0.5p = 0.55p = 0.6p = 0.65p = 0.7p = 0.75p = 0.8p = 0.85p = 0.9p = 0.95p = 1.0 Figure 4. Average magnetization per spin, (cid:104) m (cid:105) , observed withtime after a quench during the ensuing aging process, as de-scribed in Fig. 3. Like there, systems with p ≤ . behaveglassy in a p -independent manner with little discernible mag-netic ordering, while the more ferromagnetic systems becomeincreasingly more ordered. transition between the glassy and the ferromagnetic case,as consecutive snapshots of both, the valley count as wellas the magnetization, are shown in Fig. 5 for a progres-sion of times that increases by a factor of 8. In theseplots, we have also marked the zero-temperature transi-tion at p c ≈ . , which proves consistent asymptoticallywith the transition out of the glassy relaxation behavior.Finally, we can also look at the instantaneous rate ofbarrier crossing events, effectively the derivative of thevalley production, i.e, inverting the integral in Eq. (3). density p < m > V a ll e y s t = 2 t = 2 t = 2 Figure 5. Finite-time snapshots of the numbers of valleys gen-erated (top) and the corresponding magnetization per spin, (cid:104) m (cid:105) (bottom), as a function of ferromagnetic bond fraction p for three different times, taken from the data at T = 0 . J shown in Fig. 3 and Fig. 4, respectively. The vertical line at p c = 0 . indicates the zero-temperature transition found inRef. [50] between a glassy and a ferromagnetic phase. Sweeps -3 -2 -1 E v e n t R a t e p = 0.5p = 0.55p = 0.6p = 0.65p = 0.7p = 0.75p = 0.8p = 0.85p = 0.9p = 0.95p = 1.0 Figure 6. Instantaneous rate of record barrier crossing events,as defined in Fig. 1, with time after the quench, as describedin Fig. 3. Asymptotically, for larger times, that rate variesas a power-law with a seemingly hyperbolic decline, ∼ /t (dotted line), for all p < . to an almost quadratic decline, ∼ /t (dash-dotted line), for larger p . Indeed, throughout the glassy regime, the rate deceler-ates roughly hyperbolically, in accordance with the RDpredictions. [Note that this could miss a minor logarith-mic correction, such as λ ( t ) ∼ / ( t √ ln t ) , for instance,needed to get √ ln t for the valley production in Fig. 3.]For p > p c , in the ferromagnetic coarsening regime, wenotice that the rate falls off increasingly sharper, ulti-mately about as ∼ /t . Consequently, its integral stallsout into the plateaus seen in Fig. 3. Apparently, do-main mergers occur more rapidly, on a power-law scale, Figure 7. Number of valleys traversed during relaxation en-suing after a quench of SK for different bond fractions α froma high temperature T = ∞ to T = 0 . J , averaged over anensemble of trajectories for N = 2048 spins. In the range . ≤ α ≤ . , the number of valleys traversed grows loga-rithmically and largely independent of α , indicating that theregime is glassy.Figure 8. Average magnetization in the same simulationsshown in Fig. 7. According to this measurement, the systembegins to order at α c ≈ . , since a non-zero magnetization inthe long-time limit indicates that majority of the spins haveferromagnetically ordered. The transition in magnetizationshown here is far more dramatic than in the valley counts,but nevertheless affirms the same critical threshold. in coarsening ferromagnets. Despite the rapid drop inthe event rate, the average domain size manages to in-crease as a power-law [44], because later mergers expellarger amounts of excess heat, see Fig. 2. In case ofthe glass, each event expels on average a fixed amountof heat, roughly. Therefore, both valley production anddomain growth proceed similarly (logarithmically), as anintegral of the event rate, since each activation has thesame impact. Figure 9. Instantaneous average valley counts and magneti-zation as function of α at different sweep-times t = 16 , 256and 4096 from left to right, each for three different systemsizes indicated on the legend. The first row shows the aver-age number of valleys, and the second row shows the averagemagnetization. According to this data, the valley productionis time dependent as the sharpness of the transition becomesmore pronounced in the later sweeps. In contrast, the magne-tization appears to be saturated already early on, predictingthe critical threshold within 16 sweeps. Additionally, we seeno system size effects when using α as the parameter. B. Sherrington-Kirkpatrick Model
Using the valley counts defined in Sec. III B as an or-der parameter, we find a clear transition from a glassyregime to a ferromagnetic one in the mean field as well.However, unlike for EA on a cubic lattice, extending theneighborhood of each spin to all others in the case ofSK changes the dynamics, and we have to explore thecritical threshold at which the spin glass to ferromag-netic transition takes place on a different scale. Mutualconnections between all spins require the number of fer-romagnetic bonds to only slightly exceed the number ofantiferromagnetic bonds, in order to tip the system intobecoming ordered. The transition to the ferromagneticregime occurs almost immediate beyond a bond densityof p = 0 . , with a strong system size dependence, forcingus to adapt a different scale to observe it. To properlydescribe the behavior of SK, we therefore reparametrizethe bond density in terms of α via p = + α √ N . Then,within the range of ≤ α ≤ . , we can localize a tran-sition that varies only slowly with size.Similar to Fig. 3 for EA, in Fig. 7 we show the numbersof valleys found in a SK system with N = 2048 spins.There appears to be a critical threshold at α c ≈ α ≤ . , the valley production increases about aslog ( t ) , essentially uniform with bond density, given thenearly perfect overlap in the data. This is no longer casewhen α > . , where the production of valleys decreasesgradually before plateauing completely. While domainsin the sense of geometric regions of a certain length do notexist in a mean field system with long-range interactions,individual spins develop clusters of increasingly orderedlocal fields with some of their neighbors that entrench the Sweeps -5 -4 -3 -2 -1 E v e n t R a t e Figure 10. Instantaneous rates for the number of record bar-rier crossings as a function of time, for every α value. Theinstantaneous rate decreases as a power-law for all but thehighest admixture values. In the glassy regime, the decel-erations is essentially hyperbolic, while the rate drops moresharply for α > . , up to roughly t − . at α = 1 . , beyondwhich further record events become immeasurably rare. system into deeper valleys. It becomes increasingly moredifficult for the system to overcome the energy barrier offlipping the entire cluster, causing the relaxation processto evolve logarithmically [44].That said, evidence of a critical threshold suggests thatbeyond α c , the system changes its landscape dramati-cally. It exhibits an inclination to order rapidly, facil-itated by the fact that local fields of individual spinsimmediately affect all others, as the evolution of mag-netization in Fig. 8 suggests. Flat interfaces betweensuch clusters, as they may exist between domains inlow-dimensional lattices like EA, are absent here andany imbalance in size quickly erodes inferior clusters.Therefore, despite the quantitative differences pertainingto local structure between the Edwards-Anderson andSherrington-Kirkpatrick spin glass, our results suggestthat the glassy behavior in both can be attributed to thehierarchical nature of the energy landscape, and the lackof it beyond the transition to ferromagnetic order, seenboth in Fig. 4 and Fig. 8.We have also checked the evolution of valley countsacross different system sizes and found only a minimaldependence of the transition on larger size, as shown inFig. 9. While the relationship (or lack thereof) between the number of valleys encountered and the bond admix-ture exhibits time dependence, the critical threshold withregard to ordering already emerges after about two hun-dred sweeps. There is clearly an agreement between val-ley statistics and the ferromagnetic order parameter insuggesting α c ≈ . as the critical threshold.Lastly, we look at the deceleration of the rate of recordbarrier crossing events in Fig. 10. As shown in Fig. 6for the Edwards-Anderson model, the rate decays witha power of time t . While there is a steeper decelera-tion in the barrier crossing events for larger α values, thedifference between the exponents is quite subtle on thistime scale within our simulations. In the glassy regime, α < α c ≈ . , the rate clearly decays hyperbolically,whereas it falls off steeper above α c . However, for values α > . , the fall-off becomes so significant that new val-leys are not encountered beyond the first ∼ sweeps. V. CONCLUSION
Our study explores the distinction between glassy re-laxation and ordinary coarsening, which is often ignoredin the description and analysis of aging systems. Focus-ing on families of models that interpolate between eitherextreme, we not only apply measures [30, 31] that clearlyindicate the difference but also show a rather sharp tran-sition in the non-equilibrium behavior between those ex-tremes that, for the Edwards-Anderson model on a cu-bic lattice, appears to coincide with the (equilibrium)zero-temperature transition between spin glass and fer-romagnet [50]. The corresponding transition we find at asub-extensive scale in SK seems to have been unnoticed.While the distinction we are making between a coars-ening (ferromagnetic) and an aging (glassy) regime canbe seen as semantic, considering that both, algebraicas well as logarithmic growing domains, are commonlyportrayed as coarsening [44], the difference in dynamicbehavior after a quench is profound. The picture thatemerges is one of a largely convex landscape on one sidewith invariant energetic barriers in the case of coarsening,a system that despite its often complex network of frac-tal interfaces locally homogenizes rather quickly. On theother side, we find a very hierarchical landscape [36–40]with energetic (and potentially entropic) barriers thatgrow with deeper entrenchment within the landscape,rendering all but record fluctuations ineffective for re-laxation. [1] H. Frauenfelder, ed.,
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