Eur. Phys. J. B (2020) 93: 88https://doi.org/10.1140/epjb/e2020-100598-3 THE EUROPEAN

PHYSICAL JOURNAL BColloquium

Growth kinetics and aging phenomena in a frustrated system?

Manoj Kumar1, Federico Corberi2, Eugenio Lippiello3, and Sanjay Puri4,a

1 Centre for Fluid and Complex Systems, Coventry University, CV1 5FB, Coventry, UK2 Dipartimento di Fisica “E. R. Caianiello”, and INFN, Gruppo Collegato di Salerno, and CNISM, Unita di

Salerno,Universita di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy3 Department of Mathematics and Physics, University of Campania L. Vanvitelli, Viale Lincoln 5, 81100 Caserta,

Italy4 School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India

Received 13 December 2019 / Received in final form 4 March 2020Published online 14 May 2020c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature,

2020

Abstract. We study numerically the ordering kinetics in a two-dimensional Ising model with randomcoupling where the fraction of antiferromagnetic links a can be gradually tuned. We show that, uponincreasing such fraction, the behavior changes in a radical way. Small a does not prevent the system froma complete ordering, but this occurs in an extremely (logarithmically) slow manner. However, larger valuesof this parameter destroy complete ordering, due to frustration, and the evolution is comparatively faster(algebraic). Our study shows a precise correspondence between the kind of developing order, ferromagneticversus frustrated, and the speed of evolution. The aging properties of the system are studied by focusingon the scaling properties of two-time quantities, the autocorrelation and linear response functions. Wefind that the contribution of an equilibrium and an aging part to these functions occurs differently in thevarious regions of the phase diagram of the model. When quenching inside the ferromagnetic phase, thetwo-time quantities are obtained by the addition of these parts. Instead, in the paramagnetic phase, thesetwo contributions enter multiplicatively. Both of the scaling forms are shown with excellent accuracy, andthe corresponding scaling functions and exponents have been determined and discussed.

1 Introduction

The kinetics of phase ordering remains an interestingproblem in non-equilibrium statistical mechanics. Thisterm refers to the nonequilibrium evolution of a systemwhen it is rendered thermodynamically unstable by aninstantaneous quench from a high-temperature phase to alow-temperature phase [1,2]. The system develops order-ing among local regions known as domains, which growin time t with a characteristic length scale L(t) untilthe system reaches the equilibrium state. A fundamentalquantity of interest is the domain growth-law, i.e., howL(t) depends on the time t elapsed after the quench. Forpure systems, i.e., in the absence of quenched disorder,it is given by the power-law behavior L(t) ∼ t1/z withan universal exponent z that depends only on few rele-vant parameters such as the dimension of order parameterand the presence of conservation laws. In the presenceof quenched disorder, when frustration is absent or irrel-evant, the kinetics is slowed down due to presence of

? Contribution to the Topical Issue “Recent Advances in theTheory of Disordered Systems”, edited by Ferenc Igloi andHeiko Rieger.

a e-mail: purijnu@gmail.com

energy barriers [3]. Many efforts have been made [4–38]to understand if this leads to a logarithmic or to an alge-braic growth law. Various classes of disordered systemshave been investigated including, for example, Ising mod-els with random bonds [7–10,12–14,17,18,21,24,35,39–41],random-fields [19,23,25–27,29,33,42,43], and with site orbond dilution [6,7,34,44–47].

Huse and Henley [32], under the assumption that theenergy-barriers in disordered systems EB(L) scales as apower-law in L, i.e, EB(L) ∼ Lψ, predicted a logarithmicgrowth, L(t) ∼ (ln t)1/ψ with the barrier exponent ψ > 0.Nevertheless, finding logarithmic growth numerically hasremained a challenge due to a slow evolution. For example,several numerical studies [8–14,17,24,35] have found analgebraic growth with a disorder-dependent exponent. Butextensive simulations on the same system in reference [18]have found a signature of logarithmic growth at late times.Recently, Cugliandolo et al. [30,48] also have argued thatan algebraic-behavior is an intermediate regime of growth.Further, an increasing number of studies have reported aclean crossover from power-law to a logarithmic behav-ior in various systems such as Ising models with randomfields [19,29,33,37], random dilution [6,7,34,44,46,49], andpolymers (or elastic strings) in random media [48,50–52]. Moreover, a number of experimental studies on

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random-bond [53–56] and random diluted systems [49]also have reported a logarithmic growth. Therefore, it islikely that a pre-asymptotic power-law coarsening regimeexists, followed by a truly logarithmic which is sometimeshardly accessible numerically.

In addition, recently it has also been found [7] a nontriv-ial dependence of the growth law L(t) on the amount ofrandomness in diluted Ising models [6,7,46]. Specifically,for a sufficiently small fraction of diluted sites (or bonds)d, the kinetics of growth slows down upon increasing duntil a certain threshold value d∗, after which increas-ing further d produces a faster growth. In [6,7,46] it isargued that L(t) increases asymptotically in a logarith-mic way for any 0 < d < dc, where dc = 1 − pc is thepercolation threshold above which the networks becomesdisconnected, but it turns into an algebraic behavior witha temperature-dependent exponent right at dc. This is dueto the fact that the fractal topology of the network at thepercolation threshold plays an important role in softeningof the pinning energy barriers which cause the speed-ing up of the evolution. This interplay between algebraic(at d = 0 and d = dc) and logarithmic (for 0 < d < dc)growth law is responsible for the non monotonous depen-dence of the speed of growth on the dilution strength d asmentioned above.

All the systems discussed to date are such that theaddition of quenched disorder leaves the structure ofthe equilibrium states, with two free energy minima atlow temperatures, qualitatively preserved. However, thereare cases in which the effect of disorder is so strongas to alter this structure possibly. This may occur, inparticular, when the disorder is associated with frus-tration. With adding frustration, the problem becomesmore complicated as even the low-temperature equilib-rium properties in these systems are still debated [57]. Itis useful to stress that studying the off equilibrium kinet-ics of such systems may help shedding some light on theequilibrium structure as well since the coarsening behaviordiscussed before is expected to be associated to systemswith a ferromagnetic-like equilibrium phase space struc-ture whereas a different dynamical evolution characterizessystem with a mean-field glassy scenario.

This paper focuses on growth kinetics and aging in a dis-ordered system with frustration. It provides an overviewof two of our recent works [38,58], where we investigatedequilibrium and off-equilibrium dynamics in a frustratedIsing magnet. The frustration is tunable with one limitbeing the non-frustrated system, where the propertiesare better understood. Specifically, we considered a two-dimensional Ising model with both ferromagnetic andantiferromagnetic interactions, and tune the fraction a ofthe latter gradually. In an effort to stay as close as possi-ble to the ferromagnetic system, we set the strength |J+|of ferromagnetic couplings larger than that, |J−|, of theantiferromagnetic ones.

Regarding equilibrium properties, the system exhibitsdifferent phases in the low-temperature phase diagram(see Fig. 1). As a is progressively increased, one movesfrom a ferromagnetic (FM) phase (a < af ), where frus-tration plays a minor role, to a strongly frustrated para-magnetic (PM) phase (af ≤ a ≤ aa), which at T = 0 is a

Fig. 1. Schematic representation of the phase-diagram of themodel.

spin-glass phase [59–61] (shown by a thick green line inFig. 1). For even larger value of a, we enter in an antifer-romagnetic (AFM) region (a > aa) . Therefore, by tuninga one can study the system evolution in transition from aferromagnetic phase to a deeply frustrated one.

Another important feature of slowly evolving systems isthat they are characterized by the aging phenomenon [62,63], which is embodied by dynamical scaling, and is wellcharacterized by two-time quantities, such as the auto-correlation function C(t, tw) and the associated linearresponse function R(t, tw) [64], with t ≥ tw. In equilib-rium states these quantities are linearly related accordingto the usual fluctuation-dissipation theorem (FDT). Ongeneral grounds, for out-of-equilibrium states the responsefunction cannot be written in terms of C(t, tw) alone [65].However, the relation between these two quantities isexpected to be meaningful and to bear important infor-mation both on the equilibrium structure and on thedynamical properties [66,67]. The nonequilibrium scalingproperties of two-time quantities are quite well under-stood for a non-disordered system [68–70]. In particular,it is known how the contribution of degrees of freedomwhich are thermalized to the bath temperature and thoseof a genuinely out of equilibrium nature add up to forma stationary and a non-stationary term, Ceq and Cag

respectively (and similarly for the response), in terms ofwhich two-times quantities can be written. However, fordisordered systems, a similar understanding in general ismissing [65,67,71–75], though a violation of FDT has beenfound [70,76].

In reference [58], we computed these two-time functionsin the frustrated magnet and determined their scalingforms. We showed that not only C(t, tw) and R(t, tw) takedifferent scaling forms in the various regions of the phase-diagram (i.e., FM or AFM and PM), but also that Ceq andCag combine differently to form the whole correlation C(and similarly for the response function) in such differentphases of the system. Specifically, we showed how the addi-tive structure of the two-time quantities (C = Ceq +Cag)in the FM phase, turns into a multiplicative one in thePM region where C(t, tw) = Ceq(t − tw) · Cag(t, tw), andsimilarly for the response function.

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In this paper, we highlight our main results on growthkinetics and aging in a two-dimensional frustrated mag-net. This paper is organized as follows. In Section 2 weintroduce the model and describe the structure of equi-librium states and the phase diagram. In Section 3, wepresent numerical results for the growth kinetics of themodel in its different phases. Section 4 is devoted to thestudy of aging phenomena, in which we discuss the scal-ing properties of two-time quantities, and then presentdetailed numerical results for these quantities. Finally, inSection 5, we conclude the paper with a summary and adiscussion of our findings.

2 Model and phase structure

We consider the random-bond spin model given by theHamiltonian

H({si}) = −∑〈ij〉

Jijsisj , (1)

where si = ±1 are the Ising spins, and 〈ij〉 denotes near-est neighbors sites of a two-dimensional square lattice.Jij are the uncorrelated stochastic random coupling con-stants, drawn from a bimodal distribution, which takes avalue J0− ε with probability a, and J0 + ε with probability1− a, i.e.,

P (Jij) = a δJij ,J0−ε + (1− a) δJij ,J0+ε, (2)

where J0 > 0, and δ is the Kronecker function. Clearly,ε ≤ J0 corresponds to a non-frustrated case. Here, instead,we use ε > J0, meaning that the fraction a of bonds areAFM with Jij < 0 (which we also denote as J− = J0 − ε)and the remaining ones are FM with Jij > 0 (denoted asJ+ = J0 + ε).

We consider a simple case with

J0 < ε <q

q − 2J0, (3)

where q is the coordination number of the lattice. Whenthis condition holds, a spin to which at least an FM bondis attached will always lower its energy by pointing alongthe direction of the majority (if a majority exists) of spinsto which it is connected by FM bonds. For instance, evenif a spin has three antiferromagnetic and only one ferro-magnetic bond, the energy will lower by aligning it to thespin on the other end of the ferromagnetic bond. Hence,equation (3) corresponds to a ferromagnetic-always-winscondition. This choice has been made in order to stay asclose as possible to a ferromagnetic system, in order tounderstand its properties more easily. However, as we willshow soon, this does not prevent the system from exhibit-ing a frustration dominated phase. Notice that J0 = 0is the usual Edwards-Anderson spin-glass systems whereboth positive and negative bonds are of equal strength(J± = ±ε), and therefore do not obey equation (3).

Here we set J0 = 1, and choose ε = 1.25J0 (i.e. J+ =J0 + ε = 2.25 and J− = J0 − ε = −0.25), which obviously

Fig. 2. In the upper stripe, four typical bond configurationsare pictorially shown, corresponding to 0 < a < a∗, a∗ < a .ap, ap . a < 1 − a∗ and 1 − a∗ < a < 1, from left to right,respectively. FM bonds are drawn in blue, AFM ones in red.The bar below the configuration stripe describes the physicalphases of the systems as a is varied, e.g., if FM, PM, etc.

satisfies equation (3). All the numerical data are presentedfor square lattices of size 5122 with periodic boundaryconditions applied on both sides.

We start with a discussion of the T = 0 equilibriumstates of the model. The ground state problem of afrustrated Ising spin system can be exactly solved on a pla-nar graph (without periodic boundary condition), usingthe minimum-weight–perfect-matching (MWPM) algo-rithm [77,78]. However, for non-planar graphs, the kindof system considered here, we have used a highly efficientiterative windowing technique developed by Khoshbakhtand Weigel [79]. This algorithm is based on the MWPMapproach by mapping the ground state problem on thetoroidal lattices and can be used to find the exact groundstate in polynomial time up to 30002 spins on a system offull periodic boundary conditions.

In order to classify the phase structure of the model,it is useful to consider the two global order parameters,viz., the spontaneous magnetization m and the staggeredmagnetization M , defined as

m =1

N

∑i

si, M =1

N

∑i

σi, (4)

where σi = (−1)isi is the staggered spin, the index i runsover the lattice sites in such a way that two nearest neigh-bors (NNs) always have a different value of (−1)i. In otherwords, the staggered magnetization M is basically the dif-ference between the magnetization of two sub-lattices. Inthe following, we describe the physical phases of the modelas a is varied, using the values of m and M measured inthe ground states as a guide.

2.1 Ferromagnetic phase (0 ≤ a < af )

This phase can be split into the two sectors with 0 ≤ a <a∗ and a∗ ≤ a < af .

Sector 0 < a < a∗: in this region, there are very fewAFM bonds, which are basically isolated in a sea of FMones, as shown in Figure 2, where a pictorial representa-tion is provided. Therefore, also from equation (3), theground state is akin to a usual FM system, and hencewe expect |m| ' 1 and M = 0. We can see in Figure 3that this is indeed the case. The value of a∗ at which theAFM bonds starts clustering is expected to be located

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Fig. 3. Plot of the absolute value of the magnetization |m|and of the staggered magnetization |M | in the ground states atT = 0 for different values of a. The ground states are obtainedon a 5122 lattice with the periodic boundary condition (figurefrom Ref. [38]).

Fig. 4. Configurations of the GS for a 5122 lattice for differentvalues of a. Spins up are marked in black and spins down arein white (figure from Ref. [38]).

between a = 0.2 and a = 0.3 [7]. A representation of anactual ground state configuration of the model for variousvalues of a is shown in Figure 4, and clearly the picturewith a = 0.2 (in the upper left panel) confirms the abovedescription, showing a complete FM ordered structure.

Sector a∗ ≤ a < af : in this region, there is still aprevalence of FM order, which extends up to af & 0.4(see Fig. 3). This is due to the fact that the numberof FM bonds is larger than that of the AFM ones andalso because they are comparatively much stronger, beingJ+ = 9|J−|. However since AFM bonds can also coalesce,regions with down spins may be found locally, as it can beseen in Figure 4 for a = 0.3 and a = 0.4 (upper central andright panel). This is why we call this sector as defectiveferromagnet. In this extended FM region, the magnetiza-tion |m| decreases upon raising a which vanishes at thetransition point a = af , and M = 0, as it can be observedin Figure 3.

Fig. 5. Ground state configurations of the staggered spin σifor a 5122 lattice for different values of a. σi = 1 are markedin black and σi = −1 are in white.

2.2 Paramagnetic phase (af ≤ a ≤ aa)

Also in this phase, we can distinguish two subregions,namely those with af < a < ap and with ap < a < aa(Fig. 2), as we discuss below.

Sector af < a < ap: in this sector, the FM bonds stillprevail to form a sea that spans the system. However,the AFM bonds also get grouped to form sufficiently con-nected paths so as to destroy the FM state. Therefore, onehas m = 0 and also M = 0, since negative bonds are in aminority which is insufficient to establish an AFM order-ing in this sector. This is confirmed in Figure 3. Due tothe fact that m = M ' 0, we generically call this regionas paramagnetic. Further, a = ap = 0.5 is the bond perco-lation threshold above which FM bonds do not span thesystem. A ground state of the system for a = 0.5 is shownin Figure 4 (bottom row, left).

Sector ap < a < aa: in this region, there are still FMislands in a sea of AFM bonds. Therefore, m = 0, asexpected throughout this region, is observed in Figure 3.However, the presence of a spanning sea of AFM bonds isnot sufficient to establish a global AFM order even whena is so large that FM bonds are isolated. This is obviouslydue to the fact that AFM interactions are weak as com-pared to FM ones, and, indeed, we see in Figure 3 thatthe property M = 0 extends up to a = aa, where aa islocated around a & 0.95.

The development of AFM order can easily be observedby plotting the staggered spin σi instead of si, becauseplotting {si} results in a uniform grey plot in which AFMstructure cannot be seen clearly. This is done in Figure 5,where AFM structures can be easily spotted as black orwhite regions.

2.3 Antiferromagnetic phase (aa ≤ a ≤ 1)

In this region the AFM order sets in, and there are veryfew and far apart strong FM bonds, which represent akind of defects in an otherwise perfectly ordered AFMstate. Each of these produces a defect in an otherwiseantiferromagnetically ordered system. In this region, onehas m = 0 and M 6= 0, as it can be seen in Figure 3. AnAFM structure can be clearly seen from a ground stateat a = 0.97 in Figure 5, where staggered spins {σi} areplotted.

Eur. Phys. J. B (2020) 93: 88 Page 5 of 15

3 Numerical simulations for growth kinetics

3.1 Simulation details

The system is prepared in a disordered state with spinspointing randomly up or down, corresponding to T =∞.It is then quenched to a low temperature Tf at time t = 0.Here, we stress that quenching to a very small Tf canresult in sluggish kinetics, and no substantial growth ofL(t) can be detected in the simulation window. On theother hand, a larger choice of the quench temperature mayresult in Tf > Tc(a). Therefore, a reasonable choice of Tfbecomes necessary in order to study the off-equilibriumgrowth kinetics of the system.

We found, out of many Tf , the two suitable choicesTf = 0.4 and Tf = 0.75, which represent a good compro-mise between the two contrasting issues discussed above.Both these temperatures are much below the criticaltemperature Tc(a = 0) ' 2.269J+ ' 5.105 of the cleanferromagnet. From the side of AFM, Tf = 0.4 is smallerthan the critical temperature of the clean antiferromag-net Tc(a = 1) ' 2.269J− ' 0.567 while the Tf = 0.75 isabove. Coming to the PM region, since Tc(a) = 0, for val-ues of a in the PM phase, the quench is necessarily madeabove the critical temperature.

After the quenching, which occurs at t = 0, the systemhas been evolved using non-conserved dynamics [1,2] withthe Glauber transition rates:

W (si → −si) =1

2

[1− tanh

(β∆E

2

)]= [1 + exp (β∆E)]

−1, (5)

where ∆E is the change in energy resulting due to singlespin-flip (si → −si), given as

∆E = 2si∑j∈ni

Jijsj . (6)

Here, ni refers to the set of nearest neighbors of site i.Using this algorithm, the system is evolved up to t = 107

Monte-Carlo steps, each of which amounts to attemptedupdates of N(= 5122) spins in the lattice. All simulationresults have been averaged over 103 different realizationsof disorder and initial conditions.

The main observable we are interested in is the typicalsize of domains which grows as a function of time, i.e.,L(t). We define it as the inverse of the excess energy

L(t) = N [〈E(t)〉 − Eeq]−1, (7)

where 〈E(t)〉 is the average value of the energy at timet and Eeq is the average energy of the equilibrium stateat T = Tf . This can be obtained by evolving the groundstate at T = Tf until stationarity is achieved. The use ofequation (7) to determine L(t) is standard in phase-ordering kinetics and, in a non-disordered system, canbe easily understood from the fact that the excess non-equilibrium energy is stored on the interfaces (e.g., domainwalls), whose density scales as the inverse of typical

domain size L(t) [1]. Notice that as the present modelhas a frustrated (or paramagnetic) phase in the regionaf ≤ a ≤ aa, L(t) in equation (7) should prudentiallybe regarded simply as the inverse distance from theequilibrium energy.

3.2 Numerical results

We now present our numerical simulations on the growthkinetics after the temperature quench. Figures 6a and 6bshow the behavior of L(t), in the whole range of values ofa, for Tf = 0.4 and Tf = 0.75, respectively. Notice thatwe have plotted L(t)/L(t = 4) to better compare differentL(t) curves for different a.

As a general remark, an oscillating behavior of thegrowth can be seen for Tf = 0.4 in Figure 6a. This isusually interpreted [6,7,15,16] as a stop and go mecha-nism when interfaces get pinned on defects such as weakAFM bonds in an FM phase or vice versa. For instance,in an FM phase for small a, the smallest energetic barriereb encountered by a piece of an interface when it crossesfrom a single AFM bond to FM ones, is typically givenas eb = J+ − J− = 2ε. Then, the associated Arrheniustime to escape the pinned state is τ ' exp (eb/kBT ). Withε = 1.25 in our simulations, one has τ ' 518 for Tf = 0.4and τ ' 28 for Tf = 0.75. We see in Figure 6 that, forTf = 0.4, this value is very well compatible with thetime where L(t), after becoming very slow, starts growingfaster again. As Tf is raised, the stop and go mechanism,although still present, is less coherent, and the oscillationsare smeared out. This is observed at Tf = 0.75, where theoscillatory phenomenon is hinted. Moreover, the speed ofordering increases upon raising Tf .

Let us now describe the nature of kinetics in the variousphases of the system.

3.2.1 Ferromagnetic region 0 ≤ a ≤ afIn this region, starting from a = 0 (pure case) where theexpected behavior L(t) ∝ t1/2 is clearly observed (Fig. 6),the growth slows down upon rising a, but this occurs onlyup to a certain value, which we interpret as a = a∗, whichis located around a = 0.2. Upon increasing a beyond a∗,the phase-ordering process speeds up again, up to af . Thisis very well observed both for Tf = 0.4 and Tf = 0.75.The value of a∗, at these two temperatures is compara-ble, as it is expected assuming that this behavior can beascribed to the topology of the bond network alone. Themorphology of the growing domains can be observed fromFigure 7. Upon increasing a, domains become more jaggedand possibly fractal.

Next, to determine the nature of the growth, we com-pute the effective growth exponent zeff , defined as,

1

zeff(t)=d[lnL(t)]

d[ln t]. (8)

At Tf = 0.4, the oscillating nature of the curves shadowthe genuine growth law, and it is therefore almost impossi-ble to come up with any quantitative statement about theneat growth, e.g., if it is consistent with a power-law or

Page 6 of 15 Eur. Phys. J. B (2020) 93: 88

Fig. 6. Log–log plot of L(t) vs. t for different values of a (spec-ified in the legend) for the temperature quench at (a) (upperpanel) Tf = 0.4 and (b) (lower panel) Tf = 0.75. The black

dashed lines are the power-law (t1/2) fits, which correspondto the pure case for a = 0 or for a = 1. The other dashedlines in the lower panel, near to (and of the same color of) the

data for a = 0.4, 0.5, 0.6, 0.7, are the power-law fits of t1/z withz ' 5.2, 4.2, 2.93, 2.53, respectively (figure from Ref. [38]).

with a logarithm or something else. Therefore, we plot zeff

for Tf = 0.75 in Figure 8, which clearly depicts the nonmonotonous nature of growth. For a = 0, asymptoticallyzeff ' 2, as expected. When a is progressively increasedfrom a = 0 to a = 0.4 ' af , looking zeff at sufficientlylarge times (t > 104), one sees that it initially starts ris-ing until a reaches the value a∗ ' 0.2 and then decreasescontinuously, meaning that the growth slows down uptoa = a∗ and then speeds up with increasing a from a = a∗.This non-monotonous fashion of the growth mechanismhas also been found in disordered non-frustrated systems[6,7,46]. Furthermore, a positive slope or a curvature in zeff

Fig. 7. Evolution of {si}-configurations, where si = ±1 aremarked in black and white, respectively, in the FM region atdifferent times (see key) after a quench to Tf = 0.4. The valueof a is 0.1 in the upper row and 0.2 in the lower row.

Fig. 8. Log-linear plot of effective exponent zeff(t) as a func-tion of time t for a quench to Tf = 0.75, for different values of aas specified in the key. The dashed lines indicate the estimatedalgebraic growth for a = 0.4, 0.5, 0.6, and 0.7 with asymptoticvalues of zeff ' 5.2, 4.17, 2.93, and 2.53, respectively.

as function of time t can also be observed for all valuesof a in the range 0 < a ≤ 0.3, which implies a logarithmicnature of the growth. Instead, for a & 0.4 ' af , zeff(t) atlate times is showing an approximately constant behav-ior, representing an algebraic growth, L(t) ∼ t1/zeff . Thedashed lines represent our best estimate of zeff in thisregion.

The different asymptotic behavior – algebraic versuslogarithmic – observed at a = af with respect to the restof the FM region can also be interpreted upon thinkingaf as the lower limit of the PM region, where algebraicbehaviors are observed. We will comment further on thispoint below.

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Fig. 9. Evolving morphologies at different times of the stag-gered spins in the AFM region after a quench to Tf = 0.4with a = 0.97. σi = ±1 are marked in black and white,respectively.

3.2.2 Paramagnetic region af < a < aa

In this region, we expect a kind of spin-glass order atTf = 0. For a two-dimensional spin glass (correspondingto J0 = 0 in our model) it has been shown [11,80–83]that the existence of a spin-glass phase at T = 0 rules thekinetic in a long lasting pre-asymptotic regime in which analgebraic behavior of a growing length scale has been iden-tified [11]. Interestingly, we see from Figure 6b, the data isconsistent with an algebraic growth L(t) ∝ t1/z, with ana-dependent exponent, in agreement with reference [11].

The algebraic increase of L(t) can also be confirmedfrom inspection of the effective exponent zeff in Figure 8,where at the late time regime t & 103–104, this quantitystays basically constant except for some noisy behavior.Notice also that 1/zeff raises as a is increased, which, atleast partly, can be ascribed to the fact that the numberof the largest barriers, which are associated to the frac-tion of J+, keep decreasing with increasing a. A power-lawfor L(t) in this PM region, as opposed to the logarith-mic one in the FM region, can perhaps be determined bythe spin-glass structure, which has many quasi-equivalentlow-energy states that can speed up of the evolution fromlogarithmic to algebraic.

3.2.3 Antiferromagnetic region a ≥ aaIn this region with AFM order we expect a situationmirroring the one discussed in the FM region, with theobvious correspondences a = 0 ↔ a = 1, and a = af ↔a = aa. This picture agrees with what is observed in oursimulation. Considering the data in Figure 6a, one clearlyobserves the non-monotonic behavior of growth in thesame way as observed in the FM phase. Upon decreasinga from the pure antiferromagnet value a = 1, the growthkinetics quickly becomes much slower in going to a = 0.97,and then increases again until the upper limit of the PMphase is achieved at a = aa & 0.95. The evolution of thestaggered spin σi is visualized in Figure 9.

4 Aging phenomena

To study the aging phenomena of a system, it is usefulto compute the two-time quantities, namely, the autocor-relation and the response function. The autocorrelation

function is defined as

C(t, tw) =1

N

N∑i=1

[〈si(t)si(tw)〉 − 〈si(t)〉〈si(tw)〉], (9)

where tw and t are called waiting and observation time,respectively, with t ≥ tw. 〈. . . 〉 denotes average over ini-tial conditions (i.e, thermal average) as well as overthe realizations of the quenched disorder. The impulsiveauto-response function is defined as

R(t, tw) =1

N

N∑i=1

δ〈si(t)〉hδhi(tw)

∣∣∣∣h=0

, (10)

where hi(t) is a time-dependent magnetic field and 〈. . . 〉hmeans an average in the presence of such field. This quan-tity describe the (spatially average) linear response of aspin si at time t > tw when a sufficiently weak magneticfield hi(tw) is applied at the same site i at a previoustime tw. Since this quantity is very noisy one usually mea-sures the so called integrated auto-response function, alsodefined as the zero-field cooled susceptibility

χ(t, tw) =

∫ t

tw

dt′R(t, t′). (11)

This quantity has an enhanced signal/noise ratio and istherefore more suitable to numerical investigations [84].Let us now discuss the known scaling behavior of thesetwo time quantities C(t, tw) and R(t, tw) (or χ(t, tw)) indifferent systems.

4.1 Scaling behaviors of two-time quantities

Non-disordered systems: for systems without quencheddisorder, the scaling behavior of the two-time quantitiesdepends on the kind of quench. More precisely, as dis-cussed in reference [84], one has three different behaviorscorresponding to (i) a sub-critical quench to Tf < Tc, (ii)a critical quench to Tf = Tc > 0 for d > dL (lower criticaldimension) and (iii) a critical quench to Tf = Tc = 0 ford = dL. We discuss them separately below.

4.1.1 Sub-critical quench to Tf < Tc (d > dL)

In this case, for large tw, C(t, tw) and R(t, tw) arerepresented by the sum of two contributions [85],given as

C(t, tw) = Ceq(t− tw) + Cag(t, tw), (12)

R(t, tw) = Req(t− tw) +Rag(t, tw), (13)

where Ceq and Cag (and similarly forR) are an equilibriumand an aging term, respectively. The first term describethe equilibrium (or stationary) contribution formed in theinterior of domains which reach equilibrium very fast. Thesecond contribution is the remaining out-of-equilibrium

Page 8 of 15 Eur. Phys. J. B (2020) 93: 88

(or aging) contribution due to the interfaces which arevery slow in evolution. The equilibrium part is time-translational invariant (TTI), hence it depend only onthe time difference t − tw, and obeys the fluctuation-dissipation theorem

TReq(t− tw) = −dCeq(t− tw)/dt. (14)

The aging parts obey the scaling forms

Cag(t, tw) = C(L(t)

L(tw)

), (15)

and

Rag(t, tw) = L(tw)−(z+α)R(L(t)

L(tw)

), (16)

where C and R are scaling functions and α is an agingexponent [13,14,65,70,84–97].

Similarly, from equations (13,16), the integrated auto-response can also be written in an additive structure,χ(t, tw) = χeq(t− tw) + χag(t, tw), with

χag(t, tw) = L(tw)−αF(L(t)

L(tw)

). (17)

Additionally, Ceq and Cag in equation (12) have somelimiting properties given as follows. For short-time dif-ferences t − tw, Cag

∼= C(1) = qEA, where qEA is the socalled Edwards-Anderson order parameter, which for aferromagnet is simply the squared spontaneous magne-tization, qEA = m2. Instead, in the large time-differenceregime, namely with t − tw → ∞ with fixed L(t)/L(tw),Ceq(t − tw) = 0 and only the aging part contributesto C(t, tw). Concerning the scaling function C(y), it isexpected to behave as

C(y) ∼ y−λC (18)

for y � 1 [98], where λC is an autocorrelation exponent.On the other hand, for the response function, Req obeys

the FDT in equation (14) and vanishes in the large time-difference regime while, conversely, Rag vanish in theshort time-difference regime. The above features affect thebehavior of the integrated autoresponse which in the agingregime at the late times, exhibits a power-law decay of thescaling function F(y) in equation (17) as

F(y) ∼ y−λχ (19)

at large y, with the exponent λχ expected to be the sameas the exponent α in the FM phase (a < af ' 0.4) [46,90].

Let us mention that the above features are independentof Tf and hence apply down to Tf = 0, as the temperatureis an irrelevant parameter in the renormalization groupsense [99,100].

4.1.2 Critical quench to Tf = Tc > 0 (d > dL)

In this case the equilibrium and aging parts of bothC(t, tw) and R(t, tw) take the multiplicative forms [63,68,101]

C(t, tw) = Ceq(t− tw)Cag(t, tw), (20)

R(t, tw) = Req(t− tw)Rag(t, tw), (21)

where

Cag(t, tw) = C

(L(t)

L(tw)

), (22)

Rag(t, tw) = R

(L(t)

L(tw)

), (23)

with C(x) and R(x) scaling functions (different from theones of Eqs. (15) and (16)), whereas the equilibriumcontributions obey

Ceq(t− tw) = (t− tw + t0)−B (24)

and

Req(t− tw) = (t− tw + t0)−(1+A), (25)

where t0 is a microscopic time. The FDT theorem inequation (14) implies that A = B, which are furtherrelated as A = B = (d − 2 + η)/zc to the static criticalexponent η and to the dynamic critical exponent zc.

Notice that, from equations (21) and (23), unlike theprevious case of sub-critical quenches, here it is notpossible to decompose the integrated response χ intoits equilibrium and aging parts. However, it has beenshown [102] that the quantity 1 − Tfχ(t, tw), which rep-resents the distance from the equilibrium static value,scales as

1− Tfχ(t, tw) = L(tw)−zBG(L(t)

L(tw)

), (26)

where G is a scaling function, and the exponent zB isa scaling exponent. Moreover, in the short time differ-ence regime, namely letting tw become large while keepingt − tw fixed, from equations (20) and (21) one getsC(t, tw) ∝ Ceq(t− tw), and R(t, tw) ∝ Req(t− tw).

4.1.3 Quenches to Tf = Tc = 0 (d = dL)

At dL, Tc = 0 and hence Tf = 0 can be viewed also asa limiting case of a critical quench. This raises a ques-tion about which structure (additive or multiplicative) oftwo-time quantities applies in this case. In other words,whether we consider this as the limit for d → dL of aquench at the critical temperature (hence the multiplica-tive structure would apply), or the limit for d → dL of a

Eur. Phys. J. B (2020) 93: 88 Page 9 of 15

sub-critical quench at Tf = 0 (hence the additive decom-position would hold). The correct picture is the latter,since an equilibrium system without quenched disorderis perfectly ordered at Tf = 0, and therefore the scalingstructure of two-time quantities must be akin to one ofthe sub-critical quenches. In addition, since the equilib-rium state at T = 0 has no dynamics, then Ceq(t− tw) ≡ 0and C(t, tw) = Cag(t, tw). The same property is shared byχeq in scalar systems with a discrete (up-down) symme-try, while in vectorial systems with continuous symmetryχeq 6= 0 due to the presence of Goldstone modes. The dis-tinguishing feature of the quench at Tf = 0 with d = dL(with respect to those at Tf = 0 with d > dL) is the valueof the response function exponent, which turns out to beα = 0 in this case [15,84,85,91,103–108].

4.1.4 Disordered systems

In disordered systems, it is expected that the presence ofquenched disordered can modify the scaling behavior withrespect to what is known for the non-disordered case. Inspin-glass models with p-spins [109] or mean-field interac-tions [61,110,111], when quenches to a phase with qEA > 0is made, symmetry breaking occurs and, as expected, onehas an additive structure but with a response functionexponent α = 0, in contrast to α 6= 0 found in the caseof quenches below the critical point in the absence ofdisorder.

4.2 Numerical results

In the following, we present numerical simulations forthe scaling behavior of χ(t, tw) and C(t, tw) in the dif-ferent phases of the present model and analyze theirscaling behavior. The numerical results are presented fora temperature quench to Tf = 0.75. To compute χ numer-ically without applying the small perturbation we use thegeneralization of the fluctuation-dissipation theorem tonon-equilibrium states derived in references [65,67,71–75].To determine the equilibrium parts of two-time quanti-ties Ceq and χeq, we use the ground state of system atT = 0 as an initial condition, and equilibrate it at thefinal temperature of quench Tf = 0.75, and then computethe above mentioned two-time quantities. In studying thescaling behavior of two-time quantities we will also deter-mine the value of the associated exponents, introducedin Section 4.1, whose estimated values are collected inTable 1.

4.2.1 Quenches with a < af

In this FM phase symmetry breaking takes place andhence an additive structure is expected for the two-time quantities. This means that for the autocorrelation,according to equations (12) and (15), the plot of C(t, tw)−Ceq(t − tw) against L(t)/L(tw) for any given value ofa < af , should show a data collapse. Figure 10 is the plotwhich confirms this structure. Here, in panels (a) and(b), we plot C(t, tw) versus the time difference (t − tw)at a = 0.1 and a = 0.3, respectively, for different valuesof tw (see the figure caption). In these two panels, it isclear that the longer the waiting time tw, the later C(t, tw)

decays, which is the hallmark of aging. In the main frameof panel (c), we plot the aging quantity Cag(t, tw) ≡C(t, tw)−Ceq(t− tw) against L(t)/L(tw)− 1 which showsa very neat data collapse both for a = 0.1 and for a = 0.3.We point out that a good data collapse is also obtainedwhen Ceq is not subtracted as in this case Ceq(t− tw) ≈ 0.Notice that we have used L(t)/L(tw) − 1 on the x-axisin order to better show the small time-difference regime.Thus this plot proves quite convincingly that the scalingstructure described in Section 4.1.1 for a non-disorderedcase also applies to the present disordered case. How-ever, the scaling function C in equation (15) changes withvarying a. This is a clear indication of a violation of theso-called superuniversality hypothesis [4,5,30], which basi-cally amounts to the fact that scaling functions should beindependent on the presence/amount of disorder, in thepresent system.

In the inset of Figure 10c we plot Cag(t, tw) againstL(t)/L(tw) (on a double-log scale) to show the scalingfunction C(y) which is expected to decay as C(y) ∼ y−λC

for y � 1 (Eq. (18)). More precisely, we consider differenta = 0, 0.1, 0.3, but at a fix tw = 1 in order to reach theasymptotic regime of power-law decay. For a = 0, λC '1.25, as expected for the pure Ising case in two-dimension[4,5,112–114]. For a = 0.1, λC ' 1.07, and for a = 0.3,λC ' 1. Clearly, the value of λC is consistent with theYeung-Rao-Desai inequality λC ≥ d/2 [115].

Let us now analyze the response function. In panel (a)of Figure 11, we plot χ(t, tw) versus the time difference(t− tw) at a = 0.3 for different values of tw (see the figurecaption). In panel (b), we show a plot of L(tw)α[χ(t, tw)−χeq(t− tw)] against L(t)/L(tw)− 1, where the aging expo-nent α > 0 is determined from the data collapse. Noticethat the data collapses only in the region of large timeseparation where the system shows aging behavior. Thegood collapse shows that the additive combination ofequation (17) also holds here. Here also, we stress thatthe scaling function F depends quite strongly on a, a factthat invalidates superuniversality, as already noticed forthe autocorrelation function.

To explore the behavior of α as function of a we examinethe autoresponse integrated function in the aging regimeat the late times, where we expect the power-law decay[Eq. (19)] at large y. In the inset of the lower panelof Figure 11c, χag(t, tw) is plotted against L(t)/L(tw)(on a double-log scale) for different a = 0, 0.1, 0.3 at afix tw = 1 in order to reach the asymptotic regime ofpower-law decay. The dashed lines show the behaviorχag(t) ∼ L(t)−λχ for L(t) � 1. The estimated values areλχ ' 0.625, 0.6, 0.17 for a = 0, 0.1, 0.3, respectively, whichturn out to be comparable with the aging exponent α, asexpected. For a = 0, λχ = λC/z, as expected for the pureIsing case in the two dimensions [1,116].

4.2.2 Quenches with a = af

A quench is made at Tf = 0 with a = af is a case ofa critical quench and hence one expects a multiplicativescaling structure as the one discussed in Section 4.1.2.This is further due to the fact that the equilibrium mag-netization vanishes at af (Fig. 3) and hence qEA = 0 in

Page 10 of 15 Eur. Phys. J. B (2020) 93: 88

Fig. 10. (a) Plot of C(t, tw) versus time difference (t − tw)at a = 0.1 for different values of tw, drawn with differentcolors (see key). (b) As in panel (a) but for a = 0.3. (c)In the main frame, C(t, tw) − Ceq(t − tw) is plotted againstL(t)/L(tw) − 1 (on a log-log scale) for a = 0.1 (upper set ofcurves), and a = 0.3 (lower set of curves). Data for the samevalue of tw and for different a are plotted with the same col-ors. The solid back curve is the scaling function C of thepure case with a = 0. The inset in panel (c) is the plot ofCag(t, tw = 1) ≡ C(t, 1) − Ceq(t − 1) vs. L(t)/L(tw) for dif-ferent a = 0, 0.1, 0.3. The dashed lines indicate the power-lawdecay at large t, Cag(t) ∼ L(t)−λC , with λC ' 1.25, 1.07, 1 fora = 0, 0.1, 0.3, respectively.

this case. This same structure of the two-time quantitiesis expected to be observed also in our numerical quenchesto finite temperatures, provided they are sufficiently deepin temperature and limited in time in order not to exceedthe long-lasting aging stage. We already know that this isthe case with our choice of Tf as the evolution of L(t) in

Fig. 11. (a) Plot of χ(t, tw) versus time difference (t− tw) ata = 0.3 for different values of tw, drawn with different colors(see key). (b) In the main frame, L(tw)α[χ(t, tw)−χeq(t− tw)]is plotted against L(t)/L(tw) − 1 for a = 0.3. Curves for dif-ferent values of tw are drawn with different colors (see keyin the upper panel). The values of α for different a < af '0.4 are α = 0.625, 0.6, 0.4, 0.2 for a = 0, 0.1, 0.2, 0.3, respec-tively. The inset is the plot of the aging quantity χag(t, tw) ≡χ(t, tw)−χeq(t− tw) vs. L(t)/L(tw) at tw = 1 and for differenta = 0, 0.1, 0.3. The dashed lines indicate the power-law decayof χag(t) ∼ L(t)−λχ at large t, with λχ ' 0.625, 0.62, 0.17 fora = 0, 0.1, 0.3, respectively. The exponent λχ is of the sameorder as the exponent α, as expected.

Figure 6b shows no signs of convergence to an equilibriumvalue.

Let us now focus on the numerical data for this case.The main panel of Figure 12a is the unscaled plot ofC(t, tw) vs. the time-delay t − tw, for different values oftw, at a = 0.4 (see the figure caption). Scaling this dataaccording to equations (20) and (22) would require usto plot C(t, tw)/Ceq(t − tw) versus L(t)/L(tw). That iswhat we present in the main panel of Figure 12b, where anice data collapse of the curves at different tw can clearlybe seen. This is a clearcut confirmation of the multi-plicative structure in the quench of the present model ata = 0.4(' af ). The inset of Figure 12a shows the behaviorof Ceq(t− tw) on a double logarithmic scale. From equa-tion (24), by neglecting t0 for large t − tw, this quantityshould obey Ceq(t − tw) ' (t − tw)−B . This is shown bya dashed green line. Notice that Ceq decays with a verysmall exponent B ' 0.005.

Eur. Phys. J. B (2020) 93: 88 Page 11 of 15

Fig. 12. (a) In the main frame, C(t, tw) is plotted againstthe time delay (t − tw) at a = 0.4 for different values of tw,drawn with different colors (see key in lower panel). In the insetthe equilibrium correlation Ceq(t− tw) is plotted (on a log-logscale) against t − tw. The green dashed line is the algebraicform (t − tw)−0.005. (b) C(t, tw)/Ceq(t − tw) is plotted (on alog-log scale) against L(t)/L(tw) − 1 for a = 0.4. Curves fordifferent values of tw are drawn with different colors, see key.The inset of this panel is the plot of C(t, tw) vs. L(t)/L(tw) attw = 1 for a = 0.4 and the dashed lines indicate the asymptoticbehavior C(t) ∼ L(t)−λC for t� 1 with λC ' 0.67.

Coming to the asymptotic behavior of C(t, tw), fromequations (20), (22) and (24) it is expected

C(t, tw) = t−Bw (t/tw − 1)−BC

(L(t)

L(tw)

), (27)

i.e, C(t, tw) = t−Bw C(y), with C(y) ∼ y−λC for y � 1. Inorder to reliably measure λC , we look at C(t, tw = 1),namely we focus on the shortest possible waiting time.The inset of Figure 12b is the plot of this quantity vs.L(t)/L(tw) for a = 0.4. The dashed lines indicate the larget behavior C(t) ∼ L(t)−λC , with λC ' 0.67.

Next, we look at χ(t, tw), which is plotted for a = 0.4 inFigure 13a as a function of the time delay t− tw, for dif-ferent values of tw (see the figure caption). The scaling ofχ(t, tw) is expected to be of the form of equation (26). Theexponent B appearing in this scaling relation is alreadydetermined in the inset of Figure 12a, where Ceq decays

Fig. 13. (a) Plot of χ(t, tw) versus time difference (t− tw) ata = 0.4 for different values of tw, drawn with different colors(see key). (b) In the main frame, L(tw)zB [1 − Tfχ(t, tw)] isplotted against L(t)/L(tw) − 1 with zB = 0.025 for a = 0.4.Curves for different values of tw are drawn with different colors(see key in the upper panel). The inset shows the plot of 1 −Tfχ(t, tw) against L(t)/L(tw) (on a double-log scale) at a fixtw = 1 for a = 0.4 and a dashed line represent the large tbehavior 1− Tfχ(t) ∼ L(t)−λχ with λχ ' 0.013.

with a very small exponent B ' 0.005 for sufficiently large(t − tw) as shown by a dashed green line. Now recallFigure 6b, where the asymptotic growth exponent z ' 5.2for a = 0.4 (shown by a maroon dashed line). Usingthese exponents z and B, we plot L(tw)zB [1− Tfχ(t, tw)]against L(t)/L(tw)− 1 in Figure 13b. According to equa-tion (26), we should observe data collapse of this data forvalues of tw. This is indeed what one sees, which confirmsthe multiplicative scaling also for χ(t, tw) in this case.

Finally, considering the large y behavior, the responsescaling function G(y) in equation (26) should scale asymp-totically as y−λχ . As already discussed for the autocorre-lation function, in order to enter the asymptotic regime ofthe power-law decay, we look at the data for tw = 1, andplot 1−Tfχ(t, 1) against L(t)/L(tw) on a double-log scalefor a = 0.4 in the inset of Figure 13b. The dashed line isthe asymptotic behavior 1− Tfχ(t, 1) ∼ L(t)−λχ with theexponent λχ ' 0.013.

4.2.3 Quenches with a > af

As mentioned in Section 3.2.2, an algebraic behavior ofL(t) is not only observed at the critical point (a = af ),

Page 12 of 15 Eur. Phys. J. B (2020) 93: 88

Fig. 14. (a) In the main frame, C(t, tw) is plotted againstthe time delay (t − tw) at a = 0.7 for different values of tw,drawn with different colors and symbols (see key in lowerpanel). In the inset we have plotted the equilibrium cor-relation Ceq(t − tw) (on a log-log scale) for different a =0.5, 0.6, 0.7 (see key in the inset). The dashed lines are thepower-law fits of (t− tw)−B for determining the exponent B:B = 0.016, 0.05, 0.12 for a = 0.5, 0.6, 0.7, respectively. (b) Inthe main frame, C(t, tw)/Ceq(t − tw) is plotted (on a log-logscale) against L(t)/L(tw)− 1 for a = 0.5, 0.6, and 0.7. Curvesfor different values of tw are drawn with different colors (seekey), and for same tw but for different a’s are plotted withthe same colors and same symbols. In the inset of this panel,C(t, tw) vs. L(t)/L(tw) is plotted at a fix tw = 1 and fora = 0.5, 0.6, 0.7 (see key in the inset). The dashed lines arethe asymptotic behavior C(t) ∼ L(t)−λC for t� 1 with λC '0.66, 0.67, 0.67 for a = 0.5, 0.6, 0.7, respectively.

but also in the whole region af < a < aa, presumablydue to the proximity of the spin-glass phase at T = 0.Therefore, also in the case of quench with a > af , the two-time quantities are expected to show scaling. However, itis not obvious which scaling structure should apply to thiscase. On the one hand, the system could feel the criticalpoint at (a = af , T = 0), in that case one would expectbasically the same multiplicative structure observed in thequench at a = af and with the same exponents. On theother hand, the system can be influenced by the spin-glassphase with qEA > 0, by virtue of which one could guessan additive structure.

From our numerical data, we found that for all a >af , C(t, tw) < Ceq(t − tw) for any t > tw which implies

Fig. 15. (a) Plot of χ(t, tw) versus time difference (t − tw)for a = 0.7 and different values of tw, drawn with different col-ors (see key). (b) In the main frame, L(tw)zB [1− Tfχ(t, tw)] isplotted against L(t)/L(tw)− 1 with the exponent zB = 0.067,0.15, 0.3 for a = 0.5, 0.6, and 0.7, respectively. Curves for differ-ent values of tw are drawn with different colors (see key). In theinset we plot 1 − Tfχ(t, tw) against L(t)/L(tw) (on a double-log scale), for tw = 1 and a = 0.5, 0.6, 0.7 (see key in the inset).The dashed lines are the large t behavior 1−Tfχ(t) ∼ L(t)−λχ

with λχ ' 0.053, 0.13, 0.16 for a = 0.5, 0.6, 0.7, respectively.

Cag = C − Ceq < 0, a fact that rules out the additivescheme, as Cag < 0 is unphysical. Indeed, form our simula-tion data, a multiplicative structure is very well verified aswe found the data collapse of C(t, tw)/Ceq(t− tw) for var-ious values of tw. This data is shown in the main frame ofFigure 14b for different values of a = 0.5, 0.6, and 0.7. Anexcellent data collapse can be clearly seen for all a > af .The upper panel Figure 14a is the unscaled plot of C(t, tw)against the time difference (t− tw), for a = 0.7. In an insetof this figure, we show Ceq(t− tw) on a double logarithmicscale for a = 0.5, 0.6, 0.7. From this data, we determine theexponent B from the power-law fit of Ceq ∼ (t− tw)−B forlarge t − tw as shown by the dashed lines. The exponentB comes out as B = 0.016, 0.05, 0.12 for a = 0.5, 0.6, 0.7,respectively, which will be needed to scale χ(t, tw). Next,having understood how to extract the autocorrelationexponent λC , we plot C(t, 1) vs L(t)/L(tw) (on a double-log scale) in the inset of Figure 14b, for a = 0.5, 0.6, 0.7.The dashed lines give the asymptotic behavior L(t)−λC

Eur. Phys. J. B (2020) 93: 88 Page 13 of 15

Table 1. Values of the autocorrelation exponent λC , theauto-response exponent λχ, and the scaling exponentsα, zB for all considered values of a at the temperaturequench Tf = 0.75.

a α zB λC λχ

0 0.625 – 1.25 0.6250.1 0.62 – 1.07 0.60.2 0.4 – 1.05 0.420.3 0.2 – 1.0 0.170.4 – 0.025 0.67 0.0130.5 – 0.067 0.66 0.0530.6 – 0.15 0.67 0.130.7 – 0.28 0.67 0.16

with λC ' 0.66, 0.67, 0.67 for a = 0.5, 0.6, 0.7, respec-tively. Notice that the exponent λC takes a value aroundλC ' 0.67 for all values of a in the paramagnetic region,suggesting a unique value.

Now we move to analyze the response function χ(t, tw),which is shown for a = 0.7 in Figure 15a. Recalling theexponent B determined in inset of Figure 14a and thedynamic exponent z from Figure 6b, we next plottedL(tw)zB [1−Tfχ(t, tw)] versus L(t)/L(tw)− 1 in the lowerpanel Figure 15b for a = 0.5, 0.6, 0.7 and for different tw-values (see the figure caption). Clearly, from this figure,our determined values of static and dynamic exponentsz and B produce a nice data collapse for all differenta = 0.5, 0.6, and 0.7. As before, given equation (26), thisconfirms the multiplicative structure also for this caseof quench with a > af . The inset of Figure 15b dis-plays the asymptotic behavior given by the power-lawdecay 1 − Tfχ(t, tw) ∼ L(t)−λχ , with the exponent λχ '0.053, 0.13, and 0.16 for a = 0.5, 0.6, and 0.7, respec-tively. Notice that in this case the exponent λχ 6= zB, asexpected, unlike in the previous case of a quench with a .af . Furthermore, notice that both the scaling functions

C(y) and G(y) for C(t, tw) and χ(t, tw), respectively, inFigures 14b and 15b are different for different a, whichagain consistently showing a violation of superuniversality.

Finally, we find it useful to list in a Table 1 the measuredvalues of all the exponents: the autocorrelation expo-nent λC , the auto-response exponent λχ, and the scalingexponents α and zB.

5 Summary and discussion

This paper was a review of our results on growth kineticsand aging in a two-dimensional frustrated system. After aquenching has been made to a low-temperature Tf , thesystem show different phases with varying the frustra-tion strength namely the fraction of negative couplingsa. These phases are ferromagnetic for 0 ≤ a < af , param-agnetic with a zero-temperature spin-glass phase for af ≤a < aa, and an antiferromagnetic for aa ≤ a < 1. We havefound that the system shows a nontrivial behavior of boththe growth and the aging properties for quenches operatedin different phases. Our major findings are summarized asfollows.

In the FM and AFM phases, the speed of growth variesin a non-monotonic way as the amount of disorder a isincreased. Specifically, speaking of the FM phase for defi-niteness, there exists a value a∗ ∼ 0.2 where the kinetics isslower than for any other value of a. This non-monotonoustype of the growth behavior can be related to the topologyof the bond network (see Sect. 2). This result is con-sistent with what has already been found in models ofdisordered magnets without frustration [6,7,46], and henceextends its validity to the realm of frustrated systems,pointing toward a general robustness. We showed that thegrowth law L(t) is of a logarithmic type in the whole FM(0 < a < af ) as well as AFM region aa < a < 1. Instead,in the PM region we find that L(t) grows algebraically inthe whole phase, irrespective of the fact that the geomet-rical properties of the bond network greatly change as ais varied in [af , aa].

In the whole ferromagnetic region, for 0 ≤ a ≤ af , weobserved that the equilibrium and the aging parts of thetwo time quantities (autocorrelation and response func-tion) combine in an additive way to form the completecorrelation and response , i.e., C(t, tw) = Ceq(t − tw) +Cag(t, tw) (and similarly for the response function). How-ever, the scaling functions for the aging parts of bothquantities strongly depend on the amount a of frustra-tion. This shows that the property of superuniversalityis not obeyed. Further, the aging exponent α depend ona, and vanishes in the limit a → a−f . This can be inter-preted due to the fact that, in this limit, the system hasfractal structure, and therefore interfaces are free to movewithout experiencing any restoring force. The same fact,indeed, can be considered the origin of the speeding up ofthe growth for a & af .

At a = af , Tc = 0, which is in fact a critical pointof the FM phase, the additive structure breaks down,and a multiplicative structure emerges where the equilib-rium and the aging parts of two-time quantities multiply,i.e., C = Ceq · Cag. This is consistent with other spin-glass systems [117,118], where a similar behavior has beenobserved. The same multiplicative structure is found evenwhen the system is quenched in the PM region witha > af . Further, the scaling functions of both C and χ,and the exponent zB turns out to be strongly dependenton the amount a. This is further showing a violation ofsuperuniversality.

We also discussed the large-t behavior where the scalingfunctions for both the quantities C(t, tw) and χ(t, tw) aregoverned by power-law decay with exponents λC and λχ.The values of these exponents are reported in Table 1. Itis worth mentioning that in the FM phase the values ofλC are in agreement with the Yeung-Rao-Desai inequalityλC ≥ d/2 [115], and approach the lower-bound d/2 as a→a−f . For the response function, λχ ' α is found to be wellobeyed.

We conclude this paper with a final remark that ourresults for the growth kinetics and the aging proper-ties clearly distinguish the different phases of a d = 2frustrated system. In this respect, this motivates a fur-ther interest in investigating the three-dimensional caseof the frustrated system where its phase-structure can be

Page 14 of 15 Eur. Phys. J. B (2020) 93: 88

elucidated from a detailed nonequilibrium study of thegrowth and the aging quantities.

MK would like to acknowledge the support of theRoyal Society – SERB Newton International fellowship(NIF\R1\180386). The numerical simulations were performedon High-performance computing clusters at SPS, JNU, NewDelhi, and at IUAC, New Delhi. SP is grateful to the Depart-ment of Science and Technology (India) for support through aJ.C. Bose fellowship.

Author contribution statement

FC, EL and SP conceived, designed and supervised theproject. MK developed the codes, obtained the data, andperformed the analysis. All the authors contributed tomanuscript preparation.

Publisher’s Note The EPJ Publishers remain neutral withregard to jurisdictional claims in published maps and institu-tional affiliations.

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