Dynamics of fluctuation correlation in periodically driven classical system
DDynamics of fluctuation correlation in periodically driven classical system
Aritra Kundu, ∗ Atanu Rajak, † and Tanay Nag ‡ SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy Presidency University, 86/1, College Street, Kolkata 700073, India Institute f¨ur Theorie der Statistischen Physik, RWTH Aachen University, 52056 Aachen, Germany (Dated: February 26, 2021)Having established the fact that interacting classical kicked rotor systems exhibit long-livedprethermal phase with quasi-conserved average Hamiltonian before entering into chaotic heatingregime, we use spatio-temporal fluctuation correlation of kinetic energy to probe the above dynamicphases. We remarkably find the diffusive transport of fluctuation in the prethermal regime remindingus the underlying hydrodynamic picture in a generalized Gibbs ensemble with a definite tempera-ture that depends on the driving parameter and initial conditions. On the other hand, the heatingregime, characterized by a diffusive growth of kinetic energy, can sharply localize the correlation atthe fluctuation center for all time. Consequently, we attribute non-diffusive and non-localize struc-ture of correlation to the crossover regime, connecting the prethermal phase to the heating phase,where the kinetic energy displays complicated growth structure. We understand these numericalfindings using the notion of relative phase matching where prethermal phase (heating regime) refersto an effectively coupled (isolated) nature of the rotors.
In recent years periodically driven isolated systemsemerge as an exciting field of research, giving justice tothe fact that driven systems exhibit intriguing proper-ties as compared to their equilibrium counterparts [1–3].The quantum systems are studied extensively in this con-text theoretically [4–6] as well as experimentally [7–11];for example, dynamical localization [12–14], many-bodylocalization [15–20], quantum phase transitions [21, 22],Floquet topological insulator [23–31], Floquet topologi-cal superconductor [32, 33], Floquet time crystals [34–37], higher harmonic generation [38–41] are remarkablenonequilibrium phenomena. Consequently the heatinghappens to be very crucial factor as far as the stability ofthe driven systems is concerned [42–44]. The consensusso far is that the driven quantum many-body systemsheat up to an infinite-temperature state [16, 45–47] withsome exceptions [48, 49]. However, it has been shownthat heating can be suppressed for integrable systems dueto infinite number of constants of motion, as manifestedthrough the non-equilibrium steady states [13, 50, 51].On the other hand, many-body localized systems pre-vent heating for their effective local integrals of motionin the presence of interaction and disorder [15–19]. Thehigh frequency driving is another alternative route to pro-hibit the heating in the long-lived prethermal region, thatgrows exponentially with frequency, before heating up atthe infinite temperature state [52–69].Interestingly, the quasistationary prethermal state isconcomitantly described by an effective static Hamilto-nian, obtained using the Floquet-Magnus expansion, inthe high-frequency regime [55, 57, 58, 60–64]. Here arisesa very relevant question whether the classical systems ex-hibit such interesting intermediate prethermal plateau. ∗ [email protected] † [email protected] ‡ [email protected] Notably, classical kicked rotors [65, 68] and periodicallydriven classical spin chains [66, 67] demonstrate Floquetprethermalization while the static systems do not neces-sarily support a bounded spectrum. Similar to the quan-tum case, Floquet-Magnus expansion leads to an effectivestatic classical Hamiltonian describing the prethermalphase where heating is exponentially suppressed. Theprethermal phase is further characterised by generalizedGibbs ensemble (GGE) causing hydrodynamic behaviorto emerge in the above phase [70–72].The framework of fluctuating hydrodynamics becomesa convenient tool to investigate the equilibrium trans-port in classical non-linear systems [73–81]. The inte-grable (non-integrable) classical systems typically admitballistic (non-ballistic) transport [76, 82–85]. The theoryof fluctuating hydrodynamic is also employed to under-stand the transport in non-linear Fermi-Pasta-Ulam likesystems [74, 82]. Given the above background, we wouldlike to investigate the non-equilibrium dynamics of fluc-tuation correlation of the kinetic energy in classical in-teracting kicked rotor model as a probe to the hydrody-namic behavior of the problem. The motivation behindchoosing such model is that in the limit of infinite num-ber of particles, a Bose-Hubbard model can be mappedto the above model [68]. More importantly, kicked ro-tor systems can be realized in experiments using Joseph-son junctions with Bose-Einstein condensates [86]. Thetime-periodic delta function kicks can be implementedby varying the potential depth and width controlling theintensity of laser light [87, 88]. The main questions thatwe pose in this work are as follows: How does a typicalfluctuation behave in quasi-stationary prethermal states,as well as in the regime where kinetic energy grows inan unbounded manner [68]? Provided the notion of theGGE in the dynamic prethermal regime, does diffusivetransport as seen for the case of static Hamiltonian [81]persist?In this work, while numerically investigating the prop- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b agation of fluctuation through the system as a function oftime, we show that the time-dynamics of the kinetic en-ergy of the system can be divided in three different tem-poral regimes. There fluctuation correlation (3) exhibitsdistinct space-time behavior (see Fig. 1). Following theinitial transient, the system enters into the prethermalregime, characterized by almost constant kinetic energywith exponentially suppressed heating, where the fluctu-ation spreads over space diffusively as a function of time(see Fig. 2). The spatial correlation becomes Gaussianwhose variance W increases linearly with time. Oncethe system starts absorbing energy from the drive, thefluctuation becomes exponentially localized at the site ofdisturbance and temporally frozen referring to the con-stant nature of W with time (see Fig. 3). We refer thisintermediate window as a crossover region that connectsthe spatially and temporally quasi-localized behavior ofcorrelations at long time with the prethermal phase (seeFig. 4). In that quasi-localized phase, W decays to van-ishingly small values while the kinetic energy of the sys-tem grows linearly with time. Therefore, the kinetic en-ergy localization (diffusion) corresponds to the diffusion(localization) of fluctuation correlation. We qualitativelyunderstand the underlying energy absorption mechanismbased on the hydrodynamic description. This study onfluctuation correlation of kinetic energy, revealing newdynamic phases e.g., localized phase and crossover regimethat do not have any static analogue, is completely newto the best of our knowledge.We consider periodically kicked classical rotor system,being relevant in the context of many-body chaos theory,is given by [89–95]. H = N (cid:88) j =1 (cid:34) p j − κ + ∞ (cid:88) n = −∞ δ ( t − nτ ) V ( r j ) (cid:35) , (1)where stretched variable r j = φ j − φ j +1 and V ( r j ) =1 + cos( r j ). Here φ j , j = 1 , · · · , N , are the angles of therotors and p j are the corresponding angular momenta.The parameter κ denotes the interaction as well as kickstrength, and τ is the time period of delta kicks. Thesystem described in Eq. (1) can have infinite energy den-sity due to the unbounded nature of kinetic energy. Wenote that the total angular momentum of the static sys-tem [i.e., temporally uniform profile of the potential part]is an exact constant of motion, since is invariant underglobal translation φ j → φ j + α for any general α . Usingclassical Hamilton’s equations of motion, one can get thediscrete maps of φ j and p j between n -th and ( n + 1)-thkicks: p j ( n + 1) = p j ( n ) + κ ( V (cid:48) ( r j − ) − V (cid:48) ( r j )) φ j ( n + 1) = φ j ( n ) + p j ( n + 1) τ. (2)Here V (cid:48) describes derivative of V with respect to r j eval-uated after n -th kick. We consider periodic boundaryconditions φ N + i = φ i . From Eq. (2), it can be noticedthat the dynamics of the system is determined by only Di ff usive fluctuation (prethermal) Exponentially localised (crossover) log n l og W K i ne t i c E ne r g y Localised (heating) K
0, with n → ∞ , red solid line). Inter-estingly, the lifetime of prethermal (heating) regime decrease(increases) with increasing driving parameter K > K . Thedifferent regimes of W are connected by black (grey) dottedline for K ( K ). one dimenisonless parameter, K = κτ that we use for allour further calculations [65, 68].We compute here the spatio-temporal correlation ofkinetic energy fluctuations, defined by C ( i, j, t, t w ) = 14 (cid:2) (cid:104) p i ( t ) p j ( t w ) (cid:105) − (cid:104) p i ( t ) (cid:105)(cid:104) p j ( t w ) (cid:105) (cid:3) , (3)where i and j respectively represent the position of i -thand j -th rotor; t ( t w ) represents an arbitrary final time(initial waiting time). The symbol (cid:104) .. (cid:105) denotes the av-erage over the initial conditions where φ j (0) are chosenfrom a uniform distribution between − π to π , and thecorresponding momenta, p j (0) = 0 for j = 1 , · · · , N .The spatio-temporal correlation captures how a typicalsmall perturbation spreads in space x ≡ i − j and time t ≡ t − t w through the system. For the static case how-ever, the notion of two different times is not requiredto compute the correlation. In Fig.1, we refer the rela-tive width of the spatio-temporal correlations as variance W = (cid:80) N/ x = − N/ x C ( x, t ) / [ (cid:80) N/ x = − N/ C ( x, t )].The observables are averaged over 10 initial condi-tions. Otherwise specifically mentioned in our simula-tions, we fix τ = 1 and N = 2048. This makes t = n ,and we use the term interchangeably. The lifetime of theprethermal state for such systems (see Eq. (1)) increasesexponentially in 1 /K [68]. For small values of K , theprethermal state persists for astronomically large time,thus making the numerical calculation extremely costlyto probe all the dynamical phases by varying time. Inorder to circumvent this problem, we choose to tune K such that the lifetime of the prethermal state can be sub-stantially minimized and we can investigate the phasewhere fluctuations get localized within our numerical fa-cilities. However, provided the distinct nature of theseregimes, our findings would remain unaltered if one ad-dresses them by varying time only. In our numerical cal-culations, we choose both t, t w within the same phase.To begin with, we first focus on the spreading of fluc-tuation (3) in the prethermal phase that is denoted bythe green solid lines in Fig. 1. The prethermal phasecan be described by a GGE with the total energy as aquasi-conserved quantity [68]. In terms of the inversefrequency Floquet-Magnus expansion, the lowest orderterm of the Floquet Hamiltonian is the average Hamilto-nian that governs the prethermal state at high frequency,given by H ∗ = 1 τ (cid:90) τ H ( t ) dt = N (cid:88) j =1 (cid:34) p j − κτ (1 + cos ( φ j − φ j +1 )) (cid:35) . (4)Employing the notion of GGE, the composite probabilitydistributions can be written as P ∗ ( { p j , r j } ) = 1 Z ∗ N (cid:89) j =1 e − ( p j T ∗ + κV ( rj ) τT ∗ ) , (5)with Z ∗ = e − H ∗ /T ∗ being the partition function asso-ciated with GGE and T ∗ is the temperature associatedwith prethermal phase. Given the particular choice ofthe initial conditions here, the prethermal temperatureis found to be T ∗ = 0 . Kτ [68]. Moreover, this de-scription of the GGE does not depend on the number ofrotors N , thus indicating the thermodynamic stability ofthis phase.We anchor the prethermal phase with the diffusivespatio-temporal growth of kinetic energy correlation asshown in Fig. 2 (a). A relevant renormalization of x and y -axes with time yields the following scalingform of the correlation: C ( x, t ) ∼ A K t − / f (cid:0) xt − / (cid:1) ,with f ( y ) = e − y / D / √ πD , as depicted in Fig. 2(b). We thus find that the correlation at differentspace time collapse together. Here, A K denotes theamplitude of the Gaussian distribution respectively fora given value of K . The diffusion constant D is ameasure of variance W , being weakly dependent onthe parameter K , grows linearly with time W ∼ n .One can observe that the fluctuation spreads in a waysuch that correlation curves keep their area constanti.e. the sum rule (cid:80) x C ( x, t ) is approximately con-stant in the prethermal phase. The sum rule deter-mines A K = (cid:80) N/ x = − N/ C ( x, t ) ≈ (cid:80) N/ x = − N/ C ( x,
0) =
FIG. 2. (a) The space-time spreading of kinetic energy corre-lation (3) for K = 0 .
14 in the prethermal regime. (b) The dif-fusive Gaussian scaling of correlation is observed with appro-priate renormalization: C ( x, t ) = A K t − / f ( xt − / ) , f ( y ) = e − y / D / √ πD with parameters D ∼ .
727 and A K ∼ . (cid:80) Nx =1 C ( x, t ) with K .FIG. 3. The space-time spreading of kinetic energy cor-relation (3) for K = 0 .
45 (a) and K = 0 . C ( x, t ) ∼ A K α − /β Γ (cid:16) β (cid:17) e − α | x | β with α ∼ .
36 [1 . , β ∼ .
63 [0 . , A K ∼ .
16 [41 .
51] for (a) [(b)]. δ x, (cid:0) (cid:104) p (cid:105) − (cid:104) p (cid:105) (cid:1) / . K /τ δ x, , by consideringthe fact that the energy absorption is exponentially sup-pressed in the prethermal regime [68]. This supports ournumerical result of quadratic growth of A K as shown inthe inset of Fig. 2 (b). The apparent mismatch in theprefactor of A K while computing numerically might bedue to the fact that exponentially slow variation of thekinetic energy in the prethermal phase is not taken intoaccount theoretically.The energy correlations of static rotor system in hightemperature platform exhibit diffusive transport [96, 97].This is in resemblance with the present case of Floquetprethermalization where kinetic and potential energy be-have in an identical fashion reminding the validity ofequipartition theorem in the GGE picture. As a result,the correlation of total energy qualitatively follows thecorrelation of kinetic energy in the prethermal regime.We now investigate the spatio-temporal evolution ofcorrelation (3) in the intermediate crossover regime, des-ignated by the blue solid line in Fig. 1, that lies be-tween the prethermal and the heating region of kineticenergy. The system starts to absorb energy from thedrive through many-body resonance channels causing thekinetic energy to grow in sub-diffusive followed by super-diffusive manner [65, 68]. However, the probability ofthe occurrence of such resonances decreases exponentiallywith 1 /K in the high frequency limit. We find that theoscillators are maximally correlated with each other at x = 0 and falls rapidly to zero in two sides x as shownin Fig. 3 (a) and (b), for K = 0 .
45 and 0 .
7, respectively.To be precise, correlation decays stretched exponentiallyin short distances: C ( x, t ) ∼ A K e − α | x | β while it falls ex-ponentially (i.e., more rapidly than stretched exponen-tial) in long distances: C ( x, t ) ∼ e − γ | x | . Here, β and γ weakly depend on K referring to the fact that drivingparameter K can in general control the spatial spreadof fluctuation. These profiles do not change with timereferring to the fact that variance of the spatial corre-lation distribution remains constant with time W ∼ n .This allows us to differentiate from the diffusive trans-port that occurs in the prethermal phase. However, withincreasing time, one can observe that long distance corre-lation becomes more noisy leaving the spatial structuresqualitatively unaltered. These noises might be originatedfrom the complex growth structure of the kinetic energy.At the end, we discuss the time zone where the aver-age kinetic energy shows unbounded chaotic diffusion, asdenoted by the red solid line in Fig. 1, resulting in the ef-fective temperature to increase linearly with time [65, 68].The correlation of the kinetic energy is fully localized inspace and temporally frozen as shown in Fig. 4). To beprecise, the correlation is nearly a δ -function centeredaround x = 0 i.e., fluctuation gets localized at the site ofdisturbance for all time. In this regime, the system showsfully chaotic behavior in the phase space and the angles ofthe rotors become statistically uncorrelated both in spaceand time. It is noteworthy that there is no descriptionof average Hamilton exist here as the inverse frequencyFloquet-Magnus expansion does not converge [66, 67]. Incontrast to the prethermal phase, the amplitude of corre-lation peak in the crossover and heating regime increasesas K η with η >
2. On the other hand, the variancein the heating regime becomes decreasing function of n ,precisely, W → n → ∞ that is markedly differentfrom the behavior of W in remaining two earlier regimes.Finally, we stress that our findings in this heating regimedo not suffer from finite size effect suggesting the ther-modynamic stability of this phase.This is clearly noted in our study that the spatio-temporal correlation provides a deep insight to charac-terize different dynamical phases. The phase matchingbetween adjacent rotors, captured by stretched variables r j ( n ) = φ j +1 ( n ) − φ j ( n ), plays very important role indetermining the nature of spreading of the fluctuations.The time-evolution of the stretched variable, followingthe equations of motion (2), is given as r j ( n + 1) = r j ( n ) + ( p j +1 ( n + 1) − p j ( n + 1)) τ . Upon satisfying theresonance condition p j − p j +1 = 2 πm/τ with m as aninteger number, the stretched variable rotates by 2 π an-gle between two subsequent kicks. When all the rotorsgo through these resonances, their relative phase match-ing is lost and the eventually the coupled rotor system
40 20 0 20 40 x C ( x , t ) t=256t=512t=1024 FIG. 4. The correlations (3) become fully localized such asnearly a δ -function in the heating region for K = 1 . turned into an array of uncoupled independent rotor. Atthis stage, the system absorbs energy from the drive at aconstant rate in an indefinite manner. This is preciselythe case for heating up regime where fluctuation corre-lations do not spread in time and space. On the otherhand, the resonances are considered to be extremely rareevents in the prethermal regime suggesting the fact thatrelative phase matching between adjacent rotors allowsthe fluctuation to propagate diffusively throughout thesystem in time. The notion of the time independent av-erage Hamilton in the prethermal region might be relatedto the fact that all the rotors rotate with a common col-lective phase and eventually controlled by an underlyingsynchronization phenomena [39, 98].Coming to the heuristic mathematical description, theexponential suppressed heating and the validity of theconstant sum-rule in the prethermal phase suggest a hy-drodynamic diffusion picture (with diffusion constant D )for the fluctuation [99–102]: ∂ t u ( x, t ) = ∂ x (cid:18) D ∂ x u ( x, t ) + B ζ ( x, t ) (cid:19) (6)where u ( x, t ) = (cid:0) p ( x, t ) − (cid:104) p ( x, t ) (cid:105) (cid:1) such that (cid:104) p ( x, t ) (cid:105) (cid:54) = (cid:104) p ( x, t (cid:48) ) (cid:105) . The conservative noise ζ ( x, t )of strength B is delta correlated in space and time (cid:104) ζ ( x, t ) ζ ( x (cid:48) , t (cid:48) ) (cid:105) = δ xx (cid:48) δ ( t − t (cid:48) ). In the high-frequencylimit, the equilibrium fluctuation dissipation relation canbe extended to long-lived prethermal regime: B ∼ DT ∗ . Given the plausible assumption that the noisepart B of the fluctuating current increases with increas-ing K , one can understand the diffusion process in aphenomenological way. The diffusion constant D is thenconsidered to be independent of K while prethermal tem-perature is determined by K , as observed in GGE pic-ture. Moreover, the self-similar Gaussian nature of fluc-tuation correlation in the prethermal phase can be un-derstood from the solution of Eq. 6 such that C ( x, t ) ∼ (cid:104) u (0 , τ i ) (cid:105) e − x / (2 Dt ) / √ πDt . In the other limit insidethe infinite temperature heating regime, the non-diffusivetransport leaves the u ( x, t ) to be δ correlated in spacewhile almost frozen in time. Now there is an extendedcrossover region, connecting the prethermal state to heat-ing regime, where the diffusion equation does not takesuch simple form causing the system to exhibit an amal-gamated behavior.In conclusion, our study demonstrates that the fluctu-ation correlation of kinetic energy can be scrutinized toprobe different dynamic phases of classical many-bodykicked rotor system [103]. It is indeed counter intuitivethat the average kinetic energy per rotor and their spatio-temporal correlations, being derived from the formerquantity, yet yield opposite behavior. The GGE descrip-tion of prethermal phase obeys diffusive transport wherespatio-temporal correlation follows self similar Gaussianprofile. During this diffusion, the correlation curveskeep their area constant due to the quasi-conservationof the kinetic energy. In the long time limit where ki- netic energy grows diffusively, fluctuation interestinglybecomes frozen in space and time. There exists an ex-tended crossover region where kinetic energy increases ina complicated way exhibiting both sub-diffusive to super-diffusive nature. The fluctuation interestingly shows arapidly (slowly) decaying short (long) range stretched(regular) exponential localization. In this case the fluctu-ation does not have any time dynamics. Therefore, corre-lated phenomena in prethermal phase gradually assem-bles to completely uncorrelated heat up phase througha crossover region. These non-trivial phases of matterare the consequences of the driving and do not haveany static analogue. Provided the understanding on longrange quantum systems [104–109], it would be interest-ing to study the fate of the above phases along withthe crossover region in long-range classical systems. Themicroscopic understanding of hydrodynamic picture andfluctuation-dissipation relation in dynamic systems areyet to be extensively analyzed in future. 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