Butterfly Effect and Spatial Structure of Information Spreading in a Chaotic Cellular Automaton
Shuwei Liu, J. Willsher, T. Bilitewski, Jinjie Li, A. Smith, K. Christensen, R. Moessner, J. Knolle
BButterfly Effect and Spatial Structure of Information Spreading in a Chaotic Cellular Automaton
Shuwei Liu,
1, 2
J. Willsher, T. Bilitewski, Jinjie Li, A. Smith, K. Christensen,
2, 6
R. Moessner, and J. Knolle
3, 7, 2 Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187-Dresden, Germany Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom Department of Physics TQM, Technische Universit¨at M¨unchen, James-Franck-Straße 1, D-85748 Garching, Germany Center for Theory of Quantum Matter, University of Colorado, Boulder, CO, 80309, USA School of Physics and Astronomy, University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom Centre for Complexity Science, Imperial College London, London SW7 2AZ, United Kingdom Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of localinformation spreading for a stochastic Cellular Automaton in the form of a spatiotemporally resolved Hammingdistance. This decorrelator is a classical version of an Out-of-Time-Order Correlator studied in the context ofquantum many-body systems. Focusing on the one-dimensional Kauffman Cellular Automaton, we extract thescaling form of our decorrelator with an associated butterfly velocity v b and a velocity-dependent Lyapunovexponent λ ( v ) . The existence of the latter is not a given in a discrete classical system. Second, we account forthe behaviour of the decorrelator in a framework based solely on the boundary of the information spreading,including an effective boundary random walk model yielding the full functional form of the decorrelator. Inparticular, we obtain analytic results for v b and the exponent β in the scaling ansatz λ ( v ) ∼ µ ( v − v b ) β , whichis usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifyingdiagnostic of information spreading. Introduction.
A central hallmark of chaotic systems [1] istheir sensitivity to perturbations: even a small change in ini-tial conditions leads to entirely unpredictable, and large, dif-ferences in the state of the system at later times. This is pop-ularly captured by the “butterfly effect” [2], which in many-body systems encodes two distinct notions: first, the exponen-tial growth of the perturbation giving rise to the notion of theLyapunov exponent, λ , characterising the growth with time [3]; and second, information spreading in space , whereby the“effect” of the butterfly’s wingbeat is felt at a distant locationonly with a time delay given by the “ballistic” propagationspeed known as butterfly velocity, v b [4, 5].A prominent recent strand of investigation of chaos in quan-tum many-body systems revolves around the study of in-formation scrambling, where Out-of-Time-Order Correlators(OTOCs) have been employed to measure the propagation ofquantum chaos [6]. OTOCs can be understood as two-timecorrelation functions where operators are not chronologicallyordered and are a simple measure of the “footprint” of an op-erator that spreads in space [7], thus, naturally measuring thespread of information. In chaotic systems, this quantity maygrow exponentially in time, governed by a Lyapunov expo-nent λ [8]. Recently, OTOCs have been studied extensivelyas an early-to-intermediate time diagnostic of quantum chaosor information spreading in a number of different quantummodels whose time evolution can be generated by differentdynamics such as Floquet dynamics [9, 10], random unitarycircuits [11–14], and time-independent Hamiltonians such asintegrable spin chains [15, 16], generalised SYK models [17],diffusive metals [18] and Luttinger-liquids [19].In that context, analogues of OTOCs have been recently de-veloped for classical systems [20–25]. For example, for spinchains the decorrelator D ( x, t ) = 1 − (cid:104) S A ( x, t ) · S B ( x, t ) (cid:105) between two copies of spin configurations S A/B , which at t = 0 only differ locally by a small spin rotation, is a semi-classical version of an OTOC [20]. For Heisenberg magnets,this exhibits ballistic propagation with a light-cone structuregoverned by a butterfly velocity even in the high-temperatureregime without long-range magnetic order but with spin dif-fusion [20].A common feature observed in classical and quantum mod-els is the exponential growth (or decay) of the OTOCs (ana-logues) along rays of constant velocity v = dxdt quantified byvelocity-dependent Lyanupov exponents (VDLEs) [8]. For in-termediate/late times it takes the scaling form D ( x, t ) ∝ e − µ ( v − v b ) β t = e − λ ( v ) t . (1)Intriguingly, a unifying framework of VDLEs with λ ( v ) = µ ( v − v b ) β captures the spatio-temporal structure of infor-mation spreading in many-body quantum, semiclassical andclassical chaotic systems [1, 3, 5, 20, 26].In this work, we provide a basic description of suchinformation-spreading in a minimal chaotic setting: theKauffmann Cellular Automaton (KCA). Our system choiceis motivated by the fact that KCA are minimal chaotic many-body models [27] which display a rich phenomenology, in-cluding a phase transition as a function of a tuning parameter,a probability p . They exhibit universal scaling, e.g. of thedirected percolation universality class [28], and lend them-selves to analytical insight [29]. Additionally, KCA have awide range of applicability [30–36]: initially introduced tostudy fitness landscapes of biological systems and gene ex-pression [37], they now appear also in optimisation problems[38], random mapping models [39], and most pertinently inthe emergence of chaos [40].Surprisingly, despite decades of research on chaotic CA,see Ref. [41] for a review, the dynamics of chaos has only a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n been investigated in terms of the global Hamming distancebut a local diagnostic has thus far been missing. Here, weconstruct an OTOC analogue for KCA and explore the VDLEphenomenology. This analogue enables us to uncover the bal-listic spatio-temporal structure of perturbation spreading inthe chaotic phase, Fig. 1(b), in contrast to the decay of such“damage” spreading in the frozen phase, Fig. 1(a). We de-velop a full microscopic theory of the VDLE, recovering thefunctional form Eq. (1) including an analytical calculation ofthe exponent β . Thus, we provide the tools for describingthe sensitivity of chaotic many-body systems to perturbationsthrough the framework of VDLEs. KCA Model and Classical OTOC Analogue.
We focus ona generic dissipative dynamical system called an
N K model.Concretely, a local KCA is a system of N Boolean elements σ ( x, t ) = ± which evolve in discrete time steps throughrules which depend upon K nearest neighbours of each sitein 1D [41]. Our KCA system evolves under a set of (annealed)local rules { f x,t } : σ ( x, t + 1) = f x,t [ σ ( x − K, t ) , ..., σ ( x, t ) , ..., σ ( x + K, t )] , (2)which are random with probability p in space and time: f x,t = (cid:40) +1 with probability p − with probability − p. (3)In a pioneering work, Derrida and Stauffer [42] showed thatKCA display a chaotic-to-frozen phase transition controlledby the parameter p , see Fig. 1 inset. The two phases are dis-tinguished by the decay or spread of localised perturbationsdiagnosed with the global Hamming distance , H ( t ) = 12 N (cid:42)(cid:88) x | σ A ( x, t ) − σ B ( x, t ) | (cid:43) p , (4)between two copies of the system σ A/B ( x, t ) which differby a single inverted site in the initial state at t = 0 . Thismeasure is then ensemble-averaged over realisations with thesame probability p . The distance grows linearly in the chaoticphase and decays to zero in the frozen phase, see Fig. 1.Analogous to the classical Heisenberg chain [20], we takethe classical OTOC as the local distance between two copiesof the same system which only differ by a local perturbationof the initial conditions, thus leading to the decorrelator D ( x, t ) = 12 (cid:2) − (cid:104) σ A ( x, t ) · σ B ( x, t ) (cid:105) p (cid:3) . (5)It is nothing but a local Hamming distance , which is relatedto the global distance by H ( t ) = N (cid:80) x D ( x, t ) . Numerical Results.
In Fig. 1 we show the spatio-temporalevolution of the decorrelator D ( x, t ) for representative valuesof p below and above p c . First, in the frozen phase D ( x, t ) initially spreads but then decays in time and space to zero.This attenuation of the local decorrelator reflects the vanishingof the long time value of the Hamming distance in the frozen FIG. 1. Main figure: lightcone structure of the decorrelator D ( x, t ) with N = 2048 and K = 4 , with a single spin flip at the origin x = 0 when t = 0 . (a) The frozen phase and (b) the chaotic phase. Inthe chaotic phase, the two dashed lines are v b and v max , respectively.Inset: mean-field prediction (dashed line) of the equilibrium value of D = D ( x = 0 , t → ∞ ) inside the lightcone as a function of p , compared with numerical data. Close agreement is observed for p > . . phase. Second, in the chaotic phase D ( x, t ) spreads with anapparent light-cone structure. Again, from the sum rule thelong-time value D = D ( x = 0 , t → ∞ ) within the light-cone can be identified as the order parameter associated withthe phase transition, see Fig. 1 inset.Because of the locality of KCA rules, the speed of the dam-age spreading measured with D ( x, t ) is necessarily boundedby the maximum velocity v max = K but the actual spread isslower, with the butterfly velocity v b < K . To demonstratethe presence of v b , we plot the behavior of the decorrelatoralong rays of constant velocity. In Fig. 2 we see that, after FIG. 2. The decorrelator D ( x, t ) along rays of different velocity for p = 0 . which is representative for all p > p c . As ln D = − λ ( v ) t = − µ ( v − v b ) β t the slopes represents − µ ( v − v b ) β . We compare thefull D ( x, t ) numerical data (solid lines) and the prediction (dashedlines) of the boundary random walk model. a transient effect, there is a ray (blue lines, corresponding to v = 2 . ) along which D ( x = v b t, t ) is constant for t > .This establishes the presence of a butterfly velocity v b < v max and is an efficient way of extracting v b as a function of p , seeblue dots in the inset of Fig. 4.Next, we investigate whether such behaviour follows thegeneral VDLE phenomenology of Eq. (1). We note that incontrast to previous work on Heisenberg spin chains, the pres-ence of Lyapunov exponents in KCA is far from obvious. Infact, for small distances one does not expect an exponentialpickup as a function of time because the local perturbationbetween the two copies σ A/B is necessarily big because ofthe discrete and bounded nature of the inverted site. However,we find that for distances x far away from the perturbation andafter disorder-averaging one may still observe an exponentialpickup in time of the OTOC analogue. Because of the discretenature of the variables the scaling behavior only appears in asmall window around v b as the decorrelator quickly saturatesto D = D for velocities v < v b and quickly decays to D = 0 for v > v b .Again, a clear picture emerges by studying D ( x, t ) along“rays” of constant velocity v = x/t . As shown in Fig. 2,we observe an exponential decay of the OTOC in time, witha VDLE λ ( v ) . For rays with v > v b , the linear decay of ln D ( x = vt, t ) = − λ ( v ) t in the long-time limit demon-strates that the decorrelator indeed decays exponentially witha VDLE λ ( v ) .Overall, the numerical results establish the presence of abutterfly velocity v B and validate the VDLE framework inKCA systems at long times. This is a surprising featuregiven the discrete, and large, nature of “perturbations” in theBoolean network, as such framework is usually observed forcontinuous and infinitesimal perturbations [8].To quantitatively analyse the small active region around the FIG. 3. Main figure: scaled probability density of boundary posi-tions (defined in the main text) of different initial configurationsplotted against time. The black dotted line is the Gaussian pro-file predicted by the random walk model. The earliest two times( t = 50 , ) show slight deviations from the equilibrium/long-time limit. Upper inset: unscaled boundary probability density at t = 50 , , ..., . Lower inset: Raw data of boundary positionsplotted against time. The red line is the average boundary, with thevelocity v = x/t = 2 . . wavefront set by v b where the damage actively spreads, wedefine for each sample the boundary of a spreading perturba-tion as the furthest point from the centre which differs fromthe unperturbed system. As shown in Fig. 3, the probabilitydensity of boundary positions P ( x ) approaches a Gaussianprofile in the long-time limit with a width σ x ∝ √ t . We findthat its mean is in quantitative agreement with the probability-dependent butterfly velocity v b ( p ) for all p , see inset of Fig. 4.The Gaussian behavior of the boundary, therefore, motivatesa random-walk-like description of the active region. Boundary Random Walk Model.
We can now develop a mi-croscopic statistical model of the boundary dynamics startingfrom the microscopic KCA rules. We only sketch the mainidea and relegate details to the Supplementary Material. Thebasic ingredient for our random-walk model is the expectationvalue of the outwards move of a damage site in each time-step:The furthest the boundary could move outwards in one time-step is K , with probability p ( K ) = p d = 2 p (1 − p ) ; and theprobability of moving x < K steps is p ( x ) = p K − xs p d , where p s = p +(1 − p ) . From this we obtain the mean and varianceof the boundary steps and the Central Limit Theorem (CLT)allows us to obtain the full distribution (valid in the long timeand p (cid:29) p c limit) as a Gaussian.This model correctly predicts various aspects of the decor-relator. First, we obtain the long-time value of D ( x, t ) insidethe light-cone as given by D = 2 p (1 − p ) which correspondsto the spins pointing up and down with random probability p and − p , see the dashed line in the inset of Fig.1(a).Second, the model predicts the butterfly velocity, given bythe following closed-form expression in p which can be read- FIG. 4. Main figure: scaling collapse performed for a range of p according to Eq. (8). At each specific p , the data is obtained by takingthe value of D in the t → ∞ limit as permitted by computationalpower. The x -axis is rescaled to √ µ ( v − v b ) . The black dashed lineis obtained by numerically evaluating D ( x, t ) from Eq. (15) in theAppendix for p = 0 . . Inset: comparison between v b obtained fromthe full D ( x, t ) data (blue dots), from the boundary velocity (greenline) and from analytic calculation (orange line). ily expanded into a power series v b ( p ) = K − p s p d = K − − ∞ (cid:88) j =1 j (cid:18) − p (cid:19) j . (6)It agrees with the full model’s butterfly velocity at large p away from p c as shown in the inset of Fig. 4.Third, the standard deviation of the boundary distribution is σ ( t ) = t (cid:18) p s p d + p s p d (cid:19) ≡ t µ ( p ) . (7)which confirms the Gaussian form of the boundary randomwalk with a variance that scales linearly with time. As de-tailed in the Supplementary Material, the cumulative distri-bution of the boundary then allows us to obtain the completefunctional form of the OTOC as a function of v , governed byits inverse width µ ( p ) . Figure.2 shows the predicted analyticform of the decorrelator (dashed lines) compared to the nu-merical data (solid lines). This quantitative agreement in thelong-time limit confirms that the boundary controls the dy-namics of chaos spreading of the full KCA model.Most notably, in the long-time limit the analytical modelrecovers the linear decay of the decorrelator in time, describedby a VDLE for v close to v b : ln D ( x = vt, t ) = − λ ( v ) t = − µ ( p )( v − v b ) t, (8)with an exponent β = 2 . Therefore, in this regime we ex-pect a data collapse around v b by plotting ln( D ) /t against (cid:112) µ ( p )( v − v b ) . Indeed, as shown in Fig. 4, the data of thesetwo variables for a range of probabilities fall onto the singlecurve of the analytical prediction (black dashed line). Discussion.
Given the discrete nature of the dynamics, thequantitative agreement between our model and the full numer-ical simulation is somewhat surprising but highlights the uni-versal features of chaos spreading. Crucially, our analysis isonly valid after sample averaging and in the long time limitwhere the effects of the discrete perturbation and dynamicshave been smoothed out. In particular, when approaching thecritical point p c from above we see systematic deviations dueto fluctuations.Our work on a minimal classical model is inspired by recentdevelopments in the study of chaos in quantum many-bodysystems where OTOCs have become a powerful quantitativetool. One important prediction in that context is that the but-terfly velocity is bounded at high temperatures v b ( T → ∞ ) ∼ v LR [43] where v LR is the Lieb-Robinson velocity which isthe upper limit of information propagation in short-range in-teracting non-relativistic quantum systems [26, 44]. For KCAone may connect the model parameter p to a temperature T via p = e − /T / ( e /T + e − /T ) [45]. Then from Eq.6 we canobtain the full temperature dependence of the butterfly veloc-ity, and the high temperature limit is v b ( T → ∞ ) ≈ K − − T . (9)It confirms that also in our classical many-body system themaximum velocity of information spreading v b ( T → ∞ ) = K − , is always less than the maximum v max = K allowedby the local dynamics. Conclusion and Outlook.
We have constructed a local diag-nostic of information spreading for a one-dimensional randomCA in analogy with recent semi-classical versions of OTOCs.We demonstrated that it displays ballistic propagation charac-terised by a butterfly velocity and exponential growth in timecaptured by a VDLE. We developed a random walk model ofthe boundary of information spreading which permits the cal-culation of the full functional form of the classical OTOCs,including the exponent β of the VDLE.An obvious extension of our work is to consider the 2-dimensional KCA where an even richer phenomenology isexpected. Even though our boundary random walk model pro-posed here is simple, it has many physical and mathematicalaspects that remain unexplored. For example, if the randomwalk is approximated to have continuous walk distance whichstill follows the same probability distribution, can it have aclosed form solution for early times without having to invokethe CLT? Alternatively, it would be worthwhile to explore aLangevin like description of the boundary which is particu-larly useful while considering KCA in higher dimensions. Forexample, in 2D KCA, we expect a diffusion-like process forthe perturbation which could potentially percolate across thelattice. In particular, it is interesting to investigate if this fallsinto any percolation universality class where established criti-cal exponents could be linked to the exponent of the classicalOTOCs.Generally we expect our decorrelator and theoretical toolsto be applicable to other stochastic models with discrete vari-ables, which are widely used to describe the dynamics of bothquantum and classical many-body systems [46]. We expectthat different variants of the boundary theory developed hereshould be able to predict the scaling forms in these systems aswell. In particular, it would be interesting to see them appliedto models with charge or dipole conservation rules [47, 48]where conjectured critical exponents could potentially be de-rived analytically. Acknowledgements.
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In this numerical analysis we perform a study of the av-erage velocity of the boundary paths, which are observed toobey a Gaussian distribution (in the long-time limit). Thecentral value of this distribution is in agreement with theprobability-dependent butterfly velocity v b ( p ) for all p > p c ,indicating that there is only one characteristic velocity whichgoverns chaos spreading. Moreover, the width of this Gaus-sian distribution spreads with √ t dependence, indicating itsrandom-walk-like nature. A data collapse of the distribu-tions in boundary position is provided in Fig. 3, showing goodagreement with a Gaussian profile. There is residual skewnessdue to the initial perturbation which vanishes in the long-timelimit.We are motivated by this result to provide a probabilis-tic model of the boundary that predicts the butterfly veloc-ity, and later we will use this model to recover the functionalform of the VDLE λ ( v ) . This model involves calculating theexpectation value of the furthest site outwards in each time-step. The furthest the boundary could move outwards in onetime-step is K , with probability p ( K ) = p d = 2 p (1 − p ) .The probability of moving x steps is p ( x ) = p K − xs p d , where p s = p + (1 − p ) .A quick verification of this model is that the long-time valueof D ( x, t ) insider the light-cone, is given by D = 2 p (1 − p ) which corresponds the spins pointing up and down withrandom probability p and − p .Now we will determine the scaling form of the OTOCs fromthis Gaussian boundary model, which is expected to be validin the long-time limit and at high p . The long-time require-ment is to ensure the distribution profile converges to Gaus-sian under the CLT which conventionally requires t > .The requirement on high p fixes the system in the chaoticphase and away from the point of phase transition, thus re-duces finite-size effects and statistical fluctuations in numeri-cal simulation.The comparison between calculated v b and the numericalsimulation is shown in the inset of Fig. 4 which demonstratesclose agreement at high p . This model only works away fromcriticality, which is expected since close to p c there will arisecorrelations which span the system and reduce the velocity ofedge proliferation.One may evaluate moments of this probability density tocalculate the time evolution of the boundary. The expectationof the outwards movement of the boundary in a single time-step is approximated by [49]. (cid:104) ∆ x (cid:105) = K (cid:88) x = −∞ xp ( x ) = K (cid:88) x = −∞ xp K − xs p d (10)so that the spacial profile as a function of time is (cid:104) x ( t ) (cid:105) = t (cid:104) ∆ x (cid:105) .By the CLT, one expects this profile to approach a Gaus-sian at large times, with a variance given by σ ( t ) = t (cid:2) (cid:104) (∆ x ) (cid:105) − (cid:104) ∆ x (cid:105) (cid:3) . One may evaluate the moments of the probability density using the following trick (cid:104) ∆ x n (cid:105) = K (cid:88) x = −∞ x n p K − xs p d = (cid:18) dd log p s + K (cid:19) n ∞ (cid:88) x =0 e log p s x . (11)This expression may be evaluated using the geometric series,and gives the surprisingly simple expression for the butterflyvelocity v b = (cid:104) x ( t ) (cid:105) t = K − p s p d , (12)and the time-dependent width of the Gaussian G ( x, t ) σ ( t ) = t (cid:18) p s p d + p s p d (cid:19) ≡ t µ ( p ) . (13)For any one realisation, the boundary will trace a biasedrandom walk in time, and inside of its chaos spreading willbe its own scrambling region with expected value D for theOTOC. The OTOC therefore acts like the cumulative proba-bility density function of the boundary’s probability density,weighted such that it has a central value of D = 2 p (1 − p ) : D ( x, t ) = D (cid:90) ∞ x G ( x (cid:48) , t ) d x (cid:48) . (14)In our approximate model at high p and late times we maytake the Gaussian limit, therefore deriving the following formfor the OTOC in terms of the error function erf( x ) D ( x, t ) = D ( p )2 (cid:20) − erf (cid:18) x − v b ( p ) t √ σ ( t ) (cid:19)(cid:21) = p (1 − p ) (cid:104) − erf (cid:16) ( v − v b ( p )) (cid:112) µ ( p ) t (cid:17)(cid:105) (15)where we have used x = vt and σ ( t ) = t/ µ ( p ) . By takingthe series expansion of the error function in v − v b for large x we recover an exponential decay of D ( x, t ) ; in logarithmicform this is (when v > v b ) ln D ( v, t ) = ln p (1 − p ) −
12 ln µ ( v − v b ) πt − µ ( v − v b ) t. (16)Comparing this result to the general form of the scaling formspredicted by previous works on spin chains, ln D ( v, t ) ∼ − µ ( v − v b ) β t,t,