Bistability and time crystals in long-ranged directed percolation
BBistability and time crystals in long-ranged directed percolation
Andrea Pizzi, Andreas Nunnenkamp, and Johannes Knolle
2, 3, 4 Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Department of Physics, Technische Universit¨at M¨unchen, James-Franck-Straße 1, D-85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences.A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatialdimension is given by probabilistic cellular automata, which typically feature time-independent and short-rangedupdate rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a novelbistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the systemdynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, wefind that the system responds with a period larger than that of the modulation for an exponentially (in systemsize) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled bya self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections.Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way forthe study of novel non-equilibrium phases in the unexplored field of ‘Floquet probabilistic cellular automata’.
INTRODUCTION
Percolation theory describes the connectivity of networks,with applications pervading virtually any branch of science[1], including economics [2], engineering [3], neurosciences[4], social sciences [5], geoscience [6], food science [7] and,most prominently, epidemiology [8]. Among the multitude ofphenomena described by percolation, of predominant impor-tance are spreading processes, in which time plays a crucialrole and that can be studied within models of directed perco-lation (DP) [9]. Characterized by universal scalings in time[10], in their discretized versions these models are probabilis-tic cellular automata (PCA), that is, dynamical systems with astate evolving in discrete time according to a set of stochasticand generally short-ranged update rules. To account for cer-tain realistic situations, e.g., of long-distance travels in epi-demic spreading, DP has been extended to long-ranged up-dates [11, 12] leading to a change of the universal scaling ex-ponents [13].Despite their wide applicability, PCAs have surprisingly re-mained an outlier in a branch of non-equilibrium physics thathas recently experienced a tremendous amount of excitement– that of discrete time crystals (DTCs) [14–20]. In essence,DTCs are systems that, under the action of a time-periodicmodulation with period T , exhibit a periodic response at adifferent period T (cid:48) (cid:54) = T , thus breaking the discrete time-translational symmetry of the drive and of the equations ofmotion. DTCs thus extend the fundamental idea of symme-try breaking [21] to non-equilibrium phases of matter. Fol-lowing the pioneering proposals in the context of many-body-localised (MBL) systems [17, 18], DTCs have been observedexperimentally [22, 23], and their notion has been extendedbeyond MBL [24–27].More recently, Yao and collaborators have fleshed out theessential ingredients of a classical DTC phase of matter [28].Namely, in a classical DTC, many-body interactions shouldallow for an infinite autocorrelation time, which should be stable in the presence of a noisy environment at finite tem-perature, a subtle requirement that rules out the vast class oflong-known deterministic dynamical systems. Despite vari-ous efforts [28–31], an example of such a classical DTC hasmostly remained elusive, and proving an infinite autocorrela-tion time for this phase of matter is an outstanding problem.The general expectation is in fact that PCAs and other minimalmodels for noisy systems in one spatial dimension can onlyshow a transient subharmonic response because noise-inducedimperfections generically nucleate and spread, destroying trueinfinite-range symmetry breaking in time [28, 32].Here, we overcome these difficulties by introducing a sim-ple and natural generalization of DP in which the dynamicalrules are governed by power-law correlations. This leads toqualitative changes of the system behaviour and, crucially, theemergence of a new bistable phase of long-ranged DP, enabledby the ability of long-range interactions to counteract the dy-namic proliferation of defects. By adding a periodic modula-tion to the update rules, we then study a version of ‘FloquetDP’ and show that the underlying bistable phase intimatelyconnects to a stable DTC. In this novel non-equilibrium phase,the system is able to self-correct noise-induced errors and theautocorrelation time grows exponentially with the system size,thus becoming infinite in the thermodynamic limit. In anal-ogy to the one-dimensional Ising model for which, at equi-librium, long-range interactions enable a normally forbiddenfinite-temperature magnetic phase [33, 34], in our model, outof equilibrium, the long-range interactions lead to a classi-cal time-crystalline phase. Crucially, our results appear natu-rally in a minimal model of long-ranged DP, but are expectedto find applications in many different contexts of dynamicalmany-body systems.Basic understanding of new concepts has historically beenbuilt around the study of minimal models, such as the Isingmodel for magnetism at equilibrium [33, 34], the kicked trans-verse field Ising chain for DTCs [17, 18], or the prototypicalDomany-Kinzel (DK) PCA for DP [35]. In this paper, we start a r X i v : . [ c ond - m a t . s t a t - m ec h ] J un our discussion with a brief review of the DK model and thengeneralize it to include power-law interactions. We character-ize its phase diagram, and show that its long-range nature isthe key ingredient for the emergence of a novel bistable phase.Finally, we include a periodic drive for the long-ranged DPprocess and show with a careful scaling analysis that the auto-correlation time of the subharmonic response is exponential insystem size. In the thermodynamic limit, our model providestherefore the first example of a PCA behaving as a classicalDTC, which is persistent and stable to the continuous pres-ence of noise. Lastly, we conclude with a summary of ourfindings and an outlook for future research. REVIEW OF DIRECTED PERCOLATION
We consider a triangular lattice in which one dimension canbe interpreted as discrete space i and the other one as discretetime t = 1 , , , . . . , see Fig. 1. To implicitly account for thetriangular nature of the lattice, i runs over integers and half-integers at odd and even times t , respectively. We denote L thespatial system size, and are interested in the thermodynamiclimit L → ∞ . The site i at time t can be either occupiedor empty, s i,t = 0 , . For a given time t , we call generation the collection of variables { s i,t } i specifying the system state.Initially, the sites are occupied with uniform probability p > . A DP process is defined by a stochastic Markovian updaterule with which, starting from the initial generation { s i, } i , allsubsequent generations { s i,t } i are obtained one by one. Themain observable we will focus on is the global density n ( t ) (henceforth just referred to as density for brevity) defined as n ( t ) = (cid:104)(cid:104) s i,t (cid:105) i (cid:105) runs (1)where the inner and outer brackets denote average over the L sites and over R independent runs, respectively. Since n (1) = p , we will often refer to p as initial density.The simplest, and yet already remarkably rich, example ofthe above setting of DP is the DK model [35]. Here, we brieflyreview it adopting an unconventional notation that, makingexplicit use of a local density, will prove very convenient for astraightforward generalization to a model of long-ranged DP.In the DK model, the probability of site i to be occupied attime t depends on the state of its neighbours i ± / at previoustime t − . More specifically, as summarized in Fig. 1(a), site i is: (i) empty if both its neighbours were empty, (ii) occupiedwith probability q if one and just one of its neighbours wasoccupied, (iii) occupied with probability q if both its neigh-bours were occupied. To account for these possibilities in acompact fashion, we define a local density n i,t as n i,t = s i − ,t − + s i + ,t − , (2)and say that site i at time t is occupied with a probability p i,t (cid:1) (cid:1) (cid:1) (cid:2) + (cid:2) - (cid:3) -1 (cid:3)(cid:4) (cid:2) , (cid:3) (cid:1) (cid:1) (cid:1) (cid:2) (b) (cid:5) (cid:2) , (cid:3) ∼ Bernoulli ( (cid:4) (cid:2) , (cid:3) ) (cid:3) = (cid:3) +1Occupation probabilitiesLocal densities Generation at time (cid:3)(cid:4) (cid:2) , (cid:3) = (cid:4) ∀ (cid:2)(cid:3) = 1Initial condition Advance time (cid:4) (cid:2) , (cid:3) = 0 if (cid:6) (cid:2) , (cid:3) = 0 (cid:1) if (cid:6) (cid:2) , (cid:3) = 0.5 (cid:1) if (cid:6) (cid:2) , (cid:3) = 1 (cid:6) (cid:2) , (cid:3) = (cid:5) (cid:2) -½, (cid:3) -1 + (cid:5) (cid:2) +½, (cid:3) -1 (cid:6) (10 ) 10 0.5Inactive Active (cid:4) = 1 T i m e Space( (cid:2) , (cid:3) ) Inactive Active (cid:4) = 0.01 (cid:6) (cid:2) , (cid:3) FIG. 1.
Domany-Kinzel model of directed percolation . (a) Theprobability of site i to be occupied at time t depends on the occu-pation of its nearest-neighbours i ± at time t − . (b) Flowchartrepresentation of the DK model. The initial occupation probabilityis uniform p i,t =1 = p . At time t , each site i is either occupied( s i,t = 1 ) or empty ( s i,t = 0 ) with probability p i,t and − p i,t ,respectively. Time is advanced, and local densities { n i,t } i are com-puted for each site i as averages of the nearest-neighbour occupationsat previous time, and these densities determine the occupation prob-abilities for the next generation, see Eq. (3). The generations at allsubsequent times are obtained by iteration. (c,d) The density n at latetimes can be used to discern the active and inactive phases, in which n ( t = 10 ) > and ≈ , respectively. The dashed lines serve asa reference to locate the phase boundary, and are the same for initialdensities p = 1 (c) and p = 0 . (d). The insets show represen-tative single instances of the DP for the points in the ( q , q ) planemarked with a cross. Here, L = 100 and R = 10 . given by p i,t = if n i,t = 0 q if n i,t = 0 . q if n i,t = 1 . (3)In other words, the probability p i,t is a nonlinear function f q ,q ( n i,t ) of the local density n i,t , with domain { , . , } .Since n i,t only involves the nearest neighbours of site i , theDK model of DP is obviously ‘ short-ranged ’. In essence, s i,t is a Bernoullian random variable of parameter p i,t , which wecompactly denote s i,t ∼ Bernoulli ( p i,t ) . The complexity ofthis model arises from the fact that the value of the parameter p i,t is not known a priori , as it depends on the actual stateof the system at previous time t − . Equipped with a ran-dom number generator, one can obtain all the generations oneby one according to the above procedure, as schematically il-lustrated in the flowchart of Fig. 1(b). Reiterating for severalindependent runs, one finally obtains the time series of thedensity n in Eq. (1).The DK model features two dynamical phases, shown inFig. 1(c,d). In the inactive phase , for small enough probabil-ities q and q , the system eventually reaches the completelyunoccupied absorbing state, that is no percolation occurs. Inthe active phase instead, for large enough probabilities q and q , a finite fraction of sites remains occupied up to infinitetime, that is the system percolates. For small initial probabil-ity p (cid:28) , the critical line separating the two phases is char-acterized by a power-law growth of the density [36], n ∼ t θ ,with exponent θ ≈ . . As conjectured by Grassberger [37],this exponent is universal for all systems in the DP universal-ity class. Indeed, DP exemplifies how the unifying concept ofuniversality pertaining to quantum and classical many-bodysystems [38] can be extended to non-equilibrium phenomena.Important for our work is that, in the DK model, whetherthe system percolates or not depends on the parameters q and q , but not on the initial density p , at least as long as p > .Indeed, the phase boundaries for initial densities p = 0 . and p = 1 in Fig. 1(c) and Fig. 1(d), respectively, coincide. LONG-RANGED PERCOLATION AND BISTABILITY
As the vast majority of PCA, the DK model features short-ranged update rules [9]. In realistic systems, however, it isoften the case that the occupation of a site i is influenced notonly by the neighbouring sites, but also by farther sites j , withan effect decreasing with the distance r i,j between the sites.Building on an analogy with the DK model, we propose herea model for such a ‘ long-ranged ’ DP, whose protocol is ex-plained in the flowchart of Fig. 2(a). Specifically, we consideras a local density n i,t a power-law-weighted average of theprevious generation { s j,t − } j centered around site in i,t = 1 N α,L (cid:88) j s j,t − ( r i,j ) α , (4)where the normalization factor N α,L ensures n i,t = 1 if allsites j are occupied, and the adjective ‘local’ emphasizes thesite-dependence. The occupation probability p i,t then de-pends on the local density n i,t through some nonlinear func-tion f µ , that for concreteness we consider to be p i,t = µ tanh (cid:0) n i,t (cid:1) , (5)with µ ∈ (0 , a control parameter. The whole DP dynamicsis determined via the occupations s i,t ∼ Bernoulli ( p i,t ) andreiterating from one generation to the next. Note, our findingsare not contingent on the specific choice of Eqs. (4) and (5),but are rather expected to hold generally for a broad class of long-ranged forms of the densities n i,t and of functions f µ –see Section Methods for details.We emphasise that Eq. (4), Eq. (5), and the flowchart inFig. 2(a) are a natural generalization of Eq. (2), Eq. (3), andFig. 1(b), respectively. Furthermore, whereas in the DK modelthe control parameters are the probabilities q and q , the con-trol parameter is now µ . As an important difference, now thedomain of f µ accounts for several (and α -dependent) valuesof n i,t , for which the piecewise definition of p i,t as in Eq. (3)would have been unpractical, and the compact form of Eq. (5)was necessary instead.The introduction of a long-ranged local density n i,t inEq. (4) has profound implications. Arguably, the most dra-matic is the appearance of a novel bistable phase, in addi-tion to the standard active and inactive ones. In the bistablephase, the ability of the system to percolate depends on theinitial density p , see the red lines in Fig. 2(b,c). That is,the bistable phase features two basins of attraction, resultinginto an asymptotically vanishing or finite n , respectively, andseparated by some critical initial density p ,c > . To charac-terize systematically the dynamical phases of our model, weplot in Fig. 2(d,e) the long-time density n ( t = 10 ) as a suit-able order parameter in the plane of the power-law exponent α and control parameter µ . Comparing the results obtained fora large and a small initial density p , it is possible to sketch aphase diagram composed of three phases: (i) inactive – n de-cays to at long times; (ii) active – n does not decay at longtimes; (iii) bistable – n either decays or not depending on p being small or large. The existence of this bistable phase isin striking contrast with short-ranged models of DP such asthe DK model, and in fact appears only for α (cid:47) , that is,when the local densities { n i,t } i are correlated over a suffi-ciently long range. To understand the origin of this rich phe-nomenology, we study the short- and infinite-range limits ofour DP process.In the short-range limit α → ∞ , the local densities n i,t reduce to the averages of the nearest-neighbour occupations s i − ,t − and s i + ,t − , that is, Eq. (4) recasts into Eq. (2) andthe DK model is recovered. In the notation of Eq. (3), theDK parameters are q = f µ (0 . and q = f µ (1) . Therefore,we can move across the DK parameter space ( q , q ) varying µ , going from the inactive phase ( µ < µ ∞ c ) to the active one( µ > µ ∞ c ), and no bistable phase is possible. We find that thetransition happens at a critical µ ∞ c = 0 . . Note that, inthe active phase, a completely empty state ( p = 0 ) remainstrivially empty at all times. This behavior is however unstable,because any p > leads to percolation (i.e., p ,c = 0 ), andwe therefore do not classify the active phase as bistable. Atcriticality, and for p (cid:28) , the density grows as n ∼ t θ with θ = 0 . , as expected for the DP universality class [9]. SeeSupplementary Fig. S2 for details.In the infinite-range limit α → , and more generally for α ≤ , the factor N α,L in Eq. (4) diverges as L → ∞ . Corre-spondingly, spatial stochastic fluctuations are suppressed, thatis, all sites i share the same occupation probability p i,t +1 = p t and density n i,t = n ( t ) = p t . Therefore, in this limit the dy- (cid:1) (cid:2) , (cid:3) ∼ Bernoulli ( (cid:4) (cid:2) , (cid:3) ) (cid:3) = (cid:3) +1OccupationprobabilitiesLocal densities Generation at time (cid:3)(cid:4) (cid:2) , (cid:3) = (cid:4) ∀ (cid:2)(cid:3) = 1Initial condition Advance time (cid:5) (cid:2) , (cid:3) = (cid:6) ( (cid:7) (cid:2) , (cid:8) ) (cid:6) (cid:1) (cid:8) , (cid:3) -1 (cid:8) (cid:4) (cid:2) , (cid:3) = (cid:9) (cid:10) ( (cid:5) (cid:2) , (cid:3) ) T i m e Space (cid:8) ( (cid:2) , (cid:3) ) (a) D en s i t y (cid:5) (cid:4) = 1 (cid:4) = 0.01Time (cid:3) -1 -2 -3 Time (cid:3) C on t r o l pa r a m e t e r (cid:10) Power law exponent (cid:6) (cid:10) = 0.85(7) ∞ c (cid:10) c = 0.6550(8)
210 3
InactiveActiveBistable (b) (cid:4) = 1 (cid:4) = 0.01 Long - t i m e den s i t y (cid:5) ( (cid:3) = ) (cid:6) I na c t i v e BistableActive I na c t i v e InactiveActiveBistable (c) (cid:6) = 2.0, (cid:10) = 0.845 (cid:6) = 1.7, (cid:10) = 0.855 (cid:6) = 2.0, (cid:10) = 0.875 B i s t ab l e A c t i v e (c)(b)(d) (e) FIG. 2.
Long-ranged directed percolation and bistability . (a) Flowchart representation of the long-ranged DP. Starting from an initialcondition p i,t =1 = p , site i at time t is occupied ( s i,t = 1 ) with probability p i,t . Local densities { n i,t } i are computed as power-law-weighted averages of the previous generation { s i,t − } i , and the occupation probabilities are updated as p i,t = f µ ( n i,t ) . (b,c) Time evolutionof the density n for p = 1 (b) and p = 0 . (c). Three dynamical phases can be distinguished: (i) inactive – n decays to (blue); (ii) active– n does not decay to (yellow); (iii) bistable – n either decays to or not depending on whether the initial density p is small or large (red).(d,e) Long-time density n ( t = 10 ) in the plane of the power-law exponent α and control parameter µ for p = 1 (d) and p = 0 . (e). Withthe criterion used in (b,c), we discern the three phases: inactive (light), active (dark), and bistable (light or dark depending on p ). The dashedlines help locating the phases and coincide in (d) and (e), and critical values µ c and µ ∞ c of µ in the limits α → and α → ∞ , respectively, arereported (the offset of µ c from the dashed line, as well as the softening of the dashed line for α ≈ , are due to finite-size effects). Crucially,the bistable phase is present only for small enough α (cid:47) , that is for a sufficiently long-ranged DP. Single instances of the DP for the threephases are shown in the insets, as obtained for the α and µ indicated with colored dots, and corresponding to the parameters used in (b,c).Here, R = 10 and in (b,c) and (d,e), respectively, and L = 500 . namics reduces to the deterministic -dimensional recurrencerelation n ( t + 1) = f µ [ n ( t )] . (6)The system asymptotic behaviour can then be understoodfrom the analysis of the fixed points (FPs) of the equation x = f µ ( x ) , which is detailed in Section Methods. FLOQUET PERCOLATION AND TIME CRYSTALS
We have established that long-range correlated local densi-ties { n i,t } i give rise to a novel bistable phase. We now showhow, in a ‘Floquet DP’ with periodically modulated updaterules, this phase intimately relates to the emergence of a clas-sical DTC. In this phase, as we shall see, the density n dis-plays oscillations over a period larger than that of the drive and up to a time that, thanks to the long-range interactions anddespite the presence of multiple sources of noise, is exponen-tially large in the system size, a feature that would generallybe forbidden in short-ranged PCA [28]. In the thermodynamiclimit L → ∞ , these subharmonic oscillations are thereforepersistent, that is, the system autocorrelation time diverges toinfinity, breaking the time-translational symmetry and provingfor the first time a classical DTC in a Floquet PCA.In the spirit of keeping the model as simple as possible, weconsider a minimal Floquet drive in which, after every T itera-tions of the DP in Eqs. (4) and (5), empty sites are turned intooccupied ones and vice versa, making the full equations ofmotion periodic with period T . As a further source of imper-fections, adding to the underlying noisy DP, we also accountfor faulty swaps with probability p d . More explicitly, the Flo- S c a li ng c oe ff. 𝛽 Power-law exponent 𝛼 -2 𝐿
200 400 500 ∼ 𝑒 𝛽𝐿 L i f e t i m e 𝜏 S ubha r m on i c i t y 𝚽 𝑡 -1)/ 𝑇
100 100 T i m e 𝑡 / 𝑇 Space Density 𝑛 Period 𝑇 DTC ( 𝛼 = 1.4) SpaceDensity 𝑛 T i m e 𝑡 / 𝑇 Period 2 𝑇 Trivial ( 𝛼 = 1.8) 𝐿 (a) (b)(c)(d) (e) 𝐿 𝛼 ∼ 𝑒 𝜏𝑇𝑡 -1- ∼ 𝑒 𝜏𝑇𝑡 -1- FIG. 3.
Discrete time crystals in Floquet long-ranged directed percolation . (a,b) Single instances of the Floquet DP, alongside with thedensity n averaged over multiple independent runs, for L = 500 . (a) For a power-law exponent α = 1 . , n oscillates subharmonically witha period that is twice that of the drive, whereas, for α = 1 . , n eventually picks the periodicity T enforced by the drive. (c) For finite systemsizes L , the subharmonicity Φ( t ) decays as Φ( t ) ∼ exp (cid:0) − t − τT (cid:1) due to the accumulation of phase slips, and, after a few time scales τ T , thedensity n synchronises with the drive and oscillates with period T . Exponential fits (dotted lines) can be used to extrapolate the lifetime τ of the subharmonic response, on which a scaling analysis is performed in (d). For α = 1 . (blue), the lifetime τ scales exponentially withthe system size, τ ∼ e βL , whereas no such a scaling is found for α = 1 . . The scaling coefficient is again found from an exponential fit(dotted line), and plotted in (e) versus the power-law exponent α . For small α , that is long-ranged enough DP, the scaling coefficient β isfinite, indicating that in the thermodynamic limit L → ∞ the subharmonic response is persistent and a DTC with infinite autocorrelation timeemerges. On the contrary, β ≈ for large α , indicating a trivial dynamical phase in which no stable subharmonic dynamics is established.Here, we considered p = 1 , µ = 0 . , p d = 0 . , T = 20 and R = 2000 . quet drive consists of the following transformation s i, kT → (cid:40) − s i, kT with probability − p d s i, kT with probability p d . (7)In Fig. 3(a,b) we show the spatio-temporal pattern of singleinstances of the Floquet DP, alongside with the density n aver-aged over several independent runs. If the DP is short-ranged enough, the spatio-temporal pattern at long times looks simi-lar from one Floquet period to the next, that is the density n synchronises with the drive and eventually picks a periodicity T . On the contrary, for a long-ranged enough DP, the systemkeeps alternating at every period between a densely occupiedregime and a sparsely occupied one, and n oscillates with pe-riod T , that is, the system breaks the discrete time-translationsymmetry of the equations of motion.When using the tag ‘classical DTC’, special care shouldbe reserved for showing the defining features of this phase,namely its rigidity and persistence [28]. Our system is rigid inthe sense that it does not rely on fine-tuned model parameters,e.g., µ , α or the initial density p , and that noise, either in theform of the inherently stochastic underlying DP or of a smallbut non-zero Floquet defect density p d , does not qualitativelychange the results. Moreover, in the limit L → ∞ , our DTCis truly persistent. Indeed, one might expect that the accumu-lation of stochastic mistakes introduces phase slips and even-tually leads to the (possibly slow but unavoidable) destructionof the subharmonic response. Although this expectation isgenerally correct for short-ranged DP models, including ourmodel at large α , it can fail for long-ranged DP models.To show that, in the limit L → ∞ , the lifetime of our DTCis infinite, we perform a scaling analysis comparing results forincreasing system sizes L . First, we introduce an order param-eter Φ( t ) , henceforth called subharmonicity , that is defined atstroboscopic times t = 1 , T, T, . . . as Φ( t ) = ( − t − T [ n ( t ) − n ( t + T )] . (8)If the density n oscillates with the same period T as the drive,then n ( t ) = n ( t + T ) and Φ( t ) = 0 . On the contrary, if n oscillates with a doubled period T , then n ( t = 1 + kT ) is positive and negative for even and odd k , respectively, and Φ( t ) is finite and maintains a constant sign. Therefore, Φ( t ) is a suitable diagnostics to track the degree of subharmonicityof n in time, and to perform the scaling analysis.In Fig. 3(c) we show Φ( t ) for various system sizes L . Forboth α = 1 . and α = 1 . , the subharmonicity decays expo-nentially in time, Φ( t ) ∼ exp (cid:0) − t − τT (cid:1) . As shown in Fig. 3(d),these two values of α are however crucially different in howthe lifetime τ T scales with the system size. In fact, τ T is ap-proximately independent of L for α = 1 . , whereas it scalesexponentially as τ ∼ exp( βL ) for α = 1 . , for which thedecay of the subharmonicity is therefore just a finite-size ef-fect. The scaling coefficient β quantifies the time crystallinityof the system, and can thus be used to obtain a full phase dia-gram as a function of the power-law exponent α , in Fig. 3(e).We observe a phase transition between a DTC and a trivialphase at α ≈ . . That is, if the DP is sufficiently long-ranged ( α (cid:47) . ), β is finite and in the thermodynamic limit L → ∞ the subharmonic response extends up to infinite time,as required for a true DTC. In contrast, for a shorter-range DP( α (cid:39) . ), β ≈ independently of L and the subharmonicresponse is always dynamically destroyed. DISCUSSION AND CONCLUSION
We have shown that long-range DP and its Floquet variantcan give rise to a bistable phase and a DTC, respectively. Atthe core of our model in Eq. (4) and Eq. (5) is the idea thatthe occupation of a given site depends on the state of all the other sites at the previous time. In this sense, our model isreminiscent of some SIR type models of epidemic spreadingin which not only a sick site can infect a susceptible site, butseveral infected sites can also cooperate to weaken a suscepti-ble site, and finally infect it [39, 40]. This cooperation mech-anism among an infinite number of ‘parent sites’, rather thana finite one as considered in previous works on long-rangedDP [13, 41], is the key feature allowing the emergence ofthe bistable phase, that finds a transparent explanation in theinfinite-range limit α → , where it corresponds to the equa-tion x = f µ ( x ) having two stable FPs. Bistability also pro-vides intuition on the origin of the DTC, to which it is deeplyconnected. Indeed, the Floquet drive in Eq. 7 switches thesystem from a densely occupied regime to a sparsely occupiedone (and vice versa). If the underlying DP is bistable, theseregimes fall each within different basins of attraction, and cantherefore be both stabilized [25, 29]. Ultimately, this doublestabilization facilitates the establishment of the DTC with in-finite autocorrelation time. Remarkably, this mechanism doesnot rely on the equations of motion being perfectly periodic,as required for DTCs in closed MBL systems [42], and weexpect that infinite autocorrelation times could be maintainedeven in the presence of aperiodic variations of the drive.The intimate connection between bistability and DTC ishowever not a strict duality, and the boundaries of the twophases, in the equilibrium and non-equilibrium phase dia-grams, respectively, do not coincide. For instance, in our anal-ysis we found that for µ = 0 . the bistable phase extends upto α ≈ . , whereas the DTC stretches slightly farther, up to α ≈ . . The origins of this imperfect correspondence canbe traced back to two competing effects. On the one hand,bistability may not be sufficient to stabilize a DTC. This canalready be understood in the limit α → , in which the asym-metry of f µ and of its FPs does not guarantee the Floquet driv-ing to switch the density n from one basin of attraction to theother, that is, across the critical probability p ,c . This issuebecomes even more relevant for larger α , for which the asym-metry is possibly accentuated and p ,c can approach (see forinstance Supplementary Fig. S1). On the other hand, a perfectbistability may not even be necessary for a DTC to exist. Infact, for the stabilization of a DTC, it may be sufficient that, ofthe densely and sparsely occupied regimes of the underlyingDP, only one is stable, and the other is just weakly unstable(that is, metastable), meaning that the timescales of the dy-namics of the density n in the two regimes are very different.Loosely speaking, the stability of one regime might be able tocompensate for the weaker instability of the other, resulting inan overall stable DTC.While these considerations are model and parameters de-pendent, and it is ultimately up to numerics to find the bistableand the DTC phases, what is universal and far reaching hereis the concept that long-ranged DP, and PCA more gener-ally, can host novel dynamical phases such as DTCs. AsYao and collaborators recently pointed out [28], long auto-correlation times are in fact generally unexpected in -dimensional PCA, because imperfections and phase slips can N e x t den s i t y (cid:1) ( (cid:2) + ) = (cid:3) (cid:4) ( (cid:1) ( (cid:2) )) (cid:5) (b)Flow of (cid:1) ( (cid:2) ) 1PercolationNo perc.0 10.33265...(c)1Density (cid:1) ( (cid:2) )PercolationNo percolation0 1 (cid:6) c = 0.52169... (cid:6) = 0 0.5 1No percolation0 10.5Stable FPUnstable FP BistableSemi-stable FP0.4 10.5 (cid:4) (cid:6) = 0 (cid:6) = 0 (cid:6) c (cid:6) (cid:6) (cid:4) = 0.6 < (cid:4) c0 (cid:4) = (cid:4) c = 0.65508... (cid:4) = 0.8 > (cid:4) c0 FIG. 4.
Graphical fixed-points analysis . For α < , the dynamics of Eq. (6) is understood from the FPs analysis of the equation x = f µ ( x ) .(a) For µ < µ c = 0 . , the system is inactive, corresponding to a single FP x = 0 : at long times, the system ends up in the empty,absorbing state with s i = 0 for all sites i . (b) At the critical point µ = µ c , a new semi-stable FP emerges at x c = 0 . , that is unstablefrom his left and stable on his right. (c) Increasing µ above µ c , the semi-stable FP splits into an unstable FP x < x c and a stable FP x > x c .Depending on whether p < x or p > x , the system flows towards n = x = 0 or n = x > , respectively, indicating bistability. nucleate, spread and destroy the order. Our work proves thatthis fate can be avoided, and time-crystalline order estab-lished, in long-ranged PCA. These systems enable in fact an‘error correction’ mechanism, in our case intimately related tothe bistability, that would be impossible if correlations werelimited to a finite radius.In conclusion, we have studied the effects of long-rangecorrelated update rules in a model of DP, which we built froman analogy with the prototypical (but short-ranged) DK PCA.First, we proved that, beyond the standard active and inactivephases, a new bistable phase emerges in which the system atlong times is either empty or finitely occupied depending onwhether it was initially sparsely or densely occupied. Second,in a Floquet DP with periodic modulation of the update rules,we showed that this bistable phase intimately connects with aDTC phase, in which the density oscillates with a period twicethat of the drive. In this DTC phase, the autocorrelation timescales exponentially with the system size, and in the thermo-dynamic limit a robust and persistent breaking of the discretetime-translation symmetry is established.As an outlook for future research, further work on the Flo-quet DP should better assess the nature of the transition be-tween the DTC and the trivial phase, characterise more sys-tematically the phase diagram in other directions of the pa-rameter space, and, most interestingly, address the role of di-mensionality. Indeed, it is well-known that dimensionalitycan facilitate the establishment of ordered phases of matterat equilibrium, and the question whether this is the case alsoout-of-equilibrium remains open. A positive answer to this question is suggested by the fact that, in D + 1 -dimensionwith D ≥ , bistability can emerge even in short-ranged mod-els of DP [40, 43, 44]. Another interesting question regardsthe fate of chaos and damage spreading in long-ranged DP[45]. Finally, on a broader perspective, our work paves theway towards the study of non-equilibrium phases of matter inthe uncharted territory of Floquet PCA, with a potentially verybroad range of applications throughout different branches ofscience. As a timely example, Floquet PCA may provide newinsights into the understanding of seasonal epidemic spread-ing and periodic intervention efficacy. METHODS
Here, we provide further technical details on our work. InEq. (4), we considered as distance r i,j between sites i and jr i,j = Lπ (cid:12)(cid:12)(cid:12)(cid:12) tan (cid:18) π i − jL (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where the tangent accounts for periodic boundary conditionsand makes the distance of the fartherst sites with | i − j | = L/ artificially diverge. This divergence is expected to reducefinite-size effects without changing the underlying physics,that is in fact dominated by sites with | i − j | (cid:28) L , for whichwe get a natural r i,j ≈ | i − j | . Indeed, as we checked, similarresults are obtained with r i,j = min( | i − j | , L − | i − j | ) . TheKac-like normalization factor N α,L reads instead N α,L = L (cid:88) j =1 (cid:16) r ,j (cid:17) − α . (10)The phenomenology of the bistable phase can be under-stood from a graphical FPs’ analysis of the equation f µ ( x ) = x illustrated in Fig. 4, which explains the dynamics for α < .Three scenarios are possible, and interpreted in terms of theways the graph of the function f µ intersects with the bisect.(i) Inactive – if µ < µ c , the only FP is x = 0 , which isstable and corresponds to a completely empty state. The sys-tem moves towards this FP and p t t →∞ −−−→ . (ii) Critical – if µ = µ c , a new semi-stable FP emerges at x c , which is attrac-tive from its right and repulsive on its left. (iii) Bistable – if µ > µ c , the semi-stable FP splits into an unstable FP x > x and a stable FP x > x . In this case, the system will reacheither the unoccupied FP x = 0 or the finitely occupied FP x > depending whether p < x or p > x , respectively.That is, the system is bistable, and the critical initial probabil-ity separating its two basins of attraction is p ,c = x (see alsoSupplementary Fig. S1). The critical value µ c is obtained nu-merically solving for the condition of tangency between thegraph of f µ and the bisect, and gives µ c = 0 . and x c = 0 . . For µ > µ c , the FPs x and x are foundsolving for f µ ( x ) = x , and, for instance, we find we find x = 0 . and x = 0 . for µ = 0 . .The FPs’ analysis also clarifies the general features of f µ that allow for the emergence of bistability, that is in fact notcontingent on the choice of f µ made in Eq. (5). Indeed, theonly requirement is that, for some parameter(s) µ , the equa-tion f µ ( x ) = x has three FPs x < x < x , of which x and x are stable, whereas x is unstable. Put simply, f µ shouldbe a nonlinear function with a graph looking qualitatively asthat of Fig. 4(c). This condition guarantees a bistable phasefor α < , which can then possibly extend to α ≥ and, inthe presence of a Floquet drive, facilitate the establishment ofa DTC.Finally, note that higher resolution and smaller fluctuationscould be achieved in the figures throughout the paper if simu-lating larger system sizes L and/or considering a larger num-ber of independent runs R . This could, for instance, allow amore accurate characterisation of both the equilibrium and thenon-equilibrium phase diagrams of our model, which could beexplored in other directions of the parameter space for vary-ing α, µ , p d and T . This would, however, require a formidablenumerical effort, and goes therefore beyond the scope of thiswork. As a reference, for instance, the generation of Fig. 3(e)for the parameters considered therein requires a computingtime of approximately × hours per GHz core.
Acknowledgements.
We are very thankful to P. Grass-berger for insightful comments on the manuscript. J. K. thanksKim Christensen for introducing him to the theory of percola-tion. We acknowledge support from the Imperial-TUM flag-ship partnership. A. P. acknowledges support from the RoyalSociety. A. N. holds a University Research Fellowship from the Royal Society and acknowledges additional support fromthe Winton Programme for the Physics of Sustainability.
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Role of the initial density.
First, we investigate our system in yet another direction in the parameter space, that is that of theinitial occupation probability (or density) p . In the main text, we have in fact seen that the system behaviour in the bistablephase drastically depends on whether p is ‘small’ or ‘large’, and, having so far limited our examples to p = 0 . and p = 1 ,we now better assess what do ‘small’ and ‘large’ mean. In Fig. S1 we consider the entire range of p from to , and look forthe critical density p ,c separating the two basins of attraction: n does and does not decay to for p < p ,c and p > p ,c ,respectively. The critical probability at α → coincides with the unstable fixed point x of the equation x = f µ ( x ) , as explainedin the main text, and is reported in the plots as a reference. For µ = 0 . (left), we find that p ,c ≈ x for all the α (cid:47) . , whereasfor larger α the system enters the inactive phase. For µ = 0 . , we observe that p ,c changes smoothly with α from x at α = 1 to at α ≈ . , when the system enters the active phase, see also Fig. 2(d,e).As we have show here, the critical probability p ,c of the bistable phase generally falls in the bulk of the range (0 , , ultimatelybecause the FP x does. This is an important feature of our model, because it means that both the possible behaviours of thebistable phase, that is percolating and not, have a broad range of p in which they are stable. On the one hand, this means thatthe results of Fig. 1 in the main text are not contingent on the choice of p = 0 . and , but would rather be analogue for otherchoices of p < p ,c and p > p ,c . On the other hand, once the Floquet drive is included, such a broad stability region enablesthe DTC robustness to noise. I n i t i a l p r obab ili t y (cid:1) (cid:2) = 0.287(8) (cid:2) = 0.332(6) (cid:3) ( (cid:4) = 10 )10 0.5 (cid:5) = 0.8 (cid:5) = 0.9Power-law exponent (cid:6)
210 3 210 3Power-law exponent (cid:6)
InactiveBistable ActiveBistable
FIG. S1.
Role of the initial density in the bistable phase . Long-time density n ( t = 10 ) in the plane of the power-law exponent α and theinitial occupation density p , for µ > µ c = 0 . . For sufficiently small α , the system is in the bistable phase, meaning that n (10 ) canbe either finite or not depending on p , the two behaviours being separated by a critical probability p ,c . For α → , the critical probability p ,c corresponds to x , the unstable FP of the equation x = f µ ( x ) , which is reported as a reference. For µ = 0 . (left), we observe that p ,c ≈ x for α up to ≈ . , above which the system enters the inactive phase. For µ = 0 . (right), instead, the critical p ,c decreases smoothly with α ,reaching at α ≈ . , at which the system enters the active phase. Here, L = 500 and R = 100 . Short-range DK limit.
Second, we investigate in more detail the limit α → ∞ already treated in the main text. In such alimit, our model of DP recasts into the DK model, upon replacing q = f µ (0 .
5) = µ tanh(1) and q = f µ (1) = µ tanh(4) .Varying µ , one can therefore move across the DK parameter space ( q , q ) along the line q = tanh 4tanh 1 q , and therefore across thephase boundary between the active and inactive phases. In Fig. S2, we plot the time series of the density n for various valuesof the control parameter µ . Reiterating from the main text, what we find is that for µ = µ ∞ c = 0 . the density grows as apower law ∼ t θ with θ = 0 . as expected in the DP universality class. For µ > µ ∞ c the density n grows in time, whereas for µ < µ ∞ c it rather decays to , signalling the active and inactive phases, respectively.Note that the setting in which DP is studied is usually that of an initial condition with one ‘seed site’ being occupied, andan infinitely large system size L , corresponding to a density /L → . In our case, the choice of the initial condition withoccupation probabilities p corresponds to a possibly small but still finite initial density p , so that in the limit L → ∞ there willalways be infinitely many seed sites. In the active phase, the clusters originating from many of these sites will grow and expandin time, eventually merging together and leading to a saturation of n at long-times. In particular, the universal power-law growthat criticality can be observed only for p (cid:28) , and it extends for a finite time, before saturation eventually sets in. Time (cid:1) +110 -4 -5 (cid:2) D en s i t y (cid:3) ( (cid:1) ) (cid:2) = 0.857 ∼ (cid:1) (cid:4) (cid:4) (cid:3) (10 ) 10 0.5 -3 -2 -1 C r i t i c a l I na c t i v e A c t i v e FIG. S2.
Limit of short-range directed percolation . In the short-range limit α → ∞ , our model of DP maps to the DK model. In thislimit, we show the dynamics of the density n for various values of the control parameter µ . For µ < µ ∞ c ( µ > µ ∞ c ), the system does not(does) percolate, that is n does (does not) decay to at long-times. At the critical point µ = µ ∞ c ≈ . (indicated with a dotted line inthe colorbar), n grows as a power-law ∼ t θ , with θ ≈ . as expected in the DP universality class. Analogue results are obtained for otherchoices of p , since in the short-ranged DK limit no bistable phase exists. Here, L = 10 , p = 10 − and R = 10 . In the inset, we reportfor reference Fig. 1(d), highlighting the line spanned in the ( q , q ) DK parameter space when varying µµ