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 spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a 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 for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.

A model of a classical discrete time crystal satisfying the criteria of persistent subharmonic response robust against thermal noise and defects has been lacking. Here, the authors show that these criteria are satisfied in one-dimensional probabilistic cellular automata with long-range interactions and bistability.

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 of phenomena described by percolation, of predominant importance are spreading processes, in which time plays a crucial role and that can be studied within models of directed percolation (DP)^{9}. Characterized by universal scalings in time^{10}, in their discretized versions these models are probabilistic cellular automata (PCA), that is, dynamical systems with a state evolving in discrete time according to a set of stochastic and generally short-ranged update rules. To account for certain realistic situations, e.g. of long-distance travels in epidemic spreading, DP has been extended to long-ranged updates^{11,12} leading to a change of the universal scaling exponents^{13}.

Despite their wide applicability, PCAs have surprisingly remained an outlier in a branch of non-equilibrium physics that has 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-periodic modulation with period ^{21} to non-equilibrium phases of matter. Following the pioneering proposals in the context of many-body-localized (MBL) systems^{17,18}, DTCs have been observed experimentally^{22,23}, and their notion has been extended beyond MBL^{24–27}.

More recently, Yao and collaborators have fleshed out the essential ingredients of a classical DTC phase of matter^{28}. Namely, in a classical DTC, many-body interactions should allow for an infinite autocorrelation time, which should be stable in the presence of a noisy environment at finite temperature, a subtle requirement that rules out the vast class of long-known deterministic dynamical systems. Despite various efforts^{28–31}, an example of such a classical DTC has mostly remained elusive, and proving an infinite autocorrelation time robust to noise and perturbations for this phase of matter is an outstanding problem. The general expectation is in fact that PCAs and other minimal models for noisy systems in one spatial dimension can only show a transient subharmonic response because noise-induced imperfections generically nucleate and spread, destroying true infinite-range symmetry breaking in time^{28,32}.

Here we overcome these difficulties by introducing a simple and natural generalization of DP in which the dynamical rules are governed by power–law correlations. This leads to qualitative changes of the system behaviour and, crucially, the emergence of a bistable phase of long-ranged DP, enabled by the ability of long-range interactions to counteract the dynamic proliferation of defects. By adding a periodic modulation to the update rules, we then study a version of periodically driven DP and show that the underlying bistable phase intimately connects to a stable DTC. In this non-equilibrium phase, the system is able to self-correct noise-induced errors and the autocorrelation time grows exponentially with the system size, thus becoming infinite in the thermodynamic limit. In analogy to the one-dimensional Ising model for which, at equilibrium, long-range interactions enable a normally forbidden finite-temperature magnetic phase^{33,34}, in our model, out of equilibrium, the long-range interactions lead to a classical time-crystalline phase. Crucially, our results appear naturally in a minimal model of long-ranged DP but are expected to find applications in many different contexts of dynamical many-body systems.

Basic understanding of new concepts has historically been built around the study of minimal models, such as the Ising model for magnetism at equilibrium^{33,34}, the kicked transverse field Ising chain for DTCs^{17,18}, or the prototypical Domany–Kinzel (DK) PCA for DP^{35}. In this paper, we start our discussion with a brief review of the DK model and then generalize it to include power–law interactions. We characterize its phase diagram and show that its long-range nature is the key ingredient for the emergence of a bistable phase. Finally, we include a periodic drive for the long-ranged DP process and show with a careful scaling analysis that the autocorrelation time of the subharmonic response is exponential in system size. In the thermodynamic limit, our model provides therefore the first example of a PCA behaving as a classical DTC, which is persistent and stable to the continuous presence of noise. Lastly, we conclude with a summary of our findings and an outlook for future research.

We consider a triangular lattice in which one dimension can be interpreted as discrete space _{i,t} = 0,1. For a given time _{1} > 0. A DP process is defined by a stochastic Markovian update rule with which, starting from the initial generation _{1}, we will often refer to _{1} as initial density.

_{i,t} of site _{1} and _{2}. _{i,t} = 1 = _{1}. At time _{i,t} = 1) or empty (_{i,t} = 0) with probability _{i,t} and 1 − _{i,t}, respectively. Time is advanced and local densities ^{3}) >0 and ≈0, respectively. The dashed lines serve as a reference to locate the phase boundary and are the same for initial densities _{1} = 1 (_{1} = 0.01 (_{1}, _{2}) plane marked with a cross. Here ^{3}.

The simplest, and yet already remarkably rich, example of the above setting of DP is the DK model^{35}. Here, we briefly review it adopting an unconventional notation that, making explicit use of a local density, will prove very convenient for a straightforward generalization to a model of long-ranged DP.

In the DK model, the probability of site _{1} if one and just one of its neighbours was occupied, and (iii) occupied with probability _{2} if both its neighbours were occupied. To account for these possibilities in a compact fashion, we define a local density _{i,t} as_{i,t} given by_{i,t} is a nonlinear function _{i,t}, with domain {0, 0.5, 1}. Since _{i,t} only involves the nearest neighbours of site _{i,t} is a Bernoullian random variable of parameter _{i,t}, which we compactly denote _{i,t} is not known a priori, as it depends on the actual state of the system at previous time

The DK model features two dynamical phases, shown in Fig. _{1} and _{2}, the system eventually reaches the completely unoccupied absorbing state, that is, no percolation occurs. In the _{1} and _{2}, a finite fraction of sites remains occupied up to infinite time, that is, the system percolates. For small initial probability _{1} ≪ 1, the critical line separating the two phases is characterized by a power–law growth of the density^{36}, ^{θ}, with exponent ^{37}, this exponent is universal for all systems in the DP universality class. Indeed, DP exemplifies how the unifying concept of universality pertaining to quantum and classical many-body systems^{38} can be extended to non-equilibrium phenomena.

Important for our work is that, in the DK model, whether the system percolates or not depends on the parameters _{1} and _{2} but not on the initial density _{1}, at least as long as _{1} > 0. Indeed, the phase boundaries for initial densities _{1} = 0.01 and _{1} = 1 in Fig.

As the vast majority of PCA, the DK model features short-ranged update rules^{9}. In realistic systems, however, it is often the case that the occupation of a site _{i,j} between the sites. Building on an analogy with the DK model, we propose here a model for such a ‘long-ranged’ DP, whose protocol is explained in the flowchart of Fig. _{i,t} a power–law-weighted average of the previous generation _{i,t} = 1 if all sites _{i,t} then depends on the local density _{i,t} through some nonlinear function _{μ} that for concreteness we consider to be_{i,t} and of functions _{μ}—see ‘Methods’ section for details.

_{i,t=1} = _{1}, site _{i,t} = 1) with probability _{i,t}. Local densities _{1} = 1 (_{1} = 0.01 (_{1} is small or large (red). ^{3}) in the plane of _{1} = 1 (_{1} = 0.01 (_{1}). The dashed lines help locating the phases and coincide in ^{4} and 10^{2} in

We emphasize that Eqs. (_{1} and _{2}, the control parameter is now _{μ} accounts for several (and _{i,t}, for which the piecewise definition of _{i,t} as in Eq. (

The introduction of a long-ranged local density _{i,t} in Eq. (_{1}, see the red lines in Fig. _{1,c} > 0. To characterize systematically the dynamical phases of our model, we plot in Fig. ^{3}) as a suitable order parameter in the plane of the power–law exponent _{1}, it is possible to sketch a phase diagram composed of three phases: (i) inactive—_{1} being small or large. The existence of this bistable phase is in striking contrast with short-ranged models of DP such as the DK model and in fact appears only for

In the short-range limit _{i,t} reduce to the averages of the nearest-neighbour occupations _{1} = _{μ}(0.5) and _{2} = _{μ}(1). Therefore, we can move across the DK parameter space (_{1},_{2}) varying _{1} = 0) remains trivially empty at all times. This behaviour is, however, unstable, because any _{1} > 0 leads to percolation (i.e. _{1,c} = 0), and we therefore do not classify the active phase as bistable. At criticality, and for _{1} ≪ 1, the density grows as ^{θ} with ^{9}. See Supplementary Fig.

In the infinite-range limit _{i,t+1} = _{t} and density _{i,t} = _{t}. Therefore, in this limit the dynamics reduces to the deterministic 0-dimensional recurrence relation_{μ}(

We have established that long-range correlated local densities ^{28}. In the thermodynamic limit

In the spirit of keeping the model as simple as possible, we consider a minimal drive in which, after every _{d}. More explicitly, the periodic drive consists of the following transformation

In Fig.

^{βL}, whereas no such a scaling is found for _{1} = 1, _{d} = 0.02,

When using the tag ‘classical DTC’, special care should be reserved for showing the defining features of this phase, namely, its rigidity and persistence^{28}. Our system is rigid in the sense that it does not rely on fine-tuned model parameters, e.g. _{1}, and that noise, either in the form of the inherently stochastic underlying DP or of a small but non-zero drive defect density _{d}, does not qualitatively change the results. Moreover, in the limit

To show that, in the limit

In Fig.

We have shown that long-range DP and its periodically driven variant can give rise to a bistable phase and a DTC, respectively. At the core of our model in Eqs. (^{39,40}. This cooperation mechanism among an infinite number of parent sites, rather than a finite one as considered in previous works on long-ranged DP^{13,41}, is the key feature allowing the emergence of the bistable phase that finds a transparent explanation in the infinite-range limit _{μ}(^{25,29}. Ultimately, this double stabilization facilitates the establishment of the DTC with infinite autocorrelation time. Remarkably, this mechanism does not rely on the equations of motion being perfectly periodic, as required for DTCs in closed MBL systems^{42}, and we expect that infinite autocorrelation times could be maintained even in the presence of aperiodic variations of the drive (although the nomenclature should be revised in this case, since the underlying discrete time symmetry would only be present on average but not for individual realizations). This is in contrast to DTCs in closed MBL systems^{42}, in which the non-ergodic dynamics hinges on the peculiar mathematical structure of the Floquet operator, which, in turns, relies on the underlying equations being perfectly periodic.

The intimate connection between bistability and DTC is, however, not a strict duality, and the boundaries of the two phases, in the equilibrium and non-equilibrium phase diagrams, respectively, do not coincide. For instance, in our analysis we found that for _{μ} and of its FPs does not guarantee the drive to switch the density _{1,c}. This issue becomes even more relevant for larger _{1,c} can approach 0 (see for instance Supplementary Fig.

While these considerations are model and parameters dependent, and it is ultimately up to numerics to find the bistable and the DTC phases, what is universal and far reaching here is the concept that long-ranged DP, and PCA more generally, can host novel dynamical phases, such as DTCs. As Yao and collaborators recently pointed out^{28}, long autocorrelation times are in fact generally unexpected in 1 + 1-dimensional PCA, because imperfections and phase slips can nucleate, spread and destroy the order. Our work proves that this fate can be avoided, and time-crystalline order established, in long-ranged PCA. These systems enable in fact an error correction mechanism, in our case intimately related to the bistability, that would be impossible if correlations were limited to a finite radius. We may speculate that, in the physical picture of a Hamiltonian system coupled to a bath, this defect suppression would correspond to the cooling rate being larger than the heating rate.

In conclusion, we have studied the effects of long-range correlated update rules in a model of DP, which we built from an analogy with the prototypical (but short-ranged) DK PCA. First, we proved that, beyond the standard active and inactive phases, a new bistable phase emerges in which the system at long times is either empty or finitely occupied depending on whether it was initially sparsely or densely occupied. Second, in a driven DP with periodic modulation of the update rules, we showed that this bistable phase intimately connects with a DTC phase, in which the density oscillates with a period twice that of the drive. In this DTC phase, the autocorrelation time scales exponentially with the system size, and in the thermodynamic limit a robust and persistent breaking of the discrete time-translation symmetry is established.

As an outlook for future research, further work on the driven DP should better assess the nature of the transition between the DTC and the trivial phase, characterize more systematically the phase diagram in other directions of the parameter space, and, most interestingly, address the role of dimensionality. Indeed, it is well known that dimensionality can facilitate the establishment of ordered phases of matter at equilibrium, and the question whether this is the case also out of equilibrium remains open. A positive answer to this question is suggested by the fact that, in ^{40,43,44}. Another interesting question regards the fate of chaos and damage spreading in long-ranged DP ^{45}. Further research should then aim to gain analytical intuition into the problem. For instance, the critical ^{41}. Finally, on a broader perspective, our work paves the way towards the study of non-equilibrium phases of matter in the uncharted territory of driven PCA, with a potentially very broad range of applications throughout different branches of science. As a timely example, Floquet PCA may provide new insights into the understanding of seasonal epidemic spreading and periodic intervention efficacy.

Here we provide further technical details on our work. In Eq. (_{i,j} between sites _{i,j} ≈ ∣

The phenomenology of the bistable phase can be understood from a graphical FP analysis of the equation _{μ}(_{μ} intersects with the bisect. (i) Inactive—if _{0} = 0, which is stable and corresponds to a completely empty state. The system moves towards this FP and _{c}, which is attractive from its right and repulsive on its left. (iii) Bistable—if _{1} > _{0} and a stable FP _{2} > _{1}. In this case, the system will reach either the unoccupied FP _{0} = 0 or the finitely occupied FP _{2} > 0 depending whether _{1} < _{1} or _{1} > _{1}, respectively. That is, the system is bistable, and the critical initial probability separating its two basins of attraction is _{1,c} = _{1} (see also Supplementary Fig. _{μ} and the bisect and gives _{c} = 0.5216(9). For _{1} and _{2} are found solving for _{μ}(_{1} = 0.3326(5) and _{2} = 0.7890(9) for

For a power–law exponent _{μ}(_{0} = 0: at long times, the system ends up in the empty, absorbing state with state variables _{i} = 0 for all sites _{c} = 0.5216(9), that is, unstable from his left and stable on his right. _{1} < _{c} and a stable FP _{2} > _{c}. Depending on whether the initial density _{1} is <_{1} or >_{1}, the system flows towards density _{0} = 0 or _{2} > 0, respectively, indicating bistability.

The FP analysis also clarifies the general features of _{μ} that allow for the emergence of bistability, that is, in fact not contingent on the choice of _{μ} made in Eq. (_{μ}(_{0} < _{1} < _{2}, of which _{0} and _{2} are stable, whereas _{1} is unstable. Put simply, _{μ} should be a nonlinear function with a graph looking qualitatively as that of Fig.

Finally, note that higher resolution and smaller fluctuations could be achieved in the figures throughout the paper if simulating larger system sizes _{d} and ^{3} h per 3 GHz core.

We are very thankful to P. Grassberger for insightful comments on the manuscript. J.K. thanks Kim Christensen for introducing him to the theory of percolation. We acknowledge support from the Imperial-TUM flagship partnership. A.P. acknowledges support from the Royal Society and hospitality at TUM. A.N. holds a University Research Fellowship from the Royal Society and acknowledges additional support from the Winton Programme for the Physics of Sustainability.

J.K. initiated the project suggesting to investigate DTCs in long-ranged DP models and to take inspiration from the DK model. A.P. proposed the model and performed the computations. A.N. made critical contributions to the analysis of the results and the preparation of the manuscript.

Open Access funding enabled and organized by Projekt DEAL.

No data sets were generated or analysed during the current study.

The codes that support the findings of this study are available at

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at