Abstract
Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy manyparticle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature timeindependent and shortranged update rules. Here, we propose a simple cellular automaton with powerlaw interactions that gives rise to a bistable phase of longranged directed percolation whose longtime 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 selfcorrecting mechanism of the longranged interactions which compensates noiseinduced 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 nonequilibrium phases in the unexplored field of driven probabilistic cellular automata.
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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 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 shortranged update rules. To account for certain realistic situations, e.g. of longdistance travels in epidemic spreading, DP has been extended to longranged 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 nonequilibrium physics that has recently experienced a tremendous amount of excitement—that of discrete time crystals (DTCs)^{14,15,16,17,18,19,20}. In essence, DTCs are systems that, under the action of a timeperiodic modulation with period T, exhibit a periodic response at a different period \(T^{\prime} \ne T\), thus breaking the discrete timetranslational symmetry of the drive and of the equations of motion. DTCs thus extend the fundamental idea of symmetry breaking^{21} to nonequilibrium phases of matter. Following the pioneering proposals in the context of manybodylocalized (MBL) systems^{17,18}, DTCs have been observed experimentally^{22,23}, and their notion has been extended beyond MBL^{24,25,26,27}.
More recently, Yao and collaborators have fleshed out the essential ingredients of a classical DTC phase of matter^{28}. Namely, in a classical DTC, manybody 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 longknown deterministic dynamical systems. Despite various efforts^{28,29,30,31}, 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 noiseinduced imperfections generically nucleate and spread, destroying true infiniterange 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 longranged DP, enabled by the ability of longrange 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 nonequilibrium phase, the system is able to selfcorrect noiseinduced errors and the autocorrelation time grows exponentially with the system size, thus becoming infinite in the thermodynamic limit. In analogy to the onedimensional Ising model for which, at equilibrium, longrange interactions enable a normally forbidden finitetemperature magnetic phase^{33,34}, in our model, out of equilibrium, the longrange interactions lead to a classical timecrystalline phase. Crucially, our results appear naturally in a minimal model of longranged DP but are expected to find applications in many different contexts of dynamical manybody 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 longrange nature is the key ingredient for the emergence of a bistable phase. Finally, we include a periodic drive for the longranged 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.
Results
Review of DP
We consider a triangular lattice in which one dimension can be interpreted as discrete space i and the other one as discrete time t = 1, 2, 3, … , see Fig. 1. To implicitly account for the triangular nature of the lattice, i runs over integers and halfintegers at odd and even times t, respectively. We denote L the spatial system size and are interested in the thermodynamic limit L → ∞. The site i at time t can be either occupied or empty, s_{i,t} = 0,1. 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_{1} > 0. A DP process is defined by a stochastic Markovian update rule with which, starting from the initial generation \({\{{s}_{i,1}\}}_{i}\), all subsequent generations \({\{{s}_{i,t}\}}_{i}\) are obtained one by one. The main observable we will focus on is the global density n(t) (henceforth just referred to as density for brevity) defined as
where the inner and outer brackets denote average over the L sites and over R independent runs, respectively. Since n(1) = p_{1}, we will often refer to p_{1} as initial density.
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 longranged DP.
In the DK model, the probability of site i to be occupied at time t depends on the state of its neighbours i ± 1/2 at previous time t − 1. More specifically, as summarized in Fig. 1a, site i is: (i) empty if both its neighbours were empty, (ii) occupied with probability q_{1} if one and just one of its neighbours was occupied, and (iii) occupied with probability q_{2} if both its neighbours were occupied. To account for these possibilities in a compact fashion, we define a local density n_{i,t} as
and say that site i at time t is occupied with a probability p_{i,t} given by
In other words, the probability p_{i,t} is a nonlinear function \({f}_{{q}_{1},{q}_{2}}({n}_{i,t})\) of the local density n_{i,t}, with domain {0, 0.5, 1}. Since n_{i,t} only involves the nearest neighbours of site i, the DK model of DP is obviously ‘shortranged’. In essence, s_{i,t} is a Bernoullian random variable of parameter p_{i,t}, which we compactly denote \({s}_{i,t} \sim \,\text{Bernoulli}\,({p}_{i,t})\). The complexity of this 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 state of the system at previous time t − 1. Equipped with a random number generator, one can obtain all the generations one by one according to the above procedure, as schematically illustrated in the flowchart of Fig. 1b. Reiterating for several independent runs, one finally obtains the time series of the density n in Eq. (1).
The DK model features two dynamical phases, shown in Fig. 1c, d. In the inactive phase, for small enough probabilities q_{1} and q_{2}, the system eventually reaches the completely unoccupied absorbing state, that is, no percolation occurs. In the active phase instead, for large enough probabilities q_{1} and q_{2}, a finite fraction of sites remains occupied up to infinite time, that is, the system percolates. For small initial probability p_{1} ≪ 1, the critical line separating the two phases is characterized by a power–law growth of the density^{36}, n ~ t^{θ}, with exponent θ ≈ 0.31. As conjectured by Grassberger^{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 manybody systems^{38} can be extended to nonequilibrium phenomena.
Important for our work is that, in the DK model, whether the system percolates or not depends on the parameters q_{1} and q_{2} but not on the initial density p_{1}, at least as long as p_{1} > 0. Indeed, the phase boundaries for initial densities p_{1} = 0.01 and p_{1} = 1 in Fig. 1c, d, respectively, coincide.
Longranged percolation and bistability
As the vast majority of PCA, the DK model features shortranged update rules^{9}. In realistic systems, however, it is often the case that the occupation of a site i is influenced not only by the neighbouring sites but also by farther sites j, with an effect decreasing with the distance r_{i,j} between the sites. Building on an analogy with the DK model, we propose here a model for such a ‘longranged’ DP, whose protocol is explained in the flowchart of Fig. 2a. Specifically, we consider as a local density n_{i,t} a power–lawweighted average of the previous generation \({\{{s}_{j,t1}\}}_{j}\) centred around site i
where the normalization factor \({{\mathcal{N}}}_{\alpha ,L}\) ensures n_{i,t} = 1 if all sites j are occupied and the adjective ‘local’ emphasizes the site dependence. The occupation probability p_{i,t} then depends on the local density n_{i,t} through some nonlinear function f_{μ} that for concreteness we consider to be
with μ ∈ (0, 1) a control parameter. The whole DP dynamics is determined via the occupations \({s}_{i,t} \sim \,\text{Bernoulli}\,({p}_{i,t})\) and reiterating from one generation to the next. Note, our findings are not contingent on the specific choice of Eqs. (4) and (5) but are rather expected to hold generally for a broad class of longranged forms of the densities n_{i,t} and of functions f_{μ}—see ‘Methods’ section for details.
We emphasize that Eqs. (4) and (5) and the flowchart in Fig. 2a are a natural generalization of Eqs. (2) and (3) and Fig. 1b, respectively. Furthermore, whereas in the DK model the control parameters are the probabilities q_{1} and q_{2}, the control parameter is now μ. As an important difference, now the domain of f_{μ} accounts for several (and αdependent) values of 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 longranged local density n_{i,t} in Eq. (4) has profound implications. Arguably, the most dramatic is the appearance of a bistable phase, in addition to the standard active and inactive ones. In the bistable phase, the ability of the system to percolate depends on the initial density p_{1}, see the red lines in Fig. 2b, c. That is, the bistable phase features two basins of attraction, resulting into an asymptotically vanishing or finite n, respectively, and separated by some critical initial density p_{1,c} > 0. To characterize systematically the dynamical phases of our model, we plot in Fig. 2d, e the longtime density n(t = 10^{3}) as a suitable order parameter in the plane of the power–law exponent α and control parameter μ. Comparing the results obtained for a large and a small initial density p_{1}, it is possible to sketch a phase diagram composed of three phases: (i) inactive—n decays to 0 at long times; (ii) active—n does not decay at long times; (iii) bistable—n either decays or not depending on p_{1} being small or large. The existence of this bistable phase is in striking contrast with shortranged models of DP such as the DK model and in fact appears only for α ⪅ 2, that is, when the local densities \({\{{n}_{i,t}\}}_{i}\) are correlated over a sufficiently long range. To understand the origin of this rich phenomenology, we study the short and infiniterange limits of our DP process.
In the shortrange limit α → ∞, the local densities n_{i,t} reduce to the averages of the nearestneighbour occupations \({s}_{i\frac{1}{2},t1}\) and \({s}_{i+\frac{1}{2},t1}\), that is, Eq. (4) recasts into Eq. (2) and the DK model is recovered. In the notation of Eq. (3), the DK parameters are q_{1} = f_{μ}(0.5) and q_{2} = f_{μ}(1). Therefore, we can move across the DK parameter space (q_{1},q_{2}) varying μ, going from the inactive phase (\(\mu \;<\;{\mu }_{{\rm{c}}}^{\infty }\)) to the active one (\(\mu \;> \;{\mu }_{{\rm{c}}}^{\infty }\)), and no bistable phase is possible. We find that the transition happens at a critical \({\mu }_{{\rm{c}}}^{\infty }=0.85(7)\). Note that, in the active phase, a completely empty state (p_{1} = 0) remains trivially empty at all times. This behaviour is, however, unstable, because any p_{1} > 0 leads to percolation (i.e. p_{1,c} = 0), and we therefore do not classify the active phase as bistable. At criticality, and for p_{1} ≪ 1, the density grows as n ~ t^{θ} with θ = 0.3(0), as expected for the DP universality class^{9}. See Supplementary Fig. 2 for details.
In the infiniterange limit α → 0, and more generally for α ≤ 1, the factor \({{\mathcal{N}}}_{\alpha ,L}\) in Eq. (4) diverges as L → ∞. Correspondingly, spatial stochastic fluctuations are suppressed, that is, 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 dynamics reduces to the deterministic 0dimensional recurrence relation
The system asymptotic behaviour can then be understood from the analysis of the fixed points (FPs) of the equation x = f_{μ}(x), which is detailed in the ‘Methods’ section.
Driven percolation and time crystals
We have established that longrange correlated local densities \({\{{n}_{i,t}\}}_{i}\) give rise to a bistable phase. We now show how, in a driven DP with periodically modulated update rules, this phase intimately relates to the emergence of a classical DTC. In this phase, as we shall see, the density n displays oscillations over a period larger than that of the drive and up to a time that, thanks to the longrange interactions and despite the presence of multiple sources of noise, is exponentially large in the system size, a feature that would generally be forbidden in shortranged PCA^{28}. In the thermodynamic limit L → ∞, these subharmonic oscillations are therefore persistent, that is, the system autocorrelation time diverges to infinity, breaking the timetranslational symmetry and proving a classical DTC in a periodically driven PCA.
In the spirit of keeping the model as simple as possible, we consider a minimal drive in which, after every T iterations of the DP in Eqs. (4) and (5), empty sites are turned into occupied ones and vice versa, making the full equations of motion periodic with period T. As a further source of imperfections, adding to the underlying noisy DP, we also account for faulty swaps with probability p_{d}. More explicitly, the periodic drive consists of the following transformation
In Fig. 3a, b, we show the spatiotemporal pattern of single instances of the driven DP, alongside with the density n averaged over several independent runs. If the DP is shortranged enough, the spatiotemporal pattern at long times looks similar from one period to the next, that is, the density n synchronizes with the drive and eventually picks a periodicity T. On the contrary, for a longranged enough DP, the system keeps alternating at every period between a densely occupied regime and a sparsely occupied one, and n oscillates with period 2T, that is, the system breaks the discrete timetranslation symmetry of the equations of motion.
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 finetuned model parameters, e.g. μ, α or the initial density p_{1}, and that noise, either in the form of the inherently stochastic underlying DP or of a small but nonzero drive defect density p_{d}, does not qualitatively change the results. Moreover, in the limit L → ∞, our DTC is truly persistent. Indeed, one might expect that the accumulation of stochastic mistakes introduces phase slips and eventually leads to the (possibly slow but unavoidable) destruction of the subharmonic response. Although this expectation is generally correct for shortranged DP models, including our model at large α, it can fail for longranged DP models.
To show that, in the limit L → ∞, the lifetime of our DTC is infinite, we perform a scaling analysis comparing results for increasing system sizes L. First, we introduce an order parameter Φ(t), henceforth called subharmonicity, that is defined at stroboscopic times t = 1, 1 + T, 1 + 2T, … as
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 2T, 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 subharmonicity of n in time and to perform the scaling analysis.
In Fig. 3c, we show Φ(t) for various system sizes L. For both α = 1.4 and α = 1.8, the subharmonicity decays exponentially in time, \({{\Phi }}(t) \sim \exp (\frac{t1}{\tau T})\). As shown in Fig. 3d, these two values of α are, however, crucially different in how the lifetime τT scales with the system size. In fact, τT is approximately independent of L for α = 1.8, whereas it scales exponentially as \(\tau \sim \exp (\beta L)\) for α = 1.4, for which the decay of the subharmonicity is therefore just a finitesize effect. The scaling coefficient β quantifies the time crystallinity of the system and can thus be used to obtain a full phase diagram as a function of the power–law exponent α, in Fig. 3e. We observe a phase transition between a DTC and a trivial phase at α ≈ 1.7. That is, if the DP is sufficiently longranged (α ⪅ 1.7), 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 shorterrange DP (α ⪆ 1.7), β ≈ 0 independently of L and the subharmonic response is always dynamically destroyed.
Discussion
We have shown that longrange 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. (4) and (5) is the idea that the occupation of a given site depends on the state of all the other sites at the previous time. In this sense, our model is reminiscent of some SIRtype models of epidemic spreading in which not only a sick site can infect a susceptible site, but several infected sites can also cooperate to weaken a susceptible site and finally infect it^{39,40}. This cooperation mechanism among an infinite number of parent sites, rather than a finite one as considered in previous works on longranged DP^{13,41}, is the key feature allowing the emergence of the bistable phase that finds a transparent explanation in the infiniterange limit α → 0, where it corresponds to the equation x = f_{μ}(x) having two stable FPs. Bistability also provides intuition on the origin of the DTC, to which it is deeply connected. Indeed, the drive in Eq. (7) switches the system from a densely occupied regime to a sparsely occupied one (and vice versa). If the underlying DP is bistable, these regimes fall each within different basins of attraction and can therefore be both stabilized by the contractive dynamics^{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 nonergodic 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 nonequilibrium phase diagrams, respectively, do not coincide. For instance, in our analysis we found that for μ = 0.9 the bistable phase extends up to α ≈ 1.6, whereas the DTC stretches slightly farther, up to α ≈ 1.7. The origins of this imperfect correspondence can be traced back to two competing effects. On the one hand, bistability may not be sufficient to stabilize a DTC. This can already be understood in the limit α → 0, in which the asymmetry of f_{μ} and of its FPs does not guarantee the drive to switch the density n from one basin of attraction to the other, that is, across the critical probability p_{1,c}. This issue becomes even more relevant for larger α, for which the asymmetry is possibly accentuated and p_{1,c} can approach 0 (see for instance Supplementary Fig. 1). On the other hand, a perfect bistability may not even be necessary for a DTC to exist. In fact, for the stabilization of a DTC, it may be sufficient that, of the densely and sparsely occupied regimes of the underlying DP, only one is stable, and the other is just weakly unstable (that is, metastable), meaning that the time scales of the dynamics of the density n in the two regimes are very different. Loosely speaking, the stability of one regime might be able to compensate for the weaker instability of the other, resulting in an overall stable DTC. The asymmetry of the underlying DP and the mismatch between the bistable phase and the DTC highlight the purely dynamical nature of the latter, that cannot ‘piggyback’ on any underlying symmetry.
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 longranged 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 + 1dimensional PCA, because imperfections and phase slips can nucleate, spread and destroy the order. Our work proves that this fate can be avoided, and timecrystalline order established, in longranged 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 longrange correlated update rules in a model of DP, which we built from an analogy with the prototypical (but shortranged) 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 timetranslation 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 D + 1dimension with D ≥ 2, bistability can emerge even in shortranged models of DP^{40,43,44}. Another interesting question regards the fate of chaos and damage spreading in longranged DP ^{45}. Further research should then aim to gain analytical intuition into the problem. For instance, the critical α separating the various phases may be located using a field theoretical approach, which has been successful in similar contexts in the past^{41}. Finally, on a broader perspective, our work paves the way towards the study of nonequilibrium 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.
Methods
Here we provide further technical details on our work. In Eq. (4), we considered as distance r_{i,j} between sites i and j
where the tangent accounts for periodic boundary conditions and makes the distance of the farthest sites with ∣i − j∣ = L/2 artificially diverge. This divergence is expected to reduce finitesize effects without changing the underlying physics, that is, in fact dominated by sites with ∣i − j∣ ≪ L, for which we get a natural r_{i,j} ≈ ∣i − j∣. Indeed, as we checked, similar results are obtained with \({r}_{i,j}=\min ( ij ,L ij )\). The Kaclike normalization factor \({{\mathcal{N}}}_{\alpha ,L}\) reads instead
The phenomenology of the bistable phase can be understood from a graphical FP analysis of the equation f_{μ}(x) = x illustrated in Fig. 4, which explains the dynamics for α < 1. Three scenarios are possible and interpreted in terms of the ways the graph of the function f_{μ} intersects with the bisect. (i) Inactive—if \(\mu \;<\;{\mu }_{{\rm{c}}}^{0}\), the only FP is x_{0} = 0, which is stable and corresponds to a completely empty state. The system moves towards this FP and \({p}_{t}\underset{\,}{\overset{t\to \infty }{\to }}0\). (ii) Critical—if \(\mu ={\mu }_{{\rm{c}}}^{0}\), a new semistable FP emerges at x_{c}, which is attractive from its right and repulsive on its left. (iii) Bistable—if \(\mu \;> \;{\mu }_{{\rm{c}}}^{0}\), the semistable FP splits into an unstable FP x_{1} > x_{0} and a stable FP x_{2} > x_{1}. In this case, the system will reach either the unoccupied FP x_{0} = 0 or the finitely occupied FP x_{2} > 0 depending whether p_{1} < x_{1} or p_{1} > x_{1}, respectively. That is, the system is bistable, and the critical initial probability separating its two basins of attraction is p_{1,c} = x_{1} (see also Supplementary Fig. 1). The critical value \({\mu }_{{\rm{c}}}^{0}\) is obtained numerically solving for the condition of tangency between the graph of f_{μ} and the bisect and gives \({\mu }_{{\rm{c}}}^{0}=0.6550(8)\) and x_{c} = 0.5216(9). For \(\mu \;> \;{\mu }_{{\rm{c}}}^{0}\), the FPs x_{1} and x_{2} are found solving for f_{μ}(x) = x, and, for instance, we find x_{1} = 0.3326(5) and x_{2} = 0.7890(9) for μ = 0.8.
The FP analysis also clarifies the general features of f_{μ} that allow for the emergence of bistability, that is, in fact not contingent on the choice of f_{μ} made in Eq. (5). Indeed, the only requirement is that, for some parameter(s) μ, the equation f_{μ}(x) = x has three FPs x_{0} < x_{1} < x_{2}, of which x_{0} and x_{2} are stable, whereas x_{1} is unstable. Put simply, f_{μ} should be a nonlinear function with a graph looking qualitatively as that of Fig. 4c. This condition guarantees a bistable phase for α < 1, which can then possibly extend to α ≥ 1 and, in the presence of a periodic drive, facilitate the establishment of a DTC.
Finally, note that higher resolution and smaller fluctuations could be achieved in the figures throughout the paper if simulating larger system sizes L and/or considering a larger number of independent runs R. This could, for instance, allow a more accurate characterization of both the equilibrium and the nonequilibrium phase diagrams of our model, which could be explored in other directions of the parameter space for varying α, μ, p_{d} and T. This would, however, require a formidable numerical effort and goes therefore beyond the scope of this work. As a reference, for instance, the generation of Fig. 3e for the parameters considered therein requires a computing time of approximately 4 × 10^{3} h per 3 GHz core.
Data availability
No data sets were generated or analysed during the current study.
Code availability
The codes that support the findings of this study are available at https://figshare.com/articles/software/Code/13468836.
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Acknowledgements
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 ImperialTUM 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.
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J.K. initiated the project suggesting to investigate DTCs in longranged 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.
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Pizzi, A., Nunnenkamp, A. & Knolle, J. Bistability and time crystals in longranged directed percolation. Nat Commun 12, 1061 (2021). https://doi.org/10.1038/s41467021212594
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DOI: https://doi.org/10.1038/s41467021212594
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