Fig. 3: Discrete time crystals in driven long-ranged directed percolation.

a, b Single instances of the periodically driven DP, alongside with the density n averaged over multiple independent runs, for L = 500 sites. a For a power–law exponent α = 1.4, n oscillates subharmonically with a period that is twice that of the drive, whereas, for α = 1.8, n eventually picks the periodicity T enforced by the drive. c For finite system sizes L, the subharmonicity Φ(t) decays as \({{\Phi }}(t) \sim \exp (-\frac{t-1}{\tau T})\) due to the accumulation of phase slips, and, after a few time scales τT, the density n synchronizes 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 = 1.4 (blue), the lifetime τ scales exponentially with the system size, τ ~ e^βL, whereas no such a scaling is found for α = 1.8. 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 is finite, indicating that in the thermodynamic limit L → ∞ the subharmonic response is persistent and a DTC with infinite autocorrelation time emerges. On the contrary, β ≈ 0 for large α, indicating a trivial dynamical phase in which no stable subharmonic dynamics is established. Here we considered p₁ = 1, μ = 0.9, p_d = 0.02, T = 20 and R = 2000.