Absolutely Stable Time Crystals at Finite Temperature

We show that locally interacting, periodically driven (Floquet) Hamiltonian dynamics coupled to a Langevin bath support finite-temperature discrete time crystals (DTCs) with an infinite autocorrelation time. By contrast to both prethermal and many-body localized DTCs, the time crystalline order we uncover is stable to arbitrary perturbations, including those that break the time translation symmetry of the underlying drive. Our approach utilizes a general mapping from probabilistic cellular automata to open classical Floquet systems undergoing continuous-time Langevin dynamics. Applying this mapping to a variant of the Toom cellular automaton, which we dub the “$\pi$-Toom time crystal,” leads to a 2D Floquet Hamiltonian with a finite-temperature DTC phase transition. We provide numerical evidence for the existence of this transition, and analyze the statistics of the finite temperature fluctuations. Finally, we discuss how general results from the field of probabilistic cellular automata imply the existence of discrete time crystals (with an infinite autocorrelation time) in all dimensions, $d\ge 1$.

Figure 1

(a) Time crystalline order parameter (e.g., stroboscopic magnetization) as a function of the error probability. The phase transition from a discrete time crystal to the disordered phase is shown for both a continuous-time, Floquet Langevin simulation of the $\pi$-Toom model with pinning potential $v=50$, 100, as well as for a direct implementation of the $\pi$-Toom PCA. We average $(-1{)}^{\lfloor t/\tau \rfloor }⟨M_A⟩$ starting at $t=3000$ for $\sim 500$ Floquet cycles, $\sim 50$ noise realizations and system size $N=32\times 32$. (b),(c) Schematic of the translation between the discrete state space of a cellular automata and the continuous state space of a Hamiltonian model. (d) Dynamics of the Langevin $\pi$-Toom model ($v=100$, $T=7$), exhibiting robust period doubling (full line even times, light dashed line odd times) with a lifetime that grows exponentially with $L$ [57]. We infer that for the system size ($L=32$) considered in panel (a), the lifetime of the $\pi$-Toom time crystal is significantly longer that the time window over which the magnetization is averaged.

Figure 2

Error probability $P_E$ versus the ratio of the pinning potential to the temperature, $v/T$, in simulations of (a) the do-nothing ($\mathcal{I}$) and Toom CAs. The dashed-red line indicates the equilibrium estimate $P_E=\frac{1}{2}\mathrm{erfc}(\sqrt{v_I/(2T)})$. (b) The $\pi$-Toom CA. While $P_E$ apparently depends on the simulated CA, as $v$ increases the $\pi$-Toom error rate converges toward the Toom error rate. In all cases, we find an exponential decay in the error rate as a function of $v/T$. Data are obtained from a $32\times 32$ system by averaging over 25 Floquet cycles after an initial evolution of 200 Floquet cycles.

Figure 3

(a) Scaled cumulants, ${\kappa }_n/(L^{3}n!)$, of the error distribution from a Floquet Langevin simulation of the $\pi$-Toom model for $L\times L\times L$ space-time volumes. The dashed curves are fits to ${\kappa }_n/L^3=c_n-b_n{L}^{-{\mu }_n}$. As shown in the Supplemental Material [57], we find $\mu >0$ indicating convergence to a finite $c_n$. Each data point was estimated from the statistics of 3000 independent Langevin trajectories ($T=5.17$, $v=100$), with $1000L\times L\times L$ blocks sampled from each trajectory. (b),(c) The connected two-point correlations of the errors for $v=100$ and $T=5.17$. The magnitude of the error correlations for a diagonal cut along space are depicted in (b), while the real-space propagation of errors is shown in (c). The color map range is rescaled by 1.0, 0.025, and 0.004 for the left, middle, and right panels, respectively.

Figure 4

Connected two-point correlations of the errors for $v=100$ and $T=5.17$ and $T=11.94$. Along different directions, the errors display fundamentally distinct correlations, which are more prominent for higher temperature.