Cavendish Laboratory, University of Cambridge, Cambridge, UK.
School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, UK.
Nat Commun. 2021 Feb 16;12(1):1061. doi: 10.1038/s41467-021-21259-4.
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.
随机过程控制着科学中各种现实系统的时间演化。在马尔可夫图像中，具有空间维度概念的噪声多粒子系统的最小描述是概率元胞自动机，它通常具有时间独立和短程更新规则。在这里，我们提出了一个具有幂律相互作用的简单元胞自动机，它产生了长程定向渗流的双稳相，其长时间行为不仅由系统动力学决定，还由初始条件决定。在更新规则的周期性调制存在的情况下，我们发现系统对调制的响应时间比调制时间长，时间长到系统尺寸的指数倍。这种对基础动力学离散时间平移对称性的破坏是由长程相互作用的自校正机制实现的，该机制补偿了噪声引起的不完美性。因此，我们的工作为物质的经典离散时间晶体相提供了一个确凿的例子，并为探索驱动概率元胞自动机领域中的新型非平衡相铺平了道路。
