Classical discrete time crystals

Abstract

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: discrete time crystals (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions, and the breakdown of ergodicity. However, subharmonic responses [1] are also a well-known feature of classical dynamical systems ranging from predator-prey models [2] to Faraday waves [3] and AC-driven charge density waves [4]. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finitetemperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially-long time scales. Power-law correlations are present along a first-order line which terminates at a critical point. We analyze the transition by mapping it to the locked-to-sliding transition of a DCdriven charge density wave. Our work points to a classical limit for quantum time crystals, and raises several intriguing questions concerning the non-equilibrium universality class of the CDTC critical point. arXiv:1801.02628v1 [cond-mat.stat-mech]

Time crystallinity in dissipative Floquet systems

“…In terms of the density matrix of the system, the noise potentially translates to the probability distribution of the observable over individual realisations being broad, which we argue destroys the DTC. We show that while this is indeed the case for 1-d spin chains with local interactions and dissipation, in 2-d the dissipative processes can naturally lead to a narrow distribution and hence persistent time-crystalline order unlike in 1-d. We note that this broadening mechanism can be absent altogether in mean-field dynamics such as for fully-connected models [17][18][19][20].…”

Time Quasilattices in Dissipative Dynamical Systems

“…Discrete time crystals feature a period-doubled response to a periodic driving [7][8][9][10]. There should also be a sense of rigidity, in order to bring them in line with our intuition regarding spatial crystals [21]. The three cases of interest in the present context are as follows.…”

“…Despite lacking a conserved quantity which can be thought of as energy, it is still possible to define a temperaturelike external noise in dissipative systems, at least away from the superstable points. Robustness of the trajectories to finite temperature necessarily requires the interactions of a macroscopic number of degrees of freedom, not present here [21]. The next section addresses this issue.…”

“…classical time crystals, which break continuous time translation symmetry, was not ruled out by the no-go theorems applied to the quantum case [2,6,21]. Some leeway is built into requirement (v), since the concept of a true ground state requires energy to be conserved, which is not the case in any of the known examples of discrete time crystals.…”

“…Extending the crystal lattice analogy, we further identify time quasicrystals: systems in which the symmetry of a periodic driving is spontaneously broken to the symmetry of a time quasilattice, in which the stability is made rigid by the interactions between the macroscopic number of degrees of freedom of a quantum many-body state. We examine a number of recent experimental proposals concerning discrete time crystals in driven dissipative many-body systems [19][20][21], identifying that several additionally host time quasicrystals. We detail experimental signatures of these new states of matter.…”