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arXiv:2108.07272 · Physical review. B./Physical review. B

Classical approaches to prethermal discrete time crystals in one, two, and three dimensions

Andrea Pizzi, Andreas Nunnenkamp, Johannes Knolle

Published Sep 27, 2021• 44 citations
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Abstract

Prethermal discrete time crystals (pDTC) are new collective nonequilibrium phases of matter that break, up to exponentially long times, the discrete time translational symmetry of a periodic drive. Previously studied as a quantum phenomenon, this paper establishes that pDTCs also emerge for classical Hamiltonian dynamics even in their `higher-order and fractional' version. This enables efficient simulations of such phenomena in experimentally relevant settings, including a study of the interplay between dimensionality and interaction range.

DOI
10.1103/physrevb.104.094308

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Prethermal discrete time crystals (pDTC) are new collective nonequilibrium phases of matter that break, up to exponentially long times, the discrete time translational symmetry of a periodic drive. Previously studied as a quantum phenomenon, this paper establishes that pDTCs also emerge for classical Hamiltonian dynamics even in their `higher-order and fractional' version. This enables efficient simulations of such phenomena in experimentally relevant settings, including a study of the interplay between dimensionality and interaction range.
Systems subject to a high-frequency drive can spend an exponentially long time in a prethermal regime, in which novel phases of matter with no equilibrium counterpart can be realized. Because of the notorious computational challenges of quantum many-body systems, numerical investigations in this direction have remained limited to one spatial dimension, in which long-range interactions have been proven a necessity. Here, we show that prethermal nonequilibrium phases of matter are not restricted to the quantum domain. Studying the Hamiltonian dynamics of a large three-dimensional lattice of classical spins, we provide the first numerical proof of prethermal phases of matter in a system with short-range interactions. Concretely, we find higher-order as well as fractional discrete time crystals breaking the time-translational symmetry of the drive with unexpectedly large integer as well as fractional periods. Our work paves the way toward the exploration of novel prethermal phenomena by means of classical Hamiltonian dynamics with virtually no limitations on the system's geometry or size, and thus with direct implications for experiments.
Prethermal discrete time crystals (PDTCs), an emergent nonequilibrium phase of matter, have been studied in two- and higher-dimensional lattices with nearest-neighbor (NN) interactions and one-dimensional (1D) lattices with long-range interactions. However, different from prethermalization that can be observed in 1D Floquet classical spin systems with NN interactions, classical PDTCs in Floquet 1D systems with only NN interactions have not been proposed before. Here, we demonstrate the emergence of disorder-free discrete time crystals (DTCs) in 1D Floquet classical spin systems with NN interactions. We show that the thermalization time first grows exponentially as the driving frequency increases and is then saturated, which depends on the energy density of the initial state. Since thermalization of the effective Hamiltonian is slow, there is no typical prethermalization and PDTCs in the Floquet system before final thermalization. The robustness of DTC order is verified by introducing imperfect spin flip operations. Our work provides an exploration of quantum characteristics, when considering the classical counterparts of quantum phenomena, and will be helpful for further investigations of both classical and quantum prethermal systems and discrete time-crystalline order.
We demonstrate that the prethermal regime of periodically driven (Floquet), classical many-body systems can host nonequilibrium phases of matter. In particular, we show that there exists an effective Hamiltonian that captures the dynamics of ensembles of classical trajectories despite the breakdown of this description at the single trajectory level. In addition, we prove that the effective Hamiltonian can host emergent symmetries protected by the discrete time-translation symmetry of the drive. The spontaneous breaking of such an emergent symmetry leads to a subharmonic response, characteristic of time crystalline order, that survives to exponentially late times in the frequency of the drive. To this end, we numerically demonstrate the existence of classical prethermal time crystals in systems with different dimensionalities and ranges of interaction. Extensions to higher order and fractional time crystals are also discussed.
We present fresh evidence for the presence of discrete quantum time crystals in two spatial dimensions. Discrete time crystals are intricate quantum systems that break discrete time translation symmetry in driven quantum many-body systems undergoing nonequilibrium dynamics. They are stabilized by many-body localization arising from disorder. We directly target the thermodynamic limit using instances of infinite tensor network states, and we implement disorder in a translationally invariant setting by introducing auxiliary systems at each site. We discuss how such disorder can be realized in programmable quantum simulators: This gives rise to the interesting situation in which a classical tensor network simulation can contribute to devising a blueprint of a quantum simulator featuring prethermal time crystalline dynamics, one that will ultimately have to be built in order to explore the stability of this phase of matter for long times.
Discrete time crystals are periodically driven systems characterized by a response with periodicity nT, with T the period of the drive and n > 1. Typically, n is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-1/2 systems with n restricted to 2. Here, we show that a clean spin-1/2 system in the presence of long-range interactions and transverse field can sustain a huge variety of different 'higher-order' discrete time crystals with integer and, surprisingly, even fractional n > 2. We characterize these (arguably prethermal) non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.
The spontaneous breaking of time-translation symmetry has led to the discovery of a new phase of matter: the discrete time crystal. Discrete time crystals exhibit rigid subharmonic oscillations that result from a combination of many-body interactions, collective synchronization, and ergodicity breaking. This Colloquium reviews recent theoretical and experimental advances in the study of quantum and classical discrete time crystals. The breaking of ergodicity is focused upon as the key to discrete time crystals and the delaying of ergodicity as the source of numerous phenomena that share many of the properties of discrete time crystals, including the ac Josephson effect, coupled map lattices, and Faraday waves. Theoretically, there is a diverse array of strategies to stabilize time-crystalline order in both closed and open systems, ranging from localization and prethermalization to dissipation and error correction. Experimentally, many-body quantum simulators provide a natural platform for investigating signatures of time-crystalline order; recent work utilizing trapped ions, solid-state spin systems, and superconducting qubits are reviewed. Finally, this Colloquium concludes by describing outstanding challenges in the field and a vision for new directions on both the experimental and theoretical fronts.