Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase in both the presence and the absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly interacting regime along with the emergent integrals of motion in the dynamical system, which can be characterized by level statistics and the out-of-time-ordered correlators. We propose two cold atom experimental schemes to realize the clean Floquet time crystals, one by making use of dipolar gases and another by synthetic dimensions.
Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase in both the presence and the absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly interacting regime along with the emergent integrals of motion in the dynamical system, which can be characterized by level statistics and the out-of-time-ordered correlators. We propose two cold atom experimental schemes to realize the clean Floquet time crystals, one by making use of dipolar gases and another by synthetic dimensions.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry. The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization. Here, we consider a simple model for a one-dimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied. We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator. Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions. Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.
Recent works on the observation of discrete time-crystalline signatures throw up major puzzles on the necessity of localization for stabilizing such out-of-equilibrium phases. Motivated by these studies, we delve into a clean interacting Floquet system whose quasi-spectrum conforms to the ergodic Wigner-Dyson distribution, yet with an unexpectedly robust, long-lived time-crystalline dynamics in the absence of disorder or fine-tuning. We relate such behavior to a measure zero set of nonthermal Floquet eigenstates with long-range spatial correlations, which coexist with otherwise thermal states at near-infinite temperature and develop a high overlap with a family of translationally invariant, symmetry-broken initial conditions. This resembles the notion of ``dynamical scars'' that remain robustly localized throughout a thermalizing Floquet spectrum with fractured structure. We dub such a long-lived discrete time crystal formed in partially nonergodic systems, ``scarred discrete time crystal'' which is distinct by nature from those stabilized by either many-body localization or prethermalization mechanism.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry.The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization.Here, we consider a simple model for a onedimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied.We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator.Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions.Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.
This thesis is concerned with phases of matter, one of the central notions in condensed matter physics. Traditionally, condensed matter physics has been concerned with phases of matter in thermal equilibrium, which means it is coupled to a heat bath. The main interest of this thesis, however, is isolated systems, in which the system is allowed to reach a steady state on its own, without interacting with a heat bath. In such a context it is possible for the steady state to be non-thermal in character, leading to many new phenomena.A main interest of this thesis will be Floquet systems, which are systems that are periodically driven, for example by a time-oscillatory electric field. In this thesis, we will identify and charcterize phases of matter occuring in Floquet systems that are entirely new, in the sense that they have no analog in equilibrium.We introduce a “Floquet equivalence principle”, which states that Floquet topological phases with symmetry G are in one-to-one correspondence with stationary topological phases with additional symmetry. This allows us to leverage the existing literature on topological phases with symmetries to understand Floquet topological phases. Such phases can be stabilized in driven strongly disordered systems through the phenomenon of “many-body localization” (MBL). We discuss properties of Floquet phases such as the “pumping” of lower-dimensional topological phases onto the boundary at each time cycle.We then turn to spontaneous symmetry-breaking phases. We show that in Floquet systems, there is a striking new kind of such phase: the Floquet time crystal, in which the symmetry that is spontaneously broken is discrete time-translation symmetry. Such systems, though driven at frequency \\omega, respond at a fractional frequency \\omega/n. We show using analytical arguments and numerical evidence that such phases can be stabilized in driven strongly disordered systems through the phenomenon of “many-body localization” (MBL).Next, we show that both Floquet time crystals and Floquet topological phases can be stabilized even without disorder. We establish a new scenario for “pre-thermalization”, a phenomenon where the eventual thermalization of the system takes place at a rate that is exponentially small in a parameter. In the intermediate regime, before pre-thermalization, there is a quasi-stationary pre-thermal regime in which Floquet phases can be stabilized.In a slight digression, we then develop a systematic theory of stationary topological phases with discrete spatial symmetries (as opposed to the discrete temporal symmetry characterizing Floquet phases), showing that they also satisfy a “crystallineequivalence principle” relating phases of matter with spatial symmetry to phases of matter with internal symmetry. Our arguments are based on notions of “gauging spatial symmetries” as well as a viewpoint based on topological quantum field theory (TQFT).Finally, we put the Floquet equivalence principle on a systematic footing, and unify it with the crystalline equivalence principle for stationary topological phases, by invoking a powerful homotopy-theoretic viewpoint on phases of matter. The end result is a general theory of strongly correlated phases of matter with space-time symmetries.
The exotic phenomenon of time-translation-symmetry breaking under periodic driving---one of the main features of the so-called time crystals---has been shown to occur in many-body systems even in clean setups where disorder is absent. In this work, we propose a realization of this effect in few-body systems, both in the context of trapped cold atoms with strong interactions and of a circuit of superconducting qubits. We show how these two models can be treated in a fairly similar way by adopting an effective spin-chain description, to which we apply a simple driving protocol. We focus on the response of the magnetization in the presence of imperfect pulses and interactions, and show how the results can be interpreted, in the cold atomic case, in the context of experiments with trapped bosons and fermions. Furthermore, we provide a set of realistic parameters for the implementation of the superconducting circuit.
We propose the concept of a spontaneous many-body Floquet state. This is a state that, in the absence of external periodic driving, self-oscillates like in the presence of a periodic Hamiltonian, this behavior being spontaneously induced by many-body interactions. In addition, its quantum fluctuations are described by regular Floquet theory. Furthermore, it is also a time crystal, presenting long-range time-periodic order. However, this crystalline behavior is very different to that of conventional Floquet discrete time crystals: here, there is no external periodic driving, energy is conserved, and the nature of the spontaneous symmetry breaking is continuous instead of discrete. We demonstrate that spontaneous many-body Floquet states can emerge in a variety of canonical many-body problems, ranging from interacting fermions to Bose-Hubbard models. We specifically show that a spontaneous many-body Floquet state is a universal intrinsic state of a one-dimensional flowing atom condensate, both subsonic and supersonic, resulting from a dynamical phase transition and robust against external perturbations and quantum fluctuations, proposing also realistic experimental scenarios for its observation. A spontaneous many-body Floquet state not only represents a realization of a continuous time crystal, but also another paradigm in Floquet physics.