Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Abstract Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Time crystals are time-periodic self-organized structures postulated by Frank Wilczek in 2012. While the original concept was strongly criticized, it stimulated at the same time an intensive research leading to propositions and experimental verifications of discrete (or Floquet) time crystals-the structures that appear in the time domain due to spontaneous breaking of discrete time translation symmetry. The struggle to observe discrete time crystals is reviewed here together with propositions that generalize this concept introducing condensed matter like physics in the time domain. We shall also revisit the original Wilczek's idea and review strategies aimed at spontaneous breaking of continuous time translation symmetry.
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 idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an intrinsically out-of-equilibrium many-body phase of matter. We include a compendium of necessary background, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. We formalize the notion of a time-crystal as a stable, macroscopic, conservative clock --- explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. We also cover a range of related phenomena, including various types of long-lived prethermal time-crystals, and expose the roles played by symmetries -- exact and (emergent) approximate -- and their breaking. We clarify the distinctions between many-body time-crystals and other ostensibly similar phenomena dating as far back as the works of Faraday and Mathieu. En route, we encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in their ground states, together with a theorem ruling this out. We also analyze pioneering recent experiments detecting signatures of time crystallinity in a variety of different platforms, and provide a detailed theoretical explanation of the physics in each case. In all existing experiments, the system does not realize a `true' time-crystal phase, and we identify necessary ingredients for improvements in future experiments.
Experimental advances have allowed for the exploration of nearly isolated quantum many-body systems whose coupling to an external bath is very weak. A particularly interesting class of such systems is those that do not thermalize under their own isolated quantum dynamics. In this review, we highlight the possibility for such systems to exhibit new nonequilibrium phases of matter. In particular, we focus on discrete time crystals, which are many-body phases of matter characterized by a spontaneously broken discrete time-translation symmetry. We give a definition of discrete time crystals from several points of view, emphasizing that they are a nonequilibrium phenomenon that is stabilized by many-body interactions, with no analog in noninteracting systems. We explain the theory behind several proposed models of discrete time crystals, and compare several recent realizations, in different experimental contexts.
The engineering of new states of matter through Floquet driving has revolutionized the field of condensed matter physics. This technique enables the creation of hybrid topological states and ordered phases that are absent in normal systems. Crystalline structures, exemplifying spatially ordered systems under periodic driving, have been extensively studied. However, recent focus has shifted towards discrete time crystals (DTCs), periodically driven quantum many-body systems that break time translation symmetry under specific conditions. In this paper, the model of discrete time crystals is extended to allow for the formation of time-varying tesseracts, allowing for the investigation of time translational symmetry in pseudo-higher-dimensional lattice systems.