Time crystals are proposed states of matter that spontaneously break time translation symmetry. Despite much recent interest, there is no settled definition for such states and existing definitions only tangentially refer to the well-established foundations of spontaneous symmetry breaking. We offer a definition of time crystals, which treats time translation much like any other symmetry and follows the traditional recipe for defining Wigner symmetries and order parameters. Using our definition, we find: (i) systems with time independent Hamiltonians should not exhibit time translation symmetry breaking, and (ii) the many-body localized $\ensuremath{\pi}$ spin glass/Floquet time crystal can be viewed as breaking both a global internal symmetry and the time translation symmetry, as befits the two aspects of its name.
Time crystals are proposed states of matter that spontaneously break time translation symmetry. Despite much recent interest, there is no settled definition for such states and existing definitions only tangentially refer to the well-established foundations of spontaneous symmetry breaking. We offer a definition of time crystals, which treats time translation much like any other symmetry and follows the traditional recipe for defining Wigner symmetries and order parameters. Using our definition, we find: (i) systems with time independent Hamiltonians should not exhibit time translation symmetry breaking, and (ii) the many-body localized \ensuremathπ spin glass/Floquet time crystal can be viewed as breaking both a global internal symmetry and the time translation symmetry, as befits the two aspects of its name.
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.
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.
Recently, Wilczek proposed a fascinating new concept of time crystals, which spontaneously break the continuous time translation symmetry, in analogy with ordinary crystals which break the continuous spatial translation symmetry. Li et al. soon followed with a concrete proposal for an experimental realization and observation of a (space-)time crystal using trapped ions in a ring threaded by an Aharonov-Bohm flux. However, the very existence, even as a matter of principle, of time crystals is rather controversial. Bruno proved a no-go theorem for time crystals for a specific class of Hamiltonians, which covers the original proposal. A similar observation was made for a charge density wave in a superconducting ring by Nozieres. However, since these arguments were not fully general, several new realizations of time crystals, which avoid these no-go arguments, were proposed. In fact, a part of the confusion can be attributed to the lack of a precise mathematical definition of time crystals. Here, we first propose a definition of time crystals in the equilibrium, which is a natural generalization of that of ordinary crystals and can be formulated precisely also for time crystals. We then prove generally the absence of time crystals defined as such, in the ground state of an arbitrary Hamiltonian which consists only of short-range interactions. While the nature of the time crystal is less understood, it is natural to expect that time crystals would be most easily realized, if at all, at zero temperature, as in the case of ordinary crystals. Thus, we focus on the question whether time crystals can exist as the ground state, as in Wilczek’s original proposal. Naively, time crystals would be defined in terms of the ground-state expectation value of an observable O(t). If it exhibits a periodic time dependence, the system may be regarded as a time crystal. However, the very definition of the ground state Ĥ|0〉 = E0|0〉 immediately implies that the expectation value of any Heisenberg operator O(t) ≡ eiĤtO(0)e−iĤt is time-independent. Yet it is too early to reject the idea of time crystals just from this observation, since a similar argument would preclude ordinary (spatial) crystals. One might naively define crystals from a spatially modulating expectation value of the density operator ρ(~x) = e−i ~ P ·~xρ(~0)ei ~ P ·~x. The unique ground state of the Hamiltonian in a finite box is nevertheless symmetric and hence ~ P |0〉 = 0, implying that 〈0|ρ(~x)|0〉 is constant over space.
Time crystals correspond to a phase of matter where time-translational symmetry (TTS) is broken. Up to date, they are well studied in open quantum systems, where an external drive allows us to break discrete TTS, ultimately leading to Floquet time crystals. At the same time, genuine time crystals for closed quantum systems are believed to be impossible. In this study we propose a form of a Hamiltonian for which the unitary dynamics exhibits the time crystalline behavior and breaks continuous TTS. This is based on a spin-1/2 many-body Hamiltonian which has long-range multispin interactions in the form of spin strings, thus bypassing previously known no-go theorems. We show that quantum time crystals are stable to local perturbations at zero temperature. Finally, we reveal the intrinsic connection between continuous and discrete TTS, thus linking the two realms.