Significance Crystals—an extremely common and important class of states of matter—have been studied intensely and fruitfully for many years. Recently, the possibility of “time crystals,” which self-organize into regular patterns of behavior in time, was proposed theoretically and has inspired important experimental discoveries. This qualitatively new class of states of matter plausibly will lead, among other things, to better clocks. New concepts are needed to describe time (or space-time) crystals, because the most straightforward attempts lead to mathematical singularities. Here, we show how an enriched version of the simplest proposed time-crystal model can be realized as a limiting case of a conventional, fully nonsingular physical system. This firmer foundation predicts characteristic behavior and should support wide-ranging generalizations.
Significance Crystals—an extremely common and important class of states of matter—have been studied intensely and fruitfully for many years. Recently, the possibility of “time crystals,” which self-organize into regular patterns of behavior in time, was proposed theoretically and has inspired important experimental discoveries. This qualitatively new class of states of matter plausibly will lead, among other things, to better clocks. New concepts are needed to describe time (or space-time) crystals, because the most straightforward attempts lead to mathematical singularities. Here, we show how an enriched version of the simplest proposed time-crystal model can be realized as a limiting case of a conventional, fully nonsingular physical system. This firmer foundation predicts characteristic behavior and should support wide-ranging generalizations.
Crystals are widely present in our daily life in the form of solid materials whose atoms self-organize in regular patterns, such as salt, diamonds, and ice. While these arrangements tend to be periodic in space, such regularities can also arise in time. By using common tools in condensed matter theory, the authors show here that time crystals can indeed metamorphose as normal crystals do, with the difference that now the structural transformation happens in the temporal and not spatial dimension.
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 are quantum many-body systems that, due to interactions between particles, are able to spontaneously self-organize their motion in a periodic way in time by analogy with the formation of crystalline structures in space in condensed matter physics. In solid state physics properties of space crystals are often investigated with the help of external potentials that are spatially periodic and reflect various crystalline structures. A similar approach can be applied for time crystals, as periodically driven systems constitute counterparts of spatially periodic systems, but in the time domain. Here we show that condensed matter problems ranging from single particles in potentials of quasicrystal structure to many-body systems with exotic long-range interactions can be realized in the time domain with an appropriate periodic driving. Moreover, it is possible to create molecules where atoms are bound together due to destructive interference if the atomic scattering length is modulated in time.
Time crystals are exotic phases of matter characterized by a broken time-translational symmetry, such that the ground state of the system evolves in time in a periodic fashion. Even though the time-crystal concept was introduced relatively recently, a related proposal can be traced back to Louis de Broglie. In his thesis, de Broglie conjectured that elementary particles may have some sort of internal clock that rules their behavior in the microscopic world. Here, I revisit de Broglie's idea and demonstrate that a special extreme case of classical mechanics yields in a natural way time-crystal type dynamics. Remarkably, it is found that time-crystal particles are characterized by a spin 4-vector that has a purely kinematic origin and is determined by the binormal of the velocity trajectory in the Bloch sphere. The dynamics of time-crystal particles is ruled by a generalized least action principle, such that the particle dynamically probes the nearby space and moves on average towards the direction that minimizes the action. I apply the theory to the case of charged particles and find that it predicts spin precession and that it provides a simple and intuitive picture for the origin of the anomalous magnetic moment of the electron. The classical formalism is recovered as an "effective theory" valid on a coarse time scale.
Continuous time crystals represent a novel state in many-body systems that can self-organize into timeperiodic oscillations without external periodic driving. Recent experiments have achieved such systems in dissipative solid-state materials, where persistent temporal order is autonomously sustained. A decisive characteristic of time crystals is their robustness, meaning that despite various disturbances, including fluctuations in internal parameters and external noise, they can still maintain rhythmic behavior, which has scientific value and echnological application potential. Although previous studies have shown that specific experimental parameters have robustness, thare is a lack of a systematic framework for quantifying and predicting their resilience to disturbances, and the underlying physics of this robustnessis still not fully understood. The key unresolved problems include how nonlinear interactions and feedback mechanisms contribute to stability, and what the critical thresholds are for parameter variations beyond which temporal order collapses. This work addresses these gaps by systematically analyzing how internal parameters and external influences affect the oscillation period and overall stability. Internally, the dynamics are determined by dipole-dipole interactions and atomic transition strengths, which define the temporal symmetry breaking that occurs in the system. Externally, the response of the system is controlled by the strength of the optical driving field and the energy dissipation rate. A key finding is the determination of an intrinsic feedback mechanism for a dynamically stabile time crystal. This mechanism plays a role in restoring force, correcting deviations caused by minor disturbances, and maintaining the coherence of oscillatory phase. Moreover, the system displays nonlinear dynamical behavior, characterized by two different states: one is stable oscillation continuing under moderate disturbance, and the other is stronger disturbance causing dynamical phase transition, resulting in switching between disordered or dynamically unstable state and stable state. These results provide a comprehensive understanding of the various behaviors observed in continuous time crystals and lay an important theoretical foundation for utilizing their unique properties in advanced applications such as quantum information processing and precision metrology.
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.