We investigate a time-independent many-boson system, whose ground states are quasi-degenerate and become infinitely degenerate in the thermodynamic limit. Out of these quasi-degenerate ground states we construct a quantum state that evolves in time with a period that is logarithmically proportional to the number of particles, that is, T ∼ log N. This boson system in such a state is a quantum time crystal as it approaches the ground state in the thermodynamic limit. The logarithmic dependence of its period on the total particle number N makes it observable experimentally even for systems with very large number of particles. Possible experimental proposals are discussed.
We investigate a time-independent many-boson system, whose ground states are quasi-degenerate and become infinitely degenerate in the thermodynamic limit. Out of these quasi-degenerate ground states we construct a quantum state that evolves in time with a period that is logarithmically proportional to the number of particles, that is, T ∼ log N. This boson system in such a state is a quantum time crystal as it approaches the ground state in the thermodynamic limit. The logarithmic dependence of its period on the total particle number N makes it observable experimentally even for systems with very large number of particles. Possible experimental proposals are discussed.
We discuss the possibility of making a quasi time crystal. A simple two-state model is studied to clarify our definition. In a superposition of the ground state and the excited state and the probability of observation varies periodically in time during the lifetime of the excited state. The quasi time crystal is also discussed around the first order quantum phase transition, which is characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. Our results have broad validity. As an example, the one-dimensional transverse field Ising model with surface fields is shown to have similar behavior. The oscillating magnetization profile is solved exactly.
It is widely believed that ground-state time crystals are not realizable in realistic macroscopic systems. In particular, Watanabe and Oshikawa proved a theorem that implies the absence of the time-dependent long-range order (TDLRO) in the ground states of short-range many-body systems. However, this theorem does not forbid the presence of the ground-state TDLRO for macroscopic quantities. In this work, we investigate a simple bosonic model with a squeezed ground state and point out that the time-dependence of the ground-state TDLRO for the number operator is proportional to the square of the average number in the infinite-squeezing limit, or equivalently, the infinite average-number limit. This result implies the presence of the TDLRO for macroscopic boson number. We also discuss the physical implementations in optical, spin, and tight-binding systems, including the variants. We find an example with the macroscopic TDLRO whose essence is low-lying-state physics and another with marginal TDLRO at the quantum critical point. In addition, we reconsider the definition of the ground-state time crystal in terms of the Floquet picture.
Quantum field theories of many-body systems predict ``instantonic crystals'' connecting degenerate thermodynamic states with classically broken symmetries. The instantons break continuous translation symmetry of Euclidean space along the axis of imaginary time. The possibility for the emergence of ``thermodynamic quantum time crystals'' that extremize the Euclidean action of the system appeals to the imagination. However, the authors prove here that at least two general classes of ``instantonic crystals'', connecting states of Fermi systems with charge, spin, or superconducting orders can constitute neither thermodynamic equilibrium nor the ground state.
We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from H^0 to H^1 in a total evolution-time τ, and our goal is to determine how the final state of the system depends on τ. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large τ, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in τ with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.
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.
A time crystal is a state of periodically driven matter that breaks discrete time-translation symmetry.Time crystals have been demonstrated experimentally in various programmable quantum simulators, and they exemplify how nonequilibrium, driven quantum systems can exhibit intriguing and robust properties absent in systems at equilibrium.These robust driven states need to be stabilized by some mechanism, with the preeminent candidates being many-body localization and prethermalization.This introduces additional constraints that make it challenging to experimentally observe time crystallinity in naturally occurring systems.Recent theoretical work has developed the notion of prethermalization without temperature, expanding the class of time-crystal systems to explain time-crystalline observations at (or near) infinite temperature.In this work, we conclusively observe the emergence of a prethermal Uð1Þ time-crystalline state at quasi-infinite temperature in a solid-state NMR quantum emulator by verifying the requisites of prethermalization without temperature.In addition to observing the signature period-doubling behavior, we show the existence of a long-lived prethermal regime whose lifetime is significantly enhanced by strengthening an emergent Uð1Þ conservation law.Not only do we measure this enhancement through the global magnetization, but we also exploit on-site disorder to measure local observables, ruling out the possibility of many-body localization and confirming the emergence of long-range correlations.