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.
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.
A model demonstrating the existence of a thermodynamically stable quantum time-space crystal has been proposed and studied. This state is characterized by an order parameter periodic in both real and imaginary times. The average of the order parameter over phases of the oscillations vanishes but correlation functions of two or more order parameters show nondecaying oscillations. An alternative interpretation of the results is based on the concept of an operator order parameter introduced for this purpose. The model studied here has been suggested previously, in particular, for describing the pseudogap state in superconducting cuprates. Although many properties of the time-space crystal considered here are close to those of a well known DDW state, static magnetic moments oscillating at $(\ensuremath{\pi},\ensuremath{\pi})$ do not exist. Instead, $\ensuremath{\delta}$ peaks at finite energies are predicted in the cross-section of inelastic spin-polarized neutron scattering.
Temperature is a fundamental thermodynamic variable for matter. Physical observables are often found to either increase or decrease with it, or show a non-monotonic dependence with peaks signaling underlying phase transitions or anomalies. Statistical field theory has established connection between temperature and time: a quantum ensemble with inverse temperature β is formally equivalent to a dynamic system evolving along an imaginary time from 0 to iβ in the space one dimension higher. Here we report that a gas of hard-core bosons interacting with a thermal bath manifests an unexpected temperature-periodic oscillation of its macroscopic observables, arising from the microscopic origin of space-time locked translational symmetry breaking and crystalline ordering. Such a temperature crystal, supported by quantum Monte Carlo simulation, generalizes the concept of purely spatial density-wave order to the imaginary time axis for Euclidean action.
In this work we introduce boundary time crystals. Here continuous time-translation symmetry breaking occurs only in a macroscopic fraction of a many-body quantum system. After introducing their definition and properties, we analyze in detail a solvable model where an accurate scaling analysis can be performed. The existence of the boundary time crystals is intimately connected to the emergence of a time-periodic steady state in the thermodynamic limit of a many-body open quantum system. We also discuss connections to quantum synchronization.
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.
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.