Absence of Quantum Time Crystals

Abstract

In analogy with crystalline solids around us, Wilczek recently proposed the idea of "time crystals" as phases that spontaneously break the continuous time translation into a discrete subgroup. The proposal stimulated further studies and vigorous debates whether it can be realized in a physical system. However, a precise definition of the time crystal is needed to resolve the issue. Here we first present a definition of time crystals based on the time-dependent correlation functions of the order parameter. We then prove a no-go theorem that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions.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 [1][2][3]. 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 [4][5][6]. While the proposal of time crystals was quite bold, it is, on the other hand, rather natural from the viewpoint of relativity: since we live in the Lorentz invariant space-time, why don't we have time crystals if there are ordinary crystals with a long-range order in spatial directions?However, the very existence, even as a matter of principle, of time crystals is rather controversial. For example, Bruno [7] and Nozières [8] discussed some difficulties in realizing time crystals. However, since their arguments were not fully general, several new realizations of time crystals, which avoid these no-go arguments, were proposed [9,10].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 equilibrium with respect to an arbitrary Hamiltonian which consists of not-too-long-range interactions. We present two theorems: one applies only to the ground state, and the other applies to the equilibrium with an arbitrary temperature.Naively, time crystals would be defined in terms of the expectation value Ô (t) of an observableÔ(t). If Ô (t) exhibits a periodic time dependence, the system may be regarded as a time crystal. However, the very definition of eigenstatesĤ|n = E n |n immediately implies that the expectation value of any Heisenberg operatorÔ(t) ≡ e iĤtÔ (0)e −iĤt in the Gibbs equilibrium ensemble is time independent. To see this, recall that the expectation value X is defined as X ≡ 0|X|0 at zero temperature T = 0 and X ≡ tr(Xe −βĤ )/Z = n n|X|n e −βEn /Z at a finite temperature T = β −1 > 0, where |0 is the ground state and Z ≡ tr[e −βĤ ] is th...

Discrete Time Crystals: Rigidity, Criticality, and Realizations

“…Subsequent work developed more precise definitions of such time translation symmetry breaking (TTSB) [6][7][8] and ultimately led to a proof of the "absence of (equilibrium) quantum time crystals" [9]. However, this proof leaves the door open to TTSB in an intrinsically out-of-equilibrium setting, and pioneering recent work [10,11] has demonstrated that quantum systems subject to periodic driving can indeed exhibit discrete TTSB [10][11][12][13]; such systems develop persistent macroscopic oscillations at an integer multiple of the driving period, manifesting in a sub-harmonic response for physical observables.…”

Phase structure of one-dimensional interacting Floquet systems. II. Symmetry-broken phases

“…This notion of time crystal is close in spirit to the attempted definition in Ref. 23, where a no-go theorem was proved concerning spontaneous breaking of continuous time translation symmetry. This no-go theorem is inapplicable to the present systems for a number of reasons, in particular our Hamiltonians explicitly break continuous time evolution symmetry.…”

Observation of a discrete time crystal

“…The proposal of such a "time crystal" 2 for time-independent Hamiltonians has led to much discussion, with the conclusion that such structures cannot exist in the ground state or any thermal equilibrium state of a quantum mechanical system [4][5][6] . A simple intuitive explanation is that quantum equilibrium states have time-independent observables by Time (100 Floquet periods) construction; thus, time translation symmetry can only be spontaneously broken in non-equilibrium systems [7][8][9][10] .…”