Absence of Quantum Time Crystals

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.

Figure 1

Time-dependent correlation. (a) Wigner crystal of ions in a ring threaded by an Aharonov-Bohm flux, as proposed in Ref. [4] as a possible realization of a time crystal. (b) Illustration of the time dependent correlation function, should the time crystal be indeed realized as a spontaneous rotation of the density wave (crystal) in the ground state, as proposed. The density-density correlation function between ($x_1,t_1$) and ($x_2,t_2$) must exhibit an oscillatory behavior as a function of $t_1-t_2$ for fixed $x_1$ and $x_2$.