Floquet Time Crystals

We define what it means for time translation symmetry to be spontaneously broken in a quantum system and show with analytical arguments and numerical simulations that this occurs in a large class of many-body-localized driven systems with discrete time-translation symmetry.

Figure 1

The time evolution of a short-range correlated initial state satisfies TTSB-1 for $h=0.3$. Top panel: the time dependence of the disorder-averaged $⟨{\sigma }_i^x⟩$, $⟨{\sigma }_i^y⟩$, and $⟨{\sigma }_i^z⟩$ show that the former two decay rapidly while the latter displays persistent oscillations. (The spin-flip part of the Floquet operator is here taken to be applied instantaneously.) Bottom panel: The decay of the disorder-averaged magnetization, $Z(t)$, as defined in the main text, is found to decay zero on a time scale that diverges exponentially in the system size.

Figure 2

The mutual information between the $n$ left- and rightmost sites, $F_{nn}$, for $n=2$ and $n=3$. The main panel shows results for $L=12$, as well as the extrapolated value of $F_{22}$ for $L\to \infty$. To extrapolate, we fit $F_{22}(L)=F_{22}(\infty )+c{e}^{-L/\xi }$, with $F_{22}(\infty )$, $c$, and $\xi$ fit parameters. Example fits for $h=0.1$ and $h=0.9$ are shown in the inset.