Clean Floquet Time Crystals: Models and Realizations in Cold Atoms

Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase in both the presence and the absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly interacting regime along with the emergent integrals of motion in the dynamical system, which can be characterized by level statistics and the out-of-time-ordered correlators. We propose two cold atom experimental schemes to realize the clean Floquet time crystals, one by making use of dipolar gases and another by synthetic dimensions.

Figure 1

(a) Schematic plot for the Floquet ladder. Green and red lines indicate the interchain [Eq. (2)] and intrachain couplings [Eq. (3)], respectively, which alternate during the binary drive. Blue dots represent occupied sites. In the time-crystal regime, the density distribution in two chains shows rigid reduced periods $2T$. (b) DMRG result for density polarization $P(t)$ [Eq. (5)] under perturbation $ϵ=0.12$ at stroboscopic time (lines are guides to the eyes). The interaction $U$ rigidifies the $2T$ periodicity, signifying a time-crystal phenomenon. Here the lattice size is $L=80$ for each chain, $\mathrm{\Delta }=0.1$, and the open boundary condition is used.

Figure 2

(a1),(a2) Histogram of $P(t)$. In the time-crystal regime, $P(t)$ shows an envelope modulation for the amplitude of $2T$-periodic oscillations. The modulation length $N_0\equiv t_0/T$ [set by $P(t_0)$ decreasing below 10% of the initial value] scales exponentially with the system size. (b1)–(b3) Spectral weight $A(\omega )$ for temporal correlation functions, where $\omega$ carries the unit $1/T$. (We plotted $L=6$ for example, and $\mathrm{\Delta }=0.1$.) (c1)–(c3) Distribution of level spacing ratios ($L=9$). It crosses from a GOE type deep in the thermalizing regime (c1) to the Poisson limit in the time-crystal regime (c3). (d1),(d2) OTOC with site $i=1$ and for different sites $j$’s. The system size is $L=7$ with the periodic boundary condition. The initial state is that one of the two chains is fully occupied.

Figure 3

The experimental setups and phase diagrams for fermionic (a) dipolar gases ($J=0.4U$, $L\to \infty$ extrapolated using $L=4$, 5, 6, 7 data [29]) and (b) $\mathrm{SU}(N)$ particles ($J=0$, $L=10$ for ${}^{173}\mathrm{Yb}$) with open boundary conditions. The phase boundary is set to that the “envelope” height of the oscillation, as shown in the inset in Fig. 2, remains above (or below) 50% for the time-crystal (or chaotic) phase during the first 200 periods.