Critical Time Crystals in Dipolar Systems

We analyze the quantum dynamics of periodically driven, disordered systems in the presence of long-range interactions. Focusing on the stability of discrete time crystalline (DTC) order in such systems, we use a perturbative procedure to evaluate its lifetime. For 3D systems with dipolar interactions, we show that the corresponding decay is parametrically slow, implying that robust, long-lived DTC order can be obtained. We further predict a sharp crossover from the stable DTC regime into a regime where DTC order is lost, reminiscent of a phase transition. These results are in good agreement with the recent experiments utilizing a dense, dipolar spin ensemble in diamond [Nature (London) 543, 221 (2017)]. They demonstrate the existence of a novel, critical DTC regime that is stabilized not by many-body localization but rather by slow, critical dynamics. Our analysis shows that the DTC response can be used as a sensitive probe of nonequilibrium quantum matter.

Figure 1

(a) Ensemble of randomly positioned spins in 3D interacting via dipolar interactions. (b) Illustration of single-spin-flip processes. (c),(d) Energy level diagram for the second-order process of two spins flipping, in two regimes: (c) high frequencies (${\omega }_0\gg W$) and (d) low (${\omega }_0\ll W$) frequencies. The applied field flips a spin with magnitude $ϵ/{\tau }_1$, which costs energy $\sim h_I-m^{*}{\omega }_0$.

Figure 2

Phase diagram of the DTC obtained numerically (see [23] for details). Dotted lines indicate limiting behaviors of the phase boundary: At high driving frequencies, the phase boundary is linear, ${\tau }_1\propto |ϵ|$, while for low driving frequencies, it closes up as ${\tau }_1\propto 1/{ϵ}^2$; cf. Eq. (8). This is in good agreement with the experimental observations of Ref. [13].

Figure 3

Decay rate versus perturbation $ϵ$ for various ${\tau }_1$’s obtained numerically [23]. One sees a sharp rise of the decay rate as one crosses the DTC phase boundary [determined as the $ϵ$ for which $\mathrm{\Gamma }(ϵ,{\tau }_1)=1/100$], which is reminiscent of a phase transition.