Prethermal time crystals in a one-dimensional periodically driven Floquet system

Motivated by experimental observations of time-symmetry breaking behavior in a periodically driven (Floquet) system, we study a one-dimensional spin model to explore the stability of such Floquet discrete time crystals (DTCs) under the interplay between interaction and the microwave driving. For intermediate interactions and high drivings, from the time evolution of both stroboscopic spin polarization and mutual information between two ends, we show that Floquet DTCs can exist in a prethermal time regime without the tuning of strong disorder. For much weak interactions the system is a symmetry-unbroken phase, while for strong interactions it gives its way to a thermal phase. Through analyzing the entanglement dynamics, we show that large driving fields protect the prethermal DTCs from many-body localization and thermalization. Our results suggest that by increasing the spin interaction, one can drive the experimental system into optimal regime for observing a robust prethermal DTC phase.

Figure 1

Phase diagram plotted in the parameter plane (a) the phase ${\mathrm{\Omega }}_x{\tau }_1/2\pi$ and the ratio $J/{\mathrm{\Omega }}_x$ at $\lambda =4$ without disorder $W=0$ and (b) the ratio $\lambda$ and the interaction $J/{\mathrm{\Omega }}_x$ at ${\mathrm{\Omega }}_x=41.2$ with disorder $W=4$. The time evolution of spin polarization $P\left(nT\right)$ under disorder $W=4$ in (b) for (c) different interactions $J$ at $\lambda =4$ and (d) different interaction time scales at weak interaction $J=0.5$. In the intermediate interaction regime, a DTC behavior can occur in a prethermal regime of the system without the need for fine tuning and strong disorder. The green dashed boundary line is depicted by the emergence of the steady plateau of $P\left(nT\right)$ and $I\left(nT\right)$ as in Fig. 5, while the red dashed boundary line indicates the disappearance of the steady plateau of $P\left(nT\right)$ and $I\left(nT\right)$ when $J$ increases as in Fig. 2. The parameters $\epsilon =0.038$, ${\mathrm{\Omega }}_y{\tau }_2=\pi$.

Figure 2

(a) The stroboscopic spin polarization $Z(t=nT)$ and (b) the mutual information $I\left(nT\right)$ with the increasing interaction strength $J$. A prethermal DTC phase is indicated by a robust plateau of both $Z\left(nT\right)$ and $I\left(nT\right)$ at intermediate times $nT\lesssim t^{*}$ before eventually heating up to a thermal phase after $nT>t^{*}$. The inset shows an exponential growth of $t^{*}$ vs $\mathrm{\Omega }/J$. The parameters $W=4,\lambda =4,{\mathrm{\Omega }}_x=41.2,\epsilon =0.038,{\mathrm{\Omega }}_y{\tau }_2=\pi$.

Figure 3

Dynamics of the half-chain entanglement entropy $S_L\left(t\right)$ with the initial superposition states $|{\psi }_l(t=0)\rangle$ for: (a) different interactions $J$ at ${\mathrm{\Omega }}_x=41.2,\lambda =4$ and (b) different interaction time scales $\lambda$ at ${\mathrm{\Omega }}_x=0,J=0.2$, respectively. The parameters $W=4,L=20,\epsilon =0.038,{\mathrm{\Omega }}_y{\tau }_2=\pi$.

Figure 4

Dynamics of the half-chain entanglement entropy $S_L\left(t\right)$ and the mutual information $I\left(t\right)$ between spins on opposite ends of a length chain versus time under different ${\mathrm{\Omega }}_x$ with the initial superposition states $|{\psi }_l(t=0)\rangle$ for disorder strength (a) $W=0$ and (b) $W=4$, respectively. The parameters $J=3,\lambda =4,L=20,\epsilon =0.038,{\mathrm{\Omega }}_y{\tau }_2=\pi$.

Figure 5

The nonequilibrium dynamics of the mutual information between spins on opposite ends of a length chain with the initial superposition states $|{\psi }_l(t=0)\rangle$. (a) $I\left(nT\right)$ and (b) $\langle I\rangle$ versus $J$ at $\epsilon =0.038$. (c) $I\left(nT\right)$ and (d) $\langle I\rangle$ versus $\epsilon$ at $J=3$. The parameters $\lambda =4,{\mathrm{\Omega }}_x=41.2,W=4$.

Figure 6

The subharmonic Fourier response of spin polarization $P\left(nT\right)$ with the short-range correlated state ${\psi }_s(t=0)$ using $1\le n\le 100$ Floquet cycles for (a) different interactions at $\lambda =4$ and (b) different time scale at $J=0.5$. The parameters ${\mathrm{\Omega }}_x=41.2,W=4,\epsilon =0.038,L=24$.

Figure 7

(a) The spin polarization $P\left(nT\right)$ and (b) the subharmonic Fourier response with the short-range correlated state ${\psi }_s(t=0)$ using $1\le n\le 100$ Floquet cycles. The parameters $J=3,\lambda =4,W=4,\epsilon =0.038,L=24$.

Figure 8

(a) The magnitude of spin polarization $Z\left(nT\right)$ and (b) the mutual information $I\left(nT\right)$ with different chain lengths $L$. The maximal bond dimension is taken up to 400. The parameters $W=4,\lambda =4,{\mathrm{\Omega }}_x=41.2,\epsilon =0.038,{\mathrm{\Omega }}_y{\tau }_2=\pi$.

Figure 9

(a) The magnitude of spin polarization $Z\left(nT\right)$, (b) the entanglement entropy $S_L\left(nT\right)$, and (c) the mutual information $I\left(nT\right)$ at different disorder strengths. The parameters ${\mathrm{\Omega }}_x=41.2,\lambda =4,\epsilon =0.038,L=24$.