Discrete time crystal in globally driven interacting quantum systems without disorder

Time crystals in periodically driven systems have initially been studied assuming either the ability to quench the Hamiltonian between different many-body regimes, the presence of disorder or long-range interactions. Here we propose the simplest scheme to observe discrete time crystal dynamics in a one-dimensional driven quantum system of the Ising type with short-range interactions and no disorder. The system is subject only to a periodic kick by a global magnetic field, and no extra Hamiltonian quenching is performed. We analyze the emerging time crystal stabilizing mechanisms via extensive numerics as well as using an analytic approach based on an off-resonant transition model. Due to the simplicity of the driven Ising model, our proposal can be implemented with current experimental platforms including trapped ions, Rydberg atoms, and superconducting circuits.

Figure 1

(a) Schematic diagram showing the dynamics of the individual spins in our model for a small perturbation $\epsilon$ in the spin flip. The presence of interaction helps to synchronize the spins. (b) The energy spectrum of the NN Ising model for $h=0$ and the driving scheme. The energy levels are evenly spaced with a gap $\mathrm{\Delta }E=2J$. The kick operator $K_{\frac{\pi }{2}}$ only connects two eigenstates within the same degenerate manifold, while the errors in the kick operator $K_{-\epsilon \pi }$ may couple different manifolds. However, for the case of off-resonant driving as in our protocol where the driving frequency ${\omega }_d=2\pi /\tau$ is far from the energy gap $\mathrm{\Delta }E=2J$, the effect of $K_{-\epsilon \pi }$ is switched off for $\epsilon \ll {\epsilon }^{*}$ and the DTC is robust.

Figure 2

[(a) and (b)] Stroboscopic magnetization $m^x\left(n\right)$ for $N=10$ and Fourier transforms of $m^x\left(n\right)$ for various system sizes $N$. Here $\epsilon =0.08\text{and}h=0$. In the absence of spin-spin interaction as shown in (a), the Fourier peaks for various $N$ coincide. The splitting of the two peaks equals $2\epsilon$ for all $N$ while it scales approximately as ${(\epsilon /{\epsilon }^{*})}^{m_{a}N}$ from the scaling analysis in (c) and (d) in the presence of interaction. (c) Fourier peak splitting as a function of $\epsilon$ for $h=0,J\tau =0.6$. The straight lines show the linear fits for various system sizes. (d) The linear fitting parameters of (c) as a function of $N$. Straight lines show the linear fitting of the data points.

Figure 3

[(a)–(c)] Color map of the Fourier spectrum of the stroboscopic magnetization in $x$ direction with $\epsilon$, $J\tau$, and $h/J$ as the driving parameter respectively. Here $N=14$ and the Fourier transform is performed over 1000 periods of the drive. The amplitude of the Fourier peaks is divided by the maximum amplitude for better visualization. [(d)–(f)] KL divergence of the Fourier spectrum in the corresponding upper panel. The plateaus around $\mathrm{KLD}\sim 2$ indicate the DTC phase and the vertical dash lines show the approximated phase boundary. [(g)–(i)] Cuts of the Fourier spectrum color map in (a) for $\epsilon$ from three different phases. (g) $\epsilon =0.1$ and the system is in DTC phase. Inset shows a zoom-in of the main peaks around $\omega \tau =\pi$ (the Fourier transform is carried out over ${10}^5$ periods here). The two main peaks are separated by ${(\epsilon /{\epsilon }^{*})}^{m_{a}N}$ in this phase. (h) $\epsilon =0.25$ and there is no prominent peak observed. (i) $\epsilon =0.5$ and the system oscillate with the drive. One prominent peak is observed at $\omega =0$. The spectrum is folded into $[-\pi ,\pi )$ for better visualization of the main peak.

Figure 4

[(a) and (b)] Color map of the Fourier spectrum of the stroboscopic magnetization in the $x$ direction with $h/J$ as the driving parameter for various $J\tau$. Here, $N=8$ and the Fourier transform is performed over 1000 periods.

Figure 5

[(a)–(c)] Stroboscopic magnetization as a function of time for various values of $\epsilon$ and the corresponding Fourier spectrum performed over ${10}^5$ kicking periods for $N=14,h/J=0.32,\text{and}J\tau =0.6$. The initial state is prepared in one of ferromagnetic ground states of $H_0$ with $h/J=0.32$. (d) Stroboscopic magnetization from TEBD simulation [25] for $N=80,\epsilon =0.02,J\tau =0.6,\text{and}h/J=0.32$. The lines are just guide to the eyes.

Figure 6

Color map of the KLD in the $\epsilon -h/J$ and $J\tau -h/J$ parameter space for $N=14$. The top row corresponds to the nearest-neighbor interacting case ($\alpha \to \infty$) and the interaction range increases down the row. A labels the DTC phase, B labels the regime in which the system oscillates with the drive, and C labels the regime where the drive hits the second harmonic of the system.

Figure 7

Stroboscopic magnetization as a function of time and the corresponding Fourier spectrum for various $\alpha$ and ${\epsilon }_0$. Here, $N=14,h/J=0.32,\text{and}J\tau =0.6$.