Higher-order discrete time crystals in a quantum chaotic top

TL;DR

The paper investigates whether the simplest quantum chaotic system, the all-to-all coupled spin-$1/2$ kicked-top model, can host higher-order discrete time crystals. Using Floquet analysis, mean level-spacing statistics, magnetization dynamics, linear entropy, and out-of-time-ordered correlators, the authors identify robust $2$-DTC and dynamical freezing around alternating rotationally symmetric points, and uncover a $4$-DTC phase that arises in the regular regime at larger angular momentum. The $4$-DTC is explained by semiclassical phase-space structures, specifically spiral saddle-node dynamics, and is accompanied by decreasing linear entropy, signaling enhanced stability; dynamical conservation emerges for the $2$-DTC and DF phases via OTOCs but not for the $4$-DTC. These findings show higher-order DTCs in a minimal chaotic model, linking quantum dynamics to classical phase-space features and suggesting routes toward quantum simulations and metrology.

Abstract

We characterize various dynamical phases of the simplest version of the quantum kicked-top model, a paradigmatic system for studying quantum chaos. This system exhibits both regular and chaotic behavior depending on the kick strength. The existence of the $2$-DTC phase has previously been reported around the rotationally symmetric point of the system, where it displays regular dynamics. We show that the system hosts robust $2$-DTC and dynamical freezing (DF) phases around alternating rotationally symmetric points. Interestingly, we also identify $4$-DTC phases that cannot be explained by the system's $\mathbb{Z}_2$ symmetry; these phases become stable for higher values of angular momentum. We explain the emergence of higher-order DTC phases through classical phase portraits of the system, connected with spin coherent states (SCSs). The $4$-DTC phases appear for certain initial states that are close to the spiral saddle points identified in the classical picture. Moreover, the linear entropy decreases as the angular momentum increases, indicating enhanced stability of the $4$-DTC phases. We also find an emergent conservation law for both the $2$-DTC and DF phases, while dynamical conservation arises periodically for the $4$-DTC phases.

Figures (7)

• Figure 1: Variation of mean level spacing ratio as a function of $k$ and $p$. For $k\lesssim3$, the system shows regular behavior for any $p$. At integer multiples of $\pi$ in $p$, the system remains regular for any value of $k$. For other values of $p$, regular-to-chaotic crossover appears for $k\approx3$. Here, $j=1000$.
• Figure 2: (a) Variation of the average magnetization, $\langle J_z \rangle$, with stroboscopic time for $p = 2\pi$ and $k = 6$ for two different initial states (spin polarized and SCS), illustrating dynamical freezing (DF). (b) The corresponding Poincaré surface plot obtained from classical dynamics for one initial condition and $300$ time steps with $k=6$. (c) Same as (a) but for $p = \pi$, exhibiting the $2$-DTC phase. (d) The corresponding Poincaré surface plot for the $2$-DTC case, showing transitions between two degenerate states. For both (a) and (c), the initial value of $\langle J_z(0) \rangle$ is shown throughout the time evolution as a reference.
• Figure 3: Density plots of (a) the order parameter ($O$) and (b) the standard deviation ($\Delta(k,p)$) for $J = 100$ after $4000$ kicks, considering $\ket{j, -j}$ as the initial state. The order parameter distinguishes the DTC ($O \approx 0$), DF ($O \approx -0.5$), and chaotic ($O \approx 0.5$) phases, whereas $\Delta(k,p)$ captures time-crystalline behavior ($\Delta(k,p) \neq 0$) and vanishes ($\Delta(k,p) = 0$) in both the chaotic and DF phases. The values of $O$ and $\Delta(k,p)$ for different phases correspond exclusively to the initial state $\ket{j, -j}$.
• Figure 4: Variation of $\langle J_z(t) \rangle$ for (a) $k = 0.1$, (b) $k = 1.5$, and (c) $k = 3.5$ with $p = \pi/2$ and $J = 100$. Two initial states, $\ket{j, -j}$ and $\ket{\theta = 0.2, \phi = 0.5}$, are considered. Plot (b) shows a four-period oscillation of $\langle J_z(t) \rangle$ for both initial states. Fourier transforms of $\langle J_z(t) \rangle$ for $k = 0.1$, $1.5$, and $3.5$ are shown in (d), (e), and (f), respectively. Although two peaks appear, they correspond to the frequencies $f$ and $-f$. Only peaks below $f = 0.5$ are considered here.
• Figure 5: Illustration of the emergence of a spiral saddle node by tuning $k$. Two-dimensional phase-space plots are shown in (a), (b), and (c) for $k = 0.1$, $1.5$, and $3.5$, respectively, with $p = \pi/2$. The spiral saddle node appears for $k = 1.5$ at $\theta = \pi/2$ and $\phi = 0$. These are special points, as discussed in the main text. The corresponding Poincaré surface plots are shown for the same $k$ values in (d), (e), and (f), respectively. For all plots, six initial conditions and $300$ time steps are considered. For the $4$-DTC phase (see Fig. \ref{['fig4']}(b)), the initial state around $\theta \approx \pi$ visits the four islands marked in yellow (numbered in red).