Time crystallinity and finite-size effects in clean Floquet systems

A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have, however, shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether a homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify this issue showing explicitly the key differences between the two cases. On the one hand, our careful scaling analysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneous systems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars, finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic response can in fact persist over timescales that are much larger than those set by the integrability-breaking terms, with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clarifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneous systems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.

Figure 1

Subharmonicity plateaus: a careful scaling analysis. We investigate the scaling of the subharmonicity $Z\left(t\right)$ with system size, for both the MBL (a),(c) and the homogeneous (b),(d) models, and considering an initial rotation of the spins of angles $\theta =0$ (a),(b) and $\theta =\pi /6$. In all cases, the subharmonicity $Z\left(t\right)$ exhibits a crisp signature of time crystallinity: an exponentially long (in system size) plateau. In the MBL system, the pleateau's height does not decay with system size (insets), underpinning its stability and persistence in the thermodynamic limit $L\to \infty$. Contrary, in the homogeneous setting, the value of the plateau decays exponentially (insets), as can be better appreciated for larger $\theta$ (c),(d), pointing towards the onset of thermalization in the thermodynamic limit and to the impossibility of an homogeneous DTC in the strict sense. The value of the plateau's heights in (c),(d) are obtained averaging $Z\left(t\right)$ over time. In (d), the plateau begins at a time $\tau \sim 1/V\sim 20$ (wiggly line), $V$ being the magnitude of the integrability breaking terms. For the MBL setting (a),(c), results are averaged over 100 disorder realizations (one realization is plotted in faded colors for $L=10$), whereas results in the homogeneous case (b),(d) are averaged over a decade-long moving time window (one original time trace is plotted in faded colors for $L=16$).

Figure 2

Subharmonic peak: a careful scaling analysis. For both the MBL (a) and the homogeneous (b) settings, we consider the diagnostics (ii) regarding the presence of a subharmonic peak in the system response. We plot the Fourier transform $\tilde{m}\left(f\right)$ of the magnetization computed over the first ${10}^3$ Floquet periods. In light gray, we report for reference the case of a small $J_2^z$, for which the subharmonic response disappears. The robust subharmonic responses are highlighted by a peak locked to the frequency 0.5, in both cases. The difference between the homogeneous and the MBL scenarios is that the magnitude of the subharmonic peak does and does not decay with system size. Indeed, in the absence of MBL (b) we observe an exponential decay of $|\tilde{m}\left(0.5\right)|$ with system size, suggesting the disappearance of the subharmonic peak in the thermodynamic limit. Here, we considered an initial rotation of the spins $\theta =\pi /10$.

Figure 3

Floquet spectra and the origin of subharmonicity. Polar plots of the eigenstates of the Floquet operator $U_F$ for both the MBL [$L=14$, (a)] and the homogeneous [$L=19$, (b)] settings. The quasienergy and the overlap with the initial condition (for $\theta =\pi /10$) are used as angular and radial coordinates, respectively. The density of points is imprinted in the color code. Furthermore, some eigenstates are duplicated, rotated by a phase $\pi$, and plotted as circles. This way, two $\pi$-paired eigenstates are visually signalled by a dot centered in a circle. For graphical clarity, this duplication is only performed for the 100 and 20 eigenstates with the largest overlaps with the initial condition for the MBL and the homogeneous scenarios, respectively. (a) In the MBL case, all the eigenstates (or at least the considered outer ones) are $\pi$ paired, as expected. (b) In the homogeneous case, only the two outermost eigenstates (that is, those with largest overlap with the initial condition, highlighted with red arrows) are $\pi$ paired, whereas all the others are not. These two special eigenstates are approximately given by $\frac{1}{\sqrt{2}}\left(|\Uparrow \rangle \pm |\Downarrow \rangle \right)$, and their overlap with the initial condition determines the magnitude of the subharmonic response (e.g., of the plateaus’ height in Fig. 1, or of the subharmonic peak in Fig. 2).

Figure 4

Scaling of the two largest overlaps. For the homogeneous setting, we investigate the dependence of the two largest overlaps, corresponding to the $\pi$-paired eigenstates, on the system size $L$ and initial rotation $\theta$. (a) For all considered values of $\theta$, the sum of the two overlaps decays as $e^{-L/\lambda }$ (exponential fits as dotted lines). (b) The characteristic system size scale $\lambda$ of the decay is plotted versus the initial rotation $\theta$. Even for small $\theta$, we find that $\lambda$ never diverges, meaning that thermalization is eventually expected in the thermodynamic limit. The scale $\lambda$ can take remarkably large values, up to a few hundreds, and is maximal for $\theta \approx 0.1$. The scaling in the integrable limit is plotted for reference as a dashed line.