Prethermalization without Temperature

While a clean, driven system generically absorbs energy until it reaches “infinite temperature,” it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethermal regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.

Popular Summary

The periodic driving of quantum systems can reveal novel properties and phenomena. A case in point is the discovery of new states of matter known as time crystals, which exhibit ever-changing behavior that repeats in time. A major obstacle toward realizing these phases is the tendency of driven systems to heat up toward featureless infinite temperature states. One strategy to circumvent this behavior is to delay the onset of heating by engineering a long-lived transient “prethermal” regime. We present a new mechanism for getting a prethermal regime via an additional approximate conservation law such as the total magnetization of the system. This can allow for prethermal regimes in a larger class of systems, such as those that may already be at high or infinite temperature, a phenomenon we call prethermalization without temperature.

We use our framework to explain long-lived time-crystalline signatures in a recent experimental setup of nuclear magnetic resonance spins, a system that falls outside previously known mechanisms to prevent or delay thermalization. We also predict that a judiciously chosen magnetic field can enhance the lifetime of the prethermal time crystal in the nuclear magnetic resonance setup by several orders of magnitude.

Finally, while many experiments are under way to search for and understand time crystals, they are inherently limited in their observation timescales. Therefore, it is crucial to investigate theoretically how to distinguish infinitely long-lived time crystals from transient prethermal versions. We show that local autocorrelation functions evaluated from a wide range of initial states can serve that purpose.

In all, our results extend frameworks for obtaining long-lived prethermal regimes, shed light on current experiments, and provide a way to tell long-lived time crystals from prethermal ones while also showing a clear experimental pathway toward optimization of time-crystal lifetimes.

Figure 1

Survival of the total magnetization $C_{\mathrm{tot}}(nT)$ [defined in Eq. (6)], stroboscopically observed, in a chain of length $L=20$ under the NMR Floquet drive for different driving frequencies $\omega =2\pi /T_1$ (3). The detuning of the spin rotation is $\epsilon =0.1$. (a) Without a field ($h=0$) and with an approximate global flip $P_{\theta }^x$, $\theta =\pi +\epsilon$, corresponding to the presumptive parameters of the NMR experiment, the magnetization dies off quickly with little dependence on the driving frequency. (b) At half the maximum field, $h=\pi /(2T_1)$, and with $\theta =0+\epsilon$, the survival of the magnetization is enhanced. Dashed lines show the evolution with the time-averaged leading-order effective Hamiltonian. (c) With the maximal field $h=\pi /T_1$ and with an approximate global spin flip $\theta =\pi +\epsilon$, we observe a prethermal time-crystalline signal with dramatically enhanced lifetime, by more than $100\times$ compared to (a). The lifetime shows an exponential dependence on driving frequency (Fig. 2), a hallmark of prethermalization.

Figure 2

Number of driving periods needed to reach a magnetization threshold of $C_{\mathrm{tot}}=0.8$, which serves as an estimate for $t_m$ for the data in Fig. 1 with no ($h=0$) and maximal field ($h=\pi /T_1$). At large frequencies, the maximal field data shows a greatly increased $t_m$ with an exponential dependence on $\omega$. The dashed curves show additional data for longer chains ($L=24$).

Figure 3

Time dependence of the sector (labeled by the number of up spins $N_{\uparrow }$) resolved participation entropy $S_1[|\psi (nT)⟩,N_{\uparrow }]$ of the wave function $|\psi (nT)⟩$ starting from the initial state $|00100100100100100100⟩$ (i.e., in the sector with $N_{\uparrow }=6$) under the NMR Floquet drive with a frequency of $\omega =6.25$, $h{T}_1=\pi$, and an approximate global spin flip after each period $\theta =\pi +\epsilon$ with $\epsilon =0.18$. The wave function spreads quickly within one sector, before slowly spreading over several sectors.

Figure 4

Robustness of prethermal regime to drive imperfections. (a) Survival of the total magnetization $C_{\mathrm{tot}}(nT)$ in a chain of length $L=20$ under the NMR Floquet drive (3) for driving frequency $2\pi /T_1=7.5$ and magnetic fields $h$ detuned from the optimal field $h=\pi /T_1$. The detuning of the spin rotation $P_{\pi +\epsilon }^x$ is $\epsilon =0.1$. (b) Number of driving periods needed to reach a thresholds of 0.8 for the same data as in panel (a).

Figure 5

Comparison between the survival of the global magnetization $C_{\mathrm{tot}}(nT)$ and the local correlation function $C_{ii}^z=(1/2^L)\mathrm{Tr}[Z_i(nT)Z_i]$. The former can decay much more slowly when there is an approximate global U(1) conservation, while the latter decays swiftly due to fast thermalization within U(1) sectors as shown in Fig. 3. This is in contrast to a many-body localized time crystal where both local and global autocorrelators oscillate with a finite amplitude even at infinitely late times.