Prethermal quasiconserved observables in Floquet quantum systems

Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting periodically driven (Floquet) many-body phases over an exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite-temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from nonconserved ones, allowing one to single out a set of linearly independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanisms underlying the quasiconservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid-state nuclear magnetic resonance systems.

Figure 1

Typical dynamics of $\langle \mathcal{O}\left(t\right){\mathcal{O}}^{\prime }\rangle$ in a Floquet spin chain. Here we choose KDM and $\mathcal{O}={\mathcal{O}}^{\prime }$. (a)–(c) $J\tau =0.5$; (d)–(f) $J\tau =2$. (a),(d) $\mathcal{O}=X$; (b),(e) $\mathcal{O}=Y$; (c),(f) $\mathcal{O}=Z$. Different colors correspond to different system size $L$, as shown in the legend.

Figure 2

By considering the matrix $\mathrm{\Lambda }$ obtained for each $J\tau$ Trotter step, we calculate the three largest eigenvalues as a function of $J\tau$ for (a) KDM and (b) ADM. The curve color represents different eigenvalues and the curve style represents different system sizes. From the eigenvalues and their dependence on system size, we see that there are two eigenquasiconserved observables in KDM, while only one in ADM.

Figure 3

(a)–(c) The Magnus expansion given by Eq. (3) of KDM; (d)–(f) that of ADM. (a),(d) Circles show the norm of ${\mathrm{\Omega }}_m$ (normalized by $L{2}^L$). The solid line represents the linear fit. (b),(e) Infidelity $1-\langle H_{\mathrm{pre}}\left(\infty \right)H_{\mathrm{pre}}\rangle$ of infinite-time averaged $H_{\mathrm{pre}}$ evaluated up to the $m\mathrm{th}$ order. Different curves stand for $J\tau$ from 0.2 to 2, with a step of 0.2. The darker color represents smaller $J\tau$. $L=12$ is used. (c),(f) Infinite-time autocorrelation of $H_{\mathrm{pre}}$ as a function of $J\tau$ for different system sizes. Order $m=7$.

Figure 4

$D_{\mathrm{pre}}$ expansion of KDM. (a) Norm of the $m\mathrm{th}$-order term of the quasiconserved observable $D_{\mathrm{pre}}$ (normalized by $L{2}^L$). Different curves stand for $h\tau =J\tau$ from 0 to 2, in steps of 0.2. The darker color represents smaller $J\tau$. (b) Infidelity $1-\langle D_{\mathrm{pre}}\left(\infty \right)D_{\mathrm{pre}}\rangle$ of infinite-time averaged $D_{\mathrm{pre}}$ evaluated up to the $m\mathrm{th}$ order. $L=12$ is used. (c) Fidelity $\langle D_{\mathrm{pre}}\left(\infty \right)D_{\mathrm{pre}}\rangle$ evaluated to seventh order as a function of $h\tau$ for different system sizes.

Figure 5

Autocorrelation of the average Hamiltonian for the alternating dipolar model. (a) Autocorrelation as a function of $n$. Different curve stands for $J\tau$ from 0.35 to 2.27, with a step of 0.175. The darker color represents smaller $J\tau$ and the lighter color represents larger $J\tau$. We fit the autocorrelations from $n=20$ to $n=64$ to exponentially decaying function $exp(-\gamma n)$ and plot the decay rate $\gamma$ in (b). The length of the error bars corresponds to two standard deviations of the fitted decay rate. The solid curve indicates the fit to function $\gamma =aexp(-b/J\tau )+c$. The fitted coefficients $a,b,c$ are shown in the plot with the 95% confidence interval. (c) Autocorrelation vs $J\tau$ for different $n$. The lighter colors represent smaller $n$ and the darker colors represent larger $n$. For a given $n$, the autocorrelation is normalized by $\langle \overline{H}\left(n\right)\overline{H}\rangle$ at $J\tau =0.35$, i.e., the leftmost point is normalized to 1. In (a) and (b), error bars are determined from the noise in the free induction decay [see the Supplemental Material (SM) [88] for details of the experimental scheme].

Figure 6

Scaling of the critical Trotter step for KDM (${\overline{H}}^{\left(K\right)}$: blue; $D_z$: green) and ADM (${\overline{H}}^{\left(A\right)}$: red). (a) Simulated autocorrelations as a function of $J\tau$ for $L=8,9,\cdots ,17$ using exact diagonalization. The darker colors represent larger $L$ as shown in the color bar. (b) $J_c\tau$ at which the numerical autocorrelation ($L=17$) drops to half of the value under infinitely fast driving ($J\tau \to 0$). (c) $J_c\tau$ at which the experimentally measured autocorrelation drops to half of the value under the fastest driving ($J\tau =0.35$). Error bars are determined from the noise in the free induction decay (see SM [88]).