Anomalous thermalization and transport in disordered interacting Floquet systems

Local observables in generic periodically driven closed quantum systems are known to relax to values described by periodic infinite temperature ensembles. At the same time, ergodic static systems exhibit anomalous thermalization of local observables and satisfy a modified version of the eigenstate thermalization hypothesis (ETH), when disorder is present. This raises the question, how does the introduction of disorder affect relaxation in periodically driven systems? In this Rapid Communication, we analyze this problem by numerically studying transport and thermalization in an archetypal example. We find that thermalization is anomalous and is accompanied by subdiffusive transport with a disorder-dependent dynamical exponent. Distributions of matrix elements of local operators in the eigenbases of a family of effective time-independent Hamiltonians, which describe the stroboscopic dynamics of such systems, show anomalous departures from predictions of ETH, signaling that only a modified version of ETH is satisfied. The dynamical exponent is shown to be related to the scaling of the variance of these distributions with system size.

Figure 1

Probability distribution $P\left(S_{\alpha \alpha }^z\right)$ for $W=1$, 2, 3, and 9 and $L=10$, 12, 14, and 16. The red dashed line in each panel depicts a Gaussian distribution with the same standard deviation as that of the corresponding $P\left(S_{\alpha ,\alpha }^z\right)$ for $L=16$. Statistics is obtained combining all Floquet eigenstates and all disorder realizations (over 500 realizations for $L<16$ and 50 realizations for $L=16$). Other parameters are $J=1$, $\mathrm{\Delta }=0.5$, and $T=3$.

Figure 2

Top row: Probability distribution $P\left(S_{\alpha \beta }^z\right)$ for $W=1$ and 3 and $L=10$, 12, 14, and 16. Bottom row: Same as top row, but with the distributions scaled with the standard deviation of $P\left(S_{\alpha \beta }^z\right)$, ${\sigma }_{\alpha \beta }$. The dashed red lines in the bottom row denote the standard normal distribution. Other parameters are the same as in Fig. 1.

Figure 3

Left: Autocorrelation function $C_{L/2}\left(t\right)$ (top) and mean-square displacement $X^2\left(t\right)$ (bottom) as a function of time on a log-log scale. Dashed lines: $L=23$; solid lines: $L=27$. Shades indicate statistical uncertainty (in most cases smaller than the linewidth) and best fits of the underlying power law are indicated by dotted black lines. Right: The exponents $\gamma$ and $\alpha /2$ as a function of the disorder strength $W$. Error bars indicate only statistical errors and not systematic uncertainty.

Figure 4

Left: Extraction of the exponent $\delta$ from the scaling of $\sqrt{\mathcal{N}}\text{std}\left(S_{\alpha \beta }^z\right)$ with $L$. Right: Comparison of the exponent $\gamma$ to its value extracted from the relation $\gamma =1/\left(2\delta +1\right)$.