Localization, Topology, and Quantized Transport in Disordered Floquet Systems

We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.

Figure 1

Disorder average of the diagonal pumped charge plotted against disorder strength. The transition between the quantized charge regime and the trivial one $Q_d=0$ is linked to the closing of the minimum energy gap due to the disorder, shown in the inset, as highlighted by the vertical dashed line.

Figure 2

(a) Disorder averaged IPR distribution of SP Floquet states for several values of $W$ and $L=280$. The inset shows the sharp peaks almost superimposed at small IPR for $W=2J_0$ and $W=4J_0$. (b) Characteristic localization length ${\overline{\xi }}_L(W)$ as a function of disorder, for a trivial driving cycle [$J_1=4J_0+{\delta }_0\sin (\omega t)$]. The inset shows the collapse obtained by ${\overline{\xi }}_L\to {\overline{\xi }}_L/L$ and $W\to W{L}^{1/\beta }$, with $\beta \simeq 2.5$. (c) Characteristic localization length when the system exhibits topological transport [$J_1=J_0+{\delta }_0\sin (\omega t)$]. The inset shows the collapse ${\overline{\xi }}_L\to {\overline{\xi }}_L/L$ and $W\to (W-W_c)L^{1/\beta }$, with $W_c\simeq 3.5J_0$ and $\beta \simeq 2$. The parameters used in the simulation are ${\delta }_0=0.5J_0$, ${\mathrm{\Delta }}_0=1.5J_0$, $\hslash \omega =0.01J_0$.

Figure 3

${\mathrm{IPR}}_{\alpha }$ vs the time-averaged energy of the corresponding Floquet state $⟨E{⟩}_{\alpha }$. The dashed lines indicates the value $1/L$ associated to extended states; the data refer to several realizations of a chain with 200 sites and disorder strength $W=2J_0$. In the insets the same data are shown for a larger disorder. $Q_d=\pm 1$ is the charge transported when a single band is completely filled.

Figure 4

(a) Quasienergy ${\mathcal{E}}_0(\theta )$ of the many-body Floquet state with lowest energy. The winding is well defined only for $W=2J_0$ and $W=8J_0$, when $|{\mathrm{\Psi }}_0⟩$ has a nonvanishing projection on a single MB Floquet state. (b) SP quasienergy spectrum for all possible angles $\theta$ as a function of the disorder. (c) typical behaviors of SP ${\epsilon }_{\alpha }(\theta )$. When $W<W_c$ most of the states are extended and sensitive to the boundary condition (upper panel), while for $W>W_c$ many quasienergies do not contribute to the winding, being periodic in $\theta$ (lower panel). All data correspond to a single disordered chain of $L=80$ sites. The size of the dots is proportional to $n_{\alpha }(\theta )$.