Quantized charge transport in disordered Floquet topological insulators in the absence of Anderson localization

We perform a numerical study of Floquet topological insulators with temporal disorder to investigate the existence of quantized charge transport without Anderson localization. We first argue that in setups with temporal imperfections, Anderson localization cannot be expected but bulk transport is diffusive in the long-time limit. In a second step we compute the corrections to the cumulative averaged pumped charge due to the temporal disorder and show that transport is characterized by two regimes: the transient regime, represented by a plateau for uncorrelated disorder, and the long-time behavior with a common scaling law for both uncorrelated and correlated disorder. Most notably, our numerical results indicate that the dynamic corrections vanish in the long-time limit, such that quantized charge transport and diffusive bulk motion can coexist in temporally disordered Floquet topological insulators.

Figure 1

Two-dimensional tight-binding model of a DFTI with additional temporal disorder. In each period (each gray layer), the Hamiltonian is piecewise constant on a segment with duration $T/4$. The nearest-neighbor hopping in each segment is represented by the corresponding direction ${\mathit{\gamma }}_i$ (colored bonds) and the coupling constant ${\mathcal{J}}^{\left(n\right)}$. The latter changes from period to period according to Eq. (5). At the end of each period, a $\delta$ kick partially randomizes the phase.

Figure 2

Spread $\sigma \left(N\right)$ as a function of stroboscopic time $N$ at various disorder strengths $\eta$. Results are given for two different mean values of the normal distribution (5): (a) ${\mathcal{J}}_0=\pi /2$ and (b) ${\mathcal{J}}_0=0.9\times \pi /2$. The black and gray lines mark $\sigma \left(N\right)=3$ in (a) and (b), respectively, which are used to determine $N_c$. Inset: Scaling of $N_c$ (defined by $\sigma \left(N_c\right)=3)$ as a function of $\eta$ [black and gray lines refer to panel (a) and (b), respectively].

Figure 3

Left: Schematic representation of the geometry $\overline{\mathrm{\Gamma }}$ with threaded flux $\mathrm{\Phi }$ and initial occupied lattice region ${\overline{\mathrm{\Gamma }}}_{\mathrm{init}}$ (shaded red), which defines implicitly the initially occupied Floquet states (shaded blue). Right: Schematic representation of the time evolution of the initial states without temporal disorder (top) and with temporal disorder (bottom). Note that the flux was omitted for greater clarity on the right.

Figure 4

Deviation of the cumulative averaged pumped charge from unity, ${|1-\langle Q\rangle }_N|$, as a function of stroboscopic time $N$ at various disorder strengths $\eta$. Parameter sets used are: $(L_x,\eta )=(10,0.05)$ (red, dashed), $(L_x,\eta )=(20,0.05)$ (violet, solid), $(L_x,\eta )=(10,0.01)$ (green, dotted), and $(L_x,\eta )=(20,0.01)$ (blue, dot dashed). The gray line indicates a scaling behavior ${|1-\langle Q\rangle }_N|\sim N^{-1}$. Inset: Plateau value $Q_p$ as a function of disorder strength $\eta$. Here, additional $\eta$ values are included compared to the main plot.

Figure 5

Deviation of the cumulative averaged pumped charge from unity, ${|1-\langle Q\rangle }_N|$, as a function of stroboscopic time $N$ for different correlation lengths ${\tau }_c$. The disorder strength is $\eta =0.05$ and $\eta =0.005$ in (a) and (b), respectively. In both cases $L_x=10$. The gray line indicates a scaling ${|1-\langle Q\rangle }_N|\sim N^{-1}$.

Figure 6

Spread $\sigma \left(N\right)$ as a function of the stroboscopic time $N$ at various parameters $\lambda$ of the missed-kicks disorder. The horizontal black line marks $\sigma \left(N_c\right)=3$ (arbitrary value) to estimate a critical time $N_c$ for the transition to diffusive bulk transport.

Figure 7

Deviation ${|1-\langle Q\rangle }_N|$ of the cumulative averaged pumped charge from ${\langle Q\rangle }_N=1$ for the missed-kicks disorder, as a function of stroboscopic time $N$ for various system sizes $L_x$ and disorder parameters $\lambda$. The parameters are: $(L_x,\lambda )=(10,0.1)$ (red, dashed), $(L_x,\lambda )=(20,0.1)$ (violet, solid), $(L_x,\lambda )=(10,0.001)$ (green, dotted), and $(L_x,\lambda )=(20,0.001)$ (blue, dot dashed). The gray line indicates the scaling behavior $\sim N^{-1}$. Inset: Plateau value $Q_p$ of ${\langle Q\rangle }_N$ as a function of disorder strength $\lambda$. Here, additional $\lambda$ values are used compared to the main plot.