Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

We show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition.

Popular Summary

Periodic driving opens up new routes for realizing topological phenomena in many-body quantum systems. Topological phases typically exhibit protected edge or surface modes, which necessarily coexist with propagating bulk modes that are spread throughout the system. The presence of such propagating bulk modes is required in order to avoid a quantum anomaly. This situation remains true even when disorder is present. A periodically driven system, however, is not bound by the same rules. Here, we show that the special character of periodically driven systems drastically changes the relationships among topology, disorder, and localization.

We introduce a topological phase that has chiral edge modes, despite all of the bulk states of the system being localized by disorder. This topological phase is only achievable in a periodically driven system. We refer to this unique nonequilibrium phase of matter as an anomalous Floquet-Anderson insulator (AFAI). We show that the AFAI phase possesses remarkable properties and is robust over a wide swath of parameter space. Most strikingly, it exhibits nonadiabatic yet precisely quantized charge pumping at a finite driving frequency. This quantized transport is observed when all of the states near one edge of the system are filled with fermions. Crucially, we show that the quantization of the charge pump is protected by disorder and the two-dimensional nature of the system. This situation is in contrast to adiabatic quantum pumps, proposed by Thouless three decades ago, in which quantization can only be stabilized by going to the zero-frequency (adiabatic) limit.

The finite frequency quantized pumping phenomenon of the AFAI will be interesting to investigate experimentally and may find applications, for example, in the development of precision current standards.

Figure 1

The anomalous Floquet-Anderson insulator (AFAI), in a disordered two-dimensional periodically driven system with time-dependent Hamiltonian $H(t)$. In the AFAI phase all bulk states are localized, yet the system hosts chiral propagating edge states at all quasienergies. The nontrivial topology of the phase is characterized by a nonzero value of the winding number defined in Eq. (8).

Figure 2

Simple explicit model for achieving the anomalous Floquet-Anderson phase. (a) The Hamiltonian is piecewise constant, defined in five equal length segments of duration $T/5$. During steps 1–4, nearest-neighbor hopping is applied along the colored bonds as shown. The hopping strength $J$ is chosen such that a particle hops between adjacent sites with probability one during each step. In the fifth step, all hopping is turned off and a random disorder potential is applied (the same potential is used for all subsequent driving cycles). (b) Over one driving period, a particle initialized on any site in the bulk returns precisely to its original position (blue arrow). Along the upper edge, a particle initialized on site $(x,0)$ of the $A$ sublattice shifts two sites to the right (red straight arrow) and acquires a phase $e^{-i[\pi /2+V_{(x+2,0)}T/5]}$. (c) With disorder, the quasienergy spectrum consists of two nonoverlapping bands of width $2\delta V$, centered at $ϵ=\pm \pi /2T$.

Figure 3

Edge states and spectral flow in the AFAI. (a) The parent phase of the AFAI is a clean system without disorder, where all Floquet bands have Chern number zero but the winding number (2) is nonzero in all gaps. In a cylinder geometry, chiral edge states propagate along the upper and lower boundaries, only at quasienergies within the bulk gaps. (b) The corresponding spectrum, shown as a function of the conserved circumferential crystal momentum component. (c) When disorder is added, all bulk states become localized while the chiral edge modes on the cylinder persist. When all states are filled near one end of the cylinder, a quantized current flows along the edge. (d) With disorder, crystal momentum is no longer a good quantum number. However, the spectrum of states localized near the upper edge, displayed as a function of the flux ${\theta }_x$ threaded through the cylinder, clearly displays a nontrivial spectral flow. The spectral flow fully winds around the quasienergy zone, accounting for the quantized pumping in the AFAI phase.

Figure 4

Localization of Floquet states in the AFAI as a function of disorder strength, computed for the model presented in Eq. (17). We use $\lambda =\pi$ and an $L\times L$ system with periodic boundary conditions. (a) Quasienergy density of Floquet states per unit area (DOS) and level-spacing ratio (LSR), for three values of disorder strength, as indicated by the markers on the axis of (b). For all cases we take $L=70$. (b) Finite-size scaling of the localization transition. Level statistics in the delocalized regime are described by the Gaussian unitary ensemble (GUE), characterized by an average level-spacing ratio $r_{\mathrm{ext}}\approx 0.60$; in the localized regime, Poissonian level statistics give $r_{\mathrm{loc}}\approx 0.39$. These characteristic values are indicated by dashed lines.

Figure 5

Wave packet dynamics in the AFAI. Using the same model as in Fig. 4, we plot the amplitude of the transmission probability, $⟨|G_N(\mathbf{r},{\mathbf{r}}_0,ϵ){|}^2⟩$, cf. Eq. (18), obtained after a time evolution of $T_{\mathrm{fin}}=300T$ and averaged over disorder realizations. We simulate a strip of size $20\times 100$ with open boundary conditions and plot $⟨|G_N{|}^2⟩$ for several quasienergies, $\epsilon /\mathrm{\Omega }=0$, $1/16$, $1/8$, $1/4$. Panel (a) shows $⟨|G_N{|}^2⟩$ when the initial wave packet is chosen at the edge ${\mathbf{r}}_0=(96,18)$. It indicates the presence of a robust edge mode at all the given quasienergies. Panel (b) shows the probability when the initial wave packet is chosen in the bulk, $\mathbf{r}=(50,10)$. This indicates that the bulk Floquet states are localized. These simulations are carried out with a time step of $dt=T/100$.

Figure 6

Quantized charge pumping in the AFAI. (a) Cumulative average of the pumped charge per cycle in the limit of long times, ${\overline{Q}}_{\infty }$ [cf. Eq. (15)], as a function of disorder strength. For $\delta VT\gtrsim 5$, the localization length is sufficiently smaller than the system size, and ${\overline{Q}}_{\infty }$ approaches unity. The inset shows the finite-size scaling of ${\overline{Q}}_{\infty }$ for $\delta VT=8$. (b) Cumulative average of the pumped charge for $N$ periods, ${\overline{⟨Q⟩}}_{NT}/N$, as a function of $N$. The disorder strength we use is $\delta VT=8$. The approach to the quantized value can be fit to a power law $(NT{)}^{-\upsilon }$ with $\upsilon =1.72$; see the log-log plot shown in the inset. In both panels, we average the charge pumped across all the lines running parallel to the $y$ direction of the cylinder (see Fig. 3) and over 100 disorder realizations. The system size is $L_x\times L_y=50\times 50$.

Figure 7

Transition from the AFAI into a trivial phase at strong disorder. (a) Average level-spacing ratio as a function of disorder strength. On increasing disorder strength, a transition is observed between two localized phases with delocalized levels at $\delta VT\approx 40$. Here, the level-spacing ratio is averaged over all quasienergies. (b) Level-spacing ratio as a function of quasienergy and its comparison with the DOS, indicating that the entire Floquet band is delocalized. (c) Effect of finite size on the distribution of the participation ratio $P_2$ at a given disorder strength, $\delta VT=40$. The system sizes we use for the simulations are $L_x\times L_y=40\times 40$, $70\times 70$, $100\times 100$. The shape of the curve does not change, indicating a critical phase. (d) Scaling collapse of the three curves with $D_2=1.3$. For the critical phase it is expected that $⟨P_2⟩\sim L^{-D_2}$. This is confirmed in the inset which shows the finite size dependence of the average participation ratio $⟨P_2⟩$. The fractal dimension $D_2$ is extracted from the slope of the linear fit.

Figure 8

Floquet spectrum for Thouless’s quantized adiabatic charge pump. (a) Quantized adiabatic pumping in a 1D system is manifested in chiral Floquet bands that wind around the quasienergy Brillouin zone (right and left movers are shown in green and orange, respectively). (b) Outside of the adiabatic limit, $\omega >0$, counterpropagating states hybridize, and all Floquet bands obtain trivial winding numbers; quantized pumping is destroyed.

Figure 9

Quantized charge pumping in the AFAI in a single disorder realization for a system of size $50\times 50$. All parameters are the same as in Fig. 6. (a) Cumulative average of pumped charge per cycle, $⟨Q{⟩}_{NT}/N$, resolved for five different vertical cuts across the cylinder (cuts along the $y$ direction, cf. Fig. 3). The spread of asymptotic values arises from numerical error in the finite time differences used for evaluating the integral in Eq. (13). (b) The pumped charge, averaged over all longitudinal cuts in the same finite-sized system. For a thermodynamically large system, averaging over all cuts is equivalent to averaging over disorder realizations; see Fig. 6.