Anomalous random multipolar driven insulators

It is by now well established that periodically driven quantum many-body systems can realize topological nonequilibrium phases without any equilibrium counterpart. Here we show that, even in the absence of time translation symmetry, nonequilibrium topological phases of matter can exist in aperiodically driven systems for tunably parametrically long prethermal lifetimes. As a prerequisite, we first demonstrate the existence of long-lived prethermal Anderson localization in two dimensions under random multipolar driving. We then show that the localization may be topologically nontrivial with a quantized bulk orbital magnetization even though there are no well-defined Floquet operators. We further confirm the existence of this anomalous random multipolar driven insulator by detecting quantized charge pumping at the boundaries, which renders it experimentally observable.

Figure 1

Scheme of the driving protocol. (a) Sequence of driving blocks $U_0^{\pm }$ employed to realize the TM protocol up to time $2^{n}T$. (b) The elementary time evolution operator $U_0^{\pm }$ involves five driving steps. The first four steps contain hopping processes on different bonds of different colors, meanwhile a sublattice potential of amplitude $\pm \delta h$ is applied. The disorder potential appears in the fifth step, leading to Anderson localization.

Figure 2

The participation ratio for (a) TM driving and (b) 3-RMD, with varying driving rates, quickly saturates to a prethermal plateau before increasing to 0.5 (indicating delocalization at long times). The dependence of the prethermal lifetime on the driving rate is shown for (c) TM driving and (d) $n$-RMD drivings. We use parameters $\delta J=1.2J_0, h_{\max }=6J_0, \delta h=7J_0$ and system size $70\times 70$ and $40\times 40$ for TM driving and RMD, respectively. $T^{-1}$ is in units of $J_0$.

Figure 3

Dynamics and scaling of prethermal lifetime for TM driving. (a) Particle density for a single random realization in the prethermal regime at time $t\approx {10}^3{J}_0^{-1}$ for $1/T=20J_0$. (b) Quantized orbital current serving as a prethermal topological order parameter. (c) Chiral edge current at the boundary between filled and empty sites. (d) Scaling of the lifetime of the quantized current. We use parameters $\delta J=1J_0, h_{\max }=20J_0, \delta h=7J_0.T^{-1}$ is in units of $J_0$.

Figure 4

(a) Particle density for a single 3-RMD realization at time $t\approx 400J_0^{-1}$ for $1/T=9$. (b) Edge current averaged over a single block at the boundary between filled and empty sites with 3-RMD for different driving frequencies. We use parameters $\delta J=1J_0, h_{\max }=10J_0, \delta h=7J_0$. $T^{-1}$ is in units of $J_0$.

Figure 5

Evolution of the particle density under Thue-Morse driving for a single random disorder realization for $1/T=20J_0$. The times shown in the three panels correspond to $2^{15}T, 2^{25}T, 2^{35}T$. The parameters are $\delta J=1J_0, h_{\max }=20J_0, \delta h=7J_0$.

Figure 6

Dynamics of ${\overline{\mathrm{PR}}}_{L_i}/N$ for initial states prepared at $L_i$ sites away from the boundary of the cylinder of size $70\times 70$. $L_i=0$ and 34 in panels (a) and (b), respectively. We use parameters $L_x=L_y=70, \delta J=1J_0, \delta h=7J_0, h_{\max }=6J_0$. $T^{-1}$ is in units of $J_0$.

Figure 7

Scaling of the delocalization time versus driving rates for different $L_i$. We use parameters $L_x=L_y=70, \delta J=1J_0, \delta h=7J_0, h_{\max }=6J_0$. $T^{-1}$ is in units of $J_0$.

Figure 8

Dynamics of the edge current and its lifetime scaling for $L_x=L_y=70$. We use parameters $\delta J=1J_0, \delta h=7J_0, h_{\max }=6J_0$ and $T^{-1}$ in units of $J_0$.

Figure 9

Scaling of the prethermal lifetime of edge current for different system sizes $L_x=20,50,70$. We use parameters $\delta J=1J_0, \delta h=5J_0, h_{\max }=6J_0$ and $T^{-1}$ in units of $J_0$.

Figure 10

Dynamics of the edge current and its scaling for $n=0$, 1, 2, 3 RMD. We use parameters $\delta J=1J_0, h_{\max }=10J_0, \delta h=7J_0$ and system size $40\times 40$.