Disorder-induced transitions in resonantly driven Floquet topological insulators

We investigate the effects of disorder in Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are induced by resonantly driving a transition between the valence and conduction bands. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a mobility gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator. Interestingly, the effects of disorder are not necessarily adverse: we show that in the same quantum well, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet topological Anderson insulator (FTAI). We identify the conditions on the driving field necessary for observing such a transition.

Figure 1

Figure shows the schematic for obtaining the quasienergies of the driven Hamiltonian. (a) Example of a time-independent band structure in momentum space. For clarity, we show a slice of the spectrum only along a particular momentum direction. (b) Three replicas (labeled as $n=-1$, 0, and 1, representing three Floquet blocks) of the band structure. The replicas are obtained by shifting the time-independent bands by $n\mathrm{\Omega }$. This represents the spectrum from the diagonal blocks of $H_r^F$ as shown in Eq. (4). Adjacent Floquet blocks have resonant quasienergies. (c) The approximate quasienergy band structure for the Floquet Hamiltonian, obtained by projecting to quasienergies ($\epsilon$) in the range $0<\epsilon <\mathrm{\Omega }$. Clearly, the quasienergy bands correspond to the two resonant bands (blue and green) in (b). The radiation potential opens up a gap at the resonance, with the gap proportional to $|{\mathbf{V}}_{\perp }|$ [see Eq. (7)].

Figure 2

The band structure for the model Hamiltonian [defined in Eq. (1)] in different parameter regimes. (a) The original time-independent band structure for the Hamiltonian for the parameter regimes: $A/M=0.2, B/M=-0.2$. The band is topologically trivial. (b) The quasienergy band structure in the presence of driving for the same system parameters as (a), with $\mathbf{V}/M=(0,0,1)$ and $\mathrm{\Omega }/M=3$. This band structure is clearly nontrivial with edge states in the gap at the resonant quasienergy, $\epsilon =\mathrm{\Omega }/2$. They are done on a lattice with periodic boundary conditions in the $x$ direction and open boundary conditions with $L=60$ sites in the $y$ direction.

Figure 3

(a) The average localization length ${\mathrm{\Lambda }}_N/\left(2L_y\right)$ as a function of quasienergy and disorder strength. This figure shows a transition in which the mobility gap closes at $U_0/M\approx 1$. Simulations were done for system sizes $L_x\times L_y=400\times 50$, and evolved for $N=2500$ time periods. (b) The topological phase transition accompanied by the mobility gap closing. We plot the disorder-averaged Bott index, ${\overline{C}}_b(0,\mathrm{\Omega }/2)$ for states between ${\epsilon }_l=0$ and ${\epsilon }_h=\mathrm{\Omega }/2$. At small disorder, the system is topological with a Bott index 1. At $U_0/M\approx 1$, the mobility gap closes and the system becomes trivial with the index going to 0. (c) Disorder averaged Bott index as a function quasienergy, ${\overline{C}}_b(0,\epsilon )$, for disorder strength, $U_0/M=0.6$. The index takes the value ${\overline{C}}_b(0,\epsilon )\approx 1$ for $\epsilon =\mathrm{\Omega }/2$ and smoothly goes to zero for quasienergy $\epsilon$ away from $\epsilon =\mathrm{\Omega }/2$. We plot for two different system sizes, $L_x=20$ and 40.

Figure 4

Level spacing statistics for the Floquet eigenvalues at different disorder strengths. The extended Floquet states must have level repulsion, with level statistics following the GUE ensemble, in which case, the variance of the level spacing distribution is ${\sigma }^2\left(P\left(s\right)\right)=0.178$. This is in contrast to localized states which have ${\sigma }^2\left(P\left(s\right)\right)\sim 1$. The level spacing $s=\mathrm{\Delta }\epsilon /\delta$ is measured in units of average level spacing $\delta$, as defined in Sec. 3. (a), (b), and (c) compare the density of states of the Hamiltonian (in red) with the variance of $P\left(s\right)$ (in black) for disorder strengths, $U_0/M=0.3$, 1, and 2, respectively. (a) Data that correspond to two quasienergy bands with extended states (indicating that the localization length is larger than the system size). The two bands are separated by a band gap. In (b), the band gap is no longer visible, while the level statistics of states in the middle of the quasienergy zone still corresponds to extended states, and in (c) all the Floquet eigenstates are localized. (d) The level spacing distribution, $P\left(s\right)$, at a given quasienergy, $\epsilon /\mathrm{\Omega }=0.43$ and disorder strength $U_0/M=0.3$. It exactly fits with the GUE distribution. (e) $P\left(s\right)$ for $\epsilon /\mathrm{\Omega }=0.25$, at disorder strength $U_0/M=2$. This distribution has better agreement with poisson statistics, indicating localized states. All the simulations were done for systems sizes $L_x\times L_y=40\times 40$.

Figure 5

(a) Band structure of the driven system for $V_z/M=2, V_x/V_z=0.15$ in the trivial phase. The other parameters of the Hamiltonian are the same, as Fig. 2. (b) shows the typical phase diagram for the Floquet topological phase as a function of $V_x$ and $V_y$. For $V_{x,y}<V_c$, the phase is topological and otherwise trivial. The clean system phase for obtaining the FTAI phase is chosen such that $V_x>V_c$, with $V_y=0$. The initial point of the trivial driven system is schematically represented as the point (blue square) in this diagram.

Figure 6

Disorder-averaged Bott index ${\overline{C}}_b(0,\mathrm{\Omega }/2)$ as a function of disorder strength $U_0/M$. The clean system starts of as trivial, as evidenced by the band structure [see Fig. 5] and the fact that the Bott index ${\overline{C}}_b$ is 0, a small disorder. At finite disorder strength, the Hamiltonian acquires a nonzero average Bott index indicating the presence of a topological phase. The dimensions of the system considered are $L_x\times L_y=40\times 40$.

Figure 7

The time evolution of a $\delta$-function wave pack for disorder strength $U_0/M=0.7$. Figures (I) and (II) show $\overline{g_N(\epsilon =\mathrm{\Omega }/2,\mathbf{r},{\mathbf{r}}^{\prime })}$ for different choice of initial position $\mathbf{r}$. (I) shows the presence of an extended edge mode at quasienergy $\epsilon =\mathrm{\Omega }/2$. (II) On the other hand, it shows that choosing the starting wave packet in the bulk, continues to remain localized in the bulk. All simulations were carried out on a lattice of size $L_x\times L_y=40\times 200$ and for a total number of time periods, $N=5000$.

Figure 8

(a) Comparison of the Bott index, $C_b(0,\mathrm{\Omega }/2)$ for as a function of the polarization angle, $\theta$ for the clean sample and disorder strength, $U_0/M=1$. Clearly, the region where the system is topological, ${\overline{C}}_b(0,\mathrm{\Omega }/2)=2$, is larger in case of the disordered system, indicating the presence of a disorder-induced topological phase. (b) The Bott index as a function of disorder strength when the initial polarization is $\theta =0.1\frac{\pi }{4}$. A topological phase, with ${\overline{C}}_b(0,\mathrm{\Omega }/2)=2$, is induced as a function of disorder. The simulations were run for system sizes $L_x\times L_y=20\times 20$.

Figure 9

This figure illustrates the diagramatic representation of the Floquet Green function and the renormalization due to disorder. The thin line represents the bare propagator defined in Eq. (A4). The thick line represents the propagator in the presence of the drive, $G^0$, defined in Eq. (A12). (a) The $n=-1$ component $G^{-1}$ as defined in Eq. (A13). (b) The renormalization due to disorder of $G^0$. This corresponds to renormalization of the bare parameters of the time-independent part of the Hamiltonian. (c) The self-energy correction due to disorder to the radiation potential. The change of the density of states near the Floquet gap at resonance is due to the renormalization of the radiation potential.

Figure 10

We plot the ${\sigma }_z$ component of the integrand $\mathcal{I}$ defined in equation (A22) as a function of $k_x$ and $k_y$. The parameters are chosen with $V_z/M=0.1, A/M=0.2, B/M=-0.2$, and $\mathrm{\Omega }/M=3$. Clearly, the integrand is sharply peaked at the resonance circle, $d_r=d-\frac{\mathrm{\Omega }}{2}=0$ (shown in red). The large negative value of the integrand results in the negative sign for ${\mathrm{\Sigma }}_V^z$. We confirm this by a numerical integration of the ${\sigma }_z$ component over the entire Brillouin zone.

Figure 11

Typical localization length ${\mathrm{\Lambda }}_N^{\text{typ}}/\left(2a\right)$ as a function of total time of evolution, $NT$, where $a=1$ is the lattice spacing. (a) The typical bulk localization length at quasienergy $\epsilon =\mathrm{\Omega }/2$ for $L_y=10$ and 12. The disorder strength is $U_0/M=0.9$. For small times, the particle is diffusive, and at long times saturates to ${\mathrm{\Lambda }}_N^{\text{typ}}={\mathrm{\Lambda }}_{\mathrm{sat}}$. ${\mathrm{\Lambda }}_{\mathrm{sat}}$ scales with $L_y$ indicating that the state is extended.

Figure 12

Winding of ${\mathbf{V}}_{\perp }\left(\mathbf{k}\right)$ on the unit sphere [see Eqs. (C13, C14, C15)] as the momentum vector, $\mathbf{k}$ varies along the resonance circle [defined in Eq. (5)]. The black point indicates the north pole $(0,0,1)$ and the red point indicates ${\widehat{V}}_{\perp }\left({\mathbf{k}}_0\right)$ with ${\mathbf{k}}_0=(1,0)$.

Figure 13

We show the dependence on polarization of light parameterized by, $\theta =arctan(A_y/A_x)$ of the Chern number and the gap in the quasienergy band structure of the upper block.