Disordered Chern insulator with a two-step Floquet drive

We explore the physics of a Chern insulator subjected to a two-step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by the presence and chirality of edge modes in the two bulk gaps of the Floquet quasienergy spectrum, around 0 and $\pi$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically nontrivial time evolution. An extensive numerical study of the bulk topological invariants in the presence of quenched disorder reveals transitions induced by strong disorder (i) from the different topological to trivial insulator phases and (ii) from a trivial to a topological Anderson insulator phase at intermediate disorder strengths. Careful analysis of level statistics of the quasienergy spectrum indicates a “levitation-annihilation” mechanism near these transitions.

Figure 1

The levitation and annihilation mechanism for Floquet systems is shown schematically, for the case of edge modes in the (a) 0 gap and (b) $\pi$ gap. The bulk localized, the bulk delocalized, and the edge modes in the spectrum are depicted as gray, orange, and purple colors, respectively.

Figure 2

The quasienergies of the unitary operator $\mathcal{U}\left(t\right)$ can be represented on a circle parametrized by $[\epsilon -2\pi ,\epsilon ]$. The branch cut, $\epsilon$, and 0 define two disjoint parts of this circle. Eigenvectors of $\mathcal{U}\left(t\right)$ do not change between $t=1/2$ and $t=1$, but the eigenvalues change. Depending on the location of an eigenvalue at $t=1/2$, its value at $t>1/2$ drifts towards 0 along two opposite directions. The circle and the square represent the evolution of the two eigenvalues as a function of $t$.

Figure 3

Eigenvalues of the unitary operator $\mathcal{U}$ of a system with semiperiodic boundary conditions represented on a cylinder, with the momentum quantum number along the periodic direction represented by the length of the cylinder and the quasienergies represented along the circumference. (a),(b) The evolution of the eigenvalues of $\mathcal{U}\left(t\right)$. All eigenvalues are 1 at $t=0$ (left panels); the spectrum of the physical unitary operator $U=\mathcal{U}(1/2)$ is shown in the second panel from the left. The evolution of eigenvalues from $t=1/2$ to 1 can happen in two different ways, as shown in (a) and (b), depending on the choice of the branch cut.

Figure 4

The Floquet quasienergy spectrum ${\omega }_{\pm ,\mathbf{k}}$ at different critical points. (a)–(c) The spectrum at the critical points which correspond to gap closings at $\omega =0$; (d)–(f) correspond to that at $\omega =\pm \pi$. For the plots, $J=\pi /6$ and $M_A=1$.

Figure 5

Graphical representation of the phase diagram of the periodically driven Chern insulator. The red circles (blue squares) depict critical points with gap closings in the quasienergy spectrum at $\omega =0\left(\pi \right)$ accompanied by change in ${\nu }_0\left({\nu }_{\pi }\right)$ shown by the color code which also shows the $\mathcal{C}$.

Figure 6

Gap closings in the Floquet quasienergy spectra away from the high-symmetry points at (a) $\omega =0$ and (b) $\omega =\pi$. The IPR of the edge states in the (c) 0 gap and (d) $\pi$ gap on a strip of width $L$, showing the breakdown of edge states at the gap closings. We use $M_A+M_B=3J$ for (a) and (c), and $M_A+M_B=2\pi -3J$ for (b) and (d), with $J=\pi /6$.

Figure 7

Phase diagram in terms of ${\nu }_0$ and ${\nu }_{\pi }$ are shown in the presence of disorder. The red (blue) dashed lines show the critical points in the absence of disorder. The plots correspond to the $M_A-M_B=4$ line in the $M_A-M_B$ plane. For the numerics, we used a $2\times 10\times 10$ system and averaged the data over 1000 disorder realizations.

Figure 8

Robustness of edge modes (winding invariants) to disorder. The critical disorder for the transition from the topological to the trivial phase follows the same qualitative behavior as the inverse participation ratio of the zero-disorder edge state (Fig. 6), indicating that the robustness of the disorder is intimately connected to the gap in the zero-disorder Floquet quasienergy spectrum. The parameters used for the plots are the same as in Fig. 6.

Figure 9

(a) Level-spacing statistics for different disorder strengths $\left(W\right)$ for the case corresponding to (a) ${\nu }_0=1$ and (d) ${\nu }_{\pi }=-1$. As $W$ is increased, the delocalized states indicated by their level-spacing variance being that of the GUE, move towards (a) $\omega =0$ and (d) $\omega =\pm \pi$, before finally annihilating and localizing. For $W=2,P\left(\mathrm{\Delta }\omega \right)$ is shown for the (b),(e) localized and (c),(f) delocalized part of the spectrum, as indicated by the arrows. For (a)–(c), we use $M_A=-1.5,M_B=2.5$, and $J=\pi /6$, whereas for (d)–(f), we use $M_A=0.5,M_B=4.5$, and $J=\pi /6+\pi /4$. The numerics are performed on a $2\times 48\times 48$ sized system and the statistics are taken over 1000 realizations.

Figure 10

Level-spacing statistics showing the reentrant topological behavior. The different lines correspond to different disorder strengths (shown by the color bar) and the horizontal dashed value shows the GUE value. At weak disorder, there are no delocalized states as the phase is trivial; however, on increasing disorder, delocalized states appear close to $\omega =0$ accompanied by a transition to a phase with ${\nu }_0=1$, and finally, at very strong disorder, the system goes to a trivial phase again with all bulk states localized. For the plots, we use $M_A=1,M_B=2$, and $J=\pi /6$ and the statistics are taken over 1000 disorder realization for a $2\times 48\times 48$ sized system.