Counterpropagating edge states in Floquet topological insulating phases

Nonequilibrium Floquet topological phases due to periodic driving are known to exhibit rich and interesting features with no static analogs. Various known topological invariants usually proposed to characterize static topological systems often fail to fully characterize Floquet topological phases. This fact has motivated extensive studies of Floquet topological phases to better understand nonequilibrium topological phases and to explore their possible applications. Here we present a theoretically simple Floquet topological insulating system that may possess an arbitrary number of counterpropagating chiral edge states. Further investigation into our system reveals another related feature by tuning the same set of system parameters, namely, the emergence of almost flat (dispersionless) edge modes. In particular, we employ two-terminal conductance and dynamical winding numbers to characterize counterpropagating chiral edge states. We further demonstrate the robustness of such edge states against symmetry preserving disorder. Finally, we identify an emergent chiral symmetry at certain subregimes of the Brillouin zone that can explain the presence of almost flat edge modes. Our results have exposed more interesting possibilities in Floquet topological matter.

Figure 1

Parameter values are $J_1=J_2=\pi /3,M=1$. Dynamical winding number (blue), two-terminal conductance (red), and number of counterpropagating edge pairs (green) for (a) zero (b) $\pi$ quasienergy gap are shown as we vary the $J_3$ parameter. Two-terminal conductance has been calculated in a rectangular sample with $N_x=50,N_y=170$ and absorbing boundary conditions are applied in the $y$ direction $n_y\in \{1,N_y\}$. Furthermore, for two-terminal conductance the value of $\epsilon$ is chosen in the gap such that $\epsilon =0.0424\pi \approx 0$ for conductance $G^0$ in zero gap and $\epsilon =0.9780\pi \approx \pi$ for conductance $G^{\pi }$ in $\pi$ quasienergy gap.

Figure 2

Parameter values are $J_1=J_2=\pi /3,M=1,N_x=400$. States localized at the left (red) and right (green) edge for (a) $J_3=0.8\pi$, (b) $J_3=1.2\pi$, (c) $J_3=1.5\pi$, and (d) $J_3=1.8\pi$ are shown. Two-terminal conductance, dynamical winding number, and number of counterpropagating edge states around zero and $\pi$ quasienergy gap can be seen in Fig. 1 for each distinct case.

Figure 3

Parameter values are $J_1=J_2=\pi /3,M=1,N_x=N_y=50$, and $J_3=0.8\pi$. Two-terminal conductance $G^{\epsilon }$ around the zero (blue) and $\pi$ (red) quasienergy gap is averaged over 100 disorder realizations. The value of $\epsilon$ is chosen in the gap such that $\epsilon =0.0079\pi \approx 0$ for conductance $G^0$ around the zero quasienergy gap and $\epsilon =0.9960\pi \approx \pi$ for conductance $G^{\pi }$ around the $\pi$ quasienergy gap.

Figure 4

Parameter values are $M=1,J_1=\pi /2,J_2=\pi /3$, and $N_x=800$. Edge spectrum for (a) $J_3=1.5\pi$, (b) $J_3=2.5\pi$, (c) $J_3=3.5\pi$, and (d) $J_3=4.5\pi$ with open boundary conditions along the $x$ direction and edge states localized at left (red) and right (green) ends are shown.

Figure 5

Parameter values are $J_1=J_2=\pi /3,M=1,N_y=400$. Open (periodic) boundary conditions are applied in the $y\left(x\right)$ direction and states localized at the left (red) and right (green) edge for (a) $J_3=0.8\pi$, (b) $J_3=1.8\pi$ have been shown.

Figure 6

Parameter values are $J_1=\pi /2,J_2=\pi /3,M=1,N_y=800$. Open (periodic) boundary conditions are applied in the $y\left(x\right)$ direction, and states localized at left (red) and right (green) edge for (a) $J_3=3.5\pi$, (b) $J_3=4.5\pi$ have been shown.