Fig. 9. Mid-gap states for the bilayer topological system with opposite periodically modulated on-site potentials.

Density distribution of mid-gap states for Δ_↑/2π = − Δ_↓/2π = 12 MHz and $\varphi =\frac{2\pi }{3}$ with J_⊥/2π = 7 MHz, J_∥/2π = 8 MHz, $J_{\parallel }^{\prime }=0.1J_{\parallel }$, and J_× = 0.2J_∥. The left panel is for a mid-gap state in the second gap; and the right panel is for a mid-gap state in the fourth gap. Note that the first and fifth gaps in Fig. 8b are not clearly shown due to finite-size effects.