Figure 5
From: Quantum phase transition induced by real-space topology
Energy spectra for the Hamiltonian in Eq. (23) on a lattice with N = 4 for (a), (b), and N = 100 for (c), which only shows the bottom energy level of the upper band and the top energy level of the lower band for simplicity, obtained by exact diagonalization. The parameters are (a) ϕ = 0; (b) ϕ = π/4; (c) ϕ = π/4. Here, we set λ2k = 0 for all figures because it is just an on-site potential and has no influence on the band structure. We see that the zero modes appear as a level crossing in the case of (a) when no flux. In contrast, in case of (b), it seems like an avoided level-crossing happens when flux presents. But in the case of (c), we demonstrate that there is still a gap between the upper and lower bands even if N is sufficiently large, which means there is no avoided level-crossing whether the flux presents or not.
