Figure 3
From: Quantum phase transition induced by real-space topology
Energy spectra for the Hamiltonian in Eq. (3) on a lattice with N = 20 for (a), (b), (c) and N = 4 for (d), obtained by exact diagonalization. The parameters are (a) ϕ = 0, λk = 1.5; (b) ϕ = π/4, λk = 1.5; (c) ϕ = 0, λk = 0.5; (d) ϕ = π/4, λk = 0.5. We see that the zero modes do not appear in the cases of (a) and (b), no matter the flux is zero or not. In contrast, the zero modes appear as a level crossing and avoided level crossing, in the cases of (c) and (d), respectively. In the case of (d), we take a small N in order to demonstrate the avoided level-crossing clearly.
