Comments on the fractal energy spectrum of honeycomb lattice with defects

We address the energy spectrum of honeycomb lattice with various defects or impurities under a perpendicular magnetic field. We use a tight-binding Hamiltonian including interactions with the nearest neighbors and investigate its energy structure for two different choices of point defects or impurities. In the first case, we fix a unit cell consisting of 8 lattice points and survey the energy eigenvalues in the presence of up to 2 point defects. Then it turns out that the existence of the fractal energy structure, called Hofstadter's butterfly, depends on the choice of defect pairs. In the second case, we extend the size of a unit cell which contains a single point defect and up to 32 lattice points. The fractal structures indeed appear for those cases and there exist a robust gapless point in the $E=0$ eV line without depending on both the size of unit cells and the shape of lattices. Therefore we keep an immortal butterfly since such a robust point corresponds to the center of a butterfly. Consequently we predict that the presence of a butterfly in a graph is equivalent to that of fractality.