Designing single and degenerate flat bands in the kagome lattice with long-range hopping

We investigate the electronic structure of the kagome lattice model with first-, second-, and two kinds of third-nearest-neighbor hoppings. We reveal that by tuning the third-nearest-neighbor hoppings, not only single flat band but also degenerate flat band can be created on the $\mathrm{\ensuremath{\Gamma}}$-M line. We provide the detailed conditions to realize them. The coexistence of these bands can be almost realized near the fundamental band gap in graphene with triangular defects in a superhoneycomb arrangement. Furthermore, due to these flat bands, several sharp peaks appear in the optical conductivity. Our results strongly indicate that long-range electron hopping has a new possibility for designing electronic structures.