The existence/absence of Dirac cones in graphynes

We investigate the physical origin of the existence or absence of Dirac cones in graphynes by combining first-principles calculations with the downfolding method. We clarify that acetylenic linkages (-C equivalent to C-) can be reduced into an effective hopping term between vertex atoms, and thus various graphynes may be described by a unified tight-binding model on a honeycomb lattice, which is topologically equivalent to that of graphene. Critically, whether Dirac cones exist or not in graphynes is determined by the combination of hopping terms. Based on the unified model proposed, two additional graphynes are revealed to possess Dirac cones, one of which has a rectangular symmetry and contains only four vertex atoms in a unit cell.