Zero modes and the edge states of the honeycomb lattice

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.