Dirac fermions in a graphene nanodisk and a graphene corner: Texture of vortices with an unusual winding number

We analyze the zero-energy sector of the trigonal zigzag nanodisk and corner based on the Dirac theory of graphene. The zero-energy states are shown to be indexed by the edge momentum and grouped according to the irreducible representation of the trigonal symmetry group C(3v). Wave functions are explicitly constructed as holomorphic or antiholomorphic functions around the K or K' point. Each zero-energy mode is a chiral edge mode. We find a texture of magnetic vortices. It is intriguing that a vortex with the winding number 2 emerges in the state belonging to the E (doublet) representation. The realization of such a vortex is very rare.