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Dirac Fermions in Graphene Nanodisk and Graphene Corner: Texture of Vortices with 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 vary rare.