Topological Edge States and Winding Numbers in Graphene with Vacancies

We show that graphene with a vacancy, namely the removal of a neutral carbon atom, is described by a chiral massive Dirac Hamiltonian. We prove that it is topological according to the tenfold classification generalized to defects by Teo and Kane. We calculate the associated winding number and show that it is equal to 1. In agreement with the Atiyah Singer index theorem, this winding number equals the Hamiltonian's analytical index which counts the number of zero modes. This index corresponds to the topological edge state (zero mode) created around the vacancy location.