Zero Modes in Bipartite Lattices: Vacancies, Boundary Conditions and Disorder

We study bipartite lattices with vacancies (specifically graphene) and zero modes. Bipartite lattices are defined as direct sums of two sublattices A and B so that each site is connected only to sites of the other sublattice. The spectrum of single particle Hamiltonians on such lattices hosts at least |NA−NB| zero modes where NA (NB) is the number of sites on sublattice A (B). A vacancy defined by the removal of a single site, can be either of type Aor B. Creating vacancies in a bipartite lattice changes the imbalance NA−NB and therefore changes the number of zeromodes. Such zero modes in graphene are topological in nature and are spatially localized on the vacancy sites. We study the effects of boundary conditions and disorder on the appearance of these zero modes and their corresponding wave functions.