Graphene with vacancies: supernumerary zero modes

The density of states, $\varrho(E)$, of graphene is investigated within the tight binding (H\"uckel) approximation in the presence of vacancies. They induce a non-vanishing density of zero modes, $n_\text{zm}$, that act as midgap states: $\varrho(E)=n_\text{zm}\delta(E) + \text{smooth}$. As is well known, the actual number of zero modes per sample can in principle exceed the sublattice imbalance: $N_\text{zm}\geq |N_\text{A}-N_\text{B}|$, where $N_\text{A}$, $N_\text{B}$ denote the number of carbon atoms in each sublattice. In this work, we establish a stronger relation that is valid in the thermodynamic limit and that involves the concentration of zero-modes: $n_\text{zm}>|c_\text{A}-c_\text{B}|$, where $c_\text{A}$ and $c_\text{B}$ denote the concentration of vacancies per sublattice; in particular, $n_\text{zm}$ is non-vanishing even in the case of balanced disorder, $N_\text{A}/N_\text{B}=1$. Adopting terminology from benzoid graph theory, the excess modes associated with the current carrying backbone (percolation cluster) are called supernumerary. In the simplest cases such modes can be associated with structural elements of internal boundaries like dangling bonds. Our result suggest that the continuum limit of bipartite hopping models supports nontrivial "supernumerary" terms that escape the present continuum descriptions.