Density of States in Graphene with Vacancies: Midgap Power Law and Frozen Multifractality

The density of states rho(E) of graphene is investigated numerically and within the self-consistent T-matrix approximation in the presence of vacancies within the tight binding model. The focus is on compensated disorder, where the concentration of vacancies n(A) and n(B) in both sublattices is the same. Formally, this model belongs to the chiral symmetry class BDI. The nonlinear sigma model predicts for BDI a Gade-type singularity rho(E) similar to vertical bar E vertical bar(-1) exp[-vertical bar log(E)vertical bar(-1/x)]. Our numerical data are comparable to this result in a preasymptotic regime that gives way, however, at even lower energies to rho(E) similar to E-1 vertical bar log(E) vertical bar(-(x) over tilde), 1 <= (x) over tilde < 2. We take this finding as evidence that, similar to he case of dirty d-wave superconductors, generic bipartite random hopping models may also exhibit unconventional (strong-coupling) fixed points for certain kinds of randomly placed scatterers if these are strong enough. Our research suggests that graphene with (effective) vacancy disorder is a physical representative of such systems.