Vacancy-Induced Low-Energy States in Undoped Graphene

We demonstrate that a nonzero concentration n_{v} of static, randomly placed vacancies in graphene leads to a density w of zero-energy quasiparticle states at the band center ε=0 within a tight-binding description with nearest-neighbor hopping t on the honeycomb lattice. We show that w remains generically nonzero in the compensated case (exactly equal number of vacancies on the two sublattices) even in the presence of hopping disorder and depends sensitively on n_{v} and correlations between vacancy positions. For low, but not-too-low, |ε|/t in this compensated case, we show that the density of states ρ(ε) exhibits a strong divergence of the form ρ_{Dyson}(ε)∼|ε|^{-1}/[log(t/|ε|)]^{(y+1)}, which crosses over to the universal low-energy asymptotic form (modified Gade-Wegner scaling) expected on symmetry grounds ρ_{GW}(ε)∼|ε|^{-1}e^{-b[log(t/|ε|)]^{2/3}} below a crossover scale ε_{c}≪t. ε_{c} is found to decrease rapidly with decreasing n_{v}, while y decreases much more slowly.