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 $\epsilon=0$ within a tight-binding description with nearest-neighbour 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, {\em but not-too-low} $|\epsilon|/t$ in this compensated case, we show that the density of states (DOS) $\rho(\epsilon)$ exhibits a strong divergence of the form $\rho_{\rm 1D}(\epsilon) \sim |\epsilon|^{-1}/ [\log(t/|\epsilon|)]^{(y+1)}$, which crosses over to the universal low-energy asymptotic form expected on symmetry grounds $\rho_{\rm GW}(\epsilon) \sim |\epsilon|^{-1}e^{-b[\log(t/|\epsilon|)]^{2/3} }$ below a crossover scale $\epsilon_c \ll t$. $\epsilon_c$ is found to decrease rapidly with decreasing $n_v$, while $y$ decreases much more slowly.