Density of states in disordered graphene

We study two lattice models, the honeycomb lattice (HCL) and a special square lattice (SQL), both reducing to the Dirac equation in the continuum limit. In the presence of disorder (Gaussian potential disorder and random vector potential), we investigate the behavior of the density of states (DOS) numerically and analytically. While an upper bound can be derived for the DOS on the SQL at the Dirac point, which is also confirmed by numerical calculations, no such upper limit exists on the HCL in the presence of random vector potential. A careful investigation of the lowest eigenvalues indeed indicates that the DOS can possibly be divergent at the Dirac point on the HCL. In spite of sharing a common continuum limit, these lattice models exhibit different behavior.