Stability of anomalous states of a local potential in graphene

Graphene Landau levels have discrete energies consisting zero energy chiral states and non-zero energy states with mixed chirality. Each Landau level splits into discrete energies when a localized potential is present. A simple scaling analysis suggests that a localized potential can act as a strong perturbation, and that it can be even more singular in graphene than in ordinary two-dimensional systems of massful electrons. Parabolic, Coulomb, and Gaussian potentials in graphene may have anomalous boundstates whose probability density has a sharp peak inside the potential and a broad peak of size magnetic length ≤ outside the potential. The n = 0 Landau level with zero energy has only one anomalous state while the n =±1 Landau levels with non-zero energy have two (integer quantum number n is related to the quantized Landau level energies). These anomalous states can provide a new magnetospectroscopic feature in impurity cyclotron resonances of graphene. In the present work we investigate quantitatively the conditions under which the anomalous states can exist. These results may provide a guide in searching for anomalous states experimentally.