Coulomb impurity problem of graphene in magnetic fields

Analytical solutions to the Coulomb impurity problem of graphene in the absence of a magnetic field show that when the dimensionless strength of the Coulomb potential g reaches a critical value the solutions become supercritical with imaginary eigenenergies. Application of a magnetic field is a singular perturbation, and no analytical solutions are known except at a denumerably infinite set of magnetic fields. We find solutions to this problem by numerical diagonalization of the large Hamiltonian matrices. Solutions are qualitatively different from those of zero magnetic field. All energies are discrete and no complex energies are allowed. We have computed the finite-size scaling function of the probability density containing an s-wave component of the Dirac wavefunctions. This function depends on the coupling constant, regularization parameter, and the gap. In the limit of vanishing regularization parameter our findings are consistent with the expected values of the exponent ν which determines the asymptotic behavior of the wavefunction near r = 0.