Polarization charge distribution in gapped graphene: Perturbation theory and exact diagonalization analysis

We study the distribution of vacuum polarization charge induced by a Coulomb impurity in massive graphene. By analytically computing the polarization function, we show that the charge density is distributed in space in a nontrivial fashion and on a characteristic length-scale set by the effective Compton wavelength. The density crosses over from a logarithmic behavior below this scale to a power-law variation above it. Our results in the continuum limit are confirmed by explicit diagonalization of the corresponding tight-binding model on a finite-size lattice. Electron-electron interaction effects are also discussed.