We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on the exact electron Green’s function, obtained by using the operator method, and leads to results that are exact in the parameter $Z\alpha$, where $\alpha$ is the “fine-structure constant” of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge $Z_{\mathrm{eff}}$, determined by the localized induced charge. We find that an impurity with bare charge $Z=1$ remains subcritical, $Z_{\mathrm{eff}}\alpha <1/2$, for any $\alpha$, while impurities with $Z=2$, 3 and higher can become supercritical at certain values of $\alpha$.

• Received 3 September 2007

DOI:https://doi.org/10.1103/PhysRevLett.100.076803