2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155469We 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 $Zα$, where $α$ 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 $Z_{eff}$, determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, $Z_{eff} α< 1/2$, for any $α$, while impurities with $Z=2,3$ and higher can become supercritical at certain values of $α$.4 pages, 2 figuresMesoscale and Nanoscale PhysicsMaterials ScienceAtomic Physicstext