  The emerging and screening of local magnetic moments in solids has been
investigated for more than 60 years. Local vacancies as in graphene or in Heavy
Fermions can induce decoupled bound states that lead to the formation of local
moments. In this paper, we address the puzzling question how these local
moments can be screened and what determines the additionally emerging low
temperature scale. We review the initial problem for half-filled conduction
bands from two complementary perspectives: By a single-particle supercell
analysis in the uncorrelated limit and by the Lieb-Mathis theorem for systems
with a large Coulomb interaction $U$. We proof that the stable local moments
are subject to screening by three different mechanisms. Firstly the local
moments are delocalized by a finite $U$ beyond the single-particle bound state.
We find a Kosterlitz-Thouless type transition governed by an exponentially
suppressed low energy scale of a counterintuitive Kondo form with $J_{\rm eff}
\propto U^n$ for small $U$, where $n>1$ depends on the precise model. Secondly,
we show that away from half-filling the local moment phase becomes unstable and
is replaced by two types of singlet phases that are adiabatically connected. At
a critical value for the band center, the physics is governed by an
exponentially suppressed Kondo scale approaching the strong coupling phase that
is replaced by an singlet formation via antiferromagnetic RKKY interaction for
large deviation from the critical values. Thirdly, we show that the local
magnetic moment can be screened by a Kondo hole orbital at finite energy, even
though the orbital occupation is negligible: An additional low energy scale
emerges below which the localized moment is quenched. Similarities to the
experimental findings in Ce$_{1-x}$La$_x$Pd$_3$ are pointed out.
