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We study the quantization of the excess charge on $N$ localized(ultra-screened) impurities in $d$-dimensional crystalline insulating systems.Solving Dyson's equation, we demonstrate that such charges are topological, byexpressing them as winding numbers of appropriate functionals of bulk positionspace Green's functions. We discuss the ties of our topological invariant withthe nearsightedness principle of W. Kohn, stating that the electronic chargedensity at fixed chemical potential depends on the external field only locally,meaning that localized perturbations by external fields may only result inlocalized charge redistributions. We arrive at the same conclusion bydemonstrating that an adiabatic perturbation comprised of a variation ofimpurities' positions and/or strengths may only result in the change in theoccupancy of impurity-localized bound states sitting, energy-wise, close to theFermi level. Finally, we conclude by discussing the relations of thenearsightedness principle with the topological invariants characterizing theboundary charge.

Topological charge quantization on localized imperfections in crystalline insulators and the nearsightedness principle of Kohn