Random Phase Approximation for gapped systems: the role of vertex corrections and applicability of the constrained random phase approximation

The many-body theory of interacting electrons is an intrinsically difficult problem that requires simplifying assumptions. The Random Phase Approximation (RPA) provides such a simplification, it is a computationally feasible approach for determining the screening properties of materials. An important application is the constrained Random Phase Appromixation (cRPA), which is the state-of-the-art scheme for the ab-initio calculation of effective interactions in low-energy models. Here, we show explicitly that the distance to the Fermi level and the resulting short electronic propagation length justifies the use of the (c)RPA. Our analysis of electron-electron interactions provides a real space analogy to Migdal's theorem on the smallness of vertex corrections in electronphonon problems. An essential finding is that an RPA calculation based on Kohn-Sham states and the Kohn-Sham gap already includes the leading (excitonic) vertex correction in insulators.