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

TL;DR

This paper demonstrates that the Random Phase Approximation (RPA) is valid for gapped systems where electronic states are far from the Fermi level, and discusses the implications for constrained RPA calculations of effective interactions.

Contribution

The paper clarifies the conditions under which RPA is justified in gapped systems and relates vertex corrections to a real-space analogy of Migdal's theorem, impacting cRPA applications.

Findings

RPA is justified for systems with large band gaps.

Vertex corrections involve quantum tunneling across the band gap.

cRPA already includes leading local vertex corrections in insulators.

Abstract

The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the determination of electronic screening properties of the Coulomb interaction, the Random Phase Approximation (RPA) provides such a simplification. Here, we explicitly show that this approximation is justified for band structures with sizeable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equivalent to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our analysis of electron-electron interactions provides a real-space analogy to Migdal's theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained Random Phase Approximation (cRPA) calculations of effective interactions. We find that their usage of Kohn-Sham energies already accounts for the leading local (excitonic) vertex correction in insulators.