Full versus quasiparticle self-consistency in vertex-corrected GW approaches

Abstract

Using seven semiconductors and insulators with band gaps covering the range from 1 to 10 eV we systematically explore the performance of two different variants of self-consistency associated with the famous Hedin system of equations: the full self-consistency and the so-called quasiparticle approximation to it. The pros and cons of these two variants of self-consistency are sufficiently well documented in the literature for the simplest GW approximation to the Hedin equations. Our study, therefore, aims primarily at the level of theory beyond GW approximation, i.e., at the level of theory which includes vertex corrections. Whereas quasiparticle self-consistency has certain advantages at the GW level (a well-known fact), the situation becomes quite different when vertex corrections are included. In the variant with full self-consistency, vertex corrections (both for polarizability and for self-energy) systematically reduce the calculated band gaps making them closer to the experimental values. In the variant with quasiparticle self-consistency, however, an inclusion of the same diagrams has a considerably larger effect and calculated band gaps become severely underestimated. Different effects of vertex corrections in two variants of self-consistency can be related to the $Z$-factor cancellation which plays a positive role in quasiparticle self-consistency at the GW level of theory but appears to be destructive for the quasiparticle approximation when higher-order diagrams are included. The second result of our study is that we were able to reproduce the results obtained with the questaal code using our flapwmbpt code when the same variant of self-consistency (quasiparticle) and the same level of vertex corrections (for polarizability only, static approximation for screened interaction, and Tamm-Dancoff approximation for the Bethe-Salpeter equation) are used.