Dynamically screened vertex correction to GW

Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator Sigma encoding all the variety of scattering processes. In the simplest scenario of correlated electrons described by the GW approximation for the electron self-energy, a particle transfers a part of its energy to neutral excitations. Higher-order (in screened Coulomb interaction W) self-energy diagrams lead to improved electron spectral functions (SFs) by taking more complicated scattering channels into account and by adding corrections to lower order self-energy terms. However, they also may lead to unphysical negative spectral functions. The resolution of this difficulty has been demonstrated in our previous works. The main idea is to represent the self-energy operator in a Fermi golden rule form which leads to a manifestly positive definite SF and allows for a very efficient numerical algorithm. So far, the method has only been applied to the three-dimensional electron gas, which is a paradigmatic system, but a rather simple one. Here we systematically extend the method to two dimensions including realistic systems such as monolayer and bilayer graphene. We focus on one of the most important vertex function effects involving the exchange of two particles in the final state. We demonstrate that it should be evaluated with the proper screening and discuss its influence on the quasiparticle properties.