Recent Developments and Future Directions

Abstract

In this last chapter, very recent promising developments beyond GW+EDMFT by taking into account nonlocal vertex corrections beyond the RPA are described. The first development is the dual fermion and boson approach in which DMFT is used as a starting point for a perturbation expansion of the nonlocal self-energy. This approach is very general and in its lowest similar to the GW+EDMFT scheme. The second development is the TRILEX approach based on the extension of the GW+EDMFT scheme to include higher-order correlation or vertex functions in the self-consistency condition. Like the GW+EDMFT scheme, this approach maintains the basic concept of DMFT in that local projected quantities are made self-consistent. In this way, nonlocal spin fluctuations missing in the GW+EDMFT scheme are incorporated. The third development is the dynamical vertex approximation method, which can be considered as a generalization of DMFT assuming the locality of the two-particle irreducible vertex. This chapter closes with a summary.