TRILEX and GW+EDMFT approach to d-wave superconductivity in the Hubbard model

We generalize the recently introduced TRILEX approach (TRiply irreducible local EXpansion) to superconducting phases. The method treats simultaneously Mott and spin-fluctuation physics using an Eliashberg theory supplemented by local vertex corrections determined by a self-consistent quantum impurity model. We show that, in the two-dimensional Hubbard model, at strong coupling, TRILEX yields a d-wave superconducting dome as a function of doping. Contrary to the standard cluster dynamical mean field theory (DMFT) approaches, TRILEX can capture d-wave pairing using only a single-site effective impurity model. We also systematically explore the dependence of the superconducting temperature on the bare dispersion at weak coupling, which shows a clear link between strong antiferromagnetic (AF) correlations and the onset of superconductivity. We identify a combination of hopping amplitudes particularly favorable to superconductivity at intermediate doping. Finally, we study within GW+EDMFT the low-temperature d-wave superconducting phase at strong coupling in a region of parameter space with reduced AF fluctuations.