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In electronic systems with long-range Coulomb interaction, the nonlocal Fockexchange term has a band-widening effect. While this effect is included incombined many-body perturbation theory and dynamical mean field theory schemes,it is not taken into account in standard extended DMFT (EDMFT) calculations.Here, we include this instantaneous term in both approaches and investigate itseffect on the phase diagram and dynamically screened interaction. We show thatthe largest deviations between previously presented EDMFT and $GW$+EDMFTresults originate from the nonlocal Fock term, and that the quantitativedifferences are especially large in the strong-coupling limit. Furthermore, weshow that the charge-ordering phase diagram obtained in $GW$+EDMFT methods formoderate interaction values is very similar to the one predicted by dual bosonmethods that include the fermion-boson or four-point vertex.

Influence of Fock exchange in combined many-body perturbation and dynamical mean field theory