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We describe a recent implementation of the combined GW and dynamical meanfield (DMFT) method "GW+DMFT" for the two-dimensional Hubbard model withon-site and nearest-neighbor repulsion. We clarify the relation of the GW+DMFTscheme to alternative approaches in the literature, and discuss thecorresponding approximations to the free energy functional of the model.Furthermore, we describe a numerically exact technique for the self-consistentsolution of the GW+DMFT equations, namely the hybridization expansioncontinuous-time algorithm for solving the dynamical impurity model that ariseswithin the GW+DMFT scheme. We compute the low-temperature phase diagram of theextended Hubbard model, addressing the metal-insulator transition at smallintersite interactions and the transition to a charge-ordered state forstronger intersite repulsions. Within the GW+DMFT framework, as in extendedDMFT, the intersite repulsion translates into a frequency-dependence of thelocal effective interaction. We analyze this dependence by extracting acharateristic plasma frequency and show how it affects the local spectralfunction.

Screening and Non-local Correlations in the Extended Hubbard Model from Self-Consistent Combined GW and Dynamical Mean Field Theory