  We describe a recent implementation of the combined GW and dynamical mean
field (DMFT) method "GW+DMFT" for the two-dimensional Hubbard model with
on-site and nearest-neighbor repulsion. We clarify the relation of the GW+DMFT
scheme to alternative approaches in the literature, and discuss the
corresponding approximations to the free energy functional of the model.
Furthermore, we describe a numerically exact technique for the self-consistent
solution of the GW+DMFT equations, namely the hybridization expansion
continuous-time algorithm for solving the dynamical impurity model that arises
within the GW+DMFT scheme. We compute the low-temperature phase diagram of the
extended Hubbard model, addressing the metal-insulator transition at small
intersite interactions and the transition to a charge-ordered state for
stronger intersite repulsions. Within the GW+DMFT framework, as in extended
DMFT, the intersite repulsion translates into a frequency-dependence of the
local effective interaction. We analyze this dependence by extracting a
charateristic plasma frequency and show how it affects the local spectral
function.
