  We propose new approach for treatment of local and non-local interactions in
correlated electronic systems, which uses self-energy and the two-particle
irreducible vertices, obtained from (extended) dynamical mean-field theory, as
an input of two-particle irreducible functional renormalization-group (2PI-fRG)
approach. Using 2PI-fRG approach allows us to treat both, local and non-local
interactions. In case of purely local interaction the corresponding equations
have similar (although not identical) structure to the earlier developed
DMF$^2$RG approach. In a simplest truncation, neglecting scale-dependence of
the two-particle irreducible vertices, we reproduce the results for the
two-particle vertices/susceptibilities in the ladder approximation of the dual
boson or D$\Gamma$A approach; in more sophisticated truncations the method
allows us to consider non-local corrections to the self-energy, as well as the
interplay of charge- and spin correlations. treatment of vertex
renormalizations. The proposed scheme is tested on the two-dimensional standard
and extended $U$-$V$ half-filled Hubbard models. For the standard Hubbard model
we obtain non-local self-energy, which is in agreement with numerical studies;
for the extended Hubbard model we obtain the boundary of charge instability,
which agrees well with the results of the dual boson (DB) approach. We also
find that the effect of spin correlations on electron interaction in the charge
channel, not considered previously in the DB approach, only slightly reduces
critical next-nearest-neighbor interaction of charge instability of the
extended Hubbard model at the considered finite small temperature, yielding
better agreement with dynamic cluster approximation. The considered method is
rather general and can be applied to study various phenomena in
strongly-correlated electronic systems.
