  We investigate the Peierls-Feynman-Bogoliubov variational principle to map
Hubbard models with nonlocal interactions to effective models with only local
interactions. We study the renormalization of the local interaction induced by
nearest-neighbor interaction and assess the quality of the effective Hubbard
models in reproducing observables of the corresponding extended Hubbard models.
We compare the renormalization of the local interactions as obtained from
numerically exact determinant Quantum Monte Carlo to approximate but more
generally applicable calculations using dual boson, dynamical mean field
theory, and the random phase approximation. These more approximate approaches
are crucial for any application with real materials in mind. Furthermore, we
use the dual boson method to calculate observables of the extended Hubbard
models directly and benchmark these against determinant Quantum Monte Carlo
simulations of the effective Hubbard model.
