  Introducing low-energy effective Hamiltonians is usual to grasp most
correlations in quantum many-body problems. For instance, such effective
Hamiltonians can be treated at the mean-field level to reproduce some physical
properties of interest. Employing effective Hamiltonians that contain many-body
correlations renders the use of perturbative many-body techniques difficult
because of the overcounting of correlations. In this work, we develop a
strategy to apply an extension of the many-body perturbation theory starting
from an effective interaction that contains correlations beyond the mean field
level. The goal is to re-organize the many-body calculation to avoid the
overcounting of correlations originating from the introduction of correlated
effective Hamiltonians in the description. For this purpose, we generalize the
formulation of the Rayleigh-Schr\"odinger perturbation theory by including free
parameters adjusted to reproduce the appropriate limits. In particular, the
expansion in the bare weak-coupling regime and the strong-coupling limit serves
as a valuable input to fix the value of the free parameters appearing in the
resulting expression. This method avoids double counting of correlations using
beyond-mean-field strategies for the description of many-body systems. The
ground state energy of various systems relevant for ultracold atomic, nuclear,
and condensed matter physics is reproduced qualitatively beyond the domain of
validity of the standard many-body perturbation theory. Finally, our method
suggests interpreting the formal results obtained as an effective field theory
using the proposed reorganization of the many-body calculation. The results,
like ground state energies, are improved systematically by considering higher
orders in the extended many-body perturbation theory while maintaining a
straightforward polynomial expansion.
