Physical and unphysical regimes of self-consistent many-body perturbation theory

In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order $N$ and plugged into the Dyson equation, which is then solved for the propagator $G_N$. For two simple examples of fermionic models -- the Hubbard atom at half filling and its zero space-time dimensional simplified version -- we find that $G_N$ converges when $Ntoinfty$ to a limit $G_infty,$, which coincides with the exact physical propagator $G_{rm exact} ,$ at small enough coupling, while $G_infty eq G_{rm exact} ,$ at strong coupling. We also demonstrate that it is possible to discriminate between these two regimes thanks to a criterion which does not require the knowledge of $G_{rm exact} ,$, as proposed in [Rossi et al., PRB 93, 161102(R) (2016)].