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 GN . We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G_N GN converges when N→∞ N→∞ to a limit G_∞\, G∞ , which coincides with the exact physical propagator Gexact Gexact at small enough coupling, while G_∞ ≠ Gexact G∞≠Gexact at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G_∞=Gexact G∞=Gexact and G_∞≠Gexact G∞≠Gexact regimes thanks to a criterion which does not require the knowledge of Gexact Gexact , as proposed in [Phys. Rev. B 93, 161102 (2016)].