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). 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) converges when N -> infinity to a limit G(infinity), which coincides with the exact physical propagator G(exact) at small enough coupling, while G(infinity )not equal G(exact) 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(infinity )= G(exac) and G(infinity )not equal G(exact) regimes thanks to a criterion which does not require the knowledge of G(exact), as proposed in [2].