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 GN 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 GN GN converges when N∞ N→∞ to a limit G_∞\, G∞, which coincides with the exact physical propagator G₄ₗ₀₂ₓ Gexact at small enough coupling, while G_∞ ≠ G₄ₗ₀₂ₓ 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_∞=G₄ₗ₀₂ₓ G∞=Gexact and G_∞≠G₄ₗ₀₂ₓ G∞≠Gexact regimes thanks to a criterion which does not require the knowledge of G₄ₗ₀₂ₓ Gexact, as proposed in Phys. Rev. B 93, 161102 (2016).