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

Abstract

In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order $\mathcal{N}$ and plugged into the Dyson equation, which is then solved for the propagator $G_{\mathcal{N}}$. 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 $G_{\mathcal{N}}$ converges when $\mathcal{N}\to \infty$ to a limit $G_{\infty }$, which coincides with the exact physical propagator $G_{\mathrm{exact}}$ at small enough coupling, while $G_{\infty }\ne G_{\mathrm{exact}}$ at strong coupling. This follows from the findings of [Kozik, Ferrero and Georges, PRL 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the $G_{\infty }=G_{\mathrm{exact}}$ and $G_{\infty }\ne G_{\mathrm{exact}}$ regimes thanks to a criterion which does not require the knowledge of $G_{\mathrm{exact}}$, as proposed in [Rossi et al., PRB 93, 161102(R) (2016)].

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