Skeleton series and multivaluedness of the self-energy functional in zero space-time dimensions

Recently, Kozik, Ferrero and Georges have discovered numerically that for a family of fundamental models of interacting fermions, the self-energy $\Sigma[G]$ is a multi-valued functional of the fully dressed single-particle propagator G, and that the skeleton diagrammatic series $\Sigma_{\rm bold}[G]$ converges to the wrong branch above a critical interaction strength. We consider the zero space-time dimensional case, where the same mathematical phenomena appear from elementary algebra. We also find a similar phenomenology for the fully bold formalism built on fully dressed single-particle propagator and pair propagator.