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

Recently, Kozik, Ferrero and Georges 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(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 the fully dressed single-particle propagator and pair propagator.