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We generate the perturbative expansion of the single-particle Green'sfunction and related self-energy for a half-filled single-band Hubbard model ona square lattice. We invoke algorithmic Matsubara integration to evaluatesingle-particle quantities for real and Matsubara frequencies and verifyresults through comparison to existing data on the Matsubara axis. With loworder expansions at weak-coupling we observe a number of outcomes expected athigher orders: the opening of a gap, pseudogap behavior, and Fermi-surfacereconstruction. Based on low-order perturbations we consider the phase diagramthat arises from truncated expansions of the self-energy and Green's functionand their relation via the Dyson equation. From Matsubara axis data we observeinsulating behavior in direct expansions of the Green's function, while thesame order of truncation of the self-energy produces metallic behavior. Thisobservation is supported by additional calculations for real frequencies. Weattribute this difference to the order in which diagrams are implicitly summedin the Dyson series. By separating the reducible and irreducible contributionsat each order we show that the reducible diagrams implicitly summed in theDyson equation lead to incorrect physics in the half-filled Hubbard model. Ourobservations for this particular case lead us to question the utility of theDyson equation for any problem that shows a disparity between reducible andirreducible contributions to the expansion of the Green's function.

physical review. b./physical review. b

Single particle properties of the 2D Hubbard model for real frequencies  at weak coupling: Breakdown of the Dyson series for partial self-energy  expansions