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Perturbation theory using self-consistent Green's functions is one of themost widely used approaches to study many-body effects in condensed matter. Onthe basis of general considerations and by performing analytical calculationsfor the specific example of the Hubbard atom, we discuss some key features ofthis approach. We show that when the domain of the functionals that are used torealize the map between the non-interacting and the interacting Green'sfunctions is properly defined, there exists a class of self-energy functionalsfor which the self-consistent Dyson equation has only one solution, which isthe physical one. We also show that manipulation of the perturbative expansionof the interacting Green's function may lead to a wrong self-energy asfunctional of the interacting Green's function, at least for some regions ofthe parameter space. These findings confirm and explain numerical results ofKozik et al. for the widely used skeleton series of Luttinger and Ward [Phys.Rev. Lett. 114, 156402]. Our study shows that it is important to distinguishbetween the maps between sets of functions and the functionals that realizethose maps. We demonstrate that the self-consistent Green's functions approachitself is not problematic, whereas the functionals that are widely used mayhave a limited range of validity.

The self-consistent Dyson equation and self-energy functionals: failure or new opportunities?