Title: The self-consistent Dyson equation and self-energy functionals: failure or new opportunities?,
Abstract: Perturbation theory using self-consistent Green's functions is one of the
most widely used approaches to study many-body effects in condensed matter. On
the basis of general considerations and by performing analytical calculations
for the specific example of the Hubbard atom, we discuss some key features of
this approach. We show that when the domain of the functionals that are used to
realize the map between the non-interacting and the interacting Green's
functions is properly defined, there exists a class of self-energy functionals
for which the self-consistent Dyson equation has only one solution, which is
the physical one. We also show that manipulation of the perturbative expansion
of the interacting Green's function may lead to a wrong self-energy as
functional of the interacting Green's function, at least for some regions of
the parameter space. These findings confirm and explain numerical results of
Kozik 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 distinguish
between the maps between sets of functions and the functionals that realize
those maps. We demonstrate that the self-consistent Green's functions approach
itself is not problematic, whereas the functionals that are widely used may
have a limited range of validity.