The Luttinger-Ward functional $\mathrm{\ensuremath{\Phi}}[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function
G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]\ensuremath{\propto}\ensuremath{\delta}\mathrm{\ensuremath{\Phi}}[\mathbf{G}]/\ensuremath{\delta}\mathbf{G}$ is not a single-valued functional of
G: in addition to the physical solution for $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]$, there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for $\mathbf{\ensuremath{\Sigma}}$ in terms of
G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the noninteracting Green's function
G0 converges to the correct physical branch of $\mathbf{\ensuremath{\Sigma}}$ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.
We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy
Σ(k;z) specific to uniform ground states (ensemble of states) of interacting fermion systems in terms of skeleton self-energy diagrams and the interacting Green function
G(k;z). Utilising a link between the latter series expansion and the classical moment problem (of the Hamburger type), along with the associated continued-fraction expansion, we reaffirm our earlier observation (2007) that for lattice models of fermions interacting through short-range two-body potentials (i.e. for Hubbard-like models) this series is uniformly convergent for almost all wave vectors
k and complex energies
z. The limit of this series is unique. We inquire into the reasons underlying the contrary observation by Kozik et al. (2015) regarding skeleton-diagrammatic perturbation series expansion of
Σ. In doing so, we make a number of observations of general interest. Chief amongst these, we observe that contrary to general belief thermal correlation functions calculated in the energy domain (i.e. over the set of the relevant Matsubara frequencies) are generally unreliable, in contrast to those calculated in the imaginary-time domain. In an appendix we present a symbolic computational formalism and short programs for singling out topologically distinct proper self-energy diagrams that are algebraically equivalent up to determinate multiplicative constants. This complements the work presented in our previous publication (2019). [Abridged abstract]
We study the properties of the Luttinger-Ward functional (LWF) in a simplified Hubbard-type model without time or spatial dimensions, but with
N identical replicas located on a single site. The simplicity of this
(0+0)d model permits an exact solution for all
N and for both bosonic and fermionic statistics. We show that fermionic statistics are directly linked to the fact that multiple values of the noninteracting Green's function
G0 map to the same value of the interacting Green's function
G; that is, the mapping ${G}_{0}\ensuremath{\mapsto}G$ is noninjective. This implies that with fermionic statistics the
(0+0)dN-replica model has a multiply valued LWF. The number of LWF values in the fermionic model increases proportionally to the number of replicas
N, while in the bosonic model the LWF has a single value regardless of
N. We also discuss the formal connection between the
N-replica model and the
(0+1)d Hubbard atom which was used in previous studies of LWF's multivaluedness.