Non-existence of the Luttinger-Ward functional and misleading   convergence of skeleton diagrammatic series for Hubbard-like models
  The Luttinger-Ward functional $\Phi[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function $\mathbf{G}$, is found to be ill-defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\Sigma}[\mathbf{G}] \propto \delta\Phi[\mathbf{G}]/\delta \mathbf{G}$ is not a single-valued functional of $\mathbf{G}$: in addition to the physical solution for $\mathbf{\Sigma}[\mathbf{G}]$, 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 $\mathbf{\Sigma}$ in terms of $\mathbf{G}$ 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 non-interacting Green's function $\mathbf{G}_0$ converges to the correct physical branch of $\mathbf{\Sigma}$ in all cases currently accessible by diagrammatic Monte Carlo. Besides their conceptual importance, these observations have important implications for techniques based on the explicit summation of diagrammatic series. 
Evgeny Kozik, Michel Ferrero, Antoine Georges
cond-mat.str-el cond-mat.supr-con math-ph math.MP