Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

The Luttinger-Ward functional $\mathrm{\Phi }[\mathbf{G}]$, 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 $\mathbf{\Sigma }[\mathbf{G}]\propto \delta \mathrm{\Phi }[\mathbf{G}]/\delta \mathbf{G}$ 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 noninteracting Green’s function ${\mathbf{G}}_0$ converges to the correct physical branch of $\mathbf{\Sigma }$ 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.

Figure 1

Formal definition of $\mathrm{\Phi }[\mathbf{G}]$ and $\mathbf{\Sigma }[\mathbf{G}]$ as skeleton diagrammatic expansions. The bold lines represent the full interacting GF $\mathbf{G}$ and the dashed lines the interaction vertex.

Figure 2

(a) Hubbard atom: Double occupancy vs interaction at $T=0.5$, for the physical and unphysical branches (see text). (b) Anderson impurity: $|D^{(\text{exact})}-D^{(\text{found})}|$, quantifying the difference between the exact solution and that found by scheme $A$ (see text), in the $U\text{−}\delta$ plane at $T=0.5$. Black points (crosses) are converged calculations, while at the red points (circles) scheme $A$ could not converge.

Figure 3

Hubbard atom: Two solutions for $\mathbf{\Sigma }[{\mathbf{G}}^{(\text{exact})}]$ and convergence of the corresponding skeleton (bold) expansion, Fig. 1, at half filling, $T=0.5$ and various $U$.

Figure 4

2D Hubbard model: Convergence of the skeleton (bold) and bare series for the momentum dependence of $\mathbf{\Sigma }$ at the lowest Matsubara frequency at half filling, $T=0.5$. The solid (dashed) lines are $\mathrm{Re}\mathbf{\Sigma }$ ($\mathrm{Im}\mathbf{\Sigma }$), the widths express the corresponding error bars.