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

2014/07/31 by Evgeny Kozik, Michel Ferrero, Antoine Georges · 11 citations
Mathematics · Physics and Astronomy · #Anderson impurity model #Cold Atom Physics and Bose-Einstein Condensates #Computer science #Convergence (economics) #Diagrammatic reasoning #Feynman diagram #Function (biology) #Hubbard model #Impurity #Mathematics #Monte Carlo method #Physics #Physics of Superconductivity and Magnetism #Quantum Monte Carlo #Quantum and electron transport phenomena #Quantum mechanics #Series (stratigraphy) #Statistical physics #Superconductivity #Theoretical physics #cond-mat.str-el #cond-mat.supr-con #math-ph #math.MP

paper · pdf · doi:10.1103/physrevlett.114.156402

published as Phys. Rev. Lett. 114, 156402 (2015) · 5 pages, 5 figures

openalex publication_date 2015/04/15 · arxiv created 2015/04/17 · arxiv updated 2015/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The Luttinger-Ward functional \mathrm\ensuremathΦ[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\ensuremathΣ[G]\ensuremath∝\ensuremathδ\mathrm\ensuremathΦ[G]/\ensuremathδG is not a single-valued functional of G: in addition to the physical solution for \mathbf\ensuremathΣ[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\ensuremathΣ 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\ensuremathΣ 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.

Citations

• Self-energy-functional approach to systems of correlated electrons
• Electron correlations in narrow energy bands

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