Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1504.08320 · Journal of Physics A Mathematical and Theoretical

Skeleton series and multivaluedness of the self-energy functional in zero space-time dimensions

Riccardo Rossi, Félix Werner

Published Oct 29, 2015• 48 citations
arXivHTMLPDF

Abstract

Recently, Kozik, Ferrero and Georges discovered numerically that for a family of fundamental models of interacting fermions, the self-energy is a multi-valued functional of the fully dressed single-particle propagator G , and that the skeleton diagrammatic series converges to the wrong branch above a critical interaction strength. We consider the zero space-time dimensional case, where the same mathematical phenomena appear from elementary algebra. We also find a similar phenomenology for the fully bold formalism built on the fully dressed single-particle propagator and pair propagator.

DOI
10.1088/1751-8113/48/48/485202

Related papers

On the Luttinger-Ward functional and the convergence of skeleton diagrammatic series expansion of the self-energy for Hubbard-like models

Behnam Farid

We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy $Σ(\mathbf{k};z)$ specific to uniform ground states (ensemble of states) of interacting fermion systems in terms of skeleton self-energy diagrams and the interacting Green function $G(\mathbf{k};z)$. 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 $\mathbf{k}$ 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]

arXiv:2108.10903•Aug 24, 2021•1 citation

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

Evgeny Kozik, Michel Ferrero, Antoine Georges

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 $\mathbf{G}$, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]\ensuremath{\propto}\ensuremath{\delta}\mathrm{\ensuremath{\Phi}}[\mathbf{G}]/\ensuremath{\delta}\mathbf{G}$ is not a single-valued functional of $\mathbf{G}$: 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 $\mathbf{G}$ 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 $\mathbf{\ensuremath{\Sigma}}$ 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.

arXiv:1407.5687•Physical Review Letters•Apr 15, 2015•148 citations

Comment on `Non-existence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for Hubbard-like models'

R. Eder

The known analytical properties of the Green's function and self-energy rule out an ambiguity of the self-energy. The noninteracting Green's functions obtained by Kozik et al. in arXiv:1407.5687 likely have pathological properties.

arXiv:1407.6599•Jul 24, 2014•3 citations

Physical and unphysical regimes of self-consistent many-body perturbation theory

Kris Van Houcke, Evgeny Kozik, Riccardo Rossi +2 more

In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator G_N G N . We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G_N G N converges when N\to∞ N → ∞ to a limit G_∞\, G ∞ , which coincides with the exact physical propagator G_{exact} G e x a c t at small enough coupling, while G_∞ ≠ G_{exact} G ∞ ≠ G e x a c t at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G_∞=G_{exact} G ∞ = G e x a c t and G_∞≠G_{exact} G ∞ ≠ G e x a c t regimes thanks to a criterion which does not require the knowledge of G_{exact} G e x a c t , as proposed in [Phys. Rev. B 93, 161102 (2016)].

arXiv:2102.04508•SciPost Physics•May 27, 2024•9 citations

Shifted-action expansion and applicability of dressed diagrammatic schemes

Riccardo Rossi, Félix Werner, Nikolay V. Prokof'ev +1 more

While bare diagrammatic series are merely Taylor expansions in powers of interaction strength, dressed diagrammatic series, built on fully or partially dressed lines and vertices, are usually constructed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can even lead to convergence to an unphysical result [E. Kozik, M. Ferrero, and A. Georges, Phys. Rev. Lett. 114, 156402 (2015)]. Here we show that for a broad class of partially dressed diagrammatic schemes, there exists an action ${S}^{(\ensuremath{\xi})}$ depending analytically on an auxiliary complex parameter $\ensuremath{\xi}$, such that the Taylor expansion in $\ensuremath{\xi}$ of correlation functions reproduces the original diagrammatic series. The resulting applicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, analyticity of ${S}^{(\ensuremath{\xi})}$ is not granted, and we formulate a sufficient condition for converging to the correct result.

arXiv:1508.03654•Physical review. B./Physical review. B•Apr 5, 2016•72 citations

Skeleton diagrammatic expansions with screened Hubbard interaction versus multivaluedness of the Luttinger-Ward functional

Aaram J. Kim, Evgeny Kozik

Bulletin of the American Physical Society•Mar 5, 2019•0 citations
Recently, Kozik, Ferrero and Georges discovered numerically that for a family of fundamental models of interacting fermions, the self-energy is a multi-valued functional of the fully dressed single-particle propagator G , and that the skeleton diagrammatic series converges to the wrong branch above a critical interaction strength. We consider the zero space-time dimensional case, where the same mathematical phenomena appear from elementary algebra. We also find a similar phenomenology for the fully bold formalism built on the fully dressed single-particle propagator and pair propagator.
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)Σ(\mathbf{k};z)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)G(\mathbf{k};z)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\mathbf{k}k and complex energies zzz. 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 ΣΣΣ. 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]
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\mathbf{G}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\mathbf{G}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\mathbf{G}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{\mathbf{G}}_{0}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.
In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator G_N G N . We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G_N G N converges when N\to∞ N → ∞ to a limit G_∞\, G ∞ , which coincides with the exact physical propagator G_{exact} G e x a c t at small enough coupling, while G_∞ ≠ G_{exact} G ∞ ≠ G e x a c t at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G_∞=G_{exact} G ∞ = G e x a c t and G_∞≠G_{exact} G ∞ ≠ G e x a c t regimes thanks to a criterion which does not require the knowledge of G_{exact} G e x a c t , as proposed in [Phys. Rev. B 93, 161102 (2016)].
While bare diagrammatic series are merely Taylor expansions in powers of interaction strength, dressed diagrammatic series, built on fully or partially dressed lines and vertices, are usually constructed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can even lead to convergence to an unphysical result [E. Kozik, M. Ferrero, and A. Georges, Phys. Rev. Lett. 114, 156402 (2015)]. Here we show that for a broad class of partially dressed diagrammatic schemes, there exists an action ${S}^{(\ensuremath{\xi})}$ depending analytically on an auxiliary complex parameter $\ensuremath{\xi}$, such that the Taylor expansion in $\ensuremath{\xi}$ of correlation functions reproduces the original diagrammatic series. The resulting applicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, analyticity of ${S}^{(\ensuremath{\xi})}$ is not granted, and we formulate a sufficient condition for converging to the correct result.