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:2411.07226 · Physical review. B./Physical review. B

Subleading logarithmic behavior in the parquet formalism

Marcel Gievers, Richard Schmidt, Jan von Delft, Fabian B. Kugler

Published Feb 27, 2025• 3 citations
arXivHTMLPDF

Abstract

The Fermi-edge singularity in x-ray absorption spectra of metals is a paradigmatic case of a logarithmically divergent perturbation series. Prior work has thoroughly analyzed the leading logarithmic terms. Here, we investigate the perturbation theory beyond leading logarithms and formulate self-consistent equations to incorporate all leading and next-to-leading logarithmic terms. This parquet solution of the Fermi-edge singularity goes beyond the previous first-order parquet solution and sheds new light on the parquet formalism regarding logarithmic behavior. We present numerical results in the Matsubara formalism and discuss the characteristic power laws. We also show that, within the single-boson exchange framework, multi-boson exchange diagrams are needed already at the leading logarithmic level.

DOI
10.1103/physrevb.111.085151

Related papers

Fermi-edge singularity and the functional renormalization group

Kugler, F., von Delft, J.

We study the Fermi-edge singularity, describing the response of a degenerate electron system to optical excitation, in the framework of the functional renormalization group (fRG). Results for the (interband) particle-hole susceptibility from various implementations of fRG (one- and two-particle-irreducible, multi-channel Hubbard-Stratonovich, flowing susceptibility) are compared to the summation of all leading logarithmic (log) diagrams, achieved by a (first-order) solution of the parquet equations. For the (zero-dimensional) special case of the x-ray-edge singularity, we show that the leading log formula can be analytically reproduced in a consistent way from a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic structure, we show that this derivation relies on fortuitous partial cancellations special to the form of and accuracy applied to the x-ray-edge singularity and does not generalize.

•May 16, 2018•11 citations

Fermi-edge singularity and the functional renormalization group

Fabian B. Kugler, Jan von Delft

We study the Fermi-edge singularity, describing the response of a degenerate electron system to optical excitation, in the framework of the functional renormalization group (fRG). Results for the (interband) particle-hole susceptibility from various implementations of fRG (one- and two-particle-irreducible, multi-channel Hubbard-Stratonovich, flowing susceptibility) are compared to the summation of all leading logarithmic (log) diagrams, achieved by a (first-order) solution of the parquet equations. For the (zero-dimensional) special case of the x-ray-edge singularity, we show that the leading log formula can be analytically reproduced in a consistent way from a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic structure, we show that this derivation relies on fortuitous partial cancellations special to the form of and accuracy applied to the x-ray-edge singularity and does not generalize.

arXiv:1706.06872•Journal of Physics Condensed Matter•Mar 28, 2018•10 citations

Absorption-edge singularities for two coupled nonequilibrium Fermi seas

Monique Combescot, Bernard Roulet

The absorption-edge singularities for two nonequilibrium Fermi seas, coupled through their interactions with a deep level are calculated by a perturbative approach and by a parquet diagram technique. The result is quite different from the usual power law divergence, and disagrees with what can be obtained from the standard Nozi\`eres--De Dominicis method.

Physical review. B, Condensed matter•Mar 15, 2000•10 citations

Parquet approximation and one-loop renormalization group: Equivalence on the leading-logarithmic level

Jan Diekmann, Severin G. Jakobs

We investigate the functional renormalization group (FRG) flow of the two-particle vertex function of a model for x-ray absorption in metals. Concerning the appearance of logarithmic divergences, the model is prototypical for an important class of mostly zero- and one-dimensional systems which includes the Kondo model and the interacting one-dimensional Fermi gas. For our analysis, we formulate the FRG in the framework of the real-time zero-temperature formalism, in which the model was studied before with a parquet-based approach. We establish that a reasonably crafted, purely fermionic one-loop FRG approximation is fully equivalent on a detailed level to the leading-logarithmic parquet approximation. These two approximation schemes are thus found to just represent different perspectives on the same technical steps. This finding also reconfirms the traditional understanding of the capabilities of one-loop RG approximations for such models, which was recently put into question by an investigation of the x-ray-absorption model with multiloop FRG.

arXiv:2009.04761•Physical review. B./Physical review. B•Apr 29, 2021•5 citations

Leading-logarithmic approximation by one-loop renormalization group within Matsubara formalism

Jan Diekmann, Severin G. Jakobs

We demonstrate how to devise a Matsubara-formalism-based one-loop approximation to the flow of the functional renormalization group (FRG) that reproduces identically the leading-logarithmic parquet approximation. This construction of a controlled fermionic FRG approximation in a regime not accessible by perturbation theory generalizes a previous study from the real-time zero-temperature formalism to the Matsubara formalism and thus to the de facto standard framework used for condensed-matter FRG studies. Our investigation is based on a simple model for the absorption of x rays in metals. It is a core part of our construction to exploit that in a suitable leading-logarithmic approximation the values of the particle-hole susceptibility on the real- and on the imaginary-frequency axes are identical.

arXiv:2310.20648•Physical review. B./Physical review. B•Mar 18, 2024•1 citation

Auxiliary particle theory of threshold singularities in photoemission and x-ray absorption spectra: Test of a conservingT-matrix approximation

T. Schauerte, Johann Kroha, P. Wölfle

We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi\`eres et al.

arXiv:cond-mat/9912007•Physical review. B, Condensed matter•Aug 15, 2000•6 citations
The Fermi-edge singularity in x-ray absorption spectra of metals is a paradigmatic case of a logarithmically divergent perturbation series. Prior work has thoroughly analyzed the leading logarithmic terms. Here, we investigate the perturbation theory beyond leading logarithms and formulate self-consistent equations to incorporate all leading and next-to-leading logarithmic terms. This parquet solution of the Fermi-edge singularity goes beyond the previous first-order parquet solution and sheds new light on the parquet formalism regarding logarithmic behavior. We present numerical results in the Matsubara formalism and discuss the characteristic power laws. We also show that, within the single-boson exchange framework, multi-boson exchange diagrams are needed already at the leading logarithmic level.
We study the Fermi-edge singularity, describing the response of a degenerate electron system to optical excitation, in the framework of the functional renormalization group (fRG). Results for the (interband) particle-hole susceptibility from various implementations of fRG (one- and two-particle-irreducible, multi-channel Hubbard-Stratonovich, flowing susceptibility) are compared to the summation of all leading logarithmic (log) diagrams, achieved by a (first-order) solution of the parquet equations. For the (zero-dimensional) special case of the x-ray-edge singularity, we show that the leading log formula can be analytically reproduced in a consistent way from a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic structure, we show that this derivation relies on fortuitous partial cancellations special to the form of and accuracy applied to the x-ray-edge singularity and does not generalize.
We study the Fermi-edge singularity, describing the response of a degenerate electron system to optical excitation, in the framework of the functional renormalization group (fRG). Results for the (interband) particle-hole susceptibility from various implementations of fRG (one- and two-particle-irreducible, multi-channel Hubbard-Stratonovich, flowing susceptibility) are compared to the summation of all leading logarithmic (log) diagrams, achieved by a (first-order) solution of the parquet equations. For the (zero-dimensional) special case of the x-ray-edge singularity, we show that the leading log formula can be analytically reproduced in a consistent way from a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic structure, we show that this derivation relies on fortuitous partial cancellations special to the form of and accuracy applied to the x-ray-edge singularity and does not generalize.
We investigate the functional renormalization group (FRG) flow of the two-particle vertex function of a model for x-ray absorption in metals. Concerning the appearance of logarithmic divergences, the model is prototypical for an important class of mostly zero- and one-dimensional systems which includes the Kondo model and the interacting one-dimensional Fermi gas. For our analysis, we formulate the FRG in the framework of the real-time zero-temperature formalism, in which the model was studied before with a parquet-based approach. We establish that a reasonably crafted, purely fermionic one-loop FRG approximation is fully equivalent on a detailed level to the leading-logarithmic parquet approximation. These two approximation schemes are thus found to just represent different perspectives on the same technical steps. This finding also reconfirms the traditional understanding of the capabilities of one-loop RG approximations for such models, which was recently put into question by an investigation of the x-ray-absorption model with multiloop FRG.
We demonstrate how to devise a Matsubara-formalism-based one-loop approximation to the flow of the functional renormalization group (FRG) that reproduces identically the leading-logarithmic parquet approximation. This construction of a controlled fermionic FRG approximation in a regime not accessible by perturbation theory generalizes a previous study from the real-time zero-temperature formalism to the Matsubara formalism and thus to the de facto standard framework used for condensed-matter FRG studies. Our investigation is based on a simple model for the absorption of x rays in metals. It is a core part of our construction to exploit that in a suitable leading-logarithmic approximation the values of the particle-hole susceptibility on the real- and on the imaginary-frequency axes are identical.
We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi\`eres et al.