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arxiv: 2511.17697 · v3 · submitted 2025-11-21 · ❄️ cond-mat.str-el

Functional renormalization with interaction flows: A single-boson exchange perspective and application to electron-phonon systems

Aiman Al-Eryani , Marcel Gievers , Kilian Fraboulet This is my paper

Pith reviewed 2026-05-17 20:10 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords functional renormalization groupsingle-boson exchangeelectron-phonon systemsAnderson impurity modeltemperature flowretarded interactionsmultiloop fRG
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0 comments X

The pith

Single-boson exchange lets fRG regulate interactions as well as propagators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a functional renormalization group formulation that dresses both bare propagators and bare interactions with regulators. It starts from the Schwinger-Dyson and Bethe-Salpeter equations and extends the multiloop fRG approach previously used for many-fermion systems. The single-boson exchange decomposition shows that the new flow equations amount to adding a regulator directly to the bosonic propagator. This extension leaves the original structure of the equations largely unchanged. The resulting framework supports temperature flows and other schemes for models with retarded interactions such as electron-phonon systems, with tests performed on the Anderson impurity model and its phonon-coupled variant.

Core claim

Starting from Schwinger--Dyson and Bethe--Salpeter equations, we develop here an fRG formulation where both bare propagators and bare interactions can be dressed with regulators. The approach thus obtained is an extension of the multiloop fRG recently introduced for many-fermion systems. Using the single-boson exchange decomposition, we show that the underlying flow equations are simply interpreted as adding a regulator to the bosonic propagator and that such an extension scarcely changes the original structure of the flow equations.

What carries the argument

Single-boson exchange decomposition, which recasts the regulated interaction flow as a simple regulator added to the bosonic propagator.

If this is right

  • Temperature flows become feasible for models that include retarded interactions.
  • Loop convergence can be compared directly to conventional cutoff schemes in the Anderson impurity model.
  • A cutoff applied simultaneously to the propagator and the bare interaction is shown to be valid for an Anderson impurity coupled to a phonon.
  • Classes of renormalization schemes previously inaccessible with standard fRG methods are now reachable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may simplify renormalization flows in any system whose interactions are carried by a bosonic mode.
  • It could improve treatment of finite-temperature effects in models with frequency-dependent interactions.
  • Analogous regulator placements might be explored for other interaction decompositions in many-body calculations.

Load-bearing premise

The single-boson exchange decomposition remains accurate when regulators are placed on interactions.

What would settle it

Numerical flows for the Anderson impurity model in which the new interaction-regulated temperature scheme deviates strongly from established benchmarks or from conventional propagator-only cutoff results.

Figures

Figures reproduced from arXiv: 2511.17697 by Aiman Al-Eryani, Kilian Fraboulet, Marcel Gievers.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of diagrams contributing to the parquet [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagrammatic classification underlying the SBE de [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Bosonic frequency dependence of susceptibilities and fermion-boson vertices for the AIM in the magnetic and density [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Imaginary part of the self-energy from the 1 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Susceptibilities and fermion-boson vertices for the AHIM at inverse temperature [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Imaginary part of the self-energy obtained for differ [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Density susceptibility evaluated at its zeroth and [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Diagrammatic representation of the flow equations resulting from the vertex expansion of the Wetterich equation [ [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Susceptibilities and fermion-boson vertices for the HA in the magnetic and density channels at [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Susceptibilities and fermion-boson vertices at 1 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Imaginary part of the self-energy at the 1 [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
read the original abstract

The functional renormalization group (fRG) is acknowledged as a powerful tool in quantum many-body physics and beyond. On the technical side, conventional implementations of the fRG rely on regulators for bare propagators only. Starting from Schwinger--Dyson and Bethe--Salpeter equations, we develop here an fRG formulation where both bare propagators and bare interactions can be dressed with regulators. The approach thus obtained is an extension of the multiloop fRG recently introduced for many-fermion systems. Using the single-boson exchange decomposition, we show that the underlying flow equations are simply interpreted as adding a regulator to the bosonic propagator and that such an extension scarcely changes the original structure of the flow equations. Overall, we provide a framework for implementing approaches that cannot be realized with conventional fRG methods, such as temperature flows for models with retarded interactions. For concrete applications, we analyze the loop convergence of our scheme against conventional cutoff schemes for the Anderson impurity model. Finally, we devise a new temperature-flow scheme that implements a cutoff in both the propagator and the bare interaction, and demonstrate its validity on a model of an Anderson impurity coupled to a phonon.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops an extension of multiloop functional renormalization group (fRG) methods that incorporates regulators on both bare propagators and bare interactions. Starting from the Schwinger-Dyson and Bethe-Salpeter equations and employing the single-boson exchange (SBE) decomposition of the four-point vertex, the authors argue that the resulting flow equations reduce to the addition of a regulator solely to the bosonic propagator, with negligible alteration to the original flow structure. Concrete applications include tests of loop convergence against conventional cutoff schemes in the Anderson impurity model and the implementation of a new temperature-flow scheme for an Anderson impurity coupled to a phonon.

Significance. If the central claim holds, the work enables fRG flows (such as temperature flows) in systems with retarded interactions that are inaccessible to conventional propagator-only regulators. The SBE perspective is presented as preserving the simplicity of the multiloop structure, and the numerical demonstrations on impurity models provide a practical test bed. This could broaden the applicability of fRG to electron-phonon and related condensed-matter problems.

major comments (1)
  1. [Derivation using SBE decomposition and electron-phonon application] The central claim that the SBE decomposition allows the regulated interaction flow to be interpreted simply as adding a regulator to the bosonic propagator (without substantial structural change or extra vertex corrections) is load-bearing for the entire extension. This assumption is not isolated and verified in the electron-phonon application, where frequency-dependent bare interactions could mix with the bosonic self-energy flow and generate additional regulator-dependent corrections that would require re-deriving the multiloop equations. The Anderson-impurity loop-convergence tests do not address this mixing.
minor comments (1)
  1. [Abstract] The abstract states that the extension 'scarcely changes the original structure of the flow equations'; a short explicit statement of which equations remain unchanged would improve clarity for readers unfamiliar with the multiloop fRG baseline.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address the major comment below and are willing to revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Derivation using SBE decomposition and electron-phonon application] The central claim that the SBE decomposition allows the regulated interaction flow to be interpreted simply as adding a regulator to the bosonic propagator (without substantial structural change or extra vertex corrections) is load-bearing for the entire extension. This assumption is not isolated and verified in the electron-phonon application, where frequency-dependent bare interactions could mix with the bosonic self-energy flow and generate additional regulator-dependent corrections that would require re-deriving the multiloop equations. The Anderson-impurity loop-convergence tests do not address this mixing.

    Authors: We thank the referee for this insightful comment. Our derivation, based on the Schwinger-Dyson and Bethe-Salpeter equations combined with the single-boson exchange (SBE) decomposition, demonstrates that regulating the bare interaction is equivalent to regulating the bosonic propagator. This equivalence is general and does not depend on the specific form of the interaction, including frequency-dependent cases. In the electron-phonon application, the bare interaction is the phonon-mediated retarded interaction, which is incorporated as the bosonic propagator in the SBE framework. The flow equations for the vertex and self-energies follow the standard multiloop structure, with the regulator on the bosonic propagator accounting for the interaction flow. Any potential mixing with the bosonic self-energy is already included in the flow and does not generate additional vertex corrections that would necessitate re-deriving the equations. The Anderson impurity tests establish the reliability of the multiloop convergence, while the electron-phonon example validates the temperature flow for retarded interactions. To explicitly address the referee's concern and isolate this verification, we will add a dedicated subsection or appendix in the revised manuscript that derives the flow equations specifically for the electron-phonon model, confirming the absence of extra corrections. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from SDE/BSE without reduction to inputs or self-citations

full rationale

The paper begins from the standard Schwinger-Dyson and Bethe-Salpeter equations and extends the existing multiloop fRG framework via the single-boson exchange decomposition. The claim that interaction regulators reduce to a bosonic-propagator regulator follows directly from applying the decomposition to the regulated equations; this is a structural consequence of the chosen vertex parametrization rather than a redefinition or fit of the target result inside the paper. No load-bearing self-citation chains, ansatzes smuggled via prior work, or predictions that are statistically forced by construction appear in the derivation steps described. The Anderson-impurity and electron-phonon applications serve as numerical checks rather than inputs that define the central result. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the single-boson exchange decomposition when regulators act on interactions and on the assumption that the multiloop fRG structure survives the extension with only minor changes.

axioms (1)
  • domain assumption Single-boson exchange decomposition accurately represents the interaction vertex for the models under study.
    Invoked to reinterpret the flow equations as a simple regulator addition to the bosonic propagator.

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