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Multiloop functional renormalization group from single bosons

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Neglecting multi-boson exchange contributions, the multiloop single-boson exchange fRG reproduces the parquet approximation to within a few percent in the 2D Hubbard model.

desk verdict A careful, useful multiloop SBE fRG derivation and benchmark; the few-percent agreement with parquet is real but is partly a test of two approximations sharing a form-factor truncation. read the letter →

arxiv 2512.11190 v2 pith:6R5FBBZR submitted 2025-12-12 cond-mat.str-el

Kilian Fraboulet , Aiman Al-Eryani , Sarah Heinzelmann , Anna Kauch , Sabine Andergassen This is my paper · ORCID
classification cond-mat.str-el
keywords multiloopfunctionalrenormalizationgroupsingle-bosonexchangedecompositionHubbardmodelparquetapproximationYukawacouplingssusceptibilitiesrestfunctionsweakcoupling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that the single-boson exchange (SBE) approximation of the multiloop functional renormalization group, which drops the flow of the multi-boson exchange rest functions, reproduces the parquet approximation accurately once loop corrections are converged. For the two-dimensional Hubbard model at weak coupling (U=2.5, β=5), the converged SBE results differ from parquet by at most 5% in the magnetic susceptibility peak, and by ≤3% at finite doping. This matters because the rest functions are the most expensive objects to compute, so dropping them makes quantitative multiloop fRG substantially cheaper. The paper also provides a complete derivation of the multiloop SBE fRG equations in physical channels, and analyzes cutoff dependence, loop convergence, and temperature dependence.

What carries the argument

The central machinery is the single-boson exchange (SBE) decomposition of the two-particle vertex, which rewrites the vertex in terms of bosonic propagators w_X(Q), Yukawa couplings λ_X(Q,k), and rest functions M_X(Q,k,k') in three physical channels (magnetic, density, superconducting). The multiloop fRG flow equations for these objects are derived from the parquet/Bethe-Salpeter structure and are solved with a loop expansion (ℓ=1,2,3,...) together with a self-energy flow equation for the self-energy. The SBE approximation consists of discarding the flow equations for the rest functions M_X, thereby removing the most expensive part of the computation.

What would settle it

Compute the magnitude of the neglected mixed form-factor bubble contributions Π_{X,nm}(Q) at the finite incommensurate wave vector that dominates the magnetic susceptibility at finite doping, or repeat the calculation without the truncation and compare the SBE approximation to parquet; if the mixed-bubble terms are not negligible at these Q, the claimed ≤3% agreement is unlikely to survive.

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Extended reading notes

Core claim

The central claim is that the SBE approximation—setting the flow of the multi-boson exchange rest functions M_X to zero while keeping the flow of bosonic propagators and Yukawa couplings—accurately reproduces the parquet approximation at loop convergence in the weak-coupling 2D Hubbard model. Quantitatively, the relative difference in the magnetic susceptibility peak is at most 5% at half filling and ≤3% at finite doping, with frequency- and momentum-dependent vertices showing similar accuracy. Since the converged multiloop SBE fRG with the rest functions included is equivalent to the parquet approximation, this shows that the computationally expensive rest-function flow can be discarded wit

Load-bearing premise

The numerical comparison relies on setting all mixed form-factor bubble contributions to zero at every momentum, although the stated justification only guarantees they vanish at q=0; if those terms are sizable at the finite momenta that dominate the response, the claimed few-percent agreement with parquet could degrade.

Editorial extensions

If this is right

  • Multiloop fRG calculations for correlated electron systems become substantially cheaper, since the rest functions M_X, the most costly objects, no longer need to be flowed.
  • The SBE approximation provides a physically transparent picture in which the dominant fluctuations are carried by single-boson exchange processes, with multi-boson effects implicitly resummed through the flow.
  • The method opens the route to more challenging parameter regimes and more realistic models, where full parquet-equivalent multiloop fRG would be prohibitively expensive.
  • The residual cutoff dependence of the SBE approximation is small at loop convergence (few percent), and the Ω-flow converges fastest among the tested schemes.
  • The approach correctly captures the interplay of magnetic and d-wave superconducting fluctuations as temperature is lowered.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the SBE approximation already matches parquet at weak coupling, one might expect it to become less accurate at intermediate to strong coupling, where multi-boson exchange processes (the dropped rest functions) are more important; the paper does not test this regime, and the claim should not be extrapolated there.
  • The numerical implementation sets all mixed form-factor bubble contributions to zero at all momenta, not just at q=0 where they vanish; since the parquet comparison uses the same truncation, the observed agreement may in part reflect a shared systematic error rather than the intrinsic accuracy of the SBE approximation.
  • A testable extension would be to include the mixed-bubble terms selectively at the dominant finite-momentum wave vectors (or to compute their magnitude) to see whether the few-percent agreement persists; if it does, the truncation is innocent; if not, the reported accuracy is partly accidental.
  • Since the SBE approximation is a differential-equation-based resummation that generates rest-function-like contributions implicitly, it may be the basis for a cheaper parquet solver for models where the full parquet solution is numerically out of reach.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a multiloop extension of the single-boson-exchange (SBE) functional renormalization group (fRG) in physical channels and applies it to the two-dimensional Hubbard model at weak coupling. The central claim is that the "SBE approximation" — obtained by neglecting the flow of the multi-boson rest functions M_X in Eqs. (81c), (82c), (83c) — reproduces the parquet approximation at loop convergence. This is supported by direct numerical comparisons for U=2.5, β=5 at half filling and at finite doping (n=0.41), with reported differences of at most 5% in the magnetic susceptibility peak at half filling and at most 3% at finite doping. The paper also provides a detailed derivation of the flow equations, discusses cutoff dependence across three flow schemes, and presents temperature-dependent results at β=5,10,20.

Significance. If the central claim holds, the paper offers a substantial computational simplification: the expensive flow of the SBE rest functions can be omitted without sacrificing quantitative accuracy in the weak-coupling regime, enabling applications to more challenging parameter regimes and more realistic models. The manuscript's strengths are its self-contained derivation of the multiloop SBE equations in physical channels, the direct parameter-free comparison with a parquet solver rather than a fitted benchmark, the explicit treatment of loop and self-energy convergence, and the systematic cutoff-dependence analysis. The main caveat is that the parquet benchmark itself is computed within a truncated form-factor space with off-diagonal bubble contributions set to zero; the headline claim therefore needs to be qualified accordingly.

major comments (3)
  1. [Section III.E and Appendix A (Eqs. A1–A3)] The mixed-form-factor bubble truncation Π_{X,nm}=Π̇_{X,nm}=0 for n≠m is justified by the statement that these contributions 'anyway at q=0 vanish,' but the flow is evaluated for all bosonic momenta Q, including the incommensurate magnetic peaks at finite doping shown in Figs. 14–15. Since the parquet benchmark is computed with the same truncation, the reported ≤3–5% agreement demonstrates that dropping M_X is accurate within this truncated form-factor space, not necessarily that the SBE approximation reproduces the full parquet approximation. The abstract and conclusion should be qualified to avoid overstating the claim. If feasible, a numerical test with mixed-bubble terms retained, or at least an estimate of their size at finite Q, would strengthen the conclusion.
  2. [Section IV.A and Fig. 10] The text explicitly states that 'none of the three flow schemes are actually fully converged at U=2.5, β=5, and half filling for ℓ=30.' Yet Figs. 11–13 compare 'converged' SBE fRG results with the parquet approximation, using up to 36 loop corrections per self-energy iteration. The paper should quantify the difference between the 30ℓ results and the converged results for the susceptibility peaks. Without this number, the reader cannot judge whether the reported 3–5% differences between the SBE approximation and parquet are contaminated by residual loop-convergence error.
  3. [Section IV.B–IV.C and Abstract] The central claim that the SBE approximation 'accurately reproduces the parquet approximation at loop convergence' is benchmarked only at β=5 (for half filling and n=0.41) and for U up to 2.5 at β=5. The temperature-dependent results at β=10 and β=20 in Sec. IV.C are not compared with the parquet approximation. The abstract and conclusion should either restrict the validation statement to the computed parameter window or add parquet benchmarks at the lower temperatures shown in Figs. 15–17.
minor comments (5)
  1. [Section III.E] The phrase 'since that their contributions anyway at q=0 vanish' is grammatically awkward and should be rephrased, e.g., 'since these contributions vanish at Q=0.' Also clarify that the truncation is applied for all Q, not only Q=0.
  2. [Section IV.A] Typo: 'hilf-filled' should be 'half-filled' in the text preceding Fig. 7.
  3. [Fig. 11 caption] Typo: 'fom' should be 'from' in the caption.
  4. [Section V] The phrase 'rest function like contributions' should be hyphenated: 'rest-function-like contributions.'
  5. [Fig. 6 caption] The phrase 'enclose the reduced Brillouin zone' is unclear; consider 'enclosed by the Γ-X-M-Γ symmetry path.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SBE approximation is a genuine truncation compared against an independent parquet benchmark, with no fitted parameters.

full rationale

The central claim of the paper is that the SBE approximation—operationally defined by discarding the flow equations for the multi-boson rest functions M_X (Eqs. (81c), (82c), (83c))—reproduces the parquet approximation at loop convergence. This is a numerical comparison between two distinct approximations within the same fRG framework: including the M_X flow is equivalent to the parquet approximation, whereas the SBE approximation is a truncation of that same system. No parameter is fitted to the parquet benchmark. The only auxiliary quantity fixed by the model is the chemical-potential shift, determined by the filling via Eq. (108), not by the benchmark. The agreement (≤5% at U=2.5, half filling; ≤3% at finite doping) is therefore a nontrivial numerical finding. The formalism in Sec. II is built on the prior derivation in Ref. [64] (whose authors do not overlap with the present paper) and on the authors' earlier SBE-fRG works (Refs. [60,61,109]); those citations supply the equation structure, but the accuracy claim is not justified by citing them—it is substantiated by the direct comparison to the parquet approximation in Figs. 11–14. The paper also explicitly discloses the shared mixed-form-factor bubble truncation in Sec. III.E and the residual cutoff dependence in Sec. IV.A. These are limitations and correctness risks (the benchmark and the SBE approximation share a systematic truncation), but they are not circularity: the derivation chain does not reduce a prediction to a fitted input, a self-citation, or a definitional equivalence. No self-definitional reduction, fitted-input-as-prediction, imported uniqueness, or renamed known result is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard fRG/parquet background and a series of numerical truncations (grids, frequency boxes, form factors, mixed-bubble neglect). No parameters are fitted to the parquet benchmark; the SBE bosonic propagators and Yukawa couplings are a bookkeeping reparametrization of the fermionic vertex, not new physical entities.

free parameters (4)
  • Momentum grid size N_k = 16
    Coarse 16×16 Brillouin-zone grid for the self-energy and form-factor projections; truncation error not quantified and affects the central comparison.
  • Frequency box size N_w = 6 for β=5,7.5; 8 for β=10,20
    Truncation of the Matsubara-frequency dependence; outside the box, objects are replaced by high-frequency asymptotics.
  • Form-factor set = s-wave (half filling); s+d-wave (finite doping)
    Truncated-unity basis truncation; the d-wave Yukawa coupling vanishes identically in the SBE approximation, a structural limitation.
  • Convergence tolerances = ε_se=10^-3, ε_vtx=10^-4
    Numerical tolerances for self-energy iterations and loop corrections; chosen by hand and influence the reported agreement.
assumptions (6)
  • standard math Standard Bethe-Salpeter/parquet decomposition and the equivalence of converged multiloop fRG to the parquet approximation
    Taken from Refs. [84,85,88]; used as the foundation of the flow equations and as the benchmark target.
  • domain assumption SU(2) spin symmetry and translational invariance of the Hubbard model
    Invoked in Sec. II.E to reduce the spin structure to three physical channels (M, D, SC); exact for the hexagonal? for the square-lattice Hubbard model, but restricts generalization.
  • domain assumption Time-reversal symmetry (λ_X = λ̄_X)
    Eq. (60); valid for the Hubbard model without magnetic fields, but an input to the derivation.
  • domain assumption Truncated-unity form-factor expansion (s-wave at half filling; s+d at finite doping) is quantitatively sufficient at weak coupling
    Sec. III.E; relies on prior fRG practice, not proven here.
  • ad hoc to paper Mixed-bubble contributions Π_{X,nm}=0 for n≠m
    Sec. III.E; justified only by vanishing at q=0, applied at all momenta Q.
  • domain assumption High-frequency asymptotics of Ref. [63] outside the frequency box
    Sec. III.E; standard in truncated fRG implementations.

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Pith. "Pith review of Multiloop functional renormalization group from single bosons." pith.science (2026). https://pith.science/paper/6R5FBBZR

@misc{pith2026251211190,
  author       = {Pith},
  title        = {Pith review of: Multiloop functional renormalization group from single bosons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6R5FBBZR}},
  note         = {Machine review of arXiv:2512.11190}
}
read the original abstract

The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., neglecting the flow of the multi-boson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.

Figures

Figures reproduced from arXiv: 2512.11190 by the authors.

Figure 1
Figure 1. FIG. 1. Frequency and momentum conventions for the two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the diagrammatic criteria underlying the parquet and the SBE decompositions of the two-particle [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Venn diagram illustrating the connection between [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Diagrammatic representation of the SBE decomposition expressed by Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Flow chart: at each integration step of the multiloop equations, there is a self-consistency cycle between the self-energy [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The coarse grid of 16 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Static bosonic momentum dependence of the magnetic susceptibility [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: presents analogous results as [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Frequency dependencies of the susceptibilities and the Yukawa couplings in all three physical channels at bosonic [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Convergence parameters for the multiloop SBE fRG at 30 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Static bosonic momentum dependence of the suscep [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Static susceptibilities in all three physical channels [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Frequency dependencies of the susceptibilities and the Yukawa couplings in all three physical channels at bosonic [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Bosonic momentum and frequency dependencies [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Bosonic momentum dependence of the static sus [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Bosonic momentum dependence of the Yukawa [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Frequency dependencies of the susceptibilities and the Yukawa couplings in all three physical channels at bosonic [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]

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Reference graph

Works this paper leans on

140 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    [52], this flow scheme implements a multiplicative soft frequency cutoff GΛ 0 (k, ν) := ΘΛ(ν) G−1 0 (k, ν) +δµΛ ,(104) with ΘΛ(ν) = ν2 ν2 + Λ2 (105) a low-frequency cutoff function

    Ω-flow.Introduced in Ref. [52], this flow scheme implements a multiplicative soft frequency cutoff GΛ 0 (k, ν) := ΘΛ(ν) G−1 0 (k, ν) +δµΛ ,(104) with ΘΛ(ν) = ν2 ν2 + Λ2 (105) a low-frequency cutoff function. Here, the flow pa- rameter Λ flows from Λ init =∞to Λ final = 0 re- covering the fullG 0. 2.TheU-flow.This flow scheme [118] is implemented by consid...

  2. [2]

    Krien, A

    F. Krien, A. I. Lichtenstein, and G. Rohringer, Fluc- tuation diagnostic of the nodal/antinodal dichotomy in the Hubbard model at weak coupling: A parquet dual fermion approach, Phys. Rev. B102, 10.1103/phys- revb.102.235133 (2020)

    doi:10.1103/phys- 2020

  3. [3]

    C. J. Eckhardt, C. Honerkamp, K. Held, and A. Kauch, Truncated unity parquet solver, Phys. Rev. B101, 155104 (2020)

    2020

  4. [4]

    Astleithner, A

    K. Astleithner, A. Kauch, T. Ribic, and K. Held, Par- quet dual fermion approach for the Falicov-Kimball model, Phys. Rev. B101, 165101 (2020)

    2020

  5. [5]

    G. V. Astretsov, G. Rohringer, and A. N. Rubtsov, Dual parquet scheme for the two-dimensional Hubbard model: Modeling low-energy physics of high-Tc cuprates with high momentum resolution, Phys. Rev. B101, 075109 (2020)

    2020

  6. [6]

    Krien and A

    F. Krien and A. Kauch, The plain and simple parquet approximation: single-and multi-boson exchange in the two-dimensional Hubbard model, Eur. Phys. J. B95, 69 (2022)

    2022

  7. [7]

    Rohringer, Spectra of correlated many-electron sys- tems: From a one- to a two-particle description, Journal of Electron Spectroscopy and Related Phenomena241, 146804 (2020)

    G. Rohringer, Spectra of correlated many-electron sys- tems: From a one- to a two-particle description, Journal of Electron Spectroscopy and Related Phenomena241, 146804 (2020)

    2020

  8. [8]

    Ellwanger, Flow equations forNpoint functions and bound states, Z

    U. Ellwanger, Flow equations forNpoint functions and bound states, Z. Phys. C - Particles and Fields62, 503 (1994)

    1994

Show all 140 references
  1. [9]

    Wetterich, Exact evolution equation for the effective potential, Physics Letters B301, 90 (1993)

    C. Wetterich, Exact evolution equation for the effective potential, Physics Letters B301, 90 (1993)

    1993

  2. [10]

    Metzner, M

    W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Sch¨ onhammer, Functional renormalization group approach to correlated fermion systems, Rev. Mod. Phys.84, 299 (2012)

    2012

  3. [11]

    T. R. Morris, The Exact Renormalization Group and Approximate Solutions, International Journal of Mod- ern Physics A09, 2411 (1994)

    1994

  4. [12]

    Krien, A

    F. Krien, A. Valli, and M. Capone, Single-boson ex- change decomposition of the vertex function, Phys. Rev. B100, 155149 (2019)

    2019

  5. [13]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonper- turbative functional renormalization group and its ap- plications, Physics Reports910, 1 (2021), the nonper- turbative functional renormalization group and its ap- plications

    2021

  6. [14]

    De Dominicis and P

    C. De Dominicis and P. C. Martin, Stationary Entropy Principle and Renormalization in Normal and Super- fluid Systems. II. Diagrammatic Formulation, J. Math. Phys.5, 31 (1964)

    1964

  7. [15]

    De Dominicis and P

    C. De Dominicis and P. C. Martin, Stationary Entropy Principle and Renormalization in Normal and Super- fluid Systems. I. Algebraic Formulation, J. Math. Phys. 5, 14 (1964)

    1964

  8. [16]

    S. X. Yang, H. Fotso, J. Liu, T. A. Maier, K. Tomko, E. F. D’Azevedo, R. T. Scalettar, T. Pruschke, and M. Jarrell, Parquet approximation for the 4×4 Hub- bard cluster, Phys. Rev. E80, 046706 (2009)

    2009

  9. [17]

    N. E. Bickers, Self-Consistent Many-Body Theory for Condensed Matter Systems, inS´ en´ echal D., Tremblay A.-M., Bourbonnais C. (eds), Theoretical Methods for Strongly Correlated Electrons, CRM Series in Mathe- matical Physics(Springer, New York, 2004)

    2004

  10. [18]

    Rohringer, A

    G. Rohringer, A. Valli, and A. Toschi, Local electronic correlation at the two-particle level, Phys. Rev. B86, 125114 (2012)

    2012

  11. [19]

    K.-M. Tam, H. Fotso, S.-X. Yang, T.-W. Lee, J. Moreno, J. Ramanujam, and M. Jarrell, Solving the parquet equations for the Hubbard model beyond weak coupling, Phys. Rev. E87, 013311 (2013)

    2013

  12. [20]

    Kauch, F

    A. Kauch, F. H¨ orbinger, G. Li, and K. Held, Interplay between magnetic and superconducting fluctuations in the doped 2d Hubbard model (2019), arXiv:1901.09743

    2019 arXiv

  13. [21]

    Valli, T

    A. Valli, T. Sch¨ afer, P. Thunstr¨ om, G. Rohringer, S. An- dergassen, G. Sangiovanni, K. Held, and A. Toschi, Dy- namical vertex approximation in its parquet implemen- tation: Application to Hubbard nanorings, Phys. Rev. B91, 115115 (2015)

    2015

  14. [22]

    Sch¨ afer, G

    T. Sch¨ afer, G. Rohringer, O. Gunnarsson, S. Ciuchi, G. Sangiovanni, and A. Toschi, Divergent Precursors of the Mott-Hubbard Transition at the Two-Particle Level, Phys. Rev. Lett.110, 246405 (2013)

    2013

  15. [23]

    G. Li, A. Kauch, P. Pudleiner, and K. Held, The vic- tory project v1.0: An efficient parquet equations solver, Comput. Phys. Commun.241, 146 (2019)

    2019

  16. [24]

    Gunnarsson, T

    O. Gunnarsson, T. Sch¨ afer, J. P. F. LeBlanc, J. Merino, G. Sangiovanni, G. Rohringer, and A. Toschi, Parquet decomposition calculations of the electronic self-energy, Phys. Rev. B93, 245102 (2016)

    2016

  17. [25]

    Janiˇ s and V

    V. Janiˇ s and V. Pokorn´ y, Critical metal-insulator tran- sition and divergence in a two-particle irreducible vertex in disordered and interacting electron systems, Phys. Rev. B90, 045143 (2014)

    2014

  18. [26]

    Ribic, G

    T. Ribic, G. Rohringer, and K. Held, Nonlocal corre- lations and spectral properties of the Falicov-Kimball model, Phys. Rev. B93, 195105 (2016)

    2016

  19. [27]

    Sch¨ afer, S

    T. Sch¨ afer, S. Ciuchi, M. Wallerberger, P. Thunstr¨ om, O. Gunnarsson, G. Sangiovanni, G. Rohringer, and A. Toschi, Nonperturbative landscape of the mott- hubbard transition: Multiple divergence lines around the critical endpoint, Phys. Rev. B94, 235108 (2016)

    2016

  20. [28]

    Vuˇ ciˇ cevi´ c, N

    J. Vuˇ ciˇ cevi´ c, N. Wentzell, M. Ferrero, and O. Par- collet, Practical consequences of the Luttinger-Ward functional multivaluedness for cluster DMFT methods, Phys. Rev. B97, 125141 (2018)

    2018

  21. [29]

    Gunnarsson, G

    O. Gunnarsson, G. Rohringer, T. Sch¨ afer, G. Sangio- vanni, and A. Toschi, Breakdown of Traditional Many- Body Theories for Correlated Electrons, Phys. Rev. Lett.119, 056402 (2017)

    2017

  22. [30]

    Springer, P

    D. Springer, P. Chalupa, S. Ciuchi, G. Sangiovanni, and A. Toschi, Interplay between local response and vertex divergences in many-fermion systems with on-site at- traction, Phys. Rev. B101, 155148 (2020)

    2020

  23. [31]

    Chalupa, P

    P. Chalupa, P. Gunacker, T. Sch¨ afer, K. Held, and A. Toschi, Divergences of the irreducible vertex func- tions in correlated metallic systems: Insights from the Anderson impurity model, Phys. Rev. B97, 245136 (2018)

    2018

  24. [32]

    Metzner and D

    W. Metzner and D. Vollhardt, Correlated Lattice Fermions ind=∞Dimensions, Phys. Rev. Lett.62, 324 (1989)

    1989

  25. [33]

    Chalupa, T

    P. Chalupa, T. Sch¨ afer, M. Reitner, D. Springer, S. An- dergassen, and A. Toschi, Fingerprints of the Local Mo- ment Formation and its Kondo Screening in the Gen- eralized Susceptibilities of Many-Electron Problems, Phys. Rev. Lett.126, 056403 (2021). 32

    2021

  26. [34]

    Krien and A

    F. Krien and A. Valli, Parquetlike equations for the Hedin three-leg vertex, Physical Review B100, 10.1103/physrevb.100.245147 (2019)

    2019 doi

  27. [35]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen- berg, Dynamical mean-field theory of strongly corre- lated fermion systems and the limit of infinite dimen- sions, Rev. Mod. Phys.68, 13 (1996)

    1996

  28. [36]

    Krien, A

    F. Krien, A. Valli, P. Chalupa, M. Capone, A. I. Licht- enstein, and A. Toschi, Boson-exchange parquet solver for dual fermions, Phys. Rev. B102, 195131 (2020)

    2020

  29. [37]

    Krien, A

    F. Krien, A. Kauch, and K. Held, Tiling with triangles: parquet andGW γmethods unified, Phys. Rev. Res.3, 013149 (2021)

    2021

  30. [38]

    smoking gun

    S. Adler, F. Krien, P. Chalupa-Gantner, G. Sangio- vanni, and A. Toschi, Non-perturbative intertwining be- tween spin and charge correlations: A “smoking gun” single-boson-exchange result, SciPost Phys.16, 054 (2024)

    2024

  31. [39]

    A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Dual fermion approach to nonlocal correlations in the Hubbard model, Phys. Rev. B77, 033101 (2008)

    2008

  32. [40]

    Kiese, N

    D. Kiese, N. Wentzell, I. Krivenko, O. Parcollet, K. Held, and F. Krien, Embedded multi-boson ex- change: A step beyond quantum cluster theories, Phys. Rev. Res.6, 043159 (2024)

    2024

  33. [41]

    Kozik, M

    E. Kozik, M. Ferrero, and A. Georges, Nonexistence of the Luttinger-Ward Functional and Misleading Con- vergence of Skeleton Diagrammatic Series for Hubbard- Like Models, Phys. Rev. Lett.114, 156402 (2015)

    2015

  34. [42]

    Krahl and C

    H. Krahl and C. Wetterich, Functional renormalization group for d-wave superconductivity, Phys. Lett. A367, 263–267 (2007)

    2007

  35. [43]

    Baier, E

    T. Baier, E. Bick, and C. Wetterich, Temperature de- pendence of antiferromagnetic order in the Hubbard model, Phys. Rev. B70, 125111 (2004)

    2004

  36. [44]

    Diehl, H

    S. Diehl, H. Gies, J. M. Pawlowski, and C. Wet- terich, Renormalization flow and universality for ultra- cold fermionic atoms, Phys. Rev. A76, 053627 (2007)

    2007

  37. [45]

    Diehl, H

    S. Diehl, H. Gies, J. M. Pawlowski, and C. Wetterich, Flow equations for the BCS-BEC crossover, Phys. Rev. A76, 021602 (2007)

    2007

  38. [46]

    Bartosch, H

    L. Bartosch, H. Freire, J. J. R. Cardenas, and P. Kopi- etz, A functional renormalization group approach to the Anderson impurity model, Journal of Physics: Con- densed Matter21, 305602 (2009)

    2009

  39. [47]

    Strack, R

    P. Strack, R. Gersch, and W. Metzner, Renormalization group flow for fermionic superfluids at zero temperature, Phys. Rev. B78, 014522 (2008)

    2008

  40. [48]

    Friederich, H

    S. Friederich, H. C. Krahl, and C. Wetterich, Functional renormalization for spontaneous symmetry breaking in the Hubbard model, Phys. Rev. B83, 155125 (2011)

    2011

  41. [49]

    Friederich, H

    S. Friederich, H. C. Krahl, and C. Wetterich, Four-point vertex in the Hubbard model and partial bosonization, Phys. Rev. B81, 235108 (2010)

    2010

  42. [50]

    T. Denz, M. Mitter, J. M. Pawlowski, C. Wetterich, and M. Yamada, Partial bosonization for the two- dimensional Hubbard model, Phys. Rev. B101, 155115 (2020)

    2020

  43. [51]

    Obert, C

    B. Obert, C. Husemann, and W. Metzner, Low-energy singularities in the ground state of fermionic superfluids, Phys. Rev. B88, 144508 (2013)

    2013

  44. [52]

    Husemann and M

    C. Husemann and M. Salmhofer, Efficient parametriza- tion of the vertex function, Ω scheme, and thet, t ′ Hub- bard model at van Hove filling, Phys. Rev. B79, 195125 (2009)

    2009

  45. [53]

    Karrasch, R

    C. Karrasch, R. Hedden, R. Peters, T. Pruschke, K. Sch¨ onhammer, and V. Meden, A finite-frequency functional renormalization group approach to the sin- gle impurity Anderson model, J. Phys. Condens. Matter 20, 345205 (2008)

    2008

  46. [54]

    Vilardi, C

    D. Vilardi, C. Taranto, and W. Metzner, Nonsepara- ble frequency dependence of the two-particle vertex in interacting fermion systems, Phys. Rev. B96, 235110 (2017)

    2017

  47. [55]

    Wang, Y.-Y

    W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, F. Wang, F. Yang, and D.-H. Lee, Functional renormalization group and variational Monte Carlo studies of the elec- tronic instabilities in graphene near 1 4 doping, Phys. Rev. B85, 035414 (2012)

    2012

  48. [56]

    Vilardi, C

    D. Vilardi, C. Taranto, and W. Metzner, Antiferromag- netic andd-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group, Phys. Rev. B99, 104501 (2019)

    2019

  49. [57]

    Tagliavini, C

    A. Tagliavini, C. Hille, F. B. Kugler, S. Andergassen, A. Toschi, and C. Honerkamp, Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions, Sci- Post Phys.6, 009 (2019)

    2019

  50. [58]

    Hille, F

    C. Hille, F. B. Kugler, C. J. Eckhardt, Y.-Y. He, A. Kauch, C. Honerkamp, A. Toschi, and S. Ander- gassen, Quantitative functional renormalization group description of the two-dimensional Hubbard model, Phys. Rev. Res.2, 033372 (2020)

    2020

  51. [59]

    E. A. Stepanov, V. Harkov, and A. I. Lichtenstein, Con- sistent partial bosonization of the extended Hubbard model, Phys. Rev. B100, 205115 (2019)

    2019

  52. [60]

    P. M. Bonetti, A. Toschi, C. Hille, S. Andergassen, and D. Vilardi, Single-boson exchange representation of the functional renormalization group for strongly inter- acting many-electron systems, Phys. Rev. Research4, 013034 (2022)

    2022

  53. [61]

    Harkov, A

    V. Harkov, A. I. Lichtenstein, and F. Krien, Parametrizations of local vertex corrections from weak to strong coupling: Importance of the Hedin three-leg vertex, Phys. Rev. B104, 125141 (2021)

    2021

  54. [62]

    [138] for a recent overview of computational results for the 2D Hubbard model

    See Ref. [138] for a recent overview of computational results for the 2D Hubbard model

  55. [63]

    Fraboulet, S

    K. Fraboulet, S. Heinzelmann, P. M. Bonetti, A. Al- Eryani, D. Vilardi, A. Toschi, and S. Andergassen, Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling, Eur. Phys. J. B95, 202 (2022)

    2022

  56. [64]

    Gievers, E

    M. Gievers, E. Walter, A. Ge, J. von Delft, and F. B. Kugler, Multiloop flow equations for single-boson ex- change fRG, Eur. Phys. J. B95, 108 (2022)

    2022

  57. [65]

    Wentzell, G

    N. Wentzell, G. Li, A. Tagliavini, C. Taranto, G. Rohringer, K. Held, A. Toschi, and S. Andergassen, High-frequency asymptotics of the vertex function: Dia- grammatic parametrization and algorithmic implemen- tation, Physical Review B102(2020)

    2020

  58. [66]

    N. E. Bickers and S. R. White, Conserving approxima- tions for strongly fluctuating electron systems. II. Nu- merical results and parquet extension, Phys. Rev. B43, 8044 (1991)

    1991

  59. [67]

    Rohringer, H

    G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, Diagrammatic routes to 33 nonlocal correlations beyond dynamical mean field the- ory, Rev. Mod. Phys.90, 025003 (2018)

    2018

  60. [68]

    Gievers,Functional approaches to Fermi polarons in cold atomic gases and solid-state systems, Ph.D

    M. Gievers,Functional approaches to Fermi polarons in cold atomic gases and solid-state systems, Ph.D. thesis, Ludwig-Maximilians-Universit¨ at M¨ unchen (2025)

    2025

  61. [69]

    F. B. Kugler,Renormalization group approaches to strongly correlated electron systems, Ph.D. thesis, Ludwig-Maximilians-Universit¨ at M¨ unchen (2019)

    2019

  62. [70]

    Schmalian, D

    J. Schmalian, D. Pines, and B. Stojkovi´ c, Microscopic theory of weak pseudogap behavior in the underdoped cuprate superconductors: General theory and quasipar- ticle properties, Phys. Rev. B60, 667 (1999)

    1999

  63. [71]

    Sadovskii, Exact numerical calculation of the den- sity of states of the fluctuating gap model, Journal of Experimental and Theoretical Physics50, 989 (1979)

    M. Sadovskii, Exact numerical calculation of the den- sity of states of the fluctuating gap model, Journal of Experimental and Theoretical Physics50, 989 (1979)

    1979

  64. [72]

    implicitly

    By “implicitly”, we mean here that no bosons were in- troduced at the level of the path-integral by means of a Hubbard-Stratonovich transformation, i.e., we are in principle still dealing with a purely fermionic theory

  65. [73]

    Hedin, New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem, Phys

    L. Hedin, New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem, Phys. Rev.139, A796 (1965)

    1965

  66. [74]

    Al-Eryani, S

    A. Al-Eryani, S. Heinzelmann, K. Fraboulet, F. Krien, and S. Andergassen, Screening and effective RPA-like charge susceptibility in the extended Hubbard model (2024), arXiv:2412.07323

    2024 arXiv

  67. [75]

    Fukushima, J

    K. Fukushima, J. M. Pawlowski, and N. Strodthoff, Emergent hadrons and diquarks, Annals of Physics446, 169106 (2022)

    2022

  68. [76]

    Zhang and J

    Y. Zhang and J. Callaway, Extended Hubbard model in two dimensions, Phys. Rev. B39, 9397 (1989)

    1989

  69. [77]

    Al-Eryani, S

    A. Al-Eryani, S. Andergassen, and M. M. Scherer, Inter- twined fluctuations and isotope effects in the Hubbard- Holstein model on the square lattice from functional renormalization, Phys. Rev. Res.7, 043052 (2025)

    2025

  70. [78]

    Terletska, T

    H. Terletska, T. Chen, J. Paki, and E. Gull, Charge or- dering and nonlocal correlations in the doped extended Hubbard model, Phys. Rev. B97, 115117 (2018)

    2018

  71. [79]

    Terletska, T

    H. Terletska, T. Chen, and E. Gull, Charge ordering and correlation effects in the extended Hubbard model, Phys. Rev. B95, 115149 (2017)

    2017

  72. [80]

    Holstein, Studies of polaron motion: Part I

    T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Annals of Physics8, 325 (1959)

    1959

  73. [81]

    J. Paki, H. Terletska, S. Iskakov, and E. Gull, Charge order and antiferromagnetism in the extended Hubbard model, Phys. Rev. B99, 245146 (2019)

    2019

  74. [82]

    Capone, G

    M. Capone, G. Sangiovanni, C. Castellani, C. Di Castro, and M. Grilli, Phase Separation Close to the Density- Driven Mott Transition in the Hubbard-Holstein Model, Phys. Rev. Lett.92, 106401 (2004)

    2004

  75. [83]

    J. K. Freericks and M. Jarrell, Competition between Electron-Phonon Attraction and Weak Coulomb Repul- sion, Phys. Rev. Lett.75, 2570 (1995)

    1995

  76. [84]

    F. B. Kugler and J. von Delft, Multiloop functional renormalization group for general models, Phys. Rev. B97, 035162 (2018)

    2018

  77. [85]

    As an example, we can mention that the bosonic prop- agatorsw (B) r must be set toB r, instead ofUforw r, for the initial conditions of conventional implementations of the fRG flow [52]

  78. [86]

    Blaizot, J

    J.-P. Blaizot, J. M. Pawlowski, and U. Reinosa, Exact renormalization group and Φ-derivable approximations, Physics Letters B696, 523 (2011)

    2011

  79. [87]

    F. B. Kugler and J. von Delft, Multiloop Functional Renormalization Group That Sums Up All Parquet Di- agrams, Phys. Rev. Lett.120, 057403 (2018)

    2018

  80. [88]

    F. B. Kugler and J. von Delft, Derivation of exact flow equations from the self-consistent parquet relations, New Journal of Physics20, 123029 (2018)

    2018

  81. [89]

    Blaizot, J

    J.-P. Blaizot, J. M. Pawlowski, and U. Reinosa, Func- tional renormalization group and 2PI effective action formalism, Annals of Physics431, 168549 (2021)

    2021

  82. [90]

    G. Li, N. Wentzell, P. Pudleiner, P. Thunstr¨ om, and K. Held, Efficient implementation of the parquet equa- tions: Role of the reducible vertex function and its ker- nel approximation, Phys. Rev. B93, 165103 (2016)

    2016

  83. [91]

    Chen and N

    C.-X. Chen and N. Bickers, Numerical solution of par- quet equations for the Anderson impurity model, Solid State Communications82, 311 (1992)

    1992

  84. [92]

    Pudleiner, A

    P. Pudleiner, A. Kauch, K. Held, and G. Li, Competi- tion between antiferromagnetic and charge density wave fluctuations in the extended Hubbard model, Phys. Rev. B100, 075108 (2019)

    2019

  85. [93]

    Janiˇ s, A

    V. Janiˇ s, A. Kauch, and V. Pokorn´ y, Thermodynam- ically consistent description of criticality in models of correlated electrons, Phys. Rev. B95, 045108 (2017)

    2017

  86. [94]

    Thoenniss, M

    J. Thoenniss, M. K. Ritter, F. B. Kugler, J. von Delft, and M. Punk, Multiloop pseudofermion func- tional renormalization for quantum spin systems: Ap- plication to the spin-1 2 kagome Heisenberg model (2020), arXiv:2011.01268

    2020 arXiv

  87. [95]

    Kauch, P

    A. Kauch, P. Pudleiner, K. Astleithner, P. Thunstr¨ om, T. Ribic, and K. Held, Generic Optical Excitations of Correlated Systems:π-tons, Phys. Rev. Lett.124, 047401 (2020)

    2020

  88. [96]

    M. K. Ritter, D. Kiese, T. M¨ uller, F. B. Kugler, R. Thomale, S. Trebst, and J. von Delft, Benchmark cal- culations of multiloop pseudofermion fRG, Eur. Phys. J. B95, 102 (2022)

    2022

  89. [97]

    Kiese, T

    D. Kiese, T. M¨ uller, Y. Iqbal, R. Thomale, and S. Trebst, Multiloop functional renormalization group approach to quantum spin systems, Phys. Rev. Res.4, 023185 (2022)

    2022

  90. [98]

    A. Ge, N. Ritz, E. Walter, S. Aguirre, J. von Delft, and F. B. Kugler, Real-frequency quantum field theory applied to the single-impurity Anderson model, Phys. Rev. B109, 115128 (2024)

    2024

  91. [99]

    Chalupa-Gantner, F

    P. Chalupa-Gantner, F. B. Kugler, C. Hille, J. von Delft, S. Andergassen, and A. Toschi, Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impu- rity model, Phys. Rev. Res.4, 023050 (2022)

    2022

  92. [100]

    Hille, D

    C. Hille, D. Rohe, C. Honerkamp, and S. Andergassen, Pseudogap opening in the two-dimensional Hubbard model: A functional renormalization group analysis, 34 Phys. Rev. Res.2, 033068 (2020)

    2020

  93. [101]

    N. Ritz, A. Ge, E. Walter, S. Aguirre, J. von Delft, and F. B. Kugler, KeldyshQFT: A C++ codebase for real- frequency multiloop functional renormalization group and parquet computations of the single-impurity An- derson model, The Journal of Chemical Physics161, 054118 (2024)

    2024

  94. [102]

    Hille,The role of the self-energy in the functional renormalization group description of interacting Fermi systems, Ph.D

    C. Hille,The role of the self-energy in the functional renormalization group description of interacting Fermi systems, Ph.D. thesis, Universit¨ at T¨ ubingen (2020)

    2020

  95. [103]

    Heinzelmann, A

    S. Heinzelmann, A. Toschi, and S. Andergassen, En- tangled magnetic, charge, and superconducting pair- ing correlations in the two-dimensional Hubbard model: a functional renormalization-group analysis (2023), arXiv:2308.06497

    2023 arXiv

  96. [104]

    Taranto, S

    C. Taranto, S. Andergassen, J. Bauer, K. Held, A. Katanin, W. Metzner, G. Rohringer, and A. Toschi, From Infinite to Two Dimensions through the Func- tional Renormalization Group, Phys. Rev. Lett.112, 196402 (2014)

    2014

  97. [105]

    More details on the construction of the identity vertices 1r and1 r can be found in Appendix C.2 of Ref. [64]

  98. [106]

    (52b) forA ph yieldsA ↑↑ ph =A ↑↓ ph +A c↑↓ ph, while the crossing symmetry ofA ph imposes thatA c↑↓ ph = −A↑↓ ph

    Formulating Eq. (52b) forA ph yieldsA ↑↑ ph =A ↑↓ ph +A c↑↓ ph, while the crossing symmetry ofA ph imposes thatA c↑↓ ph = −A↑↓ ph. By combining these two equalities, we obtain the relationA ↑↑ ph −A ↑↓ ph =−A ↑↓ ph used in Eq. (56a)

  99. [107]

    Salmhofer and C

    M. Salmhofer and C. Honerkamp, Fermionic Renormal- ization Group Flows: Technique and Theory, Progress of Theoretical Physics105, 1 (2001)

    2001

  100. [108]

    (66) is notably dis- cussed in detail in Appendix A of Ref

    The procedure to derive Eqs. (66) is notably dis- cussed in detail in Appendix A of Ref. [109]. Regard- ing those equations, it can also be noted that our fre- quency and momentum conventions in theppchannel defined in Fig. 1 can be modified in such a way that the right-hand s...

  101. [109]

    (4.31a) in Ref

    The equalityλ c↑↓ ph = +λ↑↓ ph holds (see, e.g., Eq. (4.31a) in Ref. [64]), which is to be compared withA c↑↓ ph =−A ↑↓ ph for Eq. (56a). The definitionλ M = +λ ↑↓ ph of Eq. (59a) directly follows from the relationsλ c↑↓ ph = +λ ↑↓ ph and λ↑↑ ph =λ ↑↓ ph +λ c↑↓ ph

  102. [110]

    Lichtenstein, D

    J. Lichtenstein, D. S´ anchez de la Pe˜ na, D. Rohe, E. Di Napoli, C. Honerkamp, and S. Maier, High-performance functional Renormalization Group calculations for in- teracting fermions, Computer Physics Communications 213, 100 (2017)

    2017

  103. [111]

    Patricolo, M

    M. Patricolo, M. Gievers, K. Fraboulet, A. Al-Eryani, S. Heinzelmann, P. M. Bonetti, A. Toschi, D. Vilardi, and S. Andergassen, Single-boson exchange formulation of the Schwinger-Dyson equation and its application to the functional renormalization group, SciPost Phys.18, 078 (2025)

    2025

  104. [112]

    J. B. Profe and D. M. Kennes, TU 2FRG: a scalable approach for truncated unity functional renormalization group in generic fermionic models, Eur. Phys. J. B95, 60 (2022)

    2022

  105. [113]

    G. A. H. Schober, J. Ehrlich, T. Reckling, and C. Hon- erkamp, Truncated-Unity Functional Renormalization Group for Multiband Systems With Spin-Orbit Cou- pling, Front. Phys.6, 32 (2018)

    2018

  106. [114]

    Beyer, J

    J. Beyer, J. B. Profe, L. Klebl, T. Schwemmer, D. M. Kennes, R. Thomale, C. Honerkamp, and S. Rachel, Rashba spin-orbit coupling in the square-lattice Hub- bard model: A truncated-unity functional renormaliza- tion group study, Phys. Rev. B107, 125115 (2023)

    2023

  107. [115]

    Gneist, L

    N. Gneist, L. Classen, and M. M. Scherer, Competing instabilities of the extended Hubbard model on the tri- angular lattice: Truncated-unity functional renormal- ization group and application to moir´ e materials, Phys. Rev. B106, 125141 (2022)

    2022

  108. [116]

    While the flow equations forw X(Q),λ X(Q, k), and MX(Q, k, k′) are also valid for non-local interactions, i.e., for a non-zero fermionic bare interactionF r in Eq. (12), as long as the SBE parametrization ∇r(Q, k, k′) =λ r(Q, k)wr(Q)λr(Q, k′) applies to the B-reducible part of...

  109. [117]

    J. B. Profe, D. M. Kennes, and L. Klebl, diverge im- plements various exact renormalization group examples, SciPost Physics Codebases26(2024)

    2024

  110. [118]

    Honerkamp, D

    C. Honerkamp, D. Rohe, S. Andergassen, and T. Enss, Interaction flow method for many-fermion systems, Phys. Rev. B70, 235115 (2004)

    2004

  111. [119]

    What we are describing here amounts to neglecting the SBE rest functionafterderiving the flow equations, and is defined as SBEa approximation in Ref. [61]. It has to be distinguished from the SBEb approximation ob- tained by neglecting the rest functionbeforederiving the flow ...

  112. [120]

    For the Hubbard model with a local interaction only, this occurs exclusively at finite doping

  113. [121]

    Thus, it is possible to de- termineδµ Λ by a root finding algorithm such that the filling remains fixed at each step Λ of the flow

    and therefore invertible. Thus, it is possible to de- termineδµ Λ by a root finding algorithm such that the filling remains fixed at each step Λ of the flow. The total scale derivative ofG Λ becomes ˙GΛ(k, ν) = dGΛ dΛ δµΛ=const + dGΛ dδµΛ dδµΛ dΛ (109) in the multiloop flow eq...

  114. [122]

    Honerkamp and M

    C. Honerkamp and M. Salmhofer, Temperature-flow renormalization group and the competition between su- perconductivity and ferromagnetism, Phys. Rev. B64, 184516 (2001)

    2001

  115. [123]

    Salmhofer,Renormalization, Theoretical and Math- ematical Physics (Springer Berlin Heidelberg, 1999)

    M. Salmhofer,Renormalization, Theoretical and Math- ematical Physics (Springer Berlin Heidelberg, 1999)

    1999

  116. [124]

    A. A. Katanin, Fulfillment of Ward identities in the functional renormalization group approach, Phys. Rev. B70, 115109 (2004)

    2004

  117. [125]

    D. F. Litim, Optimisation of the exact renormalisation group, Physics Letters B486, 92 (2000)

    2000

  118. [126]

    D. F. Litim, Optimized renormalization group flows, Phys. Rev. D64, 105007 (2001)

    2001

  119. [127]

    D. F. Litim, Derivative expansion and renormalisation group flows, Journal of High Energy Physics2001, 059 (2001)

    2001

  120. [128]

    D. F. Litim, Critical exponents from optimised renor- malisation group flows, Nuclear Physics B631, 128 35 (2002)

    2002

  121. [129]

    J. M. Pawlowski, Aspects of the functional renormali- sation group, Annals of Physics322, 2831 (2007)

    2007

  122. [130]

    Zorbach, J

    N. Zorbach, J. Stoll, and J. Braun, Optimization and Stabilization of Functional Renormalization Group Flows (2024), arXiv:2401.12854

    2024 arXiv

  123. [131]

    H. Eßl, M. Reitner, G. Sangiovanni, and A. Toschi, Gen- eral Shiba mapping for on-site four-point correlation functions, Phys. Rev. Res.6, 033061 (2024)

    2024

  124. [132]

    N. D. Mermin and H. Wagner, Absence of Ferro- magnetism or Antiferromagnetism in One- or Two- Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett.17, 1133 (1966)

    1966

  125. [133]

    N. E. Bickers and D. J. Scalapino, Critical behavior of electronic parquet solutions, Phys. Rev. B46, 8050 (1992)

    1992

  126. [134]

    Y. M. Vilk and A.-M. Tremblay, Non-Perturbative Many-Body Approach to the Hubbard Model and Single-Particle Pseudogap, J. Phys.7, 1309 (1997)

    1997

  127. [135]

    Al-Eryani, Diagrammatic bosonisation, aspects of criticality, and the Hohenberg-Mermin-Wagner Theo- rem in parquet approaches (2025), arXiv:2509.15094

    A. Al-Eryani, Diagrammatic bosonisation, aspects of criticality, and the Hohenberg-Mermin-Wagner Theo- rem in parquet approaches (2025), arXiv:2509.15094

    2025

  128. [136]

    Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Progress of Theoretical Physics 30, 275 (1963)

    J. Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Progress of Theoretical Physics 30, 275 (1963)

    1963

  129. [137]

    A. A. Katanin, A. Toschi, and K. Held, Comparing per- tinent effects of antiferromagnetic fluctuations in the two- and three-dimensional Hubbard model, Phys. Rev. B80, 075104 (2009)

    2009

  130. [138]

    The (1ℓ) fRG scheme reproduces the RPA if only the flow of the bosonic propagatorsw X is considered, with λX(Q,Ω, ν) = 1 for all X and neglecting the self-energy flow

  131. [139]

    Heinzelmann,The single-boson exchange formalism and its application to the functional renormalization group, Ph.D

    S. Heinzelmann,The single-boson exchange formalism and its application to the functional renormalization group, Ph.D. thesis, Universit¨ at T¨ ubingen (2023)

    2023

  132. [140]

    M. Qin, T. Sch¨ afer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard Model: A Computational Per- spective, Annual Review of Condensed Matter Physics 13, 275 (2022)

    2022

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.