SciPost logo

SciPost Submission Page

A functional-analysis derivation of the parquet equation

by Christian J. Eckhardt, Patrick Kappl, Anna Kauch, Karsten Held

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Christian J. Eckhardt
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-11-08
Date submitted: 2023-09-18 09:39
Submitted by: Eckhardt, Christian J.
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger--Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green's functions potentially leading to a classification of these in terms of their (ir)reducible components.

List of changes

- Removed V_3 from the original action and all diagrams subsequently.
- Added explaining remark for prefactor in Tilde{G}
- Used symmetry of vertices originating from the symmetry of the 1-particle Green's function under exchange of its arguments. This leads to a simplification of the parquet equation and the Bethe-Salpeter equation and their corresponding diagrams. Extra prefactors now appear.
- Added an extra appendix for the calculation of the 2-particle Green's function by different means. This in particular illustrates the origin of the prefactors in the parquet equation and the Bethe-Salpeter equation as well as giving concrete hands on examples on how to perform calculations.
- The other appendices have also been updated to employ the symmetry of vertices more effectively.

Published as SciPost Phys. 15, 203 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-10-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2305.16050v2, delivered 2023-10-29, doi: 10.21468/SciPost.Report.8015


The revised version of the manuscript has improved in terms of presentation. I will leave a few more suggestions for improvement. After the authors have considered these suggestions, the paper can be published from my point of view.

- I repeat my suggestion to give at least one precise (with equation number) reference to a published version of the BSEs of real phi^4 theory.
- Fig. 18 contains a symbol for V_n, n \geq 3. This can be simplified since only V_4 appears in the new version.
- The meaning of the diagrams is not entirely clear due to the minimalistic character of App. D and the choice of the authors to use "somewhat ambiguous" notation in the diagrams. Why not draw amputated legs shorter than attached legs? Why not use different colors / linestyles / etc. for bare vs. full propagators? To exemplify my confusion: (i) I suppose the first term on the RHS of Fig. 1 is 1/2 V_1^a G_0^{ab} V_1^b. What is the first term on the RHS of Fig. 2? 1/2 G_1^a G_2^{ab} G_1^b can't be correct? (ii) By inserting lowest-order vertices in Fig. 6, one should obtain Fig. 12. How do the prefactors 2 become prefactors 1/2? It seems that redefining a new channel-dependent 2PI vertex equal to 1/4 of the previous channel-dependent 2PI vertex might be helpful (affecting, e.g., the last three terms on the RHS of Fig. 6 and the two terms on the RHS of Fig. 7)?
- The authors occasionally write that, e.g., G_3=0 due to V_3=0. But this also requires V_1=0, doesn't it?

- paragraph below Eq. (33): appendix Appendix -> Appendix
- Eq. (47) LHS: G_2^{ef} G_2^{ef} -> G_2^{ef} G_2^{gh}
- Eq. (47) RHS, 2nd line: G_4^{abcd} -> V_4^{abcd}
- paragraph below Fig. 15: "as one that this" -> "as one that is"

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-9-25 (Invited Report)


The revised version can be published as it is.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report or comment