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A functionalanalysis derivation of the parquet equation
by Christian J. Eckhardt, Patrick Kappl, Anna Kauch, Karsten Held
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Submission summary
Authors (as registered SciPost users):  Christian J. Eckhardt 
Submission information  

Preprint Link:  https://arxiv.org/abs/2305.16050v2 (pdf) 
Date accepted:  20231108 
Date submitted:  20230918 09:39 
Submitted by:  Eckhardt, Christian J. 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The parquet equation is an exact fieldtheoretic 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 twoparticle 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 LuttingerWard functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higherorder 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 1particle Green's function under exchange of its arguments. This leads to a simplification of the parquet equation and the BetheSalpeter equation and their corresponding diagrams. Extra prefactors now appear.
 Added an extra appendix for the calculation of the 2particle Green's function by different means. This in particular illustrates the origin of the prefactors in the parquet equation and the BetheSalpeter 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 20231029 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2305.16050v2, delivered 20231029, doi: 10.21468/SciPost.Report.8015
Report
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 lowestorder vertices in Fig. 6, one should obtain Fig. 12. How do the prefactors 2 become prefactors 1/2? It seems that redefining a new channeldependent 2PI vertex equal to 1/4 of the previous channeldependent 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?
Typos:
 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"