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Functional flows for complex effective actions

by Friederike Ihssen, Jan M. Pawlowski

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Submission summary

Authors (as registered SciPost users): Friederike Ihssen · Jan M. Pawlowski
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-08-15 11:01
Submitted by: Ihssen, Friederike
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
  • Statistical and Soft Matter Physics
Approach: Theoretical


In the present work we set up a general functional renormalisation group framework for the computation of complex effective actions. For explicit computations we consider both flows of the Wilsonian effective action and the one-particle irreducible (1PI) effective action. The latter is based on an appropriate definition of a Legendre transform for complex actions, and we show its validity by comparison to exact results in zero dimensions, as well as a comparison to results for the Wilsonian effective action. In the present implementations of the general approaches, the flow of the Wilsonian effective action has a wider range of applicability and we obtain results for the effective potential of complex fields in $\phi^4$-theories from zero up to four dimensions. These results are also compared with results from the 1PI effective action within its range of applicability. The complex effective action also allows us to determine the location of the Lee-Yang zeros for general parameter values. We also discuss the extension of the present results to general theories including QCD.

Current status:
Has been resubmitted

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2023-1-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2207.10057v2, delivered 2023-01-26, doi: 10.21468/SciPost.Report.6614


see report


see report


This manuscript discusss the application of functional renormalization group approaches to complex effective actions. The manuscript is well written and the calculations are carefully laid out. The validity of the manuscript is high, although it is rather technical and really only of interest to specialists in this field. That being said it will be a valuable resource.

There are two points that should be commented on in some more detail.

1) The authors only consider complex actions due to complex external fields. They mention that the action itself could be complex (e.g. because of complex coupling constants), but do not elaborate on this. Different from the case of external fields the complex parameters would now couple to terms that are non-linear in the fields. Does the apporach still work?

2) Open quantum systems are often described using the Lindblad appraoch, which amounts to non-hermitean “Hamiltonians”. Can the authors comment on the applicability to this situation?

Some remarks along these lines would be helpful for readers who want to build on this work and possibly apply it to such models.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2022-11-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2207.10057v2, delivered 2022-11-28, doi: 10.21468/SciPost.Report.6214


See report.


See report.


The authors show how the functional renormalization group approach (FRG) can be extended to complex effective actions. More precisely, they consider a $\phi^4$ theory with a complex external source that couples linearly to the field. Both the flow of the Wilsonian effective action and the 1PI effective action are discussed. There is a nice presentation of the derivation of the Wegner's flow, the Wilson-Polchinski flow and the flow of the 1PI effective action. The results are benchmarked by considering the zero-dimensional and four-dimensional theories and the location of the Lee-Yang zeros is also discussed. Although somewhat technical (many technical details are discussed in the appendices), the paper is clearly written and provides us with an interesting application of the FRG approach.

One of the main results of the paper is that the most efficient approach is the RG-adapted flow for the Wilsonian effective action whereas the 1PI flow is not very efficient. Since the latter is often the method of choice in practice to deal with many physical models, the authors' conclusion is somewhat disappointing. It would be interesting to discuss this issue further, for example in the conclusion. Figure 1 shows that the 1PI flow does not converge anymore way before the pole is approached, i.e. $J_y\lesssim 1$, while the RG-adapted flow converges up to $J_y\simeq 2.4$. Is there a simple way to understand why the 1PI flow does so badly?

The authors consider the case where the external source is complex but the action remains real. They point out that the bare action could also be complex (the mass or some coupling constants could be complex). In that case should we expect some complications in the FRG approach or is there no essential differences between a complex external source and a complex action?

Another (related) issue is whether the authors' results could be of interest for the study of out-of-equilibrium systems where the Hamiltonian may be non-Hermitian and the bare Wilsonian action as well as the effective action complex. In this respect, it would be interesting to discuss the possible connection with a recent paper of Grunwald {\it et al.} [SciPost Phys. 12, 179 (2022)] where a harmonic oscillator perturbed by a cubic term with imaginary coupling is studied within the FRG approach.

Requested changes

See remarks in the report.

  • validity: high
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: excellent

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