# Functional renormalization group for non-Hermitian and $\mathcal{PT}$-symmetric systems

### Submission summary

 As Contributors: Lukas Grunwald Arxiv Link: https://arxiv.org/abs/2203.08108v2 (pdf) Date accepted: 2022-05-19 Date submitted: 2022-05-05 09:48 Submitted by: Grunwald, Lukas Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We generalize the vertex expansion approach of the functional renormalization group to non-Hermitian systems. As certain anomalous expectation values might not vanish, additional terms as compared to the Hermitian case can appear in the flow equations. We investigate the merits and shortcomings of the vertex expansion for non-Hermitian systems by considering an exactly solvable $\mathcal{PT}$-symmetric non-linear toy-model and reveal, that in this model, the fidelity of the vertex expansion in a perturbatively motivated truncation schema is comparable with that of the Hermitian case. The vertex expansion appears to be a viable method for studying correlation effects in non-Hermitian systems.

Published as SciPost Phys. 12, 179 (2022)

### Author comments upon resubmission

See https://drive.google.com/file/d/16wJ2nPQbA4LBp26msYZBIsOZ2BUf3gcr/view?usp=sharing, for an updated version of the manuscript, where changes and additions (excluding corrected typos) are marked in blue.

### List of changes

Comments from referee 1:
- Added clarification, that we assume non degenerate eigenvalues only for notational convenience

Comments from referee 2:
- Added footnote clarifying the connection between our approach and the one used by Schütz and Kopietz ( J. Phys. A: Math. Gen. 39 8205)
- Added remark outlining our reasoning for the cutoff choice as well as a reference to the "principle of minimal sensitivity"
- Removed second definition of the Hamiltonian

Comments from referee 3:
- Added footnote explaining sign convention in $(j,\phi)$
- Added a footnote pointing out that the $^\star$ in $\phi^\star$ is part of the symbol and not a complex conjugation
- Changed notation in definition of Greens and Vertex functions for better clarity
- Added footnote explaining why using a numerical analytic continuation will not allow an accurate approximation of the full spectrum
- Added a comment in the caption of Fig.4 for a more precise definition of $\Delta Q$

Additionally we corrected typos in the new version of the manuscript.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 179 (2022)

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Resubmission 2203.08108v2 on 5 May 2022
Submission 2203.08108v1 on 17 March 2022

## Reports on this Submission

### Anonymous Report 3 on 2022-5-13 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2203.08108v2, delivered 2022-05-13, doi: 10.21468/SciPost.Report.5073

### Report

I would like to thank the authors for replying to my report. I do not understand the argument as to why it is not possible to obtain the full spectrum using Padé approximants to perform the analytic continuation. For a zero-space dimensional problem this is quite simple to do and this would really have improved the paper. On the other hand I do not want to further delay the publication and I therefore recommend publication of the manuscript in its present form.

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### Author:  Lukas Grunwald  on 2022-05-19  [id 2497]

(in reply to Report 3 on 2022-05-13)

Dear Referee,
thank you very much for your comment.
Pade-extrapolations are very sensitive to the details of the $\omega$-grid as well as numerical/statistical noise, leading to inaccuracies in the final result (we previously tested the Pade-extrapolations for the anharmonic oscillator in the context of Ref. [26]). Since higher order excitations only have a very small weight in the Greens function (previous argument), a reliable extraction of excitation energies will not be possible with this approach, due to the inaccuracies of the extrapolation.

We hope this further clarifies our argument given in the paper.

Kind regards,
Lukas Grunwald, Volker Meden, Dante Kennes

### Report

The authors have responded convincingly to my comments.
Hence, I now recommend publication of the manuscript in its present form.

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### Report

I have read again the manuscript and the reply of the authors to my comments. I think that the authors have properly answered all of my questions and have modified their manuscript accordingly. I now recommend publication of the
manuscript in the present form in SciPost.

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