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Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization

by Andrzej Chlebicki, Pawel Jakubczyk

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Submission summary

Authors (as Contributors): Andrzej Chlebicki · Pawel Jakubczyk
Submission information
Arxiv Link: (pdf)
Date submitted: 2021-04-01 12:53
Submitted by: Chlebicki, Andrzej
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


We employ the functional renormalization group framework at the second order in the derivative expansion to study the $O(N)$ models continuously varying the number of field components $N$ and the spatial dimensionality $d$. We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents $\nu$ and $\eta$ across a line in the $(d,N)$ plane, which passes through the point $(2,2)$. By direct numerical evaluation of $\eta(d,N)$ and $\nu^{-1}(d,N)$ as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the $(d,N)$ plane, however no evidence of discontinuous or singular first and second derivatives of these functions for $d>2$. The computed derivatives of $\eta(d,N)$ and $\nu^{-1}(d,N)$ become increasingly large for $d\to 2$ and $N\to 2$ and it is only in this limit that $\eta(d,N)$ and $\nu^{-1}(d,N)$ as obtained by us are evidently nonanalytical. By scanning the dependence of the subleading eigenvalue of the RG transformation on $N$ for $d>2$ we find no indication of its vanishing as anticipated by the Cardy-Hamber scenario. For dimensionality $d$ approaching 3 there are no signatures of the Cardy-Hamber line even as a crossover and its existence in the form of a nonanalyticity is excluded.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor,

thank you very much for making the reports available to us. We amended
the paper following the Referees' suggestions. Below we summarize the
introduced extensions and alternations and give our response to the
Referees' reports.

Sincerely yours,
Andrzej Chlebicki
Pawel Jakubczyk

List of changes

We significantly strengthened our conclusion, which is reflected in the
new Abstract as Summary, as well as changes throughout the text.
- We modified and extended the introduction and summary, restructured
Sec. 2, introduced minor changes to Sec. 3, added the last paragraph of
Sec. 4, slightly modified the end of Sec. 5.1.
- We added the new Sec. 5.3 and Sec. 6.

Reports on this Submission

Anonymous Report 3 on 2021-4-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.00782v4, delivered 2021-04-30, doi: 10.21468/SciPost.Report.2859


1) Precise numerical study of second order $O(\partial^2)$ derivative expansion of the Wetterich equation for $O(N)$-models in the $(d,N)$ plane.

2) Accurate analysis (within the approximations of point 1),of the critical exponents $\nu(d,N)$ and $\eta(d,N)$ for continuous values of $d$ and $N$.

3) Study of the sub-leading RG exponent in relation with the Cardy-Hamber line.


1) The numerical results (i.e. the core of the study) are presented in plots which can be improved in quality and in ease of reading.


The paper deals with the study of $O(N)$ models using the Non-Perturbative Renormalization Group approach based on the Wetterich equation.

The novelty of the research lies in the completeness and quality of the numerical integration of the partial differential equations that encode the RG flow at the $O(\partial^2)$ order of the derivative expansion for continuous values of the parameters $(d,N)$, and in particular around the dimensions of physical relevance $d=2,3$.

The authors investigate the Cardy-Hamber prediction in quite a detail and and claim that their results demonstrate the nonexistence of the Cardy-Hamber line in the vicinity of three dimensions in disagreement with the original work on the topic. This is clearly an important result if fully confirmed.

The work is interesting and novel and definitely deserves publication in this journal, but I first I will ask the authors if they can produce plots of better quality (more clear, more readable, better scaled and with a color-code that will function also in print). Since all the outputs of this type of study are numerical and are displayed graphically, improving this point will make the manuscript much more understandable to readers and of greater general value.

A couple of more specific questions are:
1) why is the comparison between the exact and the computed $\eta$ in Figure 5 off by an almost a constant factor? Furthermore I suggest to the authors to graph the whole range $-2\leq N \leq 2$ since the point $(0,-2)$ is was solved by Fisher long ago.
2) Can the authors display the equations they are actually solving or they are really so complicated?

Requested changes

1) Improve of the plots as discussed in the report.

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

Author:  Andrzej Chlebicki  on 2021-05-11  [id 1417]

(in reply to Report 3 on 2021-04-30)
answer to question

Dear Referee,

we would like to thank you for reviewing our paper. We are very glad about your positive recommendation.

We introduced modifications to the plots. We hope it somewhat improved their readability. Concerning the specific points from the report: - We, unfortunately, do not have an explanation why the two curves plotted in Fig.5 are shifted by roughly a constant number over a range of values of N. However, please observe that this no longer holds true for N approaching 2, which is the region of our major interest. Our study is restricted to positive values of N (as explained and illustrated in particular in Fig.1). For consistency, we, therefore, refrain from discussing N negative also in Fig. 5. We agree that N<0 (and also the range d\in (1,2]) is also interesting and would like to address it in the future. - The flow equations at the considered truncation order fit roughly one page. In the amended version of the manuscript, we added an appendix, where they are exposed.

We thank you once again for reviewing our work.

Sincerely yours, Andrzej Chlebicki Pawel Jakubczyk

Anonymous Report 2 on 2021-4-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.00782v4, delivered 2021-04-09, doi: 10.21468/SciPost.Report.2768


The authors have made significant additions to the manuscript. These fully address my comments (I thank the authors for the detailed explanations in their reply also). The new results added about the shape of the effective potential and the subleading eigenvalue are interesting and strengthen the results. Fig 10 makes the challenge to the C-H prediction very clear.

I am happy to recommend publication. I have only a few comments for the authors to consider in relation to the new version.

- On p15 the authors state “whenever the vortex excitations are irrelevant, the relevant eigenvalues of the noncompact CP and Heisenberg universality classes should coincide. In d=3, this should happen for N≳6 according to our results”.

The relationship between the O(N) model and a noncompact CP^m model is special to the case N=3, m=1, so I do not understand how the above statement can be correct.

- Instead, the relationship between O(3) and noncompact CP1 gives another argument against the claim that the exponents are analytic for all N and d>2.

The standard 2+epsilon expansion of the O(N) model, continued to d=3, can be argued to describe the CP1 model. We know that this theory is distinct from O(3). This implies that at least for N=3, the standard 2+epsilon expansion must fail to describe the O(3) model for epsilon larger than some epsilon*, with epsilon* smaller than 1. This argument was made in

This argument is independent of the C-H expansion (though consistent with the C-H picture) so appears to be a challenge to the simplest interpretation of the present numerical results.

- Regarding topological excitations in d=3, N=3, some early numerical work by Kamal and Murthy, giving hints of novel universal behavior, is There were even earlier studies, e.g., but these used a too-restrictive definition of the model, so did not see a transition in the absence of topological excitations.

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

Author:  Andrzej Chlebicki  on 2021-05-11  [id 1416]

(in reply to Report 2 on 2021-04-09)

Dear Referee,

thank you very much for reviewing our paper and preparing the second report. We are very happy about your recommendation for publication. In answer to the points raised in your second report: - The sentence mentioned by you is removed in the new version of the manuscript. - We are grateful for pointing out the paper by Nahum et al, which we find greatly inspiring (also for future investigations). We recognize the substantial weight of the argument concerning the nature of the 2+epsilon expansion and the relation between the CP^1 and O(3) models. Based on the numerical results at hand we would be able to give only very speculative proposals for resolution of this issue and we prefer to leave it as an open problem (as presented in the new version of the manuscript - see the final part of Sec. 5). - We included a reference to the work by Kamal and Murthy.

Sincerely yours, Andrzej Chlebicki Pawel Jakubczyk

Report 1 by Slava Rychkov on 2021-4-3 (Invited Report)

  • Cite as: Slava Rychkov, Report on arXiv:2012.00782v4, delivered 2021-04-03, doi: 10.21468/SciPost.Report.2753


I thank the authors for carefully revising their manuscript and for adressing my concerns. I now agree with them that their method captures vortices; my previous remark about this was incorrect. I recommend the paper for publication.

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

Author:  Andrzej Chlebicki  on 2021-05-11  [id 1415]

(in reply to Report 1 by Slava Rychkov on 2021-04-03)

Dear Professor Rychkov,

thank you very much for reviewing our paper again. We are very pleased about your positive assessment and recommendation.

Sincerely yours,
Andrzej Chlebicki
Pawel Jakubczyk

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