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

by Andrzej Chlebicki, Pawel Jakubczyk

Submission summary

As Contributors: Andrzej Chlebicki · Pawel Jakubczyk
Arxiv Link: (pdf)
Date submitted: 2021-04-01 12:53
Submitted by: Chlebicki, Andrzej
Submitted to: SciPost Physics
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:
Editor-in-charge assigned

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 2 on 2021-4-9 Invited Report


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: -

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


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

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