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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 registered SciPost users): Andrzej Chlebicki · Pawel Jakubczyk
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-05-26
Date submitted: 2021-05-11 08:27
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 of the anticipated form is excluded.

Author comments upon resubmission

Dear Editor,

thank you very much for making the reports for our paper available. In the presently resubmitted version, we addressed most of the comments/suggestions from the Referees.

We attach the summary of changes and provide a response for all three of the Referees.

We are grateful to you for handling our paper.

Sincerely yours,
Andrzej Chlebicki
Pawel Jakubczyk

List of changes

Summary of changes:
- We made minor modifications (mostly stylistic) throughout the text.
- We replaced most of the figures for better readability.
- We significantly modified the final part of Sec.5.
- We replaced Fig. 7. The previous version had been (by mistake) plotted with the epsilon expansion results from the previously cited article Kleinert 1998, which (unexpectedly) differs from the currently cited Bernreuther 1986 in epsilon^4 coefficients.
- We replaced Fig.10. The previous version had been (by mistake) plotted with a less accurate set of data points. The new points are slightly shifted down, agree even better with DE(4)/MC results, and are situated even further from e_2 = 0.
- We added the appendix.
- We added references 6, 41, and 44.

Published as SciPost Phys. 10, 134 (2021)

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