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The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap

by Jaychandran Padayasi, Abijith Krishnan, Max A. Metlitski, Ilya A. Gruzberg, Marco Meineri

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Submission summary

Authors (as registered SciPost users): Ilya Gruzberg · Abijith Krishnan · Marco Meineri · Jaychandran Padayasi
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-05-19
Date submitted: 2022-03-31 17:52
Submitted by: Gruzberg, Ilya
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


This paper studies the critical behavior of the 3d classical $\mathrm{O}(N)$ model with a boundary. Recently, one of us established that upon treating $N$ as a continuous variable, there exists a critical value $N_c > 2$ such that for $2 \leq N < N_c$ the model exhibits a new extraordinary-log boundary universality class, if the symmetry preserving interactions on the boundary are enhanced. $N_c$ is determined by a ratio of universal amplitudes in the normal universality class, where instead a symmetry breaking field is applied on the boundary. We study the normal universality class using the numerical conformal bootstrap. We find truncated solutions to the crossing equation that indicate $N_c \approx 5$. Additionally, we use semi-definite programming to place rigorous bounds on the boundary CFT data of interest to conclude that $N_c > 3$, under a certain positivity assumption which we check in various perturbative limits.

Author comments upon resubmission

Dear Editors,

We would like to thank the referee for his careful assessment of the manuscript and his useful comments. We made a few changes to the text, which we detail below, in correspondence with the four remarks by the referee.

We hope that with these modifications the paper can be accepted for publication.

J. Padayasi, A. Krishnan, M. A. Metlitski, I. A. Gruzberg, M. Meineri

List of changes

The list of changes:

1. We have modified the entry in Table 1 to refer to the new result for $N = 1$.

2. We changed the last line of the caption for Table 1 to 'values in regular font are from bootstrap techniques where the uncertainty is not rigorous'.

3. Regarding the sentence on page 11 mentioned by the referee. We have replaced the sentence with the following: ``At this length-scale, we expect the renormalization group trajectory to pass close to the normal fixed point.'' We hope this clarifies the meaning of the paragraph. We believe that making this point is important, because while it is true that Eqs. (1.1) and (3.4) have the same symmetry, what makes the model in Eq.(3.4) useful is that it is weakly coupled at intermediate scales, where the fate of the RG flow is decided. We also added the footnote 9, to make the difference between the normal fixed point and the extra-ordinary one precise: there is a set of decoupled Goldstone bosons in the latter. They don't affect bulk correlation functions at the fixed point. They are instead crucial along the flow, because they are responsible for the logarithmic decay of correlations along the boundary. This is explained in the paragraph above Eq. (3.8).

4. Regarding the (former) footnote 13 mentioned by the referee. What we had in mind involved scanning over a restricted range of values for the OPE coefficient of the displacement. This would express the bounds on $\alpha$ as a function of the range in question. However, this idea is only useful if (approximate) bounds on the size of the OPE coefficient of the displacement are known. Since this is, at the moment, an abstract thought, we decided to remove the footnote altogether.

5. We have corrected multiple (minor) typos and rephrases some sentences for clarity.

Published as SciPost Phys. 12, 190 (2022)

Reports on this Submission

Report 1 by Slava Rychkov on 2022-4-29 (Invited Report)

  • Cite as: Slava Rychkov, Report on arXiv:2111.03071v2, delivered 2022-04-29, doi: 10.21468/SciPost.Report.5000


In my first report, I was already happy with the paper and I only made some minor remarks leaving it to the authors to take them into account or not. I thank the authors for having taken my remarks into consideration, and making minor corrections to their paper which remove even those minor concerns that I had. I am now even happier with the paper and recommend it for publication without any reservation.

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

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