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Spontaneous symmetry breaking in free theories with boundary potentials

by Vladimir Procházka, Alexander Söderberg

Submission summary

As Contributors: Alexander Söderberg
Preprint link: scipost_202104_00010v2
Date accepted: 2021-07-28
Date submitted: 2021-06-24 21:50
Submitted by: Söderberg, Alexander
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

Patterns of symmetry breaking induced by potentials at the boundary of free $O(N)$-models in $d=3- \epsilon$ dimensions are studied. We show that the spontaneous symmetry breaking in these theories leads to a boundary RG flow ending with $N - 1$ Neumann modes in the IR. The possibility of fluctuation-induced symmetry breaking is examined and we derive a general formula for computing one-loop effective potentials at the boundary. Using the $\epsilon$-expansion we test these ideas in an $O(N)\oplus O(N)$-model with boundary interactions. We determine the RG flow diagram of this theory and find that it has an IR-stable critical point satisfying conformal boundary conditions. The leading correction to the effective potential is computed and we argue the existence of a phase boundary separating the region flowing to the symmetric fixed point from the region flowing to a symmetry-broken phase with a combination of Neumann and Dirchlet boundary conditions.

Published as SciPost Phys. 11, 035 (2021)



Author comments upon resubmission

Dear sci-post editors and referees,

We would like to thank the referees for a very valuable and informative feedback on our work. We have taken your comments into account and wrote an updated version of the paper. We have also answered some specific points of your reviews below.

Referee 1.

We fixed the minor points and added footnote 11 to clarify the subtraction scheme question.

Referee 2.

Main points:

1.We have added a paragraph dedicated to Mermin-Wagner theorem on page 4 and another paragraph on the epsilon->1 limit at the end of section 3 on page 12. While we do agree that in general the physical interpretation of the d=2 case as phase transition is not clear the main features discussed in section 1 such as appearance of Dirichlet mode in the IR should persist (as they do for open strings, boundary sine Gordon model etc.). Furthermore our example in section 3 includes O(N) with N=1 (and it could be that for epsilon=1 it only includes this model), in which case the broken symmetry is discrete and thus allowed by Mermin-Wagner theorem.

  1. We have updated the relevant references and added a remark in conclusions on page 13. Because the non-perturbative extrapolation of epsilon->1 limit is beyond the scope of our paper we can only offer speculative/qualitative remarks at this stage. Nevertheless we believe that even if the compact scalar results apply to our model it is entirely possible that the relevant fixed points actually correspond to Dirichlet conditions in the IR in analogy with the IR fixed point of boundary sine-Gordon models.

Minor points:

  1. We added a clarification above eq 4 and below eq 23.
  2. The equal signs in eq. 10 seem to be in order.
  3. The choice of contour is justified in the text under eq. 16. We added some further explanation in the figure 1 caption.
  4. This is now clarified by footnote 11.
  5. Fixed
  6. We added footnote 14 on page 7 and the sentence around it.
  7. We expanded the figure 2 and its caption.
  8. We added some further discussion under eq. 24 and added eq. 25

Should you require any further clarification, please let us know.

Best regards,

Alexander and Vladimir

List of changes

1. We extended the paragraph below eq. (22) with footnote 11. It contains a discussion on why we can drop the Log(M)-divergence.

2. We fixed the typo in the title of section 3.1 and in the first line in subsection 3.1.

3. We fixed the index typos in eq. (33).

4. We made the notation of A^x_y appearing in first eq. (34), and made it consistent throughout the paper.

5. We added a "(phi^2_cl, chi^2_cl)"-bracket for Xi_1 and Xi_2 to highlight that those include terms that are dependent on the terms. I.e. they're not constants which can be ignored.

6. We have added a paragraph dedicated to Mermin-Wagner theorem on page 4 and another paragraph on the epsilon->1 limit at the end of section 3 on page 12. While we do agree that in general the physical interpretation of the d=2 case as phase transition is not clear the main features discussed in section 1 such as appearance of Dirichlet mode in the IR should persist (as they do for open strings, boundary sine Gordon model etc.). Furthermore our example in section 3 includes O(N) with N=1 (and it could be that for epsilon=1 it only includes this model), in which case the broken symmetry is discrete and thus allowed by Mermin-Wagner theorem.

7. We added a discussion regarding the 2d extrapolated theory, with the relevant references, in the conclusions on page 13. Because the non-perturbative extrapolation of epsilon->1 limit is beyond the scope of our paper we can only offer speculative/qualitative remarks at this stage. Nevertheless we believe that even if the compact scalar results apply to our model it is entirely possible that the relevant fixed points actually correspond to Dirichlet conditions in the IR in analogy with the IR fixed point of boundary sine-Gordon models.

8. We clarified above eq. (4) and below eq. (23) that we chose O(N) symmetry groups as an example, but in general we can consider other continuous global symmetry groups.

7. We justified the choice of contour in the text under eq. (16). We added some further explanation in the figure 1 caption.

6. We added footnote 14 where we justify why we doesn't consider the relevant operators of normal derivatives of phi^2 and chi^2 on the boundary.

7. We expanded figure 2 and its caption.

8. Regarding the symmetries of the model we consider, we expanded the discussion under eq. (24) and added eq. (25) where the symmetry is manifest for the equal flavored case.


Reports on this Submission

Anonymous Report 2 on 2021-7-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202104_00010v2, delivered 2021-07-16, doi: 10.21468/SciPost.Report.3243

Report

The authors satisfactorily addressed my comments. While I still have some minor doubts about a couple of the answers, I feel they are not important enough to further delay publication. I am therefore happy to recommend the paper for publication.

As a small remark, in footnote 15 "ineducable representations" should probably be "irreducible representations".

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Anonymous Report 1 on 2021-7-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202104_00010v2, delivered 2021-07-04, doi: 10.21468/SciPost.Report.3184

Report

I understood the reply about the point I raised. It is good to know that this term is canceled when imposing the renormalization conditions. It is still not clear to me how a counter-term proportional to log(M) can actually be allowed, it seems to signal an IR divergence of some sort, but this is probably just a technical nuisance. Since it does not stop the authors from getting a sensible final answer, I am satisfied with the explanation in the edited version. However I realised that there is a typo: "Note that the numerator of the logarithm in (22)" below eq. (22), here "numerator" should be "denominator".

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Comments

Anonymous on 2021-07-20

Dear referees,

Many thanks for the feedback on our work. We have fixed the suggested typos.

Vladimir Prochazka

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