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Bootstrapping Mixed MN Correlators in 3D

by Stefanos R. Kousvos, Andreas Stergiou

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Submission summary

Authors (as registered SciPost users): Stefanos Robert Kousvos · Andreas Stergiou
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-04-13 15:04
Submitted by: Kousvos, Stefanos Robert
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approaches: Theoretical, Computational


The recent emergence of the modern conformal bootstrap method for the study of conformal field theories (CFTs) has enabled the revisiting of old problems in classical critical phenomena described by three-dimensional CFTs. The study of such CFTs with $O(m)^n \rtimes S_n$ global symmetry, also known as MN models, is pursued in this work. Systems of mixed correlators involving scalar operators in two different representations of the global symmetry group are considered. Isolated allowed regions are found in parameter space for various values of $m$ and $n$. These "islands" can be separated into two qualitative groups: those close to the unitarity bound and those further away. As a by-product of our analysis generic tensor structures required to bootstrap any $G^n \rtimes S_n$ theory with $G$ arbitrary are worked out.

Author comments upon resubmission

We thank both referees for their careful reading of our manuscript and constructive comments. Below we list the changes made to our draft in order to address the issues raised.

List of changes

Referee 1:

To address point 1, we have included a new paragraph on page 12 "Obtaining ...", we have also added in figure 9 the corresponding allowed regions that would result if one were to demand X bound saturation at $\Lambda=35$ instead of $\Lambda 45$.

With regards to point 2, we indeed didn't go into depth about the physical motivations in order to avoid redundancy, as the referee mentions. We have added an explicit reference as suggested by the referee to previous work that discusses the motivations (last sentence on first paragraph on page 3). We also changed the phrase "Our motivation..." to "Our goal..." in order to be more appropriate.

To address point 3, we have moved the $Z \times Z$ OPE analysis to the beginning of the paper.

Regarding point 4, the $Z \times Z$ OPE does indeed exchange itself. However, we found that this did not affect our plots. We added a paragraph discussing this (the last one on page 21).

Regarding the reference [29], it will become publicly available after a specific date.

Referee 2 :

To address point 1, we have added figure 7 which combines figures 6 and 5 so the reader may conveniently see their overlap. However, we opted to keep figure 4 and figure 5 separate, the reason for this is that we have separated the $\phi$ - $X$ and $\phi$ - $S$ plane islands into different sections in the text, we are thus afraid that combining these may actually cause some confusion.

To address point 2, we have moved the $Z \times Z$ OPE analysis to section 2.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-5-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2112.03919v2, delivered 2022-05-10, doi: 10.21468/SciPost.Report.5056


With the changes, I think the submission now achieves the criteria for publication. With this said, there are a few spots that continue to be a little bit confusing. In the penultimate paragraph of the introduction, the authors note that “there exist experimentally relevant cases” but don’t provide any further context on this. Given that the authors already mention e.g. the cubic model, it’s worth at least referring back to that. Similarly, I think that the exact motivation for studying the X operator (that in the large-n limit it corresponds to the decoupled O(m) singlets, beyond just the fact that you need it to study S) is worth noting explicitly. I will leave it up to the authors if they would like to clarify these points, as I think in the latter case especially this is something of a squishy statement.

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Author:  Stefanos Robert Kousvos  on 2022-05-18  [id 2494]

(in reply to Report 2 on 2022-05-10)

We thank the referee for their additional comments.

We will add an additional sentence to clarify the point of experimental applications:

"...experimentally relevant cases among such models." -> "...experimentally relevant cases among such models. For example, beyond the cubic ($\mathbb{Z}_2{\!}^{3} \rtimes
S_3$), $MN_{2,2}$ ($O(2)^2\rtimes S_2$) and $MN_{2,3}$ ($O(2)^3\rtimes S_3$) models mentioned earlier, there are also the so called tetragonal theories ($D_4^n\rtimes S_n$, with $D_4$ the dihedral group of 8 elements) [a few citations will be included]."

With regards to motivations for studying the X bound, we will add a sentence explaining that the X bound is of interest specifically because it displays pronounced features/kinks. More specifically, at the end of paragraph 2 on page 3 we will add the following sentence: "We note that the $\Delta_X$ bound is of interest since it displays pronounced features/kinks in parameter space."

Anonymous Report 1 on 2022-4-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2112.03919v2, delivered 2022-04-21, doi: 10.21468/SciPost.Report.4958


The authors addressed all my questions and remarks. The changes to the paper are clear and clarify most of my questions/issues. However, the modifications raise two new questions (in order of importance):


Q1) The authors added the sentence: "We checked this by looking at the $\Delta_{SZ} \sim \Delta_Z$ exclusion bound." I think there is an important misunderstanding at play. Either on my part in the interpretation of this paragraph or on your part in understanding the assumption that should be assumed and which bounds it will afffect. I could probably have been more clear in my previous remark. Let me expand on it. In remark 4.) I pointed out that without loss of generality you can impose that the external operator $Z_{\text{ext}}$ corresponds to the lowest operator of its kind. I.e. you can impose $\Delta_Z\geq \Delta_{Z_\text{ext}}$ where $\Delta_Z$ is the dimension of the first exchanged $Z$ operator. This assumption will hold regardless of any symmetry properties of $Z_{\text{ext}}$ (i.e. regardless of whether $Z \times Z$ exchanges itself). The effect of this assumption should of course be checked in a bound that is not on $\Delta_Z$ itself, since in that bound it has no effect as a strictly stronger assumption is already being tested. Adding this assumption might very well affect the position of the kinks in for example Fig. 13 and could clear up the issue of the bound not converging to clear integer values in the large $m$.

Instead the added paragraph on page 21 seems to discuss another assumption. It is also not immediately clear how that assumption was actually imposed in practice. Assuming that the external operator $Z$ itself is exchanged would require imposing a gap above it. Are you saying you found the same for the maximal allowed value for this gap as you found for $\Delta_{Z}$ when prohibiting the exchange of $Z_{\text{ext}}$?


Q2) The change to Fig. 9. leaves the reader with a new question: Why does the allowed line at saturation of the $\Lambda=35$ bound have a gap near the kink? Is there no solution because you are imposing additional constraints compared to Fig. 8? It is somewhat unexpected that the solution disappears exactly at the kink given the results when the saturation of the $\Lambda=45$ bound is demanded. Supposedly the $\Delta_X$ bound shown in Fig. 8 is most stable under the increase of $\Lambda$ in the region around the kink (usually a kink get sharper at larger derivative order but the position of the kink itself does not change significantly). This feels a bit like counter evidence to "if $\Delta_X$ were to change a little there would be no major changes in the corresponding $\Delta_\phi-\Delta_S$ plane island." How big is the change in the $\Delta_X$ bound at the kink between $\Lambda=35$ and $\Lambda=45$ ? This amount of change clearly completely changes the nature of the bounds shown in Fig. 9.

Or viewed from another perspective: This must mean that the assumptions of Fig. 9 significantly alter the $\Delta_X$ bound exactly in the region of the kink, at least at $\Lambda=35$. I suspect that if you were to compute the green region at $\Lambda=45$ you would find the same effect, i.e. the exclusion of any solutions at all near the kink! This is a bit worrying. Strictly speaking Fig. 9 seems to be studying the empty set at least around the region of interest. In order to find a solution again you would have to instead demand saturation of the $\Delta_X$ bound found with the inclusion of the additional constraints used in Fig. 9.

Also: should we find it worrisome that your constraints seem to be most powerful close to the kink? (As also can also be seen in the "pinching" of the island saturating the $\Lambda=45$ bound.) If we view these constraints as getting rid of "fake" or "GFF"-like solution is it not weird that their effect is strongest close to the supposedly physical CFT solution?


I would leave it to the authors whether they want to address these questions but I think that clearing up the misunderstanding in Q1 could lead to a worthwhile improvement of the paper.


P.S.: in depth discussion -> in-depth discussion

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Author:  Stefanos Robert Kousvos  on 2022-05-18  [id 2493]

(in reply to Report 1 on 2022-04-21)

We thank the referee for their additional comments.
With regards to Q1:
"Are you saying you found the same for the maximal allowed value for this gap as you found for ΔZ when prohibiting the exchange of Zext?"
Yes, exactly.
We also explicitly checked that e.g. the $\Delta_{SY}$ exclusion bound for $m=1000$ and $n=4$ (seen e.g. in figure 13) remains unchanged if we impose the assumption $\Delta_{SZ} \geq 1.0$.
We propose the following modification to our text:
"We checked this by looking at the $\Delta_{SZ} \sim \Delta_{Z}$ exclusion bound..." ->
"We checked this by looking at the $\Delta_{SZ} \sim \Delta_{Z}$ exclusion bound... More explicitly, we checked that if $Z$ is exchanged in the $Z \times Z$ OPE, then the exclusion bound on the second exchanged operator in this representation is identical to the exclusion bound of the first exchanged operator in the $Z$ irrep if one assumes that the external operator is not exchanged. Additionally, we checked that for e.g. $m=1000$ and $n=4$ the corresponding $\Delta_{SY}$ exclusion bound in Fig. 14 remains unchanged even after adding the assumption $\Delta_{SZ} \geq 1.0$."
With regards to Q2:
The blue island on the left in Fig. 9 reaches, in its rightmost tip, the $\Delta_\phi$ of the kink of Fig. 9. Thus, the theory at the kink is allowed, even if only barely so. We will add a clarifying comment about this before Fig. 5: "Note that the rightmost tip of the left blue island in Fig. 9 extends to the $\Delta_\phi$ of the kink of Fig. 8. Thus, the putative theory that lives at that kink is allowed under the assumptions mentioned in the caption of Fig. 9. Given that this is a marginal case, however, further numerical work with stronger numerics and more refined methods is required to provide further clarity."
We would appreciate it if the referee could confirm to us whether they believe the above proposed modifications would sufficiently clarify the confusion in our text. We would then be happy to update the arXiv submission.

Anonymous on 2022-05-24  [id 2517]

(in reply to Stefanos Robert Kousvos on 2022-05-18 [id 2493])

Q1: The check that Fig.14 remains unchanged under the assumption $\Delta_{SZ}\geq1.0$ indicates that the assumption $\Delta_{SZ}>\Delta_{SZ_\text{ext}}$ would not make a difference in the plotted region. It is therefore reasonable not to discuss this condition in this context. However, the assumption I proposed could have been imposed without any loss of generality and it could in principle still affect the bound on $ \Delta_{BB}$ in figure 15. In fact, this is the bound where this assumption is most likely to have a significant effect since there is a kink at a large value of $\Delta_{SZ_\text{ext}}$ where this assumption would result in a significant alteration of the gap assumptions.

Q2: If the left island in Fig. 9 indeed touches the $\Delta_\phi$ of the kink this makes the gap assumptions less worrisome. However, looking at Fig 8 it seems that the kink is located around 0.5184 which is in between the two (blue) allowed regions in Fig 9 (unless I am somehow reading the figures wrongly the right most tip of the left island only reaches 0.5181). The kink is of course not that sharp and made out of multiple smaller kinks so perhaps the authors are interpreting the kink to be somewhere else. If the authors have a clear $\Delta_\phi$ value in mind perhaps a line indicating it in Fig 9 would be useful.

In my opinion, it is not unlikely that Fig 9 excludes the theory the authors attempt to study, due to the fact that the saturation of the $\Delta_X$ bound found without additional assumptions is incompatible with the additional assumptions imposed in Fig 9.

If the authors have another interpretation, for the fact that the additional conditions in Fig 9 constrain the $(\Delta_S, \Delta_\phi)$-plane the most near the location of the kink (even excluding completely the $\Delta_\phi$ values closest to the sharpest point of the kink), it would be great to include it.

Side remark: It seems that Fig 8. is only referenced to from Fig 9. and not in the main text. Is this as intended?

(Comments by Referee 1)

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