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Non-Wilson-Fisher kinks of $O(N)$ numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs

by Yin-Chen He, Junchen Rong, Ning Su

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Junchen Rong
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-05-17
Date submitted: 2021-04-15 12:31
Submitted by: Rong, Junchen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


It is well established that the $O(N)$ Wilson-Fisher (WF) CFT sits at a kink of the numerical bounds from bootstrapping four point function of $O(N)$ vector. Moving away from the WF kinks, there indeed exists another family of kinks (dubbed non-WF kinks) on the curve of $O(N)$ numerical bounds. Different from the $O(N)$ WF kinks that exist for arbitary $N$ in $2<d<4$ dimensions, the non-WF kinks exist in arbitrary dimensions but only for a large enough $N>N_c(d)$ in a given dimension $d$. In this paper we have achieved a thorough understanding for few special cases of these non-WF kinks. The first case is the $O(4)$ bootstrap in 2d, where the non-WF kink turns out to be the $SU(2)_1$ Wess-Zumino-Witten (WZW) model, and all the $SU(2)_{k>2}$ WZW models saturate the numerical bound on the left side of the kink. We further carry out dimensional continuation of the 2d $SU(2)_1$ kink towards the 3d $SO(5)$ deconfined phase transition. We find the kink disappears at around $d=2.7$ dimensions indicating the $SO(5)$ deconfined phase transition is weakly first order. The second interesting observation is, the $O(2)$ bootstrap bound does not show any kink in 2d ($N_c=2$), but is surprisingly saturated by the 2d free boson CFT (also called Luttinger liquid) all the way on the numerical curve. The last case is the $N=\infty$ limit, where the non-WF kink sits at $(\Delta_\phi, \Delta_T)=(d-1, 2d)$ in $d$ dimensions. We manage to write down its analytical four point function in arbitrary dimensions, which equals to the subtraction of correlation functions of a free fermion theory and generalized free theory. An important feature of this solution is the existence of a full tower of conserved higher spin current. We speculate that a new family of CFTs will emerge at non-WF kinks for finite $N$, in a similar fashion as $O(N)$ WF CFTs originating from free boson at $N=\infty$.

Author comments upon resubmission

We would like to thank both referees for their comments on our manuscript. We have made the following changes based on the referee report.

List of changes

-We have corrected the grammar mistakes and typos pointed out by both referees.
- In response to Requested Change 14 in Report 2, we added footnote 8 on page 10.
- In response to Requested Change 15 in Report 2, we added footnote 10 on page 12.
- In response to Requested Change 16 in Report 2: It is hard to say which of the two kinks does the non-WF kink corresponds to. In fact, it would be important to study the two kinks more carefully, especially to find a distinguishing operator content feature to confirm that they indeed correspond to two separate solutions of the crossing equation. We mentioned this in footnote 11.
- In response to Requested Change 1 in Report 1, we added footnote 5 on page 7.
- In response to Requested Change 3 in Report 1, we modified Figure 3.
- In response to Requested Change 4 in Report 1: In the second to the last paragraph of section 1, we added a sentence to refer the definition of Lambda to an earlier paper. Since we did not present any explicit OPE coefficients in this paper, we figure that the convention of the block is not important therefore we will not mention it here.

Published as SciPost Phys. 10, 115 (2021)

Reports on this Submission

Anonymous Report 2 on 2021-5-11 (Invited Report)


The authors have made the suggested changes and I am happy to recommend the paper be published in its current form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report 1 by Connor Behan on 2021-5-8 (Invited Report)


The paper has undergone nice improvements and meets all the criteria to be published.

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

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