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Conformal bootstrap bounds for the $U(1)$ Dirac spin liquid and $N=7$ Stiefel liquid

by Yin-Chen He, Junchen Rong, Ning Su

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Submission summary

Authors (as registered SciPost users): Yin-Chen He · Junchen Rong
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-11-18 02:56
Submitted by: He, Yin-Chen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approaches: Theoretical, Computational


We apply the conformal bootstrap technique to study the $U(1)$ Dirac spin liquid (i.e. $N_f=4$ QED$_3$) and the newly proposed $N=7$ Stiefel liquid (i.e. a conjectured 3d non-Lagrangian CFT without supersymmetry). For the $N_f=4$ QED$_3$, we focus on the monopole operator and ($SU(4)$ adjoint) fermion bilinear operator. We bootstrap their single correlators as well as the mixed correlators between them. We first discuss the bootstrap kinks from single correlators. Some exponents of these bootstrap kinks are close to the expected values of QED$_3$, but we provide clear evidence that they should not be identified as the QED$_3$. We then provide rigorous numerical bounds for the Dirac spin liquid and the $N=7$ Stiefel liquid to be stable critical phases on the triangular and kagome lattice. For the triangular and kagome Dirac spin liquid, the rigorous lower bounds of the monopole operator's scaling dimension are $1.046$ and $1.105$, respectively. These bounds are consistent with the latest Monte Carlo results.

Current status:
Has been resubmitted

Reports on this Submission

Report 3 by Denis Karateev on 2022-5-3 (Invited Report)

  • Cite as: Denis Karateev, Report on arXiv:2107.14637v2, delivered 2022-05-03, doi: 10.21468/SciPost.Report.5021


The present paper applies the numerical conformal bootstrap technique to the system of mixed correlators in 3d and discusses the numerical results in the context of two models: $U(1)$ Dirac spin liquid (DSL) and $N=7$ Stiefel liquid (SL).
In both cases the authors obtain several rigorous bounds. In the case of DSL model the results are in a good agreement with the Monte Carlo simulations.

The authors also observe kinks on their bounds. They convincingly demonstrate that these bounds have an enhanced global symmetry and have nothing to do with the models under consideration.

The assumptions used by the authors in the conformal bootstrap setup are not enough to isolate neither of the two models. This is in contrast with the case of the 3d Ising model where only few assumptions are needed for its identification. This might be considered as an important suggestion for the future numerical studies: in order to identify a generic CFT model in the conformal bootstrap setup one needs to insert a lot of data both group-theoretic and numerical coming for example from Monte Carlo simulations.

I think this paper provides a valuable contribution. It discusses application of conformal bootstrap in the new context in 3d, derives rigorous bounds and points out various issues. These will be important in future explorations. I am, thus, happy to recommend the paper for publication.

Minor comments and suggestions:

1) The phrase ``We then provide rigorous numerical bounds for the Dirac spin liquid and the $N = 7$ Stiefel liquid to be stable critical phases on the triangular and kagome lattice'' in the abstract and the phrase ``We further study the bootstrap bounds for the DSL to be a stable critical phase on the triangular and kagome lattice'' in the conclusions are confusing.

I think the authors want say something like this: ``By requiring the critical phase to be stable we obtain rigorous numerical bounds for DSL and SL..

2) On page 2: ``In recent years, the conformal bootstrap becomes ..'' should be replaced by ``In recent years, the conformal bootstrap has become ..''

3) On page 4 (last line) there is a typo: Sec. IIA should be replaced by Sec. IIB.

4) In the captions of figure 3 and 4 one should add a sentence saying that the allowed region is below the black curve.

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

Author:  Yin-Chen He  on 2022-05-17  [id 2477]

(in reply to Report 3 by Denis Karateev on 2022-05-03)

We thank the referee for carefully reading our paper for helping us improve the readability. We have made the changes accordingly.

Anonymous Report 2 on 2021-12-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2107.14637v2, delivered 2021-12-21, doi: 10.21468/SciPost.Report.4075


This manuscript performs a numerical conformal bootstrap study of $N_f=4$ QED$_3$ and the $N=7$ Stiefel liquid. There are good motivations for studying these theories due to their applications, and the authors' study with the numerical conformal bootstrap is well-justified. However, there are many obvious omissions, which are surprising given that the author list includes widely respected conformal bootstrap experts. To give a few examples, the authors do not discuss in detail the spectrum of the theory at the kink of Fig. 1, which could reveal if the symmetry enhancement they discuss indeed occurs, they do not attempt any gap assumptions that could perhaps lead them closer to the theories of interest, and they neglect to bound important physical quantities like the central charge. As a result, the manuscript appears rushed.

The method used in this work is by now standard, the results obtained mostly negative, and the overall analysis incomplete. I do not think that this manuscript meets the standards for publication.

  • validity: ok
  • significance: low
  • originality: poor
  • clarity: ok
  • formatting: below threshold
  • grammar: below threshold

Author:  Yin-Chen He  on 2022-05-17  [id 2479]

(in reply to Report 2 on 2021-12-21)

We thank the referee for reading our paper. The results in Section B, in particular, the fact that adding mild gaps in the non-conserved current channels leads the kinks to disappear, is already a strong evidence that these kinks have nothing to do with DSL (QED3). The extra details of the EFM spectrum and quantities such as central charge is not particularly illuminating to present (although we have calculated some). One main purpose of this work is to provide rigorous bounds that are useful for future numerical/experimental study on DSL/SL (unfortunately the central charge is not a useful quantity for this purpose). We believe this goal has been achieved and clearly presented.

We also want to add that, we performed calculation of mixed correlators of monopole, fermion bilinears and four-fermion operators. Technically, it is nontrivial to perform these calculations. One useful message from our results is that (as explicitly pointed out by the third referee Dr. Karateev ), DSL (Nf=4 QED3) is unlikely to be solved numerically by bootstrapping monopoles, unless a lot of data from other methods was inserted. One could investigate how the bounds change if such (unjustified) data was inserted, but our motivation was to provide rigorous bounds for future numerical/experimental study.

Report 1 by Connor Behan on 2021-12-20 (Invited Report)

  • Cite as: Connor Behan, Report on arXiv:2107.14637v2, delivered 2021-12-20, doi: 10.21468/SciPost.Report.4069


Following previous work by the authors, which can be viewed as an exploration of the $N = 5$ Stiefel liquid, this paper turns to the case of $N = 6, 7$. The main tool employed is the numerical bootstrap for $SO(N) \times SO(N - 4)$. No kinks corresponding to these theories are found but the bounds are still illuminating. Some of them apply generally while others are constraints that must hold if the Dirac spin liquid and $N = 7$ Stiefel liquid turn out to be conformal phases (dead end CFTs). The kinks that do appear in these bounds are shown to posess enhanced symmetry, thereby ruling out one of the extant proposals for bootstrapping QED3.

This paper meets the criteria for publication but the following changes should be made for readability.

Requested changes

1. Change "becomes" to "has become" on page 2.
2. Change "demanding" to "important" or something similar on page 3.
3. Change "it is because" to "this is because" and "it is defined" to "they are defined" on page 3.
4. Change "a family of infinite number of" to "an infinite family of" on page 3.
5. Change "if there exists" to "if there exist" on page 4.
6. The last paragraph of the introduction mentions section IIA a second time insteaad of IIB.
7. The reference to string theory in the introduction is confusing because the Ising and $O(N)$ CFTs are also non-Lagrangian. Are you trying to say that the WZW model requires one to consider UV fixed points?
8. Change "was thoroughly analyzed" to "were thoroughly analyzed" on page 7.
9. Change "a priori" to "a priori reason" on page 7.
10. Change "being disappeared" to "disappearing" on page 8.
11. Change "symmetry to be irrelevant" to "symmetry be irrelevant" and "DSL to be a stable" to "DSL be a stable" on page 9.
12. Change "is no ratios" to "are no ratios" on page 10.
13. Change "manifold to be SO(N)" to "manifold SO(N)" on page 11.
14. Change "a disorder" to "a disordered" on page 11.
15. Change "is same as" to "is the same as" on page 12.

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

Author:  Yin-Chen He  on 2022-05-17  [id 2478]

(in reply to Report 1 by Connor Behan on 2021-12-20)

We thank the referee for carefully reading our paper. We appreciate the referee’s effort in helping us correcting grammar mistakes and typos. We have made the changes required. Here are some extra comments for Request 7:

By “non-Lagrangian” we actually means CFTs without Lagrangian descriptions as their UV completion, such as Argyres-Douglas theory. The $\mathcal N=2$ Supersymmetric the Argyres-Douglas theories were believed to have no Lagrangian description, until a much more recent discovery of $\mathcal N=1$ Lagrangian descriptions. By this definition, the O(N) and Ising models will not be called “non-Lagrangian”. In the 3d WZW description, the WZW Lagragian and the Stiefel liquids are both IR fixed points. There exist no direct renormalization group (RG) flow from the 3d WZW Lagrangian (which describes the symmetry breaking phase) to the Stiefel liquid fixed point (which describes the conformal phase). To find a UV Lagrangian completion requires extra work, which is currently missing for $N\ge 7$ Stiefel liquids. We thank the referee for pointing out this important confusion, we have made some changes to clarify it.

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