SciPost Submission Page
Studying the 3d Ising surface CFTs on the fuzzy sphere
by Zheng Zhou, Yijian Zou
Submission summary
Authors (as registered SciPost users):  Zheng Zhou 
Submission information  

Preprint Link:  https://arxiv.org/abs/2407.15914v3 (pdf) 
Date submitted:  20240806 14:33 
Submitted by:  Zhou, Zheng 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3D Ising CFT. We propose two schemes by cutting a boundary in the orbital space or the real space to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through stateoperator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary $o$, the displacement operator $\mathrm{D}$ and their conformal descendants and extract their scaling dimensions. We also study the onepoint and twopoint correlation functions and extract the bulktosurface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3D Ising surface CFTs nonperturbatively. Other conformal data obtained in this way also agrees with prior methods.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
1 Important and timely subject
2 Novel and creative method, new results
Weaknesses
1 Results could have been more thorough
2 Some issues with methodology (see report)
Report
This is a very nice application of the fuzzy sphere regulator of the Ising model to boundary CFT in d=3, and the results should hopefully encourage many future studies further developing the method. Many previously known results are reproduced as a test of the method, and some new results are obtained demonstrating that is has concrete advantages.
Some comments:
1 It is stated below equation (10) that the value of the speed of light is identical in the bulk CFT and the boundary CFT, and so can be calibrated by setting the stress tensor dimension to be 3. I am not exactly sure how the authors implemented this in practice since the bulk stress tensor as a state arises for the CFT on a sphere without boundary, whereas they are calculating the spectrum in the CFT with a boundary, but I am guessing that what they did was to calculate the spectrum on the sphere using the same microscopic UV parameters and assume that the rescaling factor between energy and dimension is the same for the case with a boundary. However, the presence of the boundary will modify the Hamiltonian, and therefore modify the rescaling factor. So it seems to me that they really should be directly rescaling the energies on the theory with boundary to fix the dimension of the displacement operator to be 3, and this would improve the accuracy of their results. If the authors disagree with this approach then perhaps they could explain why.
2 The method by which the authors estimate their errors is to take the difference between the extrapolated value (at infinite radius) and the last computed value. This seems like a significant overestimate of the error, and therefore it seems at least a little concerning that for example the known dimension of the operator o is at the edge of their error bars. Can the authors also estimate the error by performing the fit in different ways (perhaps by using different subsets of the finite N results) and seeing if this is consistent with the errors they choose? Moreover, it would be helpful for the authors to explain the rationale behind the powers of N in the fits in table 2 and Fig 14.
3 In equation (12), they include the descendant operator $\partial \bar{\partial}D$, but the contribution to the energies of states from total derivatives should vanish. On the other hand, as they point out, there is an operator of dimension 5.02, which is so close to 5 that it will behave nearly the same ($\sim N^{3/2}$) as the term they write down.
4  At the bottom of page 13 and top of page 14, they sometimes use $\Delta$ and sometimes use $\Delta_\phi$ for the dimension of $\phi$, but it would be better to be consistent.
5 Could the authors explain why setting the m<0 orbitals to be empty also produces the ordinary surface CFT (why 1. and 2. below equation 9 are equivalent)?
6 The authors do not do the special transition, but merely comment in the discussion on how it should be possible. Could the authors comment on why they chose not to do the special transition, and how difficult in their opinion it would have been to do (i.e. some discussion of the challenges involved if there was some reason it would have been hard). Also I think it would be good to have this discussion in the introduction in order to make it clear up front that the authors did not study the special transition.
7 At the bottom of page 28 there is a stray "ref.bbl"
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
1. The paper introduces a new, exciting application for fuzzy sphere, a recently developed powerful tool in understanding CFTs. The power of their technique is clearly illustrated and the paper has good alignment with previous results.
2. The paper provides multiple checks on their technique for studying BCFTs and gives the reader confidence in the validity of their numerical calculations.
3. Their technique is easily applicable to other BCFTs provided the bulk CFT can be placed on the fuzzy sphere practically.
Weaknesses
1. The introduction could be improved from its current state. Some quantities are not explicitly defined, and thus the introduction does not flow as well. Additionally, the penultimate paragraph could better emphasize the results of the paper. This paragraph does not state the main takeaway of the paper as directly as e.g. the second sentence of the abstract does. Additionally, this paragraph does not motivate well why two schemes are used and the takeaway from using both schemes.
2. The lack of explicit definitions continues throughout the paper and makes some sections a little hard to follow. Additionally, some sections could do a better job of signposting.
3. The grammar is not perfect (some examples are listed below, although the list is not comprehensive). This weakness is not major, but it could be improved for the final version of the paper.
Report
This paper presents a method for using the fuzzy sphere to study BCFTs, specifically the 2+1D Ising CFT with a boundary at the ordinary and normal fixed points. The paper computes scaling dimensions and OPE coefficients for boundary operators for both fixed points. Additionally, the paper provides nonperturbative calculations of the boundary central charge at both fixed points.
The paper's technique is an important development for the field of BCFT. Their technique seems applicable to other CFTs that can be studied with the fuzzy sphere.
However, the lack of definitions and clarity in some parts of the paper hinder its effectiveness in conveying its message. Upon improving clarity, this paper should be wellsuited for this journal.
Requested changes
1. General grammar/spelling comments (this list is not comprehensive, but should be helpful in guiding edits):
There is at least one instance where plural subjects are accompanied by singular verbs and vice versa  namely "there exist a set..." in the second paragraph.
In the second paragraph of the paper, "the CFT" should be "a CFT".
In the paragraph above Eq. (3), the sentence "First, acting ..." is clunky.
The paper sometimes uses the future tense where the present tense is more appropriate.
2. In the second paragraph of the paper, a few things are not defined. For example, the paper never explicitly states that we are considering 2+1D CFTs. The paper also would benefit from briefly defining scaling dimensions, spin, primaries, and descendants  the standard for papers in this field of this length seems to be to define these terms and give a brief few sentence introduction to CFTs.
3. The data in the paragraph at the start of page 4 could be placed in a table. This change would make it easier to compare previous results with their results. Additionally, they should state that the largeN calculations are for the O(N) model. Finally, the paper should consider briefly defining the $\sigma$, $\epsilon$, $\epsilon'$, and $D$ operators  or at least stating for example that $D$ is the displacement operator  as it seems that defining them is the convention in the field for papers of this length.
4. As mentioned in the previous section, directly stating the paper's significance would improve the last paragraph of page 4. Additionally, the paper should comment on why it chose to calculate quantities using two schemes and which scheme should be preferred.
5. On page 7, the paper should briefly explain why a +x direction magnetic field corresponds to a free boundary condition.
6. On page 8, is the O(2) symmetry the same as the U(1) symmetry?
7. On page 9 right before the start of section 3.1, the paper should explicitly state which method it uses (finite size scaling or Lao and Rychkov's method presented in App. B). Additionally, App. B likely should also cite Lao and Rychkov's paper and should signpost and refer back to section 3 to help the reader.
8. On page 12, the paper should state explicitly how it uses the density operator to compute one and twopoint functions  especially because using fuzzy sphere to calculate OPE coefficients is a relatively new calculation that not many readers will be familiar with. The paper should also mention that it uses finite size scaling to determine and subtract off the $\lambda$ coefficients in Eq. (17). Additionally, the paper does not clearly state if Eq. (14) is what it means by "CFT predictions" in Sec 4.1 (at least, that is what I assumed after reading the paper). If Eq. (14) is indeed what the paper means by CFT predictions, what OPE coefficient value does the paper use for the black dashed line in the plots in Fig. 6?
9. On page 17, the paper could explain the relationship between <polI> and Z a little more explicitly.
10. Appendix A should also signpost  i.e., explain its purpose and say where Eq. (25) comes from for ease of reading. Appendix C.3 could also do this (e.g., it could give an equation reference for $G_{\phi\hat \phi}$ so the reader can more easily confirm Eq. (38).
Recommendation
Ask for minor revision