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Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$

by Victor Gorbenko, Slava Rychkov, Bernardo Zan

Submission summary

As Contributors: Slava Rychkov
Arxiv Link: (pdf)
Date accepted: 2018-11-15
Date submitted: 2018-10-16 02:00
Submitted by: Rychkov, Slava
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


We study complex CFTs describing fixed points of the two-dimensional $Q$-state Potts model with $Q>4$. Their existence is closely related to the weak first-order phase transition and walking RG behavior present in the real Potts model at $Q>4$. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter $Q$, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop conformal perturbation theory around complex CFTs to compute various properties of the real walking RG flow. These properties, such as drifting scaling dimensions, appear to be common features of the QFTs with walking RG flows, and can serve as a smoking gun for detecting walking in Monte Carlo simulations. The complex CFTs discussed in this work are perfectly well defined, and can in principle be seen in Monte Carlo simulations with complexified coupling constants. In particular, we predict a pair of $S_5$-symmetric complex CFTs with central charges $c\approx 1.138 \pm 0.021 i$ describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.

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Conformal field theory (CFT) First-order phase transitions Monte-Carlo simulations Potts model

Published as SciPost Phys. 5, 050 (2018)

Author comments upon resubmission

We thanks the referees for their comments. We especially thank the second referee Sylvain Ribault for his exceptionally thorough reading of our paper and for many comments.

List of changes

I would like to bring to the attention of the authors two papers in which two loop integrals of conformal perturbation theory are calculated for nearby fixed points (they appear to be essentially the same as those in section 4.4):
• A. W. W. Ludwig and J. Cardy, Nuclear Physics B285, 687
• M. Lassig, Nuclear Physics B334, 652

OUR REPLY: Checked and one of the references added in note 36

A couple of typos were spotted:
• On page 15, next to last sentence in section 3: ”form” should be changed to ”from”

OUR REPLY: taken care

• On page 20, just before formula (4.22) an ”echo” of ”with” should be removed

OUR REPLY: taken care
1- We have no less than 7 different notations for what is essentially the same parameter: c,Q,e0,g,u,t,m. Using several notations is probably inevitable, as this parameter appears in various contexts and approaches. But 7 notations are too many, maybe Q,c,g or Q,c,t could suffice? If many notations are kept, at least there should be a synthetic table of their relations.

OUR REPLY: Table 1 created


2- Similarly, we have two notations (e,m) and (r,s) for the same parameters. The discussion of Section 2.6 on how degenerate characters emerge from various terms in the partition function would be greatly helped by using (r,s) throughout, as degenerate representations correspond to r,s being nonzero integers of the same sign. Or at least, the partition function should be rewritten in terms of (r,s) in Section 2.6.

OUR REPLY: : We kept our notation. See new footnote 11. We do not rewrite the partition function fully in terms of h_{rs} characters in this paper, see the sentence added before “Singlets” subsection 2.6.


3- The formulas (2.15) and (2.16) are standard formulas in free bosonic CFTs in the presence of a background charge. The present text gives the misleading impression that these formulas are specific to the Potts model, and gives derivations that are not very enlightening and probably unnecessary. Once we know that we should compute the functional integral (2.13), these formulas can be immediately accepted, with possibly a reference to the textbook by di Francesco et al, Chapter 9.

OUR REPLY: : we kept our discussion but added a sentence at the end of section 2.4 that this is standard.


4- If e∈2Z for electric operators (Section 2.4), why is e∈Z allowed in Zc[g,1] (2.23)-(2.19)?

OUR REPLY: : Notation for Coulomb gas charge modified to avoid confusion and footnote 8 added.

5- It is strange that the partition function (2.23) is only partially described in Section 2.5, with one piece being postponed to eq. (2.31). Some restructuring would be welcome.

OUR REPLY: : We moved that last piece to section 2.5


6- 1 is not prime: is (2.32) valid for M=1?

OUR REPLY: : yes, sentence corrected


7- The notation Oe0+2P,0 is confusing, and obscures the fact that this operator is degenerate if P≥0. A notation of the type V1,P+1 would be much better. (I used the letter V in order to avoid the confusion with the non-diagonal operators Oe,m.) Similarly, in (2.31) the notation xe0+2P,0 is confusing.

OUR REPLY: : We introduced an alternative notation for operator Oe0+2P,0


8- When discussing the operator content, in other words when grouping the terms in the partition function into Virasoro characters, it would be good to systematically indicate which contributions come from which terms with which multiplicities. Presenting all the results as a table might be appropriate.

OUR REPLY: : we streamlined and clarified the discussion of how Virasoro characters form from various contributions. However we decided against adding a table.


9- Statements like 'does not exist as a primary operator' or 'a scalar operator of such dimension is present, but it's not a primary' or 'do not exist in the spectrum' should be made more precise.
In the presence of negative multiplicities, it is hard to say anything definite about the spectrum: having statements about characters would be more accurate. The appearance of degenerate characters suggests, but does not prove, that degenerate operators exist. And null vectors are both primary and descendent.

OUR REPLY: We believe our conclusions about the degenerate operators are correct. We added footnote 17 to clarify.


10- In Figure 7, at Q=2 there seem to be two degenerate operators of type (1,2) with different dimensions.

OUR REPLY: : One of these is on a solid line (critical), another dashed (tricritical). So they live in different theories.


11- The operators ϵ,ϵ′ are actually degenerate at all Q, whereas Figure 7 suggests that they are degenerate at Q=2,3 only. The indices (r,s) are not just positions in the Kac table: they make sense at all central charges.

OUR REPLY: : We are certainly aware that indices (r,s) make sense at all central charges, as our Eq. (2.31) says exactly that. We added a sentence to the caption of Figure 7 to make sure that this is not forgotten.


\subsubsection*{Section 4 before 4.1}

The authors could be more cautious in declaring that the light fields belong to degenerate conformal families: the characters suggest it, but do not prove it. Character-wise, a degenerate representation coincides with two non-degenerate representations with multiplicities 1 and −1, and we are in a context where negative multiplicities are allowed.

OUR REPLY: : this is partly addressed in footnote 17


Moreover, fixing three-point functions of degenerate fields at arbitrary central charges using crossing symmetry is not really complicated. The authors' extrapolation from minimal models is a circuitous route, which gives correct results because the correlators are uniquely determined by differential equations and crossing relations, as the authors correctly state. But this holds for all central charges.

OUR REPLY: . While we appreciate the referee’s remark, we do not feel obliged to act on it. The method we used, while it may look circuitous, was sufficient for our purposes and it gives consistent results which we conjecture to be correct. We hope that our results will stimulate further work upgrading these conjectures to the status of a theorem. We added two references to footnote 25 to indicate that alternative procedures exist

Therefore, I recommend the following:

12- Remove the discussion of minimal models, with its odd focus on the unitary models with integer m, whereas more general minimal models have two integer parameters. (The mention of minimal models in the article's Abstract could also be removed.)

OUR REPLY: while we followed many of the referee’s suggestions, we considered and decided against following this suggestion in light of our previous reply.

13- Give the explicit formulas for the structure constants, as this would allow readers to check their subsequent expansions around c=1. (See footnote 24.) The natural parameter for writing these structure constants is not m, but rather t or g. Maybe these formulas could be given in an appendix, which would include all needed structure constants, including those of the spin operator.

OUR REPLY: : While we followed many of the referee’s suggestions, we considered and decided against following this suggestion. In footnote 24 (current footnote 30) we give an explicit reference to the paper where the expression we used was given. We tend to believe there is no need to retype this expression.


\subsubsection*{Section 4.5}

This Section gives arguments for the existence of the real flow, based on the assumption that there exists a pair of complex conjugate CFTs that are moreover related by a RG flow. There are two arguments:

# A first-order argument argument, plus a heuristic argument that higher order corrections should deform the picture without changing it qualitatively.

# An all-orders argument, which is not rigorous because regularization issues are neglected, and which essentially shows the equivalence between the existence of the real flow, and the fixed point coupling constant gFP being pure imaginary.

The all-orders argument does not really establish the existence of the real flow, but it does constitute a nontrivial consistency check. Moreover, it illustrates the properties of correlation functions in complex CFTs.

The logic of the argument, and the technical details, are not clear enough.
Moreover, while interesting, this Section is not crucial to the main results of Section 4.
I recommend that Section 4.5 be either deleted, or clarified. Suggested clarifications include:

OUR REPLY: We kept the heuristic argument, but removed the all-order arguments which we agree was not clear enough.

Referee’s suggestions 14-19 are mute since we removed the all-order argument.


\subsubsection*{Miscellaneous suggestions}

20- Does eq. (2.4) for the partition function at arbitrary Q reduces to (2.2) when Q is integer?

OUR REPLY: yes, paragraph added at the end of Section 2.1 to stress that

21- In Subsection 2.2, clarify the statement that 'we will take an intuitive approach to symmetry for non-integer Q...'. Does this symmetry for non-integer Q play any role in the subsequent analysis?

OUR REPLY: a sentence added to subsection 2.2 to give two examples.


22- Typo: 'complex plain'.



23- In Section 3, the statement that 'multiplicities stay real for any Q' could be amended to 'multiplicities stay real for any real Q'. Thinking about complex Q is indeed quite natural in this context.

OUR REPLY: done. Related change: 3rd Paragraph of section 3 somewhat expanded.


24- There is a dual use of the notation g in Section 4.1: the original g from eq. (2.6), which appears again in eq. (4.7), differs from the g in eq. (4.9).

OUR REPLY: Indeed. We had footnote 22 about that , which we moved to the main text, right after 4.9, to decrease even further the chance for confusion.


25- In Section 4.1, the description of the Im-flip is confusing. Maybe an equation would be clearer.

OUR REPLY: the description of Im-flip is modified to make it hopefully less confusing


26- In the Conclusion, state explicitly what it means to 'make sense' for a Euclidean CFT, and for a Minkowskian CFT.

OUR REPLY: Adjusted, Hopefully more clear now.

OTHER CHANGES: We also streamlined discussion of drifting critical exponents Eqs 4.27-4.31 correcting a few typos. Conclusions are unchanged.

Reports on this Submission

Anonymous Report 2 on 2018-10-30 (Invited Report)


The revised version answers all criticism put by the two referees. The paper can now be published.
A typo is spotted in reference [44] - the name of the auther is misspelled.

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Report 1 by Sylvain Ribault on 2018-10-23 (Invited Report)

  • Cite as: Sylvain Ribault, Report on arXiv:1808.04380v3, delivered 2018-10-23, doi: 10.21468/SciPost.Report.624


The authors' use of my suggestions for improvement was thorough and economical. Thorough, because they considered each suggestion, and decided which modifications to perform (if any) on a case by case basis, while carefully reporting the changes in their reply. Economical, because each change tends to be the smallest modification that could do the requested clarification: in many cases this meant adding a footnote. The resulting changes are local and perturbative, except for the all-orders argument of Section 4.5, which the authors chose to delete rather than clarify.

The resulting text is still not an easy read, but the specific sources of confusion and misunderstanding that I had identified have been eliminated.

Let me briefly comment on two of the few suggestions that the authors chose not to follow:
- I now see that my remark 10 was misguided, as indeed the critical and tricritical models with the same $Q$ have different central charges.
- My suggestion 12 to eliminate minimal models can indeed be ignored, provided readers take the authors to the letter in considering the use of minimal models as a 'helpful trick' only (page 17).

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