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On four-point connectivities in the critical 2d Potts model

by Marco Picco, Sylvain Ribault, Raoul Santachiara

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link: (pdf)
Date accepted: 2019-09-19
Date submitted: 2019-08-28 02:00
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • High-Energy Physics - Theory
Approaches: Theoretical, Computational


We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states $Q\in (0,4)$ that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the $2$ to $4$ significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases $Q=0, 3, 4$. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane $\{\Re c <13\}$, where $c$ is the central charge of the Virasoro symmetry algebra.

Ontology / Topics

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Conformal field theory (CFT) Monte-Carlo simulations Potts model Virasoro algebras

Published as SciPost Phys. 7, 044 (2019)

Author comments upon resubmission

We are grateful to the reviewers for their helpful suggestions. We have now implemented most of them. The changes are clarifications, more accurate references to the literature, and cosmetic improvements.

List of changes

Answer to the Anonymous Report 2:

1. In order to clarify this point, we have modified the beginning of Section 3.3 by using the word 'analytic' rather than 'exact' for the coefficients of the relation, and by adding the clause 'while stressing that the relation cannot be exact' at the end of the first paragraph.

2. At the end of Section 3.3, we have written dependences on Q explicitly.

3. For the basic equation (1.2), we have given [8] as a reference.

4. It is true that the structure constants of the odd CFT Eq. (2.16) are related to structure constants of Liouville theory. However, the relation is not simple or straightforward. Before and after (2.16), we have added references to the article [13], which specifically discusses the relation, rather than to works on Liouville theory.

5. We have cited the recommended Delfino-Viti article (now [6]) in Section 1.1, and added the sentence 'In particular, the three-point connectivity of the two-dimensional Potts model was found to be related to a three-point function in Liouville theory [6]'.

Answer to the Anonymous Report 1:

1. We have added clickable links to the specific versions of the cited articles that we have used in the two cases where they were missing. We disagree that journal references are needed to identify cited articles: using a search engine, titles are enough. Moreover, titles provide more useful information to the reader than journal references.

2. We have added captions to the figures and tables that did not have any. (Except a few small tables which are better treated as equations.)

3. We have added labels to the axes of the figures that did not have any.

4. There is no summary of the main results at the end of the paper because there is already a section 'introduction and summary' at the beginning.

5. We have corrected the typo 'a four-point functions encodes' on page 2. On the other hand, 'phenomenons' is a nonstandard but correct plural form of 'phenomenon'.

Further modifications:

We have added the reference [33] in Section A.1.

Reports on this Submission

Anonymous Report 2 on 2019-9-15 (Invited Report)


The authors have made the appropriate changes in the revised version of their manuscript. This is an interesting paper that contributes to understanding CFT as applied to these models. I recommend publication of the manuscript in its present form.

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Anonymous Report 1 on 2019-9-3 (Invited Report)


The resubmission takes into account the comments contained in my report.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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Anonymous on 2019-09-03  [id 587]


Editorial comment on item 1 in reply to anonymous report 1:

Indeed, titles are very helpful and Sci|Post is encouraging the inclusion of titles (see for the guidelines ). Beyond this, the authors are free to have their opinion, but they should still respect the standards of Sci|Post, in particular they should not create unnecessary work for a journal that is operating free of cost. Consequently, the authors will be expected to submit a version with all the required information (in particular the DOI for all references where one is available) prior to publication.