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

### Submission summary

 As Contributors: Sylvain Ribault Arxiv Link: https://arxiv.org/abs/1906.02566v2 (pdf) Date accepted: 2019-09-19 Date submitted: 2019-08-28 02:00 Submitted by: Ribault, Sylvain Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational High-Energy Physics - Theory Approaches: Theoretical, Computational

### Abstract

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

See full Ontology or Topics database.

Published as SciPost Phys. 7, 044 (2019)

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.

### Submission & Refereeing History

Resubmission 1906.02566v2 on 28 August 2019
Submission 1906.02566v1 on 25 June 2019

## Reports on this Submission

### 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.

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

### Report

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

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