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Non-gaussianity of the critical 3d Ising model

by Slava Rychkov, David Simmons-Duffin, Bernardo Zan

This Submission thread is now published as SciPost Phys. 2, 001 (2017)

Submission summary

As Contributors: Slava Rychkov
Arxiv Link: (pdf)
Date accepted: 2017-02-08
Date submitted: 2016-12-22 01:00
Submitted by: Rychkov, Slava
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We discuss the 4pt function of the critical 3d Ising model, extracted from recent conformal bootstrap results. We focus on the non-gaussianity Q - the ratio of the 4pt function to its gaussian part given by three Wick contractions. This ratio reveals significant non-gaussianity of the critical fluctuations. The bootstrap results are consistent with a rigorous inequality due to Lebowitz and Aizenman, which limits Q to lie between 1/3 and 1.

Ontology / Topics

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3d Ising model Conformal bootstrap Ising model

Published as SciPost Phys. 2, 001 (2017)

Submission & Refereeing History

Published as SciPost Phys. 2, 001 (2017)

Reports on this Submission

Anonymous Report 2 on 2017-1-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1612.02436v2, delivered 2017-01-30, doi: 10.21468/SciPost.Report.72


1- interesting subject
2- high precision numerical analysis
3- interesting relation with lattice results


1- too concise on some technical aspects


The paper under reviewal examines the four point function of spin fields in the critical 3d Ising model by employing the most recent numerical techniques and results found from through the conformal bootstrap approach.

In particular, the author studies the ratio between this four point function and the same quantity if the Wick theorem could be applied in this case. The deviation of this ratio from 1 tells about the non-gaussianity of the model.

The main result is shown in Fig. 1, which is very suggestive, given its qualitative similarity with respect to the result of the same analysis for the 2d critical Ising model. The authors also verify that their result is compatible with certain inequalities found by Lebowitz and Aizenman through a lattice analysis.

The paper is very interesting and well written. The results are presented in a clear way and I am sure that they will lead to further
insights in the analysis of the critical 3d Ising model.

I definitely support the publication of the paper in the present form.

Requested changes

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  • validity: high
  • significance: high
  • originality: high
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  • formatting: excellent
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Anonymous Report 1 on 2017-1-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1612.02436v2, delivered 2017-01-22, doi: 10.21468/SciPost.Report.68


1-The non-Gaussianity of the 3D Ising point is a subject of fundamental importance
2-The authors provide a very clear and quantitative measure of the non-Gaussianity of the 3D Ising universality class


1-From a technical point of view, the results presented in this short note represent a straightforward application of previous results


In this note the authors apply their previous results on the 4pt spin correlation function, obtained by conformal boostrap techniques, to measure the non-Gaussianity of the 3D Ising model. The authors presents in a very clear way the outcome of their analysis and its interpretation. I believe that the plots encoding the non-Gaussianity of 3D critical Ising model will appear on the desk of many theoretical physicists and not only.
I appreciated also the comparison of the conformal bootstrap results to rigorous inequalities obtained from the lattice model, that are nicely re-derived in the Appendix.

Requested changes

No changes are required

  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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