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Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension

by Rajat K. Panda, Roberto Verdel, Alex Rodriguez, Hanlin Sun, Ginestra Bianconi, Marcello Dalmonte

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Submission summary

Authors (as registered SciPost users): Marcello Dalmonte · Rajat Panda · Roberto Verdel
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-11-20
Date submitted: 2023-11-13 14:16
Submitted by: Panda, Rajat
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Statistical and Soft Matter Physics
Approach: Computational


We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract -- through a conventional finite-size scaling analysis with modest lattice sizes -- the critical temperature with less than $1\%$ error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.

Author comments upon resubmission

Dear Editor,

We thank you for handling our manuscript and for communicating to us the Referee report. We are also grateful to the Referee for their time in carefully reading our manuscript and providing valuable feedback. We find all points raised by the Referee valid and accurate. We believe that, after addressing the requested changes, our work has been improved.

Below, we provide a summary of the changes made in our revised manuscript.

List of changes

1. A discussion about the effect of increasing the number of batches in the computation of our estimates has been added in Appendix B.
2. An additional discussion has been added in Sec. 4, explaining the size of the error bars in Fig. 5b, according to the reply to the Referee question below.
3. A short explanatory discussion has been added in Sec. 4 on accounting for the interpolation error and estimating the error bars in $T^*$.

Published as SciPost Phys. Core 6, 086 (2023)

Reports on this Submission

Report 1 by Biagio Lucini on 2023-11-13 (Invited Report)

  • Cite as: Biagio Lucini, Report on arXiv:2308.13636v3, delivered 2023-11-13, doi: 10.21468/SciPost.Report.8108


(1) The paper provides a robust analysis of phase transitions using two different unsupervised machine learning methods.
(2) The paper contrasts strengths and weaknesses of the two approaches in a sound way.
(3) While the paper uses the Ising model as a test system, conclusions seem generalisable, with the results possibly opening new avenues of investigation in Statistical Mechanics and Quantum Field Theory at finite temperature.


In my opinion, the paper does not have any weaknesses.


The authors have addressed all the points I have raised in my first report. As the paper easily meets the criteria for acceptance of this journal and is highly relevant, I recommend acceptance of the manuscript in its current form.

Requested changes


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

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