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Unsupervised machine learning approaches to the q-state Potts model

  • Regular Article - Statistical and Nonlinear Physics
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Abstract

In this paper, we study phase transitions of the q-state Potts model through a number of unsupervised machine learning techniques, namely Principal Component Analysis (PCA), k-means clustering, Uniform Manifold Approximation and Projection (UMAP), and Topological Data Analysis (TDA). Even though in all cases we are able to retrieve the correct critical temperatures \(T_\textrm{c}(q)\), for \(q=3,4\) and 5, results show that non-linear methods as UMAP and TDA are less dependent on finite-size effects. This study may be considered as a benchmark for the use of different unsupervised machine learning algorithms in the investigation of phase transitions.

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Data availability statement

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. At this point, we note that when feeding PCA, instead of providing \(\mathbf {\sigma }= [n_1, n_2, \ldots ]\), we provide their directions, i.e., \(\mathbf {\sigma }= [(\cos \theta _1, \sin \theta _1), (\cos \theta _2, \sin \theta _2), \ldots ]\). Thus, our dataset matrix has \(2\times L^2\) columns. We emphasize that such a change does not affect the final results, just the spacial position (patterns) of the clusters in PCA space.

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Acknowledgements

A.T. acknowledges financial support from the MIUR Progetti di Ricerca di Rilevante Interesse Nazionale (PRIN) Bando 2017 - grant 2017BZPKSZ. A.T. and N.C.C. acknowledge CINECA for awarding them access to the Marconi100 supercomputer, through the ISCRA framework, within the projects AI-H-QMC - HP10BGJH1X, and IsB23 (ISCRA-HP10BF65I0). N.C.C. acknowledges financial support from the Brazilian Agency National Council for Scientific and Technological Development (CNPq), grant number 313065/2021-7. D.O.C., L.A.O., J.P.L., N.C.C., and R.R.d.S. are grateful to the Brazilian Agencies CNPq, National Council for the Improvement of Higher Education (CAPES), FAPERJ, and FAPEPI for partially funding this project.

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Authors and Affiliations

  1. International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy

    Andrea Tirelli & Natanael C. Costa

  2. Departamento de Física, Universidade Federal do Piauí, Teresina, PI, 64049-550, Brazil

    Danyella O. Carvalho, Lucas A. Oliveira & José P. de Lima

  3. Instituto de Física, Universidade Federal do Rio de Janeiro, Cx.P. 68.528, Rio de Janeiro, RJ, 21941-972, Brazil

    Lucas A. Oliveira, Natanael C. Costa & Raimundo R. dos Santos

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  1. Andrea Tirelli
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Contributions

NCC, RRS, and AT conceived the project. DOC and LAO carried out Monte Carlo simulations of the Potts model under the guidance of NCC and RRS. The codes and the results for PCA and k-means were implemented/obtained by DOC, LAO, and JPL, while AT implemented the codes and obtained results for the TDA and UMAP ones. AT and NCC wrote the manuscript, and all authors participated in the discussions during the writing process.

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Correspondence to Andrea Tirelli.

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Tirelli, A., Carvalho, D.O., Oliveira, L.A. et al. Unsupervised machine learning approaches to the q-state Potts model. Eur. Phys. J. B 95, 189 (2022). https://doi.org/10.1140/epjb/s10051-022-00453-3

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