Quantitative and interpretable order parameters for phase transitions from persistent homology

Cole, Alex; Loges, Gregory J.; Shiu, Gary

doi:10.1103/physrevb.104.104426

Quantitative and interpretable order parameters for phase transitions from persistent homology

Journal Article · Tue Sep 28 00:00:00 EDT 2021 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.104.104426· OSTI ID:1852393

Cole, Alex ^[1]; Loges, Gregory J. ^[2]; Shiu, Gary ^[2]

Univ. of Amsterdam (Netherlands); OSTI
Univ. of Wisconsin, Madison, WI (United States)

Here, we apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define “persistence” critical exponents and study how they are related to those critical exponents usually considered.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Wisconsin, Madison, WI (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0017647

OSTI ID:: 1852393

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 10 Vol. 104; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (46)

Machine learning for quantum matter Carrasquilla, Juan Advances in Physics: X, Vol. 5, Issue 1 https://doi.org/10.1080/23746149.2020.1797528	journal	January 2020
The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study Hasenbusch, Martin Journal of Physics A: Mathematical and General, Vol. 38, Issue 26 https://doi.org/10.1088/0305-4470/38/26/003	journal	June 2005
The importance of the whole: Topological data analysis for the network neuroscientist Sizemore, Ann E.; Phillips-Cremins, Jennifer E.; Ghrist, Robert Network Neuroscience, Vol. 3, Issue 3 https://doi.org/10.1162/netn_a_00073	journal	January 2019
Discovering Phases, Phase Transitions and Crossovers through Unsupervised Machine Learning: A critical examination Hu, Wenjian; Singh, Rajiv R. P.; Scalettar, Richard T. arXiv https://doi.org/10.48550/arxiv.1704.00080	text	January 2017
Machine Learning as a universal tool for quantitative investigations of phase transition Giannetti, Cinzia; Lucini, Biagio; Vadacchino, Davide arXiv https://doi.org/10.48550/arxiv.1812.06726	text	January 2018
Finding self-similar behavior in quantum many-body dynamics via persistent homology Spitz, Daniel; Berges, Jürgen; Oberthaler, Markus K. arXiv https://doi.org/10.48550/arxiv.2001.02616	text	January 2020
Persistent homology analysis of protein structure, flexibility, and folding: PERSISTENT HOMOLOGY FOR PROTEIN Xia, Kelin; Wei, Guo-Wei International Journal for Numerical Methods in Biomedical Engineering, Vol. 30, Issue 8 https://doi.org/10.1002/cnm.2655	journal	June 2014
Persistent homology and string vacua Cirafici, Michele Journal of High Energy Physics, Vol. 2016, Issue 3 https://doi.org/10.1007/JHEP03(2016)045	journal	March 2016
Topological data analysis for the string landscape Cole, Alex; Shiu, Gary Journal of High Energy Physics, Vol. 2019, Issue 3 https://doi.org/10.1007/JHEP03(2019)054	journal	March 2019
Computing Persistent Homology Zomorodian, Afra; Carlsson, Gunnar Discrete & Computational Geometry, Vol. 33, Issue 2 https://doi.org/10.1007/s00454-004-1146-y	journal	November 2004
On the Local Behavior of Spaces of Natural Images Carlsson, Gunnar; Ishkhanov, Tigran; de Silva, Vin International Journal of Computer Vision, Vol. 76, Issue 1 https://doi.org/10.1007/s11263-007-0056-x	journal	June 2007
A topological measurement of protein compressibility Gameiro, Marcio; Hiraoka, Yasuaki; Izumi, Shunsuke Japan Journal of Industrial and Applied Mathematics, Vol. 32, Issue 1 https://doi.org/10.1007/s13160-014-0153-5	journal	October 2014
Machine Learning as a universal tool for quantitative investigations of phase transitions Giannetti, Cinzia; Lucini, Biagio; Vadacchino, Davide Nuclear Physics B, Vol. 944 https://doi.org/10.1016/j.nuclphysb.2019.114639	journal	July 2019
Machine learning phases of matter Carrasquilla, Juan; Melko, Roger G. Nature Physics, Vol. 13, Issue 5 https://doi.org/10.1038/nphys4035	journal	February 2017
Learning phase transitions by confusion van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D. Nature Physics, Vol. 13, Issue 5 https://doi.org/10.1038/nphys4037	journal	February 2017
Identifying topological order through unsupervised machine learning Rodriguez-Nieva, Joaquin F.; Scheurer, Mathias S. Nature Physics, Vol. 15, Issue 8 https://doi.org/10.1038/s41567-019-0512-x	journal	May 2019
Topology of viral evolution Chan, J. M.; Carlsson, G.; Rabadan, R. Proceedings of the National Academy of Sciences, Vol. 110, Issue 46 https://doi.org/10.1073/pnas.1313480110	journal	October 2013
Droplet theory in low dimensions: Ising systems in zero field Bruce, A. D.; Wallace, D. J. Journal of Physics A: Mathematical and General, Vol. 16, Issue 8 https://doi.org/10.1088/0305-4470/16/8/018	journal	June 1983
Finite-size scaling study of the equilibrium cluster distribution of the two-dimensional Ising model Toral, R.; Wall, C. Journal of Physics A: Mathematical and General, Vol. 20, Issue 14 https://doi.org/10.1088/0305-4470/20/14/032	journal	October 1987
Phase detection with neural networks: interpreting the black box Dawid, Anna; Huembeli, Patrick; Tomza, Michal New Journal of Physics, Vol. 22, Issue 11 https://doi.org/10.1088/1367-2630/abc463	journal	November 2020
Persistent homology and non-Gaussianity Cole, Alex; Shiu, Gary Journal of Cosmology and Astroparticle Physics, Vol. 2018, Issue 03 https://doi.org/10.1088/1475-7516/2018/03/025	journal	March 2018
The persistence of large scale structures. Part I. Primordial non-Gaussianity Biagetti, Matteo; Cole, Alex; Shiu, Gary Journal of Cosmology and Astroparticle Physics, Vol. 2021, Issue 04 https://doi.org/10.1088/1475-7516/2021/04/061	journal	April 2021
Multicritical behaviour in the fully frustrated XY model and related systems Hasenbusch, Martin; Pelissetto, Andrea; Vicari, Ettore Journal of Statistical Mechanics: Theory and Experiment, Vol. 2005, Issue 12 https://doi.org/10.1088/1742-5468/2005/12/P12002	journal	December 2005
Static properties of 2D spin-ice as a sixteen-vertex model Foini, Laura; Levis, Demian; Tarzia, Marco Journal of Statistical Mechanics: Theory and Experiment, Vol. 2013, Issue 02 https://doi.org/10.1088/1742-5468/2013/02/P02026	journal	February 2013
Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition Onsager, Lars Physical Review, Vol. 65, Issue 3-4 https://doi.org/10.1103/PhysRev.65.117	journal	February 1944
Identification of emergent constraints and hidden order in frustrated magnets using tensorial kernel methods of machine learning Greitemann, Jonas; Liu, Ke; Jaubert, Ludovic D. C. Physical Review B, Vol. 100, Issue 17 https://doi.org/10.1103/PhysRevB.100.174408	journal	November 2019
Monte Carlo determination of the critical temperature for the two-dimensional XY model Weber, Hans; Minnhagen, Petter Physical Review B, Vol. 37, Issue 10 https://doi.org/10.1103/PhysRevB.37.5986	journal	April 1988
Discovering phase transitions with unsupervised learning Wang, Lei Physical Review B, Vol. 94, Issue 19 https://doi.org/10.1103/PhysRevB.94.195105	journal	November 2016
Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory Wetzel, Sebastian J.; Scherzer, Manuel Physical Review B, Vol. 96, Issue 18 https://doi.org/10.1103/PhysRevB.96.184410	journal	November 2017
Machine learning vortices at the Kosterlitz-Thouless transition Beach, Matthew J. S.; Golubeva, Anna; Melko, Roger G. Physical Review B, Vol. 97, Issue 4 https://doi.org/10.1103/PhysRevB.97.045207	journal	January 2018
Identifying quantum phase transitions with adversarial neural networks Huembeli, Patrick; Dauphin, Alexandre; Wittek, Peter Physical Review B, Vol. 97, Issue 13 https://doi.org/10.1103/PhysRevB.97.134109	journal	April 2018
Learning multiple order parameters with interpretable machines Liu, Ke; Greitemann, Jonas; Pollet, Lode Physical Review B, Vol. 99, Issue 10 https://doi.org/10.1103/PhysRevB.99.104410	journal	March 2019
Towards novel insights in lattice field theory with explainable machine learning Blücher, Stefan; Kades, Lukas; Pawlowski, Jan M. Physical Review D, Vol. 101, Issue 9 https://doi.org/10.1103/PhysRevD.101.094507	journal	May 2020
Topological phase transitions in functional brain networks Santos, Fernando A. N.; Raposo, Ernesto P.; Coutinho-Filho, Maurício D. Physical Review E, Vol. 100, Issue 3 https://doi.org/10.1103/PhysRevE.100.032414	journal	September 2019
Topological persistence machine of phase transitions Tran, Quoc Hoan; Chen, Mark; Hasegawa, Yoshihiko Physical Review E, Vol. 103, Issue 5 https://doi.org/10.1103/PhysRevE.103.052127	journal	May 2021
Inferring hidden symmetries of exotic magnets from detecting explicit order parameters Rao, Nihal; Liu, Ke; Pollet, Lode Physical Review E, Vol. 104, Issue 1 https://doi.org/10.1103/PhysRevE.104.015311	journal	July 2021
Persistent homology analysis of phase transitions Donato, Irene; Gori, Matteo; Pettini, Marco Physical Review E, Vol. 93, Issue 5 https://doi.org/10.1103/PhysRevE.93.052138	journal	May 2016
Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination Hu, Wenjian; Singh, Rajiv R. P.; Scalettar, Richard T. Physical Review E, Vol. 95, Issue 6 https://doi.org/10.1103/PhysRevE.95.062122	journal	June 2017
Unsupervised Machine Learning and Band Topology Scheurer, Mathias S.; Slager, Robert-Jan Physical Review Letters, Vol. 124, Issue 22 https://doi.org/10.1103/PhysRevLett.124.226401	journal	June 2020
Interpreting machine learning of topological quantum phase transitions Zhang, Yi; Ginsparg, Paul; Kim, Eun-Ah Physical Review Research, Vol. 2, Issue 2 https://doi.org/10.1103/PhysRevResearch.2.023283	journal	June 2020
Finding hidden order in spin models with persistent homology Olsthoorn, Bart; Hellsvik, Johan; Balatsky, Alexander V. Physical Review Research, Vol. 2, Issue 4 https://doi.org/10.1103/PhysRevResearch.2.043308	journal	December 2020
Interpretable and unsupervised phase classification Arnold, Julian; Schäfer, Frank; Žonda, Martin Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033052	journal	July 2021
Machine Learning Phases of Strongly Correlated Fermions Ch’ng, Kelvin; Carrasquilla, Juan; Melko, Roger G. Physical Review X, Vol. 7, Issue 3 https://doi.org/10.1103/PhysRevX.7.031038	journal	August 2017
Coverage in sensor networks via persistent homology de Silva, Vin; Ghrist, Robert Algebraic & Geometric Topology, Vol. 7, Issue 1 https://doi.org/10.2140/agt.2007.7.339	journal	January 2007
Finding self-similar behavior in quantum many-body dynamics via persistent homology Spitz, Daniel; Berges, Jürgen; Oberthaler, Markus SciPost Physics, Vol. 11, Issue 3 https://doi.org/10.21468/SciPostPhys.11.3.060	journal	January 2021
Detection of Phase Transition via Convolutional Neural Networks Tanaka, Akinori; Tomiya, Akio Journal of the Physical Society of Japan, Vol. 86, Issue 6 https://doi.org/10.7566/JPSJ.86.063001	journal	June 2017

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