Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: Quantitative and interpretable order parameters for phase transitions from persistent homology

Abstract

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.

Authors:
 [1];  [2];  [2]
+ Show Author Affiliations
  1. Univ. of Amsterdam (Netherlands)
  2. Univ. of Wisconsin, Madison, WI (United States)
Publication Date:
Research Org.:
Univ. of Wisconsin, Madison, WI (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1852393
Grant/Contract Number:  
SC0017647
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review. B
Additional Journal Information:
Journal Volume: 104; Journal Issue: 10; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
36 MATERIALS SCIENCE; materials science; physics; BKT transition; machine learning; spin lattice models; topology

Citation Formats

Cole, Alex, Loges, Gregory J., and Shiu, Gary. Quantitative and interpretable order parameters for phase transitions from persistent homology. United States: N. p., 2021. Web. doi:10.1103/physrevb.104.104426.
Copy to clipboard
Cole, Alex, Loges, Gregory J., & Shiu, Gary. Quantitative and interpretable order parameters for phase transitions from persistent homology. United States. https://doi.org/10.1103/physrevb.104.104426
Copy to clipboard
Cole, Alex, Loges, Gregory J., and Shiu, Gary. Tue . "Quantitative and interpretable order parameters for phase transitions from persistent homology". United States. https://doi.org/10.1103/physrevb.104.104426. https://www.osti.gov/servlets/purl/1852393.
Copy to clipboard
@article{osti_1852393,
title = {Quantitative and interpretable order parameters for phase transitions from persistent homology},
author = {Cole, Alex and Loges, Gregory J. and Shiu, Gary},
abstractNote = {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.},
doi = {10.1103/physrevb.104.104426},
journal = {Physical Review. B},
number = 10,
volume = 104,
place = {United States},
year = {Tue Sep 28 00:00:00 EDT 2021},
month = {Tue Sep 28 00:00:00 EDT 2021}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1103/physrevb.104.104426
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Computing Persistent Homology
journal, November 2004

Inferring hidden symmetries of exotic magnets from detecting explicit order parameters
journal, July 2021

Unsupervised Machine Learning and Band Topology
journal, June 2020

Finite-size scaling study of the equilibrium cluster distribution of the two-dimensional Ising model
journal, October 1987

Topological data analysis for the string landscape
journal, March 2019

Learning multiple order parameters with interpretable machines
journal, March 2019

Multicritical behaviour in the fully frustrated XY model and related systems
journal, December 2005

  • Hasenbusch, Martin; Pelissetto, Andrea; Vicari, Ettore
  • Journal of Statistical Mechanics: Theory and Experiment, Vol. 2005, Issue 12
  • DOI: 10.1088/1742-5468/2005/12/P12002

Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
journal, February 1944

Droplet theory in low dimensions: Ising systems in zero field
journal, June 1983

Persistent homology and non-Gaussianity
journal, March 2018

Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory
journal, November 2017

Machine learning vortices at the Kosterlitz-Thouless transition
journal, January 2018

Machine Learning Phases of Strongly Correlated Fermions
journal, August 2017

Persistent homology analysis of phase transitions
journal, May 2016

Interpretable and unsupervised phase classification
journal, July 2021

On the Local Behavior of Spaces of Natural Images
journal, June 2007

  • Carlsson, Gunnar; Ishkhanov, Tigran; de Silva, Vin
  • International Journal of Computer Vision, Vol. 76, Issue 1
  • DOI: 10.1007/s11263-007-0056-x

Static properties of 2D spin-ice as a sixteen-vertex model
journal, February 2013

Phase detection with neural networks: interpreting the black box
journal, November 2020

The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study
journal, June 2005

Machine learning for quantum matter
journal, January 2020

Finding self-similar behavior in quantum many-body dynamics via persistent homology
journal, January 2021

Discovering phase transitions with unsupervised learning
journal, November 2016

Machine Learning as a universal tool for quantitative investigations of phase transitions
journal, July 2019

The persistence of large scale structures. Part I. Primordial non-Gaussianity
journal, April 2021

  • Biagetti, Matteo; Cole, Alex; Shiu, Gary
  • Journal of Cosmology and Astroparticle Physics, Vol. 2021, Issue 04
  • DOI: 10.1088/1475-7516/2021/04/061

Identifying topological order through unsupervised machine learning
journal, May 2019

Identification of emergent constraints and hidden order in frustrated magnets using tensorial kernel methods of machine learning
journal, November 2019

Topology of viral evolution
journal, October 2013

  • Chan, J. M.; Carlsson, G.; Rabadan, R.
  • Proceedings of the National Academy of Sciences, Vol. 110, Issue 46
  • DOI: 10.1073/pnas.1313480110

Finding hidden order in spin models with persistent homology
journal, December 2020

Topological phase transitions in functional brain networks
journal, September 2019

  • Santos, Fernando A. N.; Raposo, Ernesto P.; Coutinho-Filho, Maurício D.
  • Physical Review E, Vol. 100, Issue 3
  • DOI: 10.1103/PhysRevE.100.032414

Interpreting machine learning of topological quantum phase transitions
journal, June 2020

Topological persistence machine of phase transitions
journal, May 2021

Coverage in sensor networks via persistent homology
journal, January 2007

Identifying quantum phase transitions with adversarial neural networks
journal, April 2018

Persistent homology analysis of protein structure, flexibility, and folding: PERSISTENT HOMOLOGY FOR PROTEIN
journal, June 2014

  • Xia, Kelin; Wei, Guo-Wei
  • International Journal for Numerical Methods in Biomedical Engineering, Vol. 30, Issue 8
  • DOI: 10.1002/cnm.2655

A topological measurement of protein compressibility
journal, October 2014

  • Gameiro, Marcio; Hiraoka, Yasuaki; Izumi, Shunsuke
  • Japan Journal of Industrial and Applied Mathematics, Vol. 32, Issue 1
  • DOI: 10.1007/s13160-014-0153-5

Monte Carlo determination of the critical temperature for the two-dimensional XY model
journal, April 1988

Detection of Phase Transition via Convolutional Neural Networks
journal, June 2017

  • Tanaka, Akinori; Tomiya, Akio
  • Journal of the Physical Society of Japan, Vol. 86, Issue 6
  • DOI: 10.7566/JPSJ.86.063001

Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination
journal, June 2017

Learning phase transitions by confusion
journal, February 2017

  • van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D.
  • Nature Physics, Vol. 13, Issue 5
  • DOI: 10.1038/nphys4037

Towards novel insights in lattice field theory with explainable machine learning
journal, May 2020

Machine learning phases of matter
journal, February 2017

  • Carrasquilla, Juan; Melko, Roger G.
  • Nature Physics, Vol. 13, Issue 5
  • DOI: 10.1038/nphys4035

The importance of the whole: Topological data analysis for the network neuroscientist
journal, January 2019

  • Sizemore, Ann E.; Phillips-Cremins, Jennifer E.; Ghrist, Robert
  • Network Neuroscience, Vol. 3, Issue 3
  • DOI: 10.1162/netn_a_00073

Persistent homology and string vacua
journal, March 2016

Machine learning for quantum matter
journal, January 2020

The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study
journal, June 2005

The importance of the whole: Topological data analysis for the network neuroscientist
journal, January 2019

  • Sizemore, Ann E.; Phillips-Cremins, Jennifer E.; Ghrist, Robert
  • Network Neuroscience, Vol. 3, Issue 3
  • DOI: 10.1162/netn_a_00073

Machine Learning as a universal tool for quantitative investigations of phase transition
text, January 2018

Finding self-similar behavior in quantum many-body dynamics via persistent homology
text, January 2020

    Sort by:
    [ × clear filter / sort ]

    Similar Records in DOE PAGES and OSTI.GOV collections: