Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination
Abstract
Here, we apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models - the square and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-one Ising (BSI) model, and the 2D XY model, and examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow exploration of different phases and symmetry-breaking, but can distinguish phase transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the 'charge' correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the 'antoencoder method', and demonstrate that it toomore »
- Authors:
-
- Univ. of California, Davis, CA (United States). Dept. of Physics and Dept. of Computer Science
- Univ. of California, Davis, CA (United States). Dept. of Physics
- Publication Date:
- Research Org.:
- Univ. of California, Davis, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
- OSTI Identifier:
- 1368102
- Grant/Contract Number:
- NA0002908; DMR-1306048
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review E
- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 6; Journal ID: ISSN 2470-0045
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; phase transitions; data analysis
Citation Formats
Hu, Wenjian, Singh, Rajiv R. P., and Scalettar, Richard T. Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination. United States: N. p., 2017.
Web. doi:10.1103/PhysRevE.95.062122.
Hu, Wenjian, Singh, Rajiv R. P., & Scalettar, Richard T. Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination. United States. https://doi.org/10.1103/PhysRevE.95.062122
Hu, Wenjian, Singh, Rajiv R. P., and Scalettar, Richard T. Mon .
"Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination". United States. https://doi.org/10.1103/PhysRevE.95.062122. https://www.osti.gov/servlets/purl/1368102.
@article{osti_1368102,
title = {Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination},
author = {Hu, Wenjian and Singh, Rajiv R. P. and Scalettar, Richard T.},
abstractNote = {Here, we apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models - the square and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-one Ising (BSI) model, and the 2D XY model, and examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow exploration of different phases and symmetry-breaking, but can distinguish phase transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the 'charge' correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the 'antoencoder method', and demonstrate that it too can be trained to capture phase transitions and critical points.},
doi = {10.1103/PhysRevE.95.062122},
journal = {Physical Review E},
number = 6,
volume = 95,
place = {United States},
year = {Mon Jun 19 00:00:00 EDT 2017},
month = {Mon Jun 19 00:00:00 EDT 2017}
}
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Works referenced in this record:
Quantum Entanglement in Neural Network States
journal, May 2017
- Deng, Dong-Ling; Li, Xiaopeng; Das Sarma, S.
- Physical Review X, Vol. 7, Issue 2
Auto-association by multilayer perceptrons and singular value decomposition
journal, September 1988
- Bourlard, H.; Kamp, Y.
- Biological Cybernetics, Vol. 59, Issue 4-5
Interplay of quantum and thermal fluctuations in a frustrated magnet
journal, September 2003
- Isakov, S. V.; Moessner, R.
- Physical Review B, Vol. 68, Issue 10
Antiferromagnetism. The Triangular Ising Net
journal, July 1950
- Wannier, G. H.
- Physical Review, Vol. 79, Issue 2
LIII. On lines and planes of closest fit to systems of points in space
journal, November 1901
- Pearson, Karl
- The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 2, Issue 11
The critical properties of the two-dimensional xy model
journal, March 1974
- Kosterlitz, J. M.
- Journal of Physics C: Solid State Physics, Vol. 7, Issue 6
Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model
journal, May 2007
- Cincio, Lukasz; Dziarmaga, Jacek; Rams, Marek M.
- Physical Review A, Vol. 75, Issue 5
Orderings of a stacked frustrated triangular system in three dimensions
journal, May 1984
- Blankschtein, Daniel; Ma, M.; Berker, A. Nihat
- Physical Review B, Vol. 29, Issue 9
Applications of Monte Carlo methods to statistical physics
journal, May 1997
- Binder, K.
- Reports on Progress in Physics, Vol. 60, Issue 5
Phase diagrams and critical behavior in Ising square lattices with nearest- and next-nearest-neighbor interactions
journal, March 1980
- Binder, K.; Landau, D. P.
- Physical Review B, Vol. 21, Issue 5
Discovering phase transitions with unsupervised learning
journal, November 2016
- Wang, Lei
- Physical Review B, Vol. 94, Issue 19
Scaling and Renormalization in Statistical Physics
book, January 2015
- Cardy, John
- Cambridge University Press
Monte Carlo study of the fcc Blume-Capel model
journal, July 1980
- Jain, A. K.; Landau, D. P.
- Physical Review B, Vol. 22, Issue 1
Theory of the First-Order Magnetic Phase Change in U
journal, January 1966
- Blume, M.
- Physical Review, Vol. 141, Issue 2
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
Phase diagrams of the spin-1 Blume-Capel film with an alternating crystal field
journal, February 2004
- Ez-Zahraouy, Hamid; Kassou-Ou-Ali, Ahmed
- Physical Review B, Vol. 69, Issue 6
Ising model for tricritical points in ternary mixtures
journal, August 1974
- Mukamel, D.; Blume, M.
- Physical Review A, Vol. 10, Issue 2
Machine Learning for Detecting Gene-Gene Interactions: A Review
journal, January 2006
- McKinney, Brett A.; Reif, David M.; Ritchie, Marylyn D.
- Applied Bioinformatics, Vol. 5, Issue 2
Zinc-blende—diamond order-disorder transition in metastable crystalline alloys
journal, June 1983
- Newman, Kathie E.; Dow, John D.
- Physical Review B, Vol. 27, Issue 12
Reducing the Dimensionality of Data with Neural Networks
journal, July 2006
- Hinton, G. E.
- Science, Vol. 313, Issue 5786
Ising‐Model Spin Correlations on the Triangular Lattice
journal, August 1964
- Stephenson, John
- Journal of Mathematical Physics, Vol. 5, Issue 8
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
Solving the quantum many-body problem with artificial neural networks
journal, February 2017
- Carleo, Giuseppe; Troyer, Matthias
- Science, Vol. 355, Issue 6325
Spin-1 lattice-gas model. I. Condensation and solidification of a simple fluid
journal, June 1975
- Lajzerowicz, Joseph; Sivardière, Jean
- Physical Review A, Vol. 11, Issue 6
Machine learning phases of matter
journal, February 2017
- Carrasquilla, Juan; Melko, Roger G.
- Nature Physics, Vol. 13, Issue 5
Identifying polymer states by machine learning
journal, March 2017
- Wei, Qianshi; Melko, Roger G.; Chen, Jeff Z. Y.
- Physical Review E, Vol. 95, Issue 3
Monte Carlo analysis of the two-dimensional XY model. II. Comparison with the Kosterlitz renormalization-group equations
journal, August 1995
- Olsson, Peter
- Physical Review B, Vol. 52, Issue 6
Self-learning Monte Carlo method
journal, January 2017
- Liu, Junwei; Qi, Yang; Meng, Zi Yang
- Physical Review B, Vol. 95, Issue 4
On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting
journal, May 1966
- Capel, H. W.
- Physica, Vol. 32, Issue 5
Colloquium : Area laws for the entanglement entropy
journal, February 2010
- Eisert, J.; Cramer, M.; Plenio, M. B.
- Reviews of Modern Physics, Vol. 82, Issue 1
The link-prediction problem for social networks
journal, January 2007
- Liben-Nowell, David; Kleinberg, Jon
- Journal of the American Society for Information Science and Technology, Vol. 58, Issue 7, p. 1019-1031
Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
journal, February 1944
- Onsager, Lars
- Physical Review, Vol. 65, Issue 3-4
Entanglement in a simple quantum phase transition
journal, September 2002
- Osborne, Tobias J.; Nielsen, Michael A.
- Physical Review A, Vol. 66, Issue 3
First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach
journal, August 2015
- Kwak, Wooseop; Jeong, Joohyeok; Lee, Juhee
- Physical Review E, Vol. 92, Issue 2
Statistical Mechanics of the Model. II. Spin-Correlation Functions
journal, February 1971
- Barouch, Eytan; McCoy, Barry M.
- Physical Review A, Vol. 3, Issue 2
A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice
journal, November 2009
- Bakr, Waseem S.; Gillen, Jonathon I.; Peng, Amy
- Nature, Vol. 462, Issue 7269
Single-atom-resolved fluorescence imaging of an atomic Mott insulator
journal, August 2010
- Sherson, Jacob F.; Weitenberg, Christof; Endres, Manuel
- Nature, Vol. 467, Issue 7311
Dynamics of rough surfaces generated by two-dimensional lattice spin models
journal, April 2007
- Brito, A. Faissal; Redinz, José Arnaldo; Plascak, J. A.
- Physical Review E, Vol. 75, Issue 4
Between order and chaos
journal, December 2011
- Crutchfield, James P.
- Nature Physics, Vol. 8, Issue 1
Numerical linked-cluster algorithms. I. Spin systems on square, triangular, and kagomé lattices
journal, June 2007
- Rigol, Marcos; Bryant, Tyler; Singh, Rajiv R. P.
- Physical Review E, Vol. 75, Issue 6
Learning To Fold Proteins Using Energy Landscape Theory
journal, August 2014
- Schafer, Nicholas P.; Kim, Bobby L.; Zheng, Weihua
- Israel Journal of Chemistry, Vol. 54, Issue 8-9
Collective Monte Carlo Updating for Spin Systems
journal, January 1989
- Wolff, Ulli
- Physical Review Letters, Vol. 62, Issue 4
Behavior of damage spreading in the two-dimensional Blume-Capel model
journal, April 2002
- Liu, Ce-Jun; Schüttler, H. -B.
- Physical Review E, Vol. 65, Issue 5
Learning thermodynamics with Boltzmann machines
journal, October 2016
- Torlai, Giacomo; Melko, Roger G.
- Physical Review B, Vol. 94, Issue 16
Ising Model for the Transition and Phase Separation in - Mixtures
journal, September 1971
- Blume, M.; Emery, V. J.; Griffiths, Robert B.
- Physical Review A, Vol. 4, Issue 3
A Fast Learning Algorithm for Deep Belief Nets
journal, July 2006
- Hinton, Geoffrey E.; Osindero, Simon; Teh, Yee-Whye
- Neural Computation, Vol. 18, Issue 7
Monte Carlo study of an inhomogeneous Blume-Capel model: A case study of the local density approximation
journal, December 2008
- Pittman, S. M.; Batrouni, G. G.; Scalettar, R. T.
- Physical Review B, Vol. 78, Issue 21
Face recognition: a convolutional neural-network approach
journal, January 1997
- Lawrence, S.; Giles, C. L.
- IEEE Transactions on Neural Networks, Vol. 8, Issue 1
Deep learning in neural networks: An overview
journal, January 2015
- Schmidhuber, Jürgen
- Neural Networks, Vol. 61
Following a moving target-Monte Carlo inference for dynamic Bayesian models
journal, February 2001
- Gilks, Walter R.; Berzuini, Carlo
- Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 63, Issue 1
Single-atom imaging of fermions in a quantum-gas microscope
journal, July 2015
- Haller, Elmar; Hudson, James; Kelly, Andrew
- Nature Physics, Vol. 11, Issue 9
Business data mining — a machine learning perspective
journal, December 2001
- Bose, Indranil; Mahapatra, Radha K.
- Information & Management, Vol. 39, Issue 3
A structural approach to relaxation in glassy liquids
journal, February 2016
- Schoenholz, S. S.; Cubuk, E. D.; Sussman, D. M.
- Nature Physics, Vol. 12, Issue 5
Ising‐Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice
journal, February 1970
- Stephenson, John
- Journal of Mathematical Physics, Vol. 11, Issue 2
Critical behavior of the spin- Blume-Capel model in two dimensions
journal, May 1998
- Xavier, J. C.; Alcaraz, F. C.; Lara, D. Penã
- Physical Review B, Vol. 57, Issue 18
Drug design by machine learning: the use of inductive logic programming to model the structure-activity relationships of trimethoprim analogues binding to dihydrofolate reductase.
journal, December 1992
- King, R. D.; Muggleton, S.; Lewis, R. A.
- Proceedings of the National Academy of Sciences, Vol. 89, Issue 23
PCA Meets RG
journal, March 2017
- Bradde, Serena; Bialek, William
- Journal of Statistical Physics, Vol. 167, Issue 3-4
Monte Carlo Statistical Methods
book, January 2004
- Robert, Christian P.; Casella, George
- Springer Texts in Statistics
Future possibilities for artificial intelligence in the practical management of hypertension
journal, July 2020
- Koshimizu, Hiroshi; Kojima, Ryosuke; Okuno, Yasushi
- Hypertension Research, Vol. 43, Issue 12
Antiferromagnetism. The Triangular Ising Net
journal, June 1973
- Wannier, G. H.
- Physical Review B, Vol. 7, Issue 11
Between Order and Chaos.
journal, January 2003
- Odell, James
- The Journal of Object Technology, Vol. 2, Issue 6
Single-atom imaging of fermions in a quantum-gas microscope
text, January 2015
- Haller, Elmar; Hudson, James; Kelly, Andrew
- arXiv
First-order phase transition and tricritical scaling behavior of the Blume-Capel model: a Wang-Landau sampling approach
text, January 2015
- Kwak, Wooseop; Jeong, Joohyeok; Lee, Juhee
- arXiv
Learning phase transitions by confusion
text, January 2016
- van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D.
- arXiv
Identifying polymer states by machine learning
text, January 2017
- Wei, Qianshi; Melko, Roger G.; Chen, Jeff Z. Y.
- arXiv
Quantum Entanglement in Neural Network States
text, January 2017
- Deng, Dong-Ling; Li, Xiaopeng; Sarma, S. Das
- arXiv
Interplay of quantum and thermal fluctuations in a frustrated magnet
text, January 2003
- Isakov, S. V.; Moessner, R.
- arXiv
Phase Diagrams of the Spin-1 Blume-Capel Film With an Alternating Crystal Field
text, January 2003
- Ez-Zahraouy, Hamid; Kassou-Ou-Ali, Ahmed
- arXiv
The Critical Behaviour of the Spin-3/2 Blume-Capel Model in Two Dimensions
text, January 1999
- Xavier, J. C.; Alcaraz, F. C.; Lara, D. Pena
- arXiv
Works referencing / citing this record:
Machine learning of frustrated classical spin models (II): Kernel principal component analysis
journal, June 2018
- Wang, Ce; Zhai, Hui
- Frontiers of Physics, Vol. 13, Issue 5
Identifying topological order through unsupervised machine learning
journal, May 2019
- Rodriguez-Nieva, Joaquin F.; Scheurer, Mathias S.
- Nature Physics, Vol. 15, Issue 8
Unsupervised machine learning for detection of phase transitions in off-lattice systems. I. Foundations
journal, November 2018
- Jadrich, R. B.; Lindquist, B. A.; Truskett, T. M.
- The Journal of Chemical Physics, Vol. 149, Issue 19
Unsupervised machine learning for detection of phase transitions in off-lattice systems. II. Applications
journal, November 2018
- Jadrich, R. B.; Lindquist, B. A.; Piñeros, W. D.
- The Journal of Chemical Physics, Vol. 149, Issue 19
Machine learning & artificial intelligence in the quantum domain: a review of recent progress
journal, June 2018
- Dunjko, Vedran; Briegel, Hans J.
- Reports on Progress in Physics, Vol. 81, Issue 7
From DFT to machine learning: recent approaches to materials science–a review
journal, May 2019
- Schleder, Gabriel R.; Padilha, Antonio C. M.; Acosta, Carlos Mera
- Journal of Physics: Materials, Vol. 2, Issue 3
Characterizing the phase diagram of finite-size dipolar Bose-Hubbard systems
journal, January 2020
- Rosson, Paolo; Kiffner, Martin; Mur-Petit, Jordi
- Physical Review A, Vol. 101, Issue 1
Accelerating lattice quantum Monte Carlo simulations using artificial neural networks: Application to the Holstein model
journal, July 2019
- Li, Shaozhi; Dee, Philip M.; Khatami, Ehsan
- Physical Review B, Vol. 100, Issue 2
Criticality and factorization in the Heisenberg chain with Dzyaloshinskii-Moriya interaction
journal, July 2019
- Yi, Tian-Cheng; You, Wen-Long; Wu, Ning
- Physical Review B, Vol. 100, Issue 2
Unsupervised learning eigenstate phases of matter
journal, August 2019
- Durr, Steven; Chakravarty, Sudip
- Physical Review B, Vol. 100, Issue 7
Real-space mapping of topological invariants using artificial neural networks
journal, March 2018
- Carvalho, D.; García-Martínez, N. A.; Lado, J. L.
- Physical Review B, Vol. 97, Issue 11
Extracting many-particle entanglement entropy from observables using supervised machine learning
journal, December 2018
- Berkovits, Richard
- Physical Review B, Vol. 98, Issue 24
Profile approach for recognition of three-dimensional magnetic structures
journal, January 2019
- Iakovlev, I. A.; Sotnikov, O. M.; Mazurenko, V. V.
- Physical Review B, Vol. 99, Issue 2
Self-organizing maps as a method for detecting phase transitions and phase identification
journal, January 2019
- Shirinyan, Albert A.; Kozin, Valerii K.; Hellsvik, Johan
- Physical Review B, Vol. 99, Issue 4
Smallest neural network to learn the Ising criticality
journal, August 2018
- Kim, Dongkyu; Kim, Dong-Hee
- Physical Review E, Vol. 98, Issue 2
Deriving the order parameters of a spin-glass model using principal component analysis
journal, June 2019
- Kiwata, Hirohito
- Physical Review E, Vol. 99, Issue 6
Machine Learning Detection of Bell Nonlocality in Quantum Many-Body Systems
journal, June 2018
- Deng, Dong-Ling
- Physical Review Letters, Vol. 120, Issue 24
Machine Learning Many-Body Localization: Search for the Elusive Nonergodic Metal
journal, December 2018
- Hsu, Yi-Ting; Li, Xiao; Deng, Dong-Ling
- Physical Review Letters, Vol. 121, Issue 24
Flotation Height Prediction under Stable and Vibration States in Air Cushion Furnace Based on Hard Division Method
journal, December 2019
- Hou, Shuai; Liu, Jianhui; Lv, Wu
- Mathematical Problems in Engineering, Vol. 2019
Machine Learning of Explicit Order Parameters: From the Ising Model to SU(2) Lattice Gauge Theory
text, January 2017
- Wetzel, Sebastian Johann; Scherzer, Manuel
- arXiv
Self-Learning Monte Carlo Method: Continuous-Time Algorithm
text, January 2017
- Nagai, Yuki; Shen, Huitao; Qi, Yang
- arXiv
Machine Learning Topological Invariants with Neural Networks
text, January 2017
- Zhang, Pengfei; Shen, Huitao; Zhai, Hui
- arXiv
Identifying Quantum Phase Transitions with Adversarial Neural Networks
text, January 2017
- Huembeli, Patrick; Dauphin, Alexandre; Wittek, Peter
- arXiv
Machine learning vortices at the Kosterlitz-Thouless transition
text, January 2017
- Beach, Matthew J. S.; Golubeva, Anna; Melko, Roger G.
- arXiv
Self-learning Monte Carlo with Deep Neural Networks
text, January 2018
- Shen, Huitao; Liu, Junwei; Fu, Liang
- arXiv
Real space mapping of topological invariants using artificial neural networks
text, January 2018
- Carvalho, D.; Garcia-Martinez, N. A.; Lado, J. L.
- arXiv
Machine Learning of Frustrated Classical Spin Models. II. Kernel Principal Component Analysis
text, January 2018
- Wang, Ce; Zhai, Hui
- arXiv
Machine learning of phase transitions in the percolation and XY models
text, January 2018
- Zhang, Wanzhou; Liu, Jiayu; Wei, Tzu-Chieh
- arXiv
Probing hidden spin order with interpretable machine learning
text, January 2018
- Greitemann, Jonas; Liu, Ke; Pollet, Lode
- arXiv
Identifying topological order through unsupervised machine learning
text, January 2018
- Rodriguez-Nieva, Joaquin F.; Scheurer, Mathias S.
- arXiv
Deep Learning Topological Invariants of Band Insulators
text, January 2018
- Sun, Ning; Yi, Jinmin; Zhang, Pengfei
- arXiv
Machine learning of quantum phase transitions
text, January 2018
- Dong, Xiao-Yu; Pollmann, Frank; Zhang, Xue-Feng
- arXiv
Self-organizing maps as a method for detecting phase transitions and phase identification
text, January 2018
- Shirinyan, Albert A.; Kozin, Valerii K.; Hellsvik, Johan
- arXiv
Extracting many-particle entanglement entropy from observables using supervised machine learning
text, January 2018
- Berkovits, Richard
- arXiv
Profile approach for recognition of three-dimensional magnetic structures
text, January 2018
- Iakovlev, I. A.; Sotnikov, O. M.; Mazurenko, V. V.
- arXiv
Unsupervised Learning Eigenstate Phases of Matter
text, January 2019
- Durr, Steven; Chakravarty, Sudip
- arXiv
Criticality and factorization in the Heisenberg chain with Dzyaloshinskii-Moriya interaction
text, January 2019
- Yi, Tian-Cheng; You, Wen-Long; Wu, Ning
- arXiv