Machine learning for many-body physics: The case of the Anderson impurity model
Abstract
We applied machine learning methods in order to find the Green's function of the Anderson impurity model, a basic model system of quantum many-body condensed-matter physics. Furthermore, different methods of parametrizing the Green's function are investigated; a representation in terms of Legendre polynomials is found to be superior due to its limited number of coefficients and its applicability to state of the art methods of solution. The dependence of the errors on the size of the training set is determined. Our results indicate that a machine learning approach to dynamical mean-field theory may be feasible.
- Authors:
-
- Columbia Univ., New York, NY (United States). Dept. of Physics
- Argonne National Lab. (ANL), Argonne, IL (United States). Materials Science Division
- Univ. of Basel (Switzerland). Inst. of Physics Chemistry; Argonne National Lab. (ANL), Argonne, IL (United States). Argonne Leadership Computing Facility
- Publication Date:
- Research Org.:
- Argonne National Laboratory (ANL), Argonne, IL (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)
- OSTI Identifier:
- 1357598
- Alternate Identifier(s):
- OSTI ID: 1180270
- Grant/Contract Number:
- AC02-06CH11357; 3F-3138
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. B, Condensed Matter and Materials Physics
- Additional Journal Information:
- Journal Volume: 90; Journal Issue: 15; Journal ID: ISSN 1098-0121
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS AND COMPUTING
Citation Formats
Arsenault, Louis-François, Lopez-Bezanilla, Alejandro, von Lilienfeld, O. Anatole, and Millis, Andrew J. Machine learning for many-body physics: The case of the Anderson impurity model. United States: N. p., 2014.
Web. doi:10.1103/PhysRevB.90.155136.
Arsenault, Louis-François, Lopez-Bezanilla, Alejandro, von Lilienfeld, O. Anatole, & Millis, Andrew J. Machine learning for many-body physics: The case of the Anderson impurity model. United States. https://doi.org/10.1103/PhysRevB.90.155136
Arsenault, Louis-François, Lopez-Bezanilla, Alejandro, von Lilienfeld, O. Anatole, and Millis, Andrew J. Fri .
"Machine learning for many-body physics: The case of the Anderson impurity model". United States. https://doi.org/10.1103/PhysRevB.90.155136. https://www.osti.gov/servlets/purl/1357598.
@article{osti_1357598,
title = {Machine learning for many-body physics: The case of the Anderson impurity model},
author = {Arsenault, Louis-François and Lopez-Bezanilla, Alejandro and von Lilienfeld, O. Anatole and Millis, Andrew J.},
abstractNote = {We applied machine learning methods in order to find the Green's function of the Anderson impurity model, a basic model system of quantum many-body condensed-matter physics. Furthermore, different methods of parametrizing the Green's function are investigated; a representation in terms of Legendre polynomials is found to be superior due to its limited number of coefficients and its applicability to state of the art methods of solution. The dependence of the errors on the size of the training set is determined. Our results indicate that a machine learning approach to dynamical mean-field theory may be feasible.},
doi = {10.1103/PhysRevB.90.155136},
journal = {Physical Review. B, Condensed Matter and Materials Physics},
number = 15,
volume = 90,
place = {United States},
year = {Fri Oct 31 00:00:00 EDT 2014},
month = {Fri Oct 31 00:00:00 EDT 2014}
}
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Works referenced in this record:
Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations
journal, May 2005
- Troyer, Matthias; Wiese, Uwe-Jens
- Physical Review Letters, Vol. 94, Issue 17
Finding Density Functionals with Machine Learning
journal, June 2012
- Snyder, John C.; Rupp, Matthias; Hansen, Katja
- Physical Review Letters, Vol. 108, Issue 25
Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning
journal, January 2012
- Rupp, Matthias; Tkatchenko, Alexandre; Müller, Klaus-Robert
- Physical Review Letters, Vol. 108, Issue 5
Machine learning of molecular electronic properties in chemical compound space
journal, September 2013
- Montavon, Grégoire; Rupp, Matthias; Gobre, Vivekanand
- New Journal of Physics, Vol. 15, Issue 9
Modeling electronic quantum transport with machine learning
journal, June 2014
- Lopez-Bezanilla, Alejandro; von Lilienfeld, O. Anatole
- Physical Review B, Vol. 89, Issue 23
Potential energy surfaces for macromolecules. A neural network technique
journal, May 1992
- Sumpter, Bobby G.; Noid, Donald W.
- Chemical Physics Letters, Vol. 192, Issue 5-6
Representing high-dimensional potential-energy surfaces for reactions at surfaces by neural networks
journal, September 2004
- Lorenz, Sönke; Groß, Axel; Scheffler, Matthias
- Chemical Physics Letters, Vol. 395, Issue 4-6
A random-sampling high dimensional model representation neural network for building potential energy surfaces
journal, August 2006
- Manzhos, Sergei; Carrington, Tucker
- The Journal of Chemical Physics, Vol. 125, Issue 8
Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces
journal, April 2007
- Behler, Jörg; Parrinello, Michele
- Physical Review Letters, Vol. 98, Issue 14
Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons
journal, April 2010
- Bartók, Albert P.; Payne, Mike C.; Kondor, Risi
- Physical Review Letters, Vol. 104, Issue 13
Correlated Lattice Fermions in Dimensions
journal, January 1989
- Metzner, Walter; Vollhardt, Dieter
- Physical Review Letters, Vol. 62, Issue 3
Hubbard model in infinite dimensions
journal, March 1992
- Georges, Antoine; Kotliar, Gabriel
- Physical Review B, Vol. 45, Issue 12
Hubbard model in infinite dimensions: A quantum Monte Carlo study
journal, July 1992
- Jarrell, M.
- Physical Review Letters, Vol. 69, Issue 1
Continuous-Time Solver for Quantum Impurity Models
journal, August 2006
- Werner, Philipp; Comanac, Armin; de’ Medici, Luca
- Physical Review Letters, Vol. 97, Issue 7
Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base
journal, April 2007
- Haule, Kristjan
- Physical Review B, Vol. 75, Issue 15
Continuous-time Monte Carlo methods for quantum impurity models
journal, May 2011
- Gull, Emanuel; Millis, Andrew J.; Lichtenstein, Alexander I.
- Reviews of Modern Physics, Vol. 83, Issue 2
Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies
journal, July 2013
- Hansen, Katja; Montavon, Grégoire; Biegler, Franziska
- Journal of Chemical Theory and Computation, Vol. 9, Issue 8
The Elements of Statistical Learning
book, January 2009
- Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome
- Springer Series in Statistics
Towards an exact solution of the Anderson model
journal, November 1980
- Wiegmann, P. B.
- Physics Letters A, Vol. 80, Issue 2-3
Analytical approximation for single-impurity Anderson model
journal, March 2010
- Krivenko, I. S.; Rubtsov, A. N.; Katsnelson, M. I.
- JETP Letters, Vol. 91, Issue 6
Dynamical properties of the Anderson impurity model within a diagrammatic pseudoparticle approach
journal, October 2004
- Kirchner, S.; Kroha, J.; Wölfle, P.
- Physical Review B, Vol. 70, Issue 16
Monte Carlo Method for Magnetic Impurities in Metals
journal, June 1986
- Hirsch, J. E.; Fye, R. M.
- Physical Review Letters, Vol. 56, Issue 23
Electronic correlation in nanoscale junctions: Comparison of the GW approximation to a numerically exact solution of the single-impurity Anderson model
journal, January 2008
- Wang, X.; Spataru, C. D.; Hybertsen, M. S.
- Physical Review B, Vol. 77, Issue 4
Numerical renormalization group method for quantum impurity systems
journal, April 2008
- Bulla, Ralf; Costi, Theo A.; Pruschke, Thomas
- Reviews of Modern Physics, Vol. 80, Issue 2
Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
journal, January 1996
- Georges, Antoine; Kotliar, Gabriel; Krauth, Werner
- Reviews of Modern Physics, Vol. 68, Issue 1
Correlated electrons in high-temperature superconductors
journal, July 1994
- Dagotto, Elbio
- Reviews of Modern Physics, Vol. 66, Issue 3
Theory of the atomic limit of the Anderson model. I. Perturbation expansions re-examined
journal, December 1978
- Haldane, F. D. M.
- Journal of Physics C: Solid State Physics, Vol. 11, Issue 24
Benchmark of a modified iterated perturbation theory approach on the fcc lattice at strong coupling
journal, August 2012
- Arsenault, Louis-François; Sémon, Patrick; Tremblay, A. -M. S.
- Physical Review B, Vol. 86, Issue 8
Orthogonal polynomial representation of imaginary-time Green’s functions
journal, August 2011
- Boehnke, Lewin; Hafermann, Hartmut; Ferrero, Michel
- Physical Review B, Vol. 84, Issue 7
A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula
journal, January 2014
- Hale, Nicholas; Townsend, Alex
- SIAM Journal on Scientific Computing, Vol. 36, Issue 1
Exact diagonalization approach to correlated fermions in infinite dimensions: Mott transition and superconductivity
journal, March 1994
- Caffarel, Michel; Krauth, Werner
- Physical Review Letters, Vol. 72, Issue 10
Block Lanczos tridiagonalization of complex symmetric matrices
conference, August 2005
- Qiao, Sanzheng; Liu, Guohong; Xu, Wei
- Optics & Photonics 2005, SPIE Proceedings
Correlated Lattice Fermions in Dimensions
journal, February 1989
- Metzner, Walter; Vollhardt, Dieter
- Physical Review Letters, Vol. 62, Issue 9
Optimization of probiotic therapeutics using machine learning in an artificial human gastrointestinal tract
journal, January 2021
- Westfall, Susan; Carracci, Francesca; Estill, Molly
- Scientific Reports, Vol. 11, Issue 1
Modeling electronic quantum transport with machine learning
text, January 2014
- Lopez-Bezanilla, Alejandro; Von Lilienfeld, O. Anatole
- American Physical Society
Machine learning of molecular electronic properties in chemical compound space
text, January 2013
- Montavon, Grégoire; Rupp, Matthias; Gobre, Vivekanand
- ETH Zurich
Towards an Exact Solution of the Anderson Model
book, August 1996
- Wiegman, P. B.
- 30 Years of the Landau Institute — Selected Papers
Fast and accurate modeling of molecular atomization energies with machine learning
text, January 2012
- Rupp, Matthias; Tkatchenko, Alexandre; Müller, Klaus-Robert
- American Physical Society
The Elements of Statistical Learning
book, January 2001
- Hastie, Trevor; Friedman, Jerome; Tibshirani, Robert
- Springer Series in Statistics
Gaussian Approximation Potentials: the accuracy of quantum mechanics, without the electrons
text, January 2009
- Bartók, Albert P.; Payne, Mike C.; Kondor, Risi
- arXiv
Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning
text, January 2011
- Rupp, Matthias; Tkatchenko, Alexandre; Müller, Klaus-Robert
- arXiv
Modeling Electronic Quantum Transport with Machine Learning
text, January 2014
- Lopez-Bezanilla, Alejandro; von Lilienfeld, O. Anatole
- arXiv
A continuous-time solver for quantum impurity models
text, January 2005
- Werner, Philipp; Comanac, Armin; De Medici, Luca
- arXiv
The numerical renormalization group method for quantum impurity systems
text, January 2007
- Bulla, Ralf; Costi, Theo; Pruschke, Thomas
- arXiv
Machine learning of molecular electronic properties in chemical compound space
text, January 2013
- Montavon, Gregoire; Rupp, Matthias; Gobre, Vivekanand
- IOP Publishing
Works referencing / citing this record:
Machine Learning, Quantum Chemistry, and Chemical Space
book, January 2017
- Ramakrishnan, Raghunathan; von Lilienfeld, O. Anatole
- Reviews in Computational Chemistry, Vol. 30
Quantum Machine Learning in Chemical Compound Space
journal, March 2018
- von Lilienfeld, O. Anatole
- Angewandte Chemie International Edition, Vol. 57, Issue 16
Feature vector clustering molecular pairs in computer simulations
journal, July 2019
- Pei, Han‐Wen; Laaksonen, Aatto
- Journal of Computational Chemistry, Vol. 40, Issue 29
Crystal structure representations for machine learning models of formation energies
journal, April 2015
- Faber, Felix; Lindmaa, Alexander; von Lilienfeld, O. Anatole
- International Journal of Quantum Chemistry, Vol. 115, Issue 16
Machine learning phases of matter
journal, February 2017
- Carrasquilla, Juan; Melko, Roger G.
- Nature Physics, Vol. 13, Issue 5
Quantum machine learning for electronic structure calculations
journal, October 2018
- Xia, Rongxin; Kais, Sabre
- Nature Communications, Vol. 9, Issue 1
Pattern Learning Electronic Density of States
journal, April 2019
- Yeo, Byung Chul; Kim, Donghun; Kim, Chansoo
- Scientific Reports, Vol. 9, Issue 1
Electronic spectra from TDDFT and machine learning in chemical space
journal, August 2015
- Ramakrishnan, Raghunathan; Hartmann, Mia; Tapavicza, Enrico
- The Journal of Chemical Physics, Vol. 143, Issue 8
Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics
journal, October 2017
- Arsenault, Louis-François; Neuberg, Richard; Hannah, Lauren A.
- Inverse Problems, Vol. 33, Issue 11
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
Deep learning and the Schrödinger equation
journal, October 2017
- Mills, Kyle; Spanner, Michael; Tamblyn, Isaac
- Physical Review A, Vol. 96, Issue 4
Accelerated continuous time quantum Monte Carlo method with machine learning
journal, July 2019
- Song, Taegeun; Lee, Hunpyo
- Physical Review B, Vol. 100, Issue 4
Analytic continuation via domain knowledge free machine learning
journal, December 2018
- Yoon, Hongkee; Sim, Jae-Hoon; Han, Myung Joon
- Physical Review B, Vol. 98, Issue 24
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
Matrix product operators for sequence-to-sequence learning
journal, October 2018
- Guo, Chu; Jie, Zhanming; Lu, Wei
- Physical Review E, Vol. 98, Issue 4
Discriminative Cooperative Networks for Detecting Phase Transitions
journal, April 2018
- Liu, Ye-Hua; van Nieuwenburg, Evert P. L.
- Physical Review Letters, Vol. 120, Issue 17
Mapping and classifying molecules from a high-throughput structural database
journal, February 2017
- De, Sandip; Musil, Felix; Ingram, Teresa
- Journal of Cheminformatics, Vol. 9, Issue 1
Quantum error correction for the toric code using deep reinforcement learning
journal, September 2019
- Andreasson, Philip; Johansson, Joel; Liljestrand, Simon
- Quantum, Vol. 3
Electronic spectra from TDDFT and machine learning in chemical space
text, January 2015
- Ramakrishnan, Raghunathan; Hartmann, Mia; Tapavicza, Enrico
- AIP Publishing
Crystal structure representations for machine learning models of formation energies
text, January 2015
- Felix, Faber,; Alexander, Lindmaa,; Anatole, von Lilienfeld, O.
- Wiley
Electronic Spectra from TDDFT and Machine Learning in Chemical Space
text, January 2015
- Ramakrishnan, Raghunathan; Hartmann, Mia; Tapavicza, Enrico
- arXiv
Comparing molecules and solids across structural and alchemical space
text, January 2016
- De, Sandip; Bartók, Albert P.; Csányi, Gábor
- arXiv
Machine Learning Phases of Strongly Correlated Fermions
text, January 2016
- Ch'ng, Kelvin; Carrasquilla, Juan; Melko, Roger G.
- arXiv
Mapping and Classifying Molecules from a High-Throughput Structural Database
preprint, January 2016
- De, Sandip; Musil, Felix; Ingram, Teresa
- arXiv
Identifying polymer states by machine learning
text, January 2017
- Wei, Qianshi; Melko, Roger G.; Chen, Jeff Z. Y.
- arXiv
Extensive deep neural networks for transferring small scale learning to large scale systems
text, January 2017
- Mills, Kyle; Ryczko, Kevin; Luchak, Iryna
- arXiv
Machine Learning Topological Invariants with Neural Networks
text, January 2017
- Zhang, Pengfei; Shen, Huitao; Zhai, Hui
- arXiv
Neural-Network Quantum States, String-Bond States, and Chiral Topological States
text, January 2017
- Glasser, Ivan; Pancotti, Nicola; August, Moritz
- arXiv
Self-learning Monte Carlo with Deep Neural Networks
text, January 2018
- Shen, Huitao; Liu, Junwei; Fu, Liang
- arXiv
Matrix Product Operators for Sequence to Sequence Learning
text, January 2018
- Guo, Chu; Jie, Zhanming; Lu, Wei
- 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
Deep Learning Topological Invariants of Band Insulators
text, January 2018
- Sun, Ning; Yi, Jinmin; Zhang, Pengfei
- arXiv
Pattern Learning Electronic Density of States
text, January 2018
- Yeo, Byung Chul; Kim, Donghun; Kim, Chansoo
- 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
Machine learning density functional theory for the Hubbard model
text, January 2018
- Nelson, James; Tiwari, Rajarshi; Sanvito, Stefano
- arXiv