Self-learning Monte Carlo method: Continuous-time algorithm

Nagai, Yuki; Shen, Huitao; Qi, Yang; Liu, Junwei; Fu, Liang

doi:10.1103/physrevb.96.161102

Self-learning Monte Carlo method: Continuous-time algorithm

Journal Article · Tue Oct 03 00:00:00 EDT 2017 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.96.161102· OSTI ID:1505622

Nagai, Yuki ^[1]; Shen, Huitao ^[2]; Qi, Yang ^[2]; Liu, Junwei ^[3]; Fu, Liang ^[2]

Japan Atomic Energy Agency, Chiba (Japan); Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); Massachusetts Institute of Technology
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); Hong Kong Univ. of Science and Technology, Hong Kong (China)

The recently introduced self-learning Monte Carlo method is a general-purpose numerical method that speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. We implement this method in the framework of a continuous-time Monte Carlo method with an auxiliary field in quantum impurity models. We introduce and train a diagram generating function (DGF) to model the probability distribution of auxiliary field configurations in continuous imaginary time, at all orders of diagrammatic expansion. Furthermore, by using DGF to propose global moves in configuration space, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.

View Accepted Manuscript (DOE)

Research Organization:: Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division

Grant/Contract Number:: SC0010526

OSTI ID:: 1505622

Alternate ID(s):: OSTI ID: 1396308

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 16 Vol. 96; ISSN 2469-9950; ISSN PRBMDO

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (44)

Monte Carlo Method for Magnetic Impurities in Metals Hirsch, J. E.; Fye, R. M. Physical Review Letters, Vol. 56, Issue 23 https://doi.org/10.1103/physrevlett.56.2521	journal	June 1986
Superconductivity from emerging magnetic moments Hoshino, Shintaro; Werner, Philipp arXiv https://doi.org/10.48550/arxiv.1503.08164	text	January 2015
Detection of phase transition via convolutional neural network Tanaka, Akinori; Tomiya, Akio arXiv https://doi.org/10.48550/arxiv.1609.09087	text	January 2016
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
New tool in the box Zdeborová, Lenka Nature Physics, Vol. 13, Issue 5 https://doi.org/10.1038/nphys4053	journal	February 2017
Localized Magnetic States in Metals Anderson, P. W. Physical Review, Vol. 124, Issue 1 https://doi.org/10.1103/PhysRev.124.41	journal	October 1961
Two-dimensional Hubbard model: Numerical simulation study Hirsch, J. E. Physical Review B, Vol. 31, Issue 7 https://doi.org/10.1103/PhysRevB.31.4403	journal	April 1985
Numerical study of the two-dimensional Hubbard model White, S. R.; Scalapino, D. J.; Sugar, R. L. Physical Review B, Vol. 40, Issue 1 https://doi.org/10.1103/PhysRevB.40.506	journal	July 1989
Continuous-time quantum Monte Carlo method for fermions Rubtsov, A. N.; Savkin, V. V.; Lichtenstein, A. I. Physical Review B, Vol. 72, Issue 3 https://doi.org/10.1103/PhysRevB.72.035122	journal	July 2005
Low-temperature density matrix renormalization group using regulated polynomial expansion Sota, Shigetoshi; Tohyama, Takami Physical Review B, Vol. 78, Issue 11 https://doi.org/10.1103/PhysRevB.78.113101	journal	September 2008
Momentum-selective metal-insulator transition in the two-dimensional Hubbard model: An 8-site dynamical cluster approximation study Werner, Philipp; Gull, Emanuel; Parcollet, Olivier Physical Review B, Vol. 80, Issue 4 https://doi.org/10.1103/PhysRevB.80.045120	journal	July 2009
Submatrix updates for the continuous-time auxiliary-field algorithm Gull, Emanuel; Staar, Peter; Fuchs, Sebastian Physical Review B, Vol. 83, Issue 7 https://doi.org/10.1103/PhysRevB.83.075122	journal	February 2011
Chebyshev matrix product state impurity solver for dynamical mean-field theory Wolf, F. Alexander; McCulloch, Ian P.; Parcollet, Olivier Physical Review B, Vol. 90, Issue 11 https://doi.org/10.1103/PhysRevB.90.115124	journal	September 2014
Numerical evaluation of Green's functions based on the Chebyshev expansion Braun, A.; Schmitteckert, P. Physical Review B, Vol. 90, Issue 16 https://doi.org/10.1103/PhysRevB.90.165112	journal	October 2014
Electronic orders in multiorbital Hubbard models with lifted orbital degeneracy Hoshino, Shintaro; Werner, Philipp Physical Review B, Vol. 93, Issue 15 https://doi.org/10.1103/PhysRevB.93.155161	journal	April 2016
Critical temperature enhancement of topological superconductors: A dynamical mean-field study Nagai, Yuki; Hoshino, Shintaro; Ota, Yukihiro Physical Review B, Vol. 93, Issue 22 https://doi.org/10.1103/PhysRevB.93.220505	journal	June 2016
Accelerated Monte Carlo simulations with restricted Boltzmann machines Huang, Li; Wang, Lei Physical Review B, Vol. 95, Issue 3 https://doi.org/10.1103/PhysRevB.95.035105	journal	January 2017
Self-learning Monte Carlo method Liu, Junwei; Qi, Yang; Meng, Zi Yang Physical Review B, Vol. 95, Issue 4 https://doi.org/10.1103/PhysRevB.95.041101	journal	January 2017
Self-learning Monte Carlo method and cumulative update in fermion systems Liu, Junwei; Shen, Huitao; Qi, Yang Physical Review B, Vol. 95, Issue 24 https://doi.org/10.1103/PhysRevB.95.241104	journal	June 2017
Self-learning quantum Monte Carlo method in interacting fermion systems Xu, Xiao Yan; Qi, Yang; Liu, Junwei Physical Review B, Vol. 96, Issue 4 https://doi.org/10.1103/PhysRevB.96.041119	journal	July 2017
Monte Carlo calculations of coupled boson-fermion systems. I Blankenbecler, R.; Scalapino, D. J.; Sugar, R. L. Physical Review D, Vol. 24, Issue 8 https://doi.org/10.1103/PhysRevD.24.2278	journal	October 1981
Recommender engine for continuous-time quantum Monte Carlo methods Huang, Li; Yang, Yi-feng; Wang, Lei Physical Review E, Vol. 95, Issue 3 https://doi.org/10.1103/PhysRevE.95.031301	journal	March 2017
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
Efficient Numerical Approach to Inhomogeneous Superconductivity: The Chebyshev-Bogoliubov–de Gennes Method Covaci, L.; Peeters, F. M.; Berciu, M. Physical Review Letters, Vol. 105, Issue 16 https://doi.org/10.1103/PhysRevLett.105.167006	journal	October 2010
Superconductivity from Emerging Magnetic Moments Hoshino, Shintaro; Werner, Philipp Physical Review Letters, Vol. 115, Issue 24 https://doi.org/10.1103/PhysRevLett.115.247001	journal	December 2015
Quantum Loop Topography for Machine Learning Zhang, Yi; Kim, Eun-Ah Physical Review Letters, Vol. 118, Issue 21 https://doi.org/10.1103/PhysRevLett.118.216401	journal	May 2017
Monte Carlo Method for Magnetic Impurities in Metals Hirsch, J. E.; Fye, R. M. Physical Review Letters, Vol. 56, Issue 23 https://doi.org/10.1103/PhysRevLett.56.2521	journal	June 1986
Continuous-Time Solver for Quantum Impurity Models Werner, Philipp; Comanac, Armin; de’ Medici, Luca Physical Review Letters, Vol. 97, Issue 7 https://doi.org/10.1103/PhysRevLett.97.076405	journal	August 2006
Quantum Entanglement in Neural Network States Deng, Dong-Ling; Li, Xiaopeng; Das Sarma, S. Physical Review X, Vol. 7, Issue 2 https://doi.org/10.1103/PhysRevX.7.021021	journal	May 2017
Continuous-time Monte Carlo methods for quantum impurity models Gull, Emanuel; Millis, Andrew J.; Lichtenstein, Alexander I. Reviews of Modern Physics, Vol. 83, Issue 2 https://doi.org/10.1103/RevModPhys.83.349	journal	May 2011
Solving the quantum many-body problem with artificial neural networks Carleo, Giuseppe; Troyer, Matthias Science, Vol. 355, Issue 6325 https://doi.org/10.1126/science.aag2302	journal	February 2017
Efficient Numerical Self-Consistent Mean-Field Approach for Fermionic Many-Body Systems by Polynomial Expansion on Spectral Density Nagai, Yuki; Ota, Yukihiro; Machida, Masahiko Journal of the Physical Society of Japan, Vol. 81, Issue 2 https://doi.org/10.1143/JPSJ.81.024710	journal	February 2012
Continuous-time auxiliary-field Monte Carlo for quantum impurity models Gull, E.; Werner, P.; Parcollet, O. EPL (Europhysics Letters), Vol. 82, Issue 5 https://doi.org/10.1209/0295-5075/82/57003	journal	May 2008
Continuous-time auxiliary field Monte Carlo for quantum impurity models Gull, Emanuel; Werner, Philipp; Parcollet, Olivier arXiv https://doi.org/10.48550/arxiv.0802.3222	text	January 2008
Sub-matrix updates for the Continuous-Time Auxiliary Field algorithm Gull, Emanuel; Staar, Peter; Fuchs, Sebastian arXiv https://doi.org/10.48550/arxiv.1010.3690	text	January 2010
Efficient Numerical Self-consistent Mean-field Approach for Fermionic Many-body Systems by Polynomial Expansion on Spectral Density Nagai, Yuki; Ota, Yukihiro; Machida, Masahiko arXiv https://doi.org/10.48550/arxiv.1105.4939	text	January 2011
Machine learning phases of matter Carrasquilla, Juan; Melko, Roger G. arXiv https://doi.org/10.48550/arxiv.1605.01735	text	January 2016
Learning phase transitions by confusion van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D. arXiv https://doi.org/10.48550/arxiv.1610.02048	text	January 2016
Self-Learning Monte Carlo Method Liu, Junwei; Qi, Yang; Meng, Zi Yang arXiv https://doi.org/10.48550/arxiv.1610.03137	text	January 2016
Quantum Entanglement in Neural Network States Deng, Dong-Ling; Li, Xiaopeng; Sarma, S. Das arXiv https://doi.org/10.48550/arxiv.1701.04844	text	January 2017
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
A continuous-time solver for quantum impurity models Werner, Philipp; Comanac, Armin; De Medici, Luca arXiv https://doi.org/10.48550/arxiv.cond-mat/0512727	text	January 2005
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

Cited By (25)

Revealing Fermionic Quantum Criticality from New Monte Carlo Techniques Xu, Xiao Yan; Liu, Zi Hong; Pan, Gaopei arXiv https://doi.org/10.48550/arxiv.1904.07355	text	January 2019
Accelerating small-angle scattering experiments on anisotropic samples using kernel density estimation Saito, Kotaro; Yano, Masao; Hino, Hideitsu Scientific Reports, Vol. 9, Issue 1 https://doi.org/10.1038/s41598-018-37345-5	journal	February 2019
Itinerant quantum critical point with fermion pockets and hotspots Liu, Zi Hong; Pan, Gaopei; Xu, Xiao Yan Proceedings of the National Academy of Sciences, Vol. 116, Issue 34 https://doi.org/10.1073/pnas.1901751116	journal	August 2019
Revealing fermionic quantum criticality from new Monte Carlo techniques Xu, Xiao Yan; Hong Liu, Zi; Pan, Gaopei Journal of Physics: Condensed Matter, Vol. 31, Issue 46 https://doi.org/10.1088/1361-648x/ab3295	journal	August 2019
Accelerating lattice quantum Monte Carlo simulations using artificial neural networks: Application to the Holstein model Li, Shaozhi; Dee, Philip M.; Khatami, Ehsan Physical Review B, Vol. 100, Issue 2 https://doi.org/10.1103/physrevb.100.020302	journal	July 2019
Restricted Boltzmann machine learning for solving strongly correlated quantum systems Nomura, Yusuke; Darmawan, Andrew S.; Yamaji, Youhei Physical Review B, Vol. 96, Issue 20 https://doi.org/10.1103/physrevb.96.205152	journal	November 2017
Real-space mapping of topological invariants using artificial neural networks Carvalho, D.; García-Martínez, N. A.; Lado, J. L. Physical Review B, Vol. 97, Issue 11 https://doi.org/10.1103/physrevb.97.115453	journal	March 2018
Self-learning Monte Carlo with deep neural networks Shen, Huitao; Liu, Junwei; Fu, Liang Physical Review B, Vol. 97, Issue 20 https://doi.org/10.1103/physrevb.97.205140	journal	May 2018
Symmetry-enforced self-learning Monte Carlo method applied to the Holstein model Chen, Chuang; Xu, Xiao Yan; Liu, Junwei Physical Review B, Vol. 98, Issue 4 https://doi.org/10.1103/physrevb.98.041102	journal	July 2018
Itinerant quantum critical point with frustration and a non-Fermi liquid Liu, Zi Hong; Xu, Xiao Yan; Qi, Yang Physical Review B, Vol. 98, Issue 4 https://doi.org/10.1103/physrevb.98.045116	journal	July 2018
Deep learning topological invariants of band insulators Sun, Ning; Yi, Jinmin; Zhang, Pengfei Physical Review B, Vol. 98, Issue 8 https://doi.org/10.1103/physrevb.98.085402	journal	August 2018
Elective-momentum ultrasize quantum Monte Carlo method Liu, Zi Hong; Xu, Xiao Yan; Qi, Yang Physical Review B, Vol. 99, Issue 8 https://doi.org/10.1103/physrevb.99.085114	journal	February 2019
Flow-based generative models for Markov chain Monte Carlo in lattice field theory Albergo, M. S.; Kanwar, G.; Shanahan, P. E. Physical Review D, Vol. 100, Issue 3 https://doi.org/10.1103/physrevd.100.034515	journal	August 2019
Smallest neural network to learn the Ising criticality Kim, Dongkyu; Kim, Dong-Hee Physical Review E, Vol. 98, Issue 2 https://doi.org/10.1103/physreve.98.022138	journal	August 2018
Policy-guided Monte Carlo: Reinforcement-learning Markov chain dynamics Bojesen, Troels Arnfred Physical Review E, Vol. 98, Issue 6 https://doi.org/10.1103/physreve.98.063303	journal	December 2018
Machine Learning Topological Invariants with Neural Networks Zhang, Pengfei; Shen, Huitao; Zhai, Hui Physical Review Letters, Vol. 120, Issue 6 https://doi.org/10.1103/physrevlett.120.066401	journal	February 2018
Discriminative Cooperative Networks for Detecting Phase Transitions Liu, Ye-Hua; van Nieuwenburg, Evert P. L. Physical Review Letters, Vol. 120, Issue 17 https://doi.org/10.1103/physrevlett.120.176401	journal	April 2018
Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons Chen, Chuang; Xu, Xiao Yan; Meng, Zi Yang Physical Review Letters, Vol. 122, Issue 7 https://doi.org/10.1103/physrevlett.122.077601	journal	February 2019
Machine learning and the physical sciences Carleo, Giuseppe; Cirac, Ignacio; Cranmer, Kyle Reviews of Modern Physics, Vol. 91, Issue 4 https://doi.org/10.1103/revmodphys.91.045002	journal	December 2019
Itinerant quantum critical point with frustration and non-Fermi-liquid Liu, Zi Hong; Xu, Xiao Yan; Qi, Yang arXiv https://doi.org/10.48550/arxiv.1706.10004	text	January 2017
Machine Learning Topological Invariants with Neural Networks Zhang, Pengfei; Shen, Huitao; Zhai, Hui arXiv https://doi.org/10.48550/arxiv.1708.09401	text	January 2017
Self-learning Monte Carlo with Deep Neural Networks Shen, Huitao; Liu, Junwei; Fu, Liang arXiv https://doi.org/10.48550/arxiv.1801.01127	text	January 2018
Real space mapping of topological invariants using artificial neural networks Carvalho, D.; Garcia-Martinez, N. A.; Lado, J. L. arXiv https://doi.org/10.48550/arxiv.1801.09655	text	January 2018
Deep Learning Topological Invariants of Band Insulators Sun, Ning; Yi, Jinmin; Zhang, Pengfei arXiv https://doi.org/10.48550/arxiv.1805.10503	text	January 2018
Flow-based generative models for Markov chain Monte Carlo in lattice field theory Albergo, M. S.; Kanwar, G.; Shanahan, P. E. arXiv https://doi.org/10.48550/arxiv.1904.12072	text	January 2019