Self-learning Monte Carlo Method Zi Yang Meng ( ) Institute of - - PowerPoint PPT Presentation

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Self-learning Monte Carlo Method Zi Yang Meng ( ) Institute of - - PowerPoint PPT Presentation

Mar 07, 2024 490 likes •814 views

Self-learning Monte Carlo Method Zi Yang Meng ( ) Institute of Physics, Chinese Academy of Sciences http://ziyangmeng.iphy.ac.cn/ Institute of Physics, Chinese Academy of Sciences Know thyself "Know thyself" (Greek:

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Zi Yang Meng ( 孟子杨 )

Institute of Physics, Chinese Academy of Sciences http://ziyangmeng.iphy.ac.cn/

Self-learning Monte Carlo Method

Institute of Physics, Chinese Academy of Sciences

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Know thyself

"Know thyself" (Greek: γν θι σεαυτόν, gnothi seauton) ῶ

  • ne of the Delphic maxims and was inscribed in the

pronaos (forecourt) of the Temple of Apollo at Delphi

Institute of Physics, Chinese Academy of Sciences

Institute of Physics, Chinese Academy of Sciences

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Delphic Maxims

"Know thyself" (Greek: γν θι σεαυτόν, gnothi seauton). Thales of Miletus (c. ῶ 624 – c. 546 BC)

Institute of Physics, Chinese Academy of Sciences

Institute of Physics, Chinese Academy of Sciences

"nothing in excess" (Greek: μηδέν άγαν). Solon of Athens (c. 638 – 558 BC) "make a pledge and mischief is nigh" (Greek: γγύα πάρα δ' τη). Ἑ ἄ

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Collaborators and References

Institute of Physics, Chinese Academy of Sciences

  • Xiao Yan Xu, IOP, CAS
  • Huitao Shen, Massachusetts Institute of Technology
  • Jiuwei Liu, Massachusetts Institute of Technology
  • Yang Qi, Massachusetts Institute of Technology & Fudan University, Shanghai
  • Liang Fu, Massachusetts Institute of Technology

➢ Self-Learning Monte Carlo Method, arXiv:1610.08376 ➢ Self-Learning Monte Carlo Method in Fermion Systems, arXiv:1611.09364 ➢ Self-Learning Determinantal Quantum Monte Carlo Method, arXiv:1612.03804

Trilogy of SLMC

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Quantum Monte Carlo simulation

 Determinantal QMC for fermions

Hubbard-like models:

 Metal-Insulator transition  Interaction effects on topological state of matter

Fermions coupled to critical bosonic mode:

 Itinerant quantum critical point  Non-Fermi-liquid  Gauge field couples to fermion

…...

 World-line QMC for bosons

Heisenberg-like models:

 Quantum magnetism  Phase transition and critical phenomena  Quantum spin liquids

Duality between DQCP and bosonic SPT:

 Deconfined quantum critical point  Bosonic SPT and its critical point

…...

Institute of Physics, Chinese Academy of Sciences

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Institute of Physics, Chinese Academy of Sciences

Basic problem

Partition function: Observables: Fock space:

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Monte Carlo simulation

  • Widely used in statistical and quantum many-body physics
  • Unbiased: statistical error
  • Universal: applies to any model without sign problem
  • Markov chain Monte Carlo is a way to do important sampling
  • Distribution of converges to the Boltzmann distribution

Institute of Physics, Chinese Academy of Sciences

  • Observable can be measured from a Markov chain
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Autocorrelation time

  • Markov process, Monte Carlo time sequence
  • Autocorrelation function

Institute of Physics, Chinese Academy of Sciences

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Institute of Physics, Chinese Academy of Sciences

Monte Carlo simulation

  • Detailed balance guarantees the Markov process converges to desired

distribution MC converges to desired distribution

  • Metropolis-Hastings algorithm: proposal – acceptance/rejection

➢ N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953) ➢ W. H. Hastings, Biometrika 57, 97 (1970)

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Metropolis algorithm: local update

  • Local update
  • Acceptance ratio

Institute of Physics, Chinese Academy of Sciences ➢ N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953)

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Topological phase transitions

Institute of Physics, Chinese Academy of Sciences

Critical slowing down

  • Dynamical relaxation time diverges at the critical point: critical

system is slow to equilibrate.

  • For 2D Ising model
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Topological phase transitions

Wolff algorithm: cluster update

  • A cluster is built from bonds
  • Probability of activating a bond is cleverly designed

➢ U. Wolff, Phys. Rev. Lett. 62, 361 (1989) Institute of Physics, Chinese Academy of Sciences

  • an ideal acceptance ratio
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Institute of Physics, Chinese Academy of Sciences ➢ Swendsen and Wang, Phys. Rev. Lett. 58, 86 (1987)

Reduce critical slowing down

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Institute of Physics, Chinese Academy of Sciences

Learn thyself

  • Step too small: small difference, high acceptance
  • Step too large: big difference, low acceptance
  • Global update: explore the low-energy configurations
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SU(4) model with Dirac fermions

SLMC: Learning+Simulating

Institute of Physics, Chinese Academy of Sciences

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SU(4) model with Dirac fermions

Trilogy I: SLMC for Bosons

  • The self-learning update: cluster is constructed using the effective model
  • The acceptance ratio:
  • The acceptance ratio can be very high, autocorrelation time can be very short
  • effective model capture the low-energy physics

Institute of Physics, Chinese Academy of Sciences

Ising transition with

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SU(4) model with Dirac fermions

  • Generate configurations with local update, at T=5 > Tc.
  • Perform linear regression
  • Generate configurations with reinforced learning at Tc

Institute of Physics, Chinese Academy of Sciences

Trilogy I: SLMC for Bosons

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SU(4) model with Dirac fermions

Institute of Physics, Chinese Academy of Sciences

Trilogy I: SLMC for Bosons

System size 40x40 at Tc

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Content

Institute of Physics, Chinese Academy of Sciences

  • Speedup of 10~20 times

Trilogy I: SLMC for Bosons

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Trilogy II: SLMC for Fermions

  • Double exchange model

Institute of Physics, Chinese Academy of Sciences

  • Computational complexity
  • Fit effective model
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Trilogy II: SLMC for Fermions

  • effective model captures the low-energy physics, RKKY interaction.
  • only need to learn from small system sizes

Institute of Physics, Chinese Academy of Sciences

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Trilogy II: SLMC for Fermions

  • Cumulative update

Institute of Physics, Chinese Academy of Sciences

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Trilogy II: SLMC for Fermions

  • Computation complexity at most
  • Speedup of
  • L=4,6,8, at L=8, 10^3 times faster.

Institute of Physics, Chinese Academy of Sciences

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Institute of Physics, Chinese Academy of Sciences

Trilogy III: SLMC for DQMC

Self-learning

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Institute of Physics, Chinese Academy of Sciences

Trilogy III: SLMC for DQMC

Complexity for getting an independent configuration: Complexity for getting an independent configuration: Fermions coupled to critical bosonic modee

 Itinerant quantum critical point  Non-Fermi-liquid

➢ arXiv:1602.07150 ➢ arXiv:1612.06075

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Institute of Physics, Chinese Academy of Sciences

Trilogy III: SLMC for DQMC

Complexity for SLMC

  • Cumulative update:
  • Detail balance:
  • Sweep Green's function:

Complexity speed up

Complexity for obtaining an independent configuration:

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Institute of Physics, Chinese Academy of Sciences

Trilogy III: SLMC for DQMC

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Institute of Physics, Chinese Academy of Sciences

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Scope of the Workshop

  • Conceptual connections of machine learning and many-body physics
  • Machine learning techniques for solving many-body physics/chemistry problems
  • Quantum algorithms and quantum hardwares for machine learning
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Institute of Physics, Chinese Academy of Sciences

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Scope of the Workshop

  • Topological classification of strongly correlated systems
  • Topological phase transitions
  • Realizations of topological orders
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