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Transforming generalized Ising models into Boltzmann machines.

Nobuyuki Yoshioka1, Yutaka Akagi1, Hosho Katsura1

  • 1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

Physical Review. E
|April 20, 2019
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Summary
This summary is machine-generated.

We developed an exact mapping for generalized Ising models to Boltzmann machines using algebraic transformations. This method simplifies complex spin interactions and significantly speeds up Monte Carlo simulations, reducing critical slowing down in lattice models.

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Area of Science:

  • Statistical Mechanics
  • Computational Physics
  • Condensed Matter Theory

Background:

  • Generalized Ising models feature complex many-spin interactions.
  • Boltzmann machines offer a simplified framework with only two-spin interactions.
  • Efficient simulation of complex models is crucial for understanding physical phenomena.

Purpose of the Study:

  • To establish an exact mapping between generalized Ising models and Boltzmann machines.
  • To leverage this mapping for enhanced computational efficiency in simulations.
  • To investigate the impact of this method on critical slowing down.

Main Methods:

  • Utilized algebraic transformations, specifically star-triangle and decoration-iteration transformations.
  • Mapped many-spin interactions to equivalent two-spin interactions using auxiliary variables.
  • Applied the mapping in conjunction with the Swendsen-Wang algorithm for Monte Carlo simulations.

Main Results:

  • An exact mapping was found from generalized Ising models to equivalent Boltzmann machines.
  • The transformation reduces the complexity of spin interactions.
  • Significant reduction in critical slowing down was observed for a Kagomé lattice model.

Conclusions:

  • The developed mapping provides a powerful tool for analyzing complex spin systems.
  • This approach offers substantial benefits for Monte Carlo simulations.
  • The method effectively mitigates critical slowing down, enabling more efficient studies of phase transitions.