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Percolation in the two-dimensional Ising model.

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  • 1University of Science and Technology of China, Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, Hefei 230026, People's Republic of China.

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This summary is machine-generated.

This study explores the Ising model using percolation theory, revealing two distinct percolation transitions at Ising criticality. These transitions show unique cluster behaviors and critical exponents, offering new geometric insights into this classic model.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Complex Systems

Background:

  • The Ising model is fundamental to understanding critical phenomena.
  • Percolation theory provides a geometric lens for studying phase transitions.
  • Extended-range interactions beyond nearest neighbors are explored.

Purpose of the Study:

  • To investigate percolation transitions in the square-lattice Ising model with extended-range bonds.
  • To characterize the critical behaviors and geometric properties at these transitions.
  • To compare findings with established theories like Fortuin-Kasteleyn random clusters.

Main Methods:

  • Constructing percolation clusters by placing bonds with probability p between parallel spins.
  • Analyzing the system at Ising criticality.
  • Observing transitions from disordered to critical to long-ranged ordered phases.
  • Estimating fractal dimensions and scaling exponents.

Main Results:

  • Two percolation transitions observed as bond probability p increases.
  • A stable critical phase identified for p_{c1} < p < p_{c2}.
  • Long-ranged percolation order with giant clusters found at higher p.
  • Critical behaviors at p_{c1} align with random cluster models and Ising spin domains.
  • Fractal dimension y_{h2}=1.9580(6) and scaling exponent y_{p2}=0.552(9) estimated at p_{c2}.

Conclusions:

  • The study reveals novel geometric properties of the 2D Ising model.
  • Percolation perspective offers valuable insights into critical phenomena.
  • Identified transitions and estimated exponents contribute to understanding complex systems.