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Ising model on a small world network.

A Pekalski1

  • 1Institute of Theoretical Physics, University of Wroclaw, plac Maxa Borna 9, 50-204 Wroclaw, Poland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 12, 2001
PubMed
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A one-dimensional Ising model on a small world network exhibits ordering at finite temperatures, even with few added links. Critical exponent beta is lower than in 2D systems and unaffected by link concentration.

Area of Science:

  • Statistical physics
  • Complex networks
  • Condensed matter physics

Background:

  • The Ising model is a fundamental tool for studying magnetism and phase transitions.
  • Small world networks exhibit unique topological properties, bridging regular and random networks.

Purpose of the Study:

  • To investigate the magnetic ordering of a one-dimensional Ising model on a small world network.
  • To analyze the impact of short-range and long-range interactions on phase transitions.

Main Methods:

  • Monte Carlo simulations were employed to study the system's behavior.
  • A one-dimensional Ising model was adapted to a small world network topology.

Main Results:

  • A small fraction of long-range links induces ordering at finite temperatures.

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  • The critical exponent beta was found to be smaller than the 2D Ising model value.
  • Beta appears independent of the concentration of added long-range links.

    • Conclusions:

    • Small world networks significantly alter the critical behavior of the Ising model.
    • The presence of even a few long-range connections is sufficient to establish magnetic order.