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Magnetic models on Apollonian networks.

Roberto F S Andrade1, Hans J Herrmann

  • 1Instituto de Física, Universidade Federal da Bahia, 40210-340 Salvador, BA, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
Summary

Researchers studied Ising models on triangular Apollonian networks, finding long-range magnetic ordering for both ferromagnetic and antiferromagnetic couplings. They observed size-dependent critical behavior but no asymptotic criticality.

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Area of Science:

  • Statistical Mechanics
  • Complex Networks
  • Condensed Matter Physics

Background:

  • Ising models are fundamental in statistical mechanics for understanding magnetism.
  • Apollonian networks, inspired by circle packing, offer unique fractal structures.
  • Investigating magnetic properties on complex networks reveals emergent phenomena.

Purpose of the Study:

  • To explore the thermodynamic and magnetic characteristics of Ising models on triangular Apollonian networks.
  • To analyze the impact of different coupling constants (ferromagnetic and antiferromagnetic) on magnetic ordering.
  • To determine the presence and nature of critical behavior in these systems.

Main Methods:

  • Formulation of the problem using transfer matrices to derive thermodynamic function maps.
  • Numerical iteration of these maps across successive network generations.
  • Analysis of magnetic ordering and correlation length for various coupling strengths.

Main Results:

  • Precise determination of thermodynamic properties for the Ising model on the network.
  • Observation of long-range magnetic ordering for both ferromagnetic and antiferromagnetic interactions.
  • Identification of size-dependent effective critical behavior in the correlation length.

Conclusions:

  • The triangular Apollonian network supports long-range magnetic order.
  • Asymptotic criticality is generally absent, with deviations attributed to finite-size effects.
  • The study provides insights into magnetism on fractal-like complex networks.

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