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This study examines the Ising model on hierarchical networks, finding magnetic ordering persists on the triangular Apollonian network but shows a transition on the diamond hierarchical lattice, influenced by chemical potential.

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Area of Science:

  • Statistical physics
  • Condensed matter physics
  • Network science

Background:

  • The Ising model is a fundamental tool for studying magnetism.
  • Hierarchical networks like the Apollonian network (AN) and diamond hierarchical lattice (DHL) offer complex structures for physical models.
  • Understanding the role of vacant sites and chemical potential is crucial for magnetic properties.

Purpose of the Study:

  • To investigate the magnetic behavior of the Ising model on the triangular Apollonian network (AN) and the diamond hierarchical lattice (DHL).
  • To analyze the influence of a chemical potential (μ) on the concentration of active magnetic sites and its effect on magnetic ordering.
  • To adapt the transfer-matrix method for hierarchical structures and derive thermodynamic quantities.

Main Methods:

  • Formulation of the statistical problem in a grand canonical ensemble.
  • Adaptation of the transfer-matrix method to derive recursion relations for hierarchical networks.
  • Numerical analysis of recursion relations to obtain thermodynamic quantities.

Main Results:

  • For the triangular Apollonian network (AN), magnetic ordering is observed at all temperatures in the μ→∞ limit.
  • For the diamond hierarchical lattice (DHL), a ferromagnetic-paramagnetic transition occurs at a finite temperature in the μ→∞ limit.
  • Sufficiently negative chemical potential values lead to the disappearance of magnetic ordering in both networks.

Conclusions:

  • The Ising model exhibits distinct magnetic behaviors on different hierarchical networks.
  • Chemical potential plays a critical role in tuning magnetic ordering, with high positive values favoring order and high negative values suppressing it.
  • The adapted transfer-matrix method provides a robust framework for analyzing such complex systems.