Annealed Ising model with site dilution on self-similar structures

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.