Ferromagnetic model on the Apollonian packing.

Area of Science:

• Physics
• Materials Science
• Network Science

Background:

• Traditional Apollonian network models link magnetic coupling to network structure, not geometry.
• Apollonian packings exhibit scale variations, necessitating new evaluation approaches.

Purpose of the Study:

• Investigate the impact of Apollonian packing geometry on magnetic model behavior.
• Analyze how geometric scaling factors affect magnetic properties.

Main Methods:

• Developed a magnetic model where coupling constants depend on circle radii (r_i) and exponent α (J_{i,j}∼1/(r_{i}+r_{j})^{α}).
• Evaluated correlation length (ξ), specific heat, magnetization, and magnetic susceptibility.

Main Results:

• The uniform model (α=0) shows infinite correlation length for all temperatures (T>0).
• An increase in α leads to a transition to an uncorrelated state (finite ξ) at α_c ≃ 0.743.
• Magnetic ordering changes only with α, not temperature, for fixed α.

Conclusions:

• Geometric features of Apollonian packings significantly alter magnetic model behavior.
• The exponent α is critical in determining the magnetic state, transitioning from ordered to disordered.
• No finite temperature phase transitions were observed; changes are driven by geometric parameter variation.