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Ising model in scale-free networks: a Monte Carlo simulation
1Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Científicas, Campus de Cantoblanco, 28049 Madrid, Spain.
Summary
This study investigates the Ising model on scale-free networks. For gamma > 3, a phase transition occurs, while for gamma <= 3, a crossover is observed, with system behavior depending on network size and degree distribution.
Area of Science:
- Statistical physics
- Complex networks
- Condensed matter physics
Background:
- The Ising model is a fundamental model in statistical mechanics.
- Scale-free networks exhibit a power-law degree distribution.
- Understanding phase transitions in networked systems is crucial.
Purpose of the Study:
- To investigate the ferromagnetic-paramagnetic transition in the Ising model on uncorrelated scale-free networks.
- To analyze the influence of the degree distribution exponent (gamma) on the transition temperature.
- To compare simulation results with analytical predictions.
Main Methods:
- Monte Carlo simulations were employed.
- The study focused on networks with a degree distribution P(k) ~ k^(-gamma).
- The ferromagnetic-paramagnetic transition temperature was studied as a function of gamma.
Main Results:
- For gamma > 3, results align with analytical calculations showing a phase transition at T(c)(gamma).
- For gamma <= 3, a size-dependent crossover temperature T(co) was observed, indicating a ferromagnetic state at infinite system size.
- For gamma = 3, T(co) scales as A ln N; for 2 < gamma < 3, T(co) scales as
N(z), with a lower exponent z than predicted.
Conclusions:
- The nature of the ferromagnetic-paramagnetic transition in the Ising model is highly dependent on the degree distribution exponent gamma.
- Analytical predictions are valid for gamma > 3, but deviations occur for gamma <= 3.
- New scaling relations for the crossover temperature in the 2 < gamma < 3 regime were established.