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Anomalous mean-field behavior of the fully connected Ising model
Louis Colonna-Romano1, Harvey Gould2, W Klein3
1Department of Physics, Clark University, Worcester, Massachusetts 01610, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 7, 2014
Summary
Finite size scaling in the fully connected Ising model reveals different critical exponents and magnetization behaviors. Hyperscaling can be restored by adjusting system size (N) or scaling at pseudocritical temperatures.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
Background:
- The fully connected Ising model, lacking an intrinsic length scale, is a fundamental model in statistical mechanics.
- Understanding its critical behavior and thermodynamic quantities is crucial for theoretical physics.
Purpose of the Study:
- To investigate the application of finite size scaling to the fully connected Ising model.
- To analyze the behavior of critical exponents, mean magnetization, and susceptibility with system size (N).
- To explore the conditions under which hyperscaling can be restored.
Main Methods:
- Finite size scaling analysis using the number of spins (N) as the scaling variable.
- Examination of thermodynamic quantities, including mean magnetization and susceptibility, at critical temperatures and spinodal points.
- Investigation of the magnetization probability distribution.
Main Results:
- Critical exponents can be obtained via finite size scaling with N.
- Mean and most probable magnetization values scale differently with N at the critical temperature.
- The magnetization distribution deviates from Gaussian behavior, even for large N.
- Hyperscaling breakdown is observed and can be restored under specific scaling conditions.
Conclusions:
- Finite size scaling in the fully connected Ising model yields results dependent on the approach to the mean-field limit.
- The model exhibits unique scaling behaviors inconsistent with standard mean-field theory predictions.
- Restoring hyperscaling requires careful consideration of system size and temperature scaling parameters.
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