Renormalization group study of marginal ferromagnetism.

Area of Science:

• Statistical Physics
• Collective Behavior
• Complex Systems

Background:

• Ferromagnetic systems provide a theoretical framework for studying collective motion and imitation in biological groups.
• Scale-free correlations in starling flock speed fluctuations challenge standard ferromagnetic models, which predict finite correlation lengths in ordered phases.

Purpose of the Study:

• To develop and analyze an effective field theory for a marginal ferromagnetic model near its zero-temperature critical point.
• To reconcile the observed scale-free correlations in biological systems with theoretical frameworks in statistical physics.

Main Methods:

• Derivation of an effective field theory for the marginal model.
• Calculation of renormalization group equations at one-loop using a momentum shell approach.
• Monte Carlo simulations to study modulus susceptibility in three dimensions.

Main Results:

• The marginal model exhibits a zero-temperature critical point where the modulus correlation length diverges.
• Renormalization group analysis reveals non-vanishing cubic and quartic vertices in the infrared limit.
• The upper critical dimension for critical exponents ν and η is found to be d_c=2, leading to free values (ν=1/2, η=0) in three dimensions.

Conclusions:

• The marginal ferromagnetic model successfully explains the anomaly of scale-free correlations in biological collective motion.
• Theoretical predictions are numerically verified by Monte Carlo simulations, confirming the adapted finite-size scaling relations for d>d_c.