[Paper Review] A note on Spontaneous Symmetry Breaking in flocks of birds

Alejandra Melfo|arXiv (Cornell University)|Feb 26, 2017

Neural dynamics and brain function12 references3 citations

TL;DR

This paper argues that long-range correlations in speed fluctuations in starling flocks arise not from criticality, but from spontaneous symmetry breaking (SSB) of continuous translational symmetry to a discrete group, leading to a phonon-like Nambu-Goldstone mode. Unlike rotational SSB, which explains orientation correlation lengths, translational SSB accounts for scale-free speed correlations without requiring fine-tuning or critical parameters.

ABSTRACT

It is suggested that the observed scale-free correlations of speed fluctuations in flocks of birds are a consequence of the spontaneous breakdown of translational symmetry to a discrete group, and not an indication that the system is near a critical point in phase space. The observed long-range correlation length could then be attributed to the presence of a phonon mode in the flock.

Motivation & Objective

• To challenge the interpretation that long-range correlations in starling flocks indicate criticality near a phase transition.
• To investigate whether spontaneous symmetry breaking (SSB) of translational symmetry—rather than rotational symmetry—can explain observed scale-free speed fluctuations.
• To reconcile conflicting interpretations: criticality vs. SSB as the origin of long-range correlations in collective motion.
• To propose that phonon modes from discrete translational SSB produce scale-free speed correlations, analogous to condensed matter systems.

Proposed method

• Analyzes two-point correlation functions of velocity and speed fluctuations from high-resolution stereometric photogrammetry data of starling flocks (100–4000 birds).
• Compares observed correlation lengths (ξ_V for orientation, ξ_S for speed) to theoretical expectations from statistical mechanics and symmetry breaking.
• Applies Goldstone's theorem to systems with broken continuous translational symmetry, showing that Nambu-Goldstone modes (phonons) emerge even when symmetry is broken to a discrete subgroup.
• Uses a position-based active-elastic model (spring-like interactions) to simulate flocks and reproduce long-range correlations without fine-tuning.
• Contrasts results from models with and without translational SSB to isolate the role of symmetry breaking in generating correlation lengths.
• Reviews theoretical foundations of Goldstone's theorem in systems with discrete spatial order, citing Watanabe and Leutwyler for validity beyond continuous symmetry.

Experimental results

Research questions

• RQ1 Can long-range correlations in speed fluctuations in starling flocks be explained by spontaneous symmetry breaking of translational symmetry rather than criticality?
• RQ2 Does the presence of a phonon-like Nambu-Goldstone mode in a discretized translational system account for scale-free speed correlations?
• RQ3 Is the observed large correlation length (ξ_S ≈ flock size) a generic feature of collective motion with distance-regulating interactions, independent of fine-tuning?
• RQ4 Can models with non-critical, non-fine-tuned parameters reproduce the long-range correlations seen in real flocks?
• RQ5 What is the role of translational symmetry breaking in collective motion, and how does it compare to rotational symmetry breaking in explaining correlation structures?

Key findings

• The correlation length for speed fluctuations (ξ_S) is found to be on the order of the flock size, indicating scale-free behavior.
• This long-range correlation in speed is not explained by rotational symmetry breaking, which only accounts for orientation correlations.
• The authors demonstrate that a position-based model with spring-like interactions (active-elastic model) reproduces both ξ_V and ξ_S at the scale of the flock without fine-tuning.
• Spontaneous breaking of continuous translational symmetry to a discrete group leads to a phonon mode that generates scale-free correlations in relative positions and thus in speed fluctuations.
• Theoretical analysis confirms that Goldstone's theorem applies even when translational symmetry is broken to a discrete subgroup, validating the existence of a massless mode.
• The study concludes that the observed correlations may stem from SSB of translational symmetry rather than criticality, offering an alternative explanation to those based on fine-tuned critical points.