Long-Range Order in a Two-Dimensional DynamicalXYModel: How Birds Fly Together

We propose a nonequilibrium continuum dynamical model for the collective motion of large groups of biological organisms (e.g., flocks of birds, slime molds, etc.) Our model becomes highly nontrivial, and different from the equilibrium model, for $d<\mathrm{}{d}_c=4\mathrm{}$; nonetheless, we are able to determine its scaling exponents exactly in $d=2\mathrm{}$ and show that, unlike equilibrium systems, our model exhibits a broken continuous symmetry even in $d=2\mathrm{}$. Our model describes a large universality class of microscopic rules, including those recently simulated by Vicsek et al.