Flocking as a continuous phase transition in self-aligning active crystals

Abstract

We study a two-dimensional crystal composed of active units governed by self-alignment. This mechanism induces a torque that aligns a particle’s orientation with its velocity and leads to a phase transition from a disordered to a flocking crystal. Here, we provide the first microscopic theory that analytically maps the crystal dynamics onto a Landau–Ginzburg model, in which the velocity-dependent effective free energy undergoes a transition from a single-well shape to a Mexican-hat profile. As confirmed by simulations, our theory quantitatively predicts the transition point and characteristic spatial velocity correlations. The continuous variation of the order parameter and the divergence of the analytically predicted correlation length imply that flocking in self-aligning active crystals corresponds to a continuous phase transition of the Berezinskii–Kosterlitz–Thouless type in two dimensions and to a second-order phase transition in three dimensions. These findings provide a theoretical foundation for the flocking phenomenon observed experimentally in active granular particles and migrating cells.