Flocking with a $q$-fold discrete symmetry: band-to-lane transition in the active Potts model

TL;DR

This paper investigates the collective motion and phase transitions in a $q$-state active Potts model, revealing a novel reorientation transition from band to lane formation driven by particle velocity, supported by analytical and simulation results.

Contribution

It introduces a coarse-grained hydrodynamic theory for the $q$-state active Potts model and uncovers a new reorientation transition not present in simpler models.

Findings

Identification of a reorientation transition from transversal bands to longitudinal lanes.

Derivation of phase diagrams for $q=4$ and $q=6$.

Microscopic simulations confirming analytical predictions.

Abstract

We study the $q$-state active Potts model (APM) on a two-dimensional lattice in which self-propelled particles have $q$ internal states corresponding to the $q$ directions of motion. A local alignment rule inspired by the ferromagnetic $q$-state Potts model and self-propulsion via biased diffusion according to the internal particle states leads to a collective motion at high densities and low noise. We formulate a coarse-grained hydrodynamic theory of the model, compute the phase diagrams of the APM for $q=4$ and $q=6$ and explore the flocking dynamics in the region, in which the high-density (polar liquid) phase coexists with the low-density (gas) phase and forms a fluctuating stripe of coherently moving particles. As a function of the particle self-propulsion velocity, a novel reorientation transition of the phase-separated profiles from transversal band motion to longitudinal lane formation is revealed, which is absent in the Vicsek model and APM for $q=2$ (active Ising model). The origin of this reorientation transition is obtained via a stability analysis: for large velocities, the transverse diffusion constant approaches zero and stabilizes lanes. Finally, we perform microscopic simulations that corroborate our analytical predictions about the flocking and reorientation transitions and validate the phase diagrams of the APM.