Polar flocks with discretized directions: the active clock model approaching the Vicsek model

TL;DR

This paper introduces a discretized active flocking model that bridges the active Potts and Vicsek models, revealing how the number of directions influences flocking behavior and phase transitions.

Contribution

It proposes the active clock model as a discretization of the Vicsek model, analyzing how varying the number of directions affects flocking phenomena and phase transitions.

Findings

Small number of directions mimics active Potts model behavior.

Large number of directions reproduces Vicsek model phenomenology.

Hydrodynamic equations are derived for the models.

Abstract

We consider the off-lattice two-dimensional $q$-state active clock model (ACM) as a natural discretization of the Vicsek model (VM) describing flocking. The ACM consists of particles able to move in the plane in a discrete set of $q$ equidistant angular directions, as in the active Potts model (APM), with an alignment interaction inspired by the ferromagnetic equilibrium clock model. We find that for a small number of directions, the flocking transition of the ACM has the same phenomenology as the APM, including macrophase separation and reorientation transition. For a larger number of directions, the flocking transition in the ACM becomes equivalent to the one of the VM and displays microphase separation and only transverse bands, i.e. no re-orientation transition. Concomitantly also the transition of the $q\to\infty$ limit of the ACM, the active XY model (AXYM), is in the same universality class as the VM. We also construct a coarse-grained hydrodynamic description for the ACM and AXYM akin to the VM.