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

¹ Center for Biophysics & Department for Theoretical Physics, Saarland University - D-66123 Saarbrücken, Germany
² INM â Leibniz Institute for New Materials - Campus D2 2, 66123 Saarbrücken, Germany

^(a) swarnajitchatterjee@gmail.com (corresponding author)
^(b) mangeat@lusi.uni-sb.de
^(c) heiko.rieger@uni-saarland.de

Received: 2 March 2022
Accepted: 10 May 2022

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 → ∞ 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.