Mean-Field Theory of Chiral Active Model B: Arrested Coarsening and Chiral Fingering Instabilities

Abstract

We derive and analyze a mean-field theory of the chiral Ising model recently introduced by Wang, Pietzonka, and J\"ulicher in "Edge Currents Shape Condensates in Chiral Active Matter", arXiv:2603.20064. Starting from the master equation for clockwise and counterclockwise rotations of 2x2 spin blocks, we first obtain spatially discrete evolution equations for the spatially resolved average magnetization. On this discrete level, we show that a chiral bias strongly affects phase coarsening: domains coarsen anisotropically, develop nearly rectangular shapes, and eventually display chirality-induced arrested coarsening. Taking the continuum limit of these equations yields an active field theory that has the structure of a relaxational Model-B-type dynamics supplemented by a chiral current that permanently drives the system out of equilibrium. The coarse graining explicitly shows how microscopic rotational bias generates tangential currents localized at interfaces. Using this continuum theory, we perform a linear stability analysis of radially symmetric clusters and identify a chiral fingering instability in which angular perturbations of the interface are amplified and eventually lead to radially asymmetric rotating states or disordered states.