Quantum Vicsek Model for Active Matter

We propose a quantum analog of the Vicsek model, consisting of an ensemble of overdamped spin-1/2 particles with ferromagnetic couplings, driven by a uniformly polarized magnetic field. The spontaneous magnetization of the spin components breaks the SO(3) (or SO(2)) symmetry, inducing an ordered phase of flocking. We derive the hydrodynamic equations, similar to those formulated by Toner and Tu, by applying a mean-field approximation to the quantum analog model up to the next leading order. Our investigation not only establishes a microscopic connection between the Vicsek model and the Toner-Tu hydrodynamics for active matter, but also aims to inspire further studies of active matter in the quantum regime.