Quantum Analog of Vicsek Model for Active Matter

Abstract

We propose a quantum model consisting of an ensemble of overdamped spin$-1/2$ particles with ferromagnetic couplings, driven by a radially homogeneous magnetic field. The spontaneous magnetization of the spin components breaks the $SO(3)$ (or $SO(2)$) symmetry, inducing an ordered phase of flocking. Our model converges to the Vicsek model in the classical limit and corresponds to the Toner-Tu model in the continuous limit. Our investigation not only elucidates the intrinsic connection between these two models, but also introduces new opportunities for exploring the mechanisms underlying flocking order and correlations at the quantum level, which maybe pave the way for a new field of research -- the quantum active matter.

Figures (2)

• Figure 1: Spins in a radially homogeneous magnetic field.
• Figure 2: Numerical simulations of Quantum Vicsek Model (QVM). The phase diagram of the order parameter $\varphi=|\langle \dot{\Vec{r}}_j \rangle _j|$ as a function of the relaxation time $\gamma_s^{-1}$ and noise amplitude $\xi=\frac{2}{\gamma m \beta}$. The average of the order parameter is taking over $10^4$ time steps, and the time interval for each step is $( \Delta t = 0.1 )$. Simulation parameters: $\rho = 0.5$, $u = 0.5$, $r_c = 1.0$.