How Continuous Symmetry Stabilizes the Ordered Phase of Polar Flocks

Abstract

We study the stability of the ordered phase of compressible polar flocks against the nucleation of counter-propagating droplets, using a combination of analytical theory, microscopic and hydrodynamic simulations. For discrete-symmetry flocks, such droplets are known to always grow and propagate, making the ordered phase metastable. We explain how, on the contrary, continuous symmetry can stabilize the ordered phase at small enough noise by destabilizing the leading edge of growing droplets. Flocking models with continuous symmetries thus have a lower critical dimension than their discrete-symmetry counterparts, in contrast to equilibrium physics.

Figures (4)

• Figure 1: Fate of localized counter-propagating perturbations introduced in the ordered phase of the active XY model. (a-c): At $T=0.34$, a ballistically growing droplet is observed. (d-f): At $T=0.31$, the perturbation eventually evaporates. (Initial circular perturbation of radius $20$ for $T = 0.34$ and $60$ for $T = 0.31$ with $\rho_d=10^2 \rho_0$, $\rho_0=3$, $D=1$, $v\simeq0.55$, $\gamma=0.1$, so that ${\rm Pe}=3.0$. System size: $L_x = 1000, ~L_y = 3000$.)
• Figure 2: Stability of counter-propagating bands in simulations of the active XY model. (a-b) At $T = 0.26$ and $\mathrm{Pe} = 1.4$, a counter-propagating band inserted in the ordered phase is unstable and evaporates. (c) Band stability in microscopic simulations (symbols) compared with the linear stability analysis of the leading front (colors). The solid line corresponds to the analytical prediction for the transition line, given in Eq. (\ref{['eq:Tc']}).
• Figure 3: (a) A domain wall (blue line) can be seen as a line under tension going through an unstable region. For a passive system with $v=0$, a transverse fluctuation $\delta m_y/\rho$ (orange line) is amplified and leads to the relaxation of the domain wall into a Goldstone mode spanning the stable manifold $|\mathbf{p}|=|p_0|$ (red line). (b) Instability profile obtained from solving Eq. (\ref{['eq:lambda']}) by a shooting method and from direct PDE integration. (c) Potential $V(\lambda)$ in Eq. (\ref{['eq:lambda']}). There is a unique value of $r$ (the tilt of the potential) such that the drive $\gamma F(p_{\rm f})$ exactly balances the potential barrier. Parameters: $v=D=\gamma=\alpha=1$, giving $r=0.32$. (d) Instability profiles near criticality obtained via a shooting method. The critical profile (${\rm Pe}=0.44$) is partially screened and given by Eq. \ref{['eq:inst']}. (Amplitude normalized to $2$, $v=\gamma=1$, $\alpha=0.11$.)
• Figure 4: (a): Comparison of the transition lines $T_c^b$ to excite a propagating band and $T_c$ to excite a propagating droplet in the hydrodynamic Eqs. \ref{['eq:rho']}-\ref{['eq:m']}. The green star and red triangle indicate the location of the snapshots shown in the right panels, which illustrate the droplet instabilities: for $T<T_c^b$ (c), the center line is unstable, like band profiles, and breaks the droplet (see Movie 5), while for $T_c^b<T<T_c$ (b) the droplet evaporates from the sides (see Movie 6). For $T>T_c$, an initial droplet grows (see Movie 7).