Topological Sound and Flocking on Curved Surfaces

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature, orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state because of the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow, the system additionally supports long-wavelength propagating sound modes that get gapped by the curvature of the underlying substrate. We analytically compute the steady-state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry-protected topological modes that get localized to special geodesics on the surface (the equator or the neck, respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.

Popular Summary

Flocking, the self-organized and spontaneous motion of a large collection of self-propelled entities, is ubiquitous. Large murmurations of starlings are a familiar example, but such collective motion can also be found on much smaller scales, such as in groups of cells advancing en masse during growth and development. In many biological processes, from cell repair in the folded intestine to the shaping of embryonic limb buds, the motion of cells takes place on curved surfaces. Understanding the physical aspects of flocking on curved surfaces therefore remains crucial to developing insight into the organizational principles of living matter.

Using a continuum hydrodynamic model, we performed analytical computations that show that flocks on a sphere take the form of a moving band that wraps around an equator. Our calculations show that similar inhomogeneous traveling structures that would be impossible in equilibrium arise in other curved geometries. Furthermore, on the sphere, we find that a finite energy is required to excite traveling sound waves. We show that such sound waves are “topologically protected” and therefore robust to scattering by impurities and disorder. The exotic unidirectional sound modes found when flocks move on a curved surface are akin to edge states in integer quantum Hall systems and to well-known equatorial waves observed in the flows of oceans or in the Earth’s atmosphere.

The existence of unidirectional propagating modes that are robust against disorder appears to be a generic property of flocking systems in curved spaces, suggesting the intriguing possibility that nature may take advantage of this mechanism to control cell motion and force transmission in curved environments.

Figure 1

(a) The normalized density profile of a polar flock on a sphere given in Eq. (12), for $\eta =0.5$ (blue), 1 (orange), and 2 (green). (b) The density and polarization profiles for $\eta =2$, now shown on the sphere. The color describes the density from the maximum (red) at the center of the polar band to ${\rho }_c$ (blue) at the poles of the sphere. The polarization also vanishes at the poles.

Figure 2

The peak (maximum) density at the equator on a sphere, as we vary the mean density ${\rho }_0$. For ${\rho }_0<{\rho }_c$, we are in the disordered phase with ${\rho }_{ss}(\pi /2)={\rho }_0$. For ${\rho }_0>{\rho }_c$, we have a polar band with the density profile given in Fig. 1. The density reaches its maximum at the center of the band and grows with ${\rho }_0$, with a slope $A_{\eta }>1$. When the convective parameter $\lambda \to 0$, $\eta \to 0$, resulting in $A_{\eta }\to 1$, and we go back to the homogeneous profile as in the flat plane.

Figure 3

The relevant (slow) sound modes. (a) $m=0$ and the gap between the two bands is closed. Panels (b) and (c) are the dispersion bands for $m=0.2$ [Eq. (25)], and we directly see that a gap has opened in the real part of the spectrum, while the imaginary part of the frequency has a single crossing line at $q_y=0$. The variables are chosen to be ${\theta }_0\simeq 78°$, $\alpha \simeq 2.03$, $\overline{\lambda }\simeq 1.03$, and $\beta ={\overline{v}}_1=1$.

Figure 4

The bulk and edge mode spectrum for the case when $s=\pm 1$ (shown for the simple case when $\mu ={\overline{v}}_1$) and $m(y)$ varying from $-1$ to $+1$.

Figure 5

(a) The normalized density profile of a polar flock on a catenoid given in Eq. (40), for $\eta =0.5$ (blue line), 1 (orange line), and 2 (green line). Note that, unlike the sphere, the density grows near the edge of the catenoid. (b) The density and polarization (for $\eta =2$), now shown on the catenoid. As before, blue corresponds to low-density regions (at the neck) and red to high-density regions.

Figure 6

A representative snapshot of the equatorial density mode on a sphere (Eq. (36)) and the localized Goldstone mode on the catenoid (Eq. (42)). For clear visualization we have chosen the perturbation ${\mathsf{a}}_6$, ${\mathsf{b}}_6=0.5$ and all other an, ${\mathsf{b}}_n=0$ ($n\ne 6$). We have also taken both $\eta =\mu /{\overline{v}}_1=2$ both cases.