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 due to 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 which 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.Comment: 15 pages, 6 figure

Defect unbinding in active nematics

We formulate the statistical dynamics of topological defects in the active nematic phase, formed in two dimensions by a collection of self-driven particles on a substrate. An important consequence of the non-equilibrium drive is the spontaneous motility of strength +1/2 disclinations. Starting from the hydrodynamic equations of active nematics, we derive an interacting particle description of defects that includes active torques. We show that activity, within perturbation theory, lowers the defect-unbinding transition temperature, determining a critical line in the temperature-activity plane that separates the quasi-long-range ordered (nematic) and disordered (isotropic) phases. Below a critical activity, defects remain bound as rotational noise decorrelates the directed dynamics of +1/2 defects, stabilizing the quasi-long-range ordered nematic state. This activity threshold vanishes at low temperature, leading to a re-entrant transition. At large enough activity, active forces always exceed thermal ones and the perturbative result fails, suggesting that in this regime activity will always disorder the system. Crucially, rotational diffusion being a two-dimensional phenomenon, defect unbinding cannot be described by a simplified one-dimensional model.Comment: 15 pages (including SI), 4 figures. Significant technical improvements without changing the result

Anomalous elasticity of a cellular tissue vertex model

Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity