Oriented Active Solids

We present a complete analysis of the linearized dynamics of active solids with uniaxial orientational order, taking into account a hitherto overlooked consequence of rotation invariance. Our predictions include a purely active response of two-dimensional orientationally ordered solids to shear, the possibility of stable active solids with quasi-long-range order in two dimensions and long-range order in three dimensions, generic instability of the solid for one sign of active forcing, and the instability of the uniaxially ordered phase in momentum-conserved systems for large active forcing irrespective of its sign.

Figure 1

An orientationally ordered solid with a sinusoidal distortion. The local direction of orientational ordering is denoted by blue bars. In passive solids, such distortions are “soft,” with the leading order restoring force due to orientational elasticity $\sim {\partial }_x^4{u}_y$ (i.e., no restoring force for a plane shear). Such fluctuations in passive systems destroy the orientationally ordered solid at arbitrarily low noise. Activity leads to an additional restoring force $\sim {\partial }_x^2{u}_y$ whose direction when $b_4>0$ is shown with bold red arrows. This restoring force, which acts as an effective shear modulus (0 in passive systems), leads to the existence of active uniaxially ordered solid.