Screw symmetry, chiral hydrodynamics, and odd instability in active cholesterics

Active cholesterics are chiral in both their structure, which has continuous screw symmetry, and their active stresses, which include contributions from torque dipoles. Both expressions of chirality give rise to curl forces in the hydrodynamics, which we derive from the active Ericksen-Leslie equations using a geometric approach. This clarifies the hydrodynamics of continuous screw symmetry and provides an example of generalized odd elastic forces that originate from an equilibrium free energy. We also discuss the nonlinear structure of the active hydrodynamics in terms of the Eulerian displacement field of the cholesteric pseudolayers. For the active instability, screw symmetry generates a contribution of chiral activity to the linearized pseudolayer hydrodynamics that is absent in materials with chiral activity but achiral structure. When the two forms are sufficiently antagonistic, this term produces a new active instability with a threshold and a characteristic wave vector distinct from those of the active Helfrich-Hurault instability in chiral active smectics. Finally, we comment on the isotropic chiral hydrodynamics of materials with three-dimensional screw symmetry.