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We demonstrate that active rotations in chemically signalling particles, suchas autochemotactic {\it E. coli} close to walls, create a route for patternformation based on a nonlinear yet deterministic instability mechanism. Forslow rotations, we find a transient persistence of the uniform state, followedby a sudden formation of clusters contingent on locking of the averagepropulsion direction by chemotaxis. These clusters coarsen, which results inphase separation into a dense and a dilute region. Faster rotations arrestphase separation leading to a global travelling wave of rotors withsynchronized roto-translational motion. Our results elucidate the physicsresulting from the competition of two generic paradigms in active matter,chemotaxis and active rotations, and show that the latter provides a tool todesign programmable self-assembly of active matter, for example to controlcoarsening.

Pattern Formation in Chemically Interacting Active Rotors with Self-Propulsion