Self-assembled topological metamaterials: Weyl points for light and sound

Soft materials such as liquid crystals, block copolymers, or colloidal particles can self-assemble into highly structured phases which replicate at the mesoscopic scale the symmetry of atomic crystals. As such, they offer an unparalleled platform to design metamaterials for light and sound. Here, we present a bottom-up approach based on self-assembly to engineer three-dimensional photonic and phononic metamaterials with topologically protected Weyl points. In addition to angular and frequency selectivity of their bulk optical response, Weyl metamaterials are endowed with topological surface states, which allows for the existence of one-way channels even in the presence of time-reversal invariance. Using a combination of group-theoretical methods and numerical simulations, we identify the general symmetry constraints that a self-assembled structure has to satisfy to host Weyl points, and describe how to achieve such constraints using a symmetry-driven pipeline for metamaterial design and discovery.