Anyonic defect braiding and spontaneous chiral symmetry breaking in dihedral liquid crystals

Dihedral liquid crystals (DLCs) are assemblies of microscopic constituent particles that exhibit k-fold discrete rotational and reflection symmetries. Generalizing the half-integer defects in nematic liquid crystals, two-dimensional DLCs can host point defects of fractional topological charge ±m/k. Starting from a generic microscopic model, we derive a unified hydrodynamic description of DLCs with aligning and anti-aligning interactions in terms of Ginzburg-Landau and Swift-Hohenberg theories for a universal complex order-parameter field. Using this, we demonstrate in both continuum and particle simulations how braiding protocols, implemented through a suitable boundary anchoring, can realize classical counterparts of anyonic exchange symmetries. The theory further predicts a novel spontaneous chiral symmetry breaking transition in anti-aligning DLCs, in quantitative agreement with particle simulations. In view of recent advances in the design and assembly of dihedral colloids, we expect that these theoretical predictions can be realized with currently available technology, promising a path to fractional topological information storage in soft matter systems.