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

Dihedral ('$k$-atic') 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 $k$-atic DLCs can host point defects of fractional topological charge $\pm m/k$. Starting from a generic microscopic model, we derive a unified hydrodynamic description of DLCs with aligning or anti-aligning short-range interactions in terms of Ginzburg-Landau and Landau-Brazovskii-Swift-Hohenberg theories for a universal complex order-parameter field. Building on this framework, we demonstrate in both particle and continuum simulations how adiabatic braiding protocols, implemented through suitable boundary conditions, can emulate anyonic exchange behavior in a classical system. Analytic solutions and simulations of the mean-field theory further predict a novel spontaneous chiral symmetry breaking transition in anti-aligning DLCs, in quantitative agreement with the patterns observed in particle simulations.