Unconventional fusion and braiding of topological defects in a lattice model

We examine non-Abelian topological defects in an Abelian lattice model in two dimensions. We first construct an exact solvable lattice model that exhibits coexisting and intertwined topological and classical orders. The anyon types of quasiparticle excitations are permuted by lattice symmetry operations like translations, rotations, and reflections. The global anyon permutation symmetry has a group structure of S3, the permutation group of three elements. Topological crystalline defects-dislocations and disclinations-change the anyon type of an orbiting quasiparticle. They exhibit multichannel order-dependent fusion rules and projective braiding operations. Their braiding and exchange statistics breaks modular invariance and violates the conventional spin-statistics theorem. We develop a framework to characterize these unconventional properties that originate from the semiclassical nature of defects.