Quantum doubles in one dimension

Non-Abelian defects that bind Majorana zero modes (or their parafermionic extensions) are central to intrinsically fault-tolerant quantum computation schemes. Underpinning their established understanding is the quantum Ising spin chain, which can be recast as a fermionic model or viewed as a standalone effective theory for the toric-code edge---both of which harbor non-Abelian defects. We generalize these notions by deriving an effective Ising-like spin chain describing the edge of quantum-double topological order. By relating Majorana and parafermion modes to anyonic strings, we develop quantum-double generalizations of non-Abelian defects. We also provide a continuum description of the spin chain that makes contact with strongly correlated systems via non-Abelian bosonization.