Transport properties of anyons in random topological environments

The quasi one-dimensional transport of Abelian and non-Abelian (Ising) anyons is studied. We consider the quantum walk of an anyon that braids around islands filled by static anyons of the same type. (a) In the case of a uniform filling, Abelian anyons follow the usual quantum-walk ballistic dispersion whereas the non-Abelian (Ising model) anyons spread as a classical diffusion process. This is due to the entanglement of the walker with topological fusion degrees of freedom. (b) In the case of a random filling, we show that non-Abelian anyons maintain their diffusive behavior while the Abelian ones stop propagating -- a phenomenon known as the Anderson localization.