Transport properties of anyons in random topological environments

The quasi-one-dimensional transport of Abelian and non-Abelian anyons is studied in the presence of a random topological background. In particular, we consider the quantum walk of an anyon that braids around islands of randomly filled static anyons of the same type. Two distinct behaviors are identified. We analytically demonstrate that all types of Abelian anyons localize purely due to the statistical phases induced by their random anyonic environment. In contrast, we numerically show that non-Abelian Ising anyons do not localize. This is due to their entanglement with the anyonic environment, which effectively induces dephasing. Our study demonstrates that localization properties strongly depend on nonlocal topological interactions, and it provides a clear distinction in the transport properties of Abelian and non-Abelian anyons.