Permutation Symmetric Critical Phases in Disordered Non-Abelian Anyonic Chains

Topological phases supporting non-abelian anyonic excitations have been proposed as candidates for topological quantum computation. We study disordered non-abelian anyonic chains based on the quantum groups $SU(2)_k$, a hierarchy that includes the $\nu=5/2$ FQH state and the proposed $\nu=12/5$ Fibonacci state, among others. We find that for odd $k$ these anyonic chains realize infinite randomness critical {\it phases} in the same universality class as the $S_k$ permutation symmetric multi-critical points of Damle and Huse (arXiv:cond-mat/0207244). Indeed, we show that the pertinent subspace of these anyonic chains actually maps to the ${Z}_k \subset S_k$ symmetric sector of the Damle-Huse model, and this ${Z}_k$ symmetry stabilizes the phase.