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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. In this paper, we study disordered non-abelian anyonic chains based on the quantum groups $SU(2)_k$, a hierarchy that includes the $ν=5/2$ FQH state and the proposed $ν=12/5$ 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 (Phys. Rev. Lett. 89, 277203 (2002)). Indeed, we show that the pertinent subspace of these anyonic chains actually sits inside the ${\mathbb Z}_k \subset S_k$ symmetric sector of the Damle-Huse model, and this ${\mathbb Z}_k$ symmetry stabilizes the phase.