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Topological phases supporting non-abelian anyonic excitations have beenproposed as candidates for topological quantum computation. In this paper, westudy 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 $k$ theseanyonic chains realize infinite randomness critical {\it phases} in the sameuniversality class as the $S_k$ permutation symmetric multi-critical points ofDamle and Huse (Phys. Rev. Lett. 89, 277203 (2002)). Indeed, we show that thepertinent subspace of these anyonic chains actually sits inside the ${\mathbbZ}_k \subset S_k$ symmetric sector of the Damle-Huse model, and this ${\mathbbZ}_k$ symmetry stabilizes the phase.

Permutation Symmetric Critical Phases in Disordered Non-Abelian Anyonic Chains