Decohering topological order (TO) is central to the many-body physics of open quantum matter and decoding transitions. We identify statistical mechanical models for decohering non-Abelian TOs, which have been crucial for understanding the error threshold of Abelian stabilizer codes. The decohered density matrix can be described by loop models, whose topological loop weight $N$ is the quantum dimension of the decohering anyon—reducing to the Ising model if $N=1$. In particular, the Rényi-$n$ moments correspond to $n$ coupled $\mathrm{O}(N)$ loop models. Moreover, by diagonalizing the density matrix at maximal error rate, we connect the fidelity between two logically distinct ground states to random $\mathrm{O}(N)$ loop and spin models. We find a remarkable stability to quantum channels which proliferate non-Abelian anyons with large quantum dimension, with the possibility of critical phases for smaller dimensions. Intuitively, this stability is due to non-Abelian anyons not admitting finite-depth string operators. We confirm our framework with exact results for Kitaev quantum double models, and with numerical simulations for the non-Abelian phase of the Kitaev honeycomb model. Our work opens up the possibility of non-Abelian TO being robust against maximally proliferating certain anyons, which can inform error-correction studies of these topological memories.

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• Open quantum systems & decoherence
• Quantum memories
• Topological order