Dissipative phase transitions and passive error correction

We classify different ways to passively protect classical and quantum information, i.e., we do not allow for syndrome measurements, in the context of local Lindblad models for spin systems. Within this family of models, we suggest that passive error correction is associated with nontrivial phases of matter and propose a definition for dissipative phases based on robust steady-state degeneracy of a Lindbladian in the thermodynamic limit. We study three thermalizing models in this context: the two-dimensional (2D) Ising model, the 2D toric code, and the 4D toric code. In the low-temperature phase, the 2D Ising model hosts a robust classical steady-state degeneracy, while the 4D toric code hosts a robust quantum steady-state degeneracy. We perturb the models with terms that violate detailed balance and observe that qualitative features remain unchanged, suggesting that $Z_2$ symmetry breaking in a Lindbladian is useful to protect a classical bit while intrinsic topological order protects a qubit.