Topologically Ordered Steady States in Open Quantum Systems

The interplay between dissipation and correlation can lead to new emergent phenomena. Here we study non-equilibrium phases of matter with robust topological degeneracy of steady states, which is a generalization of the ground-state topological degeneracy of closed systems. Specifically, we construct two representative Lindbladians using engineered dissipation, and exactly solve the steady states with topological degeneracy. We find that while the degeneracy is fragile under noise in two dimensions, it is stable in three dimensions, where a genuine many-body phase with topological degeneracy is realized. We identify universal features of dissipative topological physics such as the deconfined emergent gauge field and slow relaxation dynamics of topological defects. The transition from a topologically ordered phase to a trivial phase is also investigated via numerical simulation. Our work highlights the essential difference between ground-state topological order in closed systems and steady-state topological order in open systems.