Steady-state topological order

We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct concrete lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, dissipative gauge theory, and relaxation dynamics. We take the topological degeneracy of steady states as the defining property of steady-state topological order, and study its stability using degenerate perturbation theory. We show that while the topological degeneracy is fragile in two dimensions, it is robust against perturbations in three and higher dimensions. We unveil how and why nontrivial steady-state topological order is always endowed with unusual relaxation dynamics. Particularly, the relaxation time is algebraically divergent in the thermodynamic limit, which is indicative of a gapless Liouvillian spectrum. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes: Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows. We also investigate the topological phase transition to the trivial phase under additional noise, and find that the lifting of topological degeneracy is accompanied by gapping out the gapless modes. The topological phase transition is also diagnosed from the perspective of dissipative gauge theory, where the topologically ordered phase is identified as the deconfined phase and the trivial phase as the confined phase. Our paper offers a toolbox for investigating open-system topology of steady states.