Topological phases of matter are the center of much current interest. Traditionally such states are prepared by tuning the system’s Hamiltonian while coupling it to a generic low-temperature bath. However, this approach is often ineffective, especially in cold-atom systems. We have recently shown that topological phases can emerge much more efficiently even in the absence of a Hamiltonian, by properly engineering the interaction of the system with its environment, to directly drive the system into the desired state.

Disorder is inherent in any experimental implementation. How does it affect this behavior? Generalizing appropriate topological indexes (local Chern number, Bott index) to the dissipative case, we find that the topological phases are robust to weak disorder. Strong disorder can cause transition to localization. Surprisingly, while disorder in the dissipative dynamics leads to the same universality class as in equilibrium, we give strong evidence that disorder in the Hamiltonian part gives rise to a new critical exponent. This new universality class could be tested in cold atom experiments.