Topological phases of matter are the center of much current interest, with promising potential applications in, e.g., topologically-protected transport and quantum computing.

Traditionally such states are prepared by tuning the system's Hamiltonian while coupling it to a generic bath at very low temperatures; 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.

Here we concentrate on dissipatively-induced 2D Chern insulator (lattice quantum Hall) states. We employ open quantum systems tools to explore its transport properties, such as persistent currents and the conductivity in the steady state, in the presence of various Hamiltonians. We find that, in contrast with equilibrium systems, the usual relation between the Chern topological number and the Hall conductivity is broken. We explore the intriguing edge behaviors and elucidate under which conditions the Hall conductivity is quantized.