  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 Hamiltonian while coupling it to a generic bath at very low
temperatures; This approach is often ineffective, especially in cold-atom
systems. It was 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 their transport properties, such as persistent currents and the
conductance 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 conductance is broken. We explore the
intriguing edge behaviors and elucidate under which conditions the Hall
conductance is quantized.
