  Engineered dissipation can be employed to prepare interesting quantum many
body states in a non-equilibrium fashion. The basic idea is to obtain the state
of interest as the unique steady state of a quantum master equation,
irrespective of the initial state. Due to a fundamental competition of topology
and locality, the dissipative preparation of gapped topological phases with a
non-vanishing Chern number has so far remained elusive. Here, we study the open
quantum system dynamics of fermions on a two-dimensional lattice in the
framework of a Lindblad master equation. In particular, we discover a mechanism
to dissipatively prepare a topological steady state with non-zero Chern number
by means of short-range system bath interaction. Quite remarkably, this gives
rise to a stable topological phase in a non-equilibrium phase diagram. We
demonstrate how our theoretical construction can be implemented in a
microscopic model that is experimentally feasible with cold atoms in optical
lattices.
