  The study of non-equilibrium properties in topological systems is of
practical and fundamental importance. Here, we analyze the stationary
properties of a two-dimensional bosonic Hofstadter lattice coupled to two
thermal baths in the quantum open-system formalism. Novel phenomena appear like
chiral edge heat currents that are the out-of-equilibrium counterparts of the
zero-temperature edge currents. They support a new concept of dissipative
symmetry-protection, where a set of discrete symmetries protects topological
heat currents, differing from the symmetry-protection devised in closed systems
and zero-temperature. Remarkably, one of these currents flows opposite to the
decreasing external temperature gradient. As the starting point, we consider
the case of a single external reservoir already showing prominent results like
thermal erasure effects and topological thermal currents. Our results are
experimentally accessible with platforms like photonics systems and optical
lattices.
