  We study two-dimensional bosonic and fermionic lattice systems under
nonequilibrium conditions corresponding to a sharp gradient of temperature
imposed by two thermal baths. In particular, we consider a lattice model with
broken time-reversal symmetry that exhibits both topologically trivial and
nontrivial phases. Using a nonperturbative Green function approach, we
characterize the nonequilibrium current distribution in different parameter
regimes. For both bosonic and fermionic systems, we find chiral edge currents
that are robust against coupling to reservoirs and to the presence of defects
on the boundary or in the bulk. This robustness not only originates from
topological effects at zero temperature but, remarkably, also persists as a
result of dissipative symmetries in regimes where band topology plays no role.
Chirality of the edge currents implies that energy locally flows against the
temperature gradient without any external work input. In the fermionic case,
there is also a regime with topologically protected boundary currents, which
nonetheless do not circulate around all system edges.
