  We study adatom-covered single layers of CrSiTe$_3$ and CrGeTe$_3$ using
first-principles calculations based on hybrid functionals. We find that the
insulating ground state of a monolayer of La (Lu) deposited on single-layer
CrSiTe$_3$ (CrGeTe$_3$) carries spontaneously generated current loops around
the Cr sites, These "flux states" induce antiferromagnetically ordered orbital
moments on the Cr sites and are also associated with nontrivial topological
properties. The calculated Chern numbers for these systems are predicted to be
$\pm 1$ even in the absence of spin-orbit coupling, with sizable gaps on the
order of 100 meV. The flux states and the associated topological phases result
from spontaneous time-reversal symmetry breaking due to the presence of
nonlocal Coulomb interactions.
