  We introduce an explicit scheme to realize Chern insulating phases employing
cold atoms trapped in a state-dependent optical lattice and laser-induced
tunneling processes. The scheme uses two internal states, a ground state and a
long-lived excited state, respectively trapped in separate triangular and
honeycomb optical lattices. A resonant laser coherently coupling the two
internal states enables hopping between the two sublattices with a Peierls-like
phase factor. Although laser-induced hopping by itself does not lead to
topological bands with non-zero Chern numbers, we find that such bands emerge
when adding an auxiliary lattice that perturbs the lattice structure,
effectively turning it at low energies into a realization of the Haldane model:
A two-dimensional honeycomb lattice breaking time-reversal symmetry. We
investigate the parameters of the resulting tight-binding model using
first-principles band structure calculations to estimate the relevant regimes
for experimental implementation.
