  We describe a scheme to engineer non-Abelian gauge potentials on a square
optical lattice using laser-induced transitions. We emphasize the case of
two-electron atoms, where the electronic ground state g is laser coupled to a
metastable state e within a state-dependent optical lattice. In this scheme,
the alternating pattern of lattice sites hosting g and e states depict a
checkerboard structure, allowing for laser-assisted tunneling along both
spatial directions. In this configuration, the nuclear spin of the atoms can be
viewed as a "flavor" quantum number undergoing non-Abelian tunneling along
nearest-neighbor links. We show that this technique can be useful to simulate
the equivalent of the Haldane quantum Hall model using cold atoms trapped in
square optical lattices, offering an interesting route to realize Chern
insulators. The emblematic Haldane model is particularly suited to investigate
the physics of topological insulators, but requires, in its original form,
complex hopping terms beyond nearest-neighboring sites. In general, this
drawback inhibits a direct realization with cold atoms, using standard
laser-induced tunneling techniques. We demonstrate that a simple mapping allows
to express this model in terms of matrix hopping operators, that are defined on
a standard square lattice. This mapping is investigated for two models that
lead to anomalous quantum Hall phases. We discuss the practical implementation
of such models, exploiting laser-induced tunneling methods applied to the
checkerboard optical lattice.
