Probing a half-odd topological number sequence with cold atoms in a non-Abelian optical lattice

We propose an experimental scheme to probe the contribution of a single Dirac cone to the Hall conductivity as a half-odd topological number sequence. In the scheme, the quantum anomalous Hall effect in graphene is simulated with cold atoms trapped in an optical lattice and subjected to a laser-induced non-Abelian gauge field. By tuning the laser intensity to change the gauge flux, the energies of the four Dirac points in the first Brillouin zone are shifted from each other and the contribution of the single Dirac cone to the total atomic Hall conductivity is manifested. We also show that the atomic Hall numbers can be experimentally probed with atomic density profile measurements.