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We study ultracold color fermions with three internal states Red, Green andBlue with ${\rm SU(3)}$ symmetry in optical lattices, when color-orbit couplingand color-flip fields are present. This system corresponds to a generalizationof two-internal state fermions with ${\rm SU(2)}$ symmetry in the presence ofspin-orbit coupling and spin-flipping Zeeman fields. We investigate theeigenspectrum and Chern numbers to describe different topological phases thatemerge in the phase diagrams of color-orbit coupled fermions in opticallattices. We obtain the phases as a function of artificial magnetic,color-orbit and color-flip fields that can be independently controlled. Forfixed artificial magnetic flux ratio, we identify topological quantum phasesand phase transitions in the phase diagrams of chemical potential versuscolor-flip fields or color-orbit coupling, where the chirality and number ofmidgap edge states changes. The topologically non-trivial phases are classifiedin three groups: the first group has total non-zero chirality and exhibit onlythe quantum charge Hall effect; the second group has total non-zero chiralityand exhibit both quantum charge and quantum color Hall effects; and the thirdgroup has total zero chirality, but exhibit the quantum color Hall effect.These phases are generalizations of the quantum Hall and quantum spin Hallphases for charged spin-$1/2$ fermions. Lastly, we also describe the colordensity of states and a staircase structure in the total and color fillingfactors versus chemical potential for fixed color-orbit, color-flip andmagnetic flux ratio. We show the existence of incompressible states at rationalfilling factors precisely given by a gap-labelling theorem that relates thefilling factors to the magnetic flux ratio and topological quantum numbers.

Eigenspectrum, Chern Numbers and Phase Diagrams of Ultracold Color-orbit Coupled SU(3) Fermions in Optical Lattices