The emergence of topological color insulators in optical lattices as quantum phases of SU(3) ultra-cold neutral fermions is discussed. The Chern matrix is constructed and used to classify all insulating phases in terms of three topological invariants: the charge-charge, the color-charge and the color-color Chern numbers. The classification for SU(3) topological insulators transcends that of SU(2) systems which require only the charge-charge (charge-Hall) and spincharge (spin-Hall) Chern numbers^[1]. In order to illustrate the topological classification of the insulating phases of SU(3) fermions, phase diagrams of chemical potential and color-orbit parameter versus color-flip fields for fixed magnetic flux ratio are discussed.^[2]

• ^[1] Man Hon Yau and C. A. R. Sá de Melo, Chern-number spectrum of ultracold fermions in optical lattices tuned independently via artificial magnetic, Zeeman, and spin-orbit fields, Phys. Rev. A 99, 043625 (2019).
• ^[2] Man Hon Yau and C. A. R. Sá de Melo, Topological Color-Hall Insulators: SU(3) Fermions in Optical Lattices, arXiv:1904.12791v1 (2019).