Ultracold atoms in optical lattices are powerful quantum simulators for strongly correlated systems. Rich physics emerges in the presence of multiple atomic species. With synthetic gauge fields interacting topological states of matter can be simulated.
Here we focus on the SU(3) Hubbard model on the triangular lattice at filling n=1/3. We study the effect of spin-orbit coupling and map out the magnetic phase diagram in the Mott insulator, where we find exotic long-range orders. In the presence of a 3-sublattice staggered potential we investigate the competition between charge and magnetic order, where different intermediate phases emerge between the band insulator and the Mott insulator. An additional gauge potential (flux) stabilizes a Quantum Hall insulator at weak to intermediate interactions.