  Motivated by experiments on ultracold atoms which have realized the Haldane
model for a Chern insulator, we consider its strongly correlated Mott limit
with spin-$1/2$ fermions. We find that slave rotor mean field theory yields
gapped or gapless chiral spin liquid Mott insulators. To study competing
magnetic orders, we consider the strong coupling effective spin Hamiltonian
which includes chiral three-spin exchange. We obtain its classical phase
diagram, uncovering various chiral magnetic orders including tetrahedral, cone,
and noncoplanar spiral states which can compete with putative chiral quantum
spin liquids. We study the effect of thermal fluctuations on these states,
identifying crossovers in the spin chirality, and phase transitions associated
with lattice symmetry breaking. We also discuss analogous effective spin
Hamiltonians for correlated spin-$1/2$ bosons. Finally, we point out possible
experimental implications of our results for cold atom experiments.
