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

Competing Chiral Orders in the Topological Haldane-Hubbard Model of Spin-1/2 Fermions and Bosons