Quantum Anomalous Hall State in Ferromagnetic SrRuO$_3$ (111) Bilayers

SrRuO$_3$ heterostructures grown in the (111) direction are a rare example of thin film ferromagnets. By means of density functional theory plus dynamical mean field theory we show that the half-metallic ferromagnetic state with an ordered magnetic moment of 2$mu_{B}$/Ru survives the ultimate dimensional confinement down to a bilayer, even at elevated temperatures of 500$,$K. In the minority channel, the spin-orbit coupling opens a gap at the linear band crossing corresponding to $frac34$ filling of the $t_{2g}$ shell. We demonstrate that the respective state is Haldanes quantum anomalous Hall state with Chern number $C$=1, without an external magnetic field or magnetic impurities.