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Candidates for correlated topological insulators, originated from thespin-orbit coupling as well as Hubbard type correlation, are expected in the($111$) bilayer of perovskite-structural transition-metal oxides. Based on thefirst-principles calculation and tight-binding model, the electronic structureof a LaMnO$_3$ ($111$) bilayer sandwiched in LaScO$_3$ barriers has beeninvestigated. For the ideal undistorted perovskite structure, the Fermi energyof LaMnO$_3$ ($111$) bilayer just stays at the Dirac point, rendering asemi-metal (graphene-like) which is also a half-metal (different from graphenenor previous studied LaNiO$_3$ ($111$) bilayer). The Dirac cone can be openedby the spin-orbit coupling, giving rise to nontrivial topological bandscorresponding to the (quantized) anomalous Hall effect. For the realisticorthorhombic distorted lattice, the Dirac point moves with increasing Hubbardrepulsion (or equivalent Jahn-Teller distortion). Finally, a Mott gap opens,establishing a phase boundary between the Mott insulator and topologicalmagnetic insulator. Our calculation finds that the gap opened by spin-orbitcoupling is much smaller in the orthorhombic distorted lattice ($\sim$$1.7$meV) than the undistorted one ($\sim$$11$ meV). Therefore, to suppress thelattice distortion can be helpful to enhance the robustness of topologicalphase in perovskite ($111$) bilayers.

Topological magnetic phase in LaMnO$_3$ (111) bilayer