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Perovskite bilayers with (111)-orientation combine a honeycomb lattice as akey feature with the strongly correlated, multiorbital nature of electrons intransition metal oxides. In a systematic DFT+$U$ study of (111)-oriented(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$ superlattices, we establish trends in theevolution of ground states versus band filling in (111)-oriented(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$ superlattices, with $X$ spanning the entire$3d$ transition metal series. The competition between local quasi-cubic andglobal triangular symmetry triggers unanticipated broken symmetry phases, withmechanisms ranging from Jahn-Teller distortions, to charge-, spin-, andorbital-ordering. LaMnO$_3$, where spin-orbit coupling opens a sizable gap inthe Dirac-point Fermi surface, emerges as a topological Chern insulator.

Design of Mott and topological phases on buckled 3d-oxide honeycomb lattices