We report a density functional theory plus dynamical mean field theory (DFT+DMFT) study of an oxide heterostructure of LaNiO$_3$ (LNO) bilayers in [111] direction interleaved with four atomic monolayers of LaAlO$_3$. DFT+$U$ optimizations yield two stable solutions: a uniform structure with equivalent NiO$_6$ octahedra, as well as a bond-disproportionated (BD) structure featuring a breathing distortion. For both structures, we construct the low-energy models describing the Ni $e_g$ states by means of Wannier projections supplemented by the Kanamori interaction, and solve them by DMFT. Using the continuous-time quantum Monte Carlo algorithm in the hybridization expansion, we study the temperature range between 145 and 450 K. For the uniform and the BD structure, we find similar phase diagrams that comprise four phases: a ferromagnetic metal (FM), a paramagnetic metal (PM), an antiferro-orbitally-ordered insulator (AOI), as well as a paramagnetic insulator (PI). By calculating momentum-resolved spectral functions on a torus and a cylinder, we demonstrate that the FM phase is not a Dirac metal, while both insulating phases are topologically trivial. By a comparison with available experimental data and model DMFT studies for the two-orbital Hubbard model, we suggest that LNO bilayers are in the AOI phase at room temperature.
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