  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.
