  Despite much theoretical work, developing a comprehensive ab initio model for
twisted bilayer graphene (TBG) has proven challenging due to the inherent
trade-off between accurately describing the band structure and incorporating
the interactions within the Hamiltonian, particularly given the topological
obstruction -- so-called fragile topology -- to the description of the model in
terms of localized symmetric Wannier functions within the flat band manifold.
Here, we circumvent this obstruction by using an extended 8-orbital model, for
which localized Wannier orbitals have been formulated by Carr et al. [1]. We
constructed an extended multi-orbital Hubbard model, and performed Hartree-Fock
(HF) calculations to explore its phase diagram across commensurate fillings
from -3 to 3. We found several nearly-degenerate insulating states at charge
neutrality, all of which exhibit orbital orders. Crucially, TBG near magic
angle is known to be particle-hole asymmetric, which is naturally captured by
the single-particle band structure of our model and is reflected in the
distinction between the symmetry broken states obtained at electron and hole
dopings away from the charge neutral point. At filling -1 and +2, quantum
anomalous hall states are obtained, while for the rest of the integer fillings
away from charge neutrality, we found the system to realize metallic states
with various orbital, valley and spin orderings. We also observed that most of
the Hartree--Fock ground states exhibit a generalized valley Hund's-like rule,
resulting in valley polarization. Importantly, we show that the incorporation
of the intra-valley and inter-valley exchange interactions is crucial to
properly stabilize the ordered symmetry-broken states. In agreement with
experiments, we find significant particle-hole asymmetry, which underscores the
importance of using particle-hole asymmetric models.
