  Recent experiments have shown that magic angle twisted bilayer graphene
(MATBG) can exhibit correlated insulator behavior at half-filling. Seminal
theoretical results towards understanding this phase in MATBG has shown that
Hartree-Fock ground states (with a positive charge gap) can be exact many-body
ground states of an idealized flat band interacting (FBI) Hamiltonian. We prove
that in the absence of spin and valley degrees of freedom, the only
Hartree-Fock ground states of the FBI Hamiltonian for MATBG are two
ferromagnetic Slater determinants. Incorporating spin and valley degrees of
freedom, we provide a complete characterization of the Hartree-Fock ground
state manifold, which is generated by a ${\rm U}(4) \times {\rm U}(4)$ hidden
symmetry group acting on five elements. We also introduce new tools for ruling
out translation symmetry breaking in the Hartree-Fock ground state manifold,
which may be of independent interest.
