  Strong interactions between electrons occupying bands of opposite (or like)
topological quantum numbers (Chern$=\pm1$), and with flat dispersion, are
studied by using lowest Landau level (LLL) wavefunctions. More precisely, we
determine the ground states for two scenarios at half-filling: (i) LLL's with
opposite sign of magnetic field, and therefore opposite Chern number; and (ii)
LLL's with the same magnetic field. In the first scenario -- which we argue to
be a toy model inspired by the chirally symmetric continuum model for twisted
bilayer graphene -- the opposite Chern LLL's are Kramer pairs, and thus there
exists time-reversal symmetry ($\mathbb{Z}_2$). Turning on repulsive
interactions drives the system to spontaneously break time-reversal symmetry --
a quantum anomalous Hall state described by one particle per LLL orbital,
either all positive Chern $|++\cdots+>$ or all negative $|--\cdots->$. If
instead, interactions are taken between electrons of like-Chern number, the
ground state is an $SU(2)$ ferromagnet, with total spin pointing along an
arbitrary direction, as with the $\nu=1$ spin-$\frac{1}{2}$ quantum Hall
ferromagnet. The ground states and some of their excitations for both of these
scenarios are argued analytically, and further complimented by density matrix
renormalization group (DMRG) and exact diagonalization.
