Strong interactions between electrons occupying bands of opposite (or
like) topological quantum numbers (Chern=\pm1=±1),
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}_2ℤ2).
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+}\rangle|++⋯+⟩
or all negative |{--\cdots-}\rangle|−−⋯−⟩.
If instead, interactions are taken between electrons of like-Chern
number, the ground state is an SU(2)SU(2)
ferromagnet, with total spin pointing along an arbitrary direction, as
with the \nu=1ν=1
spin-\frac{1}{2}12
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.
Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram