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We investigate few- and many-body states in half-filled ideal topological insulator flat bands realized by two degenerate Landau levels which experience opposite magnetic fields. This serves as a toy model of flat bands in moir\'e materials in which valleys have Chern numbers C=1. We argue that although the spontaneously polarized Ising Chern magnet is a natural ground state for repulsive Coulomb interactions, it can be in reasonable energetic competition with correlated Laughlin states of excitons when short-distance corrections to interactions are included. This is because charge neutral excitons in these bands behave effectively as charged particles in ordinary Landau levels. In particular, the Ising Chern magnet is no longer the ground state once the strength of a short-range intravalley repulsion is about 30% larger than the intervalley repulsion. Remarkably, these excitonic Laughlin states feature valley number fractionalization but no charge fractionalization and a quantized charge Hall conductivity identical to the Ising magnet, ₗₘ=e^2/h, and thus cannot be distinguished from it by ordinary charge transport measurements. The Laughlin state with the highest density of excitons that can be constructed in these bands is an analog of =1/4 bosonic Laughlin state and has no valley polarization even though it spontaneously breaks time reversal symmetry.
Stefanidis et al. (Tue,) studied this question.