| Summary: | When one of the graphene layers of Magic Angle Twisted Bilayer Graphene is
nearly aligned with its hexagonal boron nitride substrate (a configuration
dubbed TBG/hBN), the active electronic bands are nearly flat, and have a Chern
number $C=\pm1$. Recent experiments demonstrated a quantum anomalous Hall
effect and spontaneous valley polarization at integer filling $\nu_T=3$ of the
conduction band in this system. Motivated by this discovery, we ask whether
fractional quantum anomalous Hall states (FQAH) could also emerge in TBG/hBN.
We focus on the range of filling fractions where valley ferromagnetism was
observed experimentally. Using exact diagonalization, we find that the ground
states at $\nu_T = \frac{10}{3}$ and $\nu_T=\frac{17}{5}$ are fractional Chern
insulator states in the flat band limit (in the hole picture, these are the
fractional quantum Hall fractions $\frac{2}{3}$ and $\frac{3}{5}$). The ground
state is either spin polarized or a spin singlet depending sensitively on band
parameters. For nominally realistic band parameters, spin polarization is
favored. Flattening the Berry curvature by changing a band parameter gives way
to the spin singlet phase. Our estimation of the charge gap in the flat band
limit shows that the FQAH state may be seen at accessible temperatures in
experiments. We also study the effect of a non-zero bandwidth and show that
there is a reasonable range of parameters in which the FQAH state is the ground
state.
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