Inter-band entangled fractional Chern states in moire systems with multiple bands

2D moire materials offer ideal condition for realization of fractional Chern insulators (FCIs) due to the presence of topological flat bands at the Fermi level. With the recent discovery of FCI states in twisted bilayer MoTe2 and pentalayer graphene, research has intensified in this field. These FCIs replicate fractional quantum Hall physics without magnetic field. Fractionally filled Chern number 1 bands replicate Landau level physics, whereas fractionally filled higher Chern number bands are expected to replicate quantum Hall multilayer physics. In this work we explore if a 2D moire system can give rise to physics beyond fractional quantum Hall physics. Working on a moire system where the Chern band are degenerate with trivial bands, we show using exact diagonalization that the answer is affirmative. Particularly, the Chern number of the fractionally filled states can be different than the filling fraction in these systems. We show that in some cases these states are adiabatically connected to a folded single Chern band, but in other cases, this adiabatic connection cannot be established without breaking rotation symmetries.