Moiré fractional Chern insulators. IV. Fluctuation-driven collapse in multiband exact diagonalization calculations on rhombohedral graphene

The fractional Chern insulators (FCIs) observed in pentalayer rhombohedral graphene/hexagonal boron nitride superlattices have a unique origin contrary to theoretical expectations: their noninteracting band structure is gapless, unlike standard FCIs and the Landau level. Hartree-Fock (HF) calculations at filling $\ensuremath{\nu}=1$ yield a gapped ground state with Chern number 1 through band mixing, identifying a possible parent state. However, many-body calculations restricted to the occupied HF band predispose the system toward FCIs and are essentially uncontrolled. In this work, we use unbiased multiband exact diagonalization (ED) to allow fluctuations into the gapless bands for two normal-ordering schemes. In the ``charge neutrality'' scheme, the weak moir\'e potential leads to theoretical proposals based on Wigner crystal-like states. However, we find that FCIs seen in one-band ED calculations are destroyed by band mixing, becoming gapless as fluctuations are included. In the ``average'' scheme, the Coulomb interaction with the periodic valence charge background sets up a stronger moir\'e potential. On small systems, FCIs at $\ensuremath{\nu}=1/3$ are destroyed in multiband calculations, while those at $\ensuremath{\nu}=2/3$ are initially strengthened. However, we do not converge to a stable FCI at $\ensuremath{\nu}=2/3$ even on the largest accessible systems. These findings question prior results obtained within projection to a single HF band. They suggest that current models do not support FCIs with correlation length small enough to be converged in current unbiased ED calculations, or do not support FCIs at all.