Collapse of FCIs from Fluctuations in Multi-band ED

The pentalayer-hBN superlattice is an unexpected starting point for fractional Chern insulators (FCIs) due to its nearly gapless single-particle band structure. Here we present the results of multi-band ED calculations preserving the large low energy Hilbert space that supports the Chern-1 Hartree-Fock band at integer filling and its excitations, the moiré phonons or (pseudo-)Goldstones. We show that in the "charge neutrality" scheme where the moiré effect is weak, 1-band ED artificially removes the phonons, and the FCI gap which appears is destroyed as multi-band fluctuations are included. In the "average" scheme where the moiré valence charge background is included, we also see the collapse of the FCI gap. We discuss our results in the context of 1-band ED and recent experimental advances.