Intraband collective excitations in fractional Chern insulators are dark

Motivated by the discovery of fractional Chern insulator (FCI) states in flat-band moiré-TMD and rhombohedral multilayer graphene, we investigate density--density correlations of flat band FCI states and examine what they imply about the light response of low-energy collective excitations. In contrast to semiconductors and band insulators, in which collective excitations often couple strongly to light, we argue that for FCI states the intraband collective excitations are anomalously dark. We show how the interband density response at long wavelengths is determined by the band's quantum geometry, while the total density response of the band's FCI state is determined by Stillinger-Lovett perfect screening in a way that leaves no room for intraband density response at long wavelengths. Further, the same conclusion follows from relating the density response to current correlations and vanishing intraband matrix elements for the current operator in flat bands, implying Stillinger-Lovett perfect screening correlations in any FCI state. Our theoretical results shed new light on density correlations in fractional Chern insulators found in ideal Chern-1 flat bands and have important implications for optical signatures of intraband collective excitations in FCIs.