Density matrices and fractional incompressible states in magic-angle twisted TMD homobilayers

The recent observations of the fractional quantum Hall effect (FQHE) in AA-stacked twisted bilayers of MoTe₂ have inspired intensive study of moiré materials in the same class. At certain twist angles many TMD homobilayers exhibit flat bands that imply strong correlations. The key property that enables fractional incompressible states appears to be the non-trivial spatial structure of the layer-dependent part of the bilayer single particle Hamiltonian that is present in systems in which the valence band maxima are at the K-point.

Here we examine the properties of band-projected density matrices in K-valley TMD homobilayers, comparing them with the properties of density matrices projected to the lowest Landau level. At long-wavelengths the band-projected density-matrix algebra is closely related to the quantum geometry of the band, but this relationship does not apply at short wavelength. To assess implications for the fractional quantum Hall effect we examine the relationship between density matrix properties and the two-body energy spectra that define Haldane pseudopotentials. We will discuss implications for the ongoing explorations of the full fractional quantum Hall phenomenology in TMD homobilayers.