Cell Natural Orbitals in Interacting Topological Bands
Topological bands exhibit obstruction to exponentially localized and symmetric Wannier functions, challenging the standard paradigm of representing projected interactions in terms of local orbitals with finite range. To faithfully capture the form factors and quantum geometry of topological bands we introduce a singular-value decomposition of the band-projected density form factors, enabling a geometry-based truncation scheme of the Hilbert space, exposing an intrinsic hierarchy on band-projected interactions that is determined by the underlying wavefunctions. This decomposition is most naturally described in terms of Cell Natural Orbitals (CNOs), as the eigenstates of the unit-cell reduced one-particle density matrix, whose occupation provide a measure of the minimal orbital complexity required to faithfully represent the band wavefunctions overlaps. The CNO decomposition identifies systematically the minimal number of local orbitals needed to reproduce short-ranged interactions while resolving the hierarchy of interaction strengths across CNO channels. Applied to magic-angle twisted bilayer graphene in the chiral limit, we find that the dominant CNO is centered at the AA site, resembling the $f$-fermion of the heavy-fermion model. The subdominant CNO channels carry progressively weaker interaction matrix elements, allowing them to be treated at the static mean-field level, while the dominant channel requires a dynamical self-energy. The formalism illustrates how variations of charge density within the unit cell generate momentum dependence in the CNO envelope function and, consequently, dispersion in the single-particle spectral function. More broadly, our results establish CNOs as a geometry-informed bridge between band topology and real-space correlations, providing a systematic framework for analyzing interactions and emergent phases in quantum materials.