Quantum weight and low-loss EELS signatures of Wannier quantum geometry in black phosphorus

Abstract

Quantum geometry is now experimentally accessible in crystalline solids, with black phosphorus providing a key platform through polarization-resolved angle-resolved photoemission spectroscopy. We develop a first-principles framework that connects the momentum-resolved quantum metric of black phosphorus to a complementary bulk observable: the direction-resolved quantum weight measurable through low-loss electron energy-loss spectroscopy (EELS). A 32-band DFT--Wannier Hamiltonian is used to compute both single-band and occupied-manifold geometric quantities from analytic momentum derivatives. We show that the raw single-band quantum metric of the top valence band is not globally meaningful in the conventional cell because folding degeneracies and intra-valence near degeneracies produce true isolated-band singularities; masked maps and occupied-manifold projectors are therefore essential. Because semilocal PBE produces near-gap semimetallic pockets and spurious subgap interpolation features, we introduce an experimentally motivated restricted quantum weight $K_{ii}(ω_c)$, which obeys the corresponding restricted Souza--Wilkens--Martin sum rule and is the appropriate quantity for low-loss EELS once the zero-loss region is excluded. The restricted in-plane quantum weight is nearly isotropic, $K_{zz}/K_{xx}=0.972\pm0.005$ (armchair/zigzag), despite the strong band-mass anisotropy and armchair-only absorption onset of black phosphorus. Orbital-resolved Hubbard--Hartree corrections leave the absolute quantum weights rigid at the sub-percent level while producing a small but resolved armchair-directed drift of $K_{zz}/K_{xx}$, approximately $+0.46\%$ per eV of $U$. These results identify low-loss EELS spectral moments as a practical probe of integrated quantum geometry in an anisotropic layered material.