Resolution of topology and geometry from momentum-resolved spectroscopies

Extracting the topology and quantum geometry encoded in Bloch wavefunctions from momentum-resolved spectroscopy remains challenging. We introduce the wavefunction form factor (WFF) matrix, which represents a spectrally isolated band projector in a finite set of spectroscopic channels, such as those selected by light polarization and photoelectron spin in angle-resolved photoemission spectroscopy. The matrix can be reconstructed from polarization- and/or spin-resolved intensity measurements, including inelastic neutron scattering. When the number of independent spectroscopic channels matches the number of target bands, band topology is encoded in “spectral nodes”—momentum-space points where the WFF determinant vanishes, providing a readout of topological invariants via a “topological selection rule”. Furthermore, when the number of independent probes exceeds the number of target bands and the WFF has no rank drop on the relevant momentum manifold, our framework yields an effective band projector. This enables the extraction of Wilson-loop spectral flow diagnosing the band topology and an effective quantum geometric tensor, providing a model-independent extraction of the probe-resolved non-Abelian Berry curvature and quantum metric.