Resolution of Topology and Geometry from Momentum-Resolved Spectroscopies

TL;DR

This paper introduces a new method to directly extract quantum geometric and topological properties of Bloch wavefunctions from momentum-resolved spectroscopic data, enabling model-independent measurements of topological invariants and quantum metrics.

Contribution

The authors develop the wavefunction form factor matrix and demonstrate its ability to encode topological invariants and quantum geometric tensors from experimental spectroscopies.

Findings

Topological invariants can be read from spectral nodes where the WFF determinant vanishes.

The framework allows extraction of Wilson loop spectra and quantum geometric tensors.

Effective band projectors can be constructed when multiple probes are used.

Abstract

Extracting the complete quantum geometric and topological character of Bloch wavefunctions from experiments remains a challenge in condensed matter physics. Here, we resolve this by introducing the "wavefunction form factor" (WFF) matrix, a quantity directly constructible from intensities in momentum- and energy-resolved spectroscopies like ARPES and INS. We demonstrate that band topology is encoded in "spectral nodes" -- momentum-space points where the WFF determinant vanishes, providing a direct readout of topological invariants via a topological selection rule. Furthermore, when the number of independent probes exceeds the number of the target bands, our framework yields an effective band projector. This enables the determination of Wilson loop spectra and the extraction of an effective quantum geometric tensor, providing a model-independent measurement of the non-Abelian Berry curvature and quantum metric as resolved by the experimental probes.