Probing quantum geometry of semiconductors by optical absorption

The quantum geometry in the momentum space of semiconductors and insulators, described by the quantum metric of the valence band Bloch state, has been an intriguing issue owing to its connection to various material properties. We show that because the Brillouin zone is a torus for materials in any dimension, the integration of quantum metric over momentum space represents an average distance between neighboring Bloch states, which can further be expressed in real space as a local quantity that we call fidelity marker. A linear response theory is introduced to generalize the fidelity marker to finite temperature, and moreover demonstrates that it can be measured as the local optical absorption rate of linearly polarized light, whose frequency dependence is described by a spectral function. A nonlocal fidelity marker that quantities the overlap between Wannier states is suggested to be a universal indicator of quantum phase transitions at which the quantum metric diverges. The ubiquity of these markers is demonstrated for a variety of topological insulators and topological phase transitions in different dimensions.