Direct measurement of the quantum metric tensor in solids

The quantum metric tensor is a central geometric quantity in modern
physics that determines the distance between nearby quantum states.
Despite numerous studies highlighting its relevance to fundamental
physical phenomena in solids, the method for measuring the complete
quantum metric tensors in real solid-state materials remains unknown.
Here, we report the first direct measurement of the full quantum metric
tensors of Bloch electrons in solids using black phosphorus as a
representative material. The key idea is to extract the momentum space
distribution of the pseudospin texture of the valence band from the
polarization dependence of angle-resolved photoemission spectroscopy
measurement. The spectroscopic probe of all possible quantum metric tensor
components is poised to significantly advance our understanding of quantum
geometric responses in a wide class of crystalline systems.