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.

Methods
Theoretical data is collected from the density functional theory calculations and electronic band calculations based on the tight-binding model. Also, the experimental data is collected from the angle-resolved photoemission spectroscopy.