Direct measurement of the quantum metric tensor in solids

The quantum metric tensor is a central geometric quantity in modern physics that is defined as the distance between nearby quantum states. Despite numerous studies highlighting its relevance to fundamental physical phenomena in solids, measuring the complete quantum metric tensors in real solid-state materials is challenging. In this work, we report a direct measurement of the full quantum metric tensors of Bloch electrons in solids using black phosphorus as a representative material. We extracted the momentum space distribution of the pseudospin texture of the valence band from the polarization dependence of angle-resolved photoemission spectroscopy measurement. Our approach is poised to advance our understanding of quantum geometric responses in a wide class of crystalline systems.