From Berry curvature to quantum metric: a new era of quantum geometry metrology for Bloch electrons in solids

Abstract: For decades, ``geometry" in band theory has largely meant Berry phase and Berry curvature-quantities that reshape semiclassical dynamics and underpin modern topological matter. Yet the full geometric content of a Bloch band is richer and encoded in the quantum geometric tensor (QGT), whose imaginary part is the Berry curvature and whose real part is the quantum metric. Here, we briefly review the recent progress in direct experimental access to the QGT in real crystalline solids using the polarization- and spin-resolved angle-resolved photoemission spectroscopy (ARPES). The extraction of the QGT in momentum space was successfully addressed by two different approaches: One is by introducing quasi-QGT that faithfully represents the QGT and is directly measurable by ARPES. The other is through pseudospin tomography in a material with simple low energy band structure, which successfully retrieved all matrix components of the quantum metric. We discuss the physical meaning of these two recent progresses, their implication/limitation, and open directions.