Quantum Geometry Phenomena in Condensed Matter Systems

Quantum geometry, which describes the geometry of Bloch wavefunctions in solids, has become a cornerstone of modern quantum condensed matter physics. The quantum geometrical tensor encodes this geometry through two fundamental components: the quantum metric (real part) and the Berry curvature (imaginary part). While the Berry curvature gained prominence through its manifestation in the intrinsic anomalous Hall effect, recent advances have revealed equally significant effects arising from the quantum metric. This includes its signatures in nonlinear transport, superfluid density of flat-band superconductors, and nonlinear optical responses. These advances underscore how quantum geometry is reshaping our understanding of condensed matter systems, with far-reaching implications for future technologies. In this review, we survey recent progress in the field, focusing on both foundational concepts and emergent phenomena in transport and optics-with particular emphasis on the pivotal role of the quantum metric.