Quantum geometry in quantum materials

Abstract Quantum geometry, characterized by the quantum geometric tensor, plays a central role in diverse physical phenomena in quantum materials. This pedagogical review introduces the concept and highlights its implications across multiple domains, including optical responses, Landau levels, fractional Chern insulators, superfluid weight, spin stiffness, exciton condensates, and electron-phonon coupling. By integrating these topics, we emphasize the broad significance of quantum geometry in understanding emergent behaviors in quantum systems and conclude with an outlook on open questions and future directions.