Quantum Geometric Advantage of the Correlated Exciton State in Non-linear Optics

The concept of quantum geometry for single-particle states has revolutionized our interpretation of several emergent properties in condensed matter. However, a description of the quantum geometry for interacting particles and an understanding of its implications are lacking. Here, we show that inherent in the non-linear optical response is a quantum geometry of the correlated electron-hole state (exciton) that arises from the interplay between geometry and interactions - distinct from the quantum geometric properties of the individual electron or hole states. We demonstrate using first principles calculations that this quantum many-body geometry significantly enhances the non-linear optical response in systems with strong excitonic effects. In the case of shift currents, the quantum many-body geometric term arises in the many-body shift vector and can be interpreted as a many-body analogue of the Berry phase. This work lays the foundation to study the quantum geometry of correlated states in experimentally observable settings.