Revealing Quantum Geometry in Nonlinear Quantum Materials

Abstract Berry curvature-related topological phenomena have been a central topic in condensed matter physics. Yet, until recently other quantum geometric quantities such as the metric and connection received only little attention due to the relatively few effects which have been documented for them.
This review gives a modern perspective how quantum geometric quantities naturally enter the nonlinear responses of quantum materials and demonstrate their deep connection with excitation energy, lifetimes, symmetry, and corresponding physical processes.
The multitude of nonlinear responses can be subdivided into nonlinear optical effects, subgap responses, and nonlinear transport phenomena.
Such a distinction by energy scales facilitates an intuitive understanding of the underlying electronic transitions, giving rise to a unified picture of the electron motion beyond linear order.
The well-known injection and shift currents constitute the main resonances in the optical regime. Exploiting their respective lifetime and symmetry dependencies, this review elucidates how these resonances can be distinguished by a corresponding quantum geometric quantity that shares the same symmetry.
This is followed by a brief exposition of the role of quasiparticle lifetimes for nonlinear subgap responses, which presents a window into the microscopic short-term dynamics as well as the ground state correlation and localization.
We conclude with an account of the anomalous motion due to the Berry curvature dipole and quantum metric dipole in nonlinear transport,
clarifying the correspondence between physical observables and the underlying mechanisms. 
This review highlights the close relationship between quantum geometry and nonlinear response, showing the way towards promising probes of quantum geometry and enabling novel avenues to characterize complex materials.