Sensing vibrations using quantum geometry of electrons

We show that the coupling of phonons with electrons can have nontrivial consequences to the quantum geometry of an electronic structure, which manifests as oscillations in the Berry curvature dipole and hence have observable nonlinear Hall signatures. Using these, we introduce a vibrational spectroscopy based on the geometry of quantum electronic structure (GQuES) making specific predictions for the transport and radiative GQuES spectra of two-dimensional materials. The selection rule presented here for the GQuES activity of a phonon allows the measurement of acoustic and optic phonons spanning sub-GHz, THz, and infrared frequencies, and is readily generalized to other dynamical excitations. GQuES can be used even for materials having trivial quantum electronic geometry, such as hexagonal boron nitride, through a proximal interaction with substrates such as graphene.