Quantum Geometry in Collective Modes

In recent years it has become increasingly appreciated that electrons in solids possess quantum geometric structure that impact the electronic properties of the system. Typically, this takes the form of a Berry curvature for single electron states, and is physically manifested as an anomalous velocity. In this talk we discuss quantum geometric properties of collective modes. We show that generally such excitations possess their own type of quantum geometric measure, closely related to an electric dipole moment, which we call the quantum geometric dipole (QGD). We will focus on two examples of this, excitons and plasmons in two-dimensional systems, and discuss some physical implications of a non-vanishing QGD.  While these realizations involve neutral excitations which may be described as two-body states,  we will also present a many-body formulation of the QGD that is independent of specific wavefunction forms. As an example, we apply the formalism to collective excitations of fractional quantum Hall states, and show that these generically carry non-zero QGD’s.