Quantum metric dependent anomalous velocity in systems subject to complex electric fields

Berry phases have long been known to significantly alter the properties of periodic systems, resulting in anomalous terms in the semiclassical equations of motion describing wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry connection have been proposed and shown to have novel effects on dynamics and transport. In this paper, we consider perturbing fields that are themselves non-Hermitian, in the form of complex external electric fields, which are realizable as gain/loss gradients. We derive the full set of semiclassical equations of motion and show that the anomalous velocity depends not only on the Berry curvature, but on the entirety of the quantum geometric tensor, including the quantum metric. This quantum metric dependent velocity appears regardless of whether the unperturbed Hamiltonian is Hermitian or not. These analytical results are compared with numerical lattice simulations, which reveal these anomalous terms even in one dimension. Our paper expands the range of phenomena expected to be detectable in experimental setups, which should be realizable in currently available metamaterials and classical wave systems, including mechanical, acoustic, and optical.