The breakdown of superflow in Bose superfluids has been of recent interest in the context of cold atomic gases. We study the analogous superflow breakdown problem for interacting fermions on a lattice, and point out the existence of two different dynamical instabilities associated with modulated charge order - a soft roton instability at strong coupling and an exciton condensation instability at intermediate coupling. These should be observable using Bragg scattering experiments in a running optical lattice. |