Excited state spectra at the superfluid-insulator transition out of paired condensates

We describe gapped single-particle and collective excitations across a superfluid to insulator quantum phase transition of particles (bosons or fermions) in a periodic potential, with an even number of particles per unit cell. We demonstrate that the dynamics is controlled by a quantum impurity problem of a localized particle interacting with the bulk critical modes. Critical exponents are determined by a renormalization group analysis. We discuss applications to spin oscillations of ultracold atoms in optical lattices, and to the electronic phases in the cuprate and related compounds.