  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.
