  The physics of long-range interacting quantum systems is currently living
through a renaissance driven by the fast progress in quantum simulators. In
these systems many paradigms of statistical physics do not apply and also the
universal long-wavelength physics gets substantially modified by the presence
of long-ranged forces. Here we explore the low-energy excitations of several
long-range interacting quantum systems, including spin models and interacting
Bose gases, in the ordered phase associated with the spontaneous breaking of
U(1) and SU(2) symmetries. Instead of the expected Goldstone modes, we find
three qualitatively different regimes, depending on the range of the
interaction. In one of these regimes the Goldstone modes are gapped, via a
generalization of the Higgs mechanism. Moreover, we show how this effect is
realized in current experiments with ultracold atomic gases in optical
cavities.
