  We apply the technique of quasi-adiabatic continuation to study systems with
continuous symmetries. We first derive a general form of Goldstone's theorem
applicable to gapped nonrelativistic systems with continuous symmetries. We
then show that for a fermionic system with a spin gap, it is possible to insert
$\pi$-flux into a cylinder with only exponentially small change in the energy
of the system, a scenario which covers several physically interesting cases
such as an s-wave superconductor or a resonating valence bond state.
