Quasi-adiabatic continuation in gapped spin and fermion systems: Goldstone's theorem and flux periodicity

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.