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We apply the technique of quasi-adiabatic continuation to study systems withcontinuous symmetries. We first derive a general form of Goldstone's theoremapplicable to gapped nonrelativistic systems with continuous symmetries. Wethen 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 energyof the system, a scenario which covers several physically interesting casessuch as an s-wave superconductor or a resonating valence bond state.Comment: 19 pages, 2 figures, final version in press at JSTA

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