Ground state wave function overlap in superconductors and superfluids

In order to elucidate the quantum ground state structure of nonrelativistic condensates, we explicitly construct the ground state wave function for multiple species of bosons, describing either superconductivity or superfluidity. Since each field Ψj carries a phase θj and the Lagrangian is invariant under rotations θj → θj + αj for independent αj, one can investigate the corresponding wave function overlap between a pair of ground states $\langle G\vert {G}^{\prime }\rangle $ differing by these phases. We operate in the infinite volume limit and use a particular prescription to define these states by utilizing the position space kernel and regulating the UV modes. We show that this overlap vanishes for most pairs of rotations, including θj → θj + mj ϵ, where mj is the mass of each species, while it is unchanged under the transformation θj → θj + qj ϵ, where qj is the charge of each species. We explain that this is consistent with the distinction between a superfluid, in which there is a nontrivial conserved number, and the superconductor, in which the electric field and conserved charge is screened, while it is compatible with a nonzero order parameter in both cases. Moreover, we find that this bulk ground state wave function overlap directly reflects the Goldstone boson structure of the effective theory and provides a useful diagnostic of its physical phase.