Ground State Wave Function Overlap in Superconductors and Superfluids

In order to elucidate the quantum ground state structure of non-relativistic condensates, we explicitly construct the ground state wave function for multiple species of bosons, describing either superconductivity or superfluidity. Since each field $\Psi_j$ carries a phase $\theta_j$ and the Lagrangian is invariant under rotations $\theta_j \to \theta_j + \alpha_j$ for independent $\alpha_j$, one can investigate the corresponding wave function overlap between a pair of ground states $\langle G|G'\rangle$ 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 $\theta_j \to \theta_j + m_j \epsilon$, where $m_j$ is the mass of each species, while it is unchanged under the transformation $\theta_j \to \theta_j + q_j \epsilon$, where $q_j$ is the charge of each species. We explain that this is consistent with the distinction between a superfluid, in which there is a non-trivial conserved number, and the superconductor, in which the electric field and conserved charge is screened, while it is compatible with a non-zero 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.