Establishing gauge invariant linear response of fermionic superfluids with pair fluctuations: A diagrammatic approach

We present a fully gauge-invariant linear response theory for ultra-cold Fermi gases undergoing BCS-Bose-Einstein Condensation (BEC) crossover with pair fluctuation effects included, especially in the superfluid phase. The gauge invariance is verified by introducing an effective external electromagnetic (EM) field. For pure BCS-type superfluids, the gauge invariance of the linear response theory can be maintained by constructing a full external EM vertex. By consistently including fluctuations of the order parameters as the self-energy effect is included in the quasi-particles, the Ward identity (WI) is satisfied as a consequence of gauge invariance. The construction of a gauge invariant vertex is non-trivial since in the broken symmetry phase the Nambu-Goldstone mode intertwines with the pairing fluctuation. We find that, under a suitable diagrammatic representation, the construction of such a vertex can be greatly simplified and allows us to build a vertex with pseudogap effects and consistent with the WI in the superfluid phase. We focus on a particular approach known as the G0Gt-matrix of pairing fluctuations, but our formalism also works for the G0G0t-matrix theory as well.