Theories of Linear Response in BCS Superfluids and How They Meet Fundamental Constraints

We address the importance of symmetry and symmetry breaking on linear response theories of fermionic BCS superfluids. The linear theory of a noninteracting Fermi gas is reviewed and several consistency constraints are verified. The challenge to formulate linear response theories of BCS superfluids consistent with density and spin conservation laws comes from the presence of a broken U(1)$_{\textrm{EM}}$ symmetry associated with electromagnetism (EM) and we discuss two routes for circumventing this. The first route follows Nambu's integral-equation approach for the EM vertex function, but this method is not specific for BCS superfluids. We focus on the second route based on a consistent-fluctuation-of-the order-parameter (CFOP) approach where the gauge transformation and the fluctuations of the order parameter are treated on equal footing. The CFOP approach allows one to explicitly verify several important constraints: The EM vertex satisfies not only a Ward identity which guarantees charge conservation but also a $Q$-limit Ward identity associated with the compressibility sum rule. In contrast, the spin degrees of freedom associated with another U(1)$_z$ symmetry are not affected by the Cooper-pair condensation that breaks only the U(1)$_{\textrm{EM}}$ symmetry. As a consequence the collective modes from the fluctuations of the order parameter only couple to the density response function but decouple from the spin response function, which reflects the different fates of the two U(1) symmetries in the superfluid phase. Our formulation lays the ground work for application to more general theories of BCS-Bose Einstein Condensation crossover both above and below $T_c$.