Spin-correlation functions in ultracold paired atomic-fermion systems: Sum rules, self-consistent approximations, and mean fields

The spin response functions measured in multicomponent fermion gases by means of rf transitions between hyperfine states are strongly constrained by the symmetry of the interatomic interactions. Such constraints are reflected in the spin f-sum rule that the response functions must obey. In particular, only if the effective interactions are not fully invariant in SU(2) spin space, are the response functions sensitive to mean field and pairing effects. We demonstrate, via a self-consistent calculation of the spin-spin correlation function within the framework of Hartree-Fock-BCS theory, how one can derive a correlation function explicitly obeying the f-sum rule. By contrast, simple one-loop approximations to the spin response functions do not satisfy the sum rule, except in special cases. As we show, the emergence of a second peak at higher frequency in the rf spectrum, as observed in a recent experiment in trapped Li-6, can be understood as the contribution from the paired fermions, with a shift of the peak from the normal particle response proportional to the square of the BCS pairing gap.