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

TL;DR

This paper investigates spin response functions in ultracold fermionic gases, emphasizing the importance of sum rules, and demonstrates how self-consistent calculations within Hartree-Fock-BCS theory can accurately capture pairing effects and satisfy physical constraints.

Contribution

It introduces a self-consistent method to compute spin correlation functions that obey sum rules, highlighting the limitations of simpler approximations and explaining experimental rf spectrum features.

Findings

Self-consistent calculations satisfy the f-sum rule.

Simple one-loop approximations fail to obey the sum rule.

The second peak in rf spectra is linked to pairing effects, shifting with the BCS gap.

Abstract

The spin response functions measured in multi-component 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. As we show, the emergence of a second peak at higher frequency in the rf spectrum, as observed in a recent experiment in trapped $^6\text{Li}$, 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.