Pairing phenomena in strongly correlated Fermi liquids

The correlated-basis-function method is extended to deal with pairing phenomena in strongly correlated Fermi liquids. With a variational ansatz for the model wave function we derive the "correlated" analog of the conventional Bardeen-Cooper-Schrieffer (or Balian-Werthamer), Anderson-Brinkman-Morel theory of pairing. A suitable (and well-controlled) set of approximations brings the theory into a form identical to the conventional theories, but with the bare interaction replaced by a weak effective interaction and the bare single-particle energies replaced by an effective single-particle spectrum. As usual, liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ provides a very stringent test of the theory, as both the interaction and the experimental facts are pretty clear. The variational estimates for the pairing interaction are improved by nonorthogonal perturbation theory. We find the expected enhancement of the attraction in $P$ waves, although the restriction to effective two-body interactions appears to be insufficient to generate $P$-wave pairing.