Theory of a Superfluid Fermi Liquid. I. General Formalism and Static Properties

The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with $S$-wave pairing, and applied to the calculation of the static properties. As a function of $\theta \equiv \frac{T}{T_c}$ these properties are determined entirely by the Landau parameters $F_0$, $F_1$, $Z_0$, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\chi$ and the density of the normal component ${\rho }_n$ are given by $\frac{\chi \left(\theta \right)}{\chi \mathrm{}\left(1\right)\mathrm{}}=(1+\frac{1}{4}{Z}_0)\frac{f\left(\theta \right)}{[1+\frac{1}{4}{Z}_{0}f(\theta \left)\right]},$ $\frac{{\rho }_n}{\rho }=(1+\frac{1}{3}{F}_1)\frac{f\left(\theta \right)}{[1+\frac{1}{3}{F}_{1}f(\theta \left)\right]},$ where the universal function $f\left(\theta \right)\equiv -{\left[\nu \mathrm{}\right(0\left)\right]}^{-1\mathrm{}}{\Sigma }_{\mathrm{p}}\left(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}}\right)$ is the "effective density of states near the Fermi surface" relative to its value $\nu \mathrm{}\left(0\right)\mathrm{}$ in the normal phase. Thus the often-quoted expression ${\rho }_n=\frac{1}{3}{\Sigma }_{\mathrm{p}}{\mathrm{p}}^2\left(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}}\right)$ is valid for an interacting system only in the limit $T\to 0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\chi$ and ${\rho }_n$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ne 0$); it is found that under most circumstances explicit expressions for $\chi$ and ${\rho }_n$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of ${\mathrm{He}}^3$ and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\lambda \mathrm{}\left(T\right)\mathrm{}$ is shown to be insensitive to such effects both in the Pippard limit and near $T_c$; however, in a London superconductor at lower temperatures the correction to $\lambda \mathrm{}\left(T\right)\mathrm{}$ should be observable and yield a direct estimate of $F_1$.