He3as an almost-localized Fermi liquid: The superfluid transition temperature

We examine the superfluid instability of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ from the standpoint of the almost-localized theory, based on the Gutzwiller approximation to the Hubbard model. Using the boson representation of Kotliar and Ruckenstein [Phys. Rev. Lett. 57, 1362 (1986)], we calculate the equal-spin two-particle vertex to first order in a loop expansion. We extract the l=1 coupling constant as a function of pressure and obtain values within 20% of the experimental ones. The choice of a cutoff proportional to the effective bandwidth leads to results for $T_c$ in reasonably good agreement with experiments on liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ under pressure.