Effective interactions in liquidHe3

The effective interaction between a pair of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atoms in liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is calculated within the Galitskii-Feynman (GF) $T$-matrix approximation. With the use of this $T$ matrix, the single-particle energy spectrum $\epsilon \mathrm{}\left(k\right)\mathrm{}$ is calculated in the Hartree-Fock (HF) limit. This continuous $\epsilon \mathrm{}\left(k\right)\mathrm{}$ is used as an input spectrum for the $T$ matrix (for both initial and intermediate states), and $\epsilon \mathrm{}\left(k\right)\mathrm{}$ and the $T$ matrix $\Gamma$ are evaluated iteratively until consistent. We believe this is the first fully iterative calculation using a continuous $\epsilon \mathrm{}\left(k\right)\mathrm{}$ in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. We find (a) that the total $E\approx -3.7\mathrm{}$ K at the observed saturated-vapor-pressure volume is much lower than values ($E\approx -1.0\mathrm{}$ K) found previously with the use of a spectrum having a gap at the Fermi surface $k=k_F$, (b) that any continuous $\epsilon \mathrm{}\left(k\right)\mathrm{}$ in $\Gamma$ leads generally to a much lower $E$ [and $\epsilon \mathrm{}\left(k\right)\mathrm{}$] than one having a gap at $k_F$, (c) that the scattering to intermediate hole states included in the GF $T$ matrix does not change $\Gamma$, $\epsilon \mathrm{}\left(k\right)\mathrm{}$, or $E$ significantly from the more usual Brueckner-Bethe-Goldstone $T$ matrix for which only scattering to intermediate particle states is allowed, (d) that the rearrangement contributions to $\epsilon \mathrm{}\left(k\right)\mathrm{}$ are negligible with the use of the GF $T$ matrix making the dynamical and statistical $\epsilon \mathrm{}\left(k\right)\mathrm{}$ the same in the $T$-matrix approximation, (e) that in the HF approximation, $m^{*}\mathrm{}\left(k\right)\mathrm{}\approx 0.9$ largely independent of $k$, and (f) that, when the energy dependence of the self-energy is included, $m^{*}\mathrm{}(k,E)\mathrm{}$ is enhanced at $k\approx k_F$ and somewhat above $k_F$, especially at smaller volumes.