We investigate the relative contribution of ladder and ring diagrams to the single-particle self-energy in fully-spin-polarized liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ ${(}^3$${\mathrm{He}}^{\mathrm{\uparrow }}$). Ladder diagrams are summed to all orders of the bare ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^3{}_{}{}^{}$He interaction using the Galitskii-Feynman-Hartree-Fock (GFHF) analysis. Previous studies of ${}_{}{}^3{}_{}{}^{}\mathrm{He}_{}^{\mathrm{\uparrow }}{}_{}{}^{}$, using GFHF analysis, have neglected the part of the GFHF self-energy coming from the correlation potential, ${\mathit{V}}_{\mathrm{co}}$. These calculations produced ground-state energies in fair agreement with values obtained from variational Monte Carlo (VMC) calculations. However, properties such as Landau parameters, which are directly related to long-range correlations, tend to differ considerably from known values. In the present work we have evaluated ${\mathit{V}}_{\mathrm{co}}$ and found it to have an appreciable effect on the single-particle excitation energies, ɛ(k) and the ground-state energy: Including ${\mathit{V}}_{\mathrm{co}}$ significantly reduces the ground-state energy. As a further refinement over previous GFHF calculations, we have used a more accurate center-of-mass momentum, P, dependence for the Galitskii-Feynman t matrix in the self-energy calculation. Again we find an undesirably large decrease in the ground-state energy. Finally, upon including a contribution from a summation of ring diagrams, we find a ground-state energy that is once again in fair agreement with the VMC values. The ring diagrams are driven by a local particle-hole interaction obtained by the method of correlated basis functions (CBF). Ring diagrams are then summed within a random-phase approximation. Our final ɛ(k) is used to calculate the particle-hole irreducible interaction ${\mathit{I}}_{\mathit{p}\mathrm{-}\mathit{h}}$. In the long-wavelength limit we find that our ${\mathit{I}}_{\mathit{p}\mathrm{-}\mathit{h}}$ is in much better agreement with the CBF ${\mathit{I}}_{\mathit{p}\mathrm{-}\mathit{h}}$ when our ɛ(k) includes contributions from ${\mathit{V}}_{\mathrm{co}}$, ${\mathrm{\Sigma }}_{\mathit{R}}$, and the refined self-energy calculation.

• Received 15 July 1991

DOI:https://doi.org/10.1103/PhysRevB.44.10239