Spin Correlations in a Magnetic Electron Gas

The density response function (χ_ρρ), the longitudinal spin-correlation function (χzz), the transverse spin-correlation function (χ_-+), and the cross-correlation function (χ_ρz) involving the density and the z component of spin density are computed by employing two generalized moment-conserving (MC) schemes for a magnetic electron gas. The two schemes differ in their treatment of the one-electron states. These functions are also computed in the random-phase approximation (RPA) including exchange processes, in two different ways, by solving the resulting integral equations by a variational method due to one of the authors. We prove that in the absence of spin-orbit interactions, ${{χ}}_{{ρ}z}({{→}}{q},{ω})={{χ}}_{z{ρ}}({{→}}{q},{ω})$, which enables us to set up a consistent MC scheme. In the paramagnetic state, only ${{χ}}_{{ρ}{ρ}}$ and ${{χ}}zz$ are independent, and in the long-wavelength static limit they yield results in accordance with the RPA scheme. The plasma dispersion law for long wavelength is also found to be identical in the MC and RPA scheme. In the ferromagnetic case, one of the MC schemes gives the same results as the RPA results for the Stoner model, and very different results for the Coulomb gas. The long-wavelength spin-wave dispersion is found to be different in the two schemes. A new nonlocal-zero-moment-conserving scheme is set up which gives the same equations as the RPA. It is thus concluded that a local-MC scheme is different from the RPA by virtue of the actual structure of the correlation functions, even though in the paramagnetic limit the results are similar for the static long-wavelength limit.