The exchange-correlation kernel in the spin channel in an electron liquid has the structure ${f}_{xc,\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{q\ensuremath{\rightarrow}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega})∕{q}^{2}+{B}^{L,T}(\ensuremath{\omega})$ in the limit of the long wavelength. Here
L denotes the longitudinal component and
T the transverse component relative to the direction of the wave vector
q. A collection of exact results for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ is obtained at limiting low and high frequency, respectively, in two dimensions. Based on these results, we further give approximate interpolations for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ at all frequencies in the paramagnetic case.
Spin correlations in an interacting electron liquid are studied in the high-frequency limit and in both two and three dimensions. The third-moment sum rule is evaluated and is used to derive exact limiting forms (at both long and short wavelengths) for the spin-antisymmetric local-field factor, ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega}).$ In two dimensions ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega})$ is found to diverge as
1/q at long wavelengths, and the spin-antisymmetric exchange-correlation kernel of time-dependent spin density functional theory diverges as
1/q2 in both two and three dimensions. These signal a failure of the local-density approximation, one that can be redressed by alternative approaches.
The density- and spin-correlation functions of the t model are calculated on a square lattice in leading order of a 1/N expansion using X operators and a Baym-Kadanoff expansion. While the spectra of both correlation functions are similar at low densities, the energy scale for density fluctuations becomes much higher than that of spin fluctuations at large densities, due to collective effects. Our results agree quite well with recent exact diagonalizations for the case N=2. Furthermore, we examine the idea that the charge correlations of strongly correlated electrons with spin can be described by those of uncorrelated spinless electrons in two dimensions. While the equal-time density correlation functions of both systems are similar, the dynamic ones behave quite differently at high densities.