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.
The high-frequency limits of the singular component $A(\ensuremath{\omega})$ of the small wave-vector expansion of the longitudinal
(L) and transverse
(T) components of the spin-resolved exchange-correlation kernel tensor ${f}_{xc,\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})=\ensuremath{-}v(q){G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here ${G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ is the spin-resolved local-field factor,
v(q) is the Coulomb interaction in momentum space, and $\ensuremath{\sigma}$ denotes spin. Particularly, the real part of $A(\ensuremath{\omega})$ is found to be logarithmically divergent at large $\ensuremath{\omega}$. The large wave-vector structure of the corresponding spin-resolved static structure factor is also established.
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.
Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel ${f}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})$ is utilized to express the correlation energy of a many-electron system in terms of
fxc. As a test of several of the available approximations for the exchange-correlation kernel, or equivalently the local-field factor, we calculate the uniform-gas correlation energy. While the random phase approximation
(fxc = 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation
[fxc = fxc(q = $0,\ensuremath{\omega}$
= 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation
[fxc = ${f}_{\mathrm{xc}}(q,\ensuremath{\omega}$
= 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence
[fxc = ${f}_{\mathrm{xc}}(q,\ensuremath{\omega})]$ in an approximate parametrization reduces it further to less than 0.02 eV. We also report the wave-vector analysis and the imaginary-frequency analysis of the correlation energy for each choice of kernel.
The phase diagram of the multichannel Kondo Hamiltonian with an
XXZ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-
21 impurity and
k>2 conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling
Jz∗>0 and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-
21 local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary
k and
s obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here,
Jz is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and
s is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for
s>1∕2 and arbitrary
k. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of
2s, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a
k-channel, spin-
21 Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-
21 local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in
1∕Jz, and explicit numerical renormalization-group calculations.
Within the framework of the time-dependent spin density functional formalism, we study the dynamical spin susceptibility [Formula: see text] of a general system of interacting electrons in the presence of a static external scalar potential. Based on the stationary property of the action functional, a variational expression for the real part of [Formula: see text] is derived: [Formula: see text]. The result is valid for a general exchange-correlation (XC) action functional. In particular, the XC factor 1(q,ω) is manifestly dependent on q and ω. The imaginary part Im [Formula: see text] is then determined through the Kramers–Kronig relation. Practical approximations for the required XC kernel are also discussed