A simple approximate expression in real and reciprocal spaces is given for the static exchange-correlation kernel of a uniform electron gas interacting with the long-range part only of the Coulomb interaction. This expression interpolates between the exact asymptotic behaviors of this kernel at small and large wave vectors which in turn requires, among other things, information from the momentum distribution of the uniform electron gas with the same interaction that have been calculated in the
G0W0 approximation. This exchange-correlation kernel as well as its complement analog associated to the short-range part of the Coulomb interaction are more local than the Coulombic exchange-correlation kernel and constitute potential ingredients in approximations for recent adiabatic connection fluctuation-dissipation and/or density functional theory approaches of the electronic correlation problem based on a separate treatment of long- and short-range interaction effects.
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where
n is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
Correlated, disordered lattice fermion systems out of equilibrium have been understudied, at least in part for lack of tools to study them. The development of a new method, the NE-DMFT+CPA, is explicated, for investigating the properties of these systems. This method combines DMFT and the CPA using the Kadanoff-Baym-Keldysh contour-time formalism to study the dynamics of interacting disordered systems out of equilibrium. The method is detailed, then applied to the case of the Anderson-Hubbard model: first to replicate well-known equilibrium results, then to study the case of the model under the effect of an interaction quench. Here the effect of disorder on the relaxation of the system is studied by the time-dependence of the energies and momentum distribution function, which display the screening of the Coulomb interaction by disorder. It is then used to examine the thermalization of the same system by using the fluctuation-dissipation theorem to extract the distribution function and thereby the effective temperature after relaxation. This allows the characterization of the effect of the interplay of the disorder strength and the Coulomb interaction on the thermalization behavior. It is found that the heating of the system due to the quench is mitigated by stronger disorder. Finally, current progress of the work to apply the method to binary alloy disorder and an interaction ramp is summarized. Binary disorder appears to have the effect of damping the initial transient of the potential energy more strongly than the comparable uniformly-disordered system. The relaxation of the energies under the effect of an interaction ramp suggests that the crossing of the kinetic energies may be affected by the form of the time-dependent interaction.