The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function
L. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential
W in the BSE for
L leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
Abstract In this paper, we theoretically investigate the effect of temperature on spin correlations in an unpolarized quasi-one-dimensional electron gas (Q1DEG). The correlations are treated dynamically within quantum version of the self-consistent mean-field approach of Singwi et al Numerical results for the ↑↑ and ↑↓ components of static structure factor and pair-correlation function, and the wave vector dependent static spin and charge susceptibilities are presented over a wide range of temperature T and electron coupling r s . We find that the recently reported (2020 J. Phys.: Condens. Matter. 32 335403) non-monotonic T -dependence of the contact pair-correlation function g ( r = 0; T ) is driven primarily by an interplay between ↑↓ correlations and thermal effects. At a given temperature, the dynamics of both ↑↑ and ↑↓ correlations is found to become significant with increasing coupling r s , manifesting unambiguously as pronounced peak at 3.5 k F (periodic oscillations) in the corresponding components of the structure factor (pair-correlation function). Analysis of static spin and charge susceptibilities reveals that an imbalance between ↑↑ and ↑↓ correlations may induce a transition to a spin-density wave (SDW) phase of wave vector ∼3.5 k F above a critical coupling for a sufficiently high T , while to a long-wavelength SDW phase at a low T . Higher the temperature, higher is the predicted critical coupling for the SDW phase. Interestingly, transition to the SDW phase is found to precede the recently predicted Wigner crystal instability in the finite- T Q1DEG. Further, if one starts with partially spin-polarized electrons, the SDW instability is found to shift to somewhat higher τ and r s . In addition, we have presented results for the free exchange-correlation energy, free correlation energy, and excess kinetic energy for the unpolarized and fully spin-polarized phases of the finite- T Q1DEG. Wherever interesting, we have compared our results with the predictions of the static version of the mean-field approach.
We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths
b and density parameters
rs. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density. It is observed that the peak in the static structure factor at
k=2kF grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ from the momentum density. It is found that ${K}_{\ensuremath{\rho}}$ increases by about 10% between wire widths
b=0.01 and
b=0.5. We also obtain ground-state properties of finite-thickness wires theoretically using the first-order random phase approximation (RPA) with exchange and self-energy contributions, which is exact in the high-density limit. Analytical expressions for the static structure factor and correlation energy are derived for $b\ensuremath{\ll}{r}_{\text{s}}<1$. It is found that the correlation energy varies as
b2 for $b\ensuremath{\ll}{r}_{\text{s}}$ from its value for an infinitely thin wire. It is observed that the correlation energy depends significantly on the wire model used (harmonic versus cylindrical confinement). The first-order RPA expressions for the structure factor, pair-correlation function, and correlation energy are numerically evaluated for several values of
b and ${r}_{\text{s}}\ensuremath{\le}1$. These are compared with the QMC results in the range of applicability of the theory.