In a recent paper [Phys.~Rev.~A {bf 89}, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density-functional theory of an inhomogeno
us, two-dimensional (2D), Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density-functional theory, and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, {em without any gradient corrections}, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor point-wise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for {em both} the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.
Density-functional theory is utilized to investigate the zero-temperature transition from a Fermi liquid to an inhomogeneous stripe, or Wigner crystal phase, predicted to occur in a one-component, spin-polarized, two-dimensional dipolar Fermi gas. Co
rrelations are treated semi-exactly within the local-density approximation using an empirical fit to Quantum Monte Carlo data. We find that the inclusion of the nonlocal contribution to the Hartree-Fock energy is crucial for the onset of an instability to an inhomogeneous density distribution. Our density-functional theory supports a transition to both a one-dimensional stripe phase, and a triangular Wigner crystal. However, we find that there is an instability first to the stripe phase, followed by a transition to the Wigner crystal at higher coupling.
The collective excitations of a zero-temperature, spin-polarized, harmonically trapped, two-dimensional dipolar Fermi gas are examined within the Thomas-Fermi von Weizsacker hydrodynamic theory. We focus on repulsive interactions, and investigate the
dependence of the excitation frequencies on the strength of the dipolar interaction and particle number. We find that the mode spectrum can be classified according to bulk modes, whose frequencies are shifted upward as the interaction strength is increased, and an infinite ladder of surface modes, whose frequencies are {em independent} of the interactions in the large particle limit. We argue quite generally that it is the {em local} character of the two-dimensional energy density which is responsible for the insensitivity of surface excitations to the dipolar interaction strength, and not the precise form of the equation of state. This property will not be found for the collective excitations of harmonically trapped, dipolar Fermi gases in one and three dimensions, where the energy density is manifestly nonlocal.
The average-density approximation is used to construct a nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas. This functional is then used to formulate a Thomas-Fermi von Weizsacker-like theory for the description of the
ground state properties of the system. The quality of the kinetic energy functional is tested by performing a fully self-consistent calculation for an ideal, harmonically confined, two-dimensional system. Good agreement with exact results are found, with the number and kinetic energy densities exhibiting oscillatory structure associated with the nonlocality of the energy functional. Most importantly, this functional shows a marked improvement over the two-dimensional Thomas-Fermi von Weizsacker theory, particularly in the vicinity of the classically forbidden region.
