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.
We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas-Fermi von Weizsacker approximation. We pay particular attention
to the construction of the two-dimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths.
