Thomas–Fermi–von Weizsäcker theory for a harmonically trapped, two-dimensional, spin-polarized dipolar Fermi gas

Abstract

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 Weizsäcker approximation. We pay particular attention to the construction of the twodimensional 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. PACS numbers: 31.15.E-, 71.10.Ca, 03.75.Ss, 05.30.Fk I. INTRODUCTIONUltra-cold, trapped dipolar quantum gases have received increasing attention over the past decade owing to the inherently interesting properties of the anisotropic, and long-range nature of the dipole-dipole interaction [1]. One of the important consequences of the anisotropy is that the interactions between the particles can be tuned from being predominantly attractive to repulsive by simply changing the 3D trapping geometry, or for dipoles confined to the 2D x-y plane, by adjusting the orientation of the dipoles relative to the z-axis [1, 2]. Therefore, novel physics in both the equilibrium and dynamic properties of such systems may be explored as a function of the strength of the interaction, the geometry of the confining potential, and the dimensionality of the system.While the degenerate dipolar Bose gas has been well studied experimentally and theoretically [1], realizing a degenerate dipolar Fermi gas in the laboratory has proven to be much more elusive. One of the reasons for this is that the path to quantum degeneracy is impeded by the Pauli principle, which forbids s-wave collisions between identical atoms. Thus, early attempts to cool both magnetic and molecular dipolar Fermi gases below degeneracy were unsuccessful [3][4][5][6]. However, in the recent work of M. Lu et al. [7], this experimental hurdle was finally overcome, resulting in the experimental realization of a spin polarized, degenerate dipolar Fermi gas. Specifically, using the method of sympathetic cooling, a mixture consisting of 161 Dy and the bosonic isotope 162 Dy, were cooled to T /T F ∼ 0.2. In addition, this group was also able to evaporatively cool a single component gas of 161 Dy down to a temperature of T /T F ∼ 0.7. This latter result is presumed to arise from the rethermalization provided by the strong dipolar scattering between the 161 Dy atoms which have a large magnetic moment (µ ∼ 10µ B ).The ability to fabricate such systems in the laboratory now opens the door for the investigation of both the equilibrium and dynamical properties of dipolar Fermi gases, and will enable contact to be made with the large body of theoretical work already in the literature [1]. Moreover, it is now r...

Density-functional theory for the crystalline phases of a two-dimensional dipolar Fermi gas

“…In our numerical calculations, we will take λ vW = 0.0184, as this is the value it interpolates to in the thermodynamic limit [20]. The vW contribution, Eq.…”

“…The T = 0 interaction energy functional, E int [ρ], for a uniform, spin-polarized 2D dFG is also known semiexactly [12,19,20]. In particular, one can decompose…”

“…Surprisingly, to our knowledge, no such investigation specifically dealing with a degenerate 2D dFG has been performed in the literature. We propose to fill this gap by presenting a DFT which will allow us to weigh in on the nature of the transition, the critical interaction strength at which the transition occurs, as well as providing a useful test of the various density functionals recently developed in the context of the 2D dFG [12,[19][20][21].…”

Testing the nonlocal kinetic energy functional of an inhomogeneous, two-dimensional degenerate Fermi gas within the average density approximation

“…We will examine the total energy, the KE, as well as the quality of the equilibrium spatial and KE densities (i.e., a point-wise comparison), in order to determine how well the ADA performs under a wide variety of confinement potentials. The inclusion of interactions will not alter the general results found in this paper, since in either the TFvW-like, KS, or TF DFT, the same level of approximation is made for the interaction energy functional, E int [ρ] (e.g., the Hartree-Fock approximation [2,12,13]). …”

Kohn-Sham approach to Fermi gas superfluidity: The bilayer of fermionic polar molecules

“…DFT approaches have been used recently to describe a Fermi dipolar system in various "single-orbital" approximations (Thomas-Fermi [28], Thomas-Fermi-Dirac [29], Thomas-Fermi-von Weizsacker [30,31]). In Ref.…”