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arXiv:cond-mat/0010193 · Physical Review A

Semiclassical theory of trapped fermionic dipoles

Krzysztof Góral, Berthold‐Georg Englert, Kazimierz Rza̧żewski

Published Feb 8, 2001• 60 citations
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Abstract

We investigate the properties of a degenerate dilute gas of neutral fermionic particles in a harmonic trap that interact via dipole-dipole forces. We employ the semiclassical Thomas-Fermi method and discuss the Dirac correction to the interaction energy. A nearly analytic as well as an exact numerical minimization of the Thomas-Fermi-Dirac energy functional are performed in order to obtain the density distribution. We determine the stability of the system as a function of the interaction strength, the particle number, and the trap geometry. We find that there are interaction strengths and particle numbers for which the gas cannot be trapped stably in a spherically symmetric trap, but both prolate and oblate traps will work successfully.

DOI
10.1103/physreva.63.033606

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We investigate the properties of a degenerate dilute gas of neutral fermionic particles in a harmonic trap that interact via dipole-dipole forces. We employ the semiclassical Thomas-Fermi method and discuss the Dirac correction to the interaction energy. A nearly analytic as well as an exact numerical minimization of the Thomas-Fermi-Dirac energy functional are performed in order to obtain the density distribution. We determine the stability of the system as a function of the interaction strength, the particle number, and the trap geometry. We find that there are interaction strengths and particle numbers for which the gas cannot be trapped stably in a spherically symmetric trap, but both prolate and oblate traps will work successfully.
We derive the density functional for the ground-state energy of a two-dimensional, spin-polarized gas of neutral fermionic atoms with magnetic-dipole interaction, in the Thomas-Fermi-Dirac approximation. For many atoms in a harmonic trap, we give analytical solutions for the single-particle spatial density and the ground-state energy, in dependence on the interaction strength, and we discuss the weak-interaction limit that is relevant for experiments. We then lift the restriction of full spin polarization and account for a time-independent inhomogeneous external magnetic field. The field strength necessary to ensure full spin polarization is derived.
We develop a mean-field treatment of a polarized trapped Fermi gas with dipole-dipole interactions. Our approach is based on self-consistent semiclassical Hartree-Fock theory that accounts for direct and exchange interactions. We discuss our procedure for numerically implementing the calculation. We study the thermodynamic and the first- and second-order correlation properties of the system. We find that the system entropy depends on the trap geometry, allowing the system to be cooled as the trap aspect ratio is increased, and that exchange interactions cause the correlation functions to be anisotropic in the low-temperature regime. We also find that many uniform gas thermodynamic predictions, for which direct interaction effects vanish, are qualitatively unreliable for trapped systems, most notably for oblate traps. We develop a simplified Hartree formalism that is applicable to anisotropic harmonic traps.
Recently a new outlook on dealing with dipolar ultracold fermions based on density functional methods has received attention. A Thomas-Fermi treatment coupled with a variational approach has been developed for a collection of fermions trapped in a harmonic potential interacting via dipole-dipole forces. In this thesis, firstly our alternative formalism for Thomas-Fermi method by performing some calculations based on the Kohn-Sham formalism which is one of the main idea of density functional theory is investigated. Furthermore, density distributions are obtained dependent to the parameters; rescaled interaction strength, dipole-dipole energy and the trap parameter which determine the trap geometry based on this theory. The thesis starts with a brief outline of the density functional theory and theory of our system, continues with calculations based on this theory, which are free of any variational assumptions for the density profile. Moreover, results of density graphics for harmonic trap will be followed by discussion of comparison and contrast with Thomas-Fermi method based on the paper of Goral et al.. These discussions are mainly about the shape of the density distribution, variation of the cloud parameters and energy behaviours according to the rescaled interaction strength. The thesis concludes with an analysis of contribution of density functional theory to this fermionic system.
We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers N=2−−22N=2--22N=2−−22 using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ETF{E}_{\mathit{TF}}ETF​ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.
We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a ddd-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.
The energy density of an interacting Fermi gas in a harmonic potential is derived within localdensity approximation framework,in which the gas is assumed at each point to be locally homogenous. Based on the derived expressions,the thermodynamic stability conditions of trapped gas at zero temperature are studied,and the analytical expressions of stability conditions for d=l,2,3 dimensions are given. Furthermore,the stability phase diagrams are plotted for different dimension.It is shown that it is always stable when the system has repulsive interactions or has no interactions in condition of certain density of particle number,no mater what the dimension is.However,the system with attractive interactions is stable only under the condition of weak interaction strength.For d=1 and d = 3 cases,the critical values of interaction strength have close relationship with particle number density,but as for d= 2 case,the corresponding critical value does not depend on the particle number density explicitly.