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Physical review. B, Condensed matter

Density profile of the weakly interacting Fermi gas confined in a potential well: Nonzero temperature

J. Oliva

Published Mar 1, 1989• 11 citations
Publisher

Abstract

A theory is developed for the low-temperature density profile of a weakly interacting normal Fermi gas confined within an arbitrary potential well. This problem is of interest in connection with magnetically confined spin-aligned atomic deuterium (${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$). The approach used is based on the statement of constancy of the chemical potential in equilibrium. This involves the internal chemical potential, a functional of the density, which we treat within the local-density approximation. We develop an accurate approximate form for the local internal chemical potential: For a given low temperature, it is taken equal to the known forms at low density (classical) and higher density (quantum); for intermediate density we use an appropriate interpolant which smoothly connects the low- and high-density forms. Finite-temperature density profiles are evaluated for the ideal Fermi gas for parameters appropriate to magnetically confined ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ for both symmetric linear and quadratic one-dimensionally varying wells. We also evaluate density profiles including the leading-order interparticle interaction effect. For conditions relevant to hypothetical ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ at moderate densities this effect is found to be small but measurable.

DOI
10.1103/physrevb.39.4204

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A theory is developed for the low-temperature density profile of a weakly interacting normal Fermi gas confined within an arbitrary potential well. This problem is of interest in connection with magnetically confined spin-aligned atomic deuterium (D\ensuremath↓{\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}D\ensuremath↓​). The approach used is based on the statement of constancy of the chemical potential in equilibrium. This involves the internal chemical potential, a functional of the density, which we treat within the local-density approximation. We develop an accurate approximate form for the local internal chemical potential: For a given low temperature, it is taken equal to the known forms at low density (classical) and higher density (quantum); for intermediate density we use an appropriate interpolant which smoothly connects the low- and high-density forms. Finite-temperature density profiles are evaluated for the ideal Fermi gas for parameters appropriate to magnetically confined D\ensuremath↓{\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}D\ensuremath↓​ for both symmetric linear and quadratic one-dimensionally varying wells. We also evaluate density profiles including the leading-order interparticle interaction effect. For conditions relevant to hypothetical D\ensuremath↓{\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}D\ensuremath↓​ at moderate densities this effect is found to be small but measurable.
A theory is developed for the density profile of a weakly interacting Bose gas confined within an arbitrary potential well at nonzero temperature. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (H\ensuremath{\downarrow}). The approach taken is conceptually simpler and of comparable accuracy as compared to previous approaches. It is based on the statement of constancy of the chemical potential in equilibrium. This in turn involves the internal chemical potential, a functional of the density, which we treat in a local-density approximation. For a given temperature we develop a form for the local internal chemical potential for a wide range of density using known low-density, intermediate-density (i.e., near the Bose-condensation density), and high-density analytic forms, with appropriate smooth interpolants connecting these three forms. Density profiles are evaluated within the Bogoliubov approximation for the local internal chemical potential for parameters appropriate to magnetically confined H\ensuremath{\downarrow} for both symmetric linear and quadratic one-dimensionally varying wells. Comparison is made to previous work. The nonanalytic region of the density profile near the Bose-condensation density is analyzed. We also calculate density profiles including the next-highest-order interaction correction (i.e., beyond Bogoliubov) in the internal chemical potential. For conditions relevant to hypothetical H\ensuremath{\downarrow} at moderate densities this leads to a small but discernable effect on the density profile.
A theory is developed for the nonuniform ground-state density profile of weakly interacting quantum gases confined within an arbitrary potential well. Both Bose-Einstein and Fermi-Dirac cases are considered. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$) and deuterium (${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$). The approach is based on a macroscopic formulation involving the principle of constancy of the chemical potential throughout the system. The main input is the internal chemical potential for which we use a local-density approximation (LDA). In turn we use for the LDA internal chemical potential standard results for the uniform interacting quantum gases. It is shown that generally, the systems mentioned develop macroscopically sharp surfaces (again, while in the gas phase). Density profiles are analytically calculated for the ideal Fermi gas and the leading-order weakly interacting Bose gas for the case of an arbitrary, symmetric power-law potential well. We also calculate the density profile including the next-highest-order correction in the interparticle interaction in the Bose case and the leading-order correction in the interparticle interaction in the Fermi case. The accuracy of the LDA density profiles is assessed. Numerical results are given appropriate for ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ and ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$. For weakly interacting ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ a notable contrast between the case of single hyperfine level occupation and the case of occupation of the lowest three hyperfine levels is pointed out.
We determine the thermodynamic properties and the spectral function for a homogeneous two-dimensional Fermi gas in the normal state using the Luttinger-Ward, or self-consistent T-matrix, approach. The density equation of state deviates strongly from that of the ideal Fermi gas even for moderate interactions, and our calculations suggest that temperature has a pronounced effect on the pressure in the crossover from weak to strong coupling, consistent with recent experiments. We also compute the superfluid transition temperature for a finite system in the crossover region. There is a pronounced pseudogap regime above the transition temperature: the spectral function shows a Bogoliubov-like dispersion with backbending, and the density of states is significantly suppressed near the chemical potential. The contact density at low temperatures increases with interaction and compares well with both experiment and zero-temperature Monte Carlo results.
We investigate the approach to the universal regime of the dilute unitary Fermi gas as the density is reduced to zero in a lattice model. To this end we study the chemical potential, superfluid order parameter and internal energy of the attractive Hubbard model in three different lattices with densities of states (DOSs) which share the same low-energy behavior of fermions in three-dimensional free space: a cubic lattice, a ``Bethe lattice'' with a semicircular DOS, and a ``lattice gas'' with parabolic dispersion and a sharp energy cutoff that ensures the normalization of the DOS. The model is solved using dynamical mean-field theory, that treats directly the thermodynamic limit and arbitrarily low densities, eliminating finite-size effects. At densities on the order of one fermion per site the lattice and its specific form dominate the results. The evolution to the low-density limit is smooth and it does not allow to define an unambiguous low-density regime. Such finite-density effects are significantly reduced using the lattice gas, and they are maximal for the three-dimensional cubic lattice. Even though dynamical mean-field theory is bound to reduce to the more standard static mean field in the limit of zero density due to the local nature of the self-energy and of the vertex functions, it compares well with accurate Monte Carlo simulations down to the lowest densities accessible to the latter.
A theory is developed for magnetically confined Fermi gas at a low temperature based on the density functional theory. The theory is illustrated by the numerical calculation of the density distributions of Fermi atoms 40 K with parameters according to DeMarco and Jin's experiment [Science, 285(1999)1703]. Our results are in close agreement with the experiment. To check the theory, we also performed calculations using our theory at a high temperature, which compared very well to the results of the classical limit. PACS: 31. 35. Ew, 05. 30. Fk, 51. 30. +i
Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms.