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Journal of Physics A Mathematical and Theoretical

Thermodynamic properties of a harmonically trapped, two-dimensional charged quantum gas in a magnetic field

Patrick Shea, Brandon P. van Zyl

Published Mar 17, 2008• 10 citations
Publisher

Abstract

We present exact analytical results for the thermodyanmic properties of a two-dimensional (2D), harmonically trapped charged quantum gas in a magnetic field. While our results are applicable to both Fermi and Bose gases, we focus our attention on trapped fermions owing to their relevant application in the density-functional theory of inhomogeneous Fermi systems. In particular, we test the Thomas–Fermi (or continuum) approximation (TFA) for the functional relation τ[ρ] using the exact ρ( r ) and show that it reproduces the local and global properties of the exact kinetic energy density τ( r ) surprisingly well. However, when we compare our exact results for various thermodynamic quantities with the TFA, we find that it misses several important features. These deviations are shown to be entirely due to the quantum mechanical properties of the system, which are not accounted for in the continuum approximation.

DOI
10.1088/1751-8113/41/13/135305

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We present exact analytical results for the thermodyanmic properties of a two-dimensional (2D), harmonically trapped charged quantum gas in a magnetic field. While our results are applicable to both Fermi and Bose gases, we focus our attention on trapped fermions owing to their relevant application in the density-functional theory of inhomogeneous Fermi systems. In particular, we test the Thomas–Fermi (or continuum) approximation (TFA) for the functional relation τ[ρ] using the exact ρ( r ) and show that it reproduces the local and global properties of the exact kinetic energy density τ( r ) surprisingly well. However, when we compare our exact results for various thermodynamic quantities with the TFA, we find that it misses several important features. These deviations are shown to be entirely due to the quantum mechanical properties of the system, which are not accounted for in the continuum approximation.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.
We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-NNN limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas–Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation τ TF [ρ] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities τ( r ) and ρ( r ) including their leading-order oscillating terms.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.
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 Weizs\"acker 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.
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.