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arXiv:1909.09918 · Random Matrices Theory and Application

Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap

Peter J. Forrester

Published Feb 11, 2020• 7 citations
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Abstract

We consider properties of the ground state density for the [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.

DOI
10.1142/s2010326321500180

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We consider properties of the ground state density for the [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.
Multipole moments for the harmonic oscillator as well as for other familiar quantum systems can be obtained without knowledge of the wavefunction by means of simple recursion relations. The third-order differential equation satisfied by the probability density is first obtained and from it, the recursion relation for the moments is directly deduced. A general and presumably novel formula giving all the multipole moments is obtained by solving the finite difference equation generated by the recursion relation.
In 1959, March and Young ( Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix γ( x , x 0 ) in terms of sum and difference variables. Here, γ( , 0 ) for the d -dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + 0 ∣/2 and ∣ − 0 ∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ( , 0 ) and the diagonal density ϱ( r ) = γ( , 0 )∣ 0 = , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d -dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).
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.
Closed-form analytic expressions are derived for the density profile of a harmonically trapped noninteracting Fermi gas in ddd dimensions. Shell structure effects are included to leading order in 1/N1/N1/N, where NNN is the number of particles. These corrections to the local density approximation scale as $\ensuremath{\delta}n/n\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=(1+1/d)/2$.
Using an integral expression which is based on the use of discretized path integrals, the one-particle density matrix is evaluated for individual fermions in a one- and three-dimensional harmonic well. In this regard, the present paper improves and extends work previously done by Handler. By deriving a closed form expression for the path integral with finite discretization, the density is expressed as a one-dimensional integral that can be evaluated numerically. In addition, the trace of this density matrix, which corresponds to the number of particles present as a function of the Fermi energy, is evaluated in closed form. For finite discretization, the function is continuous in the Fermi energy, but approaches the correct discontinuous limit as the discretization becomes large. Oscillations are observed at the discontinuities analogous to the Gibbs phenomenon of Fourier analysis. Calculations of the density show a marked improvement in the classically allowed region when the discretization is only one level beyond the simple Thomas–Fermi theory. The density also penetrates into the classically forbidden region as the discretization is increased, but the presence of negative oscillations requires a higher level of discretization for a proper description in this region.
We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a 'macroscopic' number of particles, and finite-temperatures.