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Journal of Physics A Mathematical and General

Relativistic effects when many independent Fermions are confined inddimensions

I. A. Howard, N. H. March

Published Jan 6, 2004• 7 citations
Publisher

Abstract

The study of the density of independent harmonically confined Fermions is of considerable current interest because of (a) quantum dots and (b) the experiments of DeMarco and Jin on some alkali metal vapours in magnetic traps.Our concern here is with relativistic effects, first in harmonically confined assemblies of electrons as in (a) above and secondly with a stronger type of Fermion confinement typified again by electrons but now in a quartic rather than quadratic potential.While our numerical illustrations are for dimensionalities d = 1 and 3, some analytical results are presented in d dimensions.

DOI
10.1088/0305-4470/37/3/028

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•Jan 1, 2001•0 citations
The study of the density of independent harmonically confined Fermions is of considerable current interest because of (a) quantum dots and (b) the experiments of DeMarco and Jin on some alkali metal vapours in magnetic traps.Our concern here is with relativistic effects, first in harmonically confined assemblies of electrons as in (a) above and secondly with a stronger type of Fermion confinement typified again by electrons but now in a quartic rather than quadratic potential.While our numerical illustrations are for dimensionalities d = 1 and 3, some analytical results are presented in d dimensions.
The kinetic energy of non-relativistic single-particle systems with arbitrary D -dimensional central potentials is found to be bounded from below by means of the orbital hyperangular quantum number, the dimensionality and some radial and logarithmic expectation values of the form ⟨ r k ⟩ and ⟨ r k (ln r ) m ⟩. Beyond the intrinsic physico-mathematical interest of this problem, we want to contribute to the current development of the theory of independent particles confined in spherically symmetric traps with non-standard dimensions. The latter has been motivated by the recent experimental achievements of the evaporative cooling of dilute (i.e. almost non-interacting) fermions in magnetic traps.
The search for the single-particle kinetic energy functional TS[n] continues to be of major interest for density functional theory. Since it is expected to be generally applicable, exactly solvable models are of obvious interest. Here we focus on one, which is also of interest experimentally in magnetic trapping of ultracold fermion vapours. This is the model of independent harmonically trapped fermions in two dimensions. Here, the role of the von Weizsacker inhomogeneity kinetic energy is a focal point, prompted also by the work of Delle Site (2005 J. Phys. A: Math. Gen. 38 7893).
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr