We investigate the Hubbard model, and the extended Hubbard model with and without correlated hopping, as an approximation to the average single-site (or local) entanglement of two-electron one-dimensional quantum dot chains with long range interactions. We focus on the effect of the chain length on the approximation accuracy. We also consider the particle-particle spatial entanglement of the three Hubbard models using the technique of Ref. [3] and compare the accuracy of this approximation to the numerically exact results of the nanostructure system.
Assuming that, in the narrow-band region, the exact Green's function of the Hubbard model essentially retains the two-peak structure proposed by Hubbard, we derive its consequences as rigorously as possible. When a quasielectron is added to the ground state of
N electrons in the narrow-band region of the Hubbard model, the whole system is shown to relax due to the strong interaction
I, thus yielding a relaxation energy $\ensuremath{\Delta}E$ between the quasielectron and the original
N electrons. This explains the conclusion in the preceding paper that a gap $\ensuremath{\Delta}E$ analogous to the energy gap in superconductivity appears between the quasielectron spectrum and the chemical potential for any occupation
n regardless of whether the lower band is exactly filled or not and that the system cannot be a normal metal. We further found that the ground state is, in fact, a bound state, and electrons involved do not obey the Landau theory of quasiparticles. However, the electron Green's function describes the motion of an electron added to this
N-electron ground state without the relaxation of the original
N electrons, and hence this added electron behaves exactly like a Landau quasiparticle and can be represented by the quasiparticle operator ${\mathrm{A}}_{k\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$ introduced by Luttinger and Nozi\`eres. Only if the original
N electrons are allowed to relax is the energy of the system reduced by $\ensuremath{\Delta}E$, yielding the energy gap $\ensuremath{\Delta}E$. When a quasielectron is removed from the system of
N+1 electrons, the same relaxation energy $\ensuremath{\Delta}E$ is removed, and again the gap $\ensuremath{\Delta}E$ is introduced between the chemical potential $\ensuremath{\mu}$ and the quasihole spectrum, thus creating a gap $2\ensuremath{\Delta}E$ between the quasielectron and quasihole spectra for any occupation
n. The Fermi level, lying at the center of the gap, never intersects the quasiparticle spectra and therefore, Luttinger's theorem that the volume within the Fermi surface is independent of interactions and equal to the value in the noninteracting limit does not apply to the Hubbard model in the narrow-band region. However, this conclusion does not immediately imply that the system is an insulator. Instead, it is shown that a current-carrying state analogous to the condensate state found by Fr\"ohlich appears under certain conditions. When the number of electrons
N is nearly or exactly equal to the number of atoms
Na, quasiparticlelike behavior disappears completely, making the system insulating. The only exception is the pathological limit $I=\ensuremath{\infty}$, where a metallic state with a Fermi volume twice that in the noninteracting limit is found.
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
Coupled quantum dots are an example of the ubiquitous quantum double potential well. In a typical transport experiment, each quantum dot is also coupled to a continuum of states. Our approach takes this into account by using a Green's function formalism to solve the full system. The time-dependent solution is then explored in different limiting cases. In general, a combination of coherent and incoherent behavior is observed. In the case that the coupling of each dot to the macroscopic world is equal, however, the time evolution is purely coherent.
We studied effects of random potentials and roles of electron-electron interactions in the gapless phase of coupled Hubbard chains, using a renormalization group technique. For noninteracting electrons, we obtained the localization length proportional to the number of chains, as already shown in the other approaches. For interacting electrons, the localization length is longer for stronger interactions, that is, the interactions counteract the random potentials. Accordingly, the localization length is not a simple linear function of the number of chains. This interaction effect is strongest when there is only a single chain. We also calculate the effects of interactions and random potentials on charge stiffness.