The Ground States of Large Quantum Dots in Magnetic Fields

TL;DR

This paper analyzes the ground states of large 2D quantum dots under magnetic fields, deriving exact energy and density formulas in the high-density limit, and introduces new inequalities and models for these systems.

Contribution

It establishes exact formulas for ground state energies in the high-density limit and introduces new inequalities and classical models for quantum dots in magnetic fields.

Findings

Exact energy and density formulas in the high-density limit.

Different limiting theories depending on magnetic field scaling.

New Lieb-Thirring inequality for 2D magnetic Hamiltonians.

Abstract

The quantum mechanical ground state of a 2D $N$-electron system in a confining potential $V(x)=Kv(x)$ ($K$ is a coupling constant) and a homogeneous magnetic field $B$ is studied in the high density limit $N\to\infty$, $K\to \infty$ with $K/N$ fixed. It is proved that the ground state energy and electronic density can be computed {\it exactly} in this limit by minimizing simple functionals of the density. There are three such functionals depending on the way $B/N$ varies as $N\to\infty$: A 2D Thomas-Fermi (TF) theory applies in the case $B/N\to 0$; if $B/N\to{\rm const.}\neq 0$ the correct limit theory is a modified $B$-dependent TF model, and the case $B/N\to\infty$ is described by a ``classical'' continuum electrostatic theory. For homogeneous potentials this last model describes also the weak coupling limit $K/N\to 0$ for arbitrary $B$. Important steps in the proof are the derivation of a new Lieb-Thirring inequality for the sum of eigenvalues of single particle Hamiltonians in 2D with magnetic fields, and an estimation of the exchange-correlation energy. For this last estimate we study a model of classical point charges with electrostatic interactions that provides a lower bound for the true quantum mechanical energy.