Ground states of large quantum dots in magnetic fields

Elliott H. Lieb, Jan Philip Solovej, Jakob Yngvason

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

The quantum-mechanical ground state of a two-dimensional (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→∞, K→∞ with K/N fixed. It is proved that the ground-state energy and electronic density can be computed 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→∞: A 2D Thomas-Fermi (TF) theory applies in the case B/N→0; if B/N→const≠0 the correct limit theory is a modified B-dependent TF model, and the case B/N→∞ is described by a classical continuum electrostatic theory. For homogeneous potentials this last model describes also the weak-coupling limit K/N→0 for arbitrary B. Important steps in the proof are the derivation of a 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.

Original languageEnglish (US)
Title of host publicationThe Stability of Matter
Subtitle of host publicationFrom Atoms to Stars: Fourth Edition
PublisherSpringer Berlin Heidelberg
Pages171-190
Number of pages20
ISBN (Print)3540420835, 9783540222125
DOIs
StatePublished - Jan 1 2005

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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