Ground states of large quantum dots in magnetic fields

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.