Abstract
The quantum-mechanical ground state of a two-dimensional (2D) TV-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 functional 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.
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Lieb, E.H., Solovej, J.P., Yngvason, J. (2001). Ground states of large quantum dots in magnetic fields. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_14
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DOI: https://doi.org/10.1007/978-3-662-04360-8_14
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