On Lifshits tails in magnetic fields

Simone Warzel

ISBN 978-3-89722-711-8
148 pages, year of publication: 2001
price: 40.50 €

We investigate the leading low-energy fall-off of the integrated density of states of a electrically charged quantum particle subject to a constant magnetic field and repulsive impurities randomly located according to Poisson's distribution.

This so-called magnetic Lifshits tail is determined for the case of two space dimensions with a perpendicular magnetic field and for all single-impurity potentials with either super-Gaussian, Gaussian or regular sub-Gaussian decay at infinity. While the result for regular sub-Gaussian decay coincides with the corresponding classical one, the Lifshits tail caused by super-Gaussian decay exhibits a universal quantum behaviour. As a consequence, Gaussian decay is proven to discriminate between quantum and classical tailing. We also give results for the Lifshits tail of the integrated density of states restricted to a single Landau band.

In the case of three space dimensions, the magnetic Lifshits tail is investigated for all impurity potentials with super-Gaussian or Gaussian decay. Its precise form is determined for all impurity potentials with stretched (sub-) Gaussian decay. In this case it turns out that the tail is independent of the magnetic field and coincides, up to a logarithmic acceleration, with that for one dimension and not too slowly decaying impurity potentials.

As a by-product we determine the strong-magnetic field asymptotics of the integrated density of states for the two-dimensional and three-dimensional situation.