  Physics of two-dimensional electron gases under perpendicular magnetic field
often displays three distinct stages when increasing the field amplitude: a low
field regime with classical magnetotransport, followed at intermediate field by
a Shubnikov-de Haas phase where the transport coefficients present quantum
oscillations, and, ultimately, the emergence at high field of the quantum Hall
effect with perfect quantization of the Hall resistance. A rigorous
demonstration of this general paradigm is still limited by the difficulty in
solving models of quantum Hall bars with macroscopic lateral dimensions and
smooth disorder. We propose here the exact solution of a simple model
exhibiting similarly two sharp transitions that are triggered by the
competition of cyclotron motion and potential-induced drift. As a function of
increasing magnetic field, one observes indeed three distinct phases showing
respectively fully broken, partially smeared, or perfect Landau level
quantization. This model is based on a non-rotationally invariant, inverted
two-dimensional harmonic potential, from which a full quantum solution is
obtained using 4D phase space quantization. The developed formalism unifies all
three possible regimes under a single analytical theory, as well as arbitrary
quadratic potentials, for all magnetic field values.
