Time evolution of perturbed Landau functions and quantization of the Hall conductance

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Abstract
We consider a two-dimensional strip subject to a strong perpendicular magnetic field, in the presence of a Hall electric field and of a model disorder potential. The quantum-mechanical time evolution of the conducting states in a broadened Landau band is investigated numerically. In an initial period the individual Hall velocities have rather different time evolutions. At this stage even negative Hall velocities occur. At large times all Hall velocities tend to a common asymptotic value. With increasing time the conducting states spread in space in the direction of the Hall velocity. Simultaneously a spreading in energy takes place. At large times the energy of each conducting state is equally spread over the whole energy range of the conducting states. Although the Hall velocities show a nonlinear behavior, the quantization of the Hall conductance is verified numerically and analytically by summing over all the velocities in the band. We obtain a detailed picture of the microscopic processes leading to this quantization in our system. The calculated time evolution illustrates important features which are not contained in the usual picture of the integer quantum Hall effect.