Landau levels on the 2D torus: a numerical approach

A numerical method is presented which allows to compute the spectrum of the Schroedinger operator for a particle constrained on a two dimensional flat torus under the combined action of a transverse magnetic field and any conservative force. The method employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of twelve digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform field satisfying Dirac's condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge.