Edge states and quantized Hall conductivity in a periodic potential

Abstract

The electronic structure of an infinitely long two-dimensional strip of finite width in a weak periodic potential and a strong magnetic field has been calculated. In an infinitely wide strip there are gaps within each Landau level, with the number of gaps depending on the product of the area of the unit cell and the magnetic field. For finite-width strips, edge states appear in these intra-Landau-level gaps. A relationship between the Hall conductivity when the Fermi level lies in one of these gaps and the number and dispersion of the edge states is established. Results for the electronic structure are presented for several cases and are shown to imply values for the Hall conductivity in agreement with those expected on the basis of Kubo-formula expressions for an infinite system.