Landau levels, electric dipole transitions, and the Hofstadter butterfly in finite systems

Abstract
We present the results of numerical calculations of the energy levels and eigenfunctions of finite sections of a tight-binding square lattice in the presence of a perpendicular magnetic field. The energy spectrum of such a system, plotted as a function of magnetic field, resembles the Hofstadter butterfly found for the infinite system. In high magnetic fields, each eigenstate carries a persistent current which has a chirality associated with whether the eigenstate exists in the bulk or the edge of the system. We present simulations of electric dipole transitions between the chirally distinguished states for an isotropic lattice. These transitions correspond to harmonics in the cyclotron resonance.