Lattice Green function for electrons in magnetic field

Abstract

The energy spectrum of electrons on a square lattice in an applied magnetic field composes the famous Hofstadter butterfly with a recursive internal subband structure. An effective method for calculating the Green function for such a system is proposed. The standard approach requires an explicit knowledge of the eigenstates and eigenenergies of the system; here we derive a Harper-like equation, that allows us to calculate the Green function for the lattice electrons in the field directly. The method is particularly useful in the weak-field regime, where the standard calculations are cumbersome.