Magnetic oscillations, disorder and the Hofstadter butterfly in finite systems

We present numerical calculations of a tight-binding model applied to a finite square lattice in the presence of a perpendicular magnetic field. The persistent current associated with each eigenstate is calculated, the chirality of which is determined by whether the eigenstate exists within the bulk or localised to the edges of the lattice. This treatment allows us to extract oscillations in the magnetization, which are analogous to de Haas-van Alphen oscillations. We consider the influence of short range disorder and long range potential modulations on these systems.