Effects of deterministic aperiodic and self-similar on-site potentials on the structure of the Hofstadter butterfly

With the fundamental work of Hofstadter on the combined effects of band structure and magnetic field on the electronic states in two dimensions (2D) as a starting point, we numerically study the effects on the Hofstadter butterfly of including a binary distribution of on-site potentials on a 2D lattice in the tight-binding picture. The effects of the external magnetic field are included through the so-called Peierls substitution. The problem is reduced to a one-dimensional set of difference equations when the binary distribution is constrained to be in one direction only. Besides a periodic structure, a number of aperiodically ordered distributions like the Fibonacci, Thue-Morse, and the Rudin-Shapiro sequences are considered, and the band structures presented and discussed. Also, 2D chessboard and Sierpinski carpet distributions are dealt with in some detail.