Nests and chains of Hofstadter butterflies

Source abstract

Abstract The ‘Hofstadter butterfly’, a plot of the spectrum of an electron in a two-dimensional periodic potential with a uniform magnetic field, contains subsets which resemble small, distorted images of the entire plot. We show how the sizes of these sub-images are determined, and calculate scaling factors describing their self-similar nesting, revealing an un-expected simplicity in the fractal structure of the spectrum. We also characterise semi-infinite chains of sub-images, showing one end of the chain is a result of gap closure, and the other end is at an accumulation point.