Nests and chains of Hofstadter butterflies

Satija, Indubala I and Wilkinson, Michael (2020). Nests and chains of Hofstadter butterflies. Journal of Physics A: Mathematical and Theoretical, 53(8), article no. 085703.

DOI: https://doi.org/10.1088/1751-8121/ab64a5

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.

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