Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

Abstract

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.

12 pages, 18 figures. Fractal properties of the energy levels are visualised in the supplementary video material https://www.youtube.com/watch?v=ODS3QVkPTPE