How Landau caterpillars turn into Hofstadter butterflies by tuning the periodic potential strength

Abstract

It is well-known that the spectrum of two-dimensional electrons in a perpendicular magnetic field is given by discrete flat Landau levels. By contrast, electrons in a two-dimensional tight-binding model give rise to a fractal Hofstadter butterfly spectrum. In this paper, we connect these two opposite limits by showing how a butterfly spectrum emerges from broadened Landau `caterpillars', by continuously increasing the periodic potential strength. We identify a series of topological transitions that isolate a lowest trivial band, a necessary condition for the butterfly to emerge. The resulting butterfly is topologically distinct from the Hofstadter butterfly at fluxes $ϕ>1$, due to anomalous behavior of diagonal hopping. Moreover, the hopping parameters of an effective tight-binding model, obtained by a Wannierization procedure at large magnetic field, are highly dependent on the flux, revealing that Wannier orbitals themselves change under the applied magnetic field. Our methods and results are relevant for artificial materials, such as moiré systems, where a full quantum of flux per lattice unit cell is experimentally accessible. On a theoretical level, having Wannierization methods for each specific flux allows more accurate many-body calculations in large magnetic fields.