Chasing the Hofstadter Butterfly

The experimental observation of the Hofstadter butterfly, the fascinating quantum fractal that also encodes the Chern numbers associated with quantum Hall state, continues to remain a challenging task. It may be possible to observe the fine structure of the butterfly, consisting of small gaps of the spectrum characterized by topological invariants greater than unity, with a resolution matching that of the Chern-$1$ gaps that form the skeleton of the butterfly. The tiny gaps of the butterfly emanating from a rational flux $p/q$ are found to be associated with infinity of possible solutions (of Diophantine equation )for the rational flux. Not supported by the simple square lattice nearest-neighbor hopping model of the Hofstadter system, these solutions are found to be hiding in neighborhood of these fluxes. By perturbing this simple system, it is possible to ``amplify'' these small gaps corresponding to higher Chern states where they replace the Chern $1$ gap of the Hofstadter butterfly. In other words, by tuning a parameter, it is possible to induce topological quantum phase transitions where the finer gaps become the new major gaps that dominate the spectrum. This may provide a possible pathway to see the topological landscape of the Hofstadter butterfly fractal in its entirety.