Topology and Self-Similarity of the Hofstadter Butterfly

We revisit the problem of self-similar properties of the Hofstadter butterfly spectrum, focusing on spectral as well as topological characteristics. In our studies involving any value of magnetic flux and arbitrary flux interval, we single out the most dominant hierarchy in the spectrum, which is found to be associated with an irrational number $\zeta=2+\sqrt{3}$ where nested set of butterflies describe a kaleidoscope. Characterizing an intrinsic frustration at smallest energy scale, this hidden quasicrystal encodes hierarchical set of topological quantum numbers associated with Hall conductivity and their scaling properties. This topological hierarchy maps to an {\it integral Apollonian gasket} near-$D_3$ symmetry, revealing a hidden symmetry of the butterfly as the energy and the magnetic flux intervals shrink to zero. With a periodic drive that induces phase transitions in the system, the fine structure of the butterfly is shown to be amplified making states with large topological invariants accessible experimentally.