What do Bloch Electrons in a Magnetic Field have to do with Apollonian packing of Circles ?

Abstract

Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the "Hofstadter butterfly". In this Apollonian-Butterfly-Connection, dubbed as $\mathcal{ABC}$, the integer curvatures of the circles contain in a convoluted form, the topological quantum numbers of the butterfly graph -- the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. The $\mathcal{ABC}$ unfolds as the conformal maps describing butterfly recursions are related to the conformal maps describing nesting of circles in the Apollonian packing. Mapping of butterflies to Apollonian at all scales where Farey tree hierarchy plays the central role, reveals how geometry and number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.