2020/03/11 by Indubala I. Satija, Satija, Indubala I · 1 citation
Physics and Astronomy · #Advanced Mathematical Theories and Applications #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Scientific Research and Discoveries #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2003.05464
openalex publication_date 2020/03/11 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Integral Apollonian packing, the packing of circles with integer curvatures,\nwhere every circle is tangent to three other mutually tangent circles, is shown\nto encode the fractal structure of the energy spectrum of two-dimensional Bloch\nelectrons in a magnetic field, known as the "Hofstadter butterfly". In this\nApollonian-Butterfly-Connection, dubbed as \ABC, the integer\ncurvatures of the circles contain in a convoluted form, the topological quantum\nnumbers of the butterfly graph -- the quanta of the Hall conductivity. Nesting\nproperties of these two fractals are described in terms of the Apollonian group\nand the conformal transformations. The \ABC unfolds as the conformal\nmaps describing butterfly recursions are related to the conformal maps\ndescribing nesting of circles in the Apollonian packing. Mapping of butterflies\nto Apollonian at all scales where Farey tree hierarchy plays the central role,\nreveals how geometry and number theory are intertwined in the quantum mechanics\nof Bloch electrons in a magnetic field.\n