Journal of Physics A: Mathematical and Theoretical PAPER What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles? To cite this article: Indubala I Satija 2021 J. Phys. A: Math. Theor. 54 025701 View the article online for updates and enhancements. This content was downloaded from IP address 129.6.222.101 on 16/12/2020 at 18:00 Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 54 (2021) 025701 (32pp) https://doi.org/10.1088/1751-8121/abc65c What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles? Indubala I Satija∗ Department of Physics, George Mason University, Fairfax, Virginia, United States of America E-mail:
[email protected] Received 16 June 2020, revised 26 October 2020 Accepted for publication 30 October 2020 Published 16 December 2020 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 Apol- lonian–butterfly-connection, the integer curvatures of the circles contain ina 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. In this mapping, Farey tree hierarchy plays the central role, revealing how the geometry and the number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.