A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

Abstract

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

References in corpus (6)

• Hofstadter's butterfly in moire superlattices: A fractal quantum Hall effect
• Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
• Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
• Counter-propagating edge modes and topological phases of a kicked quantum Hall system
• Chern numbers hiding in time-of-flight images
• Chern and Majorana Modes of Quasiperiodic Systems

Cited by in corpus (12)

• Kaleidoscopes of Hofstadter Butterflies and Aharonov-Bohm caging from $2^n$-root topology in decorated square lattices
• A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes
• Nests and Chains of Hofstadter Butterflies
• Renormalization and universality of the Hofstadter spectrum
• Ground state fractal crystals
• Golden mean renormalization for the almost Mathieu operator and related skew products
• Lattice random walks and quantum A-period conjecture
• Coupled-Ring Resonance and Unitary Groups
• Geometry, Number Theory and the Butterfly Spectrum of Two-Dimensional Bloch Electrons
• Asymptotic scaling and universality for skew products with factors in SL(2,R)
• Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets
• Tessellations and Descartes disk configurations