Pure & Poetic: Butterfly in the Quantum World
The Hofstadter butterfly is a fascinating two-dimensional spectral landscape – a graph of the allowed energies of an electron in a two-dimensional crystal in a magnetic field. It is a quantum fractal made up of integers, describing topological states of matter known as the integer quantum Hall states. My butterfly story tells the tale of its discovery by a graduate student named Douglas Hofstadter and discusses its number theoretical, geometrical and topological aspects . I will describe how the integers of the butterfly are convoluted in the Pythagorean triplets and the integer curvature of Apollonian gaskets, thus providing a marvelous example of a physical incarnation of abstract mathematics. The story will touch the topological roots of integers, hidden in the Berry phase – a quantum cousin of the physics behind a Focault pendulum that reveals the secret behind astonishingly precise quantization of Hall conductivity. As a coda, I will present some novel topological states obtained by simple variations of the butterfly Hamiltonian [2, 3].
 Butterfly in the Quantum World, Indubala Satija with contributions by Douglas Hofstadter, IOP Concise, Morgan and Claypool publication (2016).
 M. Lababidi, I. Satija, and E. Zhao, PRL 112, 026805 (2014).
 I. Satija and E. Zhao, PRB 94, 245128 (2016).