Alladi Ramakrishnan Hall

Pure and Poetic: Butterfly in the Quantum World

Indu Satija

Department of Physics and Astronomy, George Mason University, Fairfax VA, USA

The story of the Hofstadter butterfly is a magical occurrence in the quantum flatland of two-dimensional crystals in a magnetic field. In this drama, the magnetic flux plays the role of
Planck’s constant, linking the variables $x$ and $p$ in the butterfly Hamiltonian $H$ = cos $x$ + cos $p$,
as [$x, p$] = $i\hbar$ . It is a story of the reunion of Descartes and Pythagoras, and the tale of this quantum fractal is related to integral Apollonian gaskets. Integers rule the butterfly landscape as quantum numbers of the Hall conductivity, while irrational numbers emerge as the asymptotic magnification of these topological integers in the kaleidoscopic images of the butterfly [1].

[1] ``Butterfly in the Quantum World'', Indubala I Satija with contributions by Douglas Hofstadter,
IOP Concise, Morgan and Claypool Publication, 2016.