Lingling CAO, PhD student @ Cermics, ENPC

Hofstadter’s butterfly and Quantum Hall effectAn encounter of spectral theory with quantum mechanics

What is a Hofstadter’s butterfly ?

❖ Discovered by the physicist/cognitive scientist Douglas Hofstadter in 1976

❖ Represent the behavior of electrons under magnetic field

❖ Fractal structure

❖ Horizon: energy level ❖ Vertical: magnetic field strength ❖ Picture from Wikipedia

What is Quantum Hall effect?

❖ The conductance is quantized (can only be an integer of sth ..)

❖ Nobel prize in physics 1985: “Discovery of the quantized Hall effect”

❖ Nobel prize in physics 2016 : “Theoretical discoveries of topological phase transitions and topological phase of matter”

❖ Picture from Indu Satija

Quantum Hall effect v.s. Hofstadter’s butterfly

❖ Picture from Indu Satija

Integers are quantum numbers of Hall conductivity

Spectral theory in quantum mechanics

❖ Picture from https://www.allaboutcircuits.com/textbook/semiconductors/chpt-2/band-theory-of-solids/

❖ Quantum observable can be described by operators (resp. matrix).

❖ For some operator, its spectrum (resp. eigenvalue) represents the energy of the system.

❖ This leads to the band theory in quantum mechanics

Spectral theory v.s. Hofstadter’s butterfly

❖ The Hofstadter’s butterfly can be generated by the following operator (Harper’s operator )

❖ Varying energy (E, which is the spectrum of the operator ) and the magnetic field (alpha)

❖ Studying the spectral property of the Harper’s operator

❖ => information on Hofstadter’s butterfly structure

❖ => topological information for Quantum Hall effect

❖ => a beautiful combination of physics, operator theory and topology

A mathematician: what’s the difference between a donut and a coffee mug ?They both have one hole, right ?

Thank you !