Papilla: Hofstadter's Butterfly
Quantum Foundations arc, page 14 of 193. Back to The Pixelated Physics Universe
Who they are
Papilla's wings are made of allowed energies. Turn up the magnetic field and her pattern repeats inside itself, forever.
Papilla is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Put an electron on a 2-D lattice in a magnetic field and its allowed energies depend on the flux through each cell. When the flux ratio is a fraction p/q, the band splits into q subbands; when it is irrational, the spectrum becomes a Cantor set. Plotted against the flux, the energies form a self-similar butterfly. Douglas Hofstadter computed the spectrum in 1976; in 2013 graphene moiré superlattices showed its fingerprints in experiments.
$$ \begin{gathered} \psi_{n+1} + \psi_{n-1} + 2\cos(2\pi n\alpha - \nu)\,\psi_n = E\,\psi_n \\ \alpha = \frac{\Phi}{\Phi_0}, \qquad \Phi_0 = \frac{h}{e} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\psi_n$ | wavefunction amplitude on row n | 1 |
| $\alpha$ | magnetic flux per lattice cell in flux quanta | 1 |
| $\Phi$ | magnetic flux through one lattice cell | Wb |
| $\Phi_0$ | flux quantum h/e | Wb |
| $\nu$ | phase | rad |
| $E$ | energy in units of the hopping energy t | 1 |
Honest caveat: Those experiments saw butterfly features, not the whole fractal.
Sources
- Hofstadter (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239
- Dean et al. (2013) Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices, Nature 497, 598
- Ponomarenko et al. (2013) Cloning of Dirac fermions in graphene superlattices, Nature 497, 594
- Hunt et al. (2013) Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure, Science 340, 1427
Crossovers
Papilla meets Stilla (Anderson Localization). Stilla is pinned in place by random disorder. Papilla's lattice is perfectly ordered, and her energy bands split into fractal levels because of the magnetic flux. Order and disorder each sculpt where an electron's wave is allowed to go.
Papilla meets Secundar (Gauge Invariance). Papilla's butterfly depends only on the magnetic flux through each lattice cell. Secundar confirms that the vector potential behind it can be re-gauged freely, and the spectrum doesn't change, because only gauge-invariant quantities like flux appear in the physics.
Wear the lore
Concept keywords: Hofstadter's Butterfly, Papilla, Quantum Foundations, physics lore, Pixelated Physics