Stilla: Anderson Localization
Quantum Foundations arc, page 13 of 193. Back to The Pixelated Physics Universe
Who they are
Stilla was walking. Then the pillars got random, her own echoes pinned her in place, and she stopped spreading.
Stilla 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
In a lattice with random site energies, waves scattered along many paths interfere and can stop diffusing altogether. With enough disorder the wave is trapped in an envelope that falls off exponentially, |ψ(x)| ~ e^(−|x|/ξ), where ξ is the localization length. In 1-D all states localise; in 2-D there is no true metal (for non-interacting electrons without spin-orbit coupling); in 3-D there is a mobility edge. P. W. Anderson proposed it in 1958; in 2008 it was seen in matter waves, a Bose-Einstein condensate (Billy et al.; Roati et al.).
$$ \begin{gathered} H = \sum_i \varepsilon_i\, c_i^{\dagger} c_i - t \sum_{\langle ij\rangle} c_i^{\dagger} c_j, \qquad \varepsilon_i \in [-W/2,\, W/2] \ \text{random} \\ |\psi(x)| \sim e^{-|x|/\xi} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $H$ | Hamiltonian | J |
| $\varepsilon_i$ | random energy of site i | J |
| $c_i^{\dagger},\ c_i$ | create / remove a particle at site i | 1 |
| $t$ | hopping energy between neighbouring sites | J |
| $W$ | width of the disorder | J |
| $\psi(x)$ | wavefunction amplitude (1-D) | m^(−1/2) |
| $x$ | position | m |
| $\xi$ | localization length | m |
Honest caveat: Localization stops diffusion, not all motion, and the 2-D "no metal" result needs the conditions above.
Sources
- Anderson (1958) Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492
- Abrahams, Anderson, Licciardello & Ramakrishnan (1979) Scaling Theory of Localization, Phys. Rev. Lett. 42, 673
- Billy et al. (2008) Direct observation of Anderson localization of matter waves in a controlled disorder, Nature 453, 891
- Roati et al. (2008) Anderson localization of a non-interacting Bose–Einstein condensate, Nature 453, 895
Crossovers
Stilla meets Papilla (Hofstadter's Butterfly). 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.
Stilla meets Percolia (Percolation Threshold). Percolia asks whether randomly placed sites form a connected path. Stilla adds a quantum twist: even when a path exists, interference between scattered waves can stop an electron from spreading along it.
Wear the lore
- Stilla – Anderson Localization Physics T-Shirt
- Related infographic poster: The Copper Fermi Surface – Plate 057 Physics Poster
Concept keywords: Anderson Localization, Stilla, Quantum Foundations, physics lore, Pixelated Physics