Stilla

Stilla was walking. Then the pillars got random, her own echoes pinned her in place, and she stopped spreading.

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.

H = Σᵢ εᵢ cᵢ† cᵢ − t Σ⟨ij⟩ cᵢ† cⱼ, εᵢ random in [−W/2, W/2]

• H: Hamiltonian, energy (J)
• εᵢ: random energy of site i (J)
• cᵢ†, cᵢ: create / remove a particle at site i (dimensionless operators)
• t: hopping energy between neighbouring sites (J)
• W: width of the disorder (J)

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.).

Honest caveat: Localization stops diffusion, not all motion, and the 2-D "no metal" result needs the conditions above.

On the shirt

Front: Stilla, the pixel-art character, with the title. Back: the Anderson Hamiltonian, the exponential envelope, the 1-D/2-D/3-D summary and four sources from 1958 to 2008. Printed on a unisex tee.