• Stilla Anderson Localization pixel-art physics T-shirt, white tee, front view

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Stilla

Anderson Localization Physics

\[H = \sum_i \varepsilon_i\, c_i^{\dagger} c_i - t \sum_{\langle ij\rangle} c_i^{\dagger} c_j, \qquad \varepsilon_i \in [-W/2,\, W/2] \ \text{random}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

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.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Pour l'échelle : la taille M mesure 20 in (50.80 cm) de large, soit environ 3.143 × 1034 longueurs de Planck (lP = 1.616255 × 10−35 m). Un mètre ruban est plus rapide.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

I was walking. Then the pillars got random, my own echoes pinned me in place, and I stopped spreading.

Philip Anderson showed in 1958 that a wave hopping on a lattice with random site energies can stop diffusing altogether. Waves scattered along many paths interfere, and with enough disorder the wave is trapped inside an envelope that falls off exponentially, with localization length ξ. In one dimension all states localise. In two there is no true metal, at least without spin-orbit coupling or interactions. In three there is a mobility edge.

It has been seen for light, sound and microwaves, and in 2008 for matter waves in a Bose–Einstein condensate (Billy et al. with random disorder, Roati et al. with a quasi-periodic lattice). Localization stops diffusion, not all motion.

Equations

\[H = \sum_i \varepsilon_i\, c_i^{\dagger} c_i - t \sum_{\langle ij\rangle} c_i^{\dagger} c_j, \qquad \varepsilon_i \in [-W/2,\, W/2] \ \text{random}\]
\[|\psi(x)| \sim e^{-|x|/\xi}\]
SymbolMeaningUnit
\(H\) Hamiltonian \(\mathrm{J}\)
\(\varepsilon_i\) random energy of site i \(\mathrm{J}\)
\(c_i^{\dagger},\ c_i\) create / remove a particle at site i \(\mathrm{1}\)
\(t\) hopping energy between neighbouring sites \(\mathrm{J}\)
\(W\) width of the disorder \(\mathrm{J}\)
\(\psi(x)\) wavefunction amplitude (1-D) \(\mathrm{m^(−1/2)}\)
\(x\) position \(\mathrm{m}\)
\(\xi\) localization length \(\mathrm{m}\)

Sources

  1. Anderson (1958) Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492
  2. Abrahams, Anderson, Licciardello & Ramakrishnan (1979) Scaling Theory of Localization, Phys. Rev. Lett. 42, 673
  3. Billy et al. (2008) Direct observation of Anderson localization of matter waves in a controlled disorder, Nature 453, 891
  4. Roati et al. (2008) Anderson localization of a non-interacting Bose–Einstein condensate, Nature 453, 895

Field notes

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