Anderson Localization — Metal-Insulator Transition

Random disorder causes quantum wavefunctions to localize exponentially rather than extend across the lattice. In 1D and 2D, all states localize for any disorder (Anderson 1958). In 3D, a critical disorder W_c drives a metal-insulator transition. Shown: tight-binding wavefunction |ψ(x)|² vs. disorder strength.

Parameters

IPR = Σ|ψ|⁴:
Localization length ξ:
Regime:
Localized: ψ(x) ~ e^{-|x-x₀|/ξ}
Extended: ψ spread uniformly

IPR ~ 1/N → extended (metal)
IPR ~ const → localized (insulator)

1D: ALL states localized ∀W>0
3D: W_c ≈ 16.5t (Anderson transition)