Anderson Localization — 1D Disordered Lattice

Physics: Anderson (1958) showed that in a 1D lattice with random on-site energies ε_i ∈ [−W/2, W/2], all eigenstates are exponentially localized: |ψ(x)| ~ e^{−|x−x₀|/ξ}. The localization length ξ = 105.2 t²/W² (perturbative, near E=0). This is a quantum interference effect — random backscattering coherently confines electrons. In 2D, all states are also localized (weak localization). Only in 3D does a metal-insulator (Anderson) transition occur. The IPR = Σ|ψᵢ|⁴ measures localization (IPR→1/N for extended, →1 for localized).