Random disorder causes quantum interference of electron wavefunctions, exponentially localizing them in 1D and 2D for any disorder strength. Anderson's 1958 discovery that "absence of diffusion" can occur in disordered lattices earned the 1977 Nobel Prize.
Anderson model: H = Σᵢ εᵢ|i⟩⟨i| − t Σ⟨ij⟩ |i⟩⟨j|, where εᵢ ∈ [−W/2, W/2] uniformly random. In 1D, all eigenstates are localized for any W>0, with localization length ξ ≈ (t/W)² × 96 (band center). In 2D, all states are also localized (Abrahams scaling theory 1979), but ξ can be exponentially large at weak disorder. In 3D (not shown), a metal-insulator transition occurs at W_c ≈ 16.5t. Left: 1D wavefunction |ψ(x)|² showing exponential envelope e^{−|x|/ξ}. Right: 2D wavefunction amplitude map — bright spots = localized peaks.