Anderson Localization — Transfer Matrix 1D

In 1D tight-binding with random on-site disorder (Anderson model), ALL states are exponentially localized regardless of disorder strength — a consequence of Furstenberg's theorem on products of random matrices. The wavefunction amplitude decays as |ψ(n)| ~ exp(−|n|/ξ), where the localization length ξ is determined by the Lyapunov exponent of the transfer matrix product.

Localization length ξ
Lyapunov exponent γ=1/ξ
|ψ|² max at site
Weak disorder: ξ ≈ 96/W²