Localization length ξ: —
Anderson localization (1958): in 1D with any random potential, all eigenstates are exponentially localized — |ψ(x)| ~ e^{−|x−x₀|/ξ}.
This is exact for any disorder strength W > 0. The localization length ξ ~ 1/ln(1+W²/4t²) diverges only as W→0.
The participation ratio P = (∫|ψ|²dx)² / ∫|ψ|⁴dx measures how many sites the state occupies. Scaling theory: in d=1, the beta function β(g) = d(ln g)/d(ln L) < 0 always → all states localized. In d=3, there's a mobility edge separating localized and extended states.
The participation ratio P = (∫|ψ|²dx)² / ∫|ψ|⁴dx measures how many sites the state occupies. Scaling theory: in d=1, the beta function β(g) = d(ln g)/d(ln L) < 0 always → all states localized. In d=3, there's a mobility edge separating localized and extended states.