Quantum wavefunction localization by random disorder · Transfer matrix method
Parameters
Observables
Localization Length ξ—
log(|ψ|) slope—
Lyapunov exp γ—
Participation ratio—
Physics
In 1D, all states are localized for any disorder W>0 (Abrahams et al. 1979). The wavefunction decays as |ψ(x)| ~ e−|x−x₀|/ξ with localization length ξ ~ W−2 near band center.
The transfer matrix method propagates: (ψₙ₊₁, ψₙ) = Mₙ(ψₙ, ψₙ₋₁). The Lyapunov exponent γ = 1/ξ = lim 1/N log‖M_N⋯M_1‖. Even a tiny disorder drives ξ → finite.
Top: |ψ(x)|² on the chain. Bottom: log|ψ| showing exponential envelope. Blue line = fitted localization length.