Anderson Localization in 1D Disorder

All eigenstates localize exponentially in 1D random potentials — the localization length ξ ~ 1/disorder²

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Loc. length ξ

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Lyapunov γ = 1/ξ

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IPR (inverse part. ratio)

Disorder strength W 1.00

Energy E (Bloch band) 0.00

Chain length N 300

Display mode |ψ|²

Anderson (1958): in 1D, any amount of uncorrelated disorder localizes all quantum eigenstates. The wavefunction decays as |ψ(x)| ~ exp(-|x-x₀|/ξ) where ξ ~ (E²-4)·(12/W²)⁻¹ near band center. Transfer matrix method computes exact localization length via Lyapunov exponent.