Anderson Localization in 1D
All states localize in 1D for any disorder — localization length ξ shrinks with W
Anderson model (1958): tight-binding Hamiltonian H = Σᵢ εᵢ|i⟩⟨i| − t Σᵢ(|i⟩⟨i+1|+h.c.)
where εᵢ ∈ [−W/2, W/2] are random on-site energies.
In 1D, all eigenstates are exponentially localized: |ψ(x)|² ~ e^{−2|x−x₀|/ξ}.
Localization length (weak disorder): ξ ≈ 96t²/W² (Born approximation).
This visualization computes eigenstates via transfer matrix and shows the localization envelope.