Anderson Localization: 1D Transfer Matrix

Anderson localization: In 1D, any amount of disorder causes exponential localization of all eigenstates. The 1D Schrödinger equation with random on-site potentials ε_n ∈ [−W/2, W/2] becomes a transfer matrix problem:

[ψ_{n+1}; ψ_n] = M_n · [ψ_n; ψ_{n-1}] where M_n = [[E−ε_n, −1]; [1, 0]]

The Lyapunov exponent γ = lim (1/N) ln||M_N···M_1|| gives the inverse localization length ξ = 1/γ. By Furstenberg's theorem, γ > 0 almost surely for any W > 0 in 1D. In 3D, there's a mobility edge below which states are extended (metal-insulator transition). The upper panel shows the wavefunction amplitude decaying exponentially from the injection point.