Anderson Localization — Scaling Theory β(g)

Scaling Theory (Abrahams et al.)

The β function β(g) = d(ln g)/d(ln L) describes how conductance g scales with system size L. In d=1,2: β(g) < 0 always → all states localized. In d=3: β has a fixed point g* — above it the system is metallic, below insulating. The 1D wavefunction shows exponential localization.

Localization length: ξ ∝ (W/t)⁻² in 1D. Critical disorder in 3D: W_c ≈ 16.5 (tight-binding).