Quantum Hall Plateau — Laughlin Gauge Argument

σ_xy = νe²/h at exact integer/fractional filling

ν = 1.00
B = 5.0 T
Γ = 0.08

Laughlin's Gauge Argument (1981)

Thread a flux quantum Φ₀ = h/e adiabatically through a Hall cylinder. By gauge invariance, the Hamiltonian returns to itself, but an integer number of electrons — exactly ν — are pumped across. The transported charge Q = νe gives Hall conductance:

σ_xy = νe²/h (exact, topological — independent of disorder!)

Plateaux are protected because localized states (from disorder) exist between Landau levels; only the extended states at the center of each level carry current. As the Fermi level sweeps through localized states, ν stays integer-quantized.