Quantum Hall Effect — Laughlin Wavefunction

Fractional quantum Hall states and incompressible electron liquids at filling ν = 1/m

Number of electrons N7

Filling exponent m (ν=1/m)m=3 (ν=1/3)

Magnetic length ℓ_B1.2

FQHE State

Filling ν—

Hall resistance—

Quasi-particle charge—

Ground state deg.—

Correlation length—

Key Formula

Ψ_m = ∏_i<j(z_i−z_j)^m · e^{−Σ|z_k|²/4ℓ²}
m=odd → fermions
ν = 1/m
R_H = h/(νe²) = m·h/e²

The Laughlin wavefunction Ψ_m(z₁,...,z_N) = ∏(z_i−z_j)^m · exp(−Σ|z_k|²/4ℓ²) describes a strongly correlated electron liquid at filling fraction ν=1/m (m odd for fermions). The factor (z_i−z_j)^m creates an m-fold zero between each pair of electrons, keeping them apart via Coulomb repulsion. Quasi-particle excitations carry fractional charge e/m and obey anyonic statistics (θ=π/m). The Hall resistance is exactly quantized at R_H = h/(νe²).