Fractional Quantum Hall Effect — Laughlin State

Electrons in 2D + strong magnetic field: at filling ν=1/m, Laughlin's wavefunction describes a topological liquid with anyonic quasiparticles carrying charge e/m

1/3
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1.2
Laughlin wavefunction (1983): Ψ_m(z₁…z_N) = ∏ᵢ<ⱼ (zᵢ−zⱼ)ᵐ · exp(−Σ|zₖ|²/4ℓ_B²), where zⱼ = xⱼ+iyⱼ. The integer m=1/ν must be odd (fermion antisymmetry). At ν=1/3: each electron sees a vortex of strength m=3 attached to every other electron (flux attachment). Quasihole: inserting flux quantum at z₀ → Ψ ∝ ∏ⱼ(zⱼ−z₀)·Ψ_m. This excitation has charge e/m and anyonic statistics: braiding two quasiholes gives phase 2π/m — neither bosonic (2π) nor fermionic (π). The Hall conductance is quantized: σ_xy = νe²/h.