Fractional Quantum Hall Anyon Braiding — Statistical Phase

Filling Factor ν = 1/m

Braid Path

Loops around each other 1

Braid speed 1.0×

Wavefunction Phase

θ = 0.00 rad (0.0°)

Statistics angle / exchange

π/3

Quasiparticle charge

e/3

Total winding

0.00 × 2π

Braids completed

How it works

In a 2D electron gas at strong magnetic field and Landau-level filling factor ν = 1/m (m odd), the ground state is the Laughlin liquid. Its charged excitations are quasiholes carrying a fractional charge e* = e/m — not e or a multiple of it, unlike any ordinary particle.

Laughlin wavefunction: Ψ_m(z) = Π(zᵢ-zⱼ)^m · exp(-Σ|zᵢ|²/4)
Quasihole charge:      e* = e / m
Exchange phase:        θ_exch = π / m   (half the full-loop phase)
Full braid phase:      θ_braid = 2π / m  per loop, per quasiparticle pair

Braiding one quasiparticle in a closed loop fully around another multiplies the many-body wavefunction by e^(iθ), with θ a fraction of 2π set only by m — this is what makes them anyons: neither bosons (θ=0) nor fermions (θ=π), and the phase depends only on the topology of the path (how many times it winds), never on its exact shape or speed. This simulator renders the 2D electron gas as a plane of Landau-level electrons under a uniform field, places two localized quasihole density dips, and lets you drag one around the other; the accumulated phase is tracked live and drawn on a unit-circle phasor.

ν selector — sets m (1/3, 1/5, 1/7), changing both the quasiparticle charge e/m and the phase per loop 2π/m.
Loops — how many full windings "Braid Once" performs before stopping.
Braid speed — angular speed of the moving quasiparticle; the accumulated phase is unaffected, since it is purely topological.

Real-world relevance: fractional statistics of Laughlin quasiparticles were confirmed experimentally via shot-noise and interferometry measurements, and the non-Abelian generalization at ν=5/2 is a leading candidate qubit for topological quantum computing.

Landau-level electron gas Moving quasihole Fixed quasihole

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