Fractional Quantum Hall Anyon Braiding — Flux-Attachment Phase Integral (2D)

Filling Factor ν = 1/m

Braid Path

Loops around each other 1

Braid speed 1.0×

Line-Integral Phase ∮A·dl

θ = 0.00 rad (0.0°)

Statistics angle / exchange

π/3

Quasiparticle charge

e/3

Geometric winding

0.00 × 2π

Braids completed

How this 2D view differs

The 3D version shows a Landau-level lattice orbiting camera-side and simply multiplies the total angle traveled by 2π/m at the end. This 2D companion instead renders the composite-fermion / flux-attachment picture: the fixed (red) quasihole carries a synthetic Aharonov–Bohm flux line of strength Φ = 2π/m threading the plane, drawn as tangential field arrows whose length falls off as 1/ρ. The moving (yellow) quasiparticle's statistical phase is computed every animation frame as a genuine discretized line integral of that vector field along its actual velocity — not read off a lookup formula.

Flux-line vector potential:  A(r) = (Φ/2π) · (-Δy, Δx) / ρ²,  ρ = |r - r_fixed|
Per-frame phase increment:   dθ = (A · v) · dt        (numerical ∮A·dl)
Accumulated phase:           θ = Σ dθ  over the frame history

For any closed loop around the flux line, A·v · dt integrates to exactly
Φ · (Δφ / 2π) regardless of loop radius, speed, or shape — the phase
is a topological invariant, which this simulator now demonstrates by
actually computing the integral rather than assuming the result.

The panel also tracks the geometric winding (raw angle traveled ÷ 2π) completely separately from the integrated phase — the two are computed by unrelated code paths, so their agreement (phase ≈ winding × 2π/m) is a live cross-check of the topological statistics, not a coincidence of a shared formula.

ν selector — sets m (1/3, 1/5, 1/7), changing the flux strength Φ = 2π/m and the quasiparticle charge e/m.
Loops — how many full windings "Braid Once" performs before stopping.
Braid speed — angular speed of the orbiting quasiparticle; the integrated phase is unaffected, since A·v·dt = Φ·dφ/2π has no speed dependence once dφ is fixed.

Real-world relevance: fractional statistics of Laughlin quasiparticles were confirmed via shot-noise and interferometry measurements; the flux-attachment picture used here is Jain's composite-fermion construction, the standard way condensed-matter theory explains why these excitations are neither bosons nor fermions.

Landau-level electron gas Moving quasihole Fixed quasihole (flux source) Synthetic flux-line field A(r)