Quantum Hall Effect Simulations

About: Visualize Landau level wavefunctions ψₘ = zᵐe^(-|z|²/4l²ᵦ). Higher m values show characteristic ring patterns. Enable coherent states to see displaced wavepackets.

Landau Level Parameters

Quantum Number m:

m = 0

Magnetic Length l_B (nm):

l_B = 15 nm

Coherent Superposition

Displacement |α|:

|α| = 0.0

Phase arg(α):

arg(α) = 0.0

Enable Coherent State

Display Options

Show |ψ|²

Show Phase (m>0)

Intensity Scale:

Scale = 2.0

Zoom Level:

Zoom = 1.0x

About: Coherent state ψ = exp(-|z|²/4l² + λz/l) showing displaced Gaussian wavepacket. The probability density |ψ|² demonstrates minimum uncertainty.

Parameters

λ Real Part:

Re(λ) = 3.0

λ Imaginary Part:

Im(λ) = 0.0

Magnetic Length l (nm):

l = 15 nm

Zoom Level:

Zoom = 1.0x

Intensity Scale:

Scale = 1.0

About: Classical charged particles follow cyclotron orbits with frequency ωc = eB/m in a confined Hall bar geometry. Edge states form along boundaries.

Parameters

Magnetic Field B (T):

B = 5.0 T

Initial Velocity (×10⁴ m/s):

v = 5×10⁴ m/s

Zoom Level:

Zoom = 1.0x

Show Trajectories

Click to Add Particles

Particles: 2/20
Cyclotron freq: ωc = 0.88 THz

About: Laughlin wavefunction Ψ = ∏ᵢ<ⱼ (zᵢ - zⱼ)^p exp(-Σᵢ|zᵢ|²/4l²). Shows the probability density |Ψ|² demonstrating correlation holes and incompressible liquid structure.

Parameters

Power p (odd only):

p = 3

Number of Particles:

N = 5

Zoom Level:

Zoom = 1.2x

Intensity Scale:

Scale = 1.0

Show Particle Positions

Filling Factor: ν = 1/3
State: Incompressible quantum liquid

About: Vortices are topological defects with phase winding 2πn around cores. Color shows phase, brightness shows |ψ|². Vortex cores have suppressed amplitude.

Parameters

Winding Number n:

n = 1

Zoom Level:

Zoom = 1.0x

Background Amplitude:

Amplitude = 0.6

Click to Add Vortices

Vortices: 0 | Total Charge: 0

About: Berry phase γ = π(n₁n₂) from adiabatic vortex braiding. Gold vortex orbits white vortex, accumulating geometric phase each orbit.

Parameters

Adiabatic Speed:

Speed = 0.8

Moving Vortex n₁:

n₁ = 1

Fixed Vortex n₂:

n₂ = 1

Show Accumulated Phase

Berry Phase: γ = 0.00π
Expected/orbit: γ = π × 1
Orbits completed: 0.0

About: Quantum Hall conductivity σxy = νe²/h shows plateaus at integer/fractional filling. Disorder broadens transitions but preserves quantization.

Parameters

Field Range:

Range = 1x

Disorder Strength:

Disorder = 0.20

Temperature (K):

T = 0.10 K

Show Fractional Plateaus

Effect: Higher disorder → broader transitions
Note: Plateau values remain quantized

Quantum Hall Sims