When a two-dimensional electron gas is cooled and placed in a strong perpendicular magnetic field, its Hall conductance does not grow smoothly — it locks onto flat plateaus at exact integer multiples of the fundamental constant e²/h. These plateaus are astonishingly precise and completely immune to the messiness of real materials, because they are set by topology rather than by sample details. This experiment, discovered by Klaus von Klitzing in 1980, won the Nobel Prize and now defines the international standard of electrical resistance.
σ_xy = ν · e²/h, ν = n·h / (eB), E_N = ħω_c (N + ½), R_K = h/e² ≈ 25 812.807 Ω
The quantum Hall resistance is so reproducible — to better than one part in a billion across different materials and labs — that since 2019 the ohm is defined through h/e². Two pieces of dirty semiconductor on opposite sides of the planet will agree on it exactly, because the answer is a topological integer that physics simply cannot fudge.
What is the quantum Hall effect? A 2D electron gas in a strong field shows Hall conductance quantised in exact integer steps of e²/h, with flat plateaus and zero longitudinal resistance on them.
What are Landau levels? Discrete, highly degenerate energy levels into which electrons collapse in a magnetic field, spaced by the cyclotron energy ħω_c, each holding eB/h electrons per area.
Why is the conductance so precise? It is topological: each filled level contributes exactly one e²/h (its Chern number), an integer that cannot change continuously, so plateaus ignore disorder.
ν = n·h/(eB) — the number of filled Landau levels. On an integer plateau ν is a whole number and σ_xy = ν·e²/h.
One-dimensional chiral channels along the sample boundary where Landau levels cross the Fermi energy. They carry current one way only and cannot back-scatter, so they dissipate no energy.
On a plateau the Fermi level lies in a gap, so R_xx → 0. Between plateaus it crosses a Landau level, bulk states scatter, and R_xx spikes.
R_K = h/e² ≈ 25 812.807 Ω, the quantum of Hall resistance, now an exactly defined international resistance standard.
A clean 2D electron system (GaAs heterostructure or graphene), several tesla of field, and temperatures low enough that kT ≪ ħω_c.
Plateaus at fractional ν such as 1/3 from electron–electron interactions, with fractionally charged quasiparticles. This sim covers the integer effect.
It fills Landau levels to find ν, draws the σ_xy staircase and R_xx peaks, and animates chiral electrons circling the sample edges as you vary B, n and disorder.