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In a two-dimensional electron gas subjected to a strong perpendicular magnetic field, the energy spectrum splits into discrete Landau levels E_n = hbar omega_c (n + 1/2), where omega_c = eB/mc is the cyclotron frequency. When the Fermi level lies between Landau levels, the Hall conductance is exactly quantized: sigma_{xy} = nu e2/h, where nu is an integer equal to the number of filled Landau levels. This quantization is extraordinarily precise (~1 part in 109) and is independent of material details, disorder, or sample geometry — it is topological in origin. The integer quantum Hall effect provides the primary resistance standard and was the first example of a topological phase of matter.
The integer quantum Hall effect (IQHE), discovered by Klaus von Klitzing in 1980, occurs when a two-dimensional electron gas (2DEG) — typically at a semiconductor heterointerface like GaAs/AlGaAs — is placed in a strong perpendicular magnetic field at low temperature. The Hall resistance R_{xy} = V_H/I, instead of increasing linearly with B as in the classical Hall effect, develops a series of flat plateaus at precisely quantized values R_{xy} = h/(nu e2), where nu = 1, 2, 3, ... The longitudinal resistance R_{xx} simultaneously vanishes on each plateau. The quantization is exact to about 1 part in 109.
The starting point for understanding the IQHE is Landau quantization. A free electron in 2D in a magnetic field B has its continuous energy spectrum collapsed into discrete Landau levels at energies E_n = hbar omega_c (n + 1/2), each massively degenerate (degeneracy = eB/h per unit area). When exactly nu Landau levels are filled and the Fermi level sits in the gap between the nu-th and (nu+1)-th levels, the system is a gapped insulator in the bulk with quantized Hall conductance sigma_{xy} = nu e2/h.
The role of disorder is crucial and counterintuitive. In a clean system, Landau levels are infinitely sharp delta functions, and the Fermi level can only sit in a gap at discrete values of B — no plateaus would exist. Disorder broadens each Landau level into a band of mostly localized states (Anderson localization in 2D) with a narrow strip of delocalized states at the center. As B varies, the Fermi level sweeps through the localized states without changing the transport properties, creating the observed plateaus. The transition between plateaus (where R_{xx} peaks) occurs when E_F crosses the delocalized states.
The deep reason for the exact quantization is topology. The Hall conductance of each filled Landau level is a topological invariant — the Chern number — computed as an integral of the Berry curvature over the magnetic Brillouin zone. Chern numbers are integers by mathematical necessity (like the genus of a surface), and they cannot change under continuous deformations of the Hamiltonian that do not close the energy gap. This topological protection explains why the quantization is independent of disorder, sample geometry, and material details. The IQHE was the first experimentally realized topological phase of matter, launching the field that later produced topological insulators, topological superconductors, and the mathematical framework of topological band theory.