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In the quantum Hall effect, a two-dimensional electron system at low temperature in a strong perpendicular magnetic field has transverse conductivity of quantized magnitude $|{\sigma }_{xy}|=\nu e^2/h$. On a plateau with negligible longitudinal conductivity, quantum Hall conductivity tensor inversion gives $|{\rho }_{xy}|=h/(\nu e^2)$ and ${\rho }_{xx}\to 0$; the Hall sign depends on the carrier and orientation convention. The filling factor is $\nu =n_{2D}h/(eB)$ when all spin-resolved states are counted.

For the integer quantum Hall effect, filled Landau levels and a mobility gap give integer $\nu$. Disorder-localized states allow the density or field to vary without changing the number of extended occupied channels, producing a quantum Hall plateau rather than quantization only at isolated fields. Quantum Hall edge states provide an equivalent transport picture. In a Hall bar, separate voltage leads measure transverse and longitudinal voltages while a known current flows; sweeping the field reveals both the plateau and the resistance minimum. The von Klitzing-Dorda-Pepper quantum Hall experiment established this integer quantization.

The fractional quantum Hall effect requires electron correlations within a partially filled Landau level. An incompressible correlated state can occur at fractions such as $\nu =1/3$; a noninteracting partially filled level does not account for the associated gap. The Laughlin wavefunction at $\nu =1/m$, odd $m$, contains a factor ${\prod }_{i<j}(z_i-z_j{)}^m$, suppressing close electron approaches while retaining fermionic antisymmetry. Its charged collective excitations can carry magnitude $e/m$; the constituent electrons still have charge $-e$. The Tsui-Störmer-Gossard fractional Hall experiment observed the fractional Hall feature together with a longitudinal-resistance minimum, providing the experimental distinction from an arbitrary rational filling of an ordinary conducting state.