= Solution
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 $\rho_{xx}\to0$; 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.
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