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The fractional quantum Hall effect is quantized transverse sheet conductivity at fractional filling factors, produced by interacting electrons. The Laughlin wavefunction describes the sequence $\nu =1/m$, with odd $m$, including $\nu =1/3$.^[ref]

For an original ideal readout example, take a Hall bar on the $\nu =1/3$ plateau with negligible longitudinal sheet conductivity. The quantum Hall conductivity tensor inversion givesHere $R_K$ is the von Klitzing constant; $R_H$ denotes transverse electrical resistance, rather than the bulk Hall coefficient. Thus a fractional sheet conductivity corresponds to a larger transverse electrical resistance. The fraction labels a collective quantum state: it does not mean that each electron has acquired one third of its usual electric charge.

On an ideal integer plateau, the magnitude of transverse sheet conductivity is $N{e}^2/h$, where $N$ is a positive integer, while the longitudinal sheet conductivity vanishes. The measured transverse electrical resistance is then $h/(N{e}^2)$. Both statements matter: inverting a conductivity tensor mixes its longitudinal and transverse components, as the quantum Hall conductivity tensor inversion shows.

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.

The degeneracy of a Landau level is $eBA/h$ per resolved internal branch in area $A$. The filling factor $f=n_{s}h/(eB)$ varies continuously. Additional localized eigenstates need not carry bulk transport, allowing an integer plateau over an interval of $f$; a mobility gap need not be an empty spectral gap.^[ref]

The plot is an original phenomenological construction, not measured data or a calculation of Anderson localization. It stipulates dimensionless transverse sheet conductivity $k(f)=1+\frac{1}{2}{\sum }_{m=1}^5[1+\tanh ((f-m-1/2)/w)]$ and longitudinal sheet conductivity $s(f)=a{\sum }_{m=1}^5{\mathrm{sech}}^2((f-m-1/2)/w)$, with chosen $w=0.06$, $a=0.08$. Both are measured in units $e^2/h$. The quantum Hall conductivity tensor inversion produces $|{\rho }_{yx}|/R_K=k/(s^2+k^2)$ and ${\rho }_{xx}/R_K=s/(s^2+k^2)$. Regions near $k=N$, $s=0$ yield plateaus near $R_K/N$; the model alone does not explain their microscopic robustness. Each step represents one resolved branch, without assuming unresolved spin or valley multiplicities.

Daniel Tsui, Horst Störmer and Arthur Gossard reported the effect in 1982. They grew gallium arsenide–aluminium gallium arsenide layers by molecular beam epitaxy, separating the two-dimensional electron gas from its semiconductor doping. They cut Hall bars and made indium contacts. Low-field measurements established number density and charge carrier mobility. A perpendicular magnetic field sweep measured transverse and longitudinal sheet resistivities at several temperatures. Cooling sharpened the plateau near $3h/e^2$ and the longitudinal minimum at filling factor $1/3$; integer plateaus provided an internal comparison. The longitudinal minimum remained finite at the lowest temperature: zero-temperature vanishing was unresolved. This established anomalous transport, without directly measuring excitation electric charge.^[ref]

The two readouts test different statements. A single intersection with $R_H=3R_K$ would also occur on the smooth classical relation $R_H=B/(n_{s}e)$ at $n_{s}h/(eB)=1/3$. A quantum Hall plateau instead persists over a magnetic field interval. Comparing transverse and longitudinal signals helps distinguish that plateau from a contact-placement artefact; the quantum Hall conductivity tensor inversion quantifies the contamination when longitudinal sheet resistivity is nonzero.

Later, 1997 shot noise measurements associated with quantum tunnelling at filling factor $1/3$ tested the excitation electric charge separately, finding $e/3$. That is a different observable from a Hall plateau.^[ref]