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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.

At large inverse field, $B$ and the cyclotron frequency are small, so the orbital spacing $\hslash {\omega }_c$ becomes comparable with both disorder broadening and the Fermi occupation's thermal width. Scattering of quantum lifetime ${\tau }_q$ gives a typical half-width $\Gamma =\hslash /(2{\tau }_q)$; overlapping levels lose their density-of-states contrast. The fundamental Dingle factor and thermal damping of quantum oscillations areBoth shrink rapidly as $B$ falls. Well-resolved oscillations require a sufficiently large ${\omega }_c{\tau }_q$ and a temperature small compared with the orbital spacing. The scattering time in this broadening argument is not automatically the transport time inferred from mobility; small-angle events affect the two differently.

The transverse Hall resistance is approximately $|R_{xy}|=B/(n_{s}e)$ in the classical weak-field regime. At stronger fields the integer quantum Hall effect produces plateausaccompanied by nearly vanishing longitudinal resistance. Localized bulk states between extended Landau states allow plateau intervals, while current can be carried by quantum Hall edge states. At still larger fields and sufficiently low temperature, the cleanest samples show the fractional quantum Hall effect, with interaction-generated incompressible states at fractional fillings such as $\nu =1/3$. The sign of $R_{xy}$ depends on the contact and magnetic-field orientation; the sketch shows its magnitude.

The quantum Hall effect connects an electrical measurement to a quantum counting rule: a two-dimensional electron gas in a perpendicular magnetic field can exhibit plateaus of transverse electrical resistance. The integer case supplies a resistance scale built from the Planck constant and elementary charge, rather than the sample's dimensions. The fractional quantum Hall effect involves additional many-Electron physics.

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]