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David Thouless coauthored a theory connecting the Hall conductivity of a two-dimensional electron gas to a topological invariant. In a uniform magnetic field and periodic potential, the Kubo formula can be written so that quantization becomes explicit when the Fermi energy lies in a gap. The integer character of the response explains why continuous changes of microscopic parameters need not continuously change the Hall plateau. A transition between distinct plateaus requires leaving the conditions that protect that quantization.^[ref]

A physicist who researched superfluidity and superconductivity in thin systems.^[ref]

Duncan Haldane constructed a two-dimensional lattice model with quantized Hall conductivity without a net applied magnetic field. The Haldane model separates the topological mechanism of the quantum Hall effect from the usual formation of Landau levels in a uniform field. Changing model parameters closes and reopens an energy gap, separating phases with different Hall responses. This provides a compact theoretical example in which the global structure of occupied electronic states determines a measurable transport coefficient.^[ref]

A physicist researching condensed matter physics.^[ref]

Hall conductivity is the transverse part of the electrical response in which an applied electric field drives a perpendicular current density. In two dimensions, one convention writes $j_x={\sigma }_{xx}{E}_x+{\sigma }_{xy}{E}_y$ and $j_y=-{\sigma }_{xy}{E}_x+{\sigma }_{xx}{E}_y$.

The sign depends on the magnetic-field orientation and chosen axes. In a quantum Hall effect plateau, the longitudinal response is strongly suppressed and the Hall response takes quantized values. Electrical conductivity and electrical resistivity are inverse tensors; taking the reciprocal of a single component is generally incorrect when longitudinal and transverse responses coexist.

For one positive carrier population, define $r_H$ by $R_H=r_H/(ep)$ with $e>0$. Different statistical averages of energy-dependent scattering affect hall conductivity and longitudinal electrical conductivity, so $r_H$ need not be one. The measured Hall mobility is ${\mu }_H=\sigma R_H=r_H{\mu }_{\mathrm{drift}}$. Multiple charge carriers require a more general model even before such a correction is made.

A theoretical researcher whose joint analysis of electrons in a periodic potential helped give the quantum Hall effect a topological description. Integrating information about the quantum states across their momentum domain yields an integer contribution to Hall conductivity. The calculation relates the geometry of wavefunctions to a macroscopic electrical response.^[ref]

A researcher whose joint theory showed that Hall conductivity can be tied to an integer associated with electronic wavefunctions. The result connects an electrical measurement with topology: changing the invariant requires leaving the conditions under which the occupied states remain separated by an energy gap. This helps explain the stability of quantized transport.^[ref]

A quantum Hall state is a two-dimensional electronic state exhibiting quantized transverse transport in a strong perpendicular magnetic field. In an ideal plateau regime, the longitudinal conductivity is suppressed while the Hall conductivity takes ${\sigma }_{xy}=\nu e^2/h$. Integer and fractional values of the filling-related label $\nu$ distinguish different regimes.

Landau levels organize the single-particle spectrum, while interactions are essential to fractional states. A measured plateau reflects a robust bulk response together with the role of edges and disorder; it is not obtained simply by replacing a classical carrier density with an integer in the ordinary Hall formula.