Quantum Hall Effect

The quantum Hall effect is the quantization of transverse electrical response in effectively two-dimensional electron systems. In the integer quantum Hall effect, filled Landau levels or filled topological bands produce robust conductance plateaus. In the fractional quantum Hall effect, interactions produce correlated incompressible states with fractional response and anyonic excitations. For the Berry-geometry bridge, see Quantum Hall Geometry Preview.

Formula Hook

In common integer quantum Hall conventions,

$$\sigma_{xy} = \nu\frac{e^2}{h}, \qquad \nu\in\mathbb Z,$$

up to charge, orientation, and sign conventions. In a topological band description,

$$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{occupied }n} C_n,$$

where $C_n$ is the Chern number of an occupied band.

For a spinless Landau-level model with particle density $n$,

$$\nu = 2\pi\ell_B^2 n, \qquad \ell_B=\sqrt{\frac{\hbar}{\lvert eB\rvert}}.$$

Canonical Home

Integer Quantum Hall Effect is the canonical material home for integer plateaus, edge transport, localization, Chern response, and metrology. Fractional Quantum Hall Effect is the canonical material home for interaction-driven fractions, fractionalized excitations, composite fermions, and fractional edges. This glossary entry remains the compact lookup for both named effects. Supporting foundations include Landau Levels, Degeneracy of Landau Levels, and Chern Numbers.

Common Confusions

• The classical Hall effect is not quantized; quantization depends on quantum gaps and localization or topology.
• Integer and fractional quantum Hall effects have different microscopic explanations.
• The sign of $\sigma_{xy}$ depends on charge and orientation conventions.
• Spin, valley, layer, and disorder degrees of freedom change the plateau sequence and must be specified.

Related Entries

• Integer Quantum Hall Effect
• Fractional Quantum Hall Effect
• Aharonov–Bohm Effect
• Quantum Hall Geometry Preview
• Hamiltonian
• Density Matrix

References

• K. von Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Physical Review Letters 45, 494-497, 1980.
• D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Physical Review Letters 48, 1559-1562, 1982.
• R. B. Laughlin, “Quantized Hall conductivity in two dimensions,” Physical Review B 23, 5632-5633, 1981.
• R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., Springer, 1990.