Quantum Hall Discovery

One-Sentence Significance

The quantum Hall discoveries revealed conductance plateaus quantized with extraordinary precision, tying electronic transport to Landau levels, disorder, interactions, and topology.

What Was Measured

In a two-dimensional electron system at low temperature and strong magnetic field, the Hall resistance develops plateaus while the longitudinal resistance becomes small. In common integer quantum Hall conventions,

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

Equivalently, resistance plateaus involve the von Klitzing constant

$$R_K = \frac{h}{e^2}.$$

What It Showed

The integer quantum Hall effect showed that transport in a dirty real material could be quantized with topological robustness. The fractional quantum Hall effect later showed that interactions can produce new incompressible quantum fluids with fractional response and quasiparticles.

What It Did Not Show By Itself

The integer and fractional effects have different microscopic explanations. Integer plateaus can be understood through Landau levels, disorder localization, and topological invariants. Fractional plateaus require strong correlations and cannot be reduced to noninteracting filled single-particle levels.

The discovery also did not mean every Hall response is quantized. Quantization requires appropriate dimensionality, field or band topology, gaps, temperature regime, and sample conditions.

Modern Interpretation

The integer Hall conductance can be written in terms of Chern numbers of occupied bands or Landau levels:

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

This is one of the cleanest experimental bridges from quantum mechanics to topology in matter.

Canonical Links

• Integer Quantum Hall Effect
• Quantum Hall Effect Glossary Entry
• Landau Levels
• Degeneracy of Landau Levels
• Chern Number
• Chern Numbers

Common Myths

• The fractional quantum Hall effect is just the integer effect with fractional filling.
• Disorder always destroys quantization. In the integer effect, localization is part of plateau formation.
• The sign convention for Hall conductance is universal. It depends on charge and orientation conventions.
• Topology replaces microscopic physics. It constrains robust response but does not remove the need for a Hamiltonian and gap.

Quick Check

What quantity sets the natural resistance scale of the integer quantum Hall effect?

Solution

The natural resistance scale is the von Klitzing constant,

$$R_K = \frac{h}{e^2}.$$

Plateaus occur at rational multiples determined by the filling and convention.

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.