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The **quantum Hall effect** (or **integer quantum Hall effect**) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductivity σ undergoes certain quantum Hall transitions to take on the quantized values

where is the channel current, is the Hall voltage, *e* is the elementary charge and *h* is Planck's constant. The prefactor ν is known as the "filling factor", and can take on either integer (ν = 1, 2, 3, .. ) or fractional (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5 ...) values. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether ν is an integer or fraction respectively. The integer quantum Hall effect is very well understood, and can be simply explained in terms of single-particle orbitals of an electron in a magnetic field (see Landau quantization). The fractional quantum Hall effect is more complicated, as its existence relies fundamentally on electron–electron interactions. It is also very well understood as an integer quantum Hall effect, not of electrons but of charge-flux composites known as composite fermions. In 1988, it was proposed that there was quantum Hall effect without Landau levels. This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.^{[1]}

The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of *e ^{2}/h* to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance.

The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that Klaus von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was *exactly* quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature,^{[5]} and in the oxide ZnO-Mg_{x}Zn_{1-x}O.^{[6]}

In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values: , where *ω _{c} = eB/m* is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov–de Haas oscillations and the de Haas–van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy

The integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the first Chern numbers and are closely related to Berry's phase. A striking model of much interest in this context is the Azbel-Harper-Hofstadter model whose quantum phase diagram is the Hofstadter butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity.

Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called *composite fermions*.

- quantum Hall transitions
- Fractional quantum Hall effect
- Composite fermions
- Hall effect
- Hall probe
- Graphene
- Quantum spin Hall effect
- Coulomb potential between two current loops embedded in a magnetic field

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