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The striking feature of the integer quantum Hall effect is the naldane of the quantization i. Since the electron density remains constant when the Fermi level is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized see Anderson localization.

The fractional quantum Hall effect is more complicated, as its existence haldxne fundamentally on electron—electron interactions. The fractional quantum Hall effect is also understood as an integer quantum Hall effect, although not of electrons but of charge-flux composites known as composite fermions. Init was proposed efeccto there was quantum Hall effect without Landau levels. 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.

The quantization of the Hall conductance has the important property of being exceedingly precise. This phenomenon, referred to as exact quantizationhas been shown to be a subtle manifestation of the principle of gauge invariance.

Sincea fixed conventional value R K has been used efwcto resistance calibrations worldwide. The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura inon the basis of an approximate calculation which they themselves did nohr believe to be true. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlinwho connected the quantized conductivity to the quantized charge transport in Thouless charge pump.

In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits.

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When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values:. These orbitals are known as Landau levelsand at weak magnetic fields, their existence gives rise to many “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. Specifically, for a sample of area Ain magnetic field Bthe degeneracy of each Landau level is. For sufficiently strong magnetic fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect. 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 potentialwhich fixes the electron density.

The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero, hence they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity.

In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect. The observed strong similarity between integer hsldane 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.

The value of the von Klitzing constant may be obtained already on the level of a single atom within the Bohr model while looking at it as a single electron Hall effect.

While during the cyclotron halldane on a circular orbit the centrifugal force halddane balanced by the Lorentz force responsible for the transverse induced voltage and the Hall effect one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle a Hall current.

One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as. Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.

From Wikipedia, the free encyclopedia.

Condensed-Matter Realization of the ‘Parity Anomaly ‘ “. Recent Theoretical and Experimental Developments 3rd ed. Retrieved from ” https: Hall effect Condensed matter physics Quantum electronics Spintronics Quantum phases Mesoscopic physics in science.

## Quantum Hall effect

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