Editing Quantum Hall effect (section)

{{Short description|Electromagnetic effect in physics}}
The '''quantum Hall effect''' (or '''integer quantum Hall effect''') is a [[quantum mechanics|quantized]] version of the [[Hall effect]] which is observed in [[2DEG|two-dimensional electron systems]] subjected to low [[temperature]]s and strong [[magnetic field]]s, in which the Hall [[Electrical resistance and conductance|resistance]] {{math|''R''<sub>xy</sub>}} exhibits steps that take on the quantized values
: <math> R_{xy} = \frac{V_\text{Hall}}{I_\text{channel}} = \frac{h}{e^2\nu} , </math>
where {{math|''V''<sub>Hall</sub>}} is the [[Hall effect|Hall voltage]], {{math|''I''<sub>channel</sub>}} is the channel [[electric current|current]],  {{math|''e''}} is the [[elementary charge]] and {{math|''h''}} is the [[Planck constant]]. The divisor {{math|[[Nu (letter)|''ν'']]}} can take on either integer ({{math|''ν'' {{=}} 1, 2, 3,...}}) or fractional ({{math|''ν'' {{=}} {{sfrac|1|3}}, {{sfrac|2|5}}, {{sfrac|3|7}}, {{sfrac|2|3}}, {{sfrac|3|5}}, {{sfrac|1|5}}, {{sfrac|2|9}}, {{sfrac|3|13}}, {{sfrac|5|2}}, {{sfrac|12|5}},...}}) values. Here, {{math|[[Nu (letter)|''ν'']]}} is roughly but not exactly equal to the filling factor of [[Landau quantization|Landau levels]]. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether {{math|''ν''}} is an integer or fraction, respectively.

The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. 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]]).<ref>{{Cite journal|last=Editorial|date=2020-07-29|title=The quantum Hall effect continues to reveal its secrets to mathematicians and physicists|journal=Nature|language=en|volume=583|issue=7818|pages=659|doi=10.1038/d41586-020-02230-7|pmid=32728252|bibcode=2020Natur.583..659.|doi-access=free}}</ref>

The [[fractional quantum Hall effect]] is more complicated and still considered an open research problem.<ref name="Hansson 025005">{{cite journal|first=T.H.| last=Hansson |title=Quantum Hall physics: Hierarchies and conformal field theory techniques|journal=Reviews of Modern Physics|volume=89|issue=25005|date=April 2017| page=025005 | doi=10.1103/RevModPhys.89.025005 | arxiv=1601.01697 | bibcode=2017RvMP...89b5005H | s2cid=118614055 }}</ref> Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without [[Landau quantization|Landau levels]].<ref name=Haldane:1988 /> 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.<ref>{{cite book | first=Zyun F. | last=Ezawa | title=Quantum Hall Effects: Recent Theoretical and Experimental Developments | edition=3rd | publisher=World Scientific | year=2013 | isbn=978-981-4360-75-3}}</ref>