Editing Fractional quantum Hall effect (section)

== History and developments ==
The FQHE was experimentally discovered in 1982 by [[Daniel Chee Tsui|Daniel Tsui]] and [[Horst Ludwig Störmer|Horst Störmer]], in experiments performed on  [[heterostructure]]s made out of [[gallium arsenide]] developed by [[Arthur Gossard]].

There were several major steps in the theory of the FQHE.

*'''Laughlin states and fractionally-charged [[quasiparticle]]s''': this theory, proposed by [[Robert B. Laughlin]], is based on [[Laughlin wavefunction|accurate trial wave functions]] for the [[ground state]] at fraction <math>1/q</math> as well as its quasiparticle and quasihole excitations. The excitations have fractional charge of magnitude <math>e^*={e\over q}</math>.
*'''Fractional exchange statistics of quasiparticles''': [[Bertrand Halperin]] conjectured, and Daniel Arovas, [[John Robert Schrieffer]], and [[Frank Wilczek]] demonstrated, that the fractionally charged quasiparticle excitations of the Laughlin states are [[anyon]]s with fractional statistical angle <math>\theta = {\pi \over q}</math>; the wave function acquires phase factor of <math> e^{i \theta}</math> (together with an [[Aharonov-Bohm phase factor]]) when identical quasiparticles are exchanged in a counterclockwise sense. A recent experiment seems to give a clear demonstration of this effect.<ref>{{Cite arXiv |last1=An |first1=Sanghun |last2=Jiang |first2=P. |last3=Choi |first3=H. |last4=Kang |first4=W. |last5=Simon |first5=S. H. |last6=Pfeiffer |first6=L. N. |last7=West |first7=K. W. |last8=Baldwin |first8=K. W. |date=2011 |title=Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect |eprint=1112.3400 |class=cond-mat.mes-hall }}</ref>
*'''Hierarchy states''': this theory was proposed by [[Duncan Haldane]], and further clarified by [[Bertrand Halperin]], to explain the observed filling fractions not occurring at the Laughlin states' <math>\nu = 1/q</math>. Starting with the Laughlin states, new states at different fillings can be formed by condensing quasiparticles into their own Laughlin states. The new states and their fillings are constrained by the fractional statistics of the quasiparticles, producing e.g. <math>\nu = 2/5</math> and <math>2/7</math> states from the Laughlin <math>\nu = 1/3</math> state. Similarly constructing another set of new states by condensing quasiparticles of the first set of new states, and so on, produces a hierarchy of states covering all the odd-denominator filling fractions. This idea has been validated quantitatively,<ref>{{cite journal|first=M.|year=1994|title=Microscopic formulation of the hierarchy of quantized Hall states|journal=[[Phys. Lett. B|Physics Letters B]]|volume=336|issue=1|pages=48–53|arxiv=cond-mat/9311062|bibcode=1994PhLB..336...48G|doi=10.1016/0370-2693(94)00957-0|last=Greiter|s2cid=119433766}}</ref> and brings out the observed fractions in a natural order. Laughlin's original plasma model was extended to the hierarchy states by [[Allan H. MacDonald]] and others.<ref>{{cite journal|first1=A.H.|first2=G.C.|first3=M.W.C.|year=1985|title=Hierarchy of plasmas for fractional quantum Hall states|journal=[[Phys. Rev. B|Physical Review B]]|volume=31|issue=8|pages=5529–5532|bibcode=1985PhRvB..31.5529M|doi=10.1103/PhysRevB.31.5529|last1=MacDonald|last2=Aers|last3=Dharma-wardana|pmid=9936538}}</ref> Using methods introduced by [[Greg Moore (physicist)|Greg Moore]] and [[Nicholas Read]],<ref>{{cite journal |last1=Moore |first1=G. |last2=Read |first2=N. |title=Nonabelions in the fractional quantum Hall effect |journal=Nucl. Phys. |date=1990 |volume=B360 |issue=2 |page=362|bibcode=1991NuPhB.360..362M |doi=10.1016/0550-3213(91)90407-O |doi-access=free }}</ref> based on [[conformal field theory]] explicit wave functions can be constructed for all hierarchy states.<ref>{{cite journal |last1=Hansson |first1=T.H. |last2=Hermanns |first2=M. |last3=Simon |first3=S.H. |last4=Viefers|first4=S.F.|author4-link= Susanne Viefers |title=Quantum Hall physics: Hierarchies and conformal field theory techniques |journal=Rev. Mod. Phys. |date=2017 |volume=89 |issue=2 |page=025005 |doi=10.1103/RevModPhys.89.025005|arxiv=1601.01697 |bibcode=2017RvMP...89b5005H |s2cid=118614055 }}</ref>
*'''[[Composite fermions]]''': this theory was proposed by [[Jainendra K. Jain]], and further extended by Halperin, [[Patrick A. Lee]] and Read. The basic idea of this theory is that as a result of the repulsive interactions, two (or, in general, an even number of) vortices are captured by each electron, forming integer-charged quasiparticles called composite fermions. The fractional states of the electrons are understood as the integer [[quantum Hall effect|QHE]] of composite fermions. For example, this makes electrons at filling factors 1/3, 2/5, 3/7, etc. behave in the same way as at filling factor 1, 2, 3, etc. Composite fermions have been observed, and the theory has been verified by experiment and computer calculations. Composite fermions are valid even beyond the fractional quantum Hall effect; for example, the filling factor 1/2 corresponds to zero magnetic field for composite fermions, resulting in their Fermi sea.
Tsui, Störmer, and [[Robert B. Laughlin]] were awarded the 1998 [[Nobel Prize in Physics]] for their work.

Jain, [[James P. Eisenstein]], and [[Mordehai Heiblum]] won the 2025 [[Wolf Prize in Physics]] "for advancing our understanding of the surprising properties of two-dimensional electron systems in strong magnetic fields".<ref>{{Cite web |last=מיכל |date=2025-03-10 |title=James P. Eisenstein |url=https://wolffund.org.il/james-p-eisenstein/ |access-date=2025-03-17 |website=Wolf Foundation |language=en-US}}</ref>