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	\|image=[[File:Quantum_fractional_Hall_effect_educational_yellow.png\|430px\|Fractional Hall effect: a two-dimensional electron system forms fractionally charged excitations under a strong magnetic field.]]		\|image=[[File:Quantum_fractional_Hall_effect_educational_yellow.png\|430px\|Fractional Hall effect: a two-dimensional electron system forms fractionally charged excitations under a strong magnetic field.]]
	\|text=~~'''Fractional~~ Hall effect~~'''~~ is a ~~ScholarlyWiki page~~ in the Quantum Collection ~~about fractionally quantized~~ Hall conductance, ~~correlated~~ electron states, and emergent quasiparticles ~~in two-dimensional systems~~.		\|text=The fractional quantum Hall effect is a Book II topic in the Quantum Collection. It occurs in two-dimensional electron systems at low temperature and strong magnetic field, where the Hall conductance becomes quantized at fractional values. Unlike the integer effect, the fractional effect is caused by strong electron-electron correlations and the formation of collective quantum fluids. Its excitations can carry fractional electric charge and anyonic statistics. The topic connects Landau levels, topology, many-body wavefunctions, composite fermions, edge states, and some of the clearest experimental evidence for emergent quasiparticles.
	}}		}}
	== Overview ==		== Overview ==

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{{Short description|Fractional quantization of Hall conductance in two-dimensional electron systems}}
{{Quantum matter backlink|Condensed matter and solid-state physics}}
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|image=[[File:Quantum_fractional_Hall_effect_educational_yellow.png|430px|Fractional Hall effect: a two-dimensional electron system forms fractionally charged excitations under a strong magnetic field.]]
|text='''Fractional Hall effect''' is a ScholarlyWiki page in the Quantum Collection about fractionally quantized Hall conductance, correlated electron states, and emergent quasiparticles in two-dimensional systems.
}}
== Overview ==
The '''fractional Hall effect''' is a collective quantum phenomenon observed in very clean two-dimensional electron systems placed in a strong perpendicular magnetic field at low temperature. In this regime the Hall conductance forms plateaus at fractional values of <math>e^2/h</math>, rather than only at integer values.

The effect is a landmark example of strongly correlated quantum matter. It cannot be explained by treating electrons as independent particles filling single-particle Landau levels. Instead, interactions between electrons produce new many-body states with unusual excitations, including fractionally charged quasiparticles and anyonic exchange statistics.

== Key ideas ==
In the ordinary Hall effect, a transverse voltage appears when charge carriers move through a magnetic field. In the integer quantum Hall effect, the conductance is quantized because the electron motion is organized into [[Physics:Quantum Landau levels|Landau levels]]. The fractional Hall effect appears when a Landau level is only partially filled and electron-electron interactions dominate the physics.

The filling factor is commonly written as

<math display="block">
\nu = \frac{n h}{eB},
</math>

where <math>n</math> is the two-dimensional electron density and <math>B</math> is the magnetic field. Fractional Hall plateaus occur at values such as <math>\nu=1/3</math>, <math>2/5</math>, and many related fractions.

== Laughlin states ==
Robert Laughlin proposed a many-body wavefunction that explains the simplest fractional plateaus, especially <math>\nu=1/3</math>. The Laughlin state captures how electrons avoid one another while remaining in the lowest Landau level. This correlation lowers interaction energy and produces an incompressible quantum fluid.

The associated quasiparticle excitations can carry a fraction of the electron charge. For the <math>\nu=1/3</math> Laughlin state, the elementary quasiparticle charge is <math>e/3</math> in magnitude. These fractional charges have been probed experimentally through shot-noise and interferometry measurements.

== Composite fermions ==
Many observed fractions are described by the composite-fermion picture. In this approach, an electron moving in a strong magnetic field is treated as binding an even number of magnetic flux quanta, forming an emergent composite particle. Composite fermions experience a reduced effective magnetic field and can fill effective Landau levels.

This idea organizes prominent sequences of fractions, such as

<math display="block">
\nu = \frac{p}{2p \pm 1},
</math>

where <math>p</math> is an integer. The composite-fermion framework connects the fractional Hall effect to an effective integer quantum Hall effect of emergent quasiparticles.

== Topological order ==
Fractional Hall states are examples of [[Physics:Quantum Topological phases of matter|topological phases of matter]]. Their essential properties are not described by a local order parameter in the usual symmetry-breaking sense. Instead, they show topological order, robust edge modes, ground-state degeneracy on nontrivial geometries, and quasiparticle statistics that depend on braiding.

Some fractional Hall states support Abelian anyons, while more exotic candidates may support non-Abelian anyons. These possibilities connect the fractional Hall effect with [[Physics:Quantum anyon|anyons]], topological quantum matter, and proposals for fault-tolerant quantum information processing.

== Physical setting ==
The effect is typically observed in high-mobility semiconductor heterostructures or other two-dimensional materials where disorder is low enough for interaction-driven states to form. Low temperatures reduce thermal smearing, and strong magnetic fields separate the Landau levels.

Although the basic measurements are transport measurements, the physics is deeply many-body. Edge channels carry current along the boundary, while the bulk remains incompressible at a plateau. Deviations from the plateau reveal quasiparticle excitations, disorder effects, and transitions between quantum Hall states.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

* {{Cite journal |last1=Tsui |first1=D. C. |last2=Stormer |first2=H. L. |last3=Gossard |first3=A. C. |title=Two-Dimensional Magnetotransport in the Extreme Quantum Limit |journal=Physical Review Letters |volume=48 |issue=22 |pages=1559-1562 |year=1982 |doi=10.1103/PhysRevLett.48.1559}}
* {{Cite journal |last1=Laughlin |first1=R. B. |title=Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations |journal=Physical Review Letters |volume=50 |issue=18 |pages=1395-1398 |year=1983 |doi=10.1103/PhysRevLett.50.1395}}
* {{Cite journal |last1=Jain |first1=J. K. |title=Composite-fermion approach for the fractional quantum Hall effect |journal=Physical Review Letters |volume=63 |issue=2 |pages=199-202 |year=1989 |doi=10.1103/PhysRevLett.63.199}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum fractional Hall effect|1}}

Physics:Quantum fractional Hall effect - Revision history

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