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$Fractional Hall effect: a two-dimensional electron system forms fractionally charged excitations under a strong magnetic field.$

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 $e^{2} / h$ , 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 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

$ν = \frac{n h}{e B},$

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

Laughlin states

Robert Laughlin proposed a many-body wavefunction that explains the simplest fractional plateaus, especially $ν = 1 / 3$ . 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 $ν = 1 / 3$ Laughlin state, the elementary quasiparticle charge is $e / 3$ 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

$ν = \frac{p}{2 p \pm 1},$

where $p$ 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 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 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.

Physics:Quantum fractional Hall effect

Overview

Key ideas

Laughlin states

Composite fermions

Topological order

Physical setting

See also

Table of contents (217 articles)

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