nLab quantum Hall effect

Contents

Contents

Idea

A physical system in solid state physics consisting of electrons confined to an essentially 2-dimensional surface and subject to a perpendicular magnetic field.

In an appropriate limit of low temperature aspects of this system are described by the topological quantum field theory called Chern-Simons theory. In this limit the system has been proposed as constituting a possible implementation of topological quantum computation.

Properties

As a topological insulator

The bulk/edge behaviour in a quantum Hall effect is is that of a topological insulator. (While topological insulator materials typically show this behaviour without the need of a strong magnetic field.)

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References

General

Review:

Discussion via Newton-Cartan theory:

  • William Wolf, James Read, Nicholas Teh, Edge modes and dressing fields for the Newton-Cartan quantum Hall effect [arXiv:2111.08052]

See also:

Integral quantum Hall effect

Experiment

Original experimental detection:

Theory

While an intuitive understanding for the quantization of the Hall conductance has been given in

a theoretical derivation of the effect was obtained only much later in

with closely related results in

Review of this theory behind the quantum Hall effect:

Fractional quantum Hall effect

General

Review and survey of the FQHE:

See also:

A quick review of the description via Chern-Simons theory with further pointers is in the introduction of

  • Spencer D. Stirling, Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories, [arXiv:0807.2857]

Realization via AdS/CFT in condensed matter physics:

Experiment

Observation of the FQHE in GaASGaAS:

in graphene:

in oxide interfaces:

  • A. Tsukazaki et al.: Observation of the fractional quantum Hall effect in an oxide, Nature Materials 9 (2010) 889–893 [doi:10.1038/nmat2874]

in CdTeCdTe:

  • B. A. Piot, J. Kunc, M. Potemski, D. K. Maude, C. Betthausen, A. Vogl, D. Weiss, G. Karczewski, T. Wojtowicz: Fractional quantum Hall effect in CdTeCdTe, PhysRev B. 82 (2010) 081307 (R) [doi:10.1103/PhysRevB.82.081307, arXiv:1006.0908]

Phenomenological models

Phenomenological models for the fractional quantum Hall effect:

The original Laughlin wavefunction:

The Halperin multi-component model:

The Haldane-Halperin model:

The composite-fermion model (CF) which explains the FQHE as the integer quantum Hall effect not of the bare electrons but of quasi-particles which they form (for reasons not explained by the model):

Introducing abelian Chern-Simons theory to the picture:

Further discussion:

Discussion highlighting the lack of microscopic explanation of these phenomenological models:

“Though the Laughlin function very well approximates the true ground state at ν=1q\nu = \tfrac{1}{q}, the physical mechanism of related correlations and of the whole hierarchy of the FQHE remained, however, still obscure.”

“The so-called HH (Halperin–Haldane) model of consecutive generations of Laughlin states of anyonic quasiparticle excitations from the preceding Laughlin state has been abandoned early because of the rapid growth of the daughter quasiparticle size, which quickly exceeded the sample size.”

“the Halperin multicomponent theory and of the CF model advanced the understanding of correlations in FQHE, however, on a phenomenological level only. CFs were assumed to be hypothetical quasi-particles consisting of electrons and flux quanta of an auxiliary fictitious magnetic field pinned to them. The origin of this field and the manner of attachment of its flux quanta to electrons have been neither explained nor discussed.”

Quantum Hall effect via non-commutative geometry

Discussion of the integer quantum Hall effect via a Brillouin torus with noncommutative geometry and using the Connes-Chern character:

Generalization of BvESB94 to the fractional quantum Hall effect:

See also exposition in:

Discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory:

Anyons in quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):

Review:

Claims of experimental observation:

In string/M-theory

On geometric engineering of aspects of the quantum Hall effect on M5-brane worldvolumes via an effective noncommutative geometry induced by a constant B-field flux density:

category: physics

Last revised on August 18, 2024 at 20:00:49. See the history of this page for a list of all contributions to it.