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.)

(…)

References

General

Review:

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

  • Alessandro Giuliani, Vieri Mastropietro, Marcello Porta, Universality of the Hall conductivity in interacting electron systems, Communications in Mathematical Physics volume 349, pages 1107–1161(2017) (arXiv:1511.04047, doi:10.1007/s00220-016-2714-8)

Review of this theory behind the quantum Hall effect:

Fractional quantum Hall effect

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

J. Bellissard introduced an approach via noncommutative geometry and Connes-Chern character:

  • J. Bellissard, A. van Elst, H. Schulz Baldes, The noncommutative geometry of the quantum Hall effect, 79 pages, J. Math. Phys. 35, 5373 (1994) cond-mat/9411052 doi

In terms of Berry phase and Chern numbers in

  • Joseph E. Avron, Daniel Osadchy, Ruedi Seiler, A Topological look at the quantum Hall effect, Physics Today 56:8, doi

Anyons in the 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:

category: physics

Last revised on February 17, 2021 at 23:42:53. See the history of this page for a list of all contributions to it.