# nLab quantum Hall effect

Contents

### Context

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# 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:

• Klaus von Klitzing, The quantized Hall effect, Rev. Mod. Phys. 58 519 (1986) $[$doi:10.1103/RevModPhys.58.519$]$

• Steven M. Girvin, Introduction to the Fractional Quantum Hall Effect, Séminaire Poincaré 2 (2004) 53–74, reprinted in The Quantum Hall Effect, Progress in Mathematical Physics 45, Birkhäuser (2005) $[$pdf, doi:10.1007/3-7643-7393-8_4$]$

• David Tong, The Quantum Hall Effect (2016) $[$course webpage, pdf, pdf$]$

• The quantum Hall effect (pdf)

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)

On anyon phases (specifically in the quantum Hall effect) as Berry phases of a adiabatic transport of anyon positions:

### 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:

• Yosi Afron, Why is the Hall conductance quantized?, 2017 (pdf, pdf)

• Spyridon Michalakis, Why is the Hall conductance quantized?, Nature Reviews Physics 2, 392–393 (2020) (doi:10.1038/s42254-020-0212-6)

• S. Klevtsov, X. Ma, G. Marinescu, P. Wiegmann, Quantum Hall effect and Quillen metric Commun. Math. Phys. 349, 819–855 (2017) doi

### 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

Realization via AdS/CFT in condensed matter physics:

• Mitsutoshi Fujita, Wei Li, Shinsei Ryu, Tadashi Takayanagi, Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy, JHEP 0906:066 (2009) (arXiv:0901.0924)

### 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 June 8, 2022 at 06:39:51. See the history of this page for a list of all contributions to it.