nLab quantum Hall effect

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.)

(…)

Related concepts

• Hall effect

• spin Hall effect, quantum spin Hall effect

• Laughlin wavefunction

• braid group statistics, anyon

• topological insulator

• topological quantum computation

References

General

Review:

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

• Tapash Chakraborty, Pekka Pietiläinen: The Quantum Hall Effects – Integral and Fractional, Springer Series in Solid State Sciences (1995) [doi:10.1007/978-3-642-79319-6]

• Daijiro Yoshioka: The Quantum Hall Effect, Springer (2002) [doi:10.1007/978-3-662-05016-3]

• Benoît Douçot, Vincent Pasquier, Bertrand Duplantier, Vincent Rivasseau (eds.): The Quantum Hall Effect – Poincaré Seminar 2004, Progress in Mathematical Physics, Springer (2005) [doi:10.1007/3-7643-7393-8]

• 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]

See also:

• Wikipedia, Quantum Hall effect,

• Wikipedia Fractional quantum Hall effect

Integral quantum Hall effect

Experiment

Original experimental detection:

• Klaus von Klitzing, G. Dorda, M. Pepper: New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45 (1980) 494 [doi:10.1103/PhysRevLett.45.494]

• M. A. Paalanen, D. C. Tsui, A. C. Gossard: Quantized Hall effect at low temperatures, Phys. Rev. B 25 5566(R) (1982) [doi:10.1103/PhysRevB.25.5566]

Theory

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

• Robert B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B 23 5632(R) (1981) [doi:10.1103/PhysRevB.23.5632]

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

• Matthew Hastings, Spyridon Michalakis, Quantization of Hall conductance for interacting electrons without averaging assumptions, Commun. Math. Phys., 334:433–471, (2015) [arXiv:0911.4706, doi:10.1007/s00220-014-2167-x]

with closely related results in

• Alessandro Giuliani, Vieri Mastropietro, Marcello Porta, Universality of the Hall conductivity in interacting electron systems, Commun. Math. Phys. 349 (2017) 1107–1161 [arXiv:1511.04047, doi:10.1007/s00220-016-2714-8]

Review of this theory behind the quantum Hall effect:

• Joseph E. Avron, Daniel Osadchy, Ruedi Seiler: A topological look at the quantum Hall effect, Physics Today 56 8 (2003) 38–42 [doi:10.1063/1.1611351]

• Joseph E. Avron, Why is the Hall conductance quantized? (2017) [pdf, pdf]

• Spyridon Michalakis, Why is the Hall conductance quantized?, Nature Reviews Physics 2 (2020) 392–393 [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 (2017) 819–855 [doi:10.1007/s00220-016-2789-2]

Fractional quantum Hall effect

General

Review and survey of the FQHE:

• Horst L. Stormer: Nobel Lecture: The fractional quantum Hall effect, Rev. Mod. Phys. 71 (1999) 875 [doi:10.1103/RevModPhys.71.875]

• 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]

• Peter Fulde, §14.2 in: Correlated Electrons in Quantum Matter, World Scientific (2012) [doi:10.1142/8419, pdf]

• Bertrand I. Halperin, Jainendra K. Jain (eds.): Fractional Quantum Hall Effects – New Developments, World Scientific (2020) [doi:10.1142/11751]

• D. E. Feldman, Bertrand I. Halperin: Fractional charge and fractional statistics in the quantum Hall effects, Rep. Prog. Phys. 84 (2021) 076501 [doi:10.1088/1361-6633/ac03aa, arXiv:2102.08998]

• Tudor D. Stanescu, Effective theory of Abelian fractional quantum Hall liquids, Section 6.2.1 of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press (2020) [ISBN:9780367574116]

See also:

• Wikipedia, 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]

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]

Experiment

Observation of the FQHE in $GaAS$:

• D. C. Tsui, Horst L. Stormer, A. C. Gossard: Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48 (1982) 1559 [doi:10.1103/PhysRevLett.48.1559]

in graphene:

• Xu Du, Ivan Skachko, Fabian Duerr, Adina Luican, Eva Y. Andrei: Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene, Nature 462 192 (2009) [doi:10.1038/nature08522, arXiv:0910.2532]

• Kirill I. Bolotin, Fereshte Ghahari, Michael D. Shulman, Horst L. Stormer, Philip Kim: Observation of the Fractional Quantum Hall Effect in Graphene, Nature 462 (2009) 196–199 [doi:10.1038/nature08582, arXiv:0910.2763]

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 $CdTe$:

• 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 $CdTe$, 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:

• Robert B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50 (1983) 1395 [doi:10.1103/PhysRevLett.50.1395]

The Halperin multi-component model:

• Bertrand I. Halperin: Theory of the Quantized Hall Conductance, Helvetica Physica Acta 56 (1983) 75-102 [doi:10.5169/seals-115362, pdf]

The Haldane-Halperin model:

• F. Duncan M. Haldane: Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States, Phys. Rev. Lett. 51 (1983) 605 [doi:10.1103/PhysRevLett.51.605]

• Bertrand Halperin: Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52 (1984) 1583 [doi:10.1103/PhysRevLett.52.1583]

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):

• Jainendra K. Jain: Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199 [doi:10.1103/PhysRevLett.63.199]

• Jainendra K. Jain: Microscopic theory of the fractional quantum Hall effect, Adv. Phys. 41 (1992) 105-146 [doi:10.1080/00018739200101483]

• Jainendra K. Jain: Composite Fermions, Cambridge University Press (2007) [doi:10.1017/CBO9780511607561]

Introducing abelian Chern-Simons theory to the picture:

• Ana Lopez, Eduardo Fradkin: Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44 (1991) 5246 [doi:10.1103/PhysRevB.44.5246]

Further discussion:

• Jainendra K. Jain: A note contrasting two microscopic theories of the fractional quantum Hall effect, Indian J of Phys 88 (2014) 915-929 [doi:10.1007/s12648-014-0491-9, arXiv:1403.5415]

• C.-C Chang, Jainendra K. Jain: Microscopic origin of the next generation fractional quantum Hall effect, Phys. Rev. Lett. 92 (2004) 196806 [doi:10.1103/PhysRevLett.92.196806, arXiv:cond-mat/0404079]

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

• Janusz E. Jacak: Topological approach to electron correlations at fractional quantum Hall effect, Annals of Physics 430 (2021) 168493 [doi:10.1016/j.aop.2021.168493]

“Though the Laughlin function very well approximates the true ground state at $\nu = \tfrac{1}{q}$, 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:

• J. Bellissard, A. van Elst, H. Schulz-Baldes: The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 5373 (1994) [doi:10.1063/1.530758, cond-mat/9411052]

Generalization of BvESB94 to the fractional quantum Hall effect:

• Matilde Marcolli, Varghese Mathai: Towards the fractional quantum Hall effect: a noncommutative geometry perspective, in Noncommutative Geometry and Number Theory, Vieweg (2006) [doi:10.1007/978-3-8348-0352-8_12, arXiv:cond-mat/0502356]

See also exposition in:

• Vincent Pasquier: Quantum Hall Effect and Non-commutative Geometry, in: Quantum Spaces, Progress in Mathematical Physics 53, Birkhäuser (2007) [doi:10.1007/978-3-7643-8522-4_1, pdf]

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

• Leonard Susskind: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory [arXiv:hep-th/0101029]

• Simeon Hellerman, Leonard Susskind: Realizing the Quantum Hall System in String Theory [arXiv:hep-th/0107200]

  (relating this to M5-branes via the BFSS matrix model)

• Alexios P. Polychronakos: Quantum Hall states as matrix Chern-Simons theory, JHEP 0104:011 (2001) [doi:10.1088/1126-6708/2001/04/011, arXiv:hep-th/0103013]

• Simeon Hellerman, Mark Van Raamsdonk: Quantum Hall Physics = Noncommutative Field Theory, JHEP 0110:039 (2001) [doi:10.1088/1126-6708/2001/10/039, arXiv:hep-th/0103179]

• Eduardo Fradkin, Vishnu Jejjala, Robert G. Leigh: Non-commutative Chern-Simons for the Quantum Hall System and Duality, Nucl. Phys. B 642 (2002) 483-500 [doi:10.1016/S0550-3213(02)00616-8, arXiv:cond-mat/0205653]

• Andrea Cappelli, Ivan D. Rodriguez: Matrix Effective Theories of the Fractional Quantum Hall effect, J. Phys. A 42 (2009) 304006 [doi:10.1088/1751-8113/42/30/304006, arXiv:0902.0765]

• Zhihuan Dong, T. Senthil: Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems, Phys. Rev. B 102 (2020) 205126 [doi:10.1103/PhysRevB.102.205126, arXiv:2006.01282]

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):

• Bertrand I. Halperin: Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52 (1984) 1583 [doi:10.1103/PhysRevLett.52.1583]

  Erratum, Phys. Rev. Lett. 52 (1984) 2390 [doi:10.1103/PhysRevLett.52.2390.4]

• Daniel P. Arovas, John Robert Schrieffer, Frank Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53 (1984) 722 [doi:10.1103/PhysRevLett.53.722]

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):

• Gregory Moore, Nicholas Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. 360B(1991)362 (pdf, doi:10.1016/0550-3213(91)90407-O)

Review:

• Ady Stern, Anyons and the quantum Hall effect – A pedagogical review, Annals of Physics 323 1 (2008) 204-249 [doi:10.1016/j.aop.2007.10.008, arXiv:0711.4697]

• Menelaos Zikidis: Abelian Anyons and Fractional Quantum Hall Effect, Seminar notes (2017) [pdf, pdf]

• Ady Stern: Engineering Non-Abelian Quasi-Particles in Fractional Quantum Hall States – A Pedagogical Introduction, Ch. 9 in: Fractional Quantum Hall Effects, World Scientific (2020) 435-486 [doi:10.1142/9789811217494_0009]

• D. E. Feldman, Bertrand Halperin: Fractional charge and fractional statistics in the quantum Hall effects, Rep. Prog. Phys. 84 (2021) 076501 [doi:10.1088/1361-6633/ac03aa, arXiv:2102.08998]

Claims of experimental observation:

• H. Bartolomei, et al.: Fractional statistics in anyon collisions, Science 368 6487 (2020) 173-177 [doi:10.1126/science.aaz5601, arXiv:2006.13157]

• James Nakamura, S. Liang, G. C. Gardner, M. J. Manfra: Direct observation of anyonic braiding statistics, Nat. Phys. 16 (2020) 931–936 [doi:10.1038/s41567-020-1019-1, spire:1802928]

  exposition:

  Bob Yirka, Best evidence yet for existence of anyons, PhysOrg News (July 10, 2020) [phys.org/news/2020-07]

• P. Glidic et al.: Signature of anyonic statistics in the integer quantum Hall regime, Nature Commun. 15 6578 (2024) 1 [doi:10.1038/s41467-024-50820-0, arXiv:2401.06069]

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:

• Simeon Hellerman, Leonard Susskind: Realizing the Quantum Hall System in String Theory [arXiv:hep-th/0107200]

category: physics