nLab abelian Chern-Simons for fractional quantum Hall effect -- references

Abelian Chern-Simons for fractional quantum Hall effect

The idea of abelian Chern-Simons theory as an effective field theory exhibiting the fractional quantum Hall effect (abelian topological order) goes back to

• Steven M. Girvin, around (10.7.15) in: Summary, Omissions and Unanswered Questions, Chapter 10 of: The Quantum Hall Effect, Graduate Texts in Contemporary Physics, Springer (1986, 1990) [doi:10.1007/978-1-4612-3350-3]

• Steven M. Girvin, Allan H. MacDonald, around (10) of: Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, Phys. Rev. Lett. 58 12 (1987) (1987) 1252-1255 [doi:10.1103/PhysRevLett.58.1252]

• Shou Cheng Zhang, T. H. Hansson S. Kivelson: Effective-Field-Theory Model for the Fractional Quantum Hall Effect, Phys. Rev. Lett. 62 (1989) 82 [doi:10.1103/PhysRevLett.62.82]

and was made more explicit in:

• Xiao-Gang Wen, Anthony Zee: Quantum statistics and superconductivity in two spatial dimensions, Nuclear Physics B – Proceedings Supplements 15 (1990) 135-156 [doi:10.1016/0920-5632(90)90014-L]

• B. Blok, Xiao-Gang Wen: Effective theories of the fractional quantum Hall effect at generic filling fractions, Phys. Rev. B 42 (1990) 8133 [doi:10.1103/PhysRevB.42.8133]

• Xiao-Gang Wen: Topological Orders in Rigid States, Int. J. Mod. Phys. B 4 239 (1990) [doi:10.1142/S0217979290000139]

• Xiao-Gang Wen, Qian Niu: Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B 41 9377 (1990) [doi:10.1103/PhysRevB.41.9377]

• Jürg Fröhlich, T. Kerler: Universality in quantum Hall systems, Nuclear Physics B 354 2–3 (1991) 369-417 [doi:10.1016/0550-3213(91)90360-A]

• Jürg Fröhlich, Anthony Zee: Large scale physics of the quantum hall fluid, Nuclear Physics B 364 3 (1991) 517-540 [doi:10.1016/0550-3213(91)90275-3]

• Z. F. Ezawa, A. Iwazaki: Chern-Simons gauge theories for the fractional-quantum-Hall-effect hierarchy and anyon superconductivity, Phys. Rev. B 43 (1991) 2637 [doi:10.1103/PhysRevB.43.2637]

• A. P. Balachandran, A. M. Srivastava: Chern-Simons Dynamics and the Quantum Hall Effect [arXiv:hep-th/9111006, spire:319826]

• 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]

• Xiao-Gang Wen, Anthony Zee: Topological structures, universality classes, and statistics screening in the anyon superfluid, Phys. Rev. B 44 (1991) 274 [doi:10.1103/PhysRevB.44.274]

• Roberto Iengo, Kurt Lechner: Anyon quantum mechanics and Chern-Simons theory, Physics Reports 213 4 (1992) 179-269 [doi:10.1016/0370-1573(92)90039-3]

• Xiao-Gang Wen, Anthony Zee: Classification of Abelian quantum Hall states and matrix formulation of topological fluids, Phys. Rev. B 46 (1992) 2290 [doi:10.1103/PhysRevB.46.2290]

• Xiao-Gang Wen, Anthony Zee: Shift and spin vector: New topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69 (1992) 953, Erratum Phys. Rev. Lett. 69 3000 (1992) [doi:10.1103/PhysRevLett.69.953]

• Xiao-Gang Wen: Theory of Edge States in Fractional Quantum Hall Effects, International Journal of Modern Physics B 06 10 (1992) 1711-1762 [doi:10.1142/S0217979292000840]

• Xiao-Gang Wen, Topological orders and Edge excitations in FQH states, Advances in Physics 44 (1995) 405 [doi:10.1080/00018739500101566, arXiv:cond-mat/9506066]

  (in the context of topological order)

• A. P. Balachandran, L. Chandar, B. Sathiapalan: Chern-Simons Duality and the Quantum Hall Effect, Int. J. Mod. Phys. A11 (1996) 3587-3608 [doi:10.1142/S0217751X96001693, arXiv:hep-th/9509019]

• Jürg Fröhlich, B. Pedrini, Christoph Schweigert, Johannes Walcher: Universality in Quantum Hall Systems: Coset Construction of Incompressible States [arXiv:cond-mat/0002330]

Beware that many of these early articles actually consider Maxwell-Chern-Simons theory, which however was later claimed not to actually exhibit the claimed “anyon statistics”:

• Kurt Haller, Edwin Lim-Lombridas: Canonical quantization of $(2+1)$-dimensional QED with a topological mass term, Phys. Rev. D 46 (1992) 1737 [doi:10.1103/PhysRevD.46.1737]

• Kurt Haller, Edwin Lim-Lombridas: Maxwell-Chern-Simons Theory in Covariant and Coulomb Gauges Annals of Physics 246 1 (1996) 1-48 [doi:10.1006/aphy.1996.0019]

Early review:

• Shou Cheng Zhang: The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect, International Journal of Modern Physics B 06 01 (1992) 25-58 [doi:10.1142/S0217979292000037, pdf]

• Anthony Zee: Quantum Hall Fluids, in: Field Theory, Topology and Condensed Matter Physics, Lecture Notes in Physics 456, Springer (1995) [doi:10.1007/BFb0113369, arXiv:cond-mat/9501022]

• Xiao-Gang Wen: Topological orders and Edge excitations in FQH states, Advances in Physics 44 5 (1995) 405 [doi:10.1080/00018739500101566, arXiv:cond-mat/9506066]

Further review and exposition:

• Xiao-Gang Wen: Effective theory of fractional quantum Hall liquids, section 7.3 of: Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford Academic (2007)[doi:10.1093/acprof:oso/9780199227259.001.0001, pdf]

• Yuan-Ming Lu, Ashvin Vishwanath, part II of: Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86 (2012) 125119, Erratum Phys. Rev. B 89 (2014) 199903 [doi:10.1103/PhysRevB.86.125119, arXiv:1205.3156]

• Eduardo Fradkin, chapter 13.7 of: Field Theories of Condensed Matter Physics, Cambridge University Press (2013) [doi:10.1017/CBO9781139015509, ISBN:9781139015509, pdf]

• Edward Witten, pp 30 in: Three Lectures On Topological Phases Of Matter, La Rivista del Nuovo Cimento 39 (2016) 313-370 [doi:10.1393/ncr/i2016-10125-3, arXiv:1510.07698]

• David Tong §5 of: The Quantum Hall Effect, lecture notes (2016) [arXiv:1606.06687, course webpage, pdf, pdf]

• Josef Willsher; §2.3 of: The Chern–Simons Action & Quantum Hall Effect: Effective Theory, Anomalies, and Dualities of a Topological Quantum Fluid, PhD thesis, Imperial College London (2020) [pdf, pdf]

• Diego Bragato: Integer Quantum Hall Effect and Chern Simons Theories, MSc thesis, Milano & Utrecht (2022) [polimi:10589/216533, pdf]

• Eduardo Fradkin; section 10 of: Field Theoretic Aspects of Condensed Matter Physics: An Overview, Encyclopedia of Condensed Matter Physics (2nd ed.) 1 (2024) 27-131 [doi:10.1016/B978-0-323-90800-9.00269-9, arXiv:2301.13234]

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

• see there.

More on edge modes via the abelian WZW model/Floreanini-Jackiw theory chiral boson on the boundary of abelian Chern-Simons theory:

• Michael Levin: Protected edge modes without symmetry, Phys. Rev. X 3 021009 (2013) [doi:10.1103/PhysRevX.3.021009, arXiv:1301.7355]

• Nicola Maggiore: From Chern-Simons to Tomonaga-Luttinger, Int. J. Mod. Phys. A 33 (2018) 1850013 [doi:10.1142/S0217751X18500136, arXiv:1712.08744]

• Irais Rubalcava-Garcia, §3.6 in: Constructing the theory at the boundary, its dynamics and degrees of freedom [arXiv:2003.06241]

and in view of the bulk-edge correspondence:

• Jennifer Cano, Meng Cheng, Michael Mulligan, Chetan Nayak, Eugeniu Plamadeala, Jon Yard: Bulk-Edge Correspondence in $2+1$-Dimensional Abelian Topological Phases, Phys. Rev. B 89 (2014) 115116 [doi:10.1103/PhysRevB.89.115116, arXiv:1310.5708]

The symmetry protected situation:

• Yuan-Ming Lu, Ashvin Vishwanath: Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to $\mathbb{Z}_2$ spin liquids, Phys. Rev. B 93 (2016) 155121 [doi:10.1103/PhysRevB.93.155121, arXiv:1302.2634]

Amplification of the K-matrix formalism as being about (not the usual single-component but) multi-component FQH systems:

• Maissam Barkeshli, Xiao-Gang Wen: Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states, Phys. Rev. B 95 (2017) 085135 [doi:10.1103/PhysRevB.95.085135, arXiv:1010.4270]

• Inti Sodemann, Itamar Kimchi, Chong Wang, T. Senthil: Composite fermion duality for half-filled multicomponent Landau levels, Phys. Rev. B 95 (2017) 085135 [doi:10.1103/PhysRevB.95.085135, doi:1609.08616]

• Tian-Sheng Zeng: Fractional quantum Hall effect of Bose-Fermi mixtures, Phys. Rev. B 103 (2021) L201118 [doi:10.1103/PhysRevB.103.L201118, arXiv:2012.08203]

• Liangdong Hu, Zhao Liu, W. Zhu: Modular transformation and anyonic statistics of multi-component fractional quantum Hall states, Phys. Rev. B 108 (2023) 235121 [arXiv:2301.06427, doi:10.1103/PhysRevB.108.235121]

• Tian-Sheng Zeng: Three-component fractional quantum Hall effect in topological flat bands, Phys. Rev. B 110 (2024) 165126 [doi:10.1103/PhysRevB.110.165126, arXiv:2407.08568]

• Yasin F. Alam, Andreas Karch, Da-Chuan Lu, Ryan C. Spieler: $Sp(4,\mathbb{Z})$ actions on 3d $U(1)^2$ symmetric theories: Order-five duality and bilayer quantum Hall hierarchies [arXiv:2607.20622]

Further developments:

• Dmitriy Belov, Gregory W. Moore, §7 of: Classification of abelian spin Chern-Simons theories [arXiv:hep-th/0505235]

• Samuel Bieri, Jürg Fröhlich: Effective field theory and tunneling currents in the fractional quantum Hall effect, Annals of Physics 327 4 (2012) 959-993 [doi:10.1016/j.aop.2011.10.012, arXiv:1107.5012]

• Christian Fräßdorf: Abelian Chern-Simons Theory for the Fractional Quantum Hall Effect in Graphene, Phys. Rev. B 97 115123 (2018) [doi:10.1103/PhysRevB.97.115123, arXiv:1712.03595]

• Kristan Jensen, Amir Raz: The Fractional Hall hierarchy from duality [arXiv:2412.17761]

• Abhishek Agarwal, Dimitra Karabali, V. Parameswaran Nair: Fractional quantum Hall effect in higher dimensions, Phys. Rev. D 111 (2025) 025002 [doi:10.1103/PhysRevD.111.025002, arXiv:2410.14036]

  (higher dimensional Chern-Simons theory)

• T. H. Hansson, Rodrigo Arouca, Thomas Klein Kvorning: On the relation between fractional charge and statistics [arXiv:2412.15857]

• Paul Leask: Anyon interactions in the Chern–Simons–Landau–Ginzburg model of the fractional quantum Hall effect [arXiv:2510.04830]