nLab Laughlin wavefunction

Idea

In quantum physics a Laughlin wavefunction is a certain Ansatz for an n-particle wavefunction which is meant to capture at least aspects of ground states with anyonic properties, such as for the levels in the quantum Hall effect.

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Related concepts

• quantum Hall effect

• conformal block

References

General

The original article:

• 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]

Review:

• Steven M. Girvin, Section 2.1 of: 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]

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, Section III.D.2.c (pp. 1125) of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) [arXiv:0707.1888]

• David Tong, Section 3.1 of: The Quantum Hall Effect (2016) [course webpage, pdf, pdf]

• Roman Remme, The Laughlin Wavefunction, talk notes (2017) [pdf, pdf]

See also:

• Wikipedia, Laughlin wavefunciton

Characterization as a braid representation:

• Janusz E. Jacak, Formal derivation of the Laughlin function and its generalization for other topological phases of FQHE, Sci Rep 12 616 (2022) [doi:10.1038/s41598-021-04672-z]

A “hierarchy” of Laughlin-like states:

• 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]

• Tian Lan, Matrix formulation for non-Abelian families, Phys. Rev. B 100 241102 (2019) $[$arXiv:1908.02599, doi:10.1103/PhysRevB.100.241102$]$

Laughlin wavefunctions as conformal blocks

Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:

• Gregory Moore, Nicholas Read, Section 2.2 of: Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360 2–3 (1991) 362-396 [doi:10.1016/0550-3213(91)90407-O, pdf]

• Xiao-Gang Wen, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [doi:10.1103/PhysRevLett.66.802, pdf]

• B. Blok, Xiao-Gang Wen, Many-body systems with non-abelian statistics, Nuclear Physics B 374 3 (1992) 615-646 [doi:10.1016/0550-3213(92)90402-W]

• Xiao-Gang Wen, Yong-Shi Wu, Chiral operator product algebra hidden in certain fractional quantum Hall wave functions, Nucl. Phys. B 419 (1994) 455-479 [doi:10.1016/0550-3213(94)90340-9]

Review in the broader context of the CS-WZW correspondence:

• Daniel C. Cabra, Gerardo L. Rossini, Explicit connection between conformal field theory and 2+1 Chern-Simons theory, Mod. Phys. Lett. A 12 (1997) 1687-1697 [doi:10.1142/S0217732397001722, arXiv:hep-th/9506054]

Specifically for logarithmic CFT:

• Victor Gurarie, Michael Flohr, Chetan Nayak, The Haldane-Rezayi Quantum Hall State and Conformal Field Theory, Nucl. Phys. B 498 (1997) 513-538 [doi:10.1016/S0550-3213(97)00351-9, arXiv:cond-mat/9701212]

• Michael Flohr, §5.4 in: Bits and pieces in logarithmic conformal field theory, International Journal of Modern Physics A, 18 25 (2003) 4497-4591 [doi:10.1142/S0217751X03016859, arXiv:hep-th/0111228]

Specifically for su(2)-anyons:

• Kazusumi Ino, Modular Invariants in the Fractional Quantum Hall Effect, Nucl. Phys. B 532 (1998) 783-806 [doi:10.1016/S0550-3213(98)00598-7, arXiv:cond-mat/9804198]

• Nicholas Read, Edward Rezayi, Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level, Phys. Rev. B 59 (1999) 8084 [doi:10.1103/PhysRevB.59.8084]

• Eddy Ardonne, Kareljan Schoutens: Wavefunctions for topological quantum registers, Annals Phys. 322 (2007) 201-235 [doi:10.1016/j.aop.2006.07.015, arXiv:cond-mat/0606217]

• Ludmil Hadjiivanov, Lachezar S. Georgiev, Braiding Fibonacci anyons [arxiv:2404.01778]