nLab parafermion

Idea

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The introduction of the $Z_k$ parafermions $[$1$]$ in the context of statistical models and conformal field theory [2] is perhaps one of the most significant conceptual advances in modern theoretical physics $[$DGZ02$]$

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We caution the reader that the level-$k$ Read-Rezayi wavefunction $[$10$]$ is constructed from a conformal field theory (CFT) known as the $Z_k$ parafermions of Zamolodchikov-Fateev $[$32$]$ type which is closely related to the $SU(2)_k$ Chern-Simons theory, and which is a different object from the parafermions of the Fradkin-Kadanoff-Fendley type $[$46$]$. To avoid confusion we emphasise at this point that within this paper, all mention of parafermions will refer to Fradkin-Kadanoff-Fendley type. $[$CFRS00, p. 1$]$

In fact the Zamolodchikov-Fateev parafermions represent a critical point at the transition into a phase described by Fradkin-Kadanoff-Fendley parafermions. $[$CFRS00, p. 9$]$

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Related to the fractional superstring.

References

General

ZF type

Introducing parafermion 2d CFT:

• Alexander B. Zamolodchikov, Vladimir A. Fateev Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems, Sov. Phys. JETP 62 2 (1985) 215-225 $[$pdf, osti:5929972$]$

Further discussion:

• Xiang-Mao Ding, Mark. D. Gould, Yao-Zhong Zhang, Twisted Parafermions, Phys.Lett. B 530 (2002) 197-201 $[$arXiv:hep-th/0110165$]$

FKF type

• Eduardo Fradkin, Leo P. Kadanoff, Disorder variables and para-fermions in two-dimensional statistical mechanics, Nuclear Physics B 170 1 (1980) 1-15 $[$doi:10.1016/0550-3213(80)90472-1$]$

• Paul Fendley, Free parafermions, J. Phys. A 47 (2014) 075001 $[$arXiv:1310.6049, doi:10.1088/1751-8113/47/7/075001$]$

• Paul Fendley, Parafermionic edge zero modes in $\mathbb{Z}_n$-invariant spin chains, J. Stat. Mech. (2012) P11020 $[$doi:10.1088/1742-5468/2012/11/P11020$]$

Coset WZW-type

More general construction as coset-WZW models (2d current algebra CFT):

• Doron Gepner, New conformal field theories associated with Lie algebras and their partition functions, Nuclear Physics B 290 (1987) 10-24 [doi:10.1016/0550-3213(87)90176-3]

• Peter Bouwknegt, Bolin Han, Coupled free fermion conformal field theories [arXiv:2403.03471]

Relation to $SU(2)$-WZW model

For ZF parafermions

Relating parafermions to the affine su(2)-current algebra/WZW model (su(2)-anyons):

• T. Jayaraman, Kumar S. Narain, M. H. Sarmadi, $SU(2)_k$ WZW and parafermion models on the torus, Nuclear Physics B 343 2 (1990) 418-449 $[$doi:10.1016/0550-3213(90)90477-U$]$

• Daniel Cabra, Spinons and parafermions in fermion cosets, talk at International Seminar dedicated to the memory of D. V. Volkov, Kharkov (January 5-7, 1997) $[$arXiv:hep-th/9704015$]$

• Daniel C. Cabra, Eduardo Fradkin, G. L. Rossini, F. A. Schaposnik, Section 4 of: Non-Abelian fractional quantum Hall states and chiral coset conformal field theories, International Journal of Modern Physics A 15 30 (2000) 4857-4870 $[$doi:10.1142/S0217751X00002354, arXiv:cond-mat/9905192$]$

On boundary conditions (BCFT/D-branes) for the gauged WZW model via parafermions:

• Juan Maldacena, Gregory Moore, Nathan Seiberg, Geometrical interpretation of D-branes in gauged WZW models, JHEP 0107:046 (2001) $[$arXiv:hep-th/0105038, doi:10.1088/1126-6708/2001/07/046$]$

See also:

• Gor Sarkissian, pp. 75 in: Two-dimensional conformal field theories with defects and boundaries RTN (2016) $[$pdf$]$

For FKF parafermions

• Richard Fern, Johannes Kombe, Steven H. Simon, How $SU(2)_4$ Anyons are $\mathbb{Z}_3$ Parafermions, SciPost Phys. 3 037 (2017) $[$arXiv:1706.06098, doi:10.21468/SciPostPhys.3.6.037$]$

Relation to anyonic topological order

As models for su(2)-anyon wavefunctions:

• 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$]$

• David J. Clarke, Jason Alicea, Kirill Shtengel . Exotic non-Abelian anyons from conventional fractional quantum Hall states, Nature Communications 4 1348 (2013) $[$doi: 10.1038/ncomms2340 (2013)$]$

An integrable model for $\mathbb{Z}_N$-parafermions:

• Alexei M. Tsvelik, An integrable model with parafermion zero energy modes, Phys Rev. Lett. 113 066401 (2014) $[$arXiv:1404.2840, doi:10.1103/PhysRevLett.113.066401$]$

• Alexei M. Tsvelik, $\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect $[$arXiv:1407.4002$]$

• Daniel Borcherding, Non-Abelian quasi-particles in electronic systems, Hannover 2018 $[$doi:10.15488/4280$]$

Relation to topological phases of matter with topological order:

• Jason Alicea, Paul Fendley, Topological phases with parafermions: theory and blueprints, Annual Review of Condensed Matter Physics 7 (2016) 119-139 $[$doi:10.1146/annurev-conmatphys-031115-011336$]$

• Fernando Iemini, Christophe Mora, Leonardo Mazza, Topological phases of parafermions: a model with exactly-solvable ground states, Phys. Rev. Lett. 118 170402 (2017) $[$arXiv;1611.00832, doi:10.1103/PhysRevLett.118.170402$]$

On twisted parafermions:

• Xiang-Mao Ding, Mark Gould, and Yao-Zhong Zhang. Twisted Parafermions. (2001). arXiv:hep-th/0110165