nLab parafermion



Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication




The introduction of the Z kZ_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]]


We caution the reader that the level-kk Read-Rezayi wavefunction [[10]] is constructed from a conformal field theory (CFT) known as the Z kZ_k parafermions of Zamolodchikov-Fateev [[32]] type which is closely related to the SU(2) kSU(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]]


Related to the fractional superstring.



ZF type

Introducing parafermion 2d CFT:

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

Relation to SU(2)SU(2)-WZW model

For ZF parafermions

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

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

See also:

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

For FKF parafermions

Relation to anyonic topological order

As models for su(2)-anyon wavefunctions:

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

Relation to topological phases of matter with topological order:

On twisted parafermions:

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

Last revised on August 6, 2023 at 03:54:02. See the history of this page for a list of all contributions to it.