Abstract
We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/21/2 particle world lines of an SU(2)_44 Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the k=4k=4 Read-Rezayi quantum Hall state. We show that these braiding properties can be represented exactly using \mathbbm{Z}_3 parafermion operators.
Highlights
Systems containing particles with nontrivial exchange statistics are described by Topological Quantum Field Theories (TQFTs) [2, 5,6,7,8,9]
In this work we focus on TQFTs related to SU(2)k Chern-Simons TQFTs for the specific case of k = 4
It was previously believed that the braid group representation corresponding to the SU(2)4 type anyons was (like that of SU(2)2 or Ising anyons) not sufficiently rich to allow for universal quantum computation [13]
Summary
Braid group — Let us line up N particles living in two dimensions along a one dimensional line. The braid matrix i, which clockwise exchanges the ith particle in the sequence with the (i + 1)th particle, can only alter the intermediate fusion of the i particles (i.e, the spin value at the ith step of the diagram) The reason for this is that the first (i − 1) anyons are not moved, so their fusion at the (i − 1)th step is unchanged. There are several possible sets of values of the coefficients ck which can be strongly constrained by forcing these matrices to be both unitary and a representation of the braid group This parafermion braid representation generalises the well known braiding matrices for n = 2 Majorana case of the form [40,41,42].
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