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How SU(2)$_4$ Anyons are Z$_3$ Parafermions

by Richard Fern, Johannes Kombe, Steven H. Simon

  • Other versions of this Submission (with Reports) exist:

Submission summary

As Contributors: Steve Simon · Richard Fern
Arxiv Link:
Date submitted: 2017-10-05
Submitted by: Fern, Richard
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Condensed Matter Physics - Theory


We consider the braid group representation which describes the non-abelian braiding statistics of the spin $1/2$ particle world lines of an SU(2)$_4$ Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the $k=4$ Read-Rezayi quantum Hall state. We show that these braiding properties can be represented exactly using $\mathbbm{Z}_3$ parafermion operators.

Current status:

Editor-in-charge assigned, manuscript under review

Author comments upon resubmission

We thank the referees for their comments. All of the referee comments are well thought out and we agree with the suggested changes. We have addressed all of the comments of the referees, and we believe that our manuscript now is suitable for publication in SciPost. Detailed responses to the referee comments are given below along with description of the changes.

Referee 1:
We thank the referee for their careful reading and their comments.
(1) We have extended the discussion near Eqs. 32-34 to clarify the issue, in particular we also show how the states are orthonormal (and we state this explicitly). In addition near eqn 44 (now eqn 48) we have simplified the argument so that it is hopefully now clearer (our previous discussion of fitting parameters a and b was perhaps needlessly awkward).
(2) We have added a paragraph (Between Eq. 35 and 36) commenting on the fact that among theories SU(2)_k only k=1,2,4 could possibly be represented with the parafermion algebra. It is true that a few other cases seem to be simpler than the generic case (and might have some simplifying structure), however, they certainly will not fit a similar parafermion construction.
(3) We have changed the order of Refs 1 and 2 .

Referee 2:
We thank the referee for their careful reading and their comments.
(1) The confusion about 'avatars' of parafermions was also mentioned by Referee 3. We have attempted to straighten this out by being much more explicit about what we mean mathematically. See below the response to Referee 3 below for more details.
(2) We thank the referee for pointing us to this beautiful reference by Jones which we have now cited. In addition we have added several additional references to other related work (for example, that of Saleur 1991; and a number of recent related works).
(3) We have fixed the reference problem and added the citation to Cui and Zhang. We have also kept the arXiv number so that the order of appearance in the literature is clear.

Referee 3:
We thank the referee (Parsa Bonderson) for his careful reading and his comments.
(1) Parsa is certainly right that we were not careful about specifying exactly what we mean by "parafermion". This is particularly bad being that the literature is not very consistent in how this word is used. We have thus gone through the paper and changed some of the language to clarify precisely what we mean and exactly how we have used this word. We believe these changes should remove any confusion. Details of the changes are listed below.
(2) Parsa also points to his paper arXiv:1410.4540 as having worked out details of how ``parafermions" arise as symmetry defects of topological phases. This is certainly very closely related. We emphasize that we are agnostic about the physical mechanism for the production of these objects (indeed, defect parafermions have projective statistics, so are not quite the same as what we are considering). While we have added citations to his work, along with some additional works in the literature working along similar lines, we do not add a detailed discussion of the physics of symmetry defects since it is a bit off-topic to our main focus.

Explict changes regarding point (1):
We have explicitly stated exactly what we mean by parafermion with the addition "To avoid confusion we emphasize at this point that within this paper, all mention of parafermions will refer to Fradkin-Kadanoff-Fendley type --- that is, they are mathematical operators (to be defined precisely in Eqs. 19 and 20 below)". We also have removed a number of statements that were potentially creating confusion. Statements such as "Thus, we have proven the equivalence between SU(2)4 anyons and Z3 parafermions" (was in the conclusion now removed) which we agree is not precise and probably adds confusion. We also rephrase the key sentence in the introduction that explains the purpose of the paper to now read as: "The content of the current work is to express the braid group representation corresponding to SU(2)4 anyons explicitly in terms of Z3 parafermion operators". We believe these changes should make our usages unambiguous.

One thing we did not change is the title, "How SU(2)4 anyons are Z3 parafermions" as we still believe that this is accurate, despite the objections. If one is to discuss "How A is B" it often implies that "A is not always B", or "A is B only in some sense", or "A is a subclass of B". If our aim was to state that they are the same then "A is B", or "How A is exactly B" would probably be more appropriate. Therefore, while it is true that the title suggested by Parsa is also accurate, it is also somewhat long and unwieldy. Furthermore, changing our title between the first and second draft of a paper is likley to add confusion. So unless forced, we would prefer to stick with the original title.

List of changes

Page 1 - Changed the wording of the abstract to be more careful with regards to parafermion operators
Page 1 - Reworded a number of parts throughout the introduction and added extras which we hope should make our usage of the relevant terminology clearer
Page 4 - Added a comment to explain the "or" in Eqs. 32-34
Page 5 - Added a proof that our states are orthonormal
Page 6 - Reworked the final claculation of the proof, which hopefully clarifies matters
New references are [14], [16], [30], [31], [36], [37], [38], [39].

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