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Elaborating the phase diagram of spin-1 anyonic chains

by Eric Vernier, Jesper Lykke Jacobsen, Hubert Saleur

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 2, 004 (2017)

Submission summary

As Contributors: Jesper Lykke Jacobsen · Hubert Saleur · Eric Vernier
Arxiv Link: (pdf)
Date submitted: 2016-11-08 01:00
Submitted by: Vernier, Eric
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We revisit the phase diagram of spin-1 $su(2)_k$ anyonic chains, originally studied by Gils {\it et. al.} [Phys. Rev. B, {\bf 87} (23) (2013)]. These chains possess several integrable points, which were overlooked (or only briefly considered) so far. Exploiting integrability through a combination of algebraic techniques and exact Bethe ansatz results, we establish in particular the presence of new first order phase transitions, a new critical point described by a $Z_k$ parafermionic CFT, and of even more phases than originally conjectured. Our results leave room for yet more progress in the understanding of spin-1 anyonic chains.

Ontology / Topics

See full Ontology or Topics database.

Anyonic chains Bethe Ansatz First-order phase transitions Integrability/integrable models Parafermions
Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 2, 004 (2017)

Resubmission 1611.02236v2 on 9 February 2017

Reports on this Submission

Anonymous Report 3 on 2016-12-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1611.02236v1, delivered 2016-12-19, doi: 10.21468/SciPost.Report.53






The manuscript at hand is a technical addendum to the extensive study of the physics of spin-1 anyonic chains presented by Gils and collaborators in a paper some few years ago (Ref. [8] in the manuscript at hand). Rightfully, the authors point out that this earlier study has missed to identify two exactly solvable “Izergin-Korepin” points and elaborate that the originally determined phase diagram of Gils et al. needs to be clarified. This is, of course, an interesting and valid result that merits publication in some form. I therefore endorse publication of the manuscript at hand in SciPost.

Before the paper should be published the authors should take care of the following aspects of their representation:

* To say that “non Abelian anyons … have showed up” in a number of physical systems is a rather sloppy and incorrect wording for a paper that later on is arguing in rather rigorous (and sometimes dismissive) terms. Non-Abelian anyons are theoretically known to appear in px + i py superconductors, but have not been seen experimentally in this context. They have been proposed by Read and Green to be relevant to certain quantum Hall states, again with no experimental verification up to date. And whether the nanowires have seen non-Abelian end states should also be worded in a more tempered way.

* One should generally distinguish “Majorana fermions” from “Ising anyons”. In the present context where representation of su(2)_k are studied, one should consistently choose the latter.

* The first time the physics of a spin-1 anyonic chain has been discussed was in the context of Fibonacci anyons in Phys. Rev. Lett. 101, 050401 (2008), where e.g. the anyonic variant of the Haldane phase has been discussed. This reference should be included in the current discussion.

* It would be instructive to contrast the earlier phase diagrams, now shown in Fig. 1, with the results of the current study, shown in Fig. 8, side-by-side at the beginning of the paper, e.g. in a new figure 1.

* Figure 2 shows some numerical finite-size data with some finite-size extrapolation. However, in the current form this finite-size extrapolation appears to be dominated by the *smallest* system sizes with a rather noticeable discrepancy for the largest system sizes. There is no need to fit *all* the finite-size data, in fact it would be much more appropriate to only fit the largest system sizes for the “2j=2” data sets. The extrapolated gap would be noticeably smaller. Is this still sufficient numerical evidence for a gapped phase?

* In Figure 3 the authors use red and green labels, which are confusing colors for many people. Please replace one of the two colors.

* The section headings 4.2.1 and 4.2.2 needs to be adjusted.

* In equation (30) the scaling dimension of the relevant field is given. It would be interesting to discuss the k \to \infty limit here, where the operator becomes marginal. Can this su(2) limit be understood on general grounds?

* In Figure 6 it would be informative to also label the indicated field by their topological sector.

Requested changes


  • validity: high
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: reasonable
  • grammar: good

Anonymous Report 2 on 2016-11-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1611.02236v1, delivered 2016-11-18, doi: 10.21468/SciPost.Report.37


1- povides a comprehensive overview on integrable spin-1 chains (vertex models) related to the $su(2)_k$ anyon chains
2- conclusions on CFTs are supported by numerical work (mostly for $k=5,5$)


1- although not stated clearly, the phase diagrams and most of the results appear to be for $k>4$ only.
2- in fact, the $su(2)_4$ model is integrable for any coupling and known to be in the Ashkin-Teller class of models (see work on magnetic hard squares by Pearce&Kim (JPhys A20, 6471, '87) and, more recently, in Ref.[14] of the manuscript)


The authors identify several integrable points among the spin-1 $su(2)_k$ anyon models introduced and studied numerically earlier. Their exact results are highly illuminating and allow for a significant refinement of the phase diagrams of the $k>4$ models as compared to Ref. [8].

Requested changes

1- the range of $k$ for which results hold should be stated clearly
2- known properties of the $k=4$ model should be referred to

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2016-11-9 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:1611.02236v1, delivered 2016-11-09, doi: 10.21468/SciPost.Report.32


1. clear presentation
2. clear discussion of relation to literature on anyonic chain
3. clear discussion of special cases


1. first paragraph of introduction not very informative
2. in Sec. 2 it is stated that $k\ge 2$ is assumed. However, the used fusion rules $1\times 1=0+1+2$ suggest that $k\ge 4$ is intended.
3. The description in Fig. 2 also requires $k\ge 4$.
4. several references are incomplete, eg, in Ref 7 the page is missing
5. page number is missing in the reference stated In the abstract
6. for the discussion of $su(2)_k$ it is referred to Refs 7,8, so the paper is not self contained (and $su(2)_k$ is not text book material)


The authors study several integrable points in the spin-1 anyonic chain and use them to obtain information about the general phase diagram. They discuss in particular the relation of their findings to the original reference (Ref 8). Overall the quality of the presentation is very good.

Requested changes

1. Correct references.
2. Clarify the restriction on $k$.
3. If really $k\ge 2$ is intended, then add a discussion of the special cases $k=2$ and $k=3$.

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: perfect

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