# Elaborating the phase diagram of spin-1 anyonic chains

### Submission summary

 As Contributors: Jesper Lykke Jacobsen · Hubert Saleur · Eric Vernier Arxiv Link: https://arxiv.org/abs/1611.02236v2 (pdf) Date accepted: 2017-02-14 Date submitted: 2017-02-09 01:00 Submitted by: Vernier, Eric Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics Quantum Physics Approach: Theoretical

### Abstract

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

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Published as SciPost Phys. 2, 004 (2017)

We thank the referees for the constructive comments brought to our manuscript.
In the present version (v2), corrections were made in order to take in account of all received remarks.

In particular, the following changes were made :

### List of changes

- Regarding the remark of Report 53 that
{\it
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.}

We agree with this comment, however space and readability constraints made very inconvenient to put figures 1 and 8 side by side. We have instead moved fig. 8 to the beginning of the paper, where it can more easily be compared with fig. 1.

- Regarding the remark of Report 53 that
{\it
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? }

We have replaced the extrapolations on figure 3 (prevously fig. 2) by linear fits excluding the smallest system sizes. The extrapolated gap is indeed smaller, but still manifestly non-zero. This is in agreement with the fact that magnetic excitations (corresponding to holes in the Fermi sea) of the six-vertex model in its massive phase have a non-zero gap.

- Regarding the remark of Report 53 that
{\it In Figure 6 it would be informative to also label the indicated field by their topological sector.}

We have added on the legend of Figure 7 (previously fig. 6) information about the topological charge of all indicated levels

- In figure 4 (formerly figure 3), we have changed the green color to orange, in order to improve its readability.

- Regarding remarks of Reports 32 and 37, we have specified the range of values of $k$ for which our results hold. More precisely, most of our conclusions are formulated for generic $k\geq 4$, but when necessary we have treated separately the cases $k>4$ and $k=4$.
In addition, we have refered to the suggested litterature for the $k=4$ case, for which we thank the author of Report 37.

- We have also improved the introduction in order to meet the suggestions of Report 53