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Phase diagram of an extended parafermion chain

by Jurriaan Wouters, Fabian Hassler, Hosho Katsura, Dirk Schuricht

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Submission summary

Authors (as Contributors): Fabian Hassler · Dirk Schuricht · Jurriaan Wouters
Submission information
Arxiv Link: https://arxiv.org/abs/2106.15823v2 (pdf)
Date accepted: 2022-01-31
Date submitted: 2021-12-10 08:41
Submitted by: Schuricht, Dirk
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
Approach: Theoretical

Abstract

We study the phase diagram of an extended parafermion chain, which, in addition to terms coupling parafermions on neighbouring sites, also possesses terms involving four sites. Via a Fradkin--Kadanoff transformation the parafermion chain is shown to be equivalent to the non-chiral $\mathbb{Z}_3$ axial next-nearest neighbour Potts model. We discuss a possible experimental realisation using hetero-nanostructures. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase.

Published as SciPost Phys. Core 5, 008 (2022)



Author comments upon resubmission

While working on the revisions we realised an error in our original line of argument related to the RG relevance of the U(1)-breaking term [Eq. (70) in the revised manuscript]. Due to the previously overlooked oscillating prefactor this term becomes relevant in the range (35). In addition we extended our numerical simulations [see, eg, new Figure 9] and their analysis and revised the manuscript to take these new results into account. We would like to stress that our main results (in particular the topology of the phase diagram and the existence of a topological phase) are not affected by these revisions, which only concern the phase diagram at very negative field strength f.

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Reply to referee 1

We thank the referee for his/her very positive comments and helpful suggestions. To broaden the context, we have added some remarks in the introduction on the potential use of parafermions in quantum computation and experimental realisations. Furthermore, we added remarks on the generalisation of our results to general Z_n-symmetric models at suitable places. We also added a comment on the used bond dimensions in the DMRG simulations and its effect on the accuracy of the numerical data (see footnote 7).

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Reply to referee 2

We thank the referee for his/her very positive comments and helpful suggestions. To broaden the context, we have added some remarks in the introduction on the potential use of parafermions in quantum computation. Furthermore, we extended the discussion of Section 3 regarding the parameters in our proposed experimental realisation, and added suitable references.

To answer the specific questions:
-The bifurcation is the result of finite-size effects as was shown in Reference [68]. We have added this remark in the caption of Figure 7.
-In connection with (19) we drop the boundary terms since we are only interested in the bulk behaviour. The derived predictions for the phase diagram along f=0 is consistent with our numerical results. For completeness we also discuss the boundary terms in Appendix A.
-We attribute the increase of the central charge above c=1 when leaving the AFM phase to the KT transition when going into the Luttinger phase, which generally makes the numerical simulations demanding.

List of changes

-added a sentence at the end of the first paragraph of the in introduction as well as the References 25,26
-added three sentences at the end of the second paragraph of the introduction as well as the References 35-40, both to add a discussion of quantum computation as suggested by referee 2
-revised a sentence after (2) to include Reference [43]
-added remarks on the generalisation to Z_n symmetry at several places
-added several point in Section 3 and adapted Figure 1 accordingly
-revised Figure 2 to incorporate the changes at very negative field strength f
-added the used bond dimension in the first paragraph of Section 5
-added footnote 6 on boundary terms appearing in the duality transformation
-added footnote 7 on the quality of the numerical data
-added a remark on the observed bifurcation of the data in the caption of Figure 7
-revised Section 7 regarding the relevance of the U(1)-breaking term and the resulting phase
-dropped the previous Section 7.4 on the almost frustration-free line, since the argument is replaced by the relevance of the U(1)-breaking term given in Appendix C.3
-slightly revised Section 8
-slightly revised the conclusion
-revised the acknowledgment
-slightly revised the wording in Appendix B and the caption of Figure 15
-revised the discussion of the U(1)-breaking term in Appendix C.3
-corrected a few typos

Submission & Refereeing History

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Resubmission 2106.15823v2 on 10 December 2021

Reports on this Submission

Anonymous Report 1 on 2022-1-12 (Invited Report)

Report

The authors implemented all the suggestions from the Referees. Importantly, in the revised manuscript, they corrected an error in Appendix C and its consequences. I find their analysis convincing and I agree that this revision does not affect the main results of the paper, whose quality and clarity remain intact.

Therefore, I still recommend its publication on SciPost Physics Core.

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