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Phase diagram of the $\mathbb{Z}_3$-Fock parafermion chain with pair hopping

by Iman Mahyaeh, Jurriaan Wouters, Dirk Schuricht

This is not the current version.

Submission summary

As Contributors: Dirk Schuricht · Jurriaan Wouters
Arxiv Link: (pdf)
Date submitted: 2020-03-18 01:00
Submitted by: Schuricht, Dirk
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


We study a tight binding model of $\mathbb{Z}_3$-Fock parafermions with single-particle and pair-hopping terms. The phase diagram has four different phases: a gapped phase, a gapless phase with central charge $c=2$, and two gapless phases with central charge $c=1$. We characterise each phase by analysing the energy gap, entanglement entropy and different correlation functions. The numerical simulations are complemented by analytical arguments.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2020-9-8 Invited Report

  • Cite as: Anonymous, Report on arXiv:2003.07812v1, delivered 2020-09-08, doi: 10.21468/SciPost.Report.1969


- The paper is written in a very pedagogical way.
- The topic is very timely and the analysis of the phase diagram is useful.


- The phase diagram is incomplete even for the simple model considered by the authors. Complex phases give rise to different phases and should be discussed.
- The analysis is rather straightforward and in my opinion does not meet the impact criteria for SciPost Physics.


The authors study the phase diagram of the Z3 parafermionic chain. Using the recently developed concept of Fock parafermions and numerical approaches based on existing DMRG packages (ALPS and TenPy), they investigate the entanglement entropy and the finite-size scaling of the spectral gap of the model. They use this to identify four different phases as a function of the filling fraction and the relative strength of single-particle and pair hopping.

In my opinion, the paper is very nicely written and contains a useful discussion of the phase diagram. However, I do not think that this paper meets the impact criteria of SciPost Physics. In particular, it seems to me that the (mostly numerical) analysis is rather straightforward and does not rest on any novel techniques or approaches. Moreover, the model studied by the authors is rather a special case: it is well known that the complex phases of the hopping amplitudes have an important influence of the phase diagram.

The paper should ultimately be published, but in my opinion, it represents rather incremental progress in the field of parafermionic chains, so a more specialized journal might be more suitable.

  • validity: high
  • significance: ok
  • originality: low
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author Dirk Schuricht on 2020-09-16
(in reply to Report 2 on 2020-09-08)

We thank the referee for his/her positive remarks. We agree that our analysis used well-known techniques and approaches, however, our motivation was the study of the Fock parafermion model (15), to which end these methods proved suitable. Also we agree with the referee that the study of the effect of complex phases constitutes an interesting and important extension of our analysis. However, we believe that including such an analysis would go well beyond the aim of our work, which should be seen as an initial step in the study of Fock parafermion models. We have thus extended the discussion in the outlook to clarify the relevance of this extension for future studies.

Anonymous Report 1 on 2020-4-9 Invited Report

  • Cite as: Anonymous, Report on arXiv:2003.07812v1, delivered 2020-04-09, doi: 10.21468/SciPost.Report.1616


The authors investigate a chain of $Z_3$ Fock parafermions with both single particle and pair hopping terms. They use a mixture of analytical and numerical techniques to determine a phase diagram consisting of two gapless regions described by conformal field theories with central charge $c=1$, a gapless region with $c=2$ and a Mott-insulator gapped phase. Their analysis is careful and diligent, clearly indicating the differences between the phases and appears very useful in the understanding of parafermionic phases of matter, building nicely on work already done in the field. I recommend that this work is suitable for publication in SciPost, although a few minor changes could be made.

Requested changes

1) Equation (8) appears to be slightly wrong and inconsistent with Ref. [29] in the text [E. Cobanera and G. Ortiz, Fock parafermions and self-dual representations of the braid group, Phys. Rev. A 89, 012328 (2014), doi:10.1103/PhysRevA.89.012328, ibid. 91, 059901(E) (2015)]. I believe the power of $\omega$ should be changed from $m(m+1)/2$ to $m(m+3)/2$ for the two to agree and for consistency with Equation (18).

2) The correlators plotted in Figures (3) and (5) clearly show the algebraic decay described in the text but also seem to have oscillatory behaviour, particularly in (5). These oscillations seem to be quite strong in some cases and so some insight into the physical reason for these, and possibly even a quantitative understanding if possible, would be a useful addition.

3) While the phases themselves are very clearly explained, the phase transitions seem a little more uncertain. In particular, the authors show that the transition between the L and R phases appears to be second order, but a little more detail would be helpful. Does this transition line appear to be explained by a further conformal field theory, from which there is a flow to L and R in the two directions? By investigating the behaviour of the dynamical critical exponent and entanglement entropy for different lattice sizes as the transition is approached, this may be possible to determine. Alternatively, it may just be too difficult numerically.

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

Author Dirk Schuricht on 2020-05-13
(in reply to Report 1 on 2020-04-09)
answer to question

We thank the referee for his/her positive remarks and suggestions to improve our presentation. We have adapted the text accordingly, see the resubmitted version v2. Below let us comment on the specific points raised by the referee: 1. We thank the referee for his/her careful reading of the manuscript. Indeed the referee is correct that the exponent of $\omega$ in (8) was incorrect. Since our numerical simulations were performed starting from (18), all of our results remain correct. 2. We have added Figure 5(b) to make the oscillations more visible and also added discussions thereof at several places Secs. 5.1 and 5.2. The wave length of the oscillations as well as their power-law decay depend non-trivially on the filling fraction n and the parameter g. Thus it was not possible to obtain a qualitative understanding using the Luttinger-liquid description. 3. We fully agree with the referee that it would be very interesting to analyse the nature of the phase transitions in more detail. We have already performed numerical calculations to address this, see Figures 9, 12 and 13. Unfortunately the numerical simulations close to the phase transitions become quite delicate and demanding, thus we were not able to obtain more information on the phase transitions. We decided to focus on the characterisation of the phases and leave the further analysis of the transitions for the future, as we now explicitly mention in Section 5.5.1 and the outlook.

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