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Quantum tricriticality of incommensurate phase induced by quantum domain walls in frustrated Ising magnetism

by Zheng Zhou, Dong-Xu Liu, Zheng Yan, Yan Chen, Xue-Feng Zhang

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Submission summary

Authors (as registered SciPost users): Xue-Feng Zhang · Zheng Zhou
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-10-05 14:11
Submitted by: Zhou, Zheng
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Incommensurability plays a critical role in many strongly correlated systems. In some cases, the origin of such exotic order can be theoretically understood in the framework of 1d line-like topological excitations known as ``quantum strings''. Here we study an extended transverse field Ising model on a triangular lattice. Using the large-scale quantum Monte Carlo simulations, we find that the spatial anisotropy can stabilize an incommensurate phase out of the commensurate clock order. Our results for the structure factor and the string density exhibit a linear relationship between incommensurate ordering wave vector and the density of quantum strings, which is reminiscent of hole density in under-doped cuprate superconductors. When introducing the next-nearest-neighbour interaction, we observe a quantum tricritical point out of the incommensurate phase. After carefully analyzing the ground state energies within different string topological sectors, we conclude that this tricriticality is non-trivially caused by effective long-range inter-string interactions with two competing terms following different decaying behaviours.

Current status:
Has been resubmitted

Author comments upon resubmission

We are thankful to all reviewers for carefully reviewing the paper and providing useful comments and suggestions, which helped us to improve the manuscript.

We appreciate the positive evaluation of both Referees that 'the paper is scientifically sound and well written' and 'the problem is interesting and the results are intriguing'. We also thank their critical comments that are important to understanding the problems in concern. These points have all been properly addressed in the attached reply.

Thanks to the comments of the Referees, we have made substantial improvements to the paper. The revisions made are listed above.

We give a point-by-point response to the comments of all Referees. We believe that the changes made have improved our paper and hope that the current manuscript will be considered suitable for further consideration in SciPost Physics.

List of changes

1. To address the concern on the possible phases at $J_x<J$ (Comment 1 of Referee 1), we have stressed that the discussion in Section 2, Paragraph 2 only applies to the classical Ising limit $h=0$.
2. To address the concern on the meaning of the word `vibration' of the string (Comment 3 of Referee 2), we have added reference to related illustration in Section 2, Paragraph 5.
3. To address the question on the stripe phase (Comment 2 of Referee 2), we have added a related brief discussion to Section 3, Paragraph 2.
4. To address the concern on the width of incommensurate plateaux (Comment 5 of Referee 2), we have added a sentence to stress the finite size scaling result in Section 3, Paragraph 6.
5. To address the concern on the ansatz of the effective inter-string interaction (Comment 3 of Referee 1, and Comment 4 of Referee 2), we have added a detailed discussion in Section 4, Paragraph 5 and revised the wording in the Abstract and Conclusion.
6. To address the concern on the leading order approximation of $B$ (Comment 2 of Referee 1), we have appended a discussion in Section 4, Paragraph 7.

Reports on this Submission

Anonymous Report 2 on 2022-10-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2005.11133v3, delivered 2022-10-14, doi: 10.21468/SciPost.Report.5895


1.) Interesting relevant model for frustration and criticality

2.) Groundbreaking analytic analysis in terms of interacting strings

3.) Convincing large scale QMC simulations


1.) some discussions are missing or not detailed enough (see report)


The authors consider the transverse field Ising model on a triangular lattice,
which is an interesting model for rich critical behavior due to the interplay
of frustration and quantum effects. By considering an anisotropic coupling
it is possible to explain much of the behavior with an effective description in
terms of interacting strings, which is supported by large scale numerical simulations.

Apart from the necessary changes (below), I find the paper truly convincing. The work
meets the acceptance criteria and should be published after those changes have
been considered

Requested changes

1.) In the description of the construction of strings on page 4, I could not understand
the following sentence: "To avoid creation of triangle-rule-breaking defects (also known as
spinon topological defects), each bisector within the string can only choose left-going
or rightgoing directions" What is meant by "bisector"? Where do I see the directions in
Fig 2? I recommend that the explanation is expanded in more detail.

2.) In Eqs. (2) the energy of the quantum strings are defined. It should be
explained if there is a kinetic energy as well or why it can be neglected.

3.) In Eq. (5) the string density is defined, which appears to be quantized
in the numerical simulations. Is it a conserved quantity or is there another
explanation for this discrete behavior (finite size effect)? The change of the
peak position is argued to become continuous in the thermodynamic limit, but does
(density x length) remain quantized?

4.) The assumption of a power law interaction in Eq. (9) is not rigorously motivated
as previous referees also commented.
A discussion would be useful how important this assumed form is to the final outcome,
or if other forms of two competing interactions have also been tried. The clear evidence
of a long range attractive contribution to the interaction is surprising and interesting.
What could be the mechanism? The newly inserted paragraph does not explain why
one part is attractive.

5.) The relation to the hard-core boson model in Ref. [28] should be discussed
in more detail, which seems to follow similar physics. What is different?
Is the universal critical behavior the same?

6.) Editorial changes: Refs. [3] and [51] are identical.
Please check for spelling mistakes ("incommensurte"on p.2)
and spurious articles (remove "the" in front of QMC simulations).

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

Author:  Zheng Zhou  on 2022-10-26  [id 2954]

(in reply to Report 2 on 2022-10-14)
answer to question

We thank the Referee for the recommendation for publication as well as the useful comments and suggestions, which helped us to improve the manuscript. In the following we give a point-by-point reply to these comments. A more properly formatted version of the reply in LateX can be found attached in the PDF file.

1 . We thank the Referee for pointing out our ambiguous expression. In the revised manuscript, we have replaced the word 'bisector' by 'segment', and marked it in the Fig. 2 by the green and purple left and right-pointing arrows.

2 . In fact, the vibration of the segment is described by an XY-chain and the kinetic energy corresponds to the energy of the XY-chain E_{XY} and is included in Eq. (2). In the revised manuscript, we have made that clear by adding a sentence 'where \Delta is the energy gap and E_{XY} is the kinetic energy given by solving the effective spin-1/2 XY-chain'.

3 . The quantisation of string density is due to finite size effect. As the number of strings must be a even integer under periodic boundary condition, the string density must then be 2\mathbb{Z}/L_x. In the limit of small quantum fluctuation where the triangle-rule cannot be violated, any local operation cannot change the number of quantum strings, so the string density is also conserved. In the thermodynamic limit, the quantisation step 2/L_y becomes infinitesimall, so the string density becomes continuous.

4 . For the justification of the ansatz, there has in fact been a long debate between whether the interaction between strings should be exponential or power-law [J. Zaanen, Phys. Rev. B 40, 7391(R) (1989)]. We have tried both ansatz
for the second ansatz, the optimal parameters are calculated to be ๐œ‰_1=0.19 and ๐œ‰_2=0.89. The sum of residual squared of the power-law ansatz is 4.35ร—10^{-7} and for the exponential ansatz is 5.73ร—10^{-7}. We therefore adopted the power-law ansatz in the manuscript. The form of the ansatz does not affect our qualitative result as long as in V(r)=V_h(r)+V_{J'}(r), V_h(r) decays faster than V_{J'}(r), and V_{J'}(r) changes sign when J'=0. We also note that the mechanism of the repulsion V_h(r) is similar to the hard-core boson model, so there is no reason to expect that the form of interaction should be different. We have also added the discussion to a footnote in the main text.

For the reason of the attractive interaction, we added some further explanation to the newly added paragraph. At the presence of J', there is an additional mechanism of string interaction: apart from the repulsion from the hinder of motion when strings are nearby denoted V_h(r), the second, denoted V_{J'}(r), comes from the fact that the insertion of single string produces energy cost 3J'/2 per string length, while two adjacent strings cost energy 2J' per string length, which is different than two individual strings [Fig. 6(a)]. Therefore, when two string segments are adjacent, there is an extra energy gain of J' when J'>0, resulting in an attractive interaction, while when J'<0, this becomes an energy cost of |J'|, resulting in a repulsive interaction.

5 . In the hardcore boson model, the interaction power is calculated to be ๐›ผ=4.0(1), which is different from our result ๐›ผ=7.5(1). The difference is because of the different manners of string vibration. In the hardcore boson model, the vibration of the string is described by an XY-chain with only next-nearest-neighbour interaction, which is different from our model where the vibration is described by an XY-chain with only nearest-neighbour interaction. As the interaction comes from adjacent strings hindering their motions, different manners of string vibration result in different interaction powers. We also added the discussion above to the revised manuscript. However, the discussion of universality is beyond the scope of this work.

6 . We thank the Referee for the careful proofreading. In the revised manuscript, we have checked the text once again and corrected the mistakes.



Anonymous Report 1 on 2022-10-6 (Invited Report)


The authors have properly addressed the comments and questions I raised in the previous report. Now, I would recommend the paper for publication in the present form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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