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Weak topological insulating phases of hard-core-bosons on the honeycomb lattice

by Amrita Ghosh, Eytan Grosfeld

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Submission summary

Authors (as registered SciPost users): Amrita Ghosh · Eytan Grosfeld
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-02-15
Date submitted: 2021-01-22 18:11
Submitted by: Grosfeld, Eytan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approaches: Theoretical, Computational


We study the phases of hard-core-bosons on a two-dimensional periodic honeycomb lattice in the presence of an on-site potential with alternating sign along the different y-layers of the lattice. Using quantum Monte Carlo simulations supported by analytical calculations, we identify a weak topological insulator, characterized by a zero Chern number but non-zero Berry phase, which is manifested at either density 1/4 or 3/4, as determined by the potential pattern. Additionally, a charge-density-wave insulator is observed at 1/2-filling, whereas the phase diagram at intermediate densities is occupied by a superfluid phase. The weak topological insulator is further shown to be robust against any amount of nearest-neighbor repulsion, as well as weak next-nearest-neighbor repulsion. The experimental realization of our model is feasible in an optical lattice setup.

Author comments upon resubmission

Dear Editor,

We thank the Referees for their positive evaluation and constructive comments, and for recommending publication.

We have modified the paper according to the Referees’ comments, which we believe has improved the paper considerably. In particular, we have modified the introduction to stress the context of the results. We believe that our work offers a natural, almost minimal model for the realization of weak topological insulators (WTIs) of interacting bosons in 2D. Indeed, a study of such bosonic WTIs and their associated phase diagrams has been lacking in the literature and we believe that our model fills this important gap by studying the nucleation of these topological phases, their stability properties in the presence of interactions, and the interplay of the various competing orders. In addition, our paper opens up new pathways to the study of bosonic topological states by identifying the relevant tools and order parameters and by generating a framework for their study. (Indeed we already followed up on this paper with an additional paper and we see potential for an extensive theoretical study of weak and strong topological states of bosons.)

We hope that the paper is now ready for publication. Below we enclose a point by point response to the Referees.

Sincerely, Amrita Ghosh and Eytan Grosfeld

Response to Referee 1

  • "Show explicitly what are the symmetries of their model and its symmetry class."

We thank the Referee for this comment. The only symmetries in the model are time-reversal symmetry and mirror symmetry. The model is therefore classified in the AI mirror-symmetry protected class, which admits a topological number in 1D. We now detail the symmetry classification towards the end of section V.

  • "Prove that their phases are topologically protected. This means to prove that there does not exist a symmetry-preserving perturbation which removes the edge states, for instance by shifting their energies away from their respective gaps."

We added an appendix B that describes the edge-bulk correspondence and explains the protection of the edge states by a mirror symmetry and an emergent chiral symmetry. We added a figure that demonstrates the robustness of the edge states in the presence of mirror-symmetry preserving perturbations and their splitting when mirror symmetry is broken. Due to the emergent chiral symmetry, the average energy of the edge states stays pinned to the center of the rho=1/4 (or rho=3/4) gap.

Response to Referee 2

  • "A main character of weak topological insualtor is the existence of edge states on the edges along specific directions. Here the edge state is quasi-1D superfluid. One may calculate the single-particle correlator b^{dagger}_i b_j. The decaying behavior may reflect such information: it is insulating if the decay is exponential with the distance, and is gapless superfluid if the decay follows a power law."

We thank the referee for this comment. Calculating directly the correlation function along the edge is difficult using SSE QMC. Instead, in order to verify the superfluid nature of the edge state, we calculated the superfluid density along the y-direction for the different stripes of the new Figure 1b. We observe that the superfluid density is finite along the edges of the sample but is vanishingly small (zero in the thermodynamic limit) in the bulk. This is summarized in a new figure 7c and referred to in the text.

  • "Since the authors are studying a bosonic model, the Chern number and Berry phase for bosons should be calculated to characterize the bosonic weak topological insulator. "

We have calculated the Chern number and Berry phase for both bosons and fermions and found that they give the same value for both types of particles, see Eqns. (14,15) and the nearby discussion. We added a sentence to the introduction to stress this.

List of changes

1. Introduction modified to stress the context of the results and the importance of our study.
2. In a new Figure 1 we now label the stripes along the y-direction.
3. In a new Figure 7c we now plot the superfluid density along the y-direction for the different stripes, for the topological and non-topological phases. We discuss this in the text in Section 4.
4. In Section 5 we discuss the symmetries of the model.
5. In a new Appendix B we demonstrate the protection of the edge states via mirror and effective chiral symmetries. A new Fig. 13 demonstrates this protection in the presence of several types of perturbations.
6. We added two new references related to the symmetry class and to a new study of bosonic topological phases by us.

Published as SciPost Phys. 10, 059 (2021)

Reports on this Submission

Anonymous Report 1 on 2021-2-10 (Invited Report)


The authors have fully addressed the points I raised in the first round. I believe this paper should be published in SciPost Physics as is.

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