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Weak topological insulating phases of hardcorebosons 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:  https://arxiv.org/abs/2010.16126v3 (pdf) 
Date accepted:  20210215 
Date submitted:  20210122 18:11 
Submitted by:  Grosfeld, Eytan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study the phases of hardcorebosons on a twodimensional periodic honeycomb lattice in the presence of an onsite potential with alternating sign along the different ylayers 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 nonzero Berry phase, which is manifested at either density 1/4 or 3/4, as determined by the potential pattern. Additionally, a chargedensitywave insulator is observed at 1/2filling, 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 nearestneighbor repulsion, as well as weak nextnearestneighbor 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 timereversal symmetry and mirror symmetry. The model is therefore classified in the AI mirrorsymmetry 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 symmetrypreserving 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 edgebulk 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 mirrorsymmetry 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 quasi1D superfluid. One may calculate the singleparticle 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 ydirection 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 ydirection.
3. In a new Figure 7c we now plot the superfluid density along the ydirection for the different stripes, for the topological and nontopological 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)