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

This manuscript studies a bosonic analogue of weak topological insulating phases on the honeycomb lattice. Specifically the model studied are on a two-dimensional periodic honeycomb lattice in the presence of an on-site potential with alternating sign along the different y-direction of the lattice. Using quantum Monte Carlo simulations and analytical calculations, the authors identify a bosonic 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. They also map out the full phase diagram, including a charge-density-wave insulator at 1/2-filling and superfluid at intermediate densities. Supprisely 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 proposed model may be experimentally realized using cold atoms in an optical lattice. I find the results are interesting from both theoretical and experimental aspects, so I recommend its publication.

I have the following comments:

1: 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.

2: 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. as well as weak next-nearest-neighbor repulsion. The experimental realization of our model is feasible in an optical lattice setup.

1 - The paper is very well written and easy to follow.
2 - The results are presented in an intuitive, pedagogical manner.
3 - The authors perform a detailed study of their model, using multiple order parameters, topological invariants, as well as varying boundary conditions.

1 - The symmetry classification and topological protection of the model is insufficiently discussed (see report below).
2 - Is it not clear to what extent the work meets the acceptance criteria of Scipost Physics (specifically, the list of "Expectations"), as opposed to Scipost Physics Core.1 - The symmetry classification and topological protection of the model is insufficiently discussed (see report below).
2 - Is it not clear to what extent the work meets the acceptance criteria of Scipost Physics (specifically, the list of "Expectations"), as opposed to Scipost Physics Core.

The authors study the topological phases of hard-core bosons on a hexagonal lattice in which the onsite potential is modulated. They find that WTI phases appear once the onsite potential is larger than the nearest neighbor hopping strength, and that these phases are robust against NN repulsion as well as against weak NNN repulsion.

The paper is very well written. I enjoyed reading it. Results are presented in an intuitive, pedagogical way, making them easy to follow. There are however two points that I think the authors should address. These points are listed above, and I detail them here:

1) The authors discuss WTI phases appearing in symmetry class BDI and use a Hamiltonian that is non-interacting (I'm referring to Eq. 1, before the NN and NNN repulsion are added). However, the single-particle Hamiltonian of Eq. 1 does not belong to symmetry class BDI. It does have time-reversal symmetry T=K, meaning it is real, but there is no chiral symmetry. There is no unitary that anti-commutes with H because of the non-zero onsite modulation and chemical potential.
Consistent with this lack of chiral symmetry, the edge states discussed by the authors do not appear at E=0. In class BDI, it is not just translation symmetry, but also chiral symmetry which protects the WTI. Because of chiral symmetry all states come in +E, -E pairs (as can be seen from the bandstructures of Fig. 6). The edge states of a WTI should be pinned to the middle of the E=0 gap, such that they cannot be removed from this gap without breaking symmetries. In the authors' model however, edge states appear in the gap between bands 1 and 2 (or 3 and 4), away from E=0. What symmetry is responsible for their topological protection? Why can't they, in principle, be shifted up or down in energy such that they hybridize with the bulk states and dissapear?

2) While the work is novel and well presented, the authors should spend more time discussing if/how their paper meets the expectations of Scipost Physics (https://scipost.org/SciPostPhys/about#criteria). From my reading of the paper as it is now, it seems to me that it instead meets the acceptance criteria of Scipost Physics Core (https://scipost.org/SciPostPhysCore/about), provided that the point (1) above is addressed.

1 - Show explicitly what are the symmetries of their model and its symmetry class.
2 - 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.