The Topological Origin of Boundary Charges at Edges of One-Dimensional Crystals without Inversion Symmetry

Round 1

Reviewer 1 Report

The manuscript entitled "The Topological Origin of Boundary Charges at Edges of One-Dimensional Crystals without Inversion Symmetry" by P. Shi and his or her co-workers show the topological boundary charge at edges of a finite-length 1D crystals with or without inversion symmetry. Overall, the current manuscript is well-written and delivers important aspects of this work. However, there are a few points which should be clarified to deserve publication. Here are some comments to improve the current level of the manuscript.

1.      Clarify the method of obtaining the numerical values of the intensities or the electric fields phi_i,j(x) or E_i,j(x).

2.      The number of lattices was restricted to 20 unit cells. This number of cells might matter significantly affecting fields or the resultant edge charges. For example, the more the unit cells, the smaller values the formed bound charges. I think some arguments are necessary to answer this concern.

3.      The authors used MLWF to explain the trend of the edge charges. There are little information or references regarding how the authors calculated Wannier functions, which makes hard to evaluate the calculation was correct. More references or arguments are necessary.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Good work for topological photonics, any experimental attempt should be scheduled。

Author Response

Thank you for acknowledging our work. We are currently conducting pertinent experiments.

Reviewer 3 Report

The author study the topological origin of boundary charges s in one-dimensional photonic 1 crystals (1D PhCs) without inversion symmetry by changing the dielectric constant of the material.

1: My major concern is the model, in the discussion part of “The boundary charges without inversion symmetry”, the stacked layers A B C could be considered as two tight-binding sites per unit cell having alternating site energies, which is consistent with the Rice-Mele Model (doi.org/10.1103/PhysRevLett.49.1455). The non-degenerated edge states can also be calculated according to the R-M model, what’s the difference between the R-M model and the trimer SSH model in this work?

2: Another interesting question is the choice of the topological invariant, of course, the topological charges by the integration of the local density of states are reasonable indices, I just wanna know why not the common Zak phase or winding number, are there some conditions inapplicable to them? 

3: A reasonable optical numerical model must map a realizable optical structure, how is the PEC realized during the growth of an optical film system? Is it using metals, e.g., gold, silver, copper?

4:  Please specify the structure of the corresponding optical membrane system for the optical numerical model in the text. To verify the accuracy of the results, please use commercial optical software (e.g., fdtd, comsol ,Rsoft) to simulate the electric field in the boundary state and compare with Fig. 1b, Fig. 2b, Fig. 3a,c.

5：Why is the optimal value of the number of the unit cells 20, please explain it specifically.

6:  This paper only shows the z-component of the electric field, if the x and y components of the electric field are not zero, please add that as well.

Minor editing of English language required.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 4 Report

The authors study a one-dimensional photonic crystal without inversion symmetry and explain the topological origin of the edge states using the concept of Wannier center. I only have one minor comment.

In line 135, they mentioned that “the topological singularity of the 1st band crosses the zero frequency to reach imaginary frequency at this time”. Please add several sentences to explain the topological singularity and imaginary frequency.

I consider that, the work in the manuscript is a good research attempt for topological photonics, the authors try to deeply analyze the photonic crystal in the view of topological photonics. Anyway, the photonic crystal had been analyzed in the past decades, in simulations and experiments. However, except for photonic crystal fiber, few photonic crystal technologies and devices can be widely used. So, I figure that, if authors can push adead, plan the experiments in this revised manuscript or in the near future, that will be better, and make the photonic crystal be closer to the wide applications. And so, we look forward to any experimental attempt of photonic crystal  in the view of topological photonics.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Although the author could not match his model to a real optical structure, it does not affect the article's description of the topological edge states.