SciPost Submission Page
Geometry and Topology Tango in Ordered and Amorphous Chiral Matter
by Marcelo Guzmán, Denis Bartolo, David Carpentier
This is not the latest submitted version.
This Submission thread is now published as SciPost Phys. 12, 038 (2022)
|As Contributors:||Marcelo Guzmán|
|Date submitted:||2021-10-10 13:24|
|Submitted by:||Guzmán, Marcelo|
|Submitted to:||SciPost Physics|
Systems as diverse as mechanical structures and photonic metamaterials enjoy a common geometrical feature: a sublattice or chiral symmetry first introduced to characterize electronic insulators. We show how a real-space observable, the chiral polarization, distinguishes chiral insulators from one another and resolve long-standing ambiguities in the very concept of their bulk-boundary correspondence. We use it to lay out generic geometrical rules to engineer topologically distinct phases, and design zero-energy topological boundary modes in both crystalline and amorphous metamaterials.
Author comments upon resubmission
List of changes
We have included a detailed discussion of the matrix pencil method in Appendix D and added a new Figure (Fig. 11).
We have added a formal and a practical definition of the atomic limit in Section 3.
We have corrected some typos.
Submission & Refereeing History
Published as SciPost Phys. 12, 038 (2022)
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Reports on this Submission
Report 2 by Daniel Varjas on 2021-11-26 (Invited Report)
- Cite as: Daniel Varjas, Report on arXiv:scipost_202106_00037v2, delivered 2021-11-26, doi: 10.21468/SciPost.Report.3909
The authors have addressed most of my previous comments to my satisfaction.
1) The clarity of the presentation of the pencil-matrix method, as an approximation method to find maximally localized Wannier sates, was greatly improved by the addition of Appendix D and Fig. 11. The limits of applicability of this method are still unclear, and poses an interesting research question.
2) The definition of "atomic limit" was also clarified.
1) Presentation of the pencil matrix method is still somewhat confusing, see requested changes.
First of all, let me apologise to the Authors and the Editor for the slow report.
I find that the authors sufficiently addressed my criticism in the first report. Besides, I generally agree with the criticism of Anonymous Reviewer, and find that the changes promised by the Authors should be sufficient to address them.
I recommend publication after the changes requested by Anonymous Reviewer and myself are implemented.
1) The citation in "...taking advantge (sic!) of the so-called pencil-matrix method " still seems misleading. A cursory reading of the book didn't illuminate anything about this method beyond the definition of the term "pencil matrix". If the book contains anything relevant beyond the definition, please cite by section or page number. If not, please rephrase this sentence in a way that avoids giving the impression that this method was already introduced in the cited book. A reference to Appendix D should also be included in the main text.
Anonymous Report 1 on 2021-11-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202106_00037v2, delivered 2021-11-18, doi: 10.21468/SciPost.Report.3867
1 - The chiral polarization has the potential of being a useful quantity in determining the topological properties of disordered and/or amorphous systems.
2 - The presentation is well-structured, with simple examples helping the reader gain an intuitive understanding of the authors' results.
1 - I believe that figurative/metaphorical language should be avoided in scientific publications.
2 - The relation between this work and previous research could be made more explicit.
The authors introduce a quantity called the "chiral polarization." It is a real space quantity which helps to clarify the interplay between topology and lattice geometry in chiral-symmetric systems, and seems useful as a real-space marker that characterizes different domains in uniform, disordered, and/or amorphous systems.
The presentation is well structured. The quantities used by the authors as well as their results are introduced in a step-by-step fashion, with examples based on simple systems helping the reader to gain an intuitive understanding of the authors' work. In my opinion, this work does meet the Scipost acceptance criteria. Specifically, it opens a new pathway in an existing research direction, with clear potential for multipronged follow-up work (https://scipost.org/SciPostPhys/about).
Before publishing this work, however, I would urge the authors to address the following points.
1 - My main criticism of this work concerns the way in which the authors relate their findings to the existing body of research. I believe that this relation should be made more accurate and explicit, so as to avoid potentially misleading the readers. For example, on line 123 the authors write that the chiral polarization is "seemingly identical" to the skew polarization and the mean chiral displacement. Either they are identical, or they are not.
My current understanding (which may be wrong, please correct me if that is the case) is that the chiral polarization is in fact mathematically identical to the mean chiral displacement (MCD). They are the same quantity. One of the main novelties of this work, as far as I can understand, is that the authors compute the MCD using a specific set of states, the maximally localized states. This means they can use the MCD as a real-space indicator, which is useful in describing the topological domains of large disordered and/or amorphous systems. This is indeed an important result, but I wish it would be related to previous research in as accurate and explicit a way as possible.
Beyond line 123, I have found similar expressions in other parts of the paper. On line 328 the authors compare their results to the mean chiral displacement by using the words: "out of reach of conventional chiral displacement characterizations" and cite Ref. 52. This reference is a combined experimental/numerics paper measuring the MCD. Do the authors mean that the MCD is fundamentally unable to produce the same characterizations? This would be confusing to me, since their chiral polarization is mathematically identical to the MCD, as far as I can understand. Do they instead mean that the results of Ref. 52 suffer from finite-size and finite-time effects? If so, then please state this explicitly.
On line 445, they state "our protocol is close to the chiral displacement method." On line 747, they compare the chiral polarization with the MCD protocols by saying that they are "seemingly similar." To me it seems the authors' time-evolution protocol is literally identical to the MCD method. If so, please state this explcitly.
2 - In Fig. 6, the authors show that the chiral polarization remains a good indicator even when the sites of the regular lattice are shifted from their original positions. To me, it does make sense that the zero modes are robust to such a change, since the Hamiltonian matrix istelf remains constant. However, it is not obvious to me whether the robustness of the chiral polarization will persist for larger disorder strength. For example, if setting $|\delta x|/a=10$, the zero modes will clearly still be there, but will the chiral polarization be able to tell? I realize that such large shifts would probably require a large tight-binding model, but this seems achievable given Fig. 8.
3 - On line 369, the authors state that "The phase boundaries are then readily detected by jumps of the chiral-polarization vector". Can this statement be made quantitative? How high and how sharp should these jumps be before one can conclude that a phase boundary exists?
1 - I strongly believe that metaphorical/figurative language has no place in scientific publications. I urge the authors to remove and/or rephrase the following:
"tango" in the title, "spread frantically" on lines 53-54, "intimate interplay" on line 87, "intimate relation" on line 166, "illuminate the very definition" on line 205, "illuminates the geometrical implication" on line 278, "the frame topology and the frame geometry conspire" on lines 450-451, "The subtle tango" on line 458
2 - There are typos in the reference list. Some of the reference titles should contain capitalized words, such as Berry and Wannier. Further, journal abbreviations and formatting is inconsistent across the reference list. Some references are missing links.
3 - I found several minor typos throughout the paper text.
4 - There are two references to movies/videos, but I could not find them in the supplemental material or in the reference list.