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Amorphous topological phases protected by continuous rotation symmetry
by Helene Spring, Anton R. Akhmerov, Daniel Varjas
This Submission thread is now published as
|Authors (as registered SciPost users):||Anton Akhmerov · Helene Spring · Daniel Varjas|
|Preprint Link:||scipost_202106_00002v1 (pdf)|
|Date submitted:||2021-06-02 15:48|
|Submitted by:||Spring, Helene|
|Submitted to:||SciPost Physics|
Protection of topological surface states by reflection symmetry breaks down when the boundary of the sample is misaligned with one of the high symmetry planes of the crystal. We demonstrate that this limitation is removed in amorphous topological materials, where the Hamiltonian is invariant on average under reflection over any axis due to continuous rotation symmetry. While the local disorder caused by the amorphous structure weakens the topological protection, we demonstrate that the edge remains protected from localization. In order to classify such phases we perform a systematic search over all the possible symmetry classes in two dimensions and construct the example models realizing each of the proposed topological phases. Finally, we compute the topological invariant of these phases as an integral along a meridian of the spherical Brillouin zone of an amorphous Hamiltonian.
Published as SciPost Phys. 11, 022 (2021)
Author comments upon resubmission
List of changes
Both referees asked for a specific definition of amorphous matter — a request that we address in the newly added first paragraph of section II. Amorphous matter must be fully invariant under all Euclidean transformations and have no long-range correlations, unlike in the reference suggested by Referee 2, which we now use to clarify the distinction. Turning to the question of Referee 1 about whether residual correlations would open small gaps, we indeed expect that models lacking exact rotational symmetry would have boundary-mode localization at the surfaces that do not respect ensemble reflection symmetry.
The second referee also asked whether the disorder must respect certain symmetries exactly. This is indeed the case, as we now explain in the fourth paragraph of section III, and as was studied in PRB 89 155424. This reference also demonstrates that even topological invariants lead to edge mode localization in the presence of disorder. We rely on this observation to deduce the difference in classification between the amorphous and continuum models.
Following the suggestion of Referee 1, we have specified the expected transport signature of our amorphous systems in the abstract, to make the findings of the paper clearer from the start.
Regarding the remark that all amorphous systems are gapless in principle (point 5 of report 1) we do not think that this is generally true, and that it is possible to have insulating amorphous structures. For example, an amorphous atomic insulator, that consists of disconnected atoms with a gapped spectrum, will also feature a hard gap in the thermodynamic limit, a feature that is robust against adding weak hopping between the atoms. As such, we do not believe that it is a numerical error that we do not see a finite local density of states in the bulk gap. On the other hand, our invariants are likely still valid even if there is a finite DOS of localised states in the gap, however, we consider this question beyond the scope of the current manuscript.
Submission & Refereeing History
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- Report 2 submitted on 2021-06-29 09:47 by Anonymous
- Report 1 submitted on 2021-06-04 04:50 by Anonymous
Reports on this Submission
In their response and the new version, the authors address my questions and suggestions and have implemented a number of modifications in the manuscript. Based on these, I am happy to recommend publication of the manuscript in the present form.
- Cite as: Anonymous, Report on arXiv:scipost_202106_00002v1, delivered 2021-06-04, doi: 10.21468/SciPost.Report.3021
I thank the authors’ clarification for some of my questions. Hope that comments 3 and 4 can be addressed before my recommendation. Furthermore, I would like to comment that the structural disorder is randomly added around a cubic lattice configuration in Ref.  and when the disorder is sufficiently strong, e.g., W=L (system size), the configuration corresponds to an independent uniformly distributed sites without any correlation between sites, similar to the hard disk amorphous structures employed in the paper.