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A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states
by Chen-Shen Lee
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Submission summary
| Authors (as registered SciPost users): | Chen-Shen Lee |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2311.16801v3 (pdf) |
| Date accepted: | 2024-01-15 |
| Date submitted: | 2023-12-12 06:18 |
| Submitted by: | Lee, Chen-Shen |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The one-to-one relation between the winding number and the number of robust zero-energy edge states, known as bulk-boundary correspondence, is a celebrated feature of 1d systems with chiral symmetry. Although this property can be explained by the K-theory, the underlying mechanism remains elusive. Here, we demonstrate that, even without resorting to advanced mathematical techniques, one can prove this correspondence and clearly illustrate the mechanism using only Cauchy's integral and elementary algebra. Furthermore, our approach to proving bulk-boundary correspondence also provides clear insights into a kind of system that doesn't respect chiral symmetry but can have robust left or right zero-energy edge states. In such systems, one can still assign the winding number to characterize these zero-energy edge states.
Published as SciPost Phys. Core 7, 003 (2024)
Reports on this Submission
Anonymous Report 1 on 2024-1-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2311.16801v3, delivered 2024-01-10, doi: 10.21468/SciPost.Report.8392
Report
The author considers gapped chiral-symmetric 1D systems with an arbitrary number of bands and arbitrary but finite range hopping. They prove that the winding number is equal to the number of robust edge modes at either end of the chain, thus establishing bulk-boundary correspondence. This is done while avoiding advanced mathematical techniques, and resorting instead to basic algebra and complex analysis.
I found the work to be valid and well-presented. The paper is well-structured, starting with simple cases and gradually moving towards more complex systems. I believe that this work should be published with only minor changes, but I think it makes a better fit to Scipost Physics Core rather than Scipost Physics. The reason for this is that I consider it to be mainly a follow-up work, using a methodology similar to that of Ref. [28] and extending those previous results to the case of systems with more than two bands.
The only two minor comments I have are:
1) just above Eq. 1, I believe "antiunitary" should be replaced with "unitary"
2) In Section 6, I believe it might help the reader if there was an intuitive explanation for the main result of that section. Maybe something along the lines of: since the zero mode lives only on the B sublattice, it cannot be altered by any change occurring only on the A sublattice sites, even if that change breaks chiral symmetry.
Author: Chen-Shen Lee on 2024-01-31 [id 4297]
(in reply to Report 1 on 2024-01-10)We thank the referee for the careful review and the valuable suggestions. In the final version, we have addressed the points raised in the report. 1. Thank you sincerely for pointing out this error. The chiral operator is unitary in the single-particle basis. 2. we added an intuitive explanation followed by eq. (90).
Besides these, we found a non-obvious ambiguity concerning the notation |X/Y, j〉. We clarified it by introducing -) $|S_X/S_Y, j〉= |S_X/S_Y〉\otimes |j〉$ where $|S_X/S_Y〉$ is the basis state of the chiral operator with eigenvalue +1/-1. -) |X/Y, j〉: the vector composed of the coefficients according to the basis $|S_X/S_Y, j〉$