A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states
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 1$d$ systems with chiral symmetry. Although this property can be explained by the K-theory, the underlying mechanism remains elusive....
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| Format: | Article |
| Language: | English |
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SciPost
2024-01-01
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| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.7.1.003 |
| _version_ | 1827368586647175168 |
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| author | Chen-Shen Lee |
| author_facet | Chen-Shen Lee |
| author_sort | Chen-Shen Lee |
| collection | DOAJ |
| description | 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 1$d$ 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. |
| first_indexed | 2024-03-08T09:33:39Z |
| format | Article |
| id | doaj.art-2cc229b217ef4b92bcc0e5b337d875c7 |
| institution | Directory Open Access Journal |
| issn | 2666-9366 |
| language | English |
| last_indexed | 2024-03-08T09:33:39Z |
| publishDate | 2024-01-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj.art-2cc229b217ef4b92bcc0e5b337d875c72024-01-30T15:39:33ZengSciPostSciPost Physics Core2666-93662024-01-017100310.21468/SciPostPhysCore.7.1.003A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge statesChen-Shen LeeThe 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 1$d$ 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.https://scipost.org/SciPostPhysCore.7.1.003 |
| spellingShingle | Chen-Shen Lee A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states SciPost Physics Core |
| title | A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states |
| title_full | A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states |
| title_fullStr | A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states |
| title_full_unstemmed | A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states |
| title_short | A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states |
| title_sort | linear algebra based approach to understanding the relation between the winding number and zero energy edge states |
| url | https://scipost.org/SciPostPhysCore.7.1.003 |
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