Topology behind topological insulators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2014
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| _version_ | 1866917299323142144 |
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| author | Ray, Koushik Sen, Siddhartha |
| author_facet | Ray, Koushik Sen, Siddhartha |
| contents | In this paper topological $K$-group calculations for fiber bundles with structure group $SO(3)$ over tori are carried out to explain why topological insulators have special conducting points on their surface but are bulk insulators. It is shown that these special points are gap-less and conducting for topological reasons and follow from the $K$-group calculations. The existence of gap-less surface points is established with the help of an additional topological property of the $K$-groups which relates them to the index theorem of an operator. The index theorem relates zeros of operators to topology. For the topological insulator the relevant operator is a Dirac operator, that emerges in the problem because the system has strong spin-orbit interactions and time-reversal invariance. Calculating $K$-groups over tori require some special topological tools that are are not widely known. These are explained. We then show that the actual calculation of $K$-groups over tori becomes straightforward once a few topological results are in place. Since condensed matter systems with periodic lattices, are always bundles over tori the procedures described is of general interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_4898 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Topology behind topological insulators Ray, Koushik Sen, Siddhartha Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics In this paper topological $K$-group calculations for fiber bundles with structure group $SO(3)$ over tori are carried out to explain why topological insulators have special conducting points on their surface but are bulk insulators. It is shown that these special points are gap-less and conducting for topological reasons and follow from the $K$-group calculations. The existence of gap-less surface points is established with the help of an additional topological property of the $K$-groups which relates them to the index theorem of an operator. The index theorem relates zeros of operators to topology. For the topological insulator the relevant operator is a Dirac operator, that emerges in the problem because the system has strong spin-orbit interactions and time-reversal invariance. Calculating $K$-groups over tori require some special topological tools that are are not widely known. These are explained. We then show that the actual calculation of $K$-groups over tori becomes straightforward once a few topological results are in place. Since condensed matter systems with periodic lattices, are always bundles over tori the procedures described is of general interest. |
| title | Topology behind topological insulators |
| topic | Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/1408.4898 |