Continuous Topological Insulators Classification and Bulk Edge Correspondence
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908604603301888 |
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| author | Bal, Guillaume |
| author_facet | Bal, Guillaume |
| contents | This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable.
We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-Hörmander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous Topological Insulators Classification and Bulk Edge Correspondence Bal, Guillaume Mathematical Physics 35Q40, 35S35, 47A53, 47A60, 47G30, 81Q10 This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-Hörmander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators. |
| title | Continuous Topological Insulators Classification and Bulk Edge Correspondence |
| topic | Mathematical Physics 35Q40, 35S35, 47A53, 47A60, 47G30, 81Q10 |
| url | https://arxiv.org/abs/2412.00919 |