Bloch waves and Non-commutative Tori of Magnetic Translations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866911922872385536 |
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| author | Dereli, Tekin Popov, Todor |
| author_facet | Dereli, Tekin Popov, Todor |
| contents | We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction the magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant which is the non-commutative torus of magnetic translation in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_11093 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bloch waves and Non-commutative Tori of Magnetic Translations Dereli, Tekin Popov, Todor Mathematical Physics Strongly Correlated Electrons We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction the magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant which is the non-commutative torus of magnetic translation in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. |
| title | Bloch waves and Non-commutative Tori of Magnetic Translations |
| topic | Mathematical Physics Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2106.11093 |