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Quantum states on spheres in the presence of magnetic fields
The study of quantum states on the surface of various two-dimensional geometries in the presence of strong magnetic fields has proven vital to the theoretical understanding of the quantum Hall effect. In particular, Haldane’s seminal study of quantum states on the surface of a compact geometry, the...
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| Format: | Thesis |
| Language: | English |
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Department of Maths and Applied Maths
2020
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| _version_ | 1867611325411098624 |
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| access_status_str | Open Access |
| author | Slayen, Ruach Pillay |
| author2 | Murugan, Jeffrey |
| author_browse | Murugan, Jeffrey Slayen, Ruach Pillay |
| author_facet | Murugan, Jeffrey Slayen, Ruach Pillay |
| author_sort | Slayen, Ruach Pillay |
| collection | Thesis |
| description | The study of quantum states on the surface of various two-dimensional geometries in the presence of strong magnetic fields has proven vital to the theoretical understanding of the quantum Hall effect. In particular, Haldane’s seminal study of quantum states on the surface of a compact geometry, the sphere, in the presence of a monopole magnetic field, was key to developing an early understanding of the fractional quantum Hall effect. Most of the numerous studies undertaken of similar systems since then have been limited to cases in which the magnetic fields are everywhere constant and perpendicular to the surface on which the charged particles are confined. In this thesis, we study two novel variations of Haldane’s spherical monopole system: the 'squashed sphere’ in the presence of a monopole-like magnetic field, and the sphere in the presence of a dipole magnetic field. In both cases the magnetic field is neither perpendicular nor constant with respect to the surface on which the charged particles are confined. Furthermore, the spherical dipole system has vanishing net magnetic flux. For the 'squashed sphere’ system we find the lowest Landau level single-particle Hilbert space, and it is shown that the effect of the squashing is to localise the particles around the equator. For the spherical dipole system we find the entire single-particle Hilbert space and energy spectrum. We show that in the strong-field limit the spectrum exhibits a Landau level structure, as in the spherical monopole case. Unlike in the spherical monopole case, each Landau level is shown to be infinitely degenerate. The emergence of this Landau level structure is explained by the tendency of a strong dipole field to localise particles at the poles of the sphere. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/31113 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Department of Maths and Applied Maths |
| publisherStr | Department of Maths and Applied Maths |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/31113 Quantum states on spheres in the presence of magnetic fields Slayen, Ruach Pillay Murugan, Jeffrey Shock, Jonathan Applied mathematics The study of quantum states on the surface of various two-dimensional geometries in the presence of strong magnetic fields has proven vital to the theoretical understanding of the quantum Hall effect. In particular, Haldane’s seminal study of quantum states on the surface of a compact geometry, the sphere, in the presence of a monopole magnetic field, was key to developing an early understanding of the fractional quantum Hall effect. Most of the numerous studies undertaken of similar systems since then have been limited to cases in which the magnetic fields are everywhere constant and perpendicular to the surface on which the charged particles are confined. In this thesis, we study two novel variations of Haldane’s spherical monopole system: the 'squashed sphere’ in the presence of a monopole-like magnetic field, and the sphere in the presence of a dipole magnetic field. In both cases the magnetic field is neither perpendicular nor constant with respect to the surface on which the charged particles are confined. Furthermore, the spherical dipole system has vanishing net magnetic flux. For the 'squashed sphere’ system we find the lowest Landau level single-particle Hilbert space, and it is shown that the effect of the squashing is to localise the particles around the equator. For the spherical dipole system we find the entire single-particle Hilbert space and energy spectrum. We show that in the strong-field limit the spectrum exhibits a Landau level structure, as in the spherical monopole case. Unlike in the spherical monopole case, each Landau level is shown to be infinitely degenerate. The emergence of this Landau level structure is explained by the tendency of a strong dipole field to localise particles at the poles of the sphere. 2020-02-14T08:21:58Z 2020-02-14T08:21:58Z 2019 2020-02-14T08:18:42Z Master Thesis Masters MSc http://hdl.handle.net/11427/31113 eng application/pdf Department of Maths and Applied Maths Faculty of Science |
| spellingShingle | Applied mathematics Slayen, Ruach Pillay Quantum states on spheres in the presence of magnetic fields |
| thesis_degree_str | Master's |
| title | Quantum states on spheres in the presence of magnetic fields |
| title_full | Quantum states on spheres in the presence of magnetic fields |
| title_fullStr | Quantum states on spheres in the presence of magnetic fields |
| title_full_unstemmed | Quantum states on spheres in the presence of magnetic fields |
| title_short | Quantum states on spheres in the presence of magnetic fields |
| title_sort | quantum states on spheres in the presence of magnetic fields |
| topic | Applied mathematics |
| url | http://hdl.handle.net/11427/31113 |
| work_keys_str_mv | AT slayenruachpillay quantumstatesonspheresinthepresenceofmagneticfields |