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A duality construction for interacting quantum Hall systems
Thesis (PhD)--University of Stellenbosch, 2011.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : University of Stellenbosch
2011
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| _version_ | 1867613912355045377 |
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| access_status_str | Open Access |
| author | Kriel, Johannes Nicolaas |
| author2 | Scholtz, Frederik G. |
| author_browse | Kriel, Johannes Nicolaas Scholtz, Frederik G. |
| author_facet | Scholtz, Frederik G. Kriel, Johannes Nicolaas |
| author_sort | Kriel, Johannes Nicolaas |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (PhD)--University of Stellenbosch, 2011. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/6749 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:43:40.919Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2011 |
| publishDateRange | 2011 |
| publishDateSort | 2011 |
| publisher | Stellenbosch : University of Stellenbosch |
| publisherStr | Stellenbosch : University of Stellenbosch |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/6749 A duality construction for interacting quantum Hall systems Kriel, Johannes Nicolaas Scholtz, Frederik G. Geyer, Hendrik B. University of Stellenbosch. Faculty of Science. Dept. of Physics. Quantum Hall Duality construction Dissertations -- Physics Theses -- Physics Landau levels Thesis (PhD)--University of Stellenbosch, 2011. ENGLISH ABSTRACT: The fractional quantum Hall effect represents a true many-body phenomenon in which the collective behaviour of interacting electrons plays a central role. In contrast to its integral counterpart, the appearance of a mobility gap in the fractional quantum Hall regime is due entirely to the Coulomb interaction and is not the result of a perturbed single particle gap. The bulk of our theoretical understanding of the underlying many-body problem is based on Laughlin’s ansatz wave function and the composite fermion picture proposed by Jain. In the latter the fractional quantum Hall effect of interacting electrons is formulated as the integral quantum Hall effect of weakly interacting quasiparticles called composite fermions. The composite fermion picture provides a qualitative description of the interacting system’s low-energy spectrum and leads to a generalisation of Laughlin’s wave functions for the electron ground state. These predictions have been verified through extensive numerical tests. In this work we present an alternative formulation of the composite fermion picture within a more rigorous mathematical framework. Our goal is to establish the relation between the strongly interacting electron problem and its dual description in terms of weakly interacting quasiparticles on the level of the microscopic Hamiltonian itself. This allows us to derive an analytic expression for the interaction induced excitation gap which agrees very well with existing numerical results. We also formulate a mapping between the states of the free particle and interacting descriptions in which the characteristic Jastrow-Slater structure of the composite fermion ansatz appears naturally. Our formalism also serves to clarify several aspects of the standard heuristic construction, particularly with regard to the emergence of the effective magnetic field and the role of higher Landau levels. We also resolve a long standing issue regarding the overlap of unprojected composite fermion trial wave functions with the lowest Landau level of the free particle Hamiltonian. AFRIKAANSE OPSOMMING: Die fraksionele kwantum Hall-effek is ’n veeldeeltjie verskynsel waarin die kollektiewe gedrag van wisselwerkende elektrone ’n sentrale rol speel. In teenstelling met die heeltallige kwantum Hall-effek is die ontstaan van ’n energie gaping in die fraksionele geval nie ’n enkeldeeltjie effek nie, maar kan uitsluitlik aan die Coulomb wisselwerking toegeskryf word. Die teoretiese raamwerk waarbinne hierdie veeldeeltjie probleem verstaan word is grootliks gebaseer op Laughlin se proefgolffunksie en die komposiete-fermion beeld van Jain. In laasgenoemde word die fraksionele kwantum Hall-effek van wisselwerkende elektrone geformuleer as die heeltallige kwantum Hall-effek van swak-wisselwerkende kwasi-deeljies wat as komposiete-fermione bekend staan. Hierdie beeld lewer ’n kwalitatiewe beskrywing van die wisselwerkende sisteem se lae-energie spektrum en lei tot ’n veralgemening van Laughlin se golffunksies vir die elektron grondtoestand. Hierdie voorspellings is deur verskeie numeriese studies geverifieer. In hierdie tesis ontwikkel ons ’n alternatiewe formulering van die komposiete-fermion beeld binne ’n strenger wiskundige raamwerk. Ons doel is om die verband tussen die sterk-wisselwerkende elektron sisteem en sy duale beskrywing in terme van swak-wisselwerkende kwasi-deeltjies op die vlak van die mikroskopiese Hamilton-operator self te realiseer. Hierdie konstruksie lei tot ’n analitiese uitdrukking vir die opwekkingsenergie wat baie goed met bestaande numeriese resultate ooreenstem. Ons identifiseer ook ’n afbeelding tussen die vrye-deeltjie en wisselwerkende toestande waarbinne die Jastrow-Slater struktuur van die komposiete-fermion proefgolffunksies op ’n natuurlike wyse na vore kom. Verder werp ons formalisme nuwe lig op kwessies binne die standaard heuristiese konstruksie, veral met betrekking tot die oorsprong van die effektiewe magneetveld en die rol van ho¨er effektiewe Landau vlakke. Ons lewer ook uitspraak oor die vraagstuk van die oorvleueling van ongeprojekteerde komposiete-fermion golffunksies met die laagste Landau vlak van die vrye-deeltjie Landau probleem. Doctoral 2011-02-03T12:15:52Z 2011-03-14T08:37:38Z 2011-02-03T12:15:52Z 2011-03-14T08:37:38Z 2011-03 Thesis http://hdl.handle.net/10019.1/6749 en_ZA University of Stellenbosch 95 p. : ill. application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Quantum Hall Duality construction Dissertations -- Physics Theses -- Physics Landau levels Kriel, Johannes Nicolaas A duality construction for interacting quantum Hall systems |
| title | A duality construction for interacting quantum Hall systems |
| title_full | A duality construction for interacting quantum Hall systems |
| title_fullStr | A duality construction for interacting quantum Hall systems |
| title_full_unstemmed | A duality construction for interacting quantum Hall systems |
| title_short | A duality construction for interacting quantum Hall systems |
| title_sort | duality construction for interacting quantum hall systems |
| topic | Quantum Hall Duality construction Dissertations -- Physics Theses -- Physics Landau levels |
| url | http://hdl.handle.net/10019.1/6749 |
| work_keys_str_mv | AT krieljohannesnicolaas adualityconstructionforinteractingquantumhallsystems AT krieljohannesnicolaas dualityconstructionforinteractingquantumhallsystems |