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A duality construction for interacting quantum Hall systems

dc.contributor.advisorScholtz, Frederik G.en_ZA
dc.contributor.advisorGeyer, Hendrik B.en_ZA
dc.contributor.authorKriel, Johannes Nicolaasen_ZA
dc.contributor.otherUniversity of Stellenbosch. Faculty of Science. Dept. of Physics.en_ZA
dc.date.accessioned2011-02-03T12:15:52Zen_ZA
dc.date.accessioned2011-03-14T08:37:38Z
dc.date.available2011-02-03T12:15:52Zen_ZA
dc.date.available2011-03-14T08:37:38Z
dc.date.issued2011-03en_ZA
dc.identifier.urihttp://hdl.handle.net/10019.1/6749
dc.descriptionThesis (PhD)--University of Stellenbosch, 2011.en_ZA
dc.description.abstractENGLISH 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.en_ZA
dc.description.abstractAFRIKAANSE 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.af_ZA
dc.format.extent95 p. : ill.
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : University of Stellenboschen_ZA
dc.subjectQuantum Hallen_ZA
dc.subjectDuality constructionen_ZA
dc.subjectDissertations -- Physicsen_ZA
dc.subjectTheses -- Physicsen_ZA
dc.subjectLandau levelsen_ZA
dc.titleA duality construction for interacting quantum Hall systemsen_ZA
dc.typeThesisen_ZA
dc.rights.holderUniversity of Stellenbosch


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