Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Thesis advisor: Ziqiang Wang / Graphene has emerged as an important two dimensional electron system with novel physical properties due to its relativistic-like linear energy-momentum dispersion relation at low energy. Alongside two dimensional electron systems in semiconductor heterostructures, it has a rich set of integer and fractional quantum Hall states. Significant progresses have been made recently, but a full understanding of these states is still lacking. The prevailing approach for fractional quantum Hall effects in graphene has been the numerical exact diagonalization. In this work, we develop a fermionic Chern-Simons effective theory for Dirac fermions as a complement to the existing theories, and to bring new insights in our understanding of the phenomena. In particular, we study the possibility for quantum Hall plateaus at even-denominator filling factors. We first construct a unitary Chern-Simons transformation to attach even number of flux quanta to Dirac fermions. To deal with the four-fold spin-valley degeneracy, a set of K-matrices is introduced. At even-denominator filling factors in the zeroth Landau level, the fictitious magnetic field of the Chern-Simons field cancels the external magnetic field on average. It is shown that the Chern-Simons field mediates an effective mutual statistical interaction between composite Dirac fermions. We further show the statistical interaction and Coulomb interaction favor the formation of an exciton condensate. Quasi-particles at finite filling factors can be regarded as excita- tions above the exciton condensate, and can be described as massive Dirac fermions. This means a mass is generated dynamically for Dirac fermions. Different types of K-matrices give rise to different mass gaps. The Chern numbers associated with different massive Dirac band structures can be used to classify the K-matrices. In the last part of the thesis, we study the pairing instability of the composite Dirac fermion liquid. We show the statistical interaction drives a complex p-wave pairing among the quasi-particles. As long as the Coulomb pair breaking effect is weak, the system can develop a superconducting energy gap, thus form a fractional quantum Hall state. / Thesis (PhD) — Boston College, 2012. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Efeitos térmicos em teorias de calibre em (2+1) dimensões são estudados em espaços onde as coordenadas podem ou não comutar. No caso comutativo, a dependência com a temperatura do tensor de polarização é calculada a um laço em teorias envolvendo tanto bósons quanto férmions. Como aplicação, são calculados os processos de blindagem em tais modelos, chegando ao interessante resultado de que cargas magnéticas não sofrem tais efeitos na QED3. Uma prova válida em qualquer ordem de perturbação é desenvolvida, confirmando este comportamento. Em teorias não comutativas, são estudadas as correções a um laço ao coeficiente de Chern-Simons, sendo encontrado que não existe o fenômeno da mistura UV/IR na teoria Chern-Simons-Higgs. O comportamento assintótico de tal coeficiente é analisado no regime de altas temperaturas. Vários outros aspectos envolvendo os efeitos térmicos em teorias de Chern-Simons são explorados. / Thermal effects in (2+1)-dimensional gauge theories are studied in both commutative as well as noncommutative manifolds. In the first situation, the finite temperature polarization tensor is computed a tone loop for fermionic and bosonic couplings. As an application, the screening masses are evaluated and it is found the surprising result that magnetic charges are not screened in QED3. It is demonstrated that this result holds to any order in pertubationtheory. In the noncommutative case, the one loop correction to the Chern-Simons coefficient is studied, and it is found that there is no UV/IR mixing in the Chern-Simons-Higgs model. The asymptotic behavior of such coefficient is analised in the high temperature regime. Several other interesting aspects involving thermal effects of Chern-Simons theories are also discussed.

Efeitos térmicos em teorias de calibre em (2+1) dimensões são estudados em espaços onde as coordenadas podem ou não comutar. No caso comutativo, a dependência com a temperatura do tensor de polarização é calculada a um laço em teorias envolvendo tanto bósons quanto férmions. Como aplicação, são calculados os processos de blindagem em tais modelos, chegando ao interessante resultado de que cargas magnéticas não sofrem tais efeitos na QED3. Uma prova válida em qualquer ordem de perturbação é desenvolvida, confirmando este comportamento. Em teorias não comutativas, são estudadas as correções a um laço ao coeficiente de Chern-Simons, sendo encontrado que não existe o fenômeno da mistura UV/IR na teoria Chern-Simons-Higgs. O comportamento assintótico de tal coeficiente é analisado no regime de altas temperaturas. Vários outros aspectos envolvendo os efeitos térmicos em teorias de Chern-Simons são explorados. / Thermal effects in (2+1)-dimensional gauge theories are studied in both commutative as well as noncommutative manifolds. In the first situation, the finite temperature polarization tensor is computed a tone loop for fermionic and bosonic couplings. As an application, the screening masses are evaluated and it is found the surprising result that magnetic charges are not screened in QED3. It is demonstrated that this result holds to any order in pertubationtheory. In the noncommutative case, the one loop correction to the Chern-Simons coefficient is studied, and it is found that there is no UV/IR mixing in the Chern-Simons-Higgs model. The asymptotic behavior of such coefficient is analised in the high temperature regime. Several other interesting aspects involving thermal effects of Chern-Simons theories are also discussed.

In this dissertation we study several topics related to the geometry and physics of heterotic string compactification. After an introduction to some of the basic ideas of this field, we review the heterotic line bundle standard model construction and a complex structure mod- uli stabilization mechanism associated to certain hidden sector gauge bundles. Once this foundational material has been presented, we move on to the original research of this disser- tation. We present a scan over all known heterotic line bundle standard models to examine the frequency with which the particle spectrum is forced to change, or "jump," by the hidden sector moduli stabilization mechanism just mentioned. We find a significant percentage of forced spectrum jumping in those models where such a change of particle content is possible. This result suggests that one should consider moduli stabilization concurrently with model building, and that failing to do so could lead to misleading results. We also use state of the art techniques to study Yukawa couplings in these models. We find that a large portion of Yukawa couplings which naively would be expected to be non-zero actually vanish due to certain topological selection rules. There is no known symmetry which is responsible for this vanishing. In the final part of this dissertation, we study the Chern-Simons contribution to the superpotential of heterotic theories. This quantity is very important in determining the vacuum stability of these models. By explicitly building real bundle morphisms between vec- tor bundles over Calabi-Yau manifolds, we show that this contribution to the superpotential vanishes in many cases. However, by working with more complicated, and realistic geome- tries, we also present examples where the Chern-Simons contribution to the superpotential is non-zero, and indeed fractional. / Doctor of Philosophy / String theory is a candidate for a unified theory of all of the known interactions of nature. To be consistent, the theory needs to be formulated in 9 spatial dimensions, rather than the 3 of everyday experience. To connect string theory with reality, we need to reproduce the known physics of 3 dimensions from the 9 dimensional theory by hiding, or "compactifying," 6 directions on a compact internal space. The most common choice for such an internal space is called a Calabi-Yau manifold. In this dissertation, we study how the geometry of the Calabi-Yau manifold determines physical quantities seen in 3 dimensions such as the number of particle families, particle interactions and potential energy. The first project in this dissertation studies to what extent the process of making the Calabi-Yau manifold rigid, something which is required observationally, affects the particle spectrum seen in 3 dimensions. By scanning over a large model set, we conclude that computation of the particle spectrum and such "moduli stabilization" issues should be considered in concert, and not in isolation. We also showed that a large portion of the interactions that one would naively expect between the particles in such string models are actually absent. There is no known symmetry of the theory that accounts for this structure, which is linked to the topology of the extra spatial dimensions. In the final part of the dissertation, we show how to calculate previously unknown contributions to the potential energy of these string theory models. By linking to results from the mathematics literature, we show that these contributions vanish in many cases. However, we present examples where it is non-zero, a fact of crucial importance in understanding the vacua of heterotic string theories.

Estudamos as teorias não-relativísticas e não-comutativas de campos de spin zero e l/2 acoplado minimamente com o campo ele Chern-Simons em 2+ 1 dimensões. Na situação comutativa o modelo escalar foi usado para simular o efeito Aharonov-Bohrn na abordagem da teoria de campos. Na teoria escalar verificamos que, contrariamente ao resultado comutativo, a inclusão ele urna auto--interação quártica do campo escalar não ó necessária para garantir a renormalização ultravioleta do modelo. Entretanto, para obter um limite comutativo analítico a presença ele uma auto-interação quártica é exigida. Mostramos para o caso ele partículas ele spin 1/2 que a contribuição em um laço para a matriz ele espalhamento contendo o termo de Pauli é puramente não--planar. O termo de Pauli desempenha a mesma função ela auto-interação quártica como no caso escalar. Para valores pequenos do parâmetro da não--comutatividade determinamos as correções para o espalhamento Aharonov-Bohm e provamos que, até ordem de um laço, os modelos são livres de singularidades ultravioleta/infravermelha. / We study noncommutative nonrelativistic theories of spin 0 and 1/2 field coupled to thc Chern-Sirnons field in 2+1 dimensions. In the commutative situation the scalar model has been used to simulate the Aharonov-Bohm effect in the field theory context. We verified that, contrarily to the commutative result, the inclusion of a quartic self-interaction of the scalar field is not necessary to secure the ultraviolet renormalization of the model. However, to obtain a smooth commutative limit the presence of a quartic gauge invariant self-interaction is required. For the case of spin 1/2 particles we show that the one-loop contributions to the that scattering matrix the which contain the Pauli\'s term are purely nonplanar. Thc Pauli\'s term plays the same role of a quartic self-interaction in the scalar case. For small values of the noncommutative parameter we fix the corrections to the Aharonov-Bohm scattering and prove that up to one-loop the models are free from dangerous infrared/ultraviolet divergences.

Estudamos as teorias não-relativísticas e não-comutativas de campos de spin zero e l/2 acoplado minimamente com o campo ele Chern-Simons em 2+ 1 dimensões. Na situação comutativa o modelo escalar foi usado para simular o efeito Aharonov-Bohrn na abordagem da teoria de campos. Na teoria escalar verificamos que, contrariamente ao resultado comutativo, a inclusão ele urna auto--interação quártica do campo escalar não ó necessária para garantir a renormalização ultravioleta do modelo. Entretanto, para obter um limite comutativo analítico a presença ele uma auto-interação quártica é exigida. Mostramos para o caso ele partículas ele spin 1/2 que a contribuição em um laço para a matriz ele espalhamento contendo o termo de Pauli é puramente não--planar. O termo de Pauli desempenha a mesma função ela auto-interação quártica como no caso escalar. Para valores pequenos do parâmetro da não--comutatividade determinamos as correções para o espalhamento Aharonov-Bohm e provamos que, até ordem de um laço, os modelos são livres de singularidades ultravioleta/infravermelha. / We study noncommutative nonrelativistic theories of spin 0 and 1/2 field coupled to thc Chern-Sirnons field in 2+1 dimensions. In the commutative situation the scalar model has been used to simulate the Aharonov-Bohm effect in the field theory context. We verified that, contrarily to the commutative result, the inclusion of a quartic self-interaction of the scalar field is not necessary to secure the ultraviolet renormalization of the model. However, to obtain a smooth commutative limit the presence of a quartic gauge invariant self-interaction is required. For the case of spin 1/2 particles we show that the one-loop contributions to the that scattering matrix the which contain the Pauli\'s term are purely nonplanar. Thc Pauli\'s term plays the same role of a quartic self-interaction in the scalar case. For small values of the noncommutative parameter we fix the corrections to the Aharonov-Bohm scattering and prove that up to one-loop the models are free from dangerous infrared/ultraviolet divergences.

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Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

O objetivo desta tese é estudar alguns modelos contendo o termo de Chern-Simons, cuja presença induz várias características tais como massa topológica e spin fracionário. Iniciamos tratando o problema da equivalência entre os modelos Auto-Dual minimamente acoplado com o campo de Dirac, e o modelo de Maxwell-Chern-Simons com um acoplamento magnético com férmions. Nós mostramos que os setores fermiônicos dos modelos somente são equivalentes se adicionarmos uma auto-interação tipo Thirring. Em seguida, analisamos o efeito Aharonov-Bohm não-Abeliano para partículas de spin 0 e ½. Partindo da teoria quântica de campos relativística no gauge de Coulomb e usando o procedimento do corte intermediário, para separar o momento de integração nas regiões de baixa e alta energia, nós obtemos o limite não relativístico dos modelos. Desta forma as correções radiativas, tais como polarização do vácuo e correção de vértice, são automaticamente incluídas. Finalizando estudamos algumas / The aim of this work is to study some models with Chern-Simons term~ which induces several properties such as l-opological mass and fractional spin. We start studying the exact equivalence between the Self-Dual model minimally coupled to a Dirac field and the Maxwell-Chern-Simons model witb non-minimal magnetic coupling to fermions. Weshow that the fermions sectors of the models are equivalent only if a Thirring like interaction is included. Next we analyze the non-Abelian Aharonov-Bohm scattering for 0 and 1/2 spin particles. Starting Crom the relatistic quantum field theory and using 8 Coulomb gauge rormalism the one loop renorrnalizalion is implemented. Trough the introduction of na intermediaf) cutoff, separating the regions of high and low integration momentum the nonrelativistic Umit is derived. In this approach CJuant.um field theory vacuum polarization and anomalous magnetic momentum are automatically incorporated. Va also study some properties of the chat\'ged one spin particles interacting wit.h Chern Simons field.