O presente trabalho apresenta um anova proposta de tratamento de estruturas espirais em meios contínuos oscilatórios na vizinhança de bifurcações de Hopf supercríticas. Tais estruturas são normalmente descritas pela Equação de Cinzburg-Landau Complexa a qual usa um campo complexo associado a essas oscilações. A proposta apresentada reduz a dinâmica de espirais à interação entre os centros das mesmas. Inicialmente, comparamos numericamente as duas descrições e com os ganhos computacionais decorrentes da abordagem reduzida caracterizamos finamente as estruturas espaço-temporais formadas nesses sistemas: em vez dos estados congelados mencionados anteriormente na literatura encontrou-se uma dinâmica espaço-temporal intermitente. Esse regime ocorre em duas fases distintas: Líquido de Vórtices e Vidros de Vórtices. Esta última evolui em escalas de tempo ultralentas como fenômenos semelhantes encontrados na Mecânica Estatística, apesar de sua origem puramente determinista.

The Landau-Ginzburg (LG) B-Model is a significant feature of singularity theory and mirror symmetry. Krawitz in 2010, guided by work of Kaufmann, provided an explicit construction for the LG B-model when using diagonal symmetries of a quasihomogeneous, nondegenerate polynomial. In this thesis we discuss aspects of how to generalize the LG B-model construction to allow for nondiagonal symmetries of a polynomial, and hence nonabelian symmetry groups. The construction is generalized to the level of graded vector space and the multiplication developed up to an unknown factor. We present complete examples of nonabelian LG B-models for the polynomials x^2y + y^3, x^3y + y^4, and x^3 + y^3 + z^3 + w^2.

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Le sujet de ma thèse recouvre deux aspects. En premier lieu, l'objectif était d'étudier et d'améliorer les méthodes de calcul à une boucle pour les corrections radiatives dans le cadre des théories de champs perturbatives. En second lieu, l'objectif était d'appliquer ces techniques pour calculer les effets dominants des corrections radiatives electrofaibles au processus important de production de Higgs associé à deux quarks bottom au LHC (Large Hadron Collider) du CERN. L'étude concerne le Higgs du Modèle Standard. <br /><br />Le premier objectif est d'importance plutôt théorique. Bien que la méthode générale pour le calcul à une boucle des corrections radiatives dans le modèle standard soit, en principe, bien compris par le biais de la renormalisation, il y a un certain nombre de difficultés techniques. Celles-ci sont liées aux intégrales de boucle. Pour les processus avec plus de 4 particules externes, les expressions d'amplitude obtenues en utilisant la méthode d'analyse sont <br />extrêmement lourdes et très difficiles à manipuler. Dans cette thèse, nous avons étudié ce problème et réalisé que tout le calcul peut être facilement optimisé si l'on utilise la méthode des amplitudes d'hélicité. Un autre problème est lié aux propriétés analytiques des intégrales scalaires. Une partie importante de cette thèse est consacrée à ce problème et à l'étude des équations de Landau. Nous avons trouvé des effets significatifs en raison de singularités de Landau dans le processus de production de Higgs associé à deux quarks bottom au LHC.<br /><br />Le deuxième objectif est d'ordre pratique avec des conséquences expérimentales. On peut résumer les résultats obtenus dans cette thèse comme suit. Si la masse du Higgs est d'environ 120GeV, la correction au premier ordre dominant est petite de l'ordre d'environ -4%. Si la masse de Higgs est d'environ 160GeV, seuil de production d'une paire de W par le Higgs, les corrections électrofaibles bénéficient du couplage fort du Yukawa du top et sont amplifiées par la singularité de Landau conduisant à une importante correction d'environ 50%.

<p>Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting.</p><p>Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality.</p><p>Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.</p>

Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting. Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality. Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.

Dans ce manuscrit de thèse, nous nous intéressons au couplage entre le champ électromagnétique quantifié au sein d'un résonateur optique et la transition cyclotron d'un gaz d'électrons bidimensionnel soumis à un champ magnétique perpendiculaire. Nous montrons que ce système peut atteindre un régime de couplage ultrafort inédit, dans lequel la fréquence de Rabi du vide (quantifiant l'intensité de l'interaction lumière-matière) devient comparable ou plus grande que la fréquence de la transition cyclotron pour des facteurs de remplissage suffisamment élevés. Nos prédictions théoriques ont alors donné lieu à une vérification expérimentale spectaculaire. En outre, nous avons généralisé la théorie au cas du graphène dont les excitations de basse énergie sont convenablement décrites par un hamiltonien de Dirac sans masse. Nous montrons que si le couplage ultrafort peut également être atteint dans ce cas, des différences qualitatives importantes apparaissent par rapport au cas des fermions massifs du semiconducteur.

The thesis considers the Ginzburg-Landau process on the lattice $\Z^d$ whose potential is a bounded perturbation of the Gaussian potential. For such processes the thesis establishes the decay rate to equilibrium in the variance sense is $C_g t^{-d/2} + o\left(t^{-d/2}\right)$, for any local function $g$ that is bounded, mean zero, and having finite triple norm; $\triplenorm{g}=\sum_{x \in \Z^d} \norm{\partial_{\eta(x)}g}_\infty.$ The constant $C_g$ is computed explicitly. This extends the decay to equilibrium result of Janvresse, Landim, Quastel, and Yau [JLQY99] for zero-range process, and the related result of Landim and Yau [LY03] for Ginzburg-Landau processes. The thesis also considers additive functionals $\int_{0}^{t} g(\eta_s) ds$ of Ginzburg-Landau processes, where $g$ is a bounded, mean zero, local function having finite triple norm. A central limit is proven for $a^{-1}(t)\int_{0}^{t} g(\eta_s) ds$ with $a(t)= \sqrt{t}$ in $d \ge 3$, $a(t)=\sqrt{t \log{t}}$ in $d=2$, and $a(t)= t^{3/4}$ in $d=1$ and an explicit form of the asymptotic variance in each case. Corresponding invariance principles are also obtained. Standard arguments of Kipnis and Varadhan [KV86] are employed in the case $d \ge 3$. Martingale methods together with $L^2$ decay estimates for the semigroup associated with the process are employed to establish the result in the cases $d=1$ and $d=2$. This extends similar results for noninteracting random walks (see[CG84]), the symmetric simple exclusion processes (see [Kip87]), and the zero-range process (see [QJS02]).

The thesis considers the Ginzburg-Landau process on the lattice $\Z^d$ whose potential is a bounded perturbation of the Gaussian potential. For such processes the thesis establishes the decay rate to equilibrium in the variance sense is $C_g t^{-d/2} + o\left(t^{-d/2}\right)$, for any local function $g$ that is bounded, mean zero, and having finite triple norm; $\triplenorm{g}=\sum_{x \in \Z^d} \norm{\partial_{\eta(x)}g}_\infty.$ The constant $C_g$ is computed explicitly. This extends the decay to equilibrium result of Janvresse, Landim, Quastel, and Yau [JLQY99] for zero-range process, and the related result of Landim and Yau [LY03] for Ginzburg-Landau processes. The thesis also considers additive functionals $\int_{0}^{t} g(\eta_s) ds$ of Ginzburg-Landau processes, where $g$ is a bounded, mean zero, local function having finite triple norm. A central limit is proven for $a^{-1}(t)\int_{0}^{t} g(\eta_s) ds$ with $a(t)= \sqrt{t}$ in $d \ge 3$, $a(t)=\sqrt{t \log{t}}$ in $d=2$, and $a(t)= t^{3/4}$ in $d=1$ and an explicit form of the asymptotic variance in each case. Corresponding invariance principles are also obtained. Standard arguments of Kipnis and Varadhan [KV86] are employed in the case $d \ge 3$. Martingale methods together with $L^2$ decay estimates for the semigroup associated with the process are employed to establish the result in the cases $d=1$ and $d=2$. This extends similar results for noninteracting random walks (see[CG84]), the symmetric simple exclusion processes (see [Kip87]), and the zero-range process (see [QJS02]).

Dans ma thèse, j'ai travaillé sur les modèles de fils en ferromagnétisme. J'ai obtenu les résultats suivants :- Existence de solutions très régulières pour les équations de Landau-Lifschitz en dimension 3.- Stabilité de profils de murs avec critère optimal de stabilité pour un fil soumis à un champ magnétique.- Stabilité de profils de murs pour un fil soumis à un courant électrique, dans le cas d'un fil à section circulaire et dans le cas d'un fil à section ellipsoïdale. - Justification des modèles monodimensionnels de fils.