The XX century witnessed the quantum and relativistic revolutions in physics. The development of these two theories, namely, Quantum Mechanics and Relativity, was the inception of many crucial discoveries and technological advances. Among them, one stands out due to its uniqueness, the neutrino discovery. However, several neutrino properties are still obscure. Neutrinos are the only fundamental particles whose nature is currently unknown. Such fermions can either be different from their antiparticles, i.e., Dirac fermions, or be their own antiparticles, that is, Majorana fermions. On the other hand, the smallness of neutrino masses is a problem seemingly related to the neutrino nature; thus, as essential task consists in addressing the phenomenologically viable models in both cases. Furthermore, it is important to search for other physical process in which the neutrino nature may manifest through different experimental signatures. A rather difficult but promising method corresponds to the detection of the cosmic neutrino background, viz. neutrinos which are relics from the Big Bang. Previous works have shown that detection rates for Dirac and Majorana neutrinos can give different results. Nevertheless, this distinction was obtained considering the Standard Model framework only. Therefore, it is important to understand the consequences of having Non-Standard Interactions contributing to the detection of neutrinos from the cosmic background. Another remarkable relic predicted by Cosmology is the unidentified Dark Matter, composing ~25% of the Universe. All searches regarding the Weakly Interacting Massive Particle, one of the principal candidates for Dark Matter, have given negative results; this has compelled experiments to increase their sensitivity. Notwithstanding, neutrinos may stand in the way of such experimental searches given that they may constitute an irreducible background. In this thesis, we will address these three different phenomena, neutrino mass models, detection of the cosmic neutrino background and the neutrino background in Dark Matter searches, by considering the different characteristics in each case. In the study of neutrino mass models, we will consider models for both Majorana and Dirac neutrinos; specifically, we will probe the neutrinophilic two-Higgs-doublet model. Regarding the detection of relic neutrinos, we will analyse the consequences of the existence of the beyond Standard Model physics in the capture rate by tritium. Finally, we will scrutinize the impact of neutrinos in Direct Detection WIMP searches, by considering Standard Model plus additional interactions in the form of simplified models. / Ao longo do século XX testemunhamos as revoluções quântica e relativista que aconteceram na Física. O desenvolvimento da Mecânica quântica e da teoria da relatividade foi o prelúdio de inúmeras descobertas e avanços tecnológicos fundamentais; em particular, a descoberta dos neutrinos. No entanto, a sua total compreensão ainda é um mistério para a física de partículas. Entendidos como partículas fermiônicas fundamentais, os neutrinos possuem sua natureza desconhecida. Podendo ser diferentes de suas antipartículas, denominadas férmions de Dirac, ou também podendo ser as suas próprias antipartícula, sendo conhecidas como férmions de Majorana. Por outro lado, o valor de sua massa continua sendo um problema em aberto, supostamente relacionado à sua natureza. Portanto, é importante estudarmos modelos fenomenológicos viáveis para as duas naturezas possíves dos neutrinos. Além disso, é necessário procurar outros processos físicos cujos resultados experimentais sejam distintos de acordo com a natureza do neutrino. Um método bastante difícil, mas promissor, corresponde à detecção do fundo de neutrinos cósmicos, isto é, os neutrinos relíquia do Big Bang. Análises prévias mostraram que as taxas de detecção para neutrinos de Dirac e de Majorana resultam em valores distintos. Porém, este resultado foi obtido supondo como base o Modelo Padrão; assim, é crucial entender as possíveis consequências da existência de interações desconhecidas na detecção dos neutrinos da radiação cósmica de fundo. Outra relíquia notável prevista pela Cosmologia é a desconhecida Matéria Escura, que compõe ~25% do Universo. Todas as buscas por WIMPs (do inglês Weakly Interactive Massive Particles), um dos principais candidatos a Matéria Escura, tem dado resultados negativos. Isto tem forçado a criação de experimentos cada vez mais sensíveis. Contudo, os neutrinos poderão ser um obstáculo nessas buscas experimentais, pois estes convertir-se-ão em um fundo irredutível. Na presente tese, abordaremos estes três fenômenos diferentes, modelos de massa para os neutrinos, a detecção do fundo de neutrinos cósmicos e o fundo de neutrinos em experimentos de detecção direta de Matéria Escura, considerando as distintas características em cada caso. No estudo dos modelos de massa para os neutrinos consideraremos modelos para neutrinos de Majorana e Dirac; exploraremos modelos neutrinofílicos com dois dubletos de Higgs. Enquanto à detecção dos neutrinos relíquia, analisaremos as consequências da presença de física além do Modelo Padrão na taxa de captura pelo trítio. Finalmente, examinaremos o impacto dos neutrinos em experimentos de detecção direta de WIMPs, supondo as interações do Modelo Padrão junto com interações adicionais na forma de modelos simplificados.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-27T11:35:41Z No. of bitstreams: 1 cleberabrahaodesouza.pdf: 307341 bytes, checksum: f90c47044c5f5aabc5bbb77378b1cd58 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T20:52:19Z (GMT) No. of bitstreams: 1 cleberabrahaodesouza.pdf: 307341 bytes, checksum: f90c47044c5f5aabc5bbb77378b1cd58 (MD5) / Made available in DSpace on 2017-08-07T20:52:19Z (GMT). No. of bitstreams: 1 cleberabrahaodesouza.pdf: 307341 bytes, checksum: f90c47044c5f5aabc5bbb77378b1cd58 (MD5) Previous issue date: 2011-02-28 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A teoria relativística da gravitação é uma ferramenta fundamental para a compreensão da cosmologia moderna. Ela descreve a dinâmica do campo gravitacional ao longo do espaço-tempo. Dentre os possíveis modelos de universo, um dos mais comuns é o descrito pela métrica de Friedmann-Lemaitre-Robertson-Walker (FLRW). Esta métrica pode ser escrita em termos do tempo conforme e descreve um universo homogêneo e isotrópico. Neste trabalho investigamos os campos de Dirac não massivos neste modelo de universo. Os resultados mostram uma compatibilidade entre as equações de Dirac e a métrica de FLRW. Os efeitos do parâmetro de Immirzi junto com os férmions podem ser descritos em termos de um fluido perfeito usual intermediário entre a poeira e radiação. / The general relativity theory is fundamental tool for understanding the modern cosmology. In this work we investigated the massless Dirac fields in the Friedamm-Lamaitre-Robertson-Walker universe. The metric is written in terms of conformal time which describes a homogeneous and isotropic universe. We show that the effects of Immirzi parameter along with the fermions can be described in terms of a usual perfect fluid intermediate between dust and radiation.

Ce travail consiste en l'étude sous champ magnétique des propriétés électroniques des fermions de Dirac relativistes dans deux systèmes: graphène et isolants topologiques. Leur analogie avec la physique des hautes énergies et leurs applications potentielles ont suscité récemment de nombreux travaux. Les états électroniques sont donnés par un Hamiltonien de Dirac et la dispersion est analogue à celle des particules relativistes. La masse au repos est liée au gap du matériau avec une vitesse de Fermi remplaçant la vitesse de la lumière. Le graphène a été considéré comme un " système école " qui nous permet d'étudier le comportement relativiste des fermions de Dirac sans masse satisfaisant une dispersion linéaire. Quand un système de Dirac possède un gap non nul, nous avons des fermions de Dirac massifs. Les fermions de Dirac sans masse et massifs ont été étudiés dans le graphène épitaxié et les isolants topologiques cristallins Pb1-xSnxSe et Pb1-xSnxTe. Ces derniers systèmes sont une nouvelle classe de matériaux topologiques où les états de bulk sont isolants mais les états de surface sont conducteurs. Cet aspect particulier résulte de l'inversion des bandes de conduction et de valence du bulk ayant des parités différentes, conduisant à une transition de phase topologique. La magnéto-spectroscopie infrarouge est une technique idéale pour sonder ces matériaux de petit gap car elle fournit des informations quantitatives sur les paramètres du bulk via la quantification de Landau des états électroniques. En particulier, la transition de phase topologique est caractérisée par une mesure directe de l'indice topologique. / This thesis reports on the study under magnetic field of the electronic properties of relativistic-like Dirac fermions in two Dirac systems: graphene and topological insulators. Their analogies with high-energy physics and their potential applications have attracted great attention for fundamental research in condensed matter physics. The carriers in these two materials obey a Dirac Hamiltonian and the energy dispersion is analogous to that of the relativistic particles. The particle rest mass is related to the band gap of the Dirac material, with the Fermi velocity replacing the speed of light. Graphene has been considered as a “role model”, among quantum solids, that allows us to study the relativistic behavior of massless Dirac fermions satisfying a linear dispersion. When a Dirac system possesses a nonzero gap, we have massive Dirac fermions. Massless and massive Dirac fermions were studied in high-mobility multilayer epitaxial graphene and in topological crystalline insulators Pb1-xSnxSe and Pb1-xSnxTe. The latter system is a new class of topological materials where the bulk states are insulating but the surface states are conducting. This particular aspect results from the inversion of the lowest conduction and highest valence bulk bands having different parities, leading to a topological phase transition. Infrared magneto-spectroscopy is an ideal technique to probe these zero-gap or narrow gap materials since it provides quantitative information about the bulk parameters via the Landau quantization of the electron states. In particular, the topological phase transition can be characterized by a direct measurement of the topological index.

Inspired by graphene, I investigate the properties of several different real and artificial Dirac materials. Firstly, I consider a two-dimensional honeycomb lattice of metallic nanoparticles, each supporting localised surface plasmons, and study the quantum properties of the collective plasmons resulting from the near field dipolar interaction between the nanoparticles. I analytically investigate the dispersion, the effective Hamiltonian and the eigenstates of the collective plasmons for an arbitrary orientation of the individual dipole moments. When the polarisation points close to normal to the plane the spectrum presents Dirac cones, similar to those present in the electronic band structure of graphene. I derive the effective Dirac Hamiltonian for the collective plasmons and show that the corresponding spinor eigenstates represent chiral Dirac-like massless bosonic excitations that present similar effects to those of electrons in graphene, such as a non-trivial Berry phase and the absence of backscattering from smooth inhomogeneities. I further discuss how one can manipulate the Dirac points in the Brillouin zone and open a gap in the collective plasmon dispersion by modifying the polarisation of the localized surface plasmons, paving the way for a fully tunable plasmonic analogue of graphene. I present a phase diagram of gapless and gapped phases in the collective plasmon dispersion depending on the dipole orientation. When the inversion symmetry of the honeycomb structure is broken, the collective plasmons become gapped chiral Dirac modes with an energy-dependent Berry phase. I show that this concept can be generalised to describe many real and artificial graphene-like systems, labeling them Dirac materials with a linear gapped spectrum. I also show that biased bilayer graphene is another Dirac material with an energy dependent Berry phase, but with a parabolic gapped spectrum. I analyse the relativistic phenomenon of Klein Tunneling in both types of system. The Klein paradox is one of the most counter-intuitive results from quantum electrodynamics but it has been seen experimentally to occur in both monolayer and bilayer graphene, due to the chiral nature of the Dirac quasiparticles in these materials. The non-trivial Berry phase of pi in monolayer graphene leads to remarkable effects in transmission through potential barriers, whereas there is always zero transmission at normal incidence in unbiased bilayer graphene in the npn regime. These, and many other 2D materials have attracted attention due to their possible usefulness for the next generation of nano-electronic devices, but some of their Klein tunneling results may be a hindrance to this application. I will highlight how breaking the inversion symmetry of the system allows for results that are not possible in these system's inversion symmetrical counterparts.

Ce mémoire présente les travaux réalisés durant ces trois années de thèse. Ils se sont orientés autour de l'étude des propriétés électroniques et magnétiques des matériaux pour la spintronique. Ce domaine d'avenir doit encore trouver les matériaux adaptés qui permettraient de réaliser les concepts liés à la spintronique. Nos résultats sont les suivants. Nous avons proposé un nouveau modèle qui décrit l'arrangement magnétique dans les couches minces ferromagnétiques possédant une anisotropie uniaxiale. Le modèle apporte une meilleure description des largeurs de domaine en fonction de l'épaisseur ainsi qu'une bonne estimation de l'épaisseur critique à partir de laquelle les domaines de Weiss ne sont plus stables. L'ensemble d'équations en unité réduite découlant du modèle apporte quant à lui un outil supplémentaire aux expérimentateurs. L'étude ab-Initio fait sur les semi-Conducteurs magnétiques dilués à mis en évidence l'importance de la corrélation forte et de l'effet Jahn-Teller dans les matériaux tel que le (Ga,Mn)N et le (Zn,Cr)S. Ces calculs confirment l'ensemble des données expérimentales existantes sur le (Ga,Mn)N. La modélisation analytique apporte un complément aux calculs ab-Initio en faisant le lien entre les paramètres expérimentaux et la théorie de Vallin, très largement utilisée pour interpréter les mesures optiques du (Zn,Cr)S. Nos calculs ab-Initio ont montré que le métacinabre est un isolant topologique robuste, qui se distingue par la présence d'un cône de Dirac extrêmement anisotrope. Les effets de passivation à l'hydrogène influencent les états de surface des isolants topologiques de la série HgX (X : S, Se, Te). / This thesis contains the scientific work of three years. The main topic can be defined as follows : study of electronic and magnetic properties of materials for spintronics. That technology of the future has still to find the necessary materials to realize new concepts. Our results are the following : We propose a new model to describe the magnetic configuration in thin ferromagnetic film with uniaxial anisotropy. It gives a better description of domain widths in function of film thickness and permits to obtain a good evaluation of the critical thickness where domains of Weiss type are no longer stable. The set of equations in reduced units is a useful tool to analyze experimental data. The ab-Initio study of diluted magnetic semiconductors has demonstrated the combined effect of strong correlations and Jahn-Teller distortion in (Ga,Mn)N and (Zn,Cr)S. The calculations confirm the experimental results of (Ga,Mn)N. We develop an analytical model that is complementary to the ab-Initio calculations and permits to create a link between several experimental parameters, the ab-Initio calculations and the former theory of Vallin. It has been used to interpret the optical measurements of (Zn,Cr)S. Our ab-Initio studies show that metacinnabar is a strong topological insulator with one peculiarity, it has a highly anisotropic Dirac cone. The passivation with hydrogen atoms influences the surface states of the topological insulator in the series HgX (X : S, Se, Te). When the dangling bonds are saturated by hydrogen, the trivial surface states disappear.

A presente tese de Doutorado refere-se a problemas em teoria de campos e mecânica quântica no espaço não comutativo (NC). Abordamos alguns sistemas físicos bem estudados em física teórica, como a teoria de Maxwell na presença de fontes externas, equação de Pauli, equação de Dirac em campos externos e o espectro do átomo de hidrogênio relativístico. Como um primeiro problema estudamos a teoria de calibre U(1)* e extendemos o mapa de Seiberg-Witten para incluir uma corrente externa e formulamos equações clássicas para os campos no espaço não comutativo. Soluções no vácuo e em um campo magnético externo para uma carga estática de tamanho finito a foram determinadas. Encontramos que uma carga estática além de ser um monopolo elétrico comporta-se como um dipolo magnético e um campo magnético externo modifica o campo de Coulomb a longas distâncias bem como alguns fatores de forma eletromagnéticos, comportamentos inerentes a consideração de uma geometria NC. Nesta direção analisamos a ambiguidade no mapa de Seiberg-Witten e mostramos que, no mínimo até a ordem estudada aqui, isto é equivalente a ambiguidade de se adicionar uma solução homogênea à condição de conservação da corrente. Demandando que o momento magnético NC seja menor que o erro existente na medida do momento magnético de léptons, obtemos uma estimativa superior para o parâmetro e seu comprimento fundamental associado l. Estudamos os níveis de energia do átomo de hidrogênio relativístico no formalismo da equação de Dirac no espaço NC para o campo de Coulomb. Demonstramos que no caso relativístico a não comutatividade quebra totalmente a degenerescência dos níveis 2S1/2; 2P1/2 e 2P3/2, abrindo novos canais de transição permitidos. Por fim construímos uma equação de onda não relativística para partículas de spin 1/2 através do limite não relativístico da equação de Dirac no espaço NC. Apresentamos um modelo pseudoclássico (à-la Berezin-Marinov) cuja quantização coincide com as equações de onda não relativísticas. Através da interação entre um spin não-relativístico e o campo magnético, através da equação de Pauli no espaço NC, construímos uma generalização para o modelo de Heisenberg para dois spins acoplados no espaço NC. Em tal modelo calculamos a amplitude de probabilidade de transição entre dois estados ortogonais do tipo EPR (Einstein-Podolsky-Rosen) submetidos em um campo magnético oscilatório e mostramos que, algumas de tais transições, que são proibidas no espaço comutativo são possíveis devido a não comutatividade do espaço. / The present PhD thesis refers to problems in field theory and quantum mechanics in noncommutative (NC) space. We study some well known physical systems in theoretical physics, such as the Maxwell theory in the presence of external sources, the Pauli equation, the Dirac equation with external fields and the relativistic Hydrogen atom. First we study the U(1)* gauge theory and extend the Seiberg-Witten map to include an external current and formulate classical field equations in NC space. Solutions in the vacuum and in an external magnetic field for a static charge of finite size a is determined. We find that a static charge in NC space, besides being an electric monopole, behaves as a magnetic dipole and the external magnetic field modifies the Coulomb law at large distances, as well as some electromagnetic form factors. In this direction we analyse the arbitrariness in the Seiberg-Witten map and show that, at least to the order studied here, this is equivalent to adding a homogeneous solution to the charge conservation condition. Demanding that the NC magnetic moment be less than the existing error in the measurement of leptons magnetic moment we obtain an upper bound for the NC parameter and its associated fundamental length l. In addition we consider the energy levels of a hydrogen-like atom in the framework of a -modified, due to space noncommutativity, Dirac equation with a Coulomb field. It is shown that the noncommutativity completely breaks the degeneracy of the 2S1/2; 2P1/2 and 2P3/2 levels, allowing for new transition channels. At last, but not least, we construct a nonrelativistic wave equation for spin 1/2 particles through the nonrelativistic limit of the NC Dirac equation. We present a pseudoclassical model (à-la Berezin-Marinov) whose quantization coincides with the nonrelativistic wave equations. By extracting the interaction between a nonrelativistic spin and the magnetic field, from the obtained Pauli equation in NC space, we construct a generalization of the Heisenberg model for two coupled spins in NC space. In such model, it is calculated the transition probability amplitude between two orthogonal EPR (Einstein-Podolsky-Rosen) states submitted in the presence of an oscilatory magnetic field and we shown that some of such transitions, which are forbidden in NC space are possible due to space noncommutativity.

In dieser Dissertation beschäftigen wir uns mit angepassten Zusammenhängen und ihren (horizontalen) Dirac-Operatoren auf strikt pseudokonvexen CR-Mannigfaltigkeiten. Einen Zusammenhang nennen wir dann angepasst, wenn er die relevanten Daten parallelisiert. Wir beschreiben den Raum der angepassten Zusammenhänge, indem wir ihre Torsionstensoren studieren, von denen gewisse Teile durch die Geometrie der Mannigfaltigkeit festgelegt sind, während andere frei wählbar sind. Als Anwendung betrachten wir die Eigenschaften der Dirac-Operatoren, die zu diesen Zusammenhängen gehören. Weiter betrachten wir horizontale Dirac-Operatoren, die nur in Richtung des horizontalen Bündels H ableiten. Diese Operatoren sind besser an die Sub-Riemannsche Struktur einer CR-Mannigfaltigkeit angepasst als die vollen Dirac-Operatoren. Wir diskutieren, wann diese Operatoren formal selbstadjungiert sind und beweisen eine Weitzenböck-Typ-Formel. Wir konzentrieren uns dann auf den horizontalen Dirac-Operator zum Tanaka-Webster-Zusammenhang. Dieser ändert sich konform kovariant, wenn wir die Kontaktform konform ändern. Für diesen Operator betrachten wir weiterhin zwei Beispiele: Wir betrachten S^1-Bündel über Kähler-Mannigfaltigkeiten, insbesondere berechnen wir für Sphären einen Teil des Spektrums. Außerdem betrachten wir kompakte Quotienten der Heisenberggruppe und berechnen hier in den Dimensionen 3 und 5 das volle Spektrum. Die horizontalen Dirac-Operatoren sind nicht mehr elliptisch, sondern „elliptisch in Richtung von H“. Mithilfe des Heisenbergkalküls stellen wir fest, dass die horizontalen Dirac-Operatoren nicht hypoelliptisch sind. Im Fall des Tanaka-Webster-Zusammenhangs lässt sich aber zeigen, dass der zugehörige Operator auf gewissen Teilen des Spinorbündels hypoelliptisch ist. Dies genügt, um zu beweisen, dass er (nun auf dem gesamten Spinorbündel) ein reines Punktspektrum hat und die Eigenräume, bis auf den Kern, endlich-dimensional sind und aus glatten Eigenspinoren bestehen. / In the present thesis, we study adapted connections and their (horizontal) Dirac operators on strictly pseudoconvex CR manifolds. An adapted connection is one that parallelises the relevant data. We describe the space of adapted connections through their torsion tensors, certain parts of which are determined by the geometry of the manifold, while others may be freely chosen. As an application, we study the properties of the Dirac operators induced by these connections. We further consider horizontal Dirac operators, which only derive in the direction of the horizontal bundle H. These operators are more adapted to the essentially sub-Riemannian structure of a CR manifold than the full Dirac operators. We discuss the question of their self-adjointness and prove a Weitzenböck type formula for these operators. Focusing on the horizontal Dirac operator associated with the Tanaka-Webster connection, we show that this operator changes in a covariant way if we change the contact form conformally. Moreover, for this operator we discuss two examples: On S^1-bundles over Kähler manifolds, we can compute part of the spectrum and for compact quotients of the Heisenberg group, we determine the whole spectrum in dimensions three and five. The horizontal Dirac operators are not elliptic, but rather "elliptic in some directions". We review the Heisenberg Calculus for such operators and find that in general, the horizontal Dirac operators are not hypoelliptic. However, in the case of the Tanaka-Webster connection, the associated horizontal Dirac operator is hypoelliptic on certain parts of the spinor bundle and this is enough to prove that its spectrum consists only of eigenvalues and except for the kernel, the corresponding eigenspaces are finite-dimensional spaces of smooth sections.

Neste trabalho, apresentamos um método que permite explicitar a arbitrariedade contida nas soluções das equações de onda relativísticas, na presença de certos tipos de campos eletromagnéticos externos. Esta arbitrariedade está relacionada com a existência de uma transformação, com a qual podemos reduzir o número de variáveis presentes na equação original. Através desta transformação, criamos uma representação, a qual permite obter novos conjuntos de soluções exatas e construir a função de evolução para a equação de Klein-Gordon. Como resultado, apresentamos novos conjuntos de soluções, estacionárias e não-estacionárias, para o problema em um campo magnético constante e uniforme e a combinação deste campo com um campo elétrico longitudinal. / We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which reduces effectively the number of variables in the initial equations. Then we use the corresponding representations to construct new sets of exact solutions, which may have a physical interest, and to construct the evolution function to the Klein-Gordon equation. As resulted, we present new sets of stationary and nonstationary solutions in magnetic field and in some superpositions of electric and magnetic fields.

Submitted by Mario BC (mario@bc.ufrpe.br) on 2017-02-07T13:36:07Z No. of bitstreams: 1 Thiago Conrado de Vasconcelos.pdf: 4646767 bytes, checksum: 61c228fc4590858e8ee056ac3909187e (MD5) / Made available in DSpace on 2017-02-07T13:36:07Z (GMT). No. of bitstreams: 1 Thiago Conrado de Vasconcelos.pdf: 4646767 bytes, checksum: 61c228fc4590858e8ee056ac3909187e (MD5) Previous issue date: 2016-02-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Throughout the latest years, the interest on Spintronics has increased. The principal purposes of the eld are to detect, manipulate, create and polarize spin currents. Within this topic, it is possible to emphasize the Spin Hall E ect(SEH) and the Inverse Spin Hall E ect(ISEH). In this dissertation, we analytically investigate the universal fluctuation of the conductance of the spin in a chaotic quantum point with chiral symmetry at low temperatures. We used random matrices theory and the expansion of the diagrammatic method for that purpose. We showed that when the chirality is broken, the universal fluctuation of the conductance dispersion is in the order of rms hGf sHi 0:18e=4 and that when there is the preservation of the chiral symmetry, the universal fluctuation of the conductance dispersion occurs in the order of rms [GqsH] 0:283e=4 which coincides with the literature. We also worked on ISEH, through the analytical analysis with the semi-classic expansion of the conductance and showed that in the semi-classic limit the relation rms [GqsH] = p2 rms hGf sHi is valid. / Ao longo dos últimos anos tem aumentado o interesse pelo estudo da spintrônica. O objetivo principal deste campo é detectar, manipular, criar e polarizar correntes de spin. Dentro deste tópico, se destaca o Efeito Hall (SHE) de Spin e Efeito Hall de Spin Inverso (ISHE). Neste trabalho investigamos analiticamente a flutuação universal da condutância de spin num ponto quântico caótico com simetria quiral a baixas temperaturas. Para isso, utilizamos a teoria de matrizes aleatória e a expansão do método diagramático. Mostramos que, quando a simetria de quiralidade é quebrada, a flutuação universal da condutância tem uma dispersão na ordem de na ordem de rms[GfsH] p2 0:18 e/4 e que, quando a simetria de quiralidade é preservada, a flutuação universal da condutância ocorre na ordem de rms[GqsH] 0.283 e/4 , o que está de acordo com a literatura. Em nosso trabalho também investigamos o (ISHE), por meio de uma análise analítica utilizamos a expansão semi-clássica da condutância e mostramos que no limite semi-clássico vale a relação rms[GqIsH] = p2 rms[GfIsH].

Nesse trabalho, estudamos a equação de Schrödinger não-linear com uma função potencial delta atrativa. As soluções para essa equação tem uma componente localizada e uma dispersiva. Além de estudar o comportamento das soluções dessa equação em espaços de Sobolev clássicos, mostramos algumas propriedades do grupo unitário em espaços Lp, L2 com peso, Sobolev com peso e assim obtemos alguns resultados de boa colocação local e global das soluções. O ponto central desta tese é mostrarmos a existência de uma variedade invariante centro que irá consistir de órbitas periódicas no tempo. / In this work, we study the nonlinear Schrodinger equation with an attractive delta function potential.The solutions to this equation have a localized and a dispersive component. In addition to studying the behavior of solutions of this equation in classical Sobolev space, we show some properties for the unitary group in Lp, weighted L2 and Sobolev spaces and so we get some results of local and global well-posedness of solutions. The central theme this thesis is to show the existence of a center invariant manifold, which will consist of time-periodic orbits.