▼ Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism. Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase. Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number. Advisors/Committee Members: Hussey, Glenn, Tanaka, Kaori, Dick, Rainer, Tse, John, Wang, Jiun-Chau.

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Le phénomène de condensation d’un gaz de bosons lorsqu’il est refroidi à zéro degrés Kelvin futdécrit par Einstein en 1925 en s’appuyant sur des travaux de Bose. Depuis lors, de nombreux physiciens,mathématiciens et numériciens se sont intéressés au condensat de Bose-Einstein et à son caractère superfluide. Nous proposons dans cette étude des méthodes numériques ainsi qu’un code informatique pour la simulation d’un condensat de Bose-Einstein en rotation. Le principal modèle mathématique décrivant ce phénomène physique est une équation de Schrödinger présentant une non-linéarité cubique,découverte en 1961 : l’équation de Gross-Pitaevskii (GP). En nous appuyant sur le logiciel FreeFem++,nous nous servons d’une discrétisation spatiale en éléments-finis pour résoudre numériquement cette équation. Une méthode d’adaptation du maillage à la solution et l’utilisation d’éléments-finis d’ordre deux nous permet de résoudre finement le problème et d’explorer des configurations complexes en deux ou trois dimensions d’espace. Pour sa version stationnaire, nous avons développé une méthode de gradient de Sobolev ou une méthode de point intérieur implémentée dans la librairie Ipopt. Pour sa version instationnaire, nous utilisons une méthode de Time-Splitting combinée à un schéma de Crank-Nicolson ou une méthode de relaxation. Afin d’étudier la stabilité dynamique et thermodynamique d’un état stationnaire, le modèle de Bogoliubov-de Gennes propose une linéarisation de l’équation de Gross-Pitaevskii autour de cet état. Nous avons élaboré une méthode permettant de résoudre ce système aux valeurs et vecteurs propres, basée sur un algorithme de Newton ainsi que sur la méthode d’Arnoldi implémentée dans la librairie Arpack.

The phenomenon of condensation of a boson gas when cooled to zero degrees Kelvin was described by Einstein in 1925 based on work by Bose. Since then, many physicists, mathematicians and digitizers have been interested in the Bose-Einstein condensate and its superfluidity. We propose in this study numerical methods as well as a computer code for the simulation of a rotating Bose-Einstein condensate.The main mathematical model describing this phenomenon is a Schrödinger equation with a cubic nonlinearity, discovered in 1961: the Gross-Pitaevskii (GP) equation. By using the software FreeFem++ and a finite elements spatial discretization we solve this equation numerically. The mesh adaptation to the solution and the use of finite elements of order two allow us to solve the problem finely and to explore complex configurations in two or three dimensions of space. For its stationary version, we have developed a Sobolev gradient method or an internal point method implemented in the Ipopt library. .For its unsteady version, we use a Time-Splitting method combined with a Crank-Nicolson scheme ora relaxation method. In order to study the dynamic and thermodynamic stability of a stationary state,the Bogoliubov-de Gennes model proposes a linearization of the Gross-Pitaevskii equation around this state. We have developed a method…

Advisors/Committee Members: Danaila, Ionut (thesis director), Hecht, Frédéric (thesis director).

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The pi-striped, d-wave superconducting (SC) state, which is a type of pair density wave wherein the SC order is spatially modulated, has recently been shown to generate the key ingredients for quantum oscillations consistent with experimental observations (Zelli et al., 2011, 2012). This was accomplished with a phenomenological approach using non-self-consistent Bogoliubov-de Gennes (BdG) theory. The objective of this thesis is to explore two aspects of this approach: the addition of a charge density wave (CDW) order to the previous non-self-consistent calculations, and an attempt at stabilizing the pi-striped state in fully self-consistent BdG theory. It was found that the CDW order had a minimal effect on the Fermi surface characteristics of the pi-striped state, but that a sufficiently strong CDW degrades the Landau levels which are essential for the formation of quantum oscillations. The self-consistent mean-field calculations were unable to stabilize the pi-striped state under a range of modifications to the Hamiltonian. Free energy calculations with the modulated SC order treated as a parameter demonstrate that the pi-striped state is always less energetically favourable than the normal state for the scenarios which were considered. The results of this study constitute a basis for future, more comprehensive studies, using the BdG approach, of the stability of possible pi-striped SC phases.

Master of Science (MSc)

Advisors/Committee Members: Kallin, Catherine, Physics and Astronomy.

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In this thesis, we study an isolated vortex in an s-wave superconductor by solving the Bogoliubov-de Gennes equations self-consistently on a disc. We calculate the order parameter and supercurrent profiles, as well as the distribution of quasiparticle states. In contrast to quasi-classical treatments, the ratio <em>Δ_∞/E_F</em> between the order parameter and the Fermi energy is not assumed negligible. We study a regime where this ratio is on the order of 10^-1, relevant to high-temperature superconductors. In this regime, we find a Friedel-like oscillation in the order parameter profile at low temperatures. This oscillation is attributed to an increased level spacing of the quasiparticle states, causing a decrease of the number of states present inside the superconducting energy gap. The results are in good agreement with previously published works. In future studies, the method used in this thesis will be generalized to d-wave superconductors.

I detta examensarbete studeras en ensam virvel i en s-vågssupraledare genom att självkonsistent lösa Bogoliubov och de Gennes' ekvationer på en cylinderskiva. Vi beräknar ordningsparameter- och superströmsprofiler, samt fördelningen av kvasipartikeltillstånd. Till skillnad från i kvasiklassiska metoder så antas inte kvoten <em>Δ_∞/E_F</em> mellan ordningsparametern och Fermi-energin vara negligerbar. Vi studerar en regim där denna kvot är av storleksordningen 10^-1, vilket är fallet i högtemperatur-supraledare. Vid låga temperaturer finner vi i denna regim en Friedelliknande oscillation i ordningsparameterprofilen. Denna oscillations förklaras genom att separationen mellan kvasipartikeltillstånd ökar, vilket får som effekt att färre tillstånd ryms innanför det supraledande energigapet. Våra resultat överensstämmer väl med tidigare publicerade artikler. I framtida studier kommer metoden vi använder i detta examensarbete att generaliseras till d-vågssupraledare.

▼ The current theoretical framework for studying Majorana zero modes (MZM) in superconductors and its application for topological quantum computing is based on mean-field approximations and is derived from solutions to BdG equations. In this framework, particle number conservation is broken and non-interacting fermion Hamiltonian is used to describe physics of interest. We argue that these features in the current framework may make it insufficient for studying topological properties of MZM pertinent to quantum computing. After reviewing the current theory with an emphasis on its potential problems, we investigate physics beyond the BdG equations in a toy model and find evidence for the non-trivial role of particle number conservation in Berry phase of transporting a bound quasiparticle around a vortex in a s-wave superconductor. We then study the effect of particle number conservation and superconducting condensate on braiding MZM in vortices in chiral p-wave superconductors and find that they may have non-negligible effect on properties of MZM, suggesting the need for further study on the theoretical basis of this intriguing subject. Advisors/Committee Members: Leggett, Anthony J. (advisor), Stone, Michael (Committee Chair), Eckstein, James N. (committee member), Kwiat, Paul G. (committee member).

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Tese de mestrado em Física, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2015

Nemáticos e colestéricos são conhecidos pelas suas aplicações ópticas. O grande potencial destes materiais tem origem na ordem dos seus constituintes. Nas fases nemática e colestérica, as moléculas são livres de se movimentar no espaço, tal como num líquido convencional, mas obedecem a uma ordem orientacional de longo alcance. A orientação das moléculas define uma direção preferencial, quebrando a isotropia do espaço. Desta forma, nemáticos e colestéricos são na realidade líquidos anisotrópicos. A diferença entre as duas fases é que o colestérico é constituído por moléculas quirais. Enquanto as moléculas num nemático tendem a estar homogeneamente alinhadas em todo o espaço, a quiralidade do colestérico favorece uma torção espontânea. Num colestérico livre de constrangimentos, as moléculas tendem a torcer ao longo de uma direção, descrevendo uma hélice no espaço. O comprimento que corresponde a uma volta de 2_ na orientação das moléculas define a periodicidade do colestérico e tem o nome de pitch. Como consequência da ordem orientacional das moléculas, as propriedades elásticas, electromagnéticas e ópticas do material são também anisotrópicas. Estas propriedades macroscópicas assentam num meio que é fluido e flexível. Assim, nemáticos e colestéricos são extremamente sensíveis a estímulos externos e tornam-se ideais para aplicações tecnológicas controláveis. A realização deste conjunto de propriedades esteve na base do desenvolvimento dos ecrãs de cristal líquido (LCDs). Os ecrãs LCD dominaram o mercado dos mostradores planos durante mais de 50 anos e são ainda de uso padrão em pequenos aparelhos do quotidiano. A geometria de uma célula LCD comum é, de facto, bastante simples e consiste num cristal líquido confinado entre duas placas planas de vidro. Contudo, o confinamento de cristais líquidos no interior de superfícies curvas conduz a estruturas mais exóticas, com aplicações que vão desde bio-sensores a interruptores ópticos e janelas de privacidade. Assim, após o grande triunfo da tecnologia de ecrãs planos, os investigadores estão agora interessados no papel de geometrias confinantes mais complexas nas propriedades dos cristais líquidos. Devido aos constrangimentos topológicos, as superfícies curvas revestem-se de um interesse especial. O estado fundamental de um cristal líquido confinado no interior de gotas curvas é o resultado da competição entre a elasticidade e os constrangimentos topológicos impostos pelo confinamento. Dependendo da geometria da gota, as configurações de energia mínima podem exibir singularidades na sua orientação, designadas por defeitos topológicos. Esta é uma consequência de um importante teorema em topologia, o teorema de Poincaré-Hopf, para superfícies com característica de Euler diferente de zero. Estudos efectuados em gotas e camadas esféricas de cristal líquido mostram que os defeitos podem promover interações direcionais entre partículas, sendo estas apontadas como possíveis…

Advisors/Committee Members: Gama, Margarida Telo da, 1956-, Silvestre, Nuno Miguel, 1979-.

▼ From a mathematically rigorous foundation, energy minimizing configurations are numerically computed using the Landau-de Gennes energy expression for a variety of liquid crystal domains in order to compare them with previous experimental and numerical results as well as present new results. The method of manufactured solutions is used to validate numerical computations in two- and three-dimensional domains before computing the minimizers. Many configurations and defect structures seen in experiments were reproduced numerically and studied carefully, with three notable results: First, in the two-dimensional disc domain with normal Dirichlet boundary conditions, the 'split core' defect is shown to be a product of improper mesh refinement. Second, for all concentric cylinder domains with planar radial Dirichlet boundary conditions, biaxial regions were present, which is a marked difference from previous results using the Oseen-Frank model. Third, for a single colloidal particle with normal Dirichlet boundary conditions suspended in a cylinder with everywhere vertical boundary conditions, the core of the Saturn ring defect is shown to be uniaxial with negative scalar order parameter. These three results more accurately describe experimental and theoretical results than those using the vector-based models and motivate the use of the Landau-de Gennes model for future numerical computations.

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Tese de mestrado em Física, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2015

Desde a descoberta dos Cristais Líquidos (CLs), que a busca por aplicações destes materiais com propriedades mecânicas e electro-ópticas tão peculiares tem sido uma área de grande interesse. As suas aplicações mais conhecidas incluem aplicações à medicina – CLs podem ser usados como termómetros – e à fotónica – CLs estão na base dos LCDs (Liquid Crystal Displays) que representam cerca de 90% do mercado mundial de dispositivos de visualização. Tal como os LCDs, o fenómeno de molhagem tem sido alvo de grande estudo, uma vez que diversos fenómenos de molhagem são parte integrante do nosso dia-a-dia. Em particular, o estudo da molhagem engloba o estudo de fenómenos interfaciais, cuja compreensão ajuda a responder a questões fundamentais tais como definir uma interface e calcular uma tensão de superfície. Recentemente, o problema da molhagem por colestéricos começou a atrair atenções, não só pelas potenciais aplicações tecnológicas mas também pelo facto de ser um sistema ainda pouco explorado. O formalismo de Landau-de Gennes (LdG) tem-se provado ideal para o desenvolvimento de um entendimento teórico da física subjacente aos sistemas de CLs. Nesta dissertação, tiramos partido deste formalismo para modelar uma interface CI livre (interface entre uma fase líquido-cristalina colestérica e uma isotrópica) e a molhagem de uma superfície plana por um colestérico (interface CI na presença de uma superfície). Para sistemas simples de colestéricos é fácil deduzir um modelo analítico mas para situações mais complexas, onde o sistema apresenta distorções elásticas, precisamos de recorrer a modelos numéricos. Na presente dissertação começamos por introduzir o fenómeno de molhagem para fluidos simples. A equação de Young estabelece a condição de molhagem quando o ângulo de contacto é zero. Com o intuito de introduzir a molhagem por CLs, fazemos um resumo das propriedades dos CLs, assim como da descrição mesoscópica fenomenológica que irá ser usada para modelar cristais líquidos colestéricos – o modelo da energia livre de Landau-de Gennes. Para temperaturas menores do que uma temperatura crítica, os CLs exibem uma fase onde as moléculas apresentam um certo nível de ordenamento médio (fase ordenada). Acima dessa temperatura crítica, os CLs comportam-se como fluidos isotrópicos (fase isotrópica). Devido à simetria dos CLs, o parâmetro de ordem que caracteriza a fase líquido-cristalina ordenada é um tensor de segunda ordem. O modelo de energia livre de LdG descreve o CL em termos de duas densidades de energia livre: uma que decreve a transição entre a fase líquido-cristalina e a fase isotrópica, e uma que penaliza variações do parâmetro de ordem, i.e., penaliza distorções elásticas. Introduzimos também a densidade de energia livre associada à superfície plana que penaliza desvios da orientação molecular em relação à orientação favorecida na superfície. O conceito de defeitos topológicos em CLs é desenvolvido mencionando os dois…

Advisors/Committee Members: Gama, Margarida Telo da, 1956-, Bernardino, Nelson Fenando Rei, 1980-.

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Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq

The spin-1=2 and spin-1 Ising models, as well as the spin-1 Blume-Capel model have been studied in one-, two-, and three-dimensional lattices. A variational method based on Bogoliubov inequality for the free-energy has been employed. The trial Hamiltonians consist of clusters of free spins, pairs of spins, and a combination of free spins and pairs of spins. For the three approximations, the thermodynamic quantities of interest have been calculated, together with the critical transition temperature and the behavior close to the transition, in the latter case in order to compute the corresponding critical exponents. The results have been compared to each other as well with exact results, when available, or results coming from more reliable approximate methods. It has been noted that as more interactions are taken into account in the trial Hamiltonian, better results are obtained for the transition temperature, although the critical exponents are always the mean eld like ones.

Os modelos de Ising de spin-1=2 e spin-1, e o de Blume-Capel de spin-1 foram estudados em redes de uma, duas e três dimensões. Foi empregado o método variacional baseado na desigualdade de Bogoliubov para a energia livre. Os hamiltonianos tentativa utilizados consistem em blocos de spins livres, de pares de spins, e da combinacão de spins livres mais pares de spins. Para as três aproximacões, foram obtidas as quantidades termodinâmicas de interesse, bem como a temperatura crítica e o comportamento perto da transição, neste ultimo caso para se obter os respectivos expoentes críticos. Os resultados foram comparados entre si, bem como com os resultados exatos, quando dispon veis, ou provenientes de outras aproxima c~oes mais elaboradas. Veri ca-se que a medida que se incorpora mais intera c~oes nos hamiltonianos tentativa, melhores resultados s~ao obtidos para a temperatura de transi c~ao, embora os expoentes cr ticos continuem sempre sendo os mesmo de campo médio usual. Palavras-Chave: Modelo de Ising,Blume-Capel,desigualdade de Bogoliubov, campo médio.

Advisors/Committee Members: Plascak, João Antônio.

▼ Conventional lenses are limited by their fixed shaped and optical properties. Liquid crystal adaptive lenses (LCLAs) are a promising candidate to move beyond these limitations thanks their tuneable optical properties. A difficulty of working with LCs is that their properties are the result of an experimentally un-observable structure. Thankfully, modelling is capable of providing insight into this structure. Unfortunately, progress has been hamstrung by an over-reliance on experimentation. Further, what little modelling is being done usually involves simplified models and/or close-source software packages. This work uses a general model for thermotropic nematic liquid crystals based on Landau-de Gennes theory to study the texture of liquid crystal adaptive lenses. The most general version of this model was used, without the common simplifications such as: hard anchoring, neglecting elastic constants, or geometric symmetry. In order to find the equilibrium state for the nematic model, the Euler-Lagrange equation for the total free-energy is set to zero. This form is converted into a transient PDE in order to capture the dynamics of the system, and to evolve the texture towards its equilibrium state. The nematic model is coupled with a model for the electric field within the cell, and the two are solved simultaneously. This is accomplished by using the method of lines for temporal discretization and the finite element method for spatial discretization. The validity of the implemented model was first verified by modelling two important LCD configurations: the TN cell and the IPS cells. The TN cell was modelled with the electrodes off and with them on. In both cases the correct equilibrium texture was obtained. Modelling light propagation with cross-polarization microscopy produced the correct results, a bright cell when the electrodes were off and a dark one when they were on. Next, the IPS cell was also modelled. Again, the correct equilibrium result was obtained; a twisted texture was when the electrodes are turned off and an untwisted texture when the electrodes are turned on. Modelling light propagation resulted in the correct dark state when the electrodes were off andthe correct bright state when they were on. Having successful produced the expected texture and cross-polarization microscopy results, the model was applied to a LCAL. The literature review of this work identified a wide range of potential liquid crystal adaptive lenses. The final design was chosen using three criteria: 1) availability of published results, 2) modelling requirements, and 3) ease of manufacture. Based on these criteria, a design called the HMD cell was chosen. When modelled, the resulting texture and cross-polarization microscopy did not agree with previously published results. An investigation into the cause of these discrepancy was performed, but the cause has not yet been identified.

▼ We theoretically consider the properties of a dipolar Bose-Einstein condensate with a vortex. Our theory includes the influence of the leading order quantum fluctuation corrections which allows the condensate to stabilize into a droplet state in the regime of dominant dipole interactions. We develop numerical techniques to accurately and efficiently calculate the stationary vortex states and the quasi-particle excitations. These methods are carefully benchmarked where possible. We make a brief study of self-bound vortex droplets, considering their basic properties, and presenting a phase diagram for where they exist. We also compare our calculations to results which appeared from another group during our research. We show that their results suffer from serious numerical issues and are unreliable. We focus on studying the properties of a vortex line in an elongated dipolar Bose-Einstein condensate confined by a prolate trap. Increasing the strength of the dipole-dipole interactions relative to the short ranged contact interactions we find that the system crosses over to a self-bound vortex droplet stabilized from collapse by quantum fluctuations. We calculate the quasiparticle excitation spectrum of the vortex state, which is important in characterizing the vortex response, and assessing its stability. When the DDIs are sufficiently strong we find that the vortex is dynamically unstable to quadrupolar modes. Advisors/Committee Members: Blakie, Blair (advisor).

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Tese de mestrado, Física (Estatísatica não linear) Universidade de Lisboa, Faculdade de Ciências, 2017

For the past decade, one of the trends in liquid crystal studies has been the development of the fundamental understanding of interfacial and wetting phenomena of different liquid crystal phases. The reason to do so lays on the complexity and richness behind the characteristic anisotropic properties of liquid crystals that represent a viable path to the creation of novel applications, so far inaccessible with the use simple isotropic liquids. A pictorial example of the usefulness of these properties is that they allow a controlled assembly from surfaces of immersed particles in liquid crystals. These surfaces can be lithographic controlled substrates or interfaces. Thus, each of these systems has been subject to systematic and progressive study in order to complete the comprehension of this field. This thesis is inserted on this trend and is focused on the study of the Blue Phase (BP) interface with the Isotropic (I) and the Cholesteric (C) phases at the mesoscale with respect to their surface tension and interface roughness. Such study is developed for the triple coexistence, using the Landau-de Gennes (LdG) phenomenological model. As a first result of this study it was found at the triple phase coexistence line a hyperbolic relation between the two constant approximation LdG rescaled elastic constant and the pitch. Moreover, regardless of the position in the phase diagram, the blue phase periodicity has a linear relation with that of the cholesteric phase. For the BP-I coexistence it is observed that the interface equilibrium configuration has a zig-zag arrangement of double-twist cells and that the undulation profile has two distinct roughness behaviours, depending on the pitch. The transition between behaviours occurs due to an interface anchoring transition, in which for low pitch values cells have small deformations and have planar anchoring while for high pitch values cells are highly deformed and have homeotropic anchoring. Moreover, it is observed that it is possible to have a continuous transition between the roughness profiles of the BP-I and C-I interfaces over the triple phase line. For the BP-C coexistence it is found that the interface equilibrium configuration has a linear arrangement of undistorted double-twist cells with cholesteric layers tilted by _=4, regardless of the pitch values, and that there is no anchoring transition like in the BP-I interface. It is also observed that for low pitch values only the BP-I can coexist. An introductory study of the behaviour of rough substrate confined cholesterics is also presented, showing that for a sawtooth substrate weak planar anchoring the cholesteric layers adopt a parallel orientation at the substrate regardless of the substrate aperture.

Ao longo da última década, uma das tendências no estudo de Cristais Líquidos baseou-se no desenvolvimento da compreensão fundamental dos fenómenos interfaciais e de molhagem de diferentes fases. A razão para…

Advisors/Committee Members: Gama, Margarida Telo da, 1956-, Silvestre, Nuno.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico

Nesta tese estudamos teoricamente as configurações de vórtices em algumas nanoestruturas supercondutoras, como quadrados mesoscópicos com dois defeitos as- simétricos ou com um defeito preenchido com material metálico, anéis mesoscópicos e filmes finos submersos em diferentes tipos de materiais (metais, outros supercon- dutores, isolantes) ou com uma alta densidade de defeitos em sua interface. O estudo foi realizado mediante resolução numérica das equações de Ginzburg-Landau dependentes do tempo (TDGL) para os sistemas citados utilizando-se o método de campos de calibre com variáveis de ligação. Encontramos as configurações de vórtices para estas geometrias e calculamos as curvas de magnetização em função do campo magnético aplicado. Constatamos que as configurações de vórtices, os campos críticos e a própria maneira como os vórtices adentram o supercondutor, dependem fortemente do arranjo dos centros de aprisionamento e da geometria do sistema. No estudo de filmes finos com interfaces diversas, observamos que, devido ao contato com o material, existe um aumento ou uma supress~ao (dependendo do tipo de interface) da supercondutividade nas bordas da amostra, como revelam os resultados das curvas de magnetização, densidade de elétrons supercondutores e número de vórtices em função do campo magnético e do comprimento de extra- polação de De Gennes b. Para determinados valores de b o sistema tem uma resposta paramagnética na curva de magnetização, relacionada na captura do luxo magnético devido nas fronteiras da amostra

Advisors/Committee Members: AGUIAR, Jose Albino Oliveira de.

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Orientador: Prof. Dr. Luiz Roberto Evangelista

Coorientador: Prof. Dr. Rodolfo Teixeira de Souza

Dissertação (mestrado em Física) – Universidade Estadual de Maringá, 2017

Resumo: O perfil do parâmetro de ordem em relação a espessura e, também, em relação a temperatura da amostra formada por dois cilindros concêntricos deve ser determinado por meio dos métodos computacionais e numéricos. No entanto, para a amostra de cristal líquido confinada em geometria cilíndrica, leva-se em conta que as superfícies da mesma podem influenciar nas transições de fase nemática-isotrópica de volume. A princípio, a investigação das possíveis transições de fase é realizada por meio da teoria fenomenológica de Landau-de Gennes, pois esta é uma teoria adequada a ser aplicada aos cristais líquidos nemáticos, quando são notadas as temperaturas críticas próximas a transição nemática-isotrópica. O potencial de superfície de curto alcance complementa a descrição da densidade de energia livre da amostra cilíndrica concêntrica. O método de simulação de Monte Carlo é abordado com o intuito de obter uma descrição dos parâmetros de ordem de volume e de superfície via modelagem molecular, além de predizer quais tipos de transição de fase nemática-isotrópica (primeira ou segunda ordem) podem ocorrer no volume e próximas às superfícies da amostra. A simulação computacional leva em conta um potencial de interação entre as moléculas, conhecido como modelo de Lebwohl-Lasher. Com a inserção da densidade de energia livre de Landau-de Gennes em um funcional, a fim de descrever a energia total da amostra considerada, a equação de Euler-Lagrange e suas respectivas condições de contorno podem ser obtidas quando o método variacional, utilizado na determinação do estado de equilíbrio do sistema, é aplicado. Como se trata de uma equação diferencial, que dificilmente pode ser resolvida analiticamente, os métodos numéricos são aplicados na busca de possíveis soluções para o problema. Porém, o que estas soluções propõem são justamente os perfis dos parâmetros de ordem de volume e de superfície dependentes da espessura e temperatura da amostra. Posteriormente, os dados obtidos, tanto da simulação computacional, quanto do método numérico são analisados e comparados, mostrando que ambas abordagens apresentam resultados similares

The profile of the order parameter in relation to the thickness and also in relation to the temperature of the sample formed by two concentric cylinders must be determined by computational and numerical methods. However, for the sample of liquid crystal confined in cylindrical geometry, it is taken into account that, the surfaces of the sample can influence in the volume nematic-isotropic phase transitions. Originally, the investigation of the possible phase transitions is carried out by means of the phenomenological Landau-de Gennes theory, since this is a suitable theory to be applied to the nematic liquid crystals when the critical temperatures are close to the nematic-isotropic transition. The short-range surface potential…

Advisors/Committee Members: Evangelista, Luiz Roberto, Oliveira, Breno Ferraz de, Souza, Rodolfo Teixeira de, Yednak, César Augusto Refosco, Universidade Estadual de Maringá, Centro de Ciências Exatas, Departamento de Física, Programa de Pós-Graduação em Física.

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Ce mémoire présente une étude théorique des propriétés de localisation collective dans les superfluides de Bose désordonnés ou quasipériodiques. S'il est connu depuis Anderson que le désordre peut localiser les particules libres, comprendre ses effets dans les systèmes quantiques en interaction, où il est à l'origine de transitions de phase et d'effets de localisation non-Triviaux, représente aujourd'hui un défi majeur. En nous focalisant sur le cas d'un gaz de Bose dans le régime de faibles interactions, bien décrit par la théorie de Bogoliubov, nous étudions les transitions de localisation de ses excitations collectives dans différents contextes. Dans le cas d'un vrai désordre dans l'espace continu tout d'abord, nous développons un formalisme de désordre fort allant au-Delà des études antérieures, aboutissant à une description complète des propriétés de localisation des excitations en dimension arbitraire. Nous présentons un diagramme de localisation générique, et une interprétation microscopique de la propagation des excitations dans le désordre. Dans un second temps, nous considérons le cas d'un potentiel quasipériodique unidimensionel, aux propriétés intermédiaires entre un vrai désordre et un potentiel périodique. Notre traitement analytique et numérique du problème révèle une transition de localisation collective, que nous caractérisons et interprétons en termes de localisation dans un potentiel effectif multiharmonique. Pour finir, nous considérons le cas d'un gaz de Bose à deux composants. Nous développons le formalisme général pour étudier ces questions et décrivons la physique de base de ces systèmes qui présentent leurs propres spécificités.

In this thesis, we theoretically investigate the collective localization properties of weakly-Interacting Bose superfluids subjected to disordered or quasiperiodic potentials. While disorder has been recognized since Anderson to induce single-Particle localization, the interplay between disorder and interactions in quantum systems is today among the most challenging questions in the field, and underlies fascinating phase transitions and non-Trivial localization effetcs. Focusing on Bose gases in the weakly-Interacting regime for which the Bogoliubov theory proves a successful tool, we study the localization transitions of collective excitations in several contexts. First, in the case of a continuous true disorder, we develop a strong-Disorder formalism going beyond previous studies, providing us with a complete description of the localization behaviour of collective excitations in arbitrary dimension. A generic localization diagram is obtained and the transport of excitations in the disorder is microscopically interpreted. Secondly, we consider the case of one-Dimensional quasiperiodic potentials, which are known to display intermediate properties between periodic and disordered ones. We perform a numerical and analytical treatment of the localization problem of collective excitations, allowing us to quantitatively characterize and interpret the localization…

Advisors/Committee Members: Sanchez-Palencia, Laurent (thesis director).

▼ La théorie covariance de la fonctionnelle de la densité (CDF), basée sur un petit nombre de paramètres ajustables, a été utilisée avec succès pour décrire l’état fondamental et les états excités des noyaux de la carte nucléaire, pour A>12. Cette approche permet de décrire les systèmes nucléaires finis avec un Lagrangien hadronique universel résolu dans le cadre de l'approche Relativiste-Hartree-Fock-Bologuibov (RHFB). Ce modèle est également utilisé pour l'étude des étoiles compactes, car il peut être étendu à des densités élevées où la relativité restreinte ne peut pas être ignoré. Ce modèle peut également être étendu pour inclure la contribution des hypérons et ainsi que d'autres particules exotiques. Dans ce travail, la description et des prédictions basées sur l'approche RHFB pour les noyaux dans des conditions extrêmes de la masse, d'isospin et de température sont présentés.Dans la première partie de cette thèse, nous explorons l'apparition de nouvelles fermetures de couches sphériques pour des noyaux super-lourds, où les fermetures de couches sont caractérisées en termes de gap à deux nucléons. Les résultats dépendent légèrement des Lagrangians effectifs utilisés, mais les nombres magiques au-delà de ²⁰⁸ Pb sont prédit pour un nombre de protons Z=120 et 138, et pour un nombre de neutrons N=172, 184, 228, et 258. Les effets de couche sont sensibles à différents termes de champ de moyen, tels que le couplage spin-orbite, la masse scalaire et la masses effective, ainsi que l'interaction de tensorielle de Lorentz. Ces termes ont des poids différents dans les Lagrangians effectifs employées, expliquant les variations, somme toute petites, dans leurs prédictions. Employant le modèle RHFB le plus avancé, nous avons trouvé que le nucléide ³⁰⁴ 120 est favorisée comme étant le prochain noyau sphérique doublement magique au-delà de ²⁰⁸ Pb.Dans la deuxième partie de cette thèse, nous étudions l'apparition de nouveaux nombres magiques pour les noyaux de masse intermédiaire riches en neutrons, et nous analysons le rôle des interactions pseudo-vecteur et de tensorielle de Lorentz. Basé sur la transformation de Foldy-Wouthuysen, nous discutons en détail le rôle joué par les différents termes des interactions pseudo-vecteur et de tensorielle de Lorentz. Dans l'apparition des nouveaux nombres magiques N=16 , 32 et 34. Les noyaux ²⁴ O ⁴⁸ Si et ^52,54 Ca sont prédits avec un grand gap au niveau de Fermi et un gap d'appariement zéro (²⁴ O,⁵⁴Ca ) ou quasi-nul (⁴⁸ Si,⁵⁴ Ca), les rendant candidats pour de nouveaux nombres magiques des noyaux riches en neutrons. Nous constatons que les interactions de Lorentz pseudo-vecteur et tensorielle induisent des évolutions très spécifiques des énergies à une particule, ce qui pourrait signer la présence et la nécessité d'approches relativistes avec des interactions d'échanges de mésons.Dans la dernière partie de cette thèse, nous étudions les transitions de phase et excitations thermiques des deux noyaux stables et faiblement liés. Les prédictions de divers… Advisors/Committee Members: Margueron, Jérôme (thesis director).

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Cette thèse est consacrée à la description de la physique à une particule ainsi qu'à celle de fluides quantiques bosoniques dans des systèmes topologiques. Les deux premiers chapitres sont introductifs. Dans le premier, nous introduisons des éléments de théorie des bandes et les quantités géométriques et topologiques associées : tenseur métrique quantique, courbure de Berry, nombre de Chern. Nous discutons différents modèles et réalisations expérimentales donnant lieu à des effets topologiques. Dans le second chapitre, nous introduisons les condensats de Bose-Einstein ainsi que les excitons-polaritons de cavité.La première partie des résultats originaux discute des phénomènes topologiques à une particule dans des réseaux en nid d'abeilles. Cela permet de comparer deux modèles théoriques qui mènent à l'effet Hall quantique anormal pour les électrons et les photons dû à la présence d'un couplage spin-orbite et d'un champ Zeeman. Nous étudions aussi l'effet Hall quantique de vallée photonique à l'interface entre deux réseaux de cavités avec potentiels alternés opposés.Dans une seconde partie, nous discutons de nouveaux effets qui émergent due à la présence d'un fluide quantique interagissant décrit par l’équation de Gross-Pitaevskii dans ces systèmes. Premièrement, il est montré que les interactions spin anisotropes donnent lieu à des transitions topologiques gouvernées par la densité de particules pour les excitations élémentaires d’un condensat spineur d’exciton-polaritons.Ensuite, nous montrons que les tourbillons quantifiés d'un condensat scalaire dans un système avec effet Hall quantique de vallée, manifestent une propagation chirale le long de l'interface contrairement aux paquets d'ondes linéaires. La direction de propagation de ces derniers est donnée par leur sens de rotation donnant lieu à un transport de pseudospin de vallée protégé topologiquement, analogue à l’effet Hall quantique de spin.Enfin, revenant aux effets géométriques linéaires, nous nous sommes concentrés sur l’effet Hall anormal. Dans ce contexte, nous présentons une correction non-adiabatique aux équations semi-classiques décrivant le mouvement d’un paquet d’ondes qui s’exprime en termes du tenseur géométrique quantique. Nous proposons un protocole expérimental pour mesurer cette quantité dans des systèmes photonique radiatifs.

This thesis is dedicated to the description of both single-particle and bosonic quantum fluid Physics in topological systems. After introductory chapters on these subjects, I first discuss single-particle topological phenomena in honeycomb lattices. This allows to compare two theoretical models leading to quantum anomalous Hall effect for electrons and photons and to discuss the photonic quantum valley Hall effect at the interface between opposite staggered cavity lattices.In a second part, I present some phenomena which emerge due to the interplay of the linear topological effects with the presence of interacting bosonic quantum fluid described by mean-field Gross-Pitaevskii equation. First, I show that the…

Advisors/Committee Members: Solnyshkov, Dmitry (thesis director), Malpuech, Guillaume (thesis director).

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Les noyaux riches en neutrons de masse A=100-110 constituent une région de grand intérêt pour l'étude de la structure nucléaire loin de la stabilité. De précédentes études de cette région de masse ont déjà révélé la complexité de l'évolution de la collectivité et de la déformation dans les chaînes isotopiques de Zr, Mo, Ru et Pd. Afin d'étendre les données expérimentales sur la collectivité à des états de plus haut spin et à des noyaux plus riches en neutrons, des temps de vie d'états excités ont été mesurés dans des noyaux produits par une réaction de fusion-fission en cinématique inverse au GANIL. Les fragments de fission étaient séparés et identifiés en A et Z grâce au spectromètre magnétique de grande acceptance VAMOS tandis que le rayonnement gamma était détecté dans l'ensemble de détecteurs germanium EXOGAM. Environ vingt temps de vie d'états 2+, 4+ et 6+ ont été extraits à l'aide du plunger de Cologne. Cette expérience représente la première mesure RDDS dans des fragments de fission identifiés évènement par évènement à la fois en A et Z.Cette étude des noyaux de masse A=100-110 est complétée par des calculs auto-cohérents de champ moyen et au-delà avec la force de Gogny (D1S). La structure des états fondamentaux et excités est décrite dans le cadre du modèle de Hartree-Fock-Bogoliubov avec des contraintes sur les déformations axiale et triaxiale. Les excitations individuelles sont étudiées par des calculs bloqués et les états de haut spin sont décrits dans l'approximation du champ tournant. Enfin, la méthode de la coordonnée génératrice approchée par un hamiltonien collectif en cinq dimensions (5DCH) est appliquée aux états collectifs de basse énergie. Les résultats sont comparés aux mesures de la collectivité.

Neutron-rich nuclei of mass A=100-110 are of great interest for the study of nuclear structure far from stability. Previous experimental and theoretical studies suggest a complex evolution of deformation and collectivity in the isotopic chains of Zr, Mo, Ru and Pd.In order to extend information on the evolution of the collectivity towards higher spin states and more neutron-rich nuclei, lifetimes of excited states were measured in nuclei produced through a fusion-fission reaction in inverse kinematic at GANIL. Fission fragments were separated and identified in both A and Z with the high acceptance magnetic spectrometer VAMOS while the EXOGAM germanium detectors array was used for the coincident gamma-ray detection. Lifetimes of about twenty excited states were extracted using the plunger device of Cologne. This is the first RDDS measurement on fission fragments which are identified in A and Z on an event-by-event basis. The study of this mass region is completed by theoretical calculations using self consistent mean field and beyond mean field methods implemented with the Gogny force (D1S). The structure of the ground states and the excited states is described with Hartree-Fock-Bogoliubov calculations with constraints placed on the axial and triaxial deformations. Individual excitations are…

Advisors/Committee Members: Korten, Wolfram (thesis director).

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O método de segunda quantificação é utilizado para construir o operador espalhamento S no espaço de Fock, no contexto de Thermofield Dynamics (TFD), para o campo de Dirac sujeito a um potencial eletromagnético externo dependente do tempo. Esta descrição é baseada na abordagem construtiva do espaço de Fock, a qual é aplicada ao sistema original e a seu dual. Seguindo a prescrição de TFD, o operador S é utilizado para avaliar o processo de produção de pares elétron-pósitron à temperatura finita, e uma análise do limiar de produção é feita a partir do cálculo da probabilidade total de transição.

The second quantization methods is used to build the scattering operator S in Fock space, in the contex of Thermofield Dynamics (TFD), for the Dirac field subject to an external time-dependent electromagnetic potential. This description is based on the constructive approach to the Fock space, wich is applied to the original system and to its dual. Following TFD prescription, the operator S is used to estimate the process of electron-positron pair production at finite temperature, and an analysis of the production threshold is done based on the calculation of the total transition probability.

Advisors/Committee Members: Universidade Estadual Paulista (UNESP), Tomazelli, Jeferson de Lima [UNESP], Escobar, Bruto Max Pimentel [UNESP].

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Será explorada a versão hidrodinâmica da equação de Schrödinger não-linear e não-local, descrevendo condensados de Bose-Einstein com auto-interações de longo alcance. Tais sistemas têm despertado interesse tendo em vista a busca da realização da condensação de Bose-Einstein sem necessidade de um potencial externo confinante e nos quais as interações atômicas locais não são suficientes. Para obter a descrição hidrodinâmica, a transformação de Madelung para a função de onda será utilizada, reduzindo o problema a uma equação da continuidade e a uma equação de transporte de momentum. Esta última é similar à equação de Euler em fluidos ideais, porém contendo um potencial quântico efetivo e um termo não local, o qual advém da interação atômica. Tais equações de fluido traduzem, respectivamente, a conservação da probabilidade e do momentum total. O método hidrodinâmico permitirá o estudo de excitações elementares, entre os quais os modos de Bogoliubov, segundo uma abordagem macroscópica.

The hydrodynamic version of the Schrödinger equation nonlinear and nonlocal will be explored, describing Bose-Einstein condensates with long-range self-interactions. Such systems have aroused interest with a view to pursuing the realization of Bose-Einstein condensation without an external confining potential and in which local atomic interactions are not enough. For the hydrodynamic description, the eikonal decomposition of the wave function is used, reducing the problem to one equation of continuity and to a transport of momentum equation. The latter is similar to the Euler equation in ideal fluid but containing an effective quantum potential and a nonlocal term, which comes from the atomic interaction. Such fluid equations translate, respectively, conservation of probability and total momentum. The hydrodynamic method will allow the study of elementary excitations, including Bogoliubov modes according to a macroscopic approach.

Advisors/Committee Members: Haas, Fernando, Rizzato, Felipe Barbedo.

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La fission induite par neutron, découverte il y a plus de 70 ans, a de nombreuses applications, par exemple industrielles pour la production d'énergie, et intervient dans la nucléosynthèse. Cependant, sa description microscopique reste un problème ouvert. En effet, les degrés de liberté qui interviennent dans ce processus dynamique sont complexes. De plus, les noyaux fissiles ont un nombre élevé de nucléons en interaction (>200). Il s'agit donc d'un problème à N-corps quantique. Or, une résolution directe de ce dernier n'est pas possible à l'heure actuelle. Dans ce contexte, la description microscopique de la fission considérée ici est la suivante : la première étape consiste à déterminer un ensemble de configurations de champ moyen qui représentent différentes déformations du noyau, incluant ainsi explicitement les degrés de liberté collectifs qui leur sont associés. Dans la seconde étape, la dynamique est décrite dans cet espace de configurations en utilisant la méthode de la coordonnée génératrice dépendante du temps (TDGCM). L'approximation des recouvrements gaussiens (GOA) est alors utilisée. Cependant, elle introduit une erreur de modèle et limite les extensions comme par exemple la prise en compte explicite de degrés de liberté intrinsèques. Ce travail de thèse a pour objectif de décrire le processus de fission avec la TDGCM sans recourir à la GOA. Cela implique de résoudre l'équation de la dynamique en TDGCM appelée équation de Hill-Wheeler dépendante du temps (TD-HW). Les méthodes d'évaluations des matrices des recouvrements et du hamiltonien collectif sont présentées dans le cas d'une interaction de Gogny. La matrice des recouvrements représente la métrique de l'espace des configurations, et la matrice du hamiltonien collectif contient les couplages énergétiques entre les configurations. Les configurations sont exprimées dans des bases de particules deux à deux distinctes, introduisant des instabilités numériques dans les méthodes d'évaluation standard. Un formalisme adapté à ces bases est proposé permettant d'éliminer ces instabilités. Deux méthodes de résolution de TD-HW sont présentées. La première consiste à calculer l'opérateur d'évolution associé à l'équation de Hill-Wheeler dépendante du temps. Elle est adaptée à un faible nombre de configurations. La seconde utilise un schéma de discrétisation en temps permettant l'inclusion d'un plus grand nombre de configurations dans le modèle. Ce formalisme est ensuite appliqué à la description de la réaction de fission induite par neutron sur le plutonium 239, et une comparaison avec la TDGCM+GOA est effectuée.

Nuclear fission, where an atomic nucleus separates into two fragments while emitting a large amount of energy, is at the core of many applications in society (energy production) and national security (deterrence, non-proliferation). It is also a key ingredient of the mechanisms of formation of elements in the universe. Yet, nearly 80 years after its experimental discovery its theoretical description in terms of the basic constituents of the nucleus…

Advisors/Committee Members: Pillet, Nathalie (thesis director).

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Cette thèse est consacrée à l'étude mathématique et numérique du modèle de Hartree-Fock-Bogoliubov (HFB) pour les systèmes quantiques attractifs, qui est abondamment utilisé en physique nucléaire. Après avoir présenté le modèle et ses principales caractéristiques, nous expliquons comment le discrétiser et nous montrons des résultats de convergence. Nous examinons tout particulièrement l'algorithme de point fixe (parfois appelé Roothaan) et montrons qu'il converge ou alors oscille entre deux états dont aucun n'est solution du problème. Ceci généralise au cadre HFB des résultats de Cancès et Le Bris pour le modèle plus simple de Hartree-Fock dans le cas répulsif. Suivant ces mêmes auteurs, nous proposons un algorithme basé sur la contrainte relachée et pour lequel la convergence est garantie. Dans dernière partie de la thèse, nous illustrons le comportement de ces algorithmes par des simulations numériques pour plusieurs modèles. Dans un premier temps nous considérons un système purement gravitationnel où les particules interagissent avec le potentiel de Newton. Nos simulations montrent que la matrice d'appariement est toujours non nulle, un fait qui n'a pas encore pu être démontré rigoureusement. Nous étudions ensuite un modèle très simplifié pour la description de protons et neutrons dans le noyau atomique.

This work is devoted to the theoretical and numerical study of Hartree-Fock-Bogoliubov (HFB) theory for attractive quantum systems, which is one of the main methods in nuclear physics. We first present the model and its main properties, and then explain how to discretize it. We prove some convergence results, in particular for the simple fixed point algorithm (sometimes called Roothaan). We show that it converges, or oscillates between two states, none of them being a solution. This generalizes to the HFB case previous results of Cancès and Le Bris for the simpler Hartree-Fock model in the repulsive case. Following these authors, we also propose a relaxed constraint algorithm for which convergence is guaranteed. In the last part of the thesis, we illustrate the behavior of these algorithms by some numerical experiments. We first consider a system where the particles only interact through the Newton potential. Our numerical results show that the pairing matrix never vanishes, a fact that has not yet been proved rigorously. We then study a very simplified model for protons and neutrons in a nucleus.

Advisors/Committee Members: Lewin, Mathieu (thesis director).

▼ The crystallization and melting behavior of linear polyethylene and of a series of random ethylene/styrene copolymers was investigated using a combination of classical and temperature modulated differential scanning calorimetry. In the case of linear polyethylene and low styrene content copolymers, the temporal evolutions of the melting temperature, degree of crystallinity, and excess heat capacity were studied during crystallization. The following correlations were established: 1) the evolution of the melting temperature with time parallels that of the degree of crystallinity, 2) the excess heat capacity increases linearly with the degree of crystallinity during primary crystallization, reaches a maximum during the mixed stage and decays during secondary crystallization, 3) the rates of shift of the melting temperature and decay of the excess heat capacity lead to apparent activation energies that are very similar to these reported for the crystal ac relaxation by other techniques. Strong correlations in the time domain between the secondary crystallization and the evolution of the excess heat capacity suggest that the reversible crystallization/melting phenomenon is associated with molecular events in the melt-crystal fold interfacial region. In the case of higher styrene content copolymers, the multiple melting behavior at high temperature is investigated through studies of the overall crystallization kinetics, heating rate effects and partial melting. Low melting crystals can be classified into two categories according to their melting behavior, superheating and reorganization characteristics. Low styrene content copolymers still exhibit some chain folded lamellar structure. The shift of the low melting temperature with time in this case is tentatively explained in terms of reorganization effects. Decreasing the crystallization temperature or increasing the styrene content leads to low melting crystals more akin to fringed-micelles. These crystals exhibit a lower tendency to reorganize during heating. The shift of their melting temperature with time is attributed to a decrease in the conformational entropy of the amorphous fraction as a result of constraints imposed by primary and secondary crystals. To further understand the mechanism of formation of low melting crystals, quasi-isothermal crystallization experiments were carried out using temperature modulation. The evolution of the excess heat capacity was correlated with that of the melting behavior. On the basis of these results, it is speculated that the generation of excess heat capacity at high temperature results from reversible segmental exchange on the fold surface. On the other hand, the temporal evolution of the excess heat capacity at low temperature for high styrene content copolymers is attributed to the reversible segment attachment and detachment on the lateral surface of primary crystals. The existence of different mechanisms for the generation of excess heat capacity in different temperature ranges is consistent with the observation of… Advisors/Committee Members: Marand, Hervé L. (committeechair), Long, Timothy E. (committee member), Crawford, T. Daniel (committee member), Ducker, William A. (committee member), Esker, Alan R. (committee member).

▼ The paradigm of cosmic inflation has had great success in explaining the statistical properties of fluctuations in the Cosmic Microwave Background (CMB). In this dissertation we discuss a few avenues for modeling and constraining the inflationary universe - constraints on excited states of inflationary fluctuations, some aspects of multi-field tunneling and also constraints on and predictions from a specific model of inflation connecting Higgs physics and dark matter. First, we show that in standard single field slow roll inflation, Bogoliubov excitations of the fluctuation spectrum are tightly constrained by observations. These constraints ensure that the squeezed limit non-gaussianity obtained from such excited states cannot be large. They also rule out any significant imprints in the CMB coming from a sudden transition from kinetic energy domination to inflation. We then explore tunneling in the context of field theory, a scenario that has potential relevance to the pre-inflationary universe. We discuss subtleties involved in choosing the trajectory for tunneling out of a metastable vacuum in a multi-field potential. In particular, we use exact solutions and scaling relations to show that tunneling may happen along directions with large barriers, thus making the common intuition coming from quantum mechanical tunneling unreliable in estimating the tunneling trajectory and therefore, the bounce action. We then explore a specific model of inflation that involves the addition of a scalar singlet and fermionic dark matter to the standard Higgs inflation scenario. We show that dark matter constraints and the requirement to support successful inflation significantly constrain the available parameter space for this model. We also find that the model generically predicts a small value of the tensor-to-scalar ratio r, similar to standard Higgs inflation, though it allows for a larger range of values for the scalar spectral tilt nS. Advisors/Committee Members: Paban, Sonia (advisor), Fischler, Willy (committee member), Distler, Jacques (committee member), Kilic, Can (committee member), Shapiro, Paul R (committee member).

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No contexto da teoria de Bogoliubov determinamos as configurações de equilíbrio e as excitações coletivas de um condensado de Bose-Einstein espinorial homogêneo com spin hiperfino S = 1, na presença e na ausência de um campo magnético externo. Na tese mapeamos as configurações de equilíbrio em função dos parâmetros e m, onde q está relacionado com a intensidade do termo quadrático da energia de Zeeman, c2 é a intensidade do termo da interação átomo-átomo dependente do spin, p é a densidade do condensado e m é a magnetização por partícula. Pelo exame do comportamento dos ramos de energia das excitações coletivas como função do momento determinamos as configurações de equilíbrio estáveis e mostramos que é possível classificá-las pela miscibilidade das componentes a = O e a = f 1, que é uma consequência direta da simetria axial no espaço de spin. O exame do diagrama de fase do sistema indica que ele depende crucialmente do caráter antiferromagnético ou ferromagnético dos átomos. No limite antiferromagnético o estado fundamental é imiscível e de fase indeterminada. Por outro lado no limite ferromagnético o estado fundamental pode ser miscível e de fase determinada. Em contrapartida verificamos a dominância do termo quadrático da energia de Zeeman em ambos os casos, no limite antiferromagnético quando e no limite ferromagnético quando & > 2. Fenômenos tais como o colpaso do condensado e transições de fase são também possíveis. Este trabalho se diferencia dos demais pelo fato de levar em conta explicitamente a conservação da magnetizaçáo do sistema, que nos permitiu fazer um estudo sistemático das configurações de equilíbrio, o que pode servir de guia para futuros estudos de efeitos que ocorrem tanto nas regiões estáveis quanto nas instáveis.

In the framework of the Bogoliubov theory, we determined the equilibrium configurations and the collective excitations of a homogeneous Bose-Einstein S = 1 spinor condensate, in the presence and absence of an externa1 magnetic field. In this thesis we found the equilibrium configurations as function of the parameters and m, where q is the intensity of the quadratic term of the Zeeman energy, ca is the intensity of the spin dependent atom-atom interaction term, p is the condensate density and m the magnetization per particle. By the study of the behaviour of the collective excitation energies as function of the moment, we found the stable equilibrium configurations and we show that they can be classified by the miscibility of the components a = O and a f 1, which is a direct consequence of the axial symmetry in the spin space. Examining the phase diagram, we see that it depends on the antiferromagnetic or ferromagnetic character of the atoms. In the antiferromagnetic limit, the ground state is imiscible and with an undetermined phase. However in the ferromagnetic limit the ground state can be miscible and with a fixed phase. On the other hand, we see in both cases the dominance of the quadratic term of the Zeeman energy, wlien & > O in the antiferromagnetic limit and wlien…

Advisors/Committee Members: Emerson Jose Veloso de Passos, Vanderlei Salvador Bagnato, Frederico Firmo de Souza Cruz, Mahir Saleh Hussein, Antonio Fernando Ribeiro de Toledo Piza.

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Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions. The topological origin makes their key features independent of systems' microscopic details

therefore TDs display many universalities. Because of their strong impact on numerous material properties and their significant role in several technological applications it is of strong interest to find simple and robust mechanisms controlling the positioning and local number of TDs. There are strong evidences that in physics the fields are fundamental entities of nature and not particles. If this is the case then topological defects (TDs) might play the role of fundamental particles. An adequate testing ground to study and gain fundamental understanding of TDs are nematic liquid crystals. We present a numerical study of TDs within effectively two dimensional closed soft films exhibiting in-plane orientational ordering. Popular examples of such class of systems are liquid crystalline shells and various biological membranes. We analyze the impact of extrinsic and intrinsic curvature on positions of topological defects (TDs) in two-dimensional (2D) nematic films. We demonstrate that both these curvature contributions are commonly present and are expected to be weighted by comparable elastic constants. A simple Landau-de Gennes approach in terms of tensor nematic order parameter is used to numerically demonstrate impact of the curvatures on position of TDs on 2D ellipsoidal nematic shells. We introduce the Effective Topological Charge Cancellation mechanism controlling localised positional assembling tendency of TDs and the formation of pairs (defect,antidefect) on curved surfaces. Furthermore, we estimate a critical depinning threshold to form pairs (defect,antidefect) using the electrostatic analogy. Finally, we show how one could efficiently switch among qualitatively different structures by using a relative volume of ordered shells, which represents a relatively simple naturally accessible control parameter. In doctoral thesis, we developed theoretical model of erythrocyte membrane by using a hybrid Helfrich-Landau type mesoscopic approach, taking into account in-plane membrane ordering. We demonstrate that the derived extrinsic membrane energy term, which strongly depends on the local orientations of the molecules, is essential for the predicted broadening of the range of the relative volumes corresponding to the stable discocyte shapes, which is otherwise very narrow if only intrinsic curvature energy term dominates.

Topološki defekti se v naravi pogosto pojavijo ob zveznih faznih prehodih z zlomom simetrije. Zaradi njihovega topološkega izvora so glavne značilnosti topoloških defektov neodvisne od mikroskopskih lastnosti sistema. Topološki defekti imajo močan vpliv na številne lastnosti snovi in pomembno vlogo v nekaterih tehnoloških aplikacijah, zato obstaja interes, da odkrijemo enostavne in robustne mehanizme, ki kontrolirajo prostorsko porazdelitev in število toploških defektov v različnih sistemih. V fiziki je…

Advisors/Committee Members: Kralj, Samo.

▼ In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice. Advisors/Committee Members: Hutchinson, David (advisor).

▼ Nucleons in most nuclei tend to form pairing correlations in the ground state through the spin-singlet channels, with like-particle pairs making up the majority of these. In contrast, we know that the spin-triplet interaction is stronger than the spin- singlet interaction. This discrepancy has motivated much study in the past, and has not yet been resolved conclusively. Recent investigations into ground states of heavy (A > 100) nuclei have revealed a possibility of some nuclei near the proton dripline exhibiting spin-triplet ground states or possibly a mixed-pairing state. However, these investigations excluded some mean-field terms in the Hamiltonian. In this work, I reintroduce those terms and examine the effects of their inclusion. I also study the nature of the mixed pairing state and look into possible reasons for the aforementioned discrepancy. Additionally, I examine the dependence of my results on detailed features of the Hamiltonian. Advisors/Committee Members: Gezerlis, Alexandros (advisor).

▼ We apply the Hartree-Fock-Bogoliubov-Popov (HFBP) method to two problems. The first is the phenomenon of Anderson localisation of a BEC in a two-dimensional bichromatic lattice. The second is the case of a harmonically trapped BEC with anisotropic, long-range, dipolar interactions between particles. The HFBP method allows us to calculate the condensate properties and excitation spectrum of the system at finite temperature. For the bichromatic lattice, we examine the effects of interactions and finite temperature on the the localisation of the condensate. We also include a synthetic magnetic field and determine its effect. For the dipolar BEC, we examine the effect of the dipolar interactions on the condensate and thermal cloud as well as the excitation spectrum of the system. At finite temperature, we calculate the effects of both direct and exchange dipolar interactions of the thermal cloud on the condensate. The exchange interactions are numerically intensive to calculate and have been excluded from most previous studies. Advisors/Committee Members: Hutchinson, David (advisor).

▼ This dissertation deals with recent applications of Bogoliubov transformation to the phenomenology of quantum flavor mixing and to the study of critical phenomena in quantum field theories. The dissertation contains a brief review of canonical transformations, with a special emphasis on linear quantum canonical transformation (Bogoliubov transformation), as they appear in classical and quantum physics and their applications in superfluidity and low energy quantum chromodynamics (QCD). Then, the general quantum field theory of flavor mixing is introduced with Bogoliubov transformation, space-time conversion is considered and effects for phenomenology of flavor oscillations in time and space is presented. Furthermore, the Oscillator Representation Method, relevant to analysis of degrees-of-freedom rearrangement during phase transitions, is fully reviewed and illustrated. Nontrivial vacuum condensation, dynamical mass generation and duality are all incorporated as parts of this approach. Specific applications to phase transition in nonlinear sigma model and phi∧4 scalar quantum field theory are presented and possibilities for further method improvement are considered. A new independent variational approach, method of Symmetric Decomposition Problem, is also fully introduced, illustrated and applied to the analysis of the ground state in a variant of nonlinear sigma model. In this method, the structure of the Fock space in terms of the expectation values of given quantum operators is found and used to reformulate and exactly solve variational problem for the ground state of the above mentioned quantum field theory. Advisors/Committee Members: David Brown, Committee Member (advisor), Ronald Fulp, Committee Member (advisor), Dean Lee, Committee Member (advisor), Chueng-Ryong Ji, Committee Chair (advisor).