The aim of this thesis is to study topological soliton solutions in classical field theories, called sigma models, on a three-dimensional space. In chapter 1 we review the general field-theoretical framework of classical soliton solutions and exemplify it on the main features of the 0(3) σ-model and the Abehan Higgs model in (2+1) dimensions. In chapter 2 a U(l)-gauged 0(3) σ-model is discussed, where the behaviour of the gauge field is determined by a Chern-Simons term in the action. We find numerical solutions for radially symmetric fields and discuss those of degree one and two. They carry a non-vanishing angular momentum and can be interpreted as classical anyons. A similar model is studied in chapter 3. Here the potential is of Higgs-type and chosen to produce a Bogomol'nyi model where the energy is bounded from below by a linear combination of the topological degree of the matter fields and the local U(l)-charge. Depending on internal parameters, the solutions are solitons or vortices. We study them numerically and prove for a certain range of the matter field's vacuum value that there cannot be a 1-soliton.In chapter 4 we discuss a modified 0(3) σ-model in (3+0) dimensions. The topological stability of the solitons is here imphed by the degree of the map S(^3) → S(^2), which provides a lower boundon the potential energy of the configuration. Numerical solutions are obtained for configurations of azimuthal symmetry and the spectrum of slowly rotating solitons is approximated. Chapter 5 deals with a theory where the fields are maps IR(^2+1) → CP(^2). The Lagrangian includes a potential and a fourth-order term in the field-gradient. We find a family of static analytic solutions of degree one and study the 2-soIiton configuration numerically by using a gradient-flow equation on the moduli space of solutions. We conclude this thesis with a brief summary and give an outlook to open questions.

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Heterogeneous ferromagnetic-superconducting systems such as a regular array of ferromagnetic nano dots deposited on the top of a superconducting thin film have attracted many research teams both experimental and theoretical. The interest in these systems does not only stem from being good candidates for technological applications, but also because they represent a new class of physical systems where two competing order parameters can coexist. This work focuses on the theoretica laspects of these systems by studying the static and dynamics of few model systems. In the first part, the static properties of a superconducting thin film interacting with a ferromagnetic texture are considered within the London approximation. In particular, the ferromagnetic textures considered here are a circular dot of submicrometer size with in-plane magnetization, an elliptical dot magnetized in the direction perpendicular to the superconductor, and a ferromagnetic dot magnetized in the direction normal to the superconducting film and containing non magnetic cavities. I also consider the interaction of vortices in the superconductor with a ferromagnetic columnar defect which penetrates the supercondcting film. In each case the vector potential and magnetic field of the ferromagnet in the presence of the superconductor are calculated. Afterward the presence of vortices in the superconductor is assumed and the energy of vortex-texture system is found. The pinning potential and force supplied by the texture are then derived from the energy of interaction between the ferromagnet and superconductor. I show that if the magnetization of the ferromagnet exceeds a critical value then vortices are spontaneously created in the ground state of the system. Such spontaneous creation of vortices is possible mostly in a close vicinity of the superconducting transition temperature Ts. For every case, the threshold value of the magnetization at which vortices start to be spontaneously created in the SC is calculated as a function of the parameters of the texture geometry. The phase diagrams for transitions from vortexless regime to regimes with one or more vortices are determined for all cases. In the second problem, the transport properties of a ferromagnetic superconducting bilayer with alternating magnetization and vortex density are studied within a phenomenological model. I show that pinning forces do not appear for continuous distribution of vortices, so a discrete model for the bilayer system is constructed. Afterward, I calculate the pinning forces acting on vortices and antivortices resulting from highly inhomogeneous distribution of flux lines and prove that this system has strong transport anisotropy. In the absence of random pinning, the system displays a finite resistance for the current in the direction perpendicular to the domains while its resistance vanishes for the parallel current. The transport anisotropy strongly depends on temperature. I study this dependence and show that the ratio of parallel to perpendicular critical current is largest close to the superconducting transition temperature Ts and the vortex disappearance temperature Tv while it has a minimum in between them.

Heterogeneous ferromagnetic-superconducting systems such as a regular array of ferromagnetic nano dots deposited on the top of a superconducting thin film have attracted many research teams both experimental and theoretical. The interest in these systems does not only stem from being good candidates for technological applications, but also because they represent a new class of physical systems where two competing order parameters can coexist. This work focuses on the theoretica laspects of these systems by studying the static and dynamics of few model systems. In the first part, the static properties of a superconducting thin film interacting with a ferromagnetic texture are considered within the London approximation. In particular, the ferromagnetic textures considered here are a circular dot of submicrometer size with in-plane magnetization, an elliptical dot magnetized in the direction perpendicular to the superconductor, and a ferromagnetic dot magnetized in the direction normal to the superconducting film and containing non magnetic cavities. I also consider the interaction of vortices in the superconductor with a ferromagnetic columnar defect which penetrates the supercondcting film. In each case the vector potential and magnetic field of the ferromagnet in the presence of the superconductor are calculated. Afterward the presence of vortices in the superconductor is assumed and the energy of vortex-texture system is found. The pinning potential and force supplied by the texture are then derived from the energy of interaction between the ferromagnet and superconductor. I show that if the magnetization of the ferromagnet exceeds a critical value then vortices are spontaneously created in the ground state of the system. Such spontaneous creation of vortices is possible mostly in a close vicinity of the superconducting transition temperature Ts. For every case, the threshold value of the magnetization at which vortices start to be spontaneously created in the SC is calculated as a function of the parameters of the texture geometry. The phase diagrams for transitions from vortexless regime to regimes with one or more vortices are determined for all cases. In the second problem, the transport properties of a ferromagnetic superconducting bilayer with alternating magnetization and vortex density are studied within a phenomenological model. I show that pinning forces do not appear for continuous distribution of vortices, so a discrete model for the bilayer system is constructed. Afterward, I calculate the pinning forces acting on vortices and antivortices resulting from highly inhomogeneous distribution of flux lines and prove that this system has strong transport anisotropy. In the absence of random pinning, the system displays a finite resistance for the current in the direction perpendicular to the domains while its resistance vanishes for the parallel current. The transport anisotropy strongly depends on temperature. I study this dependence and show that the ratio of parallel to perpendicular critical current is largest close to the superconducting transition temperature Ts and the vortex disappearance temperature Tv while it has a minimum in between them.

Submitted by Maria Medeiros (maria.dilva1@ufcg.edu.br) on 2018-08-29T13:41:49Z No. of bitstreams: 1 DAMARES SANTOS SILVA BORGES - DISSERTAÇÃO (PPGF) 2018.pdf: 14903609 bytes, checksum: 19118754fe275415f701d012e1d4515e (MD5) / Made available in DSpace on 2018-08-29T13:41:49Z (GMT). No. of bitstreams: 1 DAMARES SANTOS SILVA BORGES - DISSERTAÇÃO (PPGF) 2018.pdf: 14903609 bytes, checksum: 19118754fe275415f701d012e1d4515e (MD5) Previous issue date: 2018-07-26 / Capes / Os defeitos topológicos são caracterizados como soluções estáveis de equações de movimento em uma ou mais dimensões espaciais e desempenham papel importante na ciência não-linear. Neste trabalho de dissertação, damos ênfase a defeitos em (1+1) e (2+1) dimensões espaço-temporais. No primeiro caso, abordamos configurações conhecidas como twistons (soluções topológicas tipo kink) presentes em cristais de polietileno. Nessa primeira abordagem, revisitamos trabalhos anteriores e, a partir do método de extensão, construímos novas famílias de potenciais que descrevem bem sistemas desse tipo. Apresentamos soluções topológicas analíticas e que não possuem problemas de degenerescência infinita. No segundo caso, estudamos estruturas conhecidas como skyrmions com base na sua descrição em materiais magnéticos, em que são denotados como configurações da magnetização em nanoescala e topologicamente estáveis.Recorremos novamente ao método de extensão e apresentamos um potencial, função de dois campos escalares acoplados, a partir do qual conseguimos modelar essas estruturas magnéticas. Além disso, o novo modelo de dois campos tem soluções analíticas conhecidas, permitindo análises interessantes como a determinação de uma quantidade topológica conservada, estudo das diferentes configurações da magnetização e cálculo do raio médio de matéria. / Topological defects are characterized as stable equation of motion solutions in one or more spatial dimensions and play an important role in nonlinear science. In this study, space-time (1 + 1) and (2 + 1) dimension defects are emphasized. In the first case, configurations known as twistons (kink-like topological solutions) present in polyethylene crystals are assessed. In this first approach, previous works were reviewed and new families of potentials that adequately describe these types of systems were constructed from the extension method, presenting analytical topological solutions that do not display infinite degeneracy problems. In the second case, structures known as skyrmions were studied based on their description in magnetic materials,where they are denoted as topologically stable nanoscale magnetization configurations. The extension method was applied and a potential from which such magnetic structures can be modelled, function of two coupled scalar fields was presented. In addition, the new two-field model possesses known analytical solutions, allowing for interesting analyses, such as the determination of a conserved topological quantity, the study of the different magnetization configurations and calculation of mean matter radius.

Among a number of fundamental issues, the origin of inertial mass remains one of the major open problems in particle physics. Furthermore, topological effects related to non perturbative field configurations are poorly understood in those gauge theories of direct relevance to our physical universe. Motivated by such issues, this Thesis provides a deeper understanding for the appearance of topological effects in abelian gauge field theories, also in relation to the existence of a mass gap for the gauge interactions. These effects are not accounted for when proceeding through gauge fixings as is customary in the literature. The original Topological-Physical factorisation put forth in this work enables to properly identify in topologically massive gauge theories (TMGT) a topological sector which appears under formal limits within the Lagrangian formulation. Our factorisation then allows for a straightforward quantisation of TMGT, accounting for all the topological features inherent to such dynamics. Moreover dual actions are constructed while preserving the gauge symmetry also in the presence of dielectric couplings. All the celebrated mass generation mechanisms preserving the gauge symmetry are then recovered but now find their rightful place through a network of dualities, modulo the presence of topological terms generating topological effects. In particular a dual formulation of the famous Nielsen-Olesen vortices is constructed from TMGT. Within a novel physically equivalent picture, these topological defects are interpreted as dielectric monopoles.

Submitted by Maria Medeiros (maria.dilva1@ufcg.edu.br) on 2018-08-29T13:13:33Z No. of bitstreams: 1 IVÂNDERSON OLIVEIRA MOREIRA - DISSERTAÇÃO (PPGF) 2018.pdf: 5107079 bytes, checksum: 1e0912f4ced4ad6d7239469d05856b77 (MD5) / Made available in DSpace on 2018-08-29T13:13:33Z (GMT). No. of bitstreams: 1 IVÂNDERSON OLIVEIRA MOREIRA - DISSERTAÇÃO (PPGF) 2018.pdf: 5107079 bytes, checksum: 1e0912f4ced4ad6d7239469d05856b77 (MD5) Previous issue date: 2018-07-26 / Capes / Defeitos topológicos são soluções de equações diferenciais que conectam configurações distintas de um sistema. Para modelos unidimensionais essas soluções são chamadas de kinks. Com isso, neste trabalho vamos estudar as soluções do tipo torção, denominadas twistons topológicos e também investigaremos o comportamento de fermions na presença de estruturas do tipo kink. Primeiramente estudaremos os twistons topológicos presentes no polietileno cristalino que representam uma torção de 180 na cadeia de CH2 causando também uma contração do comprimento da molécula. Pretendemos então construir um modelo efetivo de dois campos que não contenha degenerescência na energia através da aplicação do chamado Método de Extensão, buscando também obter soluções analíticas desse modelo. Após este estudo, voltaremos nossas investigações para as análises de férmions na presença de kinks com o objetivo de obter um controle da energia de limiar (gap de energia onde residem os estados ligados) usando dois campos escalares. / Topological defects are solutions of differential equations that connect distinct confi gurations of a system. For one-dimensional models these solutions are called kinks. In this work we will study the twist-like solutions, called topological twistons and also investigate the behavior of fermions in the presence of kink-like structures. First we will study the topological twistons present in the crystalline polyethylene which represent a 180 twist in the CH2 chain also causing a contraction of the length of the molecule. We intend to construct an effective model of two elds that does not contain degeneracy in the energy through the application of the so called Extension Method, also seeking to obtain analytical solutions of this model. After this study, we will turn our investigations for fermion analyzes in the presence of kinks in order to control of the threshold energy (energy gap where bound states reside) using two scalar elds.