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.

This thesis examines the behaviour of two new models exhibiting topological solitons. This analysis is predominantly numerical, but a limited collective coordinate approach is attempted where appropriate. In chapter 1 we review the field of solitons. In particular the nature of topological solitons and their associated mathematical formalism are explained. A number of models admitting solitons are defined. In chapter 2 we look at the numerical methods necessary to solve the time evolution of topological solitons in the S(^2) sigma model and the baby Skyrme model. We also examine methods for finding static solutions of the equations of motion of such models. In chapter 3 we define the RP(^2) sigma and baby Skyrme models. We examine the behaviour of these models and find them to be identical to their (S^2) counterparts for most field configurations. The topological reason for this is explained. The existence of a topological object called a defect is noted and the behaviour of solitons in the presence of a defect is examined. A collective coordinate approach is used to examine the behaviour of solitons in the presence of a defect. In chapter 4 the easy plane baby Skyrme model is defined. An ansatz for the static skyrmions is proposed and its energy found to be accurate to 1.2% for the 1-skyrmion and about 0.5% for 2 to 4-skyrmions. These skyrmions are composed of two quasi- independent soliton like objects which we name "half lumps". These objects may not exist alone. The scattering properties of these objects are examined numerically. The behaviour of these scattering processes are explained in terms of the fields and potential energy of their intermediate states in the simulation. In chapter 5 we summarise our work and propose future work in this field.

In the Skyrme model, nuclei are described as topological solitons known as Skyrmions. To make contact with nuclear data one must quantise these Skyrmions; most calculations to date have used rigid body quantisation, where the Skyrmions are allowed to rotate but remain rigid. The method reproduces some experimental results for light nuclei but there are some contradictions with data. In this thesis we study a more sophisticated quantisation scheme where the Skyrmions may deform, called vibrational quantisation, in the hope of fixing some of these problems. Vibrational quantisation is applied to the dodecahedral $B=7$ Skyrmion, which models Lithium-$7$. Using rigid body quantisation, the Skyrme model predicts a spin $\frac{7}{2}$ ground state while in reality the Lithium-$7$ nucleus ground state has spin $\frac{3}{2}$. We show that a quantisation which includes a $5$-dimensional vibrational manifold of deformed Skyrme configurations remedies this problem, giving the correct ground state spin. Further, the model leads to a robust prediction that the ground state of the nucleus has a larger root mean square matter radius than the second quantum state, in contrast with standard nuclear models. We consider the vibrational modes of the tetrahedral $B=16$ Skyrmion, to describe Oxygen-$16$. Motivated by Skyrme dynamics, a special $2$-dimensional submanifold of configurations is constructed. We study the manifold in detail by modelling it as a $6$-punctured sphere with constant negative curvature. The Schr\"odinger equation is solved on the sphere and the results give an excellent fit to the experimental energy spectrum. The model describes an energy splitting between certain states with equal spins but opposite parities, which is hard to explain in other models. We also find the first ever isospin $0$, spin-parity $0^-$ state in the Skyrme model. A method to calculate electromagnetic transition rates between states is formulated and then applied to our system. By considering a special type of Skyrme configuration, where a single Skyrmion orbits a large core, we show that the Skyrme model can reproduce a classical spin-orbit force due to the structure of the Skyrme fields. We quantise this model to try and find out if the classical picture holds quantum mechanically.

Le contrôle du magnétisme aux interfaces s’est avéré essentiel pour la spintronique et ses applications, en particulier celles basées sur des structures chirales de spin appelées skyrmions magnétiques. Ces skyrmions, décrits comme des solitons magnétiques, sont de potentiels vecteurs d’information. Dans des empilements ultraminces du type métal lourd / ferromagnétique / isolant, les skyrmions sont stabilisés par une interaction d’échange antisymétrique d’interface appelée interaction de Dzyaloshinskii-Moriya (DMI); celle-ci entre en compétition avec d’autres interactions telles que l’interaction d’échange symétrique ou l’anisotropie magnétique. Afin de contrôler ces skyrmions, les propriétés magnétiques aux interfaces doivent être ajustées finement et modulées par une excitation extérieure. Le champ électrique s’est avéré être un outil efficace pour manipuler ces propriétés d’interface. Il a notamment été montré dans un certain nombre d’études depuis 2009 qu’une différence de potentiel permet de modifier localement et de manière dynamique des propriétés telles que l’anisotropie magnéto-cristalline ou l’aimantation à saturation. Cependant, cet effet sur DMI, qu’il est crucial d’intégrer pour les systèmes avec skyrmions, n’avait pas été observé pour les films ultraminces.Cette thèse présente tout d'abord une optimisation des systèmes tricouches de type métal lourd/ferromagnétique/oxide dans lesquels peuvent exister des skyrmions. J’ai plus particulièrement étudié le système Ta/FeCoB/TaOx qui permet de énucléer des skyrmions en présence d’un faible champ magnétique appliqué perpendiculairement au plan des couches. Une étude approfondie en fonction de l’épaisseur de FeCoB et de l’état d’oxydation du TaOx a notamment été menée, permettant ainsi d’identifier les différentes zones présentant des skyrmions. D’autre part, le résultat majeur de cette thèse est la démonstration de la modulation de DMI par un champ électrique dans une tricouche Ta/FeCoB/TaOx. Des mesures de spectroscopie Brillouin sous champ électrique ont montré une très grande variation allant jusqu’à 130%. Puis, des observations complémentaires en microscopie à effet Kerr magnéto-optique ont permis de mesurer simultanément une variation monotone de DMI et de la taille des skyrmions en fonction du champ électrique avec une efficacité sans précédent. Puisque le champ électrique agit principalement sur l’interface entre le matériau ferromagnétique et l’oxyde (FeCoB/TaOx), cette étude indique l’existence d’une DMI de type Rashba, expliquant la forte sensibilité au champ électrique. Ces observations montrent également qu’un renversement du signe de l’IDM est possible, qui conduirait à une inversion de la chiralité des skyrmions. Cette manipulation dynamique de DMI permettrait de conférer un degré de contrôle supplémentaire pour le développement de mémoires ou de dispositifs logiques ou neuromorphiques à base de skyrmions. / Control of interfacial magnetism has emerged to be of paramount importance for spintronics applications specially involving chiral magnetic structures called skyrmions. Skyrmions are envisaged to be the future information carriers owing to their solitonic properties. In heavy metal/ ferromagnet/ insulator heterostructures, skyrmions are stabilized by interfacial Dzyaloshinskii-Moriya interaction which is an antisymmetric exchange and competes with other interactions like symmetric exchange and magnetic anisotropy. In order to tune skyrmions, the interfacial magnetic properties need to be modulated. One of the energy efficient tools to maneuver interfacial magnetism is electric field effect. Voltage gating has been shown, in a number of studies since 2009, to locally and dynamically tune magnetic properties like interface anisotropy and saturation magnetization. However, its effect on interfacial Dzyaloshinskii-Moriya Interaction (DMI), which is crucial for the stability of magnetic skyrmions, has been challenging to achieve and has not been reported yet for ultrathin films.This thesis demonstrates an optimization of trilayer systems consisting of a heavy metal/ ferromagnet/ oxide where skyrmions can be stabilized. In particular, I focussed on the Ta/FeCoB/TaOx system to nucleate skyrmions in the presence of very small out of plane magnetic field. Further, the different skyrmionic zones as a function of the FeCoB thickness and TaOx oxidation state are studied. We then show electric field induced modulation of interfacial DMI which forms the most important result of this thesis. We demonstrate 130% variation of DMI with electric field in Ta/FeCoB/TaOx trilayers through Brillouin Light Spectroscopy (BLS). Using polar Magneto-Optical-Kerr-Effect microscopy, we further show a monotonic variation of DMI and skyrmionic bubble size with electric field, with an unprecedented efficiency. Since the electric field acts mainly on the FeCoB/TaOx interface, this study also points at the existence of the Rashba DMI explaining its high sensitivity to an applied voltage. We anticipate through our observations that a sign reversal of DMI with electric field is possible, leading to a chirality switch. This dynamic manipulation of DMI establishes an additional degree of control to engineer programmable skyrmion based memory, logic or neuromorphic devices.

The Skyrme model enables us to approximate nuclei via topological solitons known as Skyrmions. The B = 4 Skyrmion is of particular importance as its symmetry and stability means that multiple B = 4 Skyrmions can combine with each other to form larger B = 4N Skyrmions. In this thesis we investigate the properties of these B = 4N Skyrmions and compare them with results found in the wider nuclear physics community. We go beyond rigid body quantization and develop a formalism of using vibrational quantisation to generate the energy spectrum of the Oxygen−16 nucleus. The Oxygen−16 nucleus is treated as an arrangement of four B = 4 Skyrmions, whose dynamics enable us to create a 2−dimensional manifold of B = 16 configurations. We solve the Schrödinger equation on this manifold and discover new states previously not found in the B = 16 sector of the Skyrme model. We compare these states with those found experimentally and find that there is a excellent it to the energy spectrum. In order to apply vibrational quantization to a wider range of nuclei we create a novel approximation for Skyrmions and the interactions between them. By generating Skyrmions with Gaussian sources we find analytic expressions for the pion fields and interaction energies of Skyrmions, with particular focus on the B = 1 and B = 4 Skyrmions, and show how this could be applied to vibrational quantization and the clustering of B = 4 Skyrmions. B = 4N nuclei are the only nuclei with zero spin and isospin, which means that their electric charge density is proportional to their baryon density. This simplification makes these nuclei particularly susceptible to investigation via electron scattering. We develop a classical averaging method to calculate the Patterson function and the form factor for a B = 4N nucleus and make comparisons with experimental data. We also discover a way of using the baryon density directly to approximate the locations of zeroes and stationary points of the form factor.

The work in this thesis is concerned with the numerical study of some stability and scattering properties of two CP¹ models in three dimensional space-time: The non-linear 0(3) model and its modified Skyrme version. Chapter 3 focuses principally on the Skyrme model on compactified plane, the topological sphere. Such model is obtained by supplementing the ordinary 0(3) lagrangian with both a Skyrme term and a potential term which, in the present work, has a rather general form. Under the numerical simulation the skyrmions behave stably and scatter either back-to-back or at 90 to the initial direction of motion, depending on the initial velocity. In the 0(3) limit the solitons are no longer stable and scatter at 90 irrespective of the speed. In the fourth chapter the 0(3) model is studied on a flat torus. Its solitons exhibit the usual instability but can be stabilised by the sole addition of a Skyrme term to the lagrangian. Scattering at right angles is observed in all cases considered, including skyrmions colliding at speeds that would bounce them back were they evolving in compactified plane. The periodic 0(3) model has no analytic solutions of degree one, so when a field configuration that resembles a single soliton is numerically evolved, it shrinks to become infinitely thin. Interestingly, such ansatz may be regarded as a soliton of unit topological charge in the context of the periodic skyrmion model. Chapter 5 closes with a summary and suggestions for future research.

No description available.

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.

No description available.

In this work, we present our investigations into two different systems, both far from equilibrium. We first present the relaxation and ordering processes in magnetic skyrmion systems. This is followed by a study of the behavior of many species interacting on a spatially heterogeneous lattice. Magnetic skyrmions have been a subject of great interest in recent years. They have been proposed to be at the heart of next-generation computing and information storage devices. One interesting feature of magnetic skyrmions is the presence of the non-dissipative Magnus force. The Magnus force causes the skyrmions to be deflected from their direction of motion. In this work, we examine the effect the strength of this Magnus force has on the late-time ordering behavior of magnetic skyrmions. We show that the late-time ordering also shows enhanced relaxation with an increase in the Magnus force. We also studied the behavior of magnetic skyrmions when confined to a narrow channel. We show that, like before, the Magnus force helps the system order faster while experiencing a constant drive. Interestingly, when the drive was periodic, the Magnus force inhibited the relaxation in the system. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of spatial heterogeneity on a two-dimensional lattice. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We show that, due to the non-linearity of the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156. / Doctor of Philosophy / In this work, we present our investigations into two different systems. Both of these systems are considered to be not in equilibrium. We first present is the behavior of magnetic skyrmions as the system settles into an arranged state. This is followed by a study of the behavior of multiple species interacting on a lattice where different parts of the lattice have different rules of interaction. Magnetic skyrmions are small defects that occur in specific types of magnetic materials. They have been proposed to be useful in next-generation computing devices. Similar to a curve-ball in baseball, but due to a different physical phenomenon, magnetic skyrmions follow curved paths when pushed. This effect, known as a Magnus force, causes the magnetic skyrmions to settle faster into a position relative to the other magnetic skyrmions in the system. We show that this effect also occurs when the magnetic skyrmions are being pushed through a narrow channel. If the push is periodically started and stopped, the Magnus force instead slows down the ability for magnetic skyrmions to settle into a position relative to the other magnetic skyrmions. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of changing the rules of species interaction based on where on a two-dimensional lattice the interaction occurred. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We showed that, due to the complex interaction scheme present in the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156.