M.Sc. (Physics) / The way we live has been fundamentally changed by technological innovations based on optical, electronic and magnetic materials. Without the continuous increase of scienti c understanding on phenomena that occur in materials, together with the processing and synthesis of materials, these technological revolutions would be impossible. Thus, the search of new materials is still the key driving force for the continuous blooming of modern technology...

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Na natureza há vários fenômenos envolvendo transições de fase com quebra ou restauração de simetrias. Tipicamente, mudanças de fase, são associadas com uma quebra ou restauração de simetria, que acontecem quando um determinado parâmetro de controle é variado, como por exemplo temperatura, densidade, campos externos, ou de forma dinâmica. Essas mudanças que os sistemas sofrem podem levar a formação de defeitos topológicos, tais como paredes de domínios, vórtices ou monopolos magnéticos. Nesse trabalho estudamos particularmente mudanças de fase associadas com quebras ou restaurações dinâmicas de simetria que estão associadas com formação ou destruição de defeitos do tipo paredes de domínio em modelos de campos escalares com simetria discreta. Nesses processos dinâmicos com formação ou destruição de domínios, estudamos a possibilidade de formação de estruturas do tipo oscillons, que são soluções não homogêneas e instáveis de campo, mas que podem concentrar nelas uma quantidade apreciável de energia e terem uma vida (duração) suficientemente grande para serem de importância física. Estudamos a possibilidade de formação dessas soluções em modelos de dois campos escalares interagentes em que o sistema é preparado em diferentes situações, com a dinâmica resultante nesses sistemas estudada numa rede discreta. / In nature there are several phenomena involving phase transitions with symmetries breaking or restoration. Typically,the phase state changes associated with a break or a restoration of the sistem's symmetry, that occur when a particular control parameter is varied, such as as temperature, density, external fields or dynamically. These systems undergo changes which can lead to formation of defects, such as domain walls, vortices or magnetic monopoles. In this work we study particularly phase changes, breaks or restorations, associated with dynamic symmetry that are associated with the formation or destruction of defects, as domain walls in models of scalar fields with discrete symmetry. In these dynamical processes with formation or destruction of domains we studied the possibility of forming oscilons type structures, which are not homogeneous and unstable field solutions, but can concentrate there in an appreciable amount of energy and have a lifetime (duration) large enough to have physical importance. We study the possibility of formation of oscillon kind solutions in models with scalar fields interacting in which the system is prepared for different situations, with the resulting dynamics of these systems studied in a discrete lattice.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-11-06T20:00:02Z No. of bitstreams: 1 Marcos Lima Dias.pdf: 1365744 bytes, checksum: 2aa8dc442d4e20438c570810fb9d2ab8 (MD5) / Made available in DSpace on 2017-11-06T20:00:02Z (GMT). No. of bitstreams: 1 Marcos Lima Dias.pdf: 1365744 bytes, checksum: 2aa8dc442d4e20438c570810fb9d2ab8 (MD5) Previous issue date: 2017-04-13 / CAPES / The theoretical study about the existence of magnetic monopole has begun by P. A. M. Dirac in 1931 and its main consequence has been the possibility of explaining the quantization of the electric charge. In Grand Uni cation Theories (GUT's), the magnetic monopole may appear as a topological defect by spontaneous symmetry break, in a process known as Higgs mechanism. The GUT's also predict the existence of a topological soliton which has both magnetic and electric charges: the so-called dyon. Theoretically, these particles are predicted very massive and abundant in the Universe. There are particular cases of monopole and dyon, called BPS's states, which have the smallest possible mass values, besides analytical solutions. In this work, we present some basic ideas about topology and homotopy; we describe the main points that have led to the development of the theoretical study of monopole and dyon both as topological solitons; and nally we investigate the e ects of the Lorentz-violating CPTeven terms in the Gauge (in particular, non birefringent one) and Higgs sectors on the BPS's states of the 't Hooft-Polyakov magnetic monopole and of the Julia-Zee dyon, both given in the Georgi-Glashow SO(3) model. / O estudo teórico a respeito da existência do monopolo magnético foi iniciado por P. A. M. Dirac em 1931 e teve como principal consequência a possibilidade de explicar a quantização da carga elétrica. Em Teorias de Grande Unificação (GUT's), o monopolo magnético pode surgir como um defeito topológico por meio da quebra espontânea de simetria, em um processo conhecido como mecanismo de Higgs. As GUT's preveem também a existência de um sóliton topológico que possui tanto carga magnética quanto elétrica: o chamado dyon. Teoricamente, essas partículas se apresentam muito massivas e abundantes no Universo. Hé casos particulares do monopolo e do dyon, denominados estados BPS's, que têm os menores valores possíveis de massa, cujas soluções são analíticas. Neste trabalho, apresentamos algumas ideias básicas acerca de topologia e homotopia; descrevemos os principais pontos que levaram ao desenvolvimento do estudo teórico do monopolo e do dyon como sólitons topológicos; e investigamos finalmente os efeitos dos termos CPTpares de violação da simetria de Lorentz nos setores de Gauge (em especial, n~ao birrefrigente) e de Higgs sobre os estados BPS's do monopolo magnético de't Hooft-Polyakov e do dyon de Julia-Zee, ambos dados no modelo SO (3) de Georgi-Glashow.

Dans cette thèse, notre étude porte sur les défauts topologiques en cosmologie et en physique de la matière condensée. Nous proposons d’étudier l’analogie entre les défauts qui apparaissent dans ces domaines. Par exemple, nous discutons l’analogie entre les cordes cosmiques et les disinclinaisons présentes dans les cristaux liquides. Cependant, nous concentrons nos efforts sur l’étude de la gravité et des aspects géométriques des défauts linéaires; cordes cosmiques «ondulées» et les disinclinaisons hyperboliques. Le champ gravitationnel des cordes ondulées est analogue à celui des cordes régulières mais avec un potentiel newtonien non nul. Pour cette raison, outre le fait que l’espace-temps a une géométrie conique, les cordes produisent également une traction gravitationnelle sur les objets situés dans leur voisinage. Ceci a pour conséquence: l’apparition de nouveaux effets en plus de la lentille gravitationnelle. En réalité, nous constatons que les champs massiques et non-massiques sont confinés au voisinage de la corde quand nous examinons la propagation non perpendiculaire à son axe. Cette affirmation est basée sur l’analyse des trajectoires et l’application du formalisme ondulatoire. Enfin, nous proposons la conception d’un guide d’onde optique ayant l’indice de réfraction spécifique permettant de simuler l’effet des cordes ondulées en laboratoire. Bien que la disinclinaison hyperbolique soit très similaire à la disinclinaison ordinaire, le milieu résultant est un métamatériau anisotrope dans lequel le rapport entre les permittivités ordinaire et extraordinaire est négatif. En fait, notre analyse concernant la disinclinaison hyperbolique étant très proche de la précédente pour la corde ondulée, nous appliquons alors le formalisme ondulatoire avec des traitements d’optique géométrique pour comprendre la propagation de la lumière. Nous trouvons que pour un agencement spécifique du champ directeur du cristal liquide hyperbolique, la lumière est confinée et dirigée vers l’axe du métamatériau. Plus précisément, on obtient un dispositif optique qui concentre les rayons de lumière, quelles que soient les conditions d’injection / In this thesis, we study topological defects in cosmology and condensed matter physics. We propose to investigate the analogy between defects that appear in these two realms with respect to aspects of formation and gravity. For instance, we discuss the analogy between cosmic strings and disclinations existing in liquid crystals. In particular, we concentrate our efforts on the study of gravity/geometric aspects of two linear defects: wiggly cosmic string and hyperbolic disclination. The gravitational field of wiggly strings is analogous to the one of regular strings, but with a non-vanishing Newtonian potential. For that reason, besides the spacetime has conical geometry, the string also produces gravitational pullings on objects located in its vicinity. This new fact leads to new effects in addition to the already expected gravitational lensing. In fact, we find that both massive and massless fields are confined to the string when we examine non-perpendicular propagation to the string axis. This statement is based on the analysis of trajectories and by applying the wave formalism. Finally, we propose the design of an optical waveguide having the specific refractive index likely to simulate the effects of wiggly strings in the laboratory. Even though the hyperbolic disclination is very similar to the ordinary one, the resulting medium is an anisotropic metamaterial, in which the ratio between ordinary and extraordinary permittivities is negative. In fact, as our analysis concerning the hyperbolic medium is very close to the previous one for the wiggly string, we also apply the wave formalism together with geometrical optics treatment to understand how light propagates. Interestingly, we find that for one specific director field arrangement of the hyperbolic liquid crystal metamaterial, light is confined and directed to the metamaterial axis. More specifically, we come into possession of an optical device that concentrates light rays, no matter how they are injected in it

In this thesis we study the connection between continuum quantum field theory and corresponding lattice field theory, specifically for two cases: Landau gauge fixing and ’t Hooft-Polyakov monopoles. To study non-perturbative phenomena such as the confinement mechanism of quarks and gluons and dynamical chiral symmetry breaking in Quantum Chromodynamics (QCD), there are two major approaches: the Dyson-Schwinger equations (DSEs) approach, which is based on the covariant continuum formulation, and lattice gauge theory. The strength and beauty of lattice gauge theory is due to the fact that gauge invariance is manifest and fixing a gauge is not required. In the covariant continuum formulation of gauge theories, on the other hand, one has to deal with the redundant degrees of freedom due to gauge invariance and has to fix gauge (most popularly, Landau gauge). There, the gauge-fixing machinery is based on the so-called Faddeev-Popov procedure or more generally, the Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Beyond perturbation theory this is aggravated by the existence of so-called Gribov copies, however, that satisfy the same gauge-fixing condition, but are related by gauge transformations, and are thus physically equivalent. When attempting to fix Landau gauge on the lattice to make a connection with its continuum counterpart, this ambiguity manifests itself in the Neuberger 0/0 problem that asserts that the expectation value of any physical observable will always be of the indefinite form 0/0. We explain the topological nature of this problem and how the complete cancellation of Gribov copies can be avoided in a modified lattice Landau gauge based on a new definition of gauge fields on the lattice as stereographically projected link variables. For compact U(1), where the Gribov copy problem is related to the classification the local minima of XY spin glass models, we explicitly show that there still remain Gribov copies but their number is exponentially reduced in lower dimensional models. We then formulate the corresponding Faddeev-Popov procedure on the lattice, for these models. Moreover, we explicitly demonstrate that the proposed modification circumvents the Neuberger 0/0 problem for lattices of arbitrary dimensions for compact U(1). Applied to the maximal Abelian subgroup this will avoid the perfect cancellation amongst the remaining Gribov copies for SU(N), and so the corresponding BRST formulation is also then possible for generic SU(N), in particular, for the Standard Model groups. For higher dimensional lattices, the gauge fixing conditions for both the standard and the modified lattice Landau gauges are systems of multivariate nonlinear equations, solving which in general is a highly non-trivial task. However, we show that these systems can be interpreted as systems of polynomial equations. They can then be solved exactly by computational Algebraic Geometry, the Groebner basis technique in particular, and numerically by the Polynomial Homotopy Continuation method. ’t Hooft-Polyakov monopoles play an important role in high energy physics due to their presence in grand unified theories and their usefulness in studying non-perturbative properties of quantum field theories through electric-magnetic dualities. In the second part of the thesis, we study adjoint Higgs models, which exhibit ’t Hooft-Polyakov monopoles, and have been extensively analyzed using semi-classical analysis in the continuum. However, to study them in a fully nonperturbative fashion, it is essential to put the theory on the lattice. Here, we investigate twisted C-periodic boundary conditions in SU(N) gauge field theory with an adjoint Higgs field and show that for even N with a suitable twist one can impose a non-zero magnetic charge relative to each of N − 1 residual U(1)’s in the broken phase, thereby creating ’t Hooft-Polyakov magnetic monopoles. This makes it possible then to use lattice Monte-Carlo simulations to study the properties of these monopoles in the full quantum theory and compare them with the existing results in the continuum. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2010

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-10-10T18:29:15Z No. of bitstreams: 1 DAMARES SANTOS SILVA BORGES – DISSERTAÇÃO (PPGFísica) 2018.pdf: 14903609 bytes, checksum: 19118754fe275415f701d012e1d4515e (MD5) / Made available in DSpace on 2018-10-10T18:29:15Z (GMT). No. of bitstreams: 1 DAMARES SANTOS SILVA BORGES – DISSERTAÇÃO (PPGFísica) 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.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-17T12:00:18Z No. of bitstreams: 1 arquivo total.pdf: 7270155 bytes, checksum: 94d354ad73c370abbc34bc7904858976 (MD5) / Made available in DSpace on 2016-03-17T12:00:18Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 7270155 bytes, checksum: 94d354ad73c370abbc34bc7904858976 (MD5) Previous issue date: 2015-02-25 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / This dissertation deals with localized solutions in several dimensions. Initially, we review some topics in classical theory of fields by using scalar fields both in flat and curved spacetime. Here they are: kinks, lumps, vortices, monopoles, dyons, cosmology with scalar fields and branes modeled by scalar fields. After this, we present some original results. Firstly, by introducing a real or integer parameter in the models, we study a way to go from kinks to compactons in models with standard dynamics and show how the procedure works in the braneworld scenario. Finally, by using the deformation method, we present new models that support compact structures and then study their linear stability. / Esta dissertação lida com soluções localizadas em diversas dimensões. Inicialmente, fazemos uma revisão de alguns tópicos em teoria clássica de campos usando campos escalares tanto no espaço plano como no espaço curvo. São eles: kinks, lumps, vórtices, monopolos, dyons, cosmologia com campos escalares e branas modeladas por campos escalares. A seguir apresentamos alguns resultados originais. Primeiramente, introduzindo nos modelos um parâmetro que pode ser real ou inteiro, estudamos uma maneira de ir de kinks para compactons em modelos com dinâmica padrão e mostramos como o procedimento funciona no cenário de mundo brana. Por fim, usando o método da deformação, apresentamos novos modelos que suportam estruturas compactas e estudamos sua estabilidade linear.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-12T13:49:32Z No. of bitstreams: 1 arquivototal.pdf: 884862 bytes, checksum: 8232d731ca2ae96983306136b97d4f8e (MD5) / Made available in DSpace on 2017-09-12T13:49:32Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 884862 bytes, checksum: 8232d731ca2ae96983306136b97d4f8e (MD5) Previous issue date: 2016-02-17 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work, it present a bibliographical approach about topological defects especi­ally the monopoles with absence of gauge fields. It begins by introducing a gradual recap on the major types of defects, and how their approaches change as the Lagran­gian assumes greater symmetry groups. Next, remains attention only to models that support monopoles. Firstly , it's studied the global monopole SO(3), presen­ting through an analytical approach, its main features such as asymptotic space with angular deficit and null gravitational interaction. Posteriorly, it's shown that monopole solutions can arise also in field theory with Lorentz symmetry violation. / Neste trabalho, apresenta-se uma abordagem bibliografica sobre defeitos topolOgicos corn destaque para os monopolos corn ausencia de campos de gauge. Inicia-se in­troduzindo uma recapitulacao gradual sobre os principais tipos de defeitos e, como suas abordagens variam a medida que a lagrangiana assume maiores grupos de si­metria. Ern seguida, mantem-se atencao apenas ern modelos que suportam solucOes tipo monopolo. Primeiramente, estuda-se o monopolo global SO(3), apresentando por meio de uma abordagem analitica suas principais caracteristicas, tais como espaco assintOtico corn deficit angular e interacdo gravitacional nula. Posterior­mente, mostra-se que solucOes tipo monopolo podem surgir, tambem, ern teorias de campo corn violacdo da simetria de Lorentz.

Made available in DSpace on 2015-05-14T12:14:03Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 548925 bytes, checksum: 2e6888057f2c80f2c45e7c38283dacaf (MD5) Previous issue date: 2006-12-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The principal aim this work is the thermodynamic caracterization of bidimensional systems in non-Euclidian geometries as XY models, Helium superfluids films and liquid crystal films. These systrems present local order in two dimensions, then topological defects arise naturally. The significative contribution for the free energy in this systems it must at the topological defects, then a good caracterization these systems is given by understanding the physics of topological defects. This way, we develop the content of this dissertation with the purpose of to understanding the behave of the systems described here in non-euclidian geometries, as well contributions of these geometries for the phase transitions ocurred in these systems, as we do in case of sphere, cone, hyperboloid. / O objetivo principal deste trabalho é a caracterização termodinâmica de sistemas bidimensionais em geometrias não euclidianas tais como modelos XY, filmes de Hélio superfluido, filmes finos de cristal líquido. E como em duas dimensões estes sistemas apresentam apenas ordem local, então neles surgem naturalmente os defeitos topológicos. A contribuição significativa para a energia deve-se ao defeitos topológicos, então uma boa caracterização destes sistemas se dá pelo entendimento da física dos defeitos topológicos. Sendo assim desenvolvemos o conteúdo deste trabalho de tal forma que ao final podemos entender como os sistemas descritos aqui se comportam em geometrias não euclidianas, bem como contribuições destas geometrias para transições de fase sofridas por estes, como fazemos no caso da esfera, do cone e do hiperbolóide.

Made available in DSpace on 2015-05-14T12:14:04Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 1188345 bytes, checksum: a5c721a567eb9bd7d22f6ee817eea8ff (MD5) Previous issue date: 2012-06-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present new results regarding topologically non-trivial configurations arising in some generalized classical field theories. We focus on static finite-energy vortices which arise when a spontaneous symmetry breaking of U(1) local gauge invariance takes place in some Abelian-Higgs models. First, we perform a brief review regarding the usual vortices. The usual structures emerge in three diferent scenarios: the Maxwell-Higgs electrodynamics, the Chern-Simons-Higgs electrodynamics and the Maxwell-Chern-Simons-Higgs electrodynamics. In all these cases, there are BPS topological vortices: energetically stable radially symmetric configurations satisfying a set of first order equations, their energy being proportional to their topological charge. We solve the BPS equations explicity for diferent values of the vorticity n, and we comment on the main features the numerical solutions we found engender. A posteriori, we introduce a generalized model, described by an arbitrary function K (X), where X = |Dϕ|2 gives the dynamic of the complex scalar field. Specifically, we choose K (X) = X − αX2, from which we get generalized non-BPS vortices only (the resulting model does not allow for BPS ones). These vortices have no electric field (as the Maxwell-Higgs ones), but also have a non-trivial topological charge (since they are topological). We then solve the second order Euler-Lagrange equations and comment on the main features the generalized solutions we found engender. Then, we introduce a second generalized model. The new model is specifed by two dimensionless functions of the scalar field, G(|ϕ|) and w (|ϕ|). These two functions are suposed to obey a constraint relating them to the unspecifed Higgs potential, V (|ϕ|). In this case, BPS topological vortices exist, and their numerical features can be quite similar, or quite diferent, from the standard ones.vi Finally, we adapt the second generalized model to the study of twinlike theories, which are diferent theories allowing the very same field solutions and the very same energy. In this case, G(|ϕ|), w (|ϕ|) and V (|ϕ|) are suposed to obey two new constraints. Here, there are twinlike BPS topological vortices related to diferent energy densities, all of them giving the very same total energy. / Neste trabalho, apresentamos alguns resultados novos sobre configurações de topologia não-trivial obtidas no âmbito de algumas teorias clássicas de campo generalizadas. Em particular, nos concentramos em vórtices estáticos possuidores de energia finita. Tais configurações surgem quando há a violação espontânea da invariância de gauge local U(1) no âmbito de alguns modelos Higgs-Abelianos. Primeiro, fazemos uma breve revisão sobre os vórtices usuais. As estruturas usuais surgem em três cenários diferentes: a eletrodinâmica de Maxwell-Higgs, a eletrodinâmica de Chern-Simons-Higgs e a eletrodinâmica de Maxwell-Chern-Simons-Higgs. Em todos estes casos, ocorrem vórtices BPS topológicos: configurações com simetria rotacional e energeticamente estáveis que satisfazem um conjunto de equações de primeira ordem, sua energia sendo proporcional à sua carga topológica. Resolvemos as equações BPS explicitamente para diferentes valores da vorticidade n, e enunciamos as principais características que as soluções numéricas resultantes engendram. A posteriori, introduzimos um modelo generalizado descrito por uma função arbitrária K (X), onde X = |Dϕ|2 determina a dinâmica do campo escalar complexo. Especificamente, escolhemos K (X) = X − αX2, da qual resultam apenas vórtices não-BPS generalizados (o modelo assim obtido não engendra vórtices BPS). Estes vórtices engendram campo elétrico nulo (tal como aqueles obtidos no âmbito da eletrodinâmica de Maxwell- Higgs usual), além de carga topológica não trivial (dado serem topológicos). Em seguida, resolvemos as equações de Euler-Lagrange (diferenciais de segunda ordem) e enunciamos as principais características que as soluções generalizadas resultantes engendram. Em seguida, apresentamos um segundo modelo generalizado. O novo modelo é determinado por duas funções adimensionais do campo escalar, G(|ϕ|) e w (|ϕ|). Estas funções são supostas obedecerem um vínculo que relaciona-as ao potencial de Higgs, V (|ϕ|). Neste iv contexto, vórtices BPS topológicos também existem, e as suas características numéricas podem ser muito semelhantes, ou muito diferentes, daquelas verificadas no contexto usual. Finalmente, adaptamos o segundo modelo generalizado ao estudo das teorias gêmeas, i.e., teorias diferentes com exatamente as mesmas soluções, além de exatamente a mesma energia. Neste caso, G(|ϕ|), w (|ϕ|) e V (|ϕ|) são supostas obedecerem dois novos vínculos. Aqui, existem vórtices BPS gêmeos caracterizados por diferentes densidades de energia, todas elas resultando na mesma energia total.