Made available in DSpace on 2015-03-26T13:35:05Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1289687 bytes, checksum: 1760c5305e12b5504a52ff55f7e9035d (MD5) Previous issue date: 2008-02-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / We study Heisenberg model of classical spins lying on the toroidal support. The isotropic regime is characterised by a fractional soliton solution. Whenever the torus size is very large, R &#8594; &#8734;, its charge equals unity and the soliton effectively lies on a infinite cylinder. However, for R = 0 the spherical geometry is recovered and we obtain that configuration and energy of a soliton lying on a sphere. Vortex-like configurations are also supported: in a ring torus (R > r) such excitations present no core where energy could blow up. At the limit R &#8594; &#8734; we are effectively describing it on an infinite annulus (or cylinder, equivalently), where the spins appear to be practically parallel to each other, yielding no net energy. On the other hand, in a horn torus (R = r) a singular core takes place, while for R < r (self- intersecting spindle torus) two such singularities appear. If R is further diminished until vanish we recover vortex configuration on a sphere. Other formal solutions, without topological stability, are obtained and discussed with some details. / Estudamos o modelo de Heisenberg para spins clássicos no suporte toroidal. O regime isotrópico é caracterizado por uma solução solitônica fracionária. Quando o tamanho do toro é muito grande, R &#8594; &#8734;, sua carga se iguala à unidade e o sóliton efetivamente se comporta como no caso do cilindro infinito. Entretanto, para R = 0 a geometria esférica é recobrada e obtemos a configuração e a energia de um sóliton numa esfera. Configurações tipo vórtice também são suportadas: num ring torus (R > r) tais excitações não apresentam caroço onde a energia poderia divergir. No limite R &#8594; &#8734;estamos efetivamente descrevendo-o em um annulus infinito (ou cilindro, equivalentemente), onde os spins aparecem praticamente paralelos um ao outro, não tendo energia líquida. Por outro lado, em um horn torus (R = r) um caroço singular toma lugar, enquanto para R < r (self-intersectind spindle torus) duas singularidades deste tipo aparecem. Se R é diminuído até se anular, recuperamos a configuração de vórtice na esfera. Outras soluções formais, sem estabilidade topológica, são obtidas e discutidas com alguns detalhes.

D'un point de vu fondamental, la structure micromagnétique d'un point de Bloch est prédite depuis plus de 50 ans, mais représente cependant une singularité topologique dans le cadre de la théorie du micromagnétisme. Par conséquent, une description purement micromagnétique du point deBloch s'avère difficile. Ce manuscrit décrit les outils mathématiques et un ensemble d'algorithmes permettant de combiner un modèle d'Heisenberg classique avec des simulations micromagnétiques sur la base des éléments finis. A l'aide de ces algorithmes, nous pouvonsrigoureusement étudier les propriétés caractéristiques d'un point de Bloch d'une paroi de domaine de type vortex localisée dans un nanofil cylindrique ferromagnétique. Cette thèse décrit le pinning/depinning des points de Bloch au réseau atomique ainsi que les différents modes depropagation détectables pour des parois de domaine sous l'influence d'un champ magnétique externe, qui peut conduire le système à des vitesses supermagnoniques. / The fundamental micromagnetic structure of Bloch points was predicted by micromagnetic theory more than 50 years ago, but represents a topological singularity within the theory of micromagnetism. This complicates a pure micromagnetic description. This thesis describes thenecessary mathematical background and a set of algorithms to combine a classical atomistic Heisenberg model with micromagnetism on the basis of the finite element method. By means of those algorithms the characteristic properties of Bloch points in vortex domain walls inferromagnetic solid cylindrical nanowires can be studied rigorously. ln addition to the pinning/depinning of Bloch points at the atomistic lattice the thesis reports on the different modes of propagation detectable for vortex domain walls in that system under the influence of an externalmagnetic field, which can drive the system of the domain wall and the Bloch point with supermagnonic velocities.

Neste trabalho estudamos modelos teóricos que descrevem sistemas eletrônicos fortemente correlacionados, em especial o modelo t-J, e suas aplicações a compostos de óxidos de cobre, notadamente os compostos que apresentam supercondutividade de alta temperatura crítica e o composto Sr2CuO2Cl2. No primeiro capítulo do trabalho, fazemos uma exposição de três modelos que envolvem o tratamento das interações elétron-elétron, que são os modelos de Hubbard de uma banda, o modelo de Heisenberg e o modelo t-J. Na dedução deste último fazemos uma expansão canônica do hamiltoniano de Hubbard, no limite de acoplamento forte, levando-nos a obter um novo hamiltoniano que pode ser utilizado para descrever um sistema antiferromagnético bidimensional na presen- ça de lacunas, que é exatamente o que caracteriza os compostos supercondutores de alta temperatura crítica na sua fase de baixa dopagem.Após termos obtido o hamiltoniano que descreve o modelo t-J, aplicamos à este uma descrição de polarons de spin, numa representação de holons, que são férmions sem spin, e spinons, que são bósons que carregam somente os graus de liberdade de spin. Utilizando uma função de Green para descrever a propagação do polaron pela rede, obtemos uma equação para a sua autoenergia somando uma série de diagramas de Feynman, sendo que para este cálculo utilizamos a aproxima ção de Born autoconsistente[1]. Do ponto de vista numérico demonstramos que a equação integral de Dyson resultante do tratamento anterior não requer um procedimento iterativo para sua solução, e com isto conseguimos trabalhar com sistemas com grande número de partículas. Os resultados mostram, como um aspecto novo, que o tempo de vida média do holon tem um valor bastante grande no ponto (π,0 ) da rede recíproca, perto da singularidade de Van Hove mencionada na literatura[2]. Este aspecto, e suas implicações, é amplamente discutido neste capítulo. No capítulo 3 estudamos o modelo estendido t-t'-J, com tunelamento à segundos vizinhos e a incorporação dos termos de três sítios[3]. Fazemos a mesma formulação do capítulo anterior, e discutimos as aplicações dos nossos resultados ao óxido mencionado anteriormente. Finalmente, no último capítulo apresentamos uma aplicação original do modelo t-J à uma rede retangular, levemente distorcida, e demonstramos que os resultados do capítulo 3 são reproduzidos sem necessidade de introduzir termos de tunelamento adicionais no hamiltoniano. Esta aplicação pode se tornar relevante para o estudo das fases de tiras encontradas recentemente nesses materiais de óxidos de cobre.

Cette thèse porte principalement sur l'étude de modèles de fermions en interactions contenant un couplage spin-orbite. Ces modèles (i) peuvent décrire une classe de matériaux composés d'iridates sur le réseau en nid d'abeille ou (ii) pourraient être réalisés artificiellement dans des systèmes d’atomes froids. Nous avons étudié, dans un premier temps, le système à demi-remplissage avec l'interaction de Hubbard et un couplage spin-orbite anisotrope. Nous avons trouvé plusieurs phases: la phase isolant topologique pour de faibles corrélations, et deux phases avec des ordres magnétiques frustrés, l'ordre de Néel et l'ordre spiral, dans la limite de très fortes corrélations. La transition entre les régimes de faibles et de fortes corrélations est une transition de Mott dans laquelle les excitations électroniques se fractionnent en excitations de charge et de spin. Les charges sont localisées par l'interaction. Le secteur de spin présente de fortes fluctuations qui sont modélisées par un gaz d’instantons. Nous avons ensuite exploré la physique d'un système régi au demi-remplissage par le modèle de Kitaev-Heisenberg, qui présente une phase magnétique de type zig-zag. En dopant le système, autour du quart remplissage, la structure de bande présente de nouveaux centres de symétrie en plus de la symétrie d'inversion. Le couplage de spin de Kitaev-Heisenberg favorise alors la formation de paires de Cooper dans un état triplet autour de ces centres de symétrie. La condensation de ces paires de Cooper autour de ces vecteurs d'onde non triviaux se manifeste par une modulation spatiale du paramètre d'ordre supraconducteur, comme dans la supraconductivité de Fulde–Ferrell–Larkin–Ovchinnikov (FFLO). La dernière partie de la thèse propose et étudie une implémentation des phases topologiques dite de Haldane et de Kane-Mele dans un système avec deux espèces de fermions sur le réseau en nid d'abeille, stabilisée grâce à l’interaction RKKY médiée par l’espèce rapide et qui agit sur l’espèce lente. / This thesis is dedicated largely to the study of theoretical models describing interacting fermions with a spin-orbit coupling. These models (i) can describe a class of 2D iridate materials on the honeycomb lattice or (ii) could be realized artificially in ultra-cold gases in optical lattices. We have studied, in the first part, the half-filled honeycomb lattice model with on-site Hubbard interaction and anisotropic spin-orbit coupling. We find several different phases: the topological insulator phase at weak coupling, and two frustrated magnetic phases, the Néel order and spiral order, in the limit of strong correlations. The transition between the weak and strong correlation regimes is a Mott transition, through which electrons are fractionalized into spins and charges. Charges are localized by the interactions. The spin sector exhibits strong fluctuations which are modeled by an instanton gas. Then, we have explored a system described by the Kitaev-Heisenberg spin Hamiltonian at half-filling, which exhibits a zig-zag magnetic order. While doping the system around the quarter filling, the band structure presents novel symmetry centers apart from the inversion symmetry point. The Kitaev-Heisenberg coupling favors the formation of triplet Cooper pairs around these new symmetry centers. The condensation of these pairs around these non-trivial wave vectors is manifested by the spatial modulation of the superconducting order parameter, by analogy to the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductivity. The last part of the thesis is dedicated to an implementation of the Haldane and Kane-Mele topological phases in a system composed of two fermionic species on the honeycomb lattice. The driving mechanism is the RKKY interaction induced by the fast fermion species on the slower one.

In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1

We investigate whether quantum spin rings with nearest-neighbor Heisenberg or Ising exchange interactions can host solitary states. Using complete diagonalization techniques the system is described without classical or semiclassical approximation. In this case definitions used in connection with classical solitons are not applicable, one needs to redefine what solitary objects on a quantum spin system with translational symmetry ought to be. Thus, we start our contribution by defining which quantum states possess solitary character. In addition we discuss useful observables in order to visualize solitary quantum states. Then we demonstrate for various quantum spin rings that solitary quantum states indeed exist, and that they are moving around the spin ring without changing their shape in the course of time.

Das Standart-Modell zur Beschreibung von Observablen magnetischer Moleküle ist das Heisenberg-Modell. In diesem wird der Magnetismus des Superaustausches der Elektronen durch einfache bilineare Spin-Spin-Kopplungen beschrieben. Zur genaueren Approximation experimenteller Ergebnisse können, der jeweiligen Struktur des Moleküls entsprechend, verschiedene Erweiterungen des Heisenberg-Modells verwendet werden. Diese werden, explizit für das 4-Spin-System {Ni4Mo12}, in ihren Auswirkungen auf die Hochtemperatur-Nullfeldsuszeptibilität, die Nullfeldsuszeptibilität und die Hochfeldmagnetisierung betrachtet. Die wesentlichen Erweiterungen sind dabei die Einzelionen-Anisotropie, die Dzyaloshinskii-Moriya-Anisotropie und die allgemeinen Kopplungen zweiter Ordnung. Letztere stellen eine Verallgemeinerung der bekannten biquadratischen Kopplungen dar und werden im Rahmen eines magneto-elastischen Modells hergeleitet. Dabei ergeben sich unterschiedliche Einschränkungen der Kopplungsmatrix zweiter Ordnung für starre und flexible Molekülstrukturen. Speziell für {Ni4Mo12} entsprechen die Ergebnisse numerischer Simulationen von Messwerten einer Strukturänderung im externen Magnetfeld.

Indiana University-Purdue University Indianapolis (IUPUI) / In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.

Ce mémoire s'inscrit à l'intérieur d'une réflexion systématique et critique portant sur le concept de réalité, à partir des recherches de Planck, Bohr et Heisenberg qui permirent le développement de la physique quantique; celle-là même qui allait bouleverser les fondements scientifiques acquis, sur lesquels reposait notre connaissance du monde. Cependant, la quête pour la connaissance du réel, par le biais de la physique quantique, en tant que langage sur le réel, pose le problème de ce qu'il conviendrait de nommer la réalité, vue à partir d'un langage mathématique essentiellement abstrait. Le problème de l'"accessibilité" au réel, par le biais de telles constructions mentales, est en fait une question de légitimité de l'investigation théorique dans la quête du réel et ce, en tant qu'organisation spécifique d'un langage beaucoup plus vaste sur lequel elle revient constamment s'appuyer, celle de sa raison même de penser le monde existant, de le pressentir à travers la multiplicité de ses manifestations.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.