La thèse porte sur l'étude de deux classes d'oxyde par spectroscopie d'absorption X, en mode diffraction résonante pour la magnétite et en mode absorption pour les pérovskites. La structure électronique de Fe3O4 a été étudiée au seuil K du Fe pour confirmer/réfuter le modèle d'ordre de charge à basse température. La méthode développée a permis d'obtenir une information quantitative en utilisant un grand nombre de réflexions et de confirmer la présence de l'ordre de charge (Fe2.5±δ δmax=0.12, δmin=0.04). Les pérovskites (PbZrO3 et BaZrO3) ont été étudiés pour mieux comprendre la nature géométrique de leur transition de phase. L'analyse de spectres XANES au seuil K du Zr pour PbZrO3 a infirmé le modèle "déplacement" à basse température mais il a indiqué la conservation des distorsions locales dans sa phase cubique. Une interprétation de la transition structurale ferroélectrique est proposée en terme de changement d'environnement local du Zirconium dans le cadre du modèle "sphérique".

La Théorie de la Fonctionnelle de la Densité dépendant du temps constitue un outil de choix pour l'étude des mécanismes élémentaires d'irradiation moléculaire. Mais les approximations qui lui sont inhérentes n'éliminent pas un effet non physique appelé auto-interaction, ce qui fausse complètement les propriétés d'irradiation. La voie la plus prometteuse permettant de supprimer l'auto-interaction sans introduire aucun paramètre libre supplémentaire est d'utiliser des fonctionnelles "orbitales dépendantes" (méthodes SIC). Seulement, le formalisme usuel qui en découle n'est pas hermitique, faussant dramatiquement les prédictions physiques dans le cas dynamique, et les tentatives visant à rétablir l'hermiticité connaissent toutes des pathologies indésirables. Ainsi, la question, dans le cas dépendent du temps, d'un formalisme SIC exact (TDSIC), satisfaisant les lois de conservation et numériquement manipulable reste une question ouverte. Nous proposons une nouvelle formulation purement variationnelle, contraignant l'orthonormalité et utilisant le degré de liberté de transformation unitaire. Cela permet d'écrire les équations TDSIC exactes sous une forme hermitique (dans le sous espace occupé), satisfaisant toutes les lois de conservation et menant à un schéma numérique de propagation clair. Le prix à payer est que le hamiltonien résultant est explicitement non local, ce qui est plus gourmand numériquement parlant. Cela nous a conduit à proposer, dans un deuxième temps, une approximation locale particulièrement intéressante, que nous avons baptisée "Generalized SIC-Slater". Enfin, nous proposons un ensemble de résultats numériques sur des systèmes moléculaires variés afin de soumettre les formalismes développés au verdict de la nature et les comparer aux formalismes SIC usuels.

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<p>This dissertation consists of two parts. In the first part, we discuss the boundary and domain wall theories of the generalized quantum double lattice realization of the two-dimensional topological orders based on Hopf algebras. The boundary Hamiltonian and domain wall Hamiltonian are constructed by using Hopf algebra pairings and generalized quantum double. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras, respectively. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.</p> <p><br></p> <p>In the second part, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems, and we establish weak Hopf symmetry breaking theory based on the fusion closed set of anyons. We present a thorough investigation of the quantum double model based on weak Hopf algebras, including the topological excitations and ribbon operators, and show that the vacuum sector of the model has weak Hopf symmetry. The gapped boundary and domain wall theories are also established. We show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra, and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. We also introduce the weak Hopf tensor network states, by which we solve the weak Hopf quantum double models on closed and open surfaces. Lastly, we discuss the duality of the quantum double phases.</p>

<p dir="ltr">We study the dynamics of a Bose-Einstein condensate (BEC) to find solutions that correspond to a sonic black/white hole. In such configurations, the condensate goes from subsonic to supersonic when crossing a particular region that can be understood as a horizon. This is because sound cannot go back from the supersonic to the subsonic region. Therefore, in this set-up the speed of sound plays the role same as that of the speed of light in a gravitational black/white hole, an important difference being that there are excitations that can go faster than the speed of sound and therefore can escape (enter) the sonic black (white) hole. These analog gravity models are of a particular interest in the study of the nature of black holes in simple laboratory setups. Here, the motion of the Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation (GPE). We discuss singular Stationary [1] and Self-similar [2] solutions of Gross-Pitaevskii Equation (GPE) in 2D (with Circular symmetry) and 3D (with Spherical symmetry). We use these solutions to study the approximate local speed of sound and magnitude of flow velocity of the condensate to see whether they cross, indicating the potential existence of a sonic analog of a black/white hole. We discuss different approaches to solve for these solutions, such as numerical techniques and the semi-analytical Laplace-Borel resummation of asymptotic transseries solutions. We check these Laplace-Borel resummations of the transseries to see how well they agree with numerical solutions. Particularly in the case of stationary solutions, we also study and demonstrate the utility of deep learning for these differential equations, phenomenon of resurgence in transseries, and trans-asymptotic summation. Furthermore, considering the practical limitations in achieving the singularities in the stationary condensate density, we also regularize [3] (remove the singularity from) the singular stationary solutions by putting a background spatial metric and/or artificially constructed specific kinds of external potentials and variable couplings so that this configuration can (at least as a better approximation) be achieved in a laboratory setup. We use these to study fluctuations in the density and phase of the regularized solutions and derive acoustic metrics under some approximations to study these analog gravity systems in context of Hawking radiation (temperature) through some approximations.</p>

On étudie la limite thermodynamique pour un système de particules quantiques en interaction dans un milieu aléatoire dans le formalisme de l'ensemble microcanonique. L'existence de la limite est démontrée pour l'énergie interne ainsi que pour l'entropie sous des conditions assez générales sur le modèle à une particule. Ensuite, la limite thermodynamique est étudiée pour le modèle des pièces à une dimension dans le cas des fermions et pour une densité de particules faible. On donne une caractérisation de l'état fondamental en terme d'espaces fonctionnels, on démontre sa non dégénérescence presque sûre et on décrit sa fonction d'autocorrélation. Ces résultats permettent d'obtenir une estimée de l'énergie fondamentale par particule comme fonction de la densité de particules dans la limite thermodynamique.

In the first part of this thesis, we study the duality of Wilson loops and M2-branes in AdS₄/CFT₃. We focus on supersymmetric M-theory solutions on AdS₄xY₇ that have a superconformal dual description on S³ = ?AdS⁴. We will find that the Hamiltonian function h_M for the M-theory circle plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle is supersymmetric precisely at the critical points of h_M, and moreover the value of this function at those points determines the M2-brane actions. Such a configuration determines the holographic dual of a Wilson loop for a Hopf circle in S³. We find agreement in large classes of examples between the Wilson loop and its dual M2-brane and also that the image h_M(Y₇) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h_M mapping to points where the derivative of the eigenvalue density is discontinuous. We will then move away from the three-sphere and construct gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localisation. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories defined on a general class of background three-manifold geometries. To further verify that our gravitational backgrounds are indeed dual to field theories on their boundaries, we compute Wilson loops and their M2-brane duals in this general setting. We find that the Wilson loop is given by a simple closed formula which depends on the background geometry only through the supersymmetric Killing vector field. The supergravity dual M2-brane precisely reproduces this large N field theory result. This constitutes a further check of AdS₄/CFT₃ for a very broad class of examples.

Spectral and wave function statistics of the quantum directed graph, QdG, are studied. The crucial feature of this model is that the direction of a bond (arc) corresponds to the direction of the waves propagating along it. We pay special attention to the full Neumann digraph, FNdG, which consists of pairs of antiparallel arcs between every node, and differs from the full Neumann graph, FNG, in that the two arcs have two incommensurate lengths. The spectral statistics of the FNG (with incommensurate bond lengths) is believed to be universal, i.e. to agree with that of the random matrix theory, RMT, in the limit of large graph size. However, the standard perturbative treatment of the field theoretical representation of the 2-point correlation function [1, 2] for a FNG, does not account for this behaviour. The nearest-neighbor spacing distribution of the closely related FNdG is studied numerically. An original, efficient algorithm for the generation of the spectrum of large graphs allows for the observation that the distribution approaches indeed universality at increasing graph size (although the convergence cannot be ascertained), in particular "level repulsion" is confirmed. The numerical technique employs a new secular equation which generalizes the analogous object known for undirected graphs [3, 4], and is based on an adaptation to digraphs of the idea of wave function continuity. In view of the contradiction between the field theory [2] and the strong indications of universality, a non-perturbative approach to analysing the universal limit is presented. The substitution of the FNG by the FNdG results in a field theory with fewer degrees of freedom. Despite this simplification, the attempt is inconclusive. Possible applications of this approach are suggested. Regarding the wave function statistics, a field theoretical representation for the spectral average of the wave intensity on an fixed arc is derived and studied in the universal limit. The procedure originates from the study of wave function statistics on disordered metallic grains [5] and is used in conjunction with the field theory approach pioneered in [2].

Neste trabalho nós propomos uma estratégia Euclidiana para entender o efeito Unruh. Com este objetivo, nós inicialmente o estudamos para campos livres escalares sem massa, numa forma que é normalmente apresentada aos físicos e que é mais próxima ao trabalho original de Unruh I321| . Logo em seguida, deduzimos o efeito de um ponto de vista algébrico. Com este objetivo, estudamos as propriedades e as definições de estados KMS para compreender como um estado de equilíbrio é descrito na abordagem algébrica. Apresentamos os axiomas de Wightman para campos escalares assim como os de Osterwalder-Schrader. Usamos, então, o Teorema de Bisognano-Wichmann para estes campos e concluímos, baseados no trabalho de Sewell [27], que um observador uniformemente acelerado vê o estado de vácuo dos observadores inerciais como um estado KMS, e portanto, como um estado de equilíbrio. Novamente, concluímos a existência do efeito Unruh. Finalmente estudamos algumas relações entre probabilidade e análise funcional. Este estudo é fundamental para o entendimento do trabalho de Klein e Landau [15] e de Gérard e Jakel [7]. Estes trabalhos afirmam que existe uma relação biunívoca entre certos estados KMS e certos processos estocásticos (Klein e Landau) e uma relação entre certos processos estocásticos e espaços de trajetórias generalizados (Gérard e Jakel). Usando estes trabalhos e as funções de Schwinger para campos escalares, deduzimos o efeito Unruh de uma nova maneira. Acreditamos que este trabalho mostra um ponto de vista interessante do efeito Unruh e ilustra o uso do formalismo Euclidiano em teorias quânticas dos campos. Mesmo que algumas demonstrações para uma prova completa do efeito, usando técnicas Euclidianas, não são obtidas, devido às dificuldades técnicas encontradas, acreditamos que o material apresentado neste trabalho fornece, no mínimo, uma boa estratégia para a compreensão completa deste fenômeno físico. Além disto, as técnicas que são mostradas podem ser usadas em diversos problemas, como a construção de campos interagentes a uma temperatura finita, que permanecem atuais e promissores. / This paper proposes a Euclidean strategy to understand the Unruh effect. On that ground we first study it for free massless scalar fields the way it is usually presented to pliysicists, which is closer to Unruh\'s original work [32]. Then we infer the effect from an algebraic perspective. We study the proprieties and definitions of KklS states in order to understand the description of an equilibrium state in the algebraic approach. We present the Wightman\'s as well as Osterwalder-Schrader\'s axioms for scalar fields. Then we use the Bisognano-Wichmann theorem for these fields and conclude, based on Sewell work 1271, that a uniformly accelerated observer will observe tlie vacuum state of inertial observers as a KMS state and thus as an equilibrium state. Once again we infer the existence of the Cnruh effect. Finally we study some relations between probability and functional analysis. This study is crucial for understanding the work of Klein and Landau 1151 as well as of Gérard and Jakel (71. They state there is a biunivocal relation between certain KMS states and certain stochastic processes (Klein and Landau) and a relation between certain stochastic processes and generalized path spaces (Gérard and Jakel). Lsing these works and Schwinger functions for scalar fields, we deduce tlie Unruh effect in a new way. LVe believe this work shows an interesting aspect of the Unruh effect and represents the use of Euclidean formalism in quantum field theory. Although some demonstrations for a complete proof of the Unruh effect using Euclidean techniques were not obtained due to technical difficulties we faced, we believe the material presented in this paper provides at least a good strategy for the complete understanding of this physical phenomenon. Furthermore the techniques shown, which remain current and promising, can be used in different problems, sudy as the construction of interacting fields at a finite temperature.