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Ph.D. (Theoretical Physics) / In this thesis the thermodynamic and magnetic properties of the non-degenerate Hubbard model are investigated. The underlying lattice is the bcc-lattice. The results obtained will therefore be especially applicable to systems with a single, narrow conduction band. As a check the thermodynamic properties of the model system are first calculated in two limiting cases, namely the free electron 'gas and the strong coupling limit. In this process, use is made of results related to Wick's Theorem, which are developed in an appendix. Another check is provided by the calculation of the ground state spectrum of a finite, fourpoint system. These results are obtained using standard group theory techniques. The ground state for the non-degenerate Hubbard model is solved approximatively by a variational method. Once again the necessary version of Wick's theorem is developed in an appendix. The ,results for the neutral case (i.e. a half-filled band) is in agreement with other studies on AB-lattices: It is found that the system is anti ferromagnetic for all values of the coupling constant. The quarter and three-quarter filled cases, hitherto not studied because of numerical complexity, yield a completely different picture. For increasing values of the coupling constant second order phase transitions are found, first from the para- to the ferromagnetic phase and then from the ferro- to the anti ferromagnetic phase. The only results available in the literature related to this case were obtained for an almost half-filled band in the strong-coupling limit and qualitatively support the findings of the present study. It is proposed that the simple theory used in this study be extended for use in physical systems such as Cr.

La mécanique quantique a révolutionné la compréhension du monde microscopique depuis son avènement au XXe siècle. Cependant, les propriétés de la matière condensée restent difficiles à étudier en raison d'une puissance de calcul insuffisante pour simuler numériquement les systèmes à N corps. Une approche alternative consiste à piéger des atomes froids dans des réseaux, dont le comportement est analogue à celui des électrons dans un cristal. Ce système modèle, dont les paramètres peuvent être contrôlés, permet de simuler les phénomènes étudiés.La technique usuellement employée pour confiner les atomes ultrafroids dans un réseau consiste à produire une onde stationnaire résultant de l'intérférence entre deux faisceaux contrapropageants. L'originalité du projet dans lequel s'inscrit cette thèse est de générer un potentiel sublongueur d'onde grâce à la modulation des forces de Casimir au voisinage d'une surface nanostructurée. Le confinement des atomes dans un réseau bidimensionnel avec une faible distance intersite (typiquement 50 nm) permettra ainsi de mieux appréhender les propriétés des matériaux, tels que le graphène.Le travail réalisé au cours de mon doctorat s'est ainsi articulé autour de quatre axes. Tout d'abord, le refroidissement d'atomes de Rubidium 87 a été effectué jusqu'à obtenir un condensat de Bose-Einstein. Puis, des simulations numériques ont été réalisées pour mettre en place une nouvelle méthode d'imagerie sublongueur d'onde, s'appuyant sur le couplage différencié des niveaux atomiques avec un double réseau. Ceci permettra d'activer de façon sélective les sites à détecter pour localiser les atomes avec une précision sublongueur d'onde. Un nouveau système d'imagerie a d'ailleurs été développé pour mieux résoudre les images des distributions atomiques. D'autre part, des simulations numériques ont été réalisées pour anticiper les résultats expérimentaux sur le transport adiabatique au voisinage d'une surface. Enfin, dans le cadre de ma convention CIFRE, une nouvelle architecture laser sera présentée, dans le but d'intéragir avec les atomes de potassium 40 qui seront à refroidir dans la suite du projet dans lequel s'inscrit ma thèse. / Quantum mechanics was a revolution for microscopic systems understanding. However, the study of many-body systems remains a challenge because of computation complexity. Ultracold atoms trapped in lattices offer an alternative way to simulate condensed matter properties. Indeed, their behaviour is similar to the one of electrons in crystals.The common approach for generating optical lattices is to make two laser beams interefere so that we can get a stationary wave that reproduces the potential wells of the crystalline structure. In the new ongoing project, the lattices will be produced by modulation of Casimir-Polder forces nearby a nanostructured surface. Ultracold atoms trapped in a 2D lattice with a short lattice spacing (50 nm) will enable a better understanding of material properties (e.g. graphene).The work I have done during my thesis can be split into in four parts. The first one consisted in cooling Rubidium 87 until Bose-Einstein condensate regime. Then, numerical simulations were performed to set up a new subwavelength imaging technique, based on different couplings between atomic levels with a double lattice. This will make it possible to activate the sites selectively, in order to pinpoint the atoms with subwavelength precision. Moreover, a new imaging system was developped to improve the resolution of the atomic cloud images. I did new calculations in order to predict experimental results on adiabatic atomic transport in the near field of a surface. Finally, a new laser architecture was designed in this thesis, as part of CIFRE convention, in order to cool down potassium 40 atoms, which has to be done in the future.

We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.

A cubic lattice with random parameters is reduced to a linear chain by the means of the projection technique. The continued fraction expansion (c.f.e.) approach is herein applied to the density of states. Coefficients of the c.f.e. are obtained numerically by the recursion procedure. Properties of the non-stationary second moments (correlations and dispersions) of their distribution are studied in a connection with the other evidences of transport in a one-dimensional Mori chain. The second moments and the spectral density are computed for the various degrees of disorder in the prototype lattice. The possible directions of the further development are outlined. The physical problem that is addressed in the dissertation is the possibility of the existence of a non-Anderson disorder of a specific type. More precisely, this type of a disorder in the one-dimensional case would result in a positive localization threshold. A specific type of such non-Anderson disorder was obtained by adopting a transformation procedure which assigns to the matrix expressing the physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamiltonian is then assigned to an equivalent one-dimensional tight-binding model. One of the benefits of this approach is that we are guaranteed to obtain a linear crystal with a positive localization threshold. The reason for this is the existence of a threshold in a prototype sample. The resulting linear model is found to be characterized by a correlated and a nonstationary disorder. The existence of such special disorder is associated with the absence of Anderson localization in specially constructed one-dimensional lattices, when the noise intensity is below the non-zero critical value. This work is an important step towards isolating the general properties of a non-Anderson noise. This gives a basis for understanding of the insulator to metal transition in a linear crystal with a subcritical noise.

Indiana University-Purdue University Indianapolis (IUPUI) / Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.