Cette thèse est consacrée à l'étude du modèle de Hubbard unidimensionnel et à ses généralisa- tions. Le modèle de Hubbard est un modèle fondamental de la physique de la matière condensée, décrivant des électrons en interaction sur un réseau. Il a une très riche structure physique. Malgré la simplicité de sa construction, le modèle a été appliqué dans différents problèmes comme la supra- conductivité à haute température, le magnétisme et la transition métal-isolant. A une dimension, le modèle de Hubbard est un modèle intégrable très étudié qui a servi de 'laboratoire' pour la physique de la matière condensée. Récemment, les systèmes intégrables quantiques d'une facon générale, et le modèle de Hubbard en particulier, sont apparus d'une manière surprenante dans le contexte de la correspondance AdS/CFT. Le point de contact entre ces domaines est les équations de Bethe : celles de nouveaux modèles intégrables et de modèles existants généralisés sont à priori significatifs dans l'application en dualité AdS/CFT. Dans la premiere partie de la thèse, les notions de base sur l'intégrabilité quantique sont présen- tées : formalisme de la matrice R, équation de Yang-Baxter, chaînes de spin intégrables. Dans la deuxième partie, certaines résultats fondamentaux concernant le modèle de Hubbard sont passés en revue : la solution par l'Ansatz de Bethe coordonnée, les solutions réelles des équations de Lieb-Wu etc. De plus, l'application dans la correspondance AdS/CFT est considérée. Cependant, on trouve que certaines modifications du modèle de Hubbard sont nécessaires pour reproduire les résultats de cette correspondance. Cela est une des motivations principales d'étude de modèles de Hubbard généralisés. La quatrième partie est consacrée aux généralisations du modèle de Hubbard, en se con- centrant sur les cas supersymétriques. La chapitre cinq expose les résultats obtenus dans le cadre de cette thèse sur les modèles de Hubbard généralisés, en particulier, l'Ansatz de Bethe coordonnée ainsi que les solutions réelles des équations de Bethe obtenues dans la limite thermodynamique. Les équations de Bethe obtenues sont différentes de celle de Lieb et Wu par des phases dont la manifesta- tion est un signe encourageant pour l'application en AdS/CFT contexte. Les applications possibles, notamment dans le domaine de la physique de la matière condensée, sont également considérées.

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We first study noninteracting electrons moving on corner-sharing tetrahedral lattices, which represent the conduction path of LiAlyTi2−yO4. A uniform box distribution type of disorder for the on-site energies is assumed. Using the Dyson-Mehta Delta-3 statistics as a criterion for localization, we have determined the critical disorder (Wc/t = 14.5 ± 0.25) and the mobility-edge trajectories. Then we study the Anderson-Hubbard model, which includes both interactions and disorder, using a real-space self-consistent Hartree-Fock theory. We provide a partial assessment on how the Hartree-Fock theory approximates the ground states of the Anderson-Hubbard model, using small clusters which can be solved exactly. The Hartree-Fock theory works very well in reproducing the ground-state energies and local charge densities. However, it does not work as well in representing the spin-spin correlations. To find the ground state, one needs to allow maximum degree of freedom in spins. Evidence of screening of disorder by the interactions is provided. We have applied the Hartree-Fock theory to large-scale three-dimensional simple cubic lattices. For a disorder strength of W/t = 6, weak interactions (U/t ≤ 3) enhance the density of states at the Fermi level and the low-frequency conductivity. There are no local magnetic moments, and the AC conductivity is Drude-like. With stronger interactions (U/t ≥ 4), the density of states at the Fermi level and the low-frequency conductivity are both suppressed. These are accompanied by the presence of local magnetic moments, and the conductivity becomes non-Drude-like. A metal-to-insulator transition is likely to take place at a critical Uc/t ≈ 8 – 9. We find that (i) the formation of magnetic moments is essential to the suppression of the density of states at the Fermi level, and therefore essential to the metal-insulator transition; (ii) the form of magnetic moments does not matter; and (iii) these results do not depend on the type of lattice or the type of disorder. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2009-05-26 02:20:04.652

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This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied. / QC 20111207

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (M.S.) -- Mississippi State University. Department of Physics and Astronomy. / Title from title screen. Includes bibliographical references.

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Os supercondutores de alta temperatura crítica apresentam propriedades diferentes dos supercondutores convencionais, uma destas propriedades importantes é a simetria não isotrópica do parâmetro de ordem. Neste trabalho se apresenta um modelo que mostra a presença de simetria d-wave em uma classe de supercondutores de alta Tc, assim como a condição para a existência de ondas de densidade nos estados de equilíbrio. Se analisa primeiro um sistema de três corpos, um bipolaron e dois elétrons, com o objetivo de estudar o efeito da repulsão coulombiana local e não local sobre a simetria do parâmetro de ordem para o caso exato e também o limite de acoplamento forte. Se mostra também a construção de um Hamiltoniano efetivo que desacopla a interação el etron-bipolaron com o fim de abordar o problema coletivo de infinitos corpos e vemos como o modelo proposto prediz a formação de ondas de densidade para uma região especifica dos parâmetros físicos do Hamiltoniano estudado. / High-temperature superconductors have different properties than conventional superconductors, one of these important properties is non-isotropic symmetry of the order parameter. In this work we present a model that shows the presence of symmetry d-wave in a class of superconductors of high Tc , as well as the condition for the existence of density waves in the states of equilibrium. We first analyze a three-body system, a bipolaron and two electrons, in order to study the effect of local and non-local Coulomb repulsion on the symmetry of the order parameter for the exact case and also the strong coupling limit. It is also shown the construction of an effective Hamiltonian that decouples the electron-bipolaron interaction in order to approach the collective problem of infinite bodies and we see how the proposed model predicts the formation of density waves for a specific region of the physical parameters of the studied Hamiltonian.

Neste trabalho estudamos o magnetismo no modelo de Hubbard em uma dimensão (1DMH) via Teoria do Funcional da Densidade (DFT). Dentro desse contexto, abordamos três tópicos: (i) Baseados numa recente proposta de generalização da DFT para magnetismo não colinear, estudamos numericamente por meio de diagonalização exata o comportamento magnético do 1DMH adaptado para o caso não colinear. (ii) Desenvolvemos e aplicamos o formalismo da DFT para magnetismo não colinear no espaço discreto e comparamos os resultados do cálculo do sistema Kohn-Sham (KS) com os obtidos no passo (i) através da diagonalização exata. Aqui testamos uma proposta de aproximação para o funcional de troca e correlação no estado de onda de densidade de spin (SDW). (iii) Construímos e testamos uma aproximação da densidade local (LDA) para o funcional de troca e correlação baseados na solução exata do modelo de Hubbard homogêneo, dada pelo Ansatz de Bethe (BA). Comparamos os resultados do cálculo numérico oriundos da BA-LDA com os provenientes da diagonalização exata, obtendo resultados satisfatórios tanto para o estado SDW como para vários sistemas não homogêneos (impurezas, super-redes, potencial binário e potencial degrau). O bom acordo entre a diagonalização exata e a DFT mostra que esta última pode ser aplicada com sucesso nos casos em que a diagonalização exata é inviável. Paralelamente, adaptamos o formalismo da DFT-SDW, usado no passo (i) para aplicação em cálculos ab initio de estrutura de bandas na aproximação Muffin- Tin. Objetivando, nesse úl¬timo contexto, desenvolver as equações matemáticas necessárias para a implementação numérica desse método, fornecendo assim o formalismo necessário para futuros cálculos ab initio / In this work we study the magnetism of the Hubbard Model in one dimension (1DMH) via the Density Functional Theory (DFT). In this context, we considered three different phys¬ical context topics: (i) based on a recent generalization proposed to DFT for non-collinear magnetism, we performed an exact numerical diagonalization in order to study the magnetic behavior of the 1DMH; (ii) we developed and applied the DFT formalism, using the Kohn-Sham (KS) scheme, for non-collinear magnetism in the discreet space and we compared the results with those obtained in the step (i) through exact diagonalization. In that approach we tested an approximation for exchange and correlation functional in the framework of the spin density wave state (SDW); (iii) finally, we built and tested a local density approximation (LDA) for the functional of exchange and correlation based on the exact solution of the homogeneous Hubbard model, given by the Bethe Ansatz (BA). We compared the results of BA-LDA obtained from the numerical calculation using the exact diagonalization, yielding satisfactory results for the SDW state, as well as several inhomogeneous systems (impurity, super-lattice, binary and step potential). The good agreement between exact diagonalization and DFT shows that this last can be applied with success in the cases the diagonalization exact is not viable. We also adapted the DFT-SDW formalism, used in the step (i), for applications in ab initio calculations of band structure in the Muffin-Tin approximation. Our goal, in this last context, is to develop the necessary mathematical equations for the numerical implementation of this method, thus provinding the necessary formalism for future ab initio calculations.