The two-dimensional electronic systems (2DESs) have kept surprising physicists for the last few decades. Examples include the integer and fractional quantum Hall effects, cuprate superconductivity, and graphene. This thesis is intended to develop suitable theoretical tools which can be generalized to study new types of 2DESs with strong correlation feature. The first part of this thesis describes the investigation of heterostructures made by Mott insulators. This work is mostly motivated by the significant improvement of techniques for layer-by-layer growth of transition metal oxides in the last few years. We construct a toy model based on generalized Hubbard model complemented with long-ranged Coulomb interaction, and we study it by Hartree-Fock theory, dynamical mean-field theory, and Thomas-Fermi theory. We argue that interesting 2D strongly correlated electronic systems can be created in such heterostructures under several conditions. Since these 2D systems are formed entirely due to the gap generated by electron-electron interaction, they are not addiabatically connected to a noninteracting electron states. This feature makes these 2D systems distinguish from the ones created in semiconductor heterostructures, and they may be potential systems having non-Fermi liquid behaviors. The second part of this thesis is devoted to the study of collective excitations in high-temperature superconductors. One important achievement in this work is to develop a time-dependent mean-field theory for t-U-J-V model, an effective low energy model for cuprates. The time-dependent mean-field theory is proven to be identical to the generalized random-phase approximation (GRPA) which includes both the bubble and ladder diagrams. We propose that the famous 41 meV magnetic resonance mode observed in the inelastic neutron scattering measurements is a collective mode arising from a conjugation relation, which has been overlooked in previous work, between the antiferromagnetic fluctuation and the phase fluctuation of the d-wave superconducting order parameter near momentum ([pi, pi]). Furthermore, we find that this collective mode signals the strength of the antiferromagnetic fluctuations which are responsible for the suppression of the superfluid density in the underdoped cuprates even at zero temperature. Finally, we perform a complete analysis on an effective model with parameters fitted by experimental data of Bi2212 within the GRPA scheme and conclude that the short-range antiferromagnetic interactions which are a remnant of the parent Mott-insulator are more likely the pairing mechanism of the High-T[subscript c] cuprates. / text

Les propriétés d'électrons en deux dimensions (2D) soulèvent des questions fondamentales qui ont été largement explorées au moyen des techniques théoriques de la matière condensée. L'extension de modèles classiques tel le modèle de Hubbard en 2D, en incluant par exemple plusieurs bandes électroniques, ore la possibilité d'accéder à des phénomèmes plus complexes, comme l'interaction du transport électronique et du magnétisme observé dans les composés de fermions lourds. Ces modèles sont en lien direct avec la question de couches minces couplées, les hétérostructures, qui sont depuis peu l'objet d'intenses recherches et orent la possibilité d'intéressantes applications. Dans ce contexte, nous étudions numériquement diérents syst èmes au moyen de la méthode du Monte Carlo Quantique du Déterminant. Tout d'abord, l'eet de la corrélation électronique dans un isolant de bande est évaluée, montrant en particulier l'absence d'une phase métallique intermédiaire. Un deuxième système est composé de deux bandes électroniques couplées, dans lequel l'eet de la largeur de bande de la partie corrélée est exploré de façon systématique. Finalement, nous étudions une interface métal-isolant, qui présente une phase intermédiaire surprenante lorsque le couplage à l'interface est ajusté.

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

Cold bosonic atoms trapped in optical lattices formed by standing wave interference patterns of multiple laser beams constitute excellent emulators of models of strongly correlated quantum systems of bosons. In this thesis, we develop and deploy strong-coupling expansion (i.e., an expansion in terms of the ratio of the inter-site hopping amplitude of the bosons to the strength of their interactions) techniques for studying the properties of three different instances of such systems. In the first instance, we have used strong coupling expansion techniques to calculate the density pro le for bosonic atoms trapped in an optical lattice with an overall harmonic trap at finite temperatures and large on site interaction in the presence of super fluid regions. Our results match well with quantum Monte Carlo simulations at finite temperature. We present calculations for the entropy per particle as a function of temperature which can be used to calibrate the temperature in experiments. Our calculations for the scaled density in the vacuum-to-super fluid transition agree well with the experimental data for appropriate temperatures. We also discuss issues connected with the demonstration of universal quantum critical scaling in the experiments. Experimental realizations of “atomtronic" Josephson junctions have recently been created in annular traps in relative rotation with respect to potential barriers that generate the weak links. If these devices are additionally subjected to optical lattice potentials, then they can incorporate strong-coupling Mott physics within the design, which can modify the behaviour and can allow for interesting new configurations of system generated barriers and of super fluid ow patterns. we have examined theoretically the behavior of a Bose super fluid in an optical lattice in the presence of an annular trap and a barrier across the annular region which acts as a Josephson junction. As the fluid is rotated relative to the barrier, it generates circulating super-currents until, at larger speeds of rotation, it develops phase slips which are typically accompanied by vortices. We use a finite temperature strong-coupling expansion about the mean- held solution of the Bose Hubbard model to calculate various properties of the device. In addition, we discuss some of the rich behavior that can result when there are Mott regions within the system. Rubidium-Cesium dipolar molecule formation through Feshbach resonance is an area of great current interest, for, the dipolar molecules, once formed, interact via v long range dipolar forces, leading to possibilities of novel phases. Experimentalists currently make such systems mostly using trial and error, and the resulting efficiencies for molecule formation tend to be low. With a goal to assist cold-atom experimentalists to achieve higher e ciencies of molecule formation, we have estimated the trap parameters for Rb and Cs atoms in a 3D optical lattice required to create single occupancy per site Mott phase for both the species in the same regions of the trap. We thus identify the ne tuning of the external magnetic held near Rb-Cs Feshbach resonance required to achieve highest probability for creating single Rb-Cs Feshbach molecules in the system. We have used the Falicov-Kimball model to describe the relevant system and strong-coupling expansions about the mean- held solution to calculate the density pro les for both species and efficiency for molecule formation, determined by overlapping regions of single occupancy for both Rb and Cs, up to second order in the expansion. We also calculate the entropy per particle which serves as an estimation of the temperature in the experimental system

Elétrons em bandas estreitas de energia são fortemente afetados pela interação coulombiana como também por desordem na rede. Ambos os efeitos podem levar à localização, mas de naturezas diferentes: estado isolante de Mott, induzido por correlação, e localização de Anderson, induzida por desordem. A existência da fase de Mott ´e também significativamente dependente do preenchimento da banda. Abordagens teóricas para lidar com esse tipo de sistema são usualmente baseadas no hamiltoniano de Hubbard ou modelos relacionados, incluindo desordem como uma distribuição de energias locais. Neste trabalho, utilizando a Teoria de Campo Médio Dinâmico (DMFT), estudamos o modelo de Anderson-Falicov-Kimball e sua versão de três bandas, obtida como uma simplificação do modelo de Hubbard de três bandas associado aos planos de CuO2 dos cupratos supercondutores de alta temperatura crítica, realizando nossa análise para os casos magnético e não magnético. A densidade de estados de uma partícula é obtida por medias aritmética e geométrica sobre a desordem, já que somente a última pode detectar a localização na ausência de um gap de energia. Variando as intensidades de interação coulombiana e desordem, construímos diagramas de fases para esse modelo, onde identificamos transições metal-isolante mediadas por correlação e desordem, bem como a inter-relação entre esses efeitos. Isso é feito para vários preenchimentos de banda, já que nosso principal interesse aqui é estudar como a variação da densidade de elétrons (dopagem) afeta os diagramas de fases previamente obtidos na ausência de dopagem. Para o modelo de uma banda no caso paramagnético, as informações reveladas pela densidade de estados são confirmadas pela análise das condutividades estática e dinâmica, incluindo efeitos de temperatura. Quando consideramos a solução magnética, observamos o comportamento da temperatura de Néel e podemos apresentar um diagrama de fases mais completo. Além de uma análise bastante extensa do modelo de uma banda, fazemos um estudo inicial do modelo de três bandas, focalizando comportamentos que possam vir a ser comparados ao que se observa nos óxidos supercondutores. / Electrons in narrow-band solids are strongly affected by the Coulomb interaction as well as lattice disorder. Both effects can lead to localization, but of different nature: correlation-induced Mott insulating state, and disorder-induced Anderson localization. The existence of the Mott phase is also significantly dependent on the band filling. Theoretical approaches to deal with this kind of system are usually based on the Hubbard Hamiltonian or related models, including disorder as a distribution of the on-site energies. In this work, utilizing Dynamic Mean Field Theory (DMFT), we study the Anderson- Falicov-Kimball and its three-band version, obtained as a simplification of the three-band Hubbard model associated to the CuO2 planes of high-critical-temperature cuprate superconductors, performing our analysis for the magnetic and non-magnetic cases. The one-particle density of states is obtained by both arithmetic and geometrical averages over disorder, since only the latter can detect localization in the absence of an energy gap. Varying the strengths of Coulomb interaction and disorder, we construct phase diagrams for these models, where we identify metal-insulator transitions driven by correlation and disorder, as well as the interplay between these effects. This is done for various band fillings, since our main interest here is to study how the variation of the electron density affects the phase diagrams previously obtained in the absence of doping. For the one-band model in the paramagnetic case, the picture revealed by the density of states is further checked by evaluating the static and dynamic conductivities, including temperature effects. When we consider the magnetic solution, we observe the N´eel temperature behavior, and we are able to present a more complete phase diagram. Besides a quite extensive analysis of the one-band model, we develop an initial study of the threeband model, focusing on behaviors that might be linked with what is observed on the superconducting oxides.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Critical phenomena study was for many years dominated by analysis of transitions generated by thermal fluctuations. This thermal fluctuations cease at T-0, however, quantum fluctuations does not end at zero temperature. These quantum fluctuations may, under certain conditions, trigger phase transitions. In this work the Hubbard model is used to study quantum state and quantum phase transitions in strongly correlated electron systems, considering the terms of intersite hopping and Coulomb repulsion intrasite. It was developed an algorithm based on Lanczos method to solving the Hubbard model applied in different types of lattices. Analysis of algorithms efficiency were made an was observed that the standard approaches to evaluate the properties of the ground state in the Hubbard model by Lanczos method presents convergence problems when there is a significant difference between hopping parameters and Coulomb interaction. This difference is very important since the energy convergence does not necessarily reflect in a convergence of the ground state. In this work are discussed several algorithms as standard Lanczos method, the Explicit Restarted Lanczos algorithm and the Modified Explicit Restarted Lanczos algorithm. A protocol based on these algorithms using the operator S2 as s stopping criterion was developed, since through this the operator it is possible to assess the error getting from the ground state itself. The algorithm based on the ERL provides better accuracy and it is 5 times faster compared with conventional ones. The MERL-based algorithm keeps the error at the last significant digit, and its processing time is about 2.5 times longer than the ERL-based algorithm, although it is still faster than the standard Lanczos method. These analyzes pave the way for a reliable and practical evaluation of the ground-state properties not only of the Hubbard model, but also for other manybodies quantum systems. The systems analyzed were clusters of polymeric lattice AB2 tipe, one-dimensional lattice considering nears and next nears neighbors hoppings and cluster of fcc lattice. All systems showed quantum state transitions. Through the study of the spin-spin correlations of the AB2 lattices clusters it was possible to analyze in detail the behavior of these spin-spin correlation functions between sublattices of a finite system. The analysis of one-dimensional lattice with next near neighbor made it possible to use an extrapolation method, which has determined that the quantum phase transition critical point, Uc/t = 4.7, from which the system changes from a paramagnetic behavior to a ferromagnetic behavior. In the fcc lattice clusters were examined the ground state energy as a function of the particle density showed a minimum value for all the structural sizes studied. The minimum energy decreases with increasing the interaction parameter U. It was observed that the ground state energy has a minimum at n = 0.6 for U/t = W, where W denotes the non-interacting bandwidth and the face-centered cubic structure is ferromagnetic. These results, when compared to the nickel properties, shown great similarity analysis in literature, made at finite temperature and support the results of Hirsh, which proposes that the interatomic interaction exchange is dominant to driving the system to a ferromagnetic phase. / O estudo dos fenômenos críticos foi, por muitos anos, dominado pela análise das transições geradas por flutuações térmicas. As flutuações térmicas cessam em T-0, porém flutuações quânticas não acabam na temperatura zero. Essas flutuações de caráter quântico podem, sob certas condições, desencadear transições de fase. Neste trabalho o modelo de Hubbard é utilizado para o estudo de transições de estado quântico e de fase quântica em sistemas de elétrons fortemente correlacionados, considerando os termos de hopping intersítios e de repulsão coulombiana intrasítio. Foi desenvolvido um algoritmo com base no método de Lanczos para resolver o modelo de Hubbard aplicado a diferentes tipos de rede. Foram feitas análises da eficiência de algoritmos, nelas foi possível observar que as abordagens padrão para avaliar as propriedades do estado fundamental do modelo de Hubbard através do método de Lanczos apresentam problemas de convergência quando há uma significante diferença entre os parametros de hopping e de interação coulombiana. Esta diferença é muito relevante uma vez que a convergência da energia não reflete necessariamente em uma convergência do estado fundamental. Neste trabalho são discutidos vários algoritmos como o método de Lanczos padrão, o algoritmo Explicit Restarted Lanczos e o algoritmo Modified Explicit Restarted Lanczos. Foi desenvolvido um protocolo baseado nesses algoritmos que utiliza o valor de S2 como critério de parada do método, uma vez que através dessa grandeza é possível avaliar o erro na obtenção do estado fundamental. O algoritmo baseado no ERL proporciona uma melhor precisão é 5 vezes mais rápido quando comparado com o convencional. O algoritmo baseado no MERL mantém o erro no último dígito significativo e seu tempo de processamento é cerca de 2.5 vezes mais longo do que o algoritmo baseado no ERL, embora ainda seja mais rápido do que o método Lanczos padrão. Essas análises abrem caminho para uma avaliação confiável e prática das propriedades do estado fundamental, não só do modelo de Hubbard, mas também para muitos outros sistemas quânticos de muitos corpos. Os sistemas analisados foram clusters de rede polimérica tipo AB2, de rede unidimensional considerando hoppings tanto de primeiros quanto de segundos vizinhos e clusters de rede fcc. Todos os sistemas apresentaram transições de estado quântico. Através do estudo das correlações spin-spin do cluster da rede AB2 foi possível analisar detalhadamente o comportamento das referidas funções de correlação spin-spin entre sub-redes de um sistema finito. A análise da rede unidimensional com hopping entre segundos vizinhos possibilitou utilizar um método de extrapolação, o qual determinou que o ponto crítico de transição de fase quântica, Uc/t = 4.7, a partir do qual o sistema passa de um comportamento paramagnético para um comportamento ferromagnético. Nos clusters de rede fcc foram examinadas as energias do estado fundamental em função da densidade de partícula, observando-se a existência de um valor de mínimo de energia para todas os tamanhos estruturais estudados. Os mínimos de energia diminuem com o aumento do parâmetro de interação U. Foi observado que a energia do estado fundamental tem um mínimo em a densidade eletrônica igual a 0.6 para U/t=W, em que W denota a largura de banda não-interagente e a estrutura cúbica de face centrada mostrou-se ferromagnético. Esses resultados, quando comparados com as propriedades do níquel, mostam grande semelhança com análises na literatura feitas sob temperatura finita e suportam os resultados de Hirsh, o qual propõe que a interação interatômica de exchange é dominante na condução do sistema à uma fase ferromagnética.

Elétrons em bandas estreitas de energia são fortemente afetados pela interação coulombiana como também por desordem na rede. Ambos os efeitos podem levar à localização, mas de naturezas diferentes: estado isolante de Mott, induzido por correlação, e localização de Anderson, induzida por desordem. A existência da fase de Mott ´e também significativamente dependente do preenchimento da banda. Abordagens teóricas para lidar com esse tipo de sistema são usualmente baseadas no hamiltoniano de Hubbard ou modelos relacionados, incluindo desordem como uma distribuição de energias locais. Neste trabalho, utilizando a Teoria de Campo Médio Dinâmico (DMFT), estudamos o modelo de Anderson-Falicov-Kimball e sua versão de três bandas, obtida como uma simplificação do modelo de Hubbard de três bandas associado aos planos de CuO2 dos cupratos supercondutores de alta temperatura crítica, realizando nossa análise para os casos magnético e não magnético. A densidade de estados de uma partícula é obtida por medias aritmética e geométrica sobre a desordem, já que somente a última pode detectar a localização na ausência de um gap de energia. Variando as intensidades de interação coulombiana e desordem, construímos diagramas de fases para esse modelo, onde identificamos transições metal-isolante mediadas por correlação e desordem, bem como a inter-relação entre esses efeitos. Isso é feito para vários preenchimentos de banda, já que nosso principal interesse aqui é estudar como a variação da densidade de elétrons (dopagem) afeta os diagramas de fases previamente obtidos na ausência de dopagem. Para o modelo de uma banda no caso paramagnético, as informações reveladas pela densidade de estados são confirmadas pela análise das condutividades estática e dinâmica, incluindo efeitos de temperatura. Quando consideramos a solução magnética, observamos o comportamento da temperatura de Néel e podemos apresentar um diagrama de fases mais completo. Além de uma análise bastante extensa do modelo de uma banda, fazemos um estudo inicial do modelo de três bandas, focalizando comportamentos que possam vir a ser comparados ao que se observa nos óxidos supercondutores. / Electrons in narrow-band solids are strongly affected by the Coulomb interaction as well as lattice disorder. Both effects can lead to localization, but of different nature: correlation-induced Mott insulating state, and disorder-induced Anderson localization. The existence of the Mott phase is also significantly dependent on the band filling. Theoretical approaches to deal with this kind of system are usually based on the Hubbard Hamiltonian or related models, including disorder as a distribution of the on-site energies. In this work, utilizing Dynamic Mean Field Theory (DMFT), we study the Anderson- Falicov-Kimball and its three-band version, obtained as a simplification of the three-band Hubbard model associated to the CuO2 planes of high-critical-temperature cuprate superconductors, performing our analysis for the magnetic and non-magnetic cases. The one-particle density of states is obtained by both arithmetic and geometrical averages over disorder, since only the latter can detect localization in the absence of an energy gap. Varying the strengths of Coulomb interaction and disorder, we construct phase diagrams for these models, where we identify metal-insulator transitions driven by correlation and disorder, as well as the interplay between these effects. This is done for various band fillings, since our main interest here is to study how the variation of the electron density affects the phase diagrams previously obtained in the absence of doping. For the one-band model in the paramagnetic case, the picture revealed by the density of states is further checked by evaluating the static and dynamic conductivities, including temperature effects. When we consider the magnetic solution, we observe the N´eel temperature behavior, and we are able to present a more complete phase diagram. Besides a quite extensive analysis of the one-band model, we develop an initial study of the threeband model, focusing on behaviors that might be linked with what is observed on the superconducting oxides.

Heterogenní katalýza hraje významnou roli pro zvy¹ování efektivity rùzných procesù. Vysokou katalytickou aktivitu vykazují oxidy ceru patøící k tzv. reducibilním oxidùm, je¾ snadno uvolòují èi (zpìtnì) pøijímají atomy kyslíku prostøednictvím procesù redukce a oxidace. Zámìrem pøedlo¾ené práce je vypoèítat pásové struktury a øezy plochami konstantní energie objemových a povrchových systémù oxidu ceru pomocí kvantovì-mechanické "ab initio"metody Teorie hustotního funkcionálu. Numerické simulace jsou provádìny pomocí programového balíèku Quantum ESPRESSO za u¾ití metod rovinných vln a pseudopotenciálu. Silná korelace elektronù v pøípadì atomù ceru je modelována pøidáním Hubbardova U-èlenu. Hlavní dùraz je kladen na mo¾nost porovnávání teoretických výsledkù s tìmi získanými pomocí experimentální metody úhlovì-rozli¹ené fotoelektronové spektroskopie umo¾òující pøímý zisk jak pásových struktur, tak øezù plochami konstantní energie zkoumaného materiálu. Porovnání pomù¾e jak poznat mo¾nosti a mo¾né hranice zmínìné experimentální metody, tak i roz¹íøit její teoretické zázemí vedoucí k prohloubení znalostí materiálù perspektivních pro katalýzu. Klíèová slova: Teorie hustotního funkcionálu, Hubbardùv U-èlen, pásová struktura, plocha konstantní energie, Brillouinova zóna 1