Дисертації з теми "Mott-Hubbard transition"

J’ai étudié la transition méta-isolant avec la théorie du champ moyen dynamique appliquée à deux Hamiltoniens largement employés pour décrire les systèmes d’électrons fortement corrélés : le modèle de Hubbard et le modèle d’Anderson périodique. Le scénario pour la transition dans le modèle de Hubbard a été passé en revue et l’analyse du spectre de photoémission près de la transition a été présentée en détail. La transition de Mott induite par le dopage dans le modèle d’Anderson périodique a été discutée par rapport à celle réalisée dans le modèle de Hubbard. Le résultat principal nous conduit à établir un scénario qualitativement différent pour les transitions induites par dopage avec des électrons ou avec des trous. Dans le premier cas, la transition est, comme attendue, similaire à la transition du premier ordre du modèle de Hubbard. Toutefois, dans le dernier cas, une transition du deuxième ordre a été trouvée. J’ai donc démontré que le scénario pour la transition de Mott du modèle de Hubbard n’est pas générique pour le modèle d’Anderson périodique
I study the Mott metal-insulator transition within the dynamical mean-field theory in two schematic Hamiltonians widely used to describe the strongly correlated electron systems : the Hubbard model and the periodic Anderson model. The scenario for the transition in the Hubbard model is reviewed and the analysis of the photoemission spectra near the transition is presented in detail. The doping driven Mott transition in the periodic Anderson model is discussed with respect to the one realized in the Hubbard model. The main finding is a qualitatively different scenario for electron or hole driven transitions. In the former case the transition is expectedly similar to the first order transition of the Hubbard model. However, in the latter case, a second order transition is found. Thus I demonstrate that the transition scenario of the Hubbard model is not generic for the periodic Anderson model

We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder 1 c ss U= 2. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.

Dans ces travaux, nous avons étudié la rigidité superfluide en CDMFT (théorie du champ moyen dynamique sur amas) dans les matériaux organiques supraconducteurs. Ceux-ci appartiennent à la catégorie des supraconducteurs à haute température. Nous obtenons des résultats, en accord qualitatif avec l'expérience, qui suivent la tendance suggérée par la loi dite de Homes. La température critique et la masse effective ont donc également été calculées à des fins d'analyses. Ces dernières quantités sont également en accord avec l'expérience et les travaux antérieurs.

Cette thèse se concentre sur des aspects multiorbitales des systèmes fermioniques fortement corrélés. En particulier, sur l'existence d'une différentiation orbitale dans laquelle la coexistence de caractère itinérant et localisé peut être associée à différentes orbitales. Cette problématique est examinée dans le contexte des atomes froids et des matériaux, offrant un pont entre les deux communautés.Dans la première partie de la thèse, nous donnons un aperçu du problème des corrélations fortes dans les matériaux, et nous introduisons le concept de 'transition de Mott sélective en orbitales'. Nous fournissons également les principaux outils pour comprendre comment les matériaux peuvent être simulés avec des atomes froids, et nous présentons des résultats importants liés à la transition métal-isolant de Mott. Les aspects techniques, basées sur la théorie du champ moyen dynamique sont également discutés, et la solution de deux principaux modèles de systèmes fermioniques fortement corrélés, à savoir le modèle d'Hubbard (HM) et le modèle de Falicov-Kimball (FKM), sont passés en revue.Ensuite, nous étudions en détail la physique de deux espèces fermioniques en interaction forte avec des masses différentes dans un réseau optique. Nous établissons les différentes phases (avec et sans ordre à longue portée) en termes de la force des interactions (U), du rapport des masses et de la température (T), et aussi nous discutons les variables thermodynamiques, qui sont pertinentes pour les expériences d'atomes froids. Nous montrons que dans la phase métallique (U inférieure à une valeur critique) et avec un certain degré de différence de masses, un 'crossover' apparaît entre un état métallique du type de liquide de Fermi à basse T, et un état avec différentiation orbital à haute T, où les fermions lourds se localisent tandis que les fermions légers restent itinérant. Par conséquent, nous proposons ce modèle minimal pour étudier la physique des systèmes qui présentent une différentiation orbitale avec des expériences d'atomes froids.Basé sur les propriétés du modèle étudié, nous proposons la 'chromatographie entropique' comme une nouvelle méthode pour refroidir des atomes fermioniques dans les réseaux optiques. Nous discutons son efficacité et ses limites, et fournissons quelques idées afin de les surmonter.Dans la dernière partie de la thèse, nous généralisons le modèle précédent aux matériaux corrélés à plusieurs bandes qui permet d'afficher la différentiation orbitale. Nous montrons que l'état de Mott sélectif en orbital peut être stable sous les distorsions du réseau, modélisées par une hybridation locale entre les orbitales. Cependant, l'état de Mott est caractérisé par un pseudo-gap, où les fluctuations de charge sont brusquement réduites, mais l'état reste compressible. En relation au modèle précédent, nous discutons le 'crossover' entre l'état métallique et l'état sélectif induit par des effets température, nous comparons nos résultats avec les expériences de photoémission, et nous prédisons ce qui se passerait dans les matériaux qui présentent une hybridation locale entre les bandes
This thesis focuses on multiorbital aspects of strongly correlated fermionic systems. In particular, it focuses on the existence of orbital differentiation in which coexistence of itinerant and localized character can be associated to different orbitals. This subject is discussed in the context of cold atoms and materials, providing a bridge between both communities.In the first part of the thesis, we give an insight into the problem of strong correlations in materials, and we introduce the concept of 'orbital-selective Mott transition'. We also provide the main tools to understand how materials can be simulated with cold atoms experiments, and we present important related results in the context of the metal-Mott insulator transition. The technical aspects, based on dynamical mean-field theory are also discussed, and the solution of two key models of strongly correlated fermionic systems, i.e., the Hubbard model (HM) and the Falicov-Kimball model (FKM), are reviewed.Then we study in detail the physics of two interacting fermionic species with different masses in an optical lattice. We establish the different phases (with and without long-range order) in terms of the interactions strength (U), mass ratio and temperature (T), and also discuss the thermodynamic variables, which are relevant in cold atoms experiments. We show that in the metallic phase (U below a critical value) and for some degree of mass imbalance, a crossover appears between a Fermi-liquid metallic state at low T, and an 'orbital-selective' state at higher T, where the heavy fermions effectively localize while the light species remain itinerant. Hence, we propose this minimal model for addressing orbital-selective physics with cold atoms experiments.Based on the properties of the studied model, we propose the 'entropic chromatography' as a new method for cooling fermionic atoms in optical lattices. We discuss its efficiency and limitations, and provide some ideas in order to overcome them.In the last part of the thesis we generalize the previous model to a model relevant for multiband correlated materials that can display orbital differentiation. We show that the orbital-selective Mott state can be stable under lattice distortions modeled by local hybridization between the orbitals. However, the Mott state is characterized by a pseudogap, where charge fluctuations abruptly reduce, but the state remains compressible. In connection with the previous model, we discuss the temperature-induced orbital-selective crossover in this problem, we compare our results with photoemission experiments, and predict what would happen in materials that display local hybridization between the bands

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores.
This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

Orientador: Eduardo Miranda
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin
Made available in DSpace on 2018-08-23T18:51:04Z (GMT). No. of bitstreams: 1 SuarezVillagran_MarthaYolima_D.pdf: 7321255 bytes, checksum: d76479a0e0c1143207cb4ee380a8034d (MD5) Previous issue date: 2013
Resumo: Estudamos nesta tese alguns aspectos da transição metal-isolante de Mott no caso desordenado. O modelo no qual baseamos nosso estudo é o modelo de Hubbard desordenado, que é o modelo mais simples a apresentar a transição metal-isolante de Mott. Analisamos esse modelo através da Teoria Dinâmica de Campo Médio Estatística (StatDMFT). Essa teoria é uma extensão natural da Teoria Dinâmica de Campo Médio (DMFT), que foi usada com relativo sucesso nos últimos anos para analisar a transição de Mott no caso limpo. Como no caso dessa última, a StatDMFT incorpora os efeitos de correlação eletrônica apenas nos seus aspetos locais. A desordem é tratada de maneira a incorporar todos os efeitos de localização de Anderson. Com essa técnica, analisamos a transição de Mott desordenada no caso bi-dimensional, usando o Monte Carlo quântico para resolver os problemas de impureza única de Anderson requeridos pela StatDMFT. Encontramos as linhas espinodais nas quais o metal e o isolante deixam de ser meta-estáveis. Também estudamos os padrões espaciais das flutuações de quantidades locais, como a auto-energia e a função de Green local, e mostramos como há o aparecimento de regiões metálicas dentro do isolante e viceversa. Analisamos efeitos de tamanho finito e mostramos que, em consonância com os teoremas de Imry e Ma, a transição de primeira ordem desaparece no limite termodinâmico. Analisamos as propriedades de transporte desse sistema através de um mapeamento a um sistema de resistores aleatórios clássicos e calculamos a corrente média e sua distribuição através da transição metal-isolante. Finalmente, estudamos o comportamento da parede de domínio que se forma entre o isolante e o metal no caso limpo. Isso foi feito através de um modelo de uma cadeia unidimensional conectada a reservatórios, um metálico e um isolante, cada um em uma de suas extremidades. Nesse caso, utilizamos o método da Teoria de Perturbação Iterada para a solução dos modelos de impureza única. Encontramos o comportamento da parede como função da temperatura e das interações
Abstract: In this thesis, we studied some aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through the Statistical Dynamical Mean-Field Theory (statDMFT). This theory is a natural extension of the Dynamical Mean-Field Theory (DMFT), which has been used with relative success in the last several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, the statDMFT incorporates the electronic correlation effects only incorporate Anderson localization effects.. With this technique, we analyzed the disordered two-dimensional Mott transition, using Quantum Monte Carlo to solve the associated single-impurity problems. We found the spinodal lines at which metal and insulator cease to be meta-stable. We also studied the spatial fluctuations of local quantities, such as the self-energy and the local Green¿s function, and showed the appearance of metallic regions within the insulator and vice-versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma, the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metalinsulator transition. Finally, we studied the behavior of the domain wall which forms between the metal and the insulator in the clean case. This was done by means of a model of a one-dimensional chain connected to two reservoirs, one metallic and the other insulating, each attached to one of the chain¿s ends. In this case, we used the Iterated Perturbation Theory technique in order to solve the associated singleimpurity problems. We then established the behavior of the domain wall width as a function of temperature and interactions
Doutorado
Física
Doutora em Ciências

La RPA Self-Consistante (SCRPA) est appliquée aux fonctions de corrélation particule-trou dans le Modèle de Hubbard.

Pour une fonction de Green générale à $n$ corps cette méthode se dérive à partir de l'Equation de Dyson où seules sont retenues les contributions instantanées de l'opérateur de masse. La fonction de Green est alors donnée par un système d'équations intégrales non-linéaires que l'on cherchera à résoudre de façon self-consistante. Elle satisfait, parmi d'autres théorèmes, la règle de somme pondérée par l'énergie. Pour les fonctions de Green à une et à deux particules, la SCRPA obéit à un principe variationnel. Dans le Modèle de Hubbard les fonctions de corrélation de charge et de spin sont calculées en SCRPA. En négligeant les densités connectées à deux corps, nous obtenons une théorie self-consistante plus simple, la RPA renormalisée. Les deux méthodes sont étudiées et comparées à la RPA standard.

Nous établissons et résolvons numériquement les équations de la RPA renormalisée pour les fonctions de corrélations de densité de charge dans le Modèle de Hubbard à une dimension. Les susceptibilités de charge et de spin longitudinal, la distribution des impulsions et plusieurs propriétés du fondamental sont évaluées et comparées aux résultats exacts. Dans la limite du couplage fort de la bande à moitié remplie, la RPA renormalisée possède une solution analytique qui est, à un facteur près, en accord avec le développement pour fortes interactions de l'ansatz de Bethe. Comme prévu, des particularités liées à la dimension spatiale $1$, par exemple un comportement de liquide de Luttinger, n'ont pas pu être retrouvées. Or, la description fournie par notre méthode pourrait être assez réaliste en dimensions plus élevées.

Une partie de ces travaux a été publié dans
"Dyson Equation Approach to Many-Body Greens Functions and
Self-Consistent RPA, First Application to the Hubbard Model"
Steffen Schäfer, Peter Schuck, Phys. Rev. B 59, 1712-1733 (1999).

In this thesis we consider a variational wave function approach as a possible route to describe the competition between disorder and strong electron-electron interaction in two dimensions. In particular we aim to obtain a transparent and physically intuitive understanding of the competition between these two localizing forces within the simplest model where they both are active, namely the disordered Hubbard model at half filling and in a square lattice. Our approach is based on an approximate form of the ground-state wave function, which we believe contains the physically relevant ingredients for a correct description of both the Mott and the Anderson insulators, where electrons are localized by the Coulomb repulsion and by disorder, respectively. For strongly interacting fermionic systems, a standard variational wave function is constructed by a correlation term acting on a Slater determinant, the latter being an uncorrelated metallic state. Previous variational calculations showed that a long-range density-density correlation factor, so called Jastrow factor, is needed to correctly describe the Mott insulator [9]. This term, which is collective by definition, correlates spatially charge uctuations, thus preventing their free motion that would otherwise imply metallic conductance. For this reason, our variational wave function does include such a term. Anderson localization is instead mostly a matter of single-particle wave functions, hence it pertains to the uncorrelated Slater determinant which the Jastrow factor acts onto. We consider both the case in which we enforce paramagnetism in the wave function and the case in which we allow for magnetic ordering. Summarizing briefly our results, we find that, when the variational wave function is forced to be paramagnetic, the Anderson insulator to Mott insulator transition is continuous. This transition can be captured by studying several quantities. In particular, a novel one that we have identified and that is easily accessible variationally is the disconnected density-density fluctuation at long wavelength, defined by lim where ^nq is the Fourier transform of the charge density at momentum q, (...) denotes quantum average at fixed disorder and the overbar represents the average over disorder configurations. We find that Ndisc q!0 is everywhere finite in the Anderson insulator and vanishes critically at the Mott transition, staying zero in the Mott insulator. When magnetism is allowed and the hopping only connects nearest neighbor sites, upon increasing interaction the paramagnetic Anderson insulator first turns antiferromagnetic and finally the magnetic and compressible Anderson insulator gives way to an incompressible antiferromagnetic Mott insulator. The optimized uncorrelated Slater determinant is always found to be the eigenstate of a disordered non-interacting effective Hamiltonian, which suggests that the model is never metallic. Finally, when magnetism is frustrated by a next to nearest neighbor hopping, the overall sequence of phases does not change. However, the paramagnetic to magnetic transition within the Anderson insulator basin of stability turns first order. Indeed, within the magnetically ordered phase, we find many almost degenerate paramagnetic states with well defined local moments. This is suggestive of an emerging glassy behavior when the competition between disorder and strong correlation is maximum.

J'étudie en détail la solution d'un modèle simplifié d'électrons fortement corrélés, à savoir le modèle de Hubbard dimérisé. Ce modèle est la réalisation la plus simple d'un problème de cluster DMFT. Je fournis une description détaillée des solutions dans une région de coexistence où l'on trouve deux états (méta) stables des équations DMFT, l'un métallique et l'autre isolant. De plus, je décris en détail comment ces états disparaissent à leurs lignes critiques respectives. Je clarifie le rôle clé joué par la corrélation intra-dimère, qui agit ici en complément des corrélations de Coulomb.Je passe en revue la question importante du passage continue entre unisolant Mott et un isolant Peierls où je caractérise une variété de régimes physiques. Dans un subtil changement de la structure électronique, lesbandes de Hubbard évoluent des bandes purement incohérentes (Mott) à desbandes purement cohérentes (Peierls) à travers un état inattendu au caractère mixte. Je trouve une température d'appariement singulet T* en-dessous de laquelle les électrons localisés à chaque site atomique peuvent se lier dans un singulet et minimiser leur entropie. Ceci constitue un nouveau paradigme d'un isolant de Mott paramagnétique.Enfin, je discute la pertinence de mes résultats pour l'interprétation de différentes études expérimentales sur VO₂. Je présente plusieurs arguments qui me permettent d'avancer la conclusion que la phase métallique, à vie longue (métastable) induite dans les expériences pompe-sonde, et l'état métallique métastable M₁, thermiquement activé dans des nano-domaines, sont identiques. De plus, ils peuvent tous être qualitativement décrits, dans le cadre de notre modèle, par un métal corrélé dimérisé
We study in detail the solution of a basic strongly correlated model,namely, the dimer Hubbard model. This model is the simplest realization ofa cluster DMFT problem.We provide a detailed description of the solutions in the ``coexistentregion'' where two (meta)stable states of the DMFT equations are found, onea metal and the other an insulator. Moreover, we describe in detail howthese states break down at their respective critical lines. We clarify thekey role played by the intra-dimer correlation, which here acts in additionto the onsite Coulomb correlations.We review the important issue of the Mott-Peierls insulator crossoverwhere we characterize a variety of physical regimes. In a subtle change inthe electronic structure the Hubbard bands evolve from purely incoherent(Mott) to purely coherent (Peierls) through a state with unexpected mixedcharacter. We find a singlet pairing temperature T* below which thelocalized electrons at each atomic site can bind into a singlet and quenchtheir entropy, this uncovers a new paradigm of a para-magnetic Mottinsulator.Finally, we discuss the relevance of our results for the interpretation ofvarious experimental studies in VO₂. We present a variety of argumentsthat allow us to advance the conclusion that the long-lived (meta-stable)metallic phase, induced in pump-probe experiments, and the thermallyactivated M₁ meta-stable metallic state in nano-domains are the same.In fact, they may all be qualitatively described by the dimerizedcorrelated metal state of our model

In this dissertation I present results for Bose-Hubbard model in 2D obtained with quantum Monte Carlo simulations by the Worm algorithm and Mean Field techniques. First I study a uniform system and I describe how the Mottinsulator-to-superfluid transition for a majority component is affected by the presence of a minority superfluid one. Then I focus on magnetic-like ground states known as checkerboard phase which can be obtained in the strongly interacting regime. I am able to establish constraints on parameters relevant for experimental setups which allows the considered phase to manifest.

This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. This thesis can be divided into five different parts. In Chapter 1, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials and a short overview of the experiments on ultracold atoms in an optical lattice. In Chapter 2 we develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this poten¬tial, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator(MI), density-wave(DW), and supersolid (SS) phases in the plane of the chemical potential and on-site repulsion ; we present phase diagrams for representative values of , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI ,DW ,and SSphases. We explore the implications of our study for experiments on cold-atom dipolar con¬densates in optical lattices in a confining potential. In Chapter3 we present an extensive study of Mottinsulator( MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with har¬monic traps. For this we develop an inhomogeneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees quantitatively with quan¬tum Monte Carlo(QMC) simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional(3D) systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also generalize our inhomogeneous mean-field theory to study BH models with har¬monic traps and(a) two species of bosons or(b) spin-1bosons. With two species of bosons we obtain rich phase diagrams with a variety of SF and MI phases and as¬sociated shells, when we include a quadratic confining potential. For the spin-1BH model we show, in a representative case, that the system can display alternating shells of polar SF and MI phases; and we make interesting predictions for experi¬ments in such systems. . In Chapter 4 we carry out an extensive study of the phase diagrams of the ex-tended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane ¬insulator(HI) phases. We show that the critical exponents and central charge for the HI-DW,MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) uni¬versality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different bound¬ary conditions lead to different phase diagrams.. In Chapter 5 we obtain the excitation spectra of the following three generalized of Bose-Hubbard(BH) models:(1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we show how to obtain excitation spectra by using the random phase approximation (RPA). We compare the results of our work with earlier studies of related models and discuss implications for experiments.

The thesis presents theoretical and numerical studies on electronic screening and Friedel Oscillations (FO) in correlated fermionic systems in the presence of external inhomogeneity. These spatial oscillations in the electronic densities appear in the neighbourhood of the impurity due to quantum scattering from the impurity. They are observed in metals at low temperatures. The interacting system has been modeled by the Hubbard Hamiltonian which has been further solved numerically using the Real space Dynamical Mean-Field Theory (R-DMFT). We study one-, two-, and three- dimensional finite lattice systems with periodic boundary conditions. The inhomogeneous potential has been modeled by a single impurity, two impurities and an extended inhomogeneity. Different approximations accounting for the electronic correlations have also been discussed in the course of the thesis. The effects of electronic correlations, particularly at the Mott metal to insulator transition, on the behaviour of FO have been explored. According to our numerical studies, the oscillations are damped with the interactions, disappear at the Mott transition and completely beyond it. At finite temperature the oscillations are damped and slowly disappear as we increase the temperature of the system. At half-filling the period and phase of the oscillations remain unaltered by the interactions. The Friedel sum rule for the interacting system has been investigated. The variation of the screening charge around the neighbourhood of the impurity called ``Neighbourhood screening charge'' or ``N-screening charge'' with interaction, temperature is studied for different impurity potentials. It is further seen that the interactions weaken the screening effects. The spectral functions in the presence of an external inhomogeneity and electronic interactions have been studied at different lattice sites. In the presence of the interaction a sign of resonance appears in the spectral function at the impurity site. In the presence of two impurities interaction weakens the interference effects on the oscillations. Finally the one-body scattering formalism is modified to describe analytically the effects of correlations in many body systems modeled by a momentum independent self-energy with semi circular density of states. Effects of electronic correlations in the scattering phase-shift and spectral functions are obtained in our calculations. The thesis provides predictive numerical results which can motivate future experiments.
Rozprawa doktorska zawiera analizę teoretyczną oraz obliczenia numeryczne dotyczące ekranowania elektronowego oraz oscylacji Friedel'a (FO) w silnie skorelowanych układach fermionowych w obecności niejednorodnego potencjału zewnętrznego. Przestrzenne oscylacje gęstości elektronowej pojawiają się w sąsiedztwie domieszki na skutek kwantowego rozpraszania na domieszce. Oscylacje te obserwuje się w niskich temperaturach w układach metalicznych. Układ oddziałujący modelowany jest w rozprawie za pomocą hamiltonianu Hubbarda, który został rozwiązany numerycznie w przybliżeniu dynamicznej teorii pola średniego w zastosowaniu do układu niejednorodnego przestrzennie (RDMFT). Rozważane są jedno, dwu i trójwymiarowe układy sieciowe z periodycznymi warunkami brzegowymi. Zewnętrzny potencjał niejednorodny modelowany jest za pomocą pojedynczej domieszki, dwóch domieszek lub niejednorodności obejmującej większą liczbę węzłów sieci. W rozprawie rozważane są różne modele uwzględniające oddziaływania elektronowe. Badano efekty tych oddziaływań na oscylacje Friedla, w szczególności blisko przejścia Motta metal-izolator. Wyniki obliczeń numerycznych zaprezentowanych w rozprawie pokazują, że oscylacje Friedla są tłumione przez oddziaływania elektronowe, zanikają wokół przejścia Motta i są całkowicie wygaszone w fazie izolatora. Wzrost temperatury układu powoduje tłumienie oscylacji. Okres i faza oscylacji nie zależą od wielkości oddziaływań, gdy gęstość cząstek odpowiada sytuacji w której pasmo jest w połowie zapełnienie. W rozprawie badano również spełnienie reguły sum Friedla w przypadku układów oddziałujących. Ponadto badano zmianę ładunku ekranującego wokół domieszki określaną jako ładunek ekranowania wokół domieszki (ang. "Neighbour screening charge" lub "N-screening charge") w zależności od oddziaływania, temperatury i dla różnych wartości potencjałów na domieszce. Badania w rozprawie pokazują, że oddziaływania osłabiają efekty ekranowania. Przedstawiono funkcje spektralne w obecności zewnętrznej niejednorodności i oddziaływań elektronowych na różnych węzłach sieci. W obecności dwóch domieszek oddziaływania osłabiają wpływ efektów interferencyjnych domieszek na oscylacje. Formalizm rozpraszania jednociałowego zmodyfikowano w celu opisu efektów korelacji w wielociałowych układach poprzez wprowadzenie energii własnej niezależnej od pędu odpowiadającej eliptycznej gęstości stanów. Rezultaty obliczeń uwidaczniają wpływ korelacji elektronowych na przesunięcie fazowe i funkcję spektralną. Rozprawa zawiera rezultaty numeryczne, które mogą stanowić motywację do przyszłych badań eksperymentalnych.

Through our whole work we study the XY model with all its entirety, a particular spin model where spins are confined in a plane. We try to bring out a good understanding of this model with all different types of phases and phase transition, it undergoes in critical situations. We conceive of these external conditions from very different physical models like High Tc Superconductor, Ultracold atoms in optical lattice which are in focus of recent research. Firstly we model high Tc Superconductors with very simple 2D XY model to get an idea about the diamagnetic response exhibited by these materials when kept in a external magnetic field. This modeling is reasonable because most of the physics of cuprate High Tc Superconductors are governed by their 2D copper oxide planes which insists us to consider 2D models. Later we shifted to a more realistic 3D anisotropic XY model , as the coupling between cuprates plane may have a considerable role in devising physics of those materials. We particularly focus on the 2D to 3D crossover effect on magnetisation showed by these models, with keeping an eye on how all these can be relate to the experimentally acquired magnetisation profile of High Tc Supercondutors. On the second project we investigate on the phase diagram of a fully frustrated 2-leg ladder Bose Hubbard model. After mapping it properly to a classical model, a bi-layer Fully Frustrated XY model on square lattice, we found that the frustration leads to the emergence of a new phase "Chiral Mott insulator(CMI)" sandwiched between "Chiral Superfluid(CSF)" and "regular Mott insulator(MI)" phase. We divide the whole report into four parts. The first chapter is basically contain-ing introductory part comprising the motivation. In the second chapter we discuss various types of phases and phase transitions of the 2D & 3D XY models. We try to address their critical behaviors. In the third chapter and onwards we consider our model in external magnetic field and observe magnetisation in these systems. Here we specially focus on 2D to 3D crossover effect on magtisation measurement. Lastly in the fourth chapter we bring out a correspondence of XY model with the 2 leg ladder fully frustrated Bose Hubbard Model. There we report the emergence of a new phase, Chiral Mott Insulator(CMI) due to frustration in system.