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The quest for obtaining higher Tc superconductivity led to the discovery of cuprates about 20 years ago. Since then, they continue to puzzle the scientific community with their bizarre properties like non-BCS superconductivity, pseudo gap, Fermi arcs, linear T resistivity etc. Since these materials show unusually high Tc, a novel mechanism is at play and strong correlations are believed to play an important role. The theme of this thesis work is to study physics of such strongly correlated systems in two dimensions at T = 0 along with development of new theoretical tools necessary for the study. The focus of the thesis is on the ground state studies of strongly correlated models like t-J and Hubbard models using variational Monte Carlo (VMC) and renormalized mean field theory (RMFT). The general method is to propose a variational wave function, motivated by the physics ideas, to be a candidate ground state of the system. Methods to efficiently evaluate the ground state energy and minimizing it with respect to the variational parameters are developed in this work. Antiferromagnetism-superconductivity competition and electron-hole asymmetry in the extended t-J model is investigated. The main result of this work is that increasing the magnitude of the next neighbor hopping (t') on hole doped side strengthen superconductivity while it stabilizes antiferromagnetism on the electron doped side. It is also shown that it is possible to characterize the T = 0 phase diagram with just one parameter called as Fermi Surface Convexity Parameter (FSCP). Next, the possibility of phase separation in the t-J model on a square lattice is investigated using local RMFT technique. It is found that for certain doping, the system phase separates into regions with antiferromagnetic and superconducting orders. Next, the role played by crystalline anisotropy of orthorhombic YBCO cuprates on their properties is examined using anisotropic tx-ty-J model and this ground state study suggests that the anisotropies seen in their properties are plausible solely due to the crystalline anisotropy. A new general method to study strongly correlated systems with singlet ground states is developed and tested in this thesis work. The last part of the thesis explores the possibility of high Tc superconductivity in graphene which is a intermediate coupling resonating valence bond (RVB) system. It is found that undoped graphene is not a superconductor, consistent with the experiments. On doping, the ground state of graphene is found to be a superconductor with “d+id” symmetry whose strength shows a dome as a function of doping which is reminiscent of RVB physics.

In dieser Arbeit werden die dynamischen Strukturen und Mannigfaltigkeiten in geschlossenen chaotischen Systemen untersucht. Das Wissen um diese dynamischen Strukturen (und Mannigfaltigkeiten) ist von Bedeutung, da sie uns einen ersten Überblick über die Dynamik des Systems geben, dass heisst, mit ihrer Hilfe sind wir in der Lage, das System zu charakterisieren und eventuell sogar seine Dynamik vorherzusagen. Die Visualisierung der dynamischen Strukturen, speziell in geschlossenen chaotischen Systemen, ist ein schwieriger und oft langer Prozess. Hier werden wir die sogenannte 'Leaking-Methode' (an Beispielen einfacher mathematischer Modelle wie der Bäcker- oder der Sinus Abbildung) vorstellen, mit deren Hilfe wir die Möglichkeit haben, Teile der Mannigfaltigkeiten des chaotischen Sattels des Systems zu visualisieren. Vergleiche zwischen den gewonnenen Strukturen und Strukturen die durch chemische oder biologische Reaktionen hervorgerufen werden, werden anhand eines kinematischen Modells des Golfstroms durchgeführt. Es wird gezeigt, dass mittels der Leaking-Methode dynamische Strukturen auch in Umweltsystemen sichtbar gemacht werden können. Am Beispiel eines realistischen Modells des Mittelmeeres erweitern wir die Leaking-Methode zur sogenannten 'Exchange-Methode'. Diese erlaubt es den Transport zwischen zwei Regionen zu charakterisieren, die Transport-Routen und Austausch-Bassins sichtbar zu machen und die Austausch-Zeiten zu berechnen. Austausch-Bassins und Zeiten werden für die nördliche und südliche Region des westlichen Mittelmeeres präsentiert. Weiterhin werden Mischungseigenschaften im Erdmantel charakterisiert und die geometrischen Eigenschaften von Mannigfaltigkeiten in einem 3dimensionalen mathematischen Modell (ABC-Abbildung) untersucht. / In this thesis, dynamical structures and manifolds in closed chaotic flows will be investigated. The knowledge about the dynamical structures (and manifolds) of a system is of importance, since they provide us first information about the dynamics of the system - means, with their help we are able to characterize the flow and maybe even to forecast it`s dynamics. The visualization of such structures in closed chaotic flows is a difficult and often long-lasting process. Here, the so-called 'Leaking-method' will be introduced, in examples of simple mathematical maps as the baker- or sine-map, with which we are able to visualize subsets of the manifolds of the system`s chaotic saddle. Comparisons between the visualized manifolds and structures traced out by chemical or biological reactions superimposed on the same flow will be done in the example of a kinematic model of the Gulf Stream. It will be shown that with the help of the leaking method dynamical structures can be also visualized in environmental systems. In the example of a realistic model of the Mediterranean Sea, the leaking method will be extended to the 'exchange-method'. The exchange method allows us to characterize transport between two regions, to visualize transport routes and their exchange sets and to calculate the exchange times. Exchange times and sets will be shown and calculated for a northern and southern region in the western basin of the Mediterranean Sea. Furthermore, mixing properties in the Earth mantle will be characterized and geometrical properties of manifolds in a 3dimensional mathematical model (ABC map) will be investigated.

Cette thèse se situe dans le cadre de l'analyse et de la commande des systèmes multidimensionnels. Ce sont des systèmes où l'information se propage dans plusieurs directions indépendantes les unes des autres (par exemple une dimension d'espace et une de temps). Les contributions présentées dans ce mémoire portent d'une part sur la commande des systèmes 2D discrets ou continus, à retards constants ou variables, et d'autre part sur la synthèse de loi de commande par retour d'état robuste des systèmes nD hybrides incertains dont l'incertitude est décrite sous forme de représentation rationnelle implicite (ILFR). Les travaux présentés utilisent deux approches, l'une basée sur le polynôme caractéristique et l'autre sur les techniques de Lyapunov. Pour les systèmes 2D à retards discrets ou continus nous avons utilisé l'approche basée sur des fonctionnelles de Lyapunov. Des conditions suffisantes de stabilité et de stabilisation par retour d'état, dépendantes du retard, sont établies. Outre la notion de stabilité, la notion de performance du type H∞ est traitée afin de résoudre le problème de rejet de perturbations pour cette classe de systèmes. Nous avons ensuite proposé un cadre assez général pour l'analyse en stabilité des systèmes nD hybrides, en utilisant la S-procédure, permettant l'obtention de conditions sous forme de LMIs faciles à exploiter numériquement. Nous avons également proposé des conditions de stabilité et de stabilisation robustes pour les systèmes nD hybrides incertains dont l'incertitude est du type LFR implicites. / This thesis deals with the analysis and the control of multidimensional systems. These systems can be defined as the classe of systems where the information is propagated in several independent direction. For instance, a 2D system with a dimension corresponding to space and, the other one to time. The contributions presented in this work focuses, on one hand, on the control of 2D discrete or continuous systems with constant or variable delays and on the other hand, on the synthesis of robust state feedback controllers for nD hybrid uncertain systems including parameter uncertainties complying with an implicit linear fractional representation (ILFR). Two approaches are used. One is based on the characteristic polynomial and the other on Lyapunov techniques. Sufficient conditions for stability and stabilization are established for 2D delayed discrete or continuous systems using Lyapunov approach. Conditions to insure both stability and a prescribed H∞ performance level are given for this class of systems. Then, a general framework for the establishment of computationally tractable LMI conditions to analyse the stability of nD hybrid systems is proposed. Robust stability and stabilization conditions are then established for nD hybrid uncertain systems. The uncertainties comply with an (ILFR) description.

Two dimensional (2D) systems with low carrier density is an outstanding platform for studying a wide spectrum of physics. These include both classical and quantum effects, arising from disorder, Coulomb interactions and even non-trivial topological properties of band-structure. In this thesis, we have explored the physics at low carrier number density in GaAs/AlGaAs heterostructure and bilayer graphene, by investigating in a larger phase space using a variety of electrical measurement tools. A two-dimensional electron system (2DES) formed in a GaAs/AlGaAs heterostructure offers an avenue to build a variety of mesoscopic devices, primarily because its surface gates can very effectively control its carrier density profile. In the first half of the thesis, we study the relevance of disorder in two kinds of devices made in a 2DES. A very strong negative gate voltage not only reduces the carrier density of the 2DES, but also drives it to a disordered state. In this state, we explore a new direction in parameter space by increasing in-plane electric field and investigating its magneto-resistance (MR). At sufficiently strong gate voltage and source-drain bias, we discover a remarkably linear MR. Its enormous magnitude and weak temperature dependence indicate that this is a classical effect of disorder. In another study, we examine a specially designed dual-gated device that can induce low number density in a periodic pattern. By applying appropriate gate voltages, we demonstrate the formation of an electrostatically tunable quantum dot lattice and study the impact of disorder on it. This work is important in paving way for solid state based platform for experimental simulations of artificial solids. The most striking property of bilayer graphene is the ability to open its band gap by a perpendicular electric field, giving the prospects of enabling a large set of de-vice applications. However, despite a band gap, a number of transport mechanisms are still active at very low densities that range from hopping transport through bulk to topologically protected 1D transport at the edges or along 1D crystal dislocations. In the second half of the thesis, we have used higher order statistical moment of resistance/conductance fluctuations, namely the variance of the fluctuations, to complement averaged resistance/conductance, and study and infer the dominant transport mechanism at low densities in a gapped bilayer graphene. Our results show possible evidence of percolative transport and topologically protected edge transport at different ranges of low number densities. We also explore the same phase space by studying its mesoscopic conductance fluctuations at very low temperatures. This is the first of its kind systematic experiment in a dual-gated bilayer graphene device. Its conductance fluctuations have several anomalous features suggesting non-universal behaviour which is at odds with conventional disordered systems.