abstract: This dissertation aims to study and understand relevant issues related to the electronic, spin and valley transport in two-dimensional Dirac systems for different given physical settings. In summary, four key findings are achieved. First, studying persistent currents in confined chaotic Dirac fermion systems with a ring geometry and an applied Aharonov-Bohm flux, unusual whispering-gallery modes with edge-dependent currents and spin polarization are identified. They can survive for highly asymmetric rings that host fully developed classical chaos. By sustaining robust persistent currents, these modes can be utilized to form a robust relativistic quantum two-level system. Second, the quantized topological edge states in confined massive Dirac fermion systems exhibiting a remarkable reverse Stark effect in response to an applied electric field, and an electrically or optically controllable spin switching behavior are uncovered. Third, novel wave scattering and transport in Dirac-like pseudospin-1 systems are reported. (a), for small scatterer size, a surprising revival resonant scattering with a peculiar boundary trapping by forming unusual vortices is uncovered. Intriguingly, it can persist in arbitrarily weak scatterer strength regime, which underlies a superscattering behavior beyond the conventional scenario. (b), for larger size, a perfect caustic phenomenon arises as a manifestation of the super-Klein tunneling effect. (c), in the far-field, an unexpected isotropic transport emerges at low energies. Fourth, a geometric valley Hall effect (gVHE) originated from fractional singular Berry flux is revealed. It is shown that gVHE possesses a nonlinear dependence on the Berry flux with asymmetrical resonance features and can be considerably enhanced by electrically controllable resonant valley skew scattering. With the gVHE, efficient valley filtering can arise and these phenomena are robust against thermal fluctuations and disorder averaging. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2017

Comprendre les transitions de phase quantiques dans les systèmes de fermions itinérants en interaction est crucial pour faire progresser notre connaissance de la criticité quantique. Cet intérêt est motivé par des expériences sur des matériaux fortement corrélés. L'attention récente s'est portée sur les matériaux bidimensionnels \((2D)\), tels que le graphène, les surfaces d'isolants topologiques et certains liquides de spin. Ces matériaux sont caractérisés par une dispersion de Dirac pseudo-relativiste. Dans ce mémoire, nous étudions les points critiques quantiques dans les systèmes de Dirac en calculant les dimensions d'échelle des bilinéaires de charge \(2\) à travers diverses classes d'universalité de Gross-Neveu, incluant Gross-Neveu, chiral Ising Gross-Neveu, chiral XY Gross-Neveu, et chiral d'Heisenberg Gross-Neveu. Nous utilisons la méthode d'expansion en grand \(N\) pour calculer les dimensions anormales en termes de \(1/N\). Ces dimensions d'échelle sont essentielles pour comprendre les transitions de phase quantiques d'un semimétal de Dirac à une phase isolante, comme observé dans des systèmes tels que le modèle \(t-V\) et des matériaux semblables au graphène. De plus, nous proposons un opérateur dual dans une théorie bosonique, qui est une combinaison de doublets monopôles invariants de jauge pour les bilinéaires dans le modèle d'Heisenberg Gross-Neveu, basé sur des conjectures précédentes. / Understanding quantum phase transitions in systems of interacting itinerant fermions is crucial for advancing our knowledge of quantum criticality. This interest is driven by experiments on strongly correlated materials. Recent focus has been on two-dimensional \((2D)\) materials, such as graphene, surfaces of topological insulators, and certain spin liquids. These materials are characterized by a pseudo-relativistic Dirac dispersion in their freely moving fermions, which lack classical analogs. In this thesis, we study the quantum critical points in Dirac systems by computing the scaling dimensions of charge \(2\) bilinears across various Gross-Neveu universality classes, including Gross-Neveu, chiral Ising Gross-Neveu, chiral XY Gross-Neveu, and chiral Heisenberg Gross-Neveu. We utilize the large \(N\) expansion method to compute the anomalous dimensions in terms of \(1/N\). These scaling dimensions are instrumental in understanding the quantum phase transitions from a Dirac semimetal to an insulating phase, as observed in systems like the \(t-V\) model and graphene-like materials. Additionally, we propose a dual operator in a bosonic theory, which is a combination of gauge-invariant monopole doublets for bilinears in the Gross-Neveu Heisenberg model, based on previous conjectures.