This thesis is devoted to quantum confinement effects in low-dimensional Dirac materials. We propose a variety of schemes in which massless Dirac fermions, which are notoriously diffcult to manipulate, can be trapped in a bound state. Primarily we appeal for the use of external electromagnetic fields. As a consequence of this endeavor, we find several interesting condensed matter analogues to effects from relativistic quantum mechanics, as well as entirely new effects and a possible novel state of matter. For example, in our study of the effective Coulomb interaction in one dimension, we demonstrate how atomic collapse may arise in carbon nanotubes or graphene nanoribbons, and describe the critical importance of the size of the band gap. Meanwhile, inspired by groundbreaking experiments investigating the effects of strain, we propose how to confine the elusive charge carriers in so-called velocity barriers, which arise due to a spatially inhomogeneous Fermi velocity triggered by a strained lattice. We also present a new and beautiful quasi-exactly solvable model of quantum mechanics, showing the possibilities for confinement in magnetic quantum dots are not as stringent as previously thought. We also reveal that Klein tunnelling is not as pernicious as widely believed, as we show bound states can arise from purely electrostatic means at the Dirac point energy. Finally, we show from an analytical solution to the quasi-relativistic two-body problem, how an exotic same-particle paring can occur and speculate on its implications if found in the laboratory.

Thesis advisor: Kevin S. Bedell / One of the many achievements of renowned physicist L.D. Landau was the formulation of Fermi Liquid Theory (FLT). Originally debuted in the 1950s, FLT has seen abundant success in understanding degenerate Fermi systems and is still used today when trying to understand the physics of a new interacting Fermi system. Of its many advantages, FLT excels in explaining why interacting Fermi systems behave like their non-interacting counterparts, and understanding transport phenomena without cumbersome and confusing mathematics. In this work, FLT is applied to systems whose low energy excitations obey the massless Dirac equation; i.e. the energy dispersion is linear in momentum, ε α ρ, as opposed to the normal quadratic, ε α ρ². Such behavior is seen in numerous, seemingly unrelated, materials including graphene, high T[subscript]c superconductors, Weyl semimetals, etc. While each of these materials possesses its own unique properties, it is their low energy behavior that provides the justification for their grouping into one family of materials called Dirac materials (DM). As will be shown, the linear spectrum and massless behavior leads to profound differences from the normal Fermi liquid behavior in both equilibrium and transport phenomena. For example, with mass having no meaning, we see the usual effective mass relation from FLT being replaced by an effective velocity ratio. Additionally, as FLT in d=2 has been poorly studied in the past, and since the most famous DM in graphene is a d=2 system, a thorough analysis of FLT in d=2 is presented. This reduced dimensionality leads to substantial differences including undamped collective modes and altered quasiparticle lifetime. In chapter 3, we apply the Virial theorem to DM and obtain an expression for the total average ground state energy $E=\frac{B}{r_s}$ where $B$ is a constant independent of density and $r_s$ is a dimensionless parameter related to the density of the system: the interparticle spacing $r$ is related to $r_s$ through $r=ar_s$ where $a$ is a characterstic length of the system (for example, in graphene, $a=1.42$ \AA). The expression derived for $E$ is unusual in that it's typically impossible to obtain a closed form for the energy with all interactions included. Additionally, the result allows for easy calculation of various thermodynamic quantities such as the compressibility and chemical potential. From there, we use the Fermi liquid results from the previous chapter and obtain an expression for $B$ in terms of constants and Fermi liquid parameters $F_0^s$ and $F_1^s$. When combined with experimental results for the compressibility, we find that the Fermi liquid parameters are density independent implying a unitary like behavior for DM. In chapter 4, we discuss the alleged universal KSS lower bound in DM. The bound, $\frac{\eta}{s}\geq\frac{\hbar}{4\pi k_B}$, was derived from high energy/string theory considerations and was conjectured to be obeyed by all quantum liquids regardless of density. The bound provides information on the interactions in the quantum liquid being studied and equality indicates a nearly perfect quantum fluid. Since its birth, the bound has been highly studied in various systems, mathematically broken, and poorly experimented on due to the difficult nature of measuring viscosity. First, we provide the first physical example of violation by showing $\frac{\eta}{s}\rightarrow 0$ as $T\rightarrow T_c$ in a unitary Fermi gas. Next, we determine the bound in DM in d=2,3 and show unusual behavior that isn't seen when the bound is calculated for normal Fermi systems. Finally we conclude in chapter 5 and discuss the outlook and other avenues to explore in DM. Specifically, it must be pointed out that the physics of what happens near charge neutrality in DM is still poorly understood. Our work in understanding the Fermi liquid state in DM is necessary in understanding DM as a whole. Such a task is crucial when we consider the potential in DM, experimentally, technologically, and purely for our understanding. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

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Ces dernières années, de nouveaux matériaux bidimensionnels aux propriétés surprenantes ont été découverts, le plus connu étant le graphène. Dans ces matériaux, les électrons du niveau de Fermi ont une masse apparente nulle, et peuvent être décrits par l’équation de Dirac sans masse. Un tel phénomène apparaît dans des situations très générales, pour les matériaux bidimensionnels ayant une structure périodique en « nid d’abeille ». De plus, la prise en compte d’interactions mène à des équations de Dirac non linéaires. Ces équations apparaissent également dans l’étude des paquets d’ondes lumineuses dans certaines fibres optiques. Le but de cette thèse est d’étudier l’existence et la stabilité de solutions stationnaires de ces équations avec termes non linéaires sous-critiques et critiques, et de montrer qu’ils sont la limite de solutions stationnaires de l’équation de Schrödinger non linéaire à potentiel périodique dans certains régimes de paramètres. Du point de vue mathématique, on devra résoudre les équations d’Euler-Lagrange de fonctionnelles d'énergie fortement indéfinies faisant intervenir l’opérateur de Dirac. Il s’agira en particulier d’étudier le cas des non-linéarités avec exposant critique, encore mal comprises pour ce type de fonctionnelle, et qui apparaissent naturellement en optique non linéaire. Les résultats de cette thèse pourraient avoir un impact important en physique, en particulier en physique du solide et optique non linéaire. / Recently, new two-dimensional materials possessing unique properties have been discovered, the most famous being the graphene. In this materials, electrons at the Fermi level behave as massless particles and can be described by the massless Dirac equation. This phenomenon is quite general, and it is a common features of "honeycomb" periodic structures. Moreover, taking into account interaction leads to non-linear Dirac equations, which also appear in the description of light propagation in particular waveguides. The aim of the thesis is to study existence and stability of stationary solutions for those equations with both sub-critical and critical nonlinearities, and to show that they are limit of stationary solutions to the Schroedinger equation with honeycomb potential, for a suitable choice of parameters. This amounts to solving the Euler-Lagrange equation for strongly indefinite energy functionals, involving the Dirac operator. We will deal with critical nonlinearities, which are still poorly understood, and appear naturally in non-linear optics. This results may have an impact on the understanding some solid state or nonlinear optics systems.

Les matériaux quantiques ont récemment introduit en physique de la matière condensée pour unifier tous les matériaux dans lesquels les fortes corrélations électroniques gouvernent les propriétés physiques du système (e.g. les isolants de Mott) et les matériaux dont les propriétés électroniques sont déterminées par la géométrie de la fonction d’onde (e.g. matériaux de Dirac). Ces matériaux montrent des propriétés émergentes résultantes de l’intrication de différents degrés de libertés : la charge, le spin et le moment orbital, donnant lieu aux propriétés topologiques des électrons. L’étude de ces interactions et des compétitions entre les degrés de liberté pertinents nécessite l’utilisation de techniques pompe-sonde ultra-rapides. Particulièrement, les pulses laser femtosecondes interagissent uniquement avec les électrons pour les placer dans un état hors-équilibre décrit par des distributions de type non Fermi-Dirac. La dynamique subséquente implique de nombreux processus, avec un temps de relaxation relié aux constantes de couplage. De plus, dans les techniques résolues en temps, la lumière peut agir comme un paramètre externe, différent des paramètres thermodynamiques, pour explorer le diagramme de phase. Cela nous donne l’opportunité de stabiliser de nouveaux états inaccessibles par des chemins thermiques quasi-adiabatiques ou de manipuler les propriétés physiques des systèmes.Dans cette thèse, nous avons réalisé différentes expériences dans le but d’étudier les propriétés à l’équilibre et hors équilibre de deux matériaux corrélés: BaCo₁₋ₓNiₓS₂ et (V₁₋ₓMₓ)₂O₃.La première partie de ce projet a été dédiée principalement à l’étude de BaNiS₂, le précurseur métallique de la transition de Mott dans BaCo₁₋ₓNiₓS₂ . En utilisant l’ARPES, nous avons étudié la structure de bandes électroniques de BaNiS₂ dans toute la zone de Brillouin. L’expérience, combinée avec des calculs théoriques, révèle un nouveau type de cône de Dirac bidimensionel à caractère orbitalaire d et induit par les corrélations. Le croisement des bandes est protégé par les symétries particulières de la structure cristalline. Nous avons aussi mesuré la structure de bandes de l’isolant de Mott BaCoS₂ dans ses phases magnétique et non magnétiques.Dans la seconde partie, nous avons étudié la dynamique électronique hors équilibre de BaNiS₂ et (V₁₋ₓMx)₂O₃. Grâce à des mesures tr-ARPES et tr-Réflectivité, nous avons observé une renormalisation non thermique et ultra-rapide du cône de Dirac dans BaNiS₂. Ce phénomène est purement provoqué par les excitations électroniques et est stabilisé par l’intéraction entre les électrons et les phonons. De plus, en utilisant différentes techniques pompe-sonde (tr-XRD basé sur XFEL et tr-Réflectivité) nous avons aussi exploré des phases hors-équilibre du matériau prototype de Mott-Hubbard (V₁₋ₓMx)₂O₃ appartenant à différentes parties de son diagramme de phase. Nos résultats montrent une phase transitoire non thermique se développant immédiatement après la photoexcitation ultra-rapide et durant quelques picosecondes dans les phases métallique et isolantes. Cette phase transitoire est accompagné par une distorsion structural qui correspond à un durcissement du réseau et est marqué par un “blue shift” du mode phononique A₁g. Nos résultats soulignent l’importance du remplissage des orbitales aussi bien que des effets important des forts couplages électron-réseau sélectifs dans les matériaux fortement corrélés. / Quantum materials is a new term in condensed matter physics that unifies all materials in which strong electronic correlation governs physical properties of the system (e.g. Mott insulators) and materials whose electronic properties are determined by the geometry of the electronic wave function (e.g. Dirac materials). These materials show emergent properties– that is, properties that only appear by intricate interactions among many degrees of freedom, such as charge, spin and orbital, giving rise to topological properties of electrons. The study of these interactions and competitions between the relevant degrees of freedom demands applying ultrafast pump-probe techniques. Particularly, femtosecond laser pulses act only on the electrons and set them to an out-of-equilibrium state inexplicable by the Fermi-Dirac distribution. The ensuing dynamics involves various processes and the rate at which the relaxation occurs is related to the coupling constants. Moreover, in time-resolved pump-probe techniques light can act as an additional external parameter to change of the phase diagram – different from thermodynamic parameters. It gives us the opportunity of stabilizing new states inaccessible by quasi-adiabatic thermal pathways or eventually manipulating the physical properties of the systems.In this thesis, we performed different experiments in order to study the equilibrium and out-of-equilibrium properties of two correlated compounds: BaCo₁₋ₓNiₓS₂ and (V₁₋ₓMₓ)₂O₃.The first part of the project was mainly devoted to the study of BaNiS₂ that is the metallic precursor of the Mott transition in BaCo₁₋ₓNiₓS₂. By applying ARPES, we studied the electronic band structure of BaNiS₂ in its entire Brillouin zone. These results combined with some theoretical calculations give evidence of a novel correlation-induced and two-dimensional Dirac cone with d-orbital character. The band crossing is protected by the specific symmetries of the crystal structure. We also investigated the electronic band structure of the Mott insulator BaCoS₂ in its magnetic and nonmagnetic phases.In the second part, we studied the out-of-equilibrium electron dynamics of BaNiS₂ and (V₁₋ₓMx)₂O₃. By means of tr-ARPES and tr-reflectivity measurements, we observed an ultrafast and non-thermal renormalization of the Dirac cone in BaNiS₂ . This phenomenon is purely provoked by the electronic excitation and is stabilized by the interplay between the electrons and phonons. Moreover, by applying various pump-probe techniques (XFEL-based tr-XRD and tr-Reflectivity) we also explored the out-of-equilibrium phases of the prototype Mott-Hubbard material (V₁₋ₓMx)₂O₃ in different parts of its phase diagram. Our results show a transient non-thermal phase developing immediately after ultrafast photoexcitation and lasting few picoseconds in both metallic and insulating phases. This transient phase is followed by a structural distortion that corresponds to a lattice hardening and is marked by a “blue shift” of the A₁g phonon mode. These results underline the importance of the orbital filling as well as the strong effect of the selective electron-lattice coupling in the strongly correlated materials.