acase@tulane.edu / 1 / Jinyu Liu

This thesis presents a study of two-dimensional electron systems in GaAs-(Al,Ga)As heterojunctions and quasi-two-dimensional electron and hole systems in graphite within the quantum Hall effect regime of low temperature and high magnetic field. This thesis covers three main sets of experimental work as well as details of the experimental methods (chapter 2) used and the background theory behind the observed results (chapter 1). The first experimental results presented in this thesis in chapter 3 focus on contactless measurement of the equilibrium magnetisation of sample A2268, a ten layer multiple quantum well sample. Fitting the shape of dHvA oscillations at various temperatures to different models for the density of states, various properties of the system can be estimated, such as the shape of the disorder-broadened density of states and the presence of a background density of states between the Landau levels. Chapter 4 focuses on measurements of the decay of induced circulating currents in the quasi-dissipationless quantum Hall regime in two samples, V0049 and T73. The induced current is measured via contactless measurement of the associated magnetic moment. The magnitude of the induced current is found to be affected by the sweep rate of the magnetic field and also the distance of approach. The decay of the induced currents is observed at several temperatures and for different magnetic field sweep rates and distances of approach. Decays are observed for up to several days at time, far longer than previously possible. Information about the rate of decay can be used to build a picture of the decay mechanisms present in the quantum Hall regime. The presence of a power-law decay regime indicates many decay mechanisms contribute to the decay of a circulating current in the quasi-dissipationless quantum Hall regime. Chapter 5 focuses on both contactless magnetometry and transport experiments carried out on a graphite sample. The experiments aim to confirm or dispute recent claims of Dirac fermions in graphite. Experiments are carried out at temperatures in the range 30 mK to ~4 K and at two different angles to the applied magnetic field. Phase analysis of both Shubnikov de Haas and de Haas van Alphen oscillations is used to distinguish between normal and Dirac fermions. Observation of quantum Hall effect displays the presence of a half-integer quantum Hall staircase similar to that observed in graphene.

Dans cette thèse, nous avons étudié les propriétés électroniques de WS2 et WSe2 par µ-PL, spectroscopie Raman, absorption optique inter bande et µ-PL résolue en temps combinées avec des champs magnétiques intenses. Nous montrons que l'émission de l'exciton par rapport au trion dans les monocouches de WS2 et WSe2 est fonction de la puissance du laser utilisé pour l'excitation de la µ-PL. De plus, nous montrons que l'intensité de l'émission du trion peut être contrôlée indépendamment en utilisant une énergie d'excitation plus basse que la bande interdite. Il s'agit d'une preuve du contrôle de la densité de porteurs dans ces systèmes 2D. Nous avons également étudié la diffusion Raman en résonance dans une monocouche de WS2. Nous observons un mode acoustique (2LA), seulement 4cm-1 en-dessous du mode E12g. Nous montrons qu'en fonction du rapport des intensité et la largeur de ligne de chacun de ces deux pics, toute analyse qui néglige la présence de la mode 2LA peut conduire à une estimation incorrecte du nombre de couche. Les propriétés électroniques de chaque vallée d'une monocouche de WSe2 ont été sondées par µ-PL via l'étude de l'émission et de la polarisation des excitons neutres et chargés. Nous montrons que le temps de diffusion de l'exciton entre les vallées de K+ et K- est de l'ordre de plusieurs ps. Enfin, grâce à la magnéto-spectroscopie, nous mettons en évidence différents types de porteurs de charges entre la monocouche et le cristal massif. Nous montrons que dans la monocouche, les porteurs de charge se comportent comme des fermions massifs Dirac, tandis que dans le monocristal de WSe2 nous observons un comportement excitonique, décrit par le modèle de l'atome d'hydrogène / In this thesis, we have studied tungsten dichalcogenides (WS2 and WSe2) by means of steady-state µ-photoluminescence (µ-PL) and Raman spectroscopy, optical interband absorption and time-resolved µ-PL techniques in the visible spectral range combined with high magnetic fields. We demonstrate that the ratio between the trion and exciton emission can be tuned by varying the power of the laser used for excitation of the µ-PL in ungated monolayer WS2 and WSe2 samples. Moreover, the intensity of the trion emission can be independently tuned using additional sub band gap illumination. This is a direct evidence that we can control the density of carriers in a 2D system. We have investigated the resonant Raman scattering in a WS2 monolayer. We observe a second order longitudinal acoustic mode (2LA) at only 4cm-1 below the first order E12g mode. We demonstrate, that depending on the intensity ratio and the respective line widths of these two peaks, any analysis which neglects the presence of the 2LA mode can lead to a potentially incorrect assignment for the number of layers. The valley dynamics in monolayer WSe2 has been probed by monitoring the emission and polarization dynamics of neutral and charged excitons in µ-PL. We demonstrate that the exciton inter valley scattering between the K+ and K- valleys is in the order of several picoseconds. Finally, using magneto-spectroscopy studies, we reveal the very different nature of carriers in monolayer and bulk dichalcogenides. We demonstrate that in monolayer WSe2, the carriers behave as massive Dirac fermions, while in bulk WSe2 we observe a distinctly excitonic behavior which is best described within the hydrogen model

The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

Ce travail consiste en l'étude sous champ magnétique des propriétés électroniques des fermions de Dirac relativistes dans deux systèmes: graphène et isolants topologiques. Leur analogie avec la physique des hautes énergies et leurs applications potentielles ont suscité récemment de nombreux travaux. Les états électroniques sont donnés par un Hamiltonien de Dirac et la dispersion est analogue à celle des particules relativistes. La masse au repos est liée au gap du matériau avec une vitesse de Fermi remplaçant la vitesse de la lumière. Le graphène a été considéré comme un " système école " qui nous permet d'étudier le comportement relativiste des fermions de Dirac sans masse satisfaisant une dispersion linéaire. Quand un système de Dirac possède un gap non nul, nous avons des fermions de Dirac massifs. Les fermions de Dirac sans masse et massifs ont été étudiés dans le graphène épitaxié et les isolants topologiques cristallins Pb1-xSnxSe et Pb1-xSnxTe. Ces derniers systèmes sont une nouvelle classe de matériaux topologiques où les états de bulk sont isolants mais les états de surface sont conducteurs. Cet aspect particulier résulte de l'inversion des bandes de conduction et de valence du bulk ayant des parités différentes, conduisant à une transition de phase topologique. La magnéto-spectroscopie infrarouge est une technique idéale pour sonder ces matériaux de petit gap car elle fournit des informations quantitatives sur les paramètres du bulk via la quantification de Landau des états électroniques. En particulier, la transition de phase topologique est caractérisée par une mesure directe de l'indice topologique. / This thesis reports on the study under magnetic field of the electronic properties of relativistic-like Dirac fermions in two Dirac systems: graphene and topological insulators. Their analogies with high-energy physics and their potential applications have attracted great attention for fundamental research in condensed matter physics. The carriers in these two materials obey a Dirac Hamiltonian and the energy dispersion is analogous to that of the relativistic particles. The particle rest mass is related to the band gap of the Dirac material, with the Fermi velocity replacing the speed of light. Graphene has been considered as a “role model”, among quantum solids, that allows us to study the relativistic behavior of massless Dirac fermions satisfying a linear dispersion. When a Dirac system possesses a nonzero gap, we have massive Dirac fermions. Massless and massive Dirac fermions were studied in high-mobility multilayer epitaxial graphene and in topological crystalline insulators Pb1-xSnxSe and Pb1-xSnxTe. The latter system is a new class of topological materials where the bulk states are insulating but the surface states are conducting. This particular aspect results from the inversion of the lowest conduction and highest valence bulk bands having different parities, leading to a topological phase transition. Infrared magneto-spectroscopy is an ideal technique to probe these zero-gap or narrow gap materials since it provides quantitative information about the bulk parameters via the Landau quantization of the electron states. In particular, the topological phase transition can be characterized by a direct measurement of the topological index.

Cette thèse présente une étude sur les hétérostructures à base de mercure, cadmium et tellure (HgCdTe ou MCT) pour l'émission et la détection de radiations Térahertz (THz). En raison de leurs propriétés physiques spécifiques, les hétérostructures à base de HgCdTe devraient en effet jouer un rôle important dans les futurs dispositifs Térahertz. Parmi les autres propriétés remarquables de ces structures, les puits quantiques de HgTe/CdTe à l'épaisseur critique (environ 6,3 nm) présentent un état sans gap caractérisé par la relation de dispersion linéaire propre aux fermions Dirac sans masse. Lorsque la largeur du puits quantique dépasse la valeur critique, la structure de la bande s’inverse. Dans ce cas, ces puits deviennent des isolants topologiques bidimensionnels qui passionnent la communauté scientifique depuis une décennie. Cette inversion de bande peut être brisée en variant plusieurs paramètres physiques tels que le champ magnétique ou la température. Ces transitions de phases topologiques pourraient être très intéressantes en vue d’applications à l’électronique haute fréquence et à basse consommation d'énergie.Dans ce travail, l’accent est mis sur des dispositifs munis de grilles et présentant une structure de bande inversée. Premièrement, nous mettons en évidence la possibilité de détecter la lumière incidente Térahertz à des températures cryogéniques. Nous rapportons également une amélioration de la détection Térahertz au voisinage de la transition de phase topologique induite par le champ magnétique et proche du point de neutralité de charge. Deuxièmement, nous observons sans ambiguïté la transition de phase induite par la température entre l'état isolant topologique et l'état isolant de l’effet Hall quantique, par des expériences de magnéto-transport. Ensuite, en utilisant la technique de détection Térahertz non résonnante, nous avons retracé avec succès les niveaux de Landau du puits et défini précisément le champ magnétique critique correspondant à la transition de phase quantique. Nous avons constaté que cette technique Térahertz peut être utilisée dans chaque échantillon avec grille sans besoin de quatre contacts de mesure ni de traitement de données mathématiques.En ce qui concerne les émetteurs Térahertz, nous présentons ici nos résultats sur l'émission stimulée d'hétérostructures de HgCdTe dans leur état semi-conducteur conventionnel à des fréquences supérieures à 30 THz. Nous discutons des mécanismes physiques impliqués et des voies prometteuses vers le domaine de fréquence entre 5 et 15 THz. Malgré le fait que les principaux matériaux pour les lasers solides à grandes longueurs d'ondes sont des hétérostructures basées sur les semi-conducteurs III-V, leurs bandes Reststrahlen rendent cette gamme de fréquences inaccessible pour les lasers à base de III-V (y compris les lasers à cascade quantique) même à des températures cryogéniques. Étant donné que la bande d'absorption du réseau cristallin dans les hétérostructures à base de Hg1-xCdxTe est décalée vers des longueurs d'onde plus grandes, ces composés (avec x <0,21) semblent être très prometteurs en tant que lasers solides Térahertz. / This thesis presents an investigation of mercury-cadmium-telluride (HgCdTe or MCT) based heterostructures for emission and detection of Terahertz (THz) radiations. Due to their specific physical properties, HgCdTe-based heterostructures are indeed expected to play an important role in future terahertz systems. Among other remarkable properties, HgTe/CdTe-based quantum wells (QWs) at the critical thickness (about 6.3 nm), exhibit a gapless state characterized by the linear energy-momentum law of massless Dirac fermions. When the QW width exceeds this critical value, the energy band structure becomes inverted. In this case, these QWs are shown to be two-dimensional topological insulators that attract since the last decade a great fundamental interest. This band inversion can be broken by varying several external physical parameters as magnetic field or temperature. These so-called topological phase transitions could be of high interest for future low-energy consumption and high frequency electronics.Here, focusing on gated devices presenting inverted band ordering, we first evidence the possibility to detect THz incident light at cryogenic temperatures. We also report on an enhancement of the terahertz photoconductive response in the vicinity of the magnetic field driven topological phase transition and close to the charge neutrality point. Secondly, we observed unambiguously the temperature driven phase transition between the topological insulator state and the usual quantum Hall insulator state by magneto-transport experiments. Then, using the non-resonant THz detection technique, we successfully imaged the QWs Landau levels and defined precisely the critical magnetic field corresponding to the quantum phase transition. We found that this THz technique can be used in every gated sample without need neither for four contacts devices nor mathematical data processing.Regarding terahertz emitters, we present here our results on stimulated emission of HgCdTe heterostructures in their conventional semiconductor state above 30 THz, discussing the physical mechanisms involved and promising routes towards the 5–15 THz frequency domain. Despite the fact that the leading materials for long wavelength solid-state lasers are heterostructures based on III-V semiconductors, their Reststrahlen bands makes this frequency range inaccessible for III-V-based lasers (including quantum cascade lasers) even at cryogenic temperatures. Since the lattice absorption band in Hg1-xCdxTe-based heterostructures is shifted to longer wavelengths, these compounds with (x<0.21) seem to be very promising as interband solid-state THz lasers.

Dans cette thèse, j’étudie la compressibilité électronique de deux isolants topologiques tridimensionnels : Le tellurure de mercure (HgTe) sous contrainte et le séléniure de bismuth (Bi2Se3).Je présente des mesures d’admittance électronique à basse température résolues en phase sur une large gamme de fréquence. Cela permet d’extraire la capacité quantique associé à la densité d’états et la résistivité des matériaux étudiés.Nous montrons qu’un isolant topologique intrinsèque présente une réponse dominée par les états de surface topologiques sur une large gamme d’énergie qui s’étend au-delà du gap de transport du matériau massif. Ce régime, appelé « écrantage de Dirac », est caractérisé par une compressibilité électronique proportionnelle à l’énergie de surface et une haute mobilité.Dans la suite, nous nous intéressons à la limite de ce régime. Nous observons qu’à haute énergie et sous l’influence de forts champs électriques perpendiculaires, des états excités massifs de surface sont peuplés ce qui se manifeste expérimentalement de différentes façons : Une chute dans la constante de diffusion électronique, un pic de conductivité ainsi que l’apparition d’un deuxième type de porteurs en magnéto-transport et de métastabilité dans la relation charge-tension.Un modèle théorique basé sur un traitement quasi-relativiste du Hamiltonien de surface est présenté. Il permet d’identifier la dépendance en énergie et champ électrique des états massifs de surface.Cette thèse est complémenté par des résultats expérimentaux sur Bi2Se3 obtenu par croissance sur nitrure de bore mettent en évidence l’importance de la pureté des interfaces d’isolants topologiques. / This thesis discusses the electronic compressibility of two representative three dimensional topological insulators: Strained mercury telluride (HgTe) and bismuth selenide (Bi2Se3).I present low temperature phase-sensitive electron admittance data over a broad frequency range. This allows to extract the quantum capacitance related to the density of states and the resistivity of the investigated materials.We show that the response of an intrinsic topological insulator is dominated by topological surface states over a large energy range exceeding the bulk material’s transport gap. This regime, named “Dirac screening” is characterized by an electron compressibility proportional to the surface Fermi level and a high mobility.Subsequently, we investigate the limits of this regime. At high energy and large perpendicular electric fields we observe the population of excited massive surface states. Experimentally, these manifest themselves in multiple signatures: A drop in the electronic diffusion constant, a peak in the conductivity, appearance of a second carrier type in magneto-transport and meta-stability in the charge-voltage relation.A theoretical model based on a quasi-relativistic treatment of the surface Hamiltonian is presented. It allows to identify the electric field and energy dependence of the massive surface states.This thesis is complemented by experimental results on Bi2Se3 grown on boron nitride, where we demonstrate the importance of clean surfaces for the study of electronic properties in topological insulators.

Nanoelectronics and spintronics are concerned with writing, transporting, and reading information stored in electronic charge and spin degrees of freedom at the nanoscale. Past few years have shown that two spintronics effects discovered in the 19th century, namely anisotropic magnetoresistance and anomalous Hall effect, can be used also for sensing antiferromagnetism which opened the field of antiferromagnetic spintronics. The more than a century of controversial studies of these effects have shown their relativistic spin-orbit coupling and spin-polarisation symmetry breaking origin. However, a complete understanding of these effects and a fully predictive theory capable of identifying novel suitable antiferromagnetic materials are still lacking. Here, by extending modern symmetry and topology concepts in condensed matter physics, we have further developed the theory of anisotropic magnetoresistance and spontaneous Hall effect. Our approach is based on magnetic symmetry and topology analysis of antiferromagnetic energy bands, Bloch spectral functions, and Berry curvatures calculated from the state-of-the- art first-principle theory. This guided us to the prediction of two novel, previously unanticipated effects: relativistic metal-insulator transition from antiferromagnetic Dirac fermions, and crystal Hall...

Hard-core bosons are versatile and useful in describing several physical systems due to their one-to-one mapping with spin-1/2 operators. We propose two frameworks where hard-core boson mapping not only reduces the complexity of the original problem, but also captures important features of the physics of the original system that would have implied high-computational procedures with not much profound insight in the mechanisms behind its behavior. The first case study comprising part i is an approach to the description of the phases 2D Lattice Gauge Theories, the Quantum 6-Vertex Model and the Quantum Dimer Model using one fluctuating electric string as an 1D precursor of the whole 2D systems[HAMS19]. Both models and consequently the string are described by the Rokhsar-Kivelson Hamiltonian with parameter v measuring the competition of potential versus kinetic terms. The string can be mapped one-to-one onto a 1D system of hard-core bosons that can be solved exactly for the Quantum 6-Vertex Model, and offers footprints of the phase diagram of the Quantum Dimer Model in the region close to the Rokhsar-Kivelson point v = 1, especially when |v| ≤ 1. The second case study we have discussed in part ii is an extension of higher-dimensional bosonization techniques in Landau Fermi liquids to the case of nodal semimetals where the Fermi surface shrinks to a point, so the description of particle-hole interactions as fluctuations of the Fermi surface is not available [MS20]. Additionaly, we focus our analysis on the Q = 0 sector where the electron and the hole have opposite momenta ±k, so they are mapped into a hard-core boson located at a site k in the reciprocal lattice. To test our extension we calculate nonperturbative corrections to the optical conductivity of 2D Dirac fermions with electron-electron interactins described as a Coulomb potential, obtaining results consistent to the literature and the experimental reports where corrections are small even in strong coupling regimes. Part iii discusses further ideas derived from parts i and ii, including a brief discussion on addressing the weak coupling instability in bilayer graphene using the bosonization extension that offers a picture of hard-core bosons describing Q = 0 excitons that undergo a Bose-Einstein condensation resulting in a ground state adiabatically disconnected from the noninteracting case.:1 Introduction 1 1.1 Quantum link models and fluctuating electric strings 2 1.2 Bosonization of Particle-hole excitations in 2D Dirac fermions 7 1.3 Structure of the document 11 i. Quantum link models and fluctuating electric strings 2. A Brief Introduction to Lattice Gauge Theories 15 2.1 Continuous formulation of U(1) gauge theories 15 2.1.1 Gauge field equations 16 2.1.2 Gauss’ law as generator of the gauge transformations 18 2.2 U(1) gauge theories on a lattice 19 2.2.1 Gauge field Hamiltonian 20 2.2.2 Cylindrical algebra from LGT 20 2.2.3 Generator of gauge transformations 21 2.3 Abelian Quantum Link Model 22 2.3.1 Quantum Link Models (QLMs) with S = 1 / 2 23 2.3.2 ’t Hooft operators and winding number sectors 24 2.3.3 Construction of the QLM Hamiltonian 26 2.4 Conclusions 28 3. Electric string in Q6VM as a XXZ chain 29 3.1 Realization of the Q6VM in the S = 1 / 2 QLM 31 3.2 Mapping the electric string to the XXZ chain 32 3.3 Phases of the electric string from the XXZ chain 33 3.3.1 v > 1: FM insulator 34 3.3.2 v = 1: RK point 36 3.3.3 −1 < v < 1: Gapless phase 36 3.3.4 v ≤ −1: KT transition and AFM insulator 37 3.4 Numerical approach: Drude Weight and system size effects 38 3.5 Summary and Discussion 40 4. Electric line in the QDM as a hard-core boson two-leg ladder 41 4.1 Realization of the QDM in the S = 1/ 2 QLM 42 4.2 Construction of an electric string in the QDM 43 4.3 Mapping the electric string in QDM to a two-leg ladder 45 4.3.1 QLM in a triangular lattice 45 4.3.2 From the triangular lattice to the two-leg ladder 45 4.3.3 Construction of the 1D bosonic Hamiltonian 46 4.4 Phases of the electric string from the bosonic two-leg ladder 48 4.4.1 Left Hand Side (LHS) of the Rokhsar-Kivelson (RK) point: Charge Density Wave (CDW) states 48 4.4.2 Right Hand Side (RHS) of the RK point: phase-separated states 50 4.5 Numerical approach: Drude Weight and system size effects 51 4.6 Summary and Discussion 52 ii Bosonization of particle-hole excitations in 2D Dirac fermions 5 Graphene in a nutshell 57 5.1 Origin of the hexagonal structure 57 5.1.1 Hybrid orbitals in C 58 5.1.2 Honeycomb lattice 60 5.2 Tight-binding approach 61 5.2.1 Hopping and overlapping matrices in Nearest Neighbor (NN) approximation 62 5.2.2 Dispersion relation for π electrons 62 5.3 Effective 2D Dirac Fermion Hamiltonian 64 5.4 Electron-electron interactions 65 6 Bosonization of the Q = 0 continuum of Dirac Fermions 67 6.1 Effective Hamiltonian and Hilbert space 69 6.2 Effective Heisenberg Hamiltonian 70 6.3 Quadratic Bosonic Hamiltonian 71 6.4 Connection to diagramatic perturbation theory 73 6.5 Parametrization of the reciprocal space 74 6.5.1 Coordinate transformation 74 6.5.2 Polar parametrization 75 6.5.3 Angular momentum channels 75 6.6 Discussion and Summary 76 7 Non-perturbative corrections to the Optical Conductivity of 2D Dirac Fermions 77 7.1 Optical Conductivity 79 7.1.1 Bosonized current operator and susceptibility 79 7.1.2 Susceptibility in terms of the eigenstates 80 7.1.3 Regularization of the Lehman representation 81 7.2 Numerical approach: IR regularization and system size effects 82 7.2.1 Discretization size dependence 82 7.2.2 Dependence on the IR cutoff 83 7.2.3 Comparison of numerical results with corrections from first order perturbation theory 84 7.2.4 Optical conductivity for several coupling constants 85 7.3 Discussion and Summary 86 iii Weak coupling instability, New Perspectives & Conclusions 8 Weak coupling instability in bilayer graphene from a bosonization picture 91 8.1 Band structure of Bernal-stacked bilayer graphene 92 8.2 Generalization of the effective Hamiltonian of graphene 93 8.2.1 Density of states in monolayer and bilayer graphene 94 8.2.2 Projection onto Q = 0 sector and effective Heisenberg pseudospin Hamiltonian 95 8.2.3 Zeeman vortex coordinates and HCB operators 95 8.2.4 Bogoliubov-Valatin basis 97 8.3 Interaction potentials 97 8.4 BCS instability in pseudospin picture 99 8.5 Numerical procedure 101 8.5.1 Numerical BCS instability 101 8.5.2 Functional form of the instability 101 8.5.3 Comparison to the instability from BCS theory 105 8.6 Conclusions 105 9 Conclusions 107 iv Appendices A. Yang & Yang’s expressions of ground state energy of XXZ Chain using Bethe Ansatz 115 A.1 Bethe Ansatz 115 A.2 Explicit formulas for f ( ∆, 0 ) 116 B. Kadanoff-Baym (KB) self-consistent Hartree-Fock (SCHF) approximation 119 B.1 Details of connection to perturbation theory 119 B.1.1 Bare and dressed fermion propagators 119 B.1.2 Bethe-Salpeter ladder 120 B.1.3 Particle-hole propagator and comparison to HP boson propagator 121 C, Optical Conductivity from Pseudospin precession 123 C.1 Minimal coupling and band (electron-hole) basis 123 C.2 Equations of motion of charge and pseudospin densities 124 C.3 Optical Conductivity from Fermi-Dirac distributions at finite temperature 124 D. Momentum space reparametrization 127 D.1 General coordinate transformations on the continuum limit 127 D.2 Polar re-discretization 129 D.3 Angular momentum channels 130 D.4 Selection of the radial parametrization 130 Bibliography 133