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

Im Rahmen des Funktionalintegralzugangs zur Quanteneichfeldtheorie wird in der vorliegenden Arbeit eine Quantisierungsprozedur in Termen eichinvarianter Felder vorgeschlagen und am Beispiel zwei- und vierdimensionaler abelscher Modelle (Thirring-Modell und QED) sowie der One-Flavour QCD konkret realisiert. Dazu wird die Algebra der aus der eichabhängigen Feldkonfiguration der zugrunde liegenden Quantenfeldtheorie gebildeten eichinvarianten Grassmann-Algebra-wertigen Differentialformen, welche die Struktur einer Z_2-graduierten Differentialalgebra trägt, näher untersucht. Danach erfolgt die Implementierung eines geeignet gewählten Satzes eichinvarianter Felder sowie bestimmter algebraischer Relationen in das Funktionalintegral, wodurch die ursprüngliche eichabhängige Feldkonfiguration ausintegriert werden kann. Diese als "Reduktion des Funktionalintegrals" bezeichnete Prozedur führt schließlich auf eine effektive bosonisierte (Quanten-) Theorie wechselwirkender eichinvarianter, und damit physikalischer Felder. Die vorgestellte Prozedur kann als allgemeines Bosonisierungsschema für Quantenfeldtheorien in beliebigen Raum-Zeit-Dimensionen angesehen werden. Die physikalische Auswertung der erhaltenen effektiven Theorien wird am Beispiel der Berechnung der chiralen Anomalie sowie bestimmter Vakuum-Erwartungswerte im Rahmen der untersuchten abelschen Modelle demonstriert. Wie sich dabei zeigt, wird man mit einer Reihe neuartiger Phänomene und Probleme konfrontiert, die bei geeigneter Behandlung tiefere Einblicke in nichtperturbative Fragestellungen erlauben. / Within the thesis a new procedure, called "reduction of the functional integral", is developed for formulating quantum field theories in terms of gauge invariant quantities (physical observable fields). It provides a new way for the construction and analysis of effective field theoretical models. Starting with a detailed mathematical analysis of the algebra of Grassmann--algebra valued gauge invariants, the procedure is applied to the two--dimensional Thirring--model, the four--dimensional spinor QED and the one--flavor QCD in four dimensions. For each of these three models an effective theory of interacting bosonic gauge invariant fields was deduced on the quantum level. Apart from this more theoretical considerations, first steps on the way to an analysis of the obtained effective models towards their application in various physical problems are performed. In the case of the two Abelian models a new approach to the bosonisation scheme and the calculation of the chiral anomaly in two and four dimensions were obtained, giving some deeper insight into the nature of the bosonisation phenomenon as well as the nature of anomalies, respectively. Moreover, the investigation of current--current expectation values shows that the suggested procedure can be viewed as a new way towards a non--perturbative formulation and understanding of quantum field theories.

Im Rahmen des Funktionalintegralzugangs zur Quanteneichfeldtheorie wird in der vorliegenden Arbeit eine Quantisierungsprozedur in Termen eichinvarianter Felder vorgeschlagen und am Beispiel zwei- und vierdimensionaler abelscher Modelle (Thirring-Modell und QED) sowie der One-Flavour QCD konkret realisiert. Dazu wird die Algebra der aus der eichabhängigen Feldkonfiguration der zugrunde liegenden Quantenfeldtheorie gebildeten eichinvarianten Grassmann-Algebra-wertigen Differentialformen, welche die Struktur einer Z_2-graduierten Differentialalgebra trägt, näher untersucht. Danach erfolgt die Implementierung eines geeignet gewählten Satzes eichinvarianter Felder sowie bestimmter algebraischer Relationen in das Funktionalintegral, wodurch die ursprüngliche eichabhängige Feldkonfiguration ausintegriert werden kann. Diese als "Reduktion des Funktionalintegrals" bezeichnete Prozedur führt schließlich auf eine effektive bosonisierte (Quanten-) Theorie wechselwirkender eichinvarianter, und damit physikalischer Felder. Die vorgestellte Prozedur kann als allgemeines Bosonisierungsschema für Quantenfeldtheorien in beliebigen Raum-Zeit-Dimensionen angesehen werden. Die physikalische Auswertung der erhaltenen effektiven Theorien wird am Beispiel der Berechnung der chiralen Anomalie sowie bestimmter Vakuum-Erwartungswerte im Rahmen der untersuchten abelschen Modelle demonstriert. Wie sich dabei zeigt, wird man mit einer Reihe neuartiger Phänomene und Probleme konfrontiert, die bei geeigneter Behandlung tiefere Einblicke in nichtperturbative Fragestellungen erlauben. / Within the thesis a new procedure, called "reduction of the functional integral", is developed for formulating quantum field theories in terms of gauge invariant quantities (physical observable fields). It provides a new way for the construction and analysis of effective field theoretical models. Starting with a detailed mathematical analysis of the algebra of Grassmann--algebra valued gauge invariants, the procedure is applied to the two--dimensional Thirring--model, the four--dimensional spinor QED and the one--flavor QCD in four dimensions. For each of these three models an effective theory of interacting bosonic gauge invariant fields was deduced on the quantum level. Apart from this more theoretical considerations, first steps on the way to an analysis of the obtained effective models towards their application in various physical problems are performed. In the case of the two Abelian models a new approach to the bosonisation scheme and the calculation of the chiral anomaly in two and four dimensions were obtained, giving some deeper insight into the nature of the bosonisation phenomenon as well as the nature of anomalies, respectively. Moreover, the investigation of current--current expectation values shows that the suggested procedure can be viewed as a new way towards a non--perturbative formulation and understanding of quantum field theories.