Simulationen von Gittereichtheorien spielen eine grundlegende Rolle bei First-Principles-Rechnungen im Kontext der Hochenergiephysik. Diese Arbeit zielt darauf ab, aktuelle Simulationsmethoden für First-Principle-Berechnungen zu verbessern und diese Methoden auf relevante physikalische Modelle anzuwenden. Wir gehen dieses Problem mit drei verschiedenen Ansätzen an: maschinelles Lernen, Quantencomputing und Tensornetzwerke. Im Rahmen des maschinellen Lernens haben wir eine Methode zur Schätzung thermodynamischer Observablen in Gitterfeldtheorien entwickelt. Genauer gesagt verwenden wir tiefe generative Modelle, um den absoluten Wert der freien Energie abzuschätzen. Wir haben die Anwendbarkeit unserer Methode durch die Untersuchung eines Spielzeugmodells demonstriert. Unser Ansatz erzeugt genauere Messungen im Vergleich mit dem Standard-Markov-Ketten-Monte-Carlo-Verfahren, wenn wir einen Phasenübergangspunkt überqueren. Im Kontext des Quantencomputings ist es unser Ziel, die aktuellen Algorithmen für Quantensimulationen zu verbessern. In dieser Arbeit haben wir uns mit zwei Themen moderner Quantencomputer befasst: der Quantenrauschunterdrückung und dem Design guter parametrischer Quantenschaltkreise. Wir haben eine Minderungsroutine zum Auslesen von Bit-Flip-Fehlern entwickelt, die Quantensimulationen drastisch verbessern kann. Wir haben auch eine dimensionale Aussagekraftanalyse entwickelt, die überflüssige Parameter in parametrischen Quantenschaltkreisen identifizieren kann. Darüber hinaus zeigen wir, wie man Expressivitätsanalysen mit Quantenhardware effizient umsetzen kann. Im Kontext des Tensornetzwerks haben wir ein Quantenbindungsmodell U(1) und 2+1-Dimensionen in einer Leitergeometrie mit DMRG untersucht. Unser Ziel ist es, die Eigenschaften des Grundzustands des Modells in einem endlichen chemischen Potential zu analysieren. Wir haben unterschiedliche Windungszahlsektoren beobachtet, als wir chemisches Potential in das System eingebracht haben. / Simulations of lattice gauge theories play a fundamental role in first principles calculations in the context of high energy physics. This thesis aims to improve state-of-the-art simulation methods for first-principle calculations and apply those methods to relevant physical models. We address this problem using three different approaches: machine learning, quantum computing, and tensor networks. In the context of machine learning, we have developed a method to estimate thermodynamic observables in lattice field theories. More precisely, we use deep generative models to estimate the absolute value of the free energy. We have demonstrated the applicability of our method by studying a toy model. Our approach produces more precise measurements in comparison with the standard Markov chain Monte Carlo method when we cross a phase transition point. In the context of quantum computing, our goal is to improve the current algorithms for quantum simulations. In this thesis, we have addressed two issues on modern quantum computers: the quantum noise mitigation and the design of good parametric quantum circuits. We have developed a mitigation routine ffor read-out bit-flip errors that can drastically improve quantum simulations. We have also developed a dimensional expressiveness analysis that can identify superfluous parameters in parametric quantum circuits. In addition, we show how to implement expressivity analysis using quantum hardware efficiently. In the context of the tensor network, we have studied a quantum bond model U(1) and 2+1 dimensions in a ladder geometry with DMRG. Our goal is to analyze the properties of the ground state of the model in a finite chemical potential. We have observed different winding number sectors when we have introduced chemical potential in the system.

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden. / This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

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