Un ordinateur quantique (OQ), i.e. utilisant les ressources de la physique Q, superposition et intrication, pourrait fournir un gain exponentiel de temps de calcul. Une simulation utilisant ces ressources est appelée simulation Q (SQ). L’avantage des SQs sur les simulations classiques est bien établi au niveau théorique, i.e. software. Leur avantage pratique requiert un hardware Q. L’OQ, sous-entendu universel (cf. plus bas), n’a pas encore vu le jour, mais les efforts en ce sens sont croissants et variés. Aussi la SQ a-t-elle déjà été illustrée par de nombreuses expériences de principe, grâce à des calculateurs ou simulateurs Qs de taille réduite. Les marches Qs (MQs) sont des schémas de SQ particulièrement étudiés, étant des briques élémentaires pour concevoir n’importe quel algorithme Q, i.e. pour le calcul Q universel. La présente thèse est un pas de plus vers une simulation des théories Qs des champs basée sur les MQs à temps discret (MQTD). En effet, il est montré, dans certains cas, comment les MQTD peuvent simuler, au continu, l'action d'un champ de jauge Yang-Mills sur de la matière fermionique, et la rétroaction de cette-dernière sur la dynamique du champ de jauge. Les schémas proposés préservent l’invariance de jauge au niveau de la grille d’espace-temps, i.e. pas seulement au continu. Il est proposé (i) des équations de Maxwell sur grille, compatibles avec la conservation du courant sur la grille, et (ii) une courbure non-abélienne définie sur la grille. De plus, il est montré comment cette matière fermionique à base de MQTD peut être couplée à des champs gravitationnels relativistes du continu, i.e. des espaces-temps courbes, en dimension 1+2. / A quantum (Q) computer (QC), i.e. utilizing the resources of Q physics, superposition of states and entanglement, could fournish an exponential gain in computing time. A simulation using such resources is called a Q simulation (QS). The advantage of QSs over classical ones is well established at the theoretical, i.e. software level. Their practical benefit requires their implementation on a Q hardware. The QC, i.e. the universal one (see below), has not seen the light of day yet, but the efforts in this direction are both growing and diverse. Also, QS has already been illustrated by numerous experimental proofs of principle, thanks too small-size and specific-task Q computers or simulators. Q walks (QWs) are particularly-studied QS schemes, being elementary bricks to conceive any Q algorithm, i.e. to achieve so-called universal Q computation. The present thesis is a step more towards a simulation of Q field theories based on discrete-time QWs (DTQWs). Indeed, it is shown, in certain cases, how DTQWs can simulate, in the continuum, the action of Yang-Mills gauge fields on fermionic matter, and the retroaction of the latter on the gauge-field dynamics. The suggested schemes preserve gauge invariance on the spacetime lattice, i.e. not only in the continuum. In the (1+2)D Abelian case, consistent lattice equivalents to both Maxwell’s equations and the current conservation are suggested. In the (1+1)D non-Abelian case, a lattice version of the non-Abelian field strength is suggested. Moreover, it is shown how this fermionic matter based on DTQWs can be coupled to relativistic gravitational fields of the continuum, i.e. to curved spacetimes, in several spatial dimensions.

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

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.

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.