Dans cette thèse, nous construisons un modèle de liaisons fortes tridimensionnel minimal pour les cuprates supraconducteurs monocouches à base de lanthane. Celui-ci prend en compte huit orbitales, dont deux d'entre elles impliquent les ions oxygène apicaux. L'optimisation des paramètres microscopiques permet de reproduire presque parfaitement la bande de conduction tridimensionnelle telle qu'elle a été obtenue à partir des calculs DFT. Nous discutons la façon dont chacun des paramètres entrant en jeu dans ce modèle multi-bandes influence la bande de conduction, et nous montrons que la forme particulière de sa dispersion contraint les valeurs des paramètres. Nous mettons alors en évidence que la détermination standard d'un modèle effectif à une bande au travers d'un traitement perturbatif converge lentement en raison de la valeur relativement faible du gap de transfert de charges. A ce stade, cela nous permet, en revanche, de lever le voile sur l'origine microscopique des amplitudes de saut des électrons au sein des plans et en-dehors des plans. Une approche alternative au calcul des paramètres microscopiques de saut du modèle effectif de liaisons fortes est présentée et mise à contribution. Il en résulte que l'accord avec la DFT est préservé à condition que les amplitudes de saut de plus longue portée soient conservées. Une comparaison avec les modèles existants est également effectuée. La surface de Fermi, mettant en exergue des domaines décalés qui alternent en taille et en forme, est comparée à l'expérience. De plus, la densité d'états du modèle est aussi calculée. Une analyse plus approfondie du modèle est réalisée au travers d'une étude en couplage faible des instabilités magnétiques. Les calculs sont effectués sur de grandes cellules et nous avons trouvé une compétition parmi plusieurs instabilités magnétiques tridimensionnelles dans la région d’intérêt du dopage en trous accessible expérimentalement. Bien qu'à notre connaissance cela ne semble pas avoir été évoqué expérimentalement, nous montrons à l'issue de notre étude, que la tendance du modèle à former des ondes de densité de spin incommensurables tridimensionnelles est la plus forte à proximité du dopage 1/8. / In this thesis, we construct a minimal three-dimensional tight-binding model for single-layer La-based cuprate superconductors. It entails eight orbitals, two of them involving apical oxygen ions. Parameter optimization allows to almost perfectly reproduce the three-dimensional conduction band as obtained from DFT. We discuss how each parameter entering this multiband model influences it, and show that the peculiar form of its dispersion severely constraints the parameter values. We then evidence that standard perturbative derivation of an effective one-band model is poorly converging because of the comparatively small value of the charge transfer gap. Yet, this allows us to unravel the microscopical origin of the in-plane and out-of-plane hopping amplitudes. An alternative approach to the computation of the tight-binding parameters of the effective model is presented and worked out. It results that the agreement with DFT is preserved provided longer-ranged hopping amplitudes are retained. A comparison with existing models is performed, too. The Fermi surface, showing staggered pieces alternating in size and shape, is compared to experiment. The density of states is calculated as well. The model is further analyzed through a weak coupling study of magnetic instabilities. It is performed on large clusters and competition between several three-dimensional magnetic instabilities in the hole-doping region of experimental interest is found. We show that the tendency to form a three-dimensional incommensurate spin density wave is strongest in the vicinity of 1/8 doping.

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

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À l’heure actuelle, les modèles photochimiques utilisés dans la correction des effets de proximitéoptique (OPC) en photolithographie sont devenus complexes et moins physiques afin de permettre decapturer rapidement le maximum d’effets optiques et chimiques. La question de la stabilité de tels modèlespurement empiriques est devenue d’actualité. Dans ce mémoire, nous avons étudié la stabilité desmodèles photochimiques actuels en examinant les différentes causes d’instabilité vis-à-vis des paramètresdu procédé. Dans la suite, nous avons développé une méthode perturbative permettant d’évaluer le critèrede la stabilité. L’obtention de modèles simples et stables nous conduit à séparer les effets optiques desautres effets chimiques. De plus, les approximations utilisées dans la modélisation des systèmes optiquesopérant à grande ouverture numérique entraînent des erreurs résiduelles pouvant dégrader la précisionet la stabilité des modèles OPC. Ainsi, nous nous sommes intéressés à étudier les limites de validitéde l’approximation de Kirchhoff, méthode qui, jusqu’à présent, est la plus utilisée dans la modélisationdu champ proche d’un masque. D’autres méthodes semi-rigoureuses, permettant de modéliser les effetstopographiques, ont été également évaluées. Ces méthodes approchées permettent de gagner en précisionmais dégradent le temps de calcul. Nous avons ainsi proposé différentes façons de corriger les effetstopographiques du masque, tout en gardant l’approximation de Kirchhoff dans la modélisation de la partieoptique. Parmi les méthodes proposées, nous exploitons celle permettant de réduire les erreurs liéesaux effets topographiques du masque par l’intermédiaire d’un second modèle empirique. Nous montronsque pour garantir une précision adéquate, il est nécessaire d’augmenter la complexité du modèle en rajoutantdes termes additionnels. Enfin, pour garantir la stabilité numérique du modèle empirique, nousintroduirons une nouvelle méthode approchée hybride rapide et précise, la méthode des multi-niveaux,permettant d’inclure les effets topographiques par décomposition multi-niveaux du masque fin et discuteronsses avantages et ses limites. / At present, common resist models utilized in photolithography to correct for optical proximity effects(OPC) became complex and less physical in order to capture the maximum of optical and chemical effectsin shorter times. The question on the stability of such models, purely empirical, become topical. In thisthesis, we study the stability of existing OPC models by examining the origins of model instability towardsprocess parameters. Thus, we have developed a perturbative method in order to evaluate the stabilitycriterion. However, achieving stable and simple models needs a separation between optical and otherchemical effects. Besides, multiple approximations, widely utilized in the modeling of optical systemsoperating at high numerical aperture, lead to residual errors which can degrade OPC model accuracyand stability. Thus, we were interested to study the limits of validity of the Kirchhoff approximation,a method which, so far, is the most commonly used in mask near-field modeling. Other semi-rigorousmethods for mask topography effect modeling were also evaluated. These approximate methods canimprove the accuracy but degrades the run time. We then suggested different techniques to correct formask topography effects, while keeping the Kirchhoff approximation in the modeling of the optical part.Among them, we showed that errors due to mask topography effects can be partially captured by asecond empirical model. However, in order to ensure a good accuracy, it is necessary to increase themodel complexity by using more additional empirical terms. Finally, in order to achieve a numericalstability of the empirical model, we introduced a new hybrid fast and accurate method, the multi-levelmethod, which allows us to correct for mask topography effects through a multi-level decomposition ofthe thin mask and discussed its advantages and drawbacks.

La majeure partie des communications mondiales est transportée par des systèmes transocéaniques à fibre optique. Il est estimé que d'ici 2020 le trafic de données atteindra 4.3 ZB par an. Afin de faire face à cette demande, différentes technologies sont actuellement étudiées pour augmenter la capacité des systèmes de transmission très longue distance. Avec l'avènement des circuits intégrés à haute vitesse, des formats de modulation avancés et des techniques de traitement de signal numérique (DSP) peuvent être utilisés pour maximiser l'efficacité spectrale de transmission. Par ailleurs, la capacité des systèmes modernes est fortement limitée par les effets non-linéaires de type Kerr dans la fibre. Ainsi, la première partie de ce travail est axée sur l’étude de la performance et des gains réalisables par des techniques DSP à faible complexité pour mitiger les effets non-linéaires monocanal. En outre, l’utilisation des formats de modulation multiniveaux à haute efficacité spectrale au-delà de 16QAM a pris de l'ampleur pour augmenter le débit de transmission des systèmes, notamment avec l’introduction des formats QAM avec mise en forme probabiliste (PCS-QAM), plus performants que les formats QAM classiques. La deuxième partie de ce travail présente donc une comparaison théorique ainsi qu’expérimentale du format PCS-64QAM avec d’autres formats à haute efficacité spectrale pour les distances transatlantiques. La mise en œuvre d’un format PCS-64QAM conçu pour les distances transpacifiques est également abordée. Enfin, la dernière partie de ce travail concrétise les résultats des travaux menés dans les deux sections précédentes en présentant plusieurs records de transmission / Global data traffic is expected to reach up to 4.3 ZB per year by 2020. With the majority of the global communications being transported on submarine point-to-point fiber-optic systems, different cutting-edge technologies have been under research to cope with this unprecedented traffic growth. Continuous advances in high-speed integrated circuits have allowed the use of advanced modulation formats and digital signal processing (DSP) techniques to maximize the transmission spectral efficiency. With mitigation of fiber linear effects efficiently carried out by DSP with relative low-complexity, the capacity of modern fiber optic systems rests limited by fiber nonlinearities. To this extent, in the first part of this work, the performance and achievable benefits of low-complexity DSP techniques aiming to mitigate fiber Kerr nonlinear effects are investigated. Besides nonlinear compensation techniques, the use of multi-level modulation formats beyond 16QAM and high symbol rate channels have gained momentum to increase the system spectral efficiency. One of the major breakthroughs in the recent years, has been the introduction of QAM-based probabilistic constellation shaping (PCS-QAM), which has proven to outperform regular QAM formats. In this sense, in the second part of this work, the practical achievable rate increase brought by PCS-QAM for transoceanic distances is investigated. A theoretical and experimental comparison with other high-capacity formats is performed, and the design of a PCS-QAM for trans-Pacific distances is addressed. Finally, in the last section, several transmission records using the two above techniques are reported

In der Gitterquantenchromodynamik sind der Einsatz von unphysikalisch großen Quarkmassen und die Extrapolation von Ergebnissen zu physikalischen Massen signifikante systematische Fehlerquellen. In dieser Arbeit wird die praktische Durchführbarkeit numerischer Simulationen der Quantenchromodynamik mit physikalisch leichten up und down Quarkmassen unter Verwendung der Wilson twisted mass Diskretisierung untersucht. Simulationen im Regime physikalisch leichter Quarkmassen sind jedoch einerseits numerisch sehr aufwendig, können andererseits aber auch durch das Auftreten großer Diskretisierungsartefakte nicht praktikabel sein. Anhand von Simulationen mit massendegenerierten dynamischen up und down Quarks wird dargestellt dass die Erweiterung der twisted mass Fermionwirkung durch den Sheikholeslami-Wohlert Term es ermöglicht physikalisch leichte Quarkmassen zu erreichen. Es wird gezeigt, dass die Simulationen stabil sind und dass die Parameter der diskretisierten Theorie so gewählt werden können, dass das geladene Pion seine physikalische Masse annimmt. Ferner wird dargestellt, dass auch die Parameter für eine Simulation mit dynamischen massendegenerierten up und down quarks sowie nichtdegenerierten strange und charm Quarks schrittweise auf ihre physkalischen Werte gesetzt werden können. Um das Verhalten von Observablen bei physikalischer Quarkmasse zu untersuchen, werden Massen und Zerfallskonstanten von pseudoskalaren Mesonen mit up, down sowie strange und charm Valenzquarks berechnet. Die Ergebnisse stimmen größtenteils überein mit den phänomenologischen Werten, obwohl weder Kontinuumslimes noch die Extrapolation zu unendlichem Volumen durchgeführt werden. Renormierte leichte, strange und charm Quarkmassen werden über Interpolationen in hadronischen Observablen berechnet und stimmen ebenso größtenteils mit phänomenologischen Werten und anderen Ergebnissen aus der Gitter-QCD überein. / In computer simulations of Lattice Quantum Chromodynamics, the usage of unphysically large quark masses and the subsequent extrapolation of results to the physical value of the quark masses are major sources of systematic uncertainty. In this thesis, the feasibility and practicality of numerical simulations of Quantum Chromodynamics with physically light up and down quarks using the Wilson twisted mass quark discretisation are explored. Working in this regime is complicated firstly by the numerical expense of these simulations and secondly by the presence of potentially large lattice artefacts. The twisted mass discretisation is affected by an unphysical mass difference between the charged and neutral pions, rendering simulations at the physical charged pion mass infeasible if this mass splitting is too large. With the aim of reducing it, the Sheikholeslami-Wohlert term is added to the twisted mass fermion action and simulations with mass degenerate up and down quarks are then performed as a proof of concept. It is demonstrated that these simulations are stable and that the parameters of the lattice theory can be successfully tuned to correspond to the physical charged pion mass. Subsequently, the parameter tuning for simulations with mass degenerate up and down quarks as well as strange and charm quarks is explored and it is shown that it can be carried out in steps. As benchmark observables, the masses and decay constants of pseudoscalar mesons with light, strange and charm valence quarks are calculated and seen to largely reproduce their phenomenological values, even though continuum and infinite volume extrapolations are not performed. Light, strange and charm quark mass estimates are determined based on this data and also seen to coincide with phenomenological and other lattice determinations.

Made available in DSpace on 2015-03-04T18:51:21Z (GMT). No. of bitstreams: 1 Iury.pdf: 2129454 bytes, checksum: 6a7aff5ca085814119b9518b0aab9bef (MD5) Previous issue date: 2010-08-05 / Conselho Nacional de Desenvolvimento Cientifico e Tecnologico / This work aims at developing computational models capable of furnishing more realistic and less costly computationally for the problem of electrokinetic remediation of polluted clayey soils. Innovative results are obtained by improving the multiscale models previously developed by Lima and co-workers through the construction of perturbations of the local microscopic problems in conjuction with more realistic boundary conditions at the electrodes and with the development of precise estimates for the assymptotic behavior of the macroscopic solution. Considering the aliance of such techniques within the framework of the homogenization method of periodic structures we discretize the macroscopic model by the finite element method numerical simulations of an electroosmose experiment capable of predicting more realistic scenarios of electrokinetic remediation. / Este trabalho objetiva o desenvolvimento de modelos computacionais capazes de construir simulações numéricas mais realistas e menos custosas computacionalmente para o problema de descontaminação de solos argilosos por técnicas de eletrocinética. Resultados inovadores são obtidos aprimorando-se os modelos multi-escala desenvolvidos anteriormente por Lima e colaboradores via construção de soluções perturbativas dos problemas locais microscópicos aliada à condições de contorno mais realistas nos eletrodos e ao desenvolvimento de estimativas precisas para o comportamento assintótico da solução macroscópica. Por intermédio da conjunção destas técnicas imersas no contexto da teoria de homogeneização de estruturas periódicas discretizamos o modelo macroscópico pelo método dos elementos finitos e construimos simulações numéricas de um experimento de eletroosmose capazes de predizer cenários mais realistas em eletrorremediação de solos.

When calculating scattering amplitudes in gauge and gravitational theories one encounters infrared (IR) divergences associated with massless fields. These are known to be artifacts of constructing a quantum field theory starting with free fields, and the assumption that in the asymptotic limit (i.e. well before and after a scattering event) the incoming and outgoing states are non-interacting. In 1937, Bloch and Nordsieck provided a technical procedure eliminating the IR divergences in the cross-sections. However, this did not address the source of the problem: A detailed analysis reveals that, in quantum electrodynamics (QED) and in perturbative quantum gravity (PQG), the interactions cannot be ignored even in the asymptotic limit. This is due to the infinite range of the massless force-carrying bosons. By taking these asymptotic interactions into account, one can find a picture changing operator that transforms the free Fock states into asymptotically interacting Faddeev- Kulish (FK) states. These FK states are charged (massive) particles surrounded by a “cloud” of soft photons (gravitons) and will render all scattering processes infrared finite already at an S-matrix level. Recently it has been found that the FK states are closely related to asymptotic symmetries. In the case of QED the FK states are eigenstates of the large gauge transformations – U(1) transformations with a non-vanishing transformation parameter at infinity. For PQG the FK states are eigenstates of the Bondi-Metzner-Sachs (BMS) transformations – the asymptotic symmetry group of an asymptotically flat spacetime. It also appears that the FK states are related the Wilson lines in the Mandelstam quantization scheme. This would allow one to obtain the physical FK states through geometrical or symmetry arguments. We attempt to clarify this relation and present a derivation of the FK states in PQG from the gravitational Wilson line in the eikonal approximation, a result that is novel to this thesis.