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Hard-core bosons in phase diagrams of 2D Lattice Gauge Theories and Bosonization of Dirac FermionsMantilla Serrano, Sebastian Felipe 27 February 2023 (has links)
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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Helicity of Quarks and Gluons at Small Bjorken xTawabutr, Yossathorn January 2022 (has links)
No description available.
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Quark Dynamics and Constituent Masses in Heavy Quark SystemsSouchlas, Nicholas 20 July 2009 (has links)
No description available.
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Topics In Effective Field Theories for the Strong InteractionThapaliya, Arbin 23 September 2016 (has links)
No description available.
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Hard scattering cross sections and parton distribution functions at the LHCKovačíková, Petra 19 August 2013 (has links)
Über einen Mellinraumzugang werden Methoden zur Auswertung von Wirkunsquerschnitten für verschiedene Prozesse mit Hadronen im Anfangszustand entwickelt. Die Arbeit geschieht im Hinblick auf drei Prozesse, für die die analyischen Ergebnisse für perturbative QCD Korrekturen zu “next-to-next-to-leading order” bekannt sind; diese sind: die Produktion der Vektorbosonen Z0 und W± über einen Drell-Yan-Prozess in der “narrow width”-Näherung, die Produktion eines Standardmodell-Higgs-Bosons über die Fusion zweier Gluonen im Grenzfall schwerer Top-Quark-Massen und die tiefinelastische Lepton-Hadron-Streuung über neutrale und geladene Ströme. Die Implementierung der Mellinraumtechniken erfolgt in dem c++ Paket sbp. Das Programm ermöglicht auf elegante Weise eine schnelle und präzise Auswertung von inklusiven Wirkungsquerschnitten. Wir vergleichen sbp mit den herkömmlichen Impulsraumtechniken, und präsentieren Studien der asymptotischen Konvergenz den perturbativen Reihen und von Skalenabhängigkeiten. Als Anwendung untersuchen wir welchen Einfluss die Behandlung der Faktorisierungs- und Renormierungsskala auf den Wirkungsquerschnitt hat. / In this thesis we will explore a Mellin space approach to the evaluation of precision cross-sections at hadron colliders. We consider three processes with known analytic results for perturbative QCD corrections up to the next-to-next-to-leading order, namely: the production of vector bosons Z0, W± via the Drell-Yan mechanism in the narrow width approximation; the production of the standard model Higgs boson via gluon-gluon fusion using the large top quark mass limit and the neutral and charged current deep inelastic lepton-hadron scattering. We develop a c++ package sbp that implements the Mellin space technique. The resulting program provides an elegant, fast and accurate solution for the evaluation of inclusive cross sections. We compare our program with available results that use standard momentum space techniques. We present studies of asymptotic convergence and scale dependence of the perturbative series. We use the package to study different treatments of factorisation and renormalisation scales in cross sections.
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Etude de la stabilité et de la précision des modèles utilisés dans la correction des effets de proximité optique en photolithographie / Study of the impact of different physical parameters during OPC model creation for 65nm and 45nm technologiesSaied, Mazen 30 September 2011 (has links)
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.
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Advanced modulation formats and nonlinear mitigation for spectral efficient optical transmission systems / Formats de modulation avancés et compensation de non linéarités pour les systèmes de transmission par fibre optique à haute efficacité spectraleFernandez de Jauregui Ruiz, Ivan 12 April 2018 (has links)
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
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Modelos multi-escala localmente perturbativos para o transporte de solutos iônicos em meios porosos argilosos / Locally perturbative multiscale methods for ionic solute transport in clayly soilsIgreja, Iury Higor Aguiar da 05 August 2010 (has links)
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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.
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Asymptotic Symmetries and Faddeev-Kulish states in QED and GravityGaharia, David January 2019 (has links)
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.
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Factorisations des processus exclusifs en chromodynamique quantique perturbativeSegond, Mathieu 07 December 2007 (has links) (PDF)
Le travail effectué dans cette thèse présente diverses études théoriques et phénoménologiques du processus de production exclusive de mésons vecteurs rho neutres polarisés longitudinalement dans les collisions entre photons virtuels, dans le cadre de la chromodynamique quantique (QCD). La virtualité des photons permet de situer notre approche dans le secteur perturbatif de la théorie. Les régimes cinématiques envisagés mènent à l'utilisation d'outils théoriques variés qui font apparaître différentes propriétés de factorisation de l'amplitude de diffusion: deux types de factorisation colinéaire (à courte distance) pour ce processus sont discutés au chapitre 1, faisant apparaître –suivant la polarisation des photons virtuels et le régime cinématique considéré- des Amplitudes de Distribution Généralisées (GDA) ou des Amplitudes de Distribution de Transition (TDA), outils communément utilisés dans la description des processus exclusifs. Nous introduisons dans le Chapitre 2 de manière autocohérente les bases du formalisme BFKL valide dans la limite à haute énergie (limite de Regge) de QCD, en vue de son utilisation phénoménologique détaillée dans le Chapitre 3: l'amplitude de diffusion du processus est décrite dans ce formalisme en exploitant la factorisation dans l'espace bidimensionnel des impulsions transverses, ou kT-factorisation. Nous prédisons la valeur de la section efficace du processus à l'ordre de Born de la resommation BFKL et nous discutons de son observation possible auprès du futur collisionneur linéaire international (ILC). Nous obtenons également les sections efficaces différentielles du processus sans transfert d'impulsion avec resommation BFKL complète à l'ordre des logarithmes dominants (Leading-Order) ainsi qu'à l'ordre suivant (Next-to-Leading-Order) afin d'établir un test fin du processus d'échange d'un Poméron dur décrit par le formalisme BFKL, observable au futur collisionneur linéaire ILC.
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