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This dissertation documents the results of two related efforts. Firstly, a model of nucleon-nucleon (NN) interactions is developed based on scalar field theory. Secondly, the relativistic 2-body Bethe-Salpeter equation (BSE) is generalized to handle inelastic processes in the ladder approximation. Scalar field theory describes the behavior of scalar particles, particles with spin 0. In the present work scalar field theory is used to describe NN interactions mediated by pion exchange. The scalar theory is applied to nucleons despite the fact that nucleons are fermions, spin 1/2 particles best described by fourcomponent Dirac spinor fields. Nevertheless, the scalar theory is shown to give a good fit to experiment for the total cross sections for several reactions [1]. The results are consistent with more elaborate spinor models involving one boson exchange (OBE). The results indicate that the spin and isospin of nucleons can to some extent be ignored under certain conditions. Being able to ignore spin and isospin greatly reduces the complexity of the model. A limitation of the scalar theory is that it does not distinguish between particle and anti-particle. Consequently one must decide how to interpret the s-channel diagrams generated by the theory, diagrams which involve particle creation and annihilation. The issue is resolved by extending the scalar theory to include electric charge, and formulating NN interactions in terms of complex scalar fields, which are able to describe both particles and anti-particles. A generalized Bethe-Salpeter equation (GBSE) is developed to handle inelastic processes in the ladder approximation. The GBSE, formulated using the scalar theory, is new, and introduces a systematic method for analyzing families of coupled reactions. A formalism is developed centered around the amplitude matrix M' defined for a given Lagrangian. M' gives the amplitudes of a family of reactions that arise from the Lagrangian. The formalism demonstrates how these amplitudes, to 2nd order, segregate into independent groups of coupled BSE's. The GBSE formalism is applied to the coupled BSE (CBSE) of Faassen and Tjon (FT) [2] for the reaction N+N->N+Delta, showing that the CBSE is missing a coupling channel, and in the expansion, under counts ladder diagrams. A proof is given of the equivalence of the series of ladder diagrams generated by M' and the S-matrix. A section on future work discusses several projects for further development and application of the GBSE.

Submitted by Cristian Gutierrez (cristian@ift.unesp.br) on 2016-11-25T17:35:07Z No. of bitstreams: 1 Cristian_Gutierrez_PhD_Thesis.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-11-30T13:24:29Z (GMT) No. of bitstreams: 1 gutierrezgomez_cl_dr_ift.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) / Made available in DSpace on 2016-11-30T13:24:29Z (GMT). No. of bitstreams: 1 gutierrezgomez_cl_dr_ift.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) Previous issue date: 2016-10-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O trabalho apresentado nessa tese foi dedicado em explorar soluções de estado ligado para a equação de Bethe-Salpeter, obtidas diretamente no espaço de Minkowski. Para isso, consideramos um procedimento que combina a representação integral de Nakanishi para a amplitude Bethe-Salpeter, desenvolvido por N. Nakanishi na década de sessenta, em conjunto com a projeção da amplitude de Bethe-Salpeter no plano nulo, também conhecida como a projeção na frente de luz. Este método, além de permitir calcular as energias de ligação, que são acessíveis a partir de cálculos bem conhecidos no espaço Euclidiano, permite que se obtenha a amplitude Bethe-Salpeter no espaço de Minkowski e a função de onda de valência na frente de luz. A verificação da validade desse procedimento foi confirmada através de comparação da amplitude de Bethe-Salpeter obtida diretamente no espaço Euclidiano com a amplitude correspondente derivada da equação de Bethe-Salpeter, usando a representação integral de Nakanishi, uma vez a rotação de Wick é realizada. O sucesso dessa abordagem, quando aplicado ao problema do estado ligado de duas partículas escalares trocando uma outra partícula escalar no estado fundamental, assim como o estudo correspondente no limite de energia zero, nos motivou a ampliar a aplicação do procedimento para o estudo de outros problemas de interesse. Em particular, o método foi estendido para o estudo de sistemas com duas dimensões espaciais e uma temporal (2+1), considerando o interesse crescente que surgiu em Física da matéria condensada, onde podemos destacar o caso de elétrons de Dirac no grafeno. Nessa análise preliminar, nos restringimos ao modelo escalar que nos permitiu acessar as principais dificuldades que deverão ser enfrentadas ao estudar o problema do estado ligado entre dois férmions. Dessa forma, este tratamento pode ser considerado como um primeiro passo para a implementação de um método mais realístico em um problema fermiônico. Os cálculos anteriores que consideramos em nossos estudos foram realizados através da aproximação de escada para o kernel de interação irredutível para os estados de onda-s. Portanto, uma das extensões que exploramos nesta tese foi o efeito de se introduzir a contribuição de ordem seguinte no kernel de interação, conhecida como a contribuição de escada-cruzada (cross-ladder). Os efeitos nas energias de ligação e na função de onda na frente de luz é foram analisados de forma detalhada, através dos resultados apresentados. Um estudo particularmente interessante, que foi extensivamente estudado nesta tese, se refere ao problema do espectro da equação Bethe-Salpeter para o estado ligado escalar-escalar. O espectro de estados excitados foi obtido com a abordagem da representação integral Nakanishi, sendo comparado com o obtido no espaço Euclidiano. Além disso, as raçoes excitado/fundamental do espectro relativístico foram reduzidas para às não-relativístico através da escolha de energias de ligação pequenas e considerando a massa do bóson trocado sendo próxima de zero. A função de onda de valência na frente de luz e a função de onda no parâmetro de impacto são apresentadas mostrando as principais características dos estados excitados conhecidos da estrutura não relativística. Na análise do espectro, também são estudadas as amplitudes de momentum-transverso para o estado fundamental e o primeiro estado excitado, que podem ser obtidos, de forma equivalente, no espaço de Minkowski assim como no espaço Euclidiano. Finalmente, focamos o estudo nos fatores de forma eletromagnéticos elásticos na abordagem da Bethe-Salpeter. Consciente de que o cálculo correto dos fatores de forma deve ser feito no espaço de Minkowski, o fator de forma elástico foi calculado levando-se em consideração a aproximação de impulso padrão. Além disso, foi também estudado o efeito da contribuição de ordem superior no fator de forma. / The work presented in this thesis was dedicated in exploring bound-state solutions of the Bethe-Salpeter equation directly in the Minkowski space. For that, we consider a method that combines the Nakanishi integral representation for the Bethe-Salpeter amplitude, developed by Noboru Nakanishi in the sixties, together with the projection of the Bethe-Salpeter amplitude onto the null-plane, also known as the light-front projection. This approach, besides of allowing to compute the binding energies, which are accessible from the usual Euclidean calculation, enables to obtain the Bethe-Salpeter amplitude in the Minkowski space and the light-front wave function. The feasibility of such an approach is further verified by comparing the Bethe-Salpeter amplitude obtained directly in the Euclidean space with the corresponding amplitude obtained by solving the Bethe-Salpeter equation, using the Nakanishi integral representation, once the Wick rotation is performed to this latter. The success of the approach when applied to study the bound state problem of two-scalar particles exchanging another scalar particle in the ground state, as well as the corresponding study at the zero-energy limit, has encouraged us to extend this method to another interesting problems. In particular, we start by extending the method to study problems in (2+1) dimensions due to the increasing interest in the condensed-matter physics, like the study of Dirac electrons in graphene. In this initial examination we restrict to the scalar model, which enables us to access to the main difficulties that we will face when studying the fermion-fermion bound state problem. Hence, this calculation can be considered as the first step towards the implementation of the method to real fermionic problems. The previous calculations have been performed by considering the ladder approximation for the irreducible interacting kernel for s-wave states. Therefore, one of the extensions that is explored in this thesis is the effect of introducing the next contribution in the interacting kernel, known as the scalar-scalar cross-ladder contribution. The effects in the eigenvalues and the light-front wave functions are analyzed in detail, by considering the computed results. A particular interesting subject, extensively studied in this thesis, is concerned to the spectrum of the Bethe-Salpeter equation for the scalar-scalar bound-state problem. The spectrum of excited states obtained with the Nakanishi integral representation approach is compared with that obtained in the Euclidean calculation. Besides, the ratio energies excited/ground of the relativistic spectrum is reduced to the non-relativistic one by choosing small binding energies and the mass of the exchanged boson approaching to zero. The valence light-front wave function and the impact-parameter space valence wave function are displayed, revealing the main features of excited states known from the non-relativistic framework. In the analysis of the spectrum, we also studied the transverse-momentum amplitudes for the ground and first-excited state, which can be equivalently obtained in the Minkowski or Euclidean spaces. Finally, we focus on the study of electromagnetic elastic form factors within the Bethe-Salpeter approach. Aware that the correct calculation of form factors should be performed in the Minkowski space, the calculation of the elastic form factor is carried out with the standard impulse approximation and in addition the effect of the next contribution to the form factor is studied.

Nesta tese obtemos, de um ponto de vista matemáticamente rigoroso, a parte inferior do espectro de energia-momento de dois modelos de teorias quânticas de campo com tempo imaginário em redes de dimensão $d+1$ (resultados explícitos para o caso $d = 3$ e matrizes de Dirac) que contém férmions: um modelo puramente fermiônico com interação quártica nos campos fermiônicos de $N$ componentes (modelo de Quatro-Férmions) e um modelo de cromodinâmica quântica. Para o modelo de Quatro-Férmions, $\\kappa$ é o parâmetro de hopping, $M_0$ é a massa bare dos férmions e $\\lambda$ é o parâmetro de interação. Uma expansão de polímeros garante a existência das funções de correlação no limite termodinâmico, na região onde $|\\frac{\\kappa}|$ é pequeno. A análise do espectro é baseada em representações espectrais para funções de correlação de dois e quatro férmions. A análise das funções de correlação adequadas é simplificada pelo uso de simetrias, em particular, de uma {\\em nova} simetria de Reflexão Temporal que aparece no nível das funções de correlação. A determinação do espectro é executada através de um estudo detalhado das taxas de decaimento das funções de correlação. Até próximo ao limiar de três partículas, o espectro de energia e momento exibe curvas de dispersão isoladas que são identificadas com partículas e estados ligados de duas partículas. No subespaço de uma partícula, o espectro consiste em uma curva de dispersão isolada. A massa da partícula é de ordem $-\\ln \\kappa$. O espectro de duas partículas aparece como soluções de uma equação de Bethe-Salpeter, resolvida primeiro em uma aproximação em escada. O espectro de duas partículas contém uma banda de duas partículas livres de largura finita. A existência de estados ligados acima ou abaixo da banda de duas partículas depende do fato do modelo apresentar ou não dominação gaussiana. Um parâmetro $\\aleph$ é dado para medir a dominação gaussiana. Para $\\aleph=0$, nenhum estado ligado ocorre. Para $\\aleph>0$, o estado ligado ocorre abaixo da banda de duas partículas. Para $\\aleph<0$, o estado ligado aparecem acima desta banda. Os resultados obtidos nesta aproximação em escada podem ser estendidos para o modelo completo através de um controle rigoroso das contribuições que diferenciam essas duas situações. Em uma segunda parte, idéias análogas são aplicadas para analisar o espectro do modelo de cromodinâmica quântica. Em particular, nós mostramos a existência dos pentaquarks no regime de acoplamento forte (acoplamento entre as plaquetas $0 <\\beta= \\frac{g^2_0} \\ll \\kappa $). O modelo possui simetria de calibre $SU(3)_c$ e de sabor $SU(2)_f$. Os pentaquark revelados são superposições de estados ligados de mésons e bárions. Apenas estados com um número ímpar de férmions e abaixo do limiar de energia meson-bárion são considerados. O pentaquark é determinado usando uma aproximação em escada para uma equação Bethe-Salpeter. Na ordem dominante em $\\beta$, a massa deste estado é aproximadamente $-5 \\ln\\kappa$ e sua energia de ligação é de ordem $\\textrm(\\kappa^2)$. O estado mais fortemente ligado tem isospin $I=\\frac$. Para $I=\\frac$ não há estados ligados. Estes resultados mostram uma dependência nos spins dos méson e bárion. Esta análise mostra que um potencial de troca de quark-anti-quark de $\\textrm(\\kappa^2)$ é a interação dominante, mas não há uma interpretação de troca de mésons. / In this thesis, we obtain, from a mathematically rigorous point of view, the low-lying energy-momentum spectrum of two $3+1$ dimensional imaginary time lattice quantum filed theory with fermion fields (we give explicit results for the case $d = 3$ and Dirac matrices): a pure fermionic model with quartic interaction in the $N$-component fermion field and a quantum chromodynamics model. For the Four-Fermion model, $\\kappa$ denotes the hopping parameter, $M_0$ the fermion bare mass and $\\lambda$ the interaction parameter. A polymer expansion show the existence of the model correlation functions in the thermodynamic limit, in the region where $|\\frac{\\kappa}|$ is small enough. The analysis of the spectrum is based on spectral representations of two- and four- point correlation functions. The analysis of such adequate correlation functions is simplified by the help of symmetries, in particular, by a {\\em new} Time Reflection symmetry, which appear in the level of correlation functions. The exact determination of the spectrum is done using a detailed study of the decay rates of the correlations. Up to near the 3 particle threshold, the energy-momentum spectrum exhibits isolated dispersion curves that are identified as particles and bound states. In the one-particle subspace, the spectrum consist in just a isolated dispersion curve. The mass of the associated particle is of order $-\\ln \\kappa$. The two-particle spectrum shows up as solutions of a Bethe-Salpeter equation, which is solved first in a ladder approximation. The two-particle spectrum contains a two free particles band of finite width. The existence of bound states above or below the band depends on wherever the model Gaussian domination holds. A parameter $\\aleph$ is given to measure the Gaussian domination. For $\\aleph=0$, no bound state occurs. For $\\aleph>0$, a bound state appears bellow the two-particles band. For $\\aleph<0$, the bound state appears above this band. The result obtained in this ladder approximation can be extended to the full model by a rigorous control of the contributions that differ these two cases. In a second part, analog ideas are applied to analyze the spectrum of a quantum chromodynamics model. In particular, we show the existence of pentaquarks in the strong coupling regime (plaquette coupling $0 <\\beta= \\frac{g^2_0} \\ll \\kappa $). The model has a $SU(3)_c$ gauge symmetry and a $SU(2)_f$ flavor symmetry. The reveled pentaquarks are superpositions of meson-baryon bound states. Only states with an odd number of fermions and bellow the meson-baryon threshold are considered. The pentaquark are determined using a ladder approximation to the Bethe-Salpeter equation. In the dominant order in $\\beta$, the bound state mass is $\\approx -5 \\ln\\kappa$ and the binding energy is of order $\\textrm(\\kappa^2)$. The most strongly bounded bound state has isospin $I=\\frac$. For $I=\\frac$, there is no bound state. These results shows a dependence in the spins of the meson and baryon. This analysis show that a $\\textrm(\\kappa^2)$ quark-anti-quark exchange potential is the dominant interaction, although there is not a meson exchange interpretation.

Cette thèse se propose d'explorer les mérites d'une famille d'approches de simulation quantique ab initio, les théories de perturbation à N-corps, pour l'exploration des propriétés électroniques et optiques de systèmes organiques. Nous avons étudié en particulier l'approximation dite de GW et l'équation de Bethe-Salpeter, très largement utilisées dès les années soixante pour les semiconducteurs de volume, mais dont l'utilisation pour les systèmes organiques moléculaires est très limitée. L'étude de quelques cas d'intérêt pour le photovoltaïque organique, et en particulier de petites molécules pour lesquelles sont disponibles des données expérimentales ou des résultats issus d'approches de chimie quantique, nous ont permis de valider ces approches issues de la physique du solide.Ce doctorat s'inscrit dans le cadre du développement d'un outil de simulation quantique spécifique (le projet FIESTA) dont l'objectif est de combiner les formalismes GW et Bethe-Salpeter avec les techniques de la chimie quantique, c'est-à-dire en particulier l'utilisation de bases localisées analytiques (bases gaussiennes) et des approches de type «résolution de l'identité» pour le traitement des intégrales Coulombiennes. Ce code est aujourd'hui massivement parallélisé, permettant, au delà des études de validation présentées dans ce travail de thèse, l'étude de systèmes complexes comprenant plusieurs centaines d'atomes. En cours de développement, l'incorporation d'approches hybrides combinant mécanique quantique et écrantage à longue portée par des approches modèles de milieu polarisable m'a permis d'une part de me familiariser avec le code et le développement méthodologique, et permet d'autre part d'envisager l'étude de systèmes réalistes en couplage avec leur environnement.Le manuscrit s‘ouvre sur une introduction au photovoltaïque organique afin de mettre en lumière les questionnements spécifiques qui requièrent le développement de nouveaux outils théoriques à la fois fiables en terme de précision et suffisamment efficaces pour traiter des systèmes de grande taille. Le premier chapitre est d'ordre méthodologique et rappelle les fondements des techniques ab initio de type champ-moyen (Hartree, Hartree-Fock et théorie de la fonctionnelle de la densité). En partant des principes de la photoémission, les théories de perturbation à N-corps et la notion de quasi-particule sont ensuite introduites, conduisant aux équations de Hedin et aux approximations GW et COHSEX. De même, à partir de la compréhension d'une expérience d'optique, le traitement des interactions électron-trou est présenté, menant à l'équation de Bethe-Salpeter. Le chapitre 2 introduit brièvement les spécificités techniques liées à l'implémentation des formalismes GW et Bethe-Salpeter. Les propriétés analytiques des bases gaussiennes et les principes mathématiques derrière les techniques de type «résolution de l'identité» et «déformation de contour», sont brièvement décrites. Le troisième chapitre présente les résultats scientifiques obtenus durant cette thèse. Le cas paradigmatique d'un polypeptide model nous permettra de discuter des spécificités de l'approche GW appliquée à des systèmes moléculaires afin d'obtenir des énergies de quasiparticule de bonne qualité. De même, l'utilisation de l'équation de Bethe-Salpeter pour l'obtention du spectre optique de ce système sera présentée, ainsi que le cas d'une famille de colorants d'importance pour les cellules de Graetzel (les coumarines). Finalement, nous explorons dans le cas du fullerène C60 et du graphène le calcul des termes de couplage électron-phonon dans le cadre de l'approche GW, c'est-à-dire au delà des approches standards de type théorie de la fonctionnelle de la densité. Notre étude vise à vérifier si une approximation statique et à écrantage constant au premier ordre permet de garder la qualité des résultats GW pour un coût numérique réduit. Après la conclusion, les appendices donnent le détail de certaines dérivations. / The present thesis aims at exploring the properties and merits of the ab initio Green's function many-body perturbation theory (MBPT) GW and Bethe-Salpeter formalisms, in order to provide a well-grounded and accurate description of the electronic and optical properties of condensed matter systems. While these approaches have been developed for extended inorganic semiconductors and extensively tested on this class of systems since the 60 s, the present work wants to assess their quality for gas phase organic molecules, where systematic studies still remain scarce. By means of small isolated study case molecules, we want to progress in the development of a theoretical framework, allowing an accurate description of complex organic systems of interest for organic photovoltaic devices. This represents the main motivation of this scientific project and we profit here from the wealth of experimental or high-level quantum chemistry reference data, which is available for these small, but paradigmatic study cases.This doctoral thesis came along with the development of a specific tool, the FIESTA package, which is a Gaussian basis implementation of the GW and Bethe-Salpeter formalisms applying resolution of the identity techniques with auxiliary bases and a contour deformation approach to dynamical correlations. Initially conceived as a serial GW code, with limited basis sets and functionalities, the code is now massively parallel and includes the Bethe-Salpeter formalism. The capacity to perform calculations on several hundreds of atoms to moderate costs clearly paves the way to enlarge our studies from simple model molecules to more realistic organic systems. An ongoing project related to the development of discrete polarizable models accounting for the molecular environment allowed me further to become more familiar with the actual implementation and code structure.The manuscript at hand is organized as follows. In an introductory chapter, we briefly present the basic mechanisms characterizing organic solar cells, accentuating the properties which seek for an accurate theoretical description in order to provide some insight into the factors determining solar cell efficiencies. The first chapter of the main part is methodological, including a discussion of the principle features and approximations behind standard mean-field techniques (Hartree, Hartree-Fock, density functional theory). Starting from a description of photoemission experiments, the MBPT and quasiparticle ideas are introduced, leading to the so-called Hedin's equations, the GW method and the COHSEX approach. In order to properly describe optical experiments, electron-hole interactions are included on top of the description of inter-electronic correlations. In this context, the Bethe-Salpeter formalism is introduced, along with an excursus on time-dependent density functional theory. Chapter 2 briefly presents the technical specifications of the GW and Bethe-Salpeter implementation in the FIESTA package. The properties of Gaussian basis sets, the ideas behind the resolution of the identity techniques and finally the contour deformation approach to dynamical correlations are discussed. The third chapter deals with the results obtained during this doctoral thesis. On the electronic structure level, a recent study on a paradigmatic dipeptide molecule will be presented. Further, also its optical properties will be explored, together with an in-depth discussion of charge-transfer excitations in a family of coumarin molecules. Finally, by means of the Buckminster fullerene C60 and the two-dimensional semi-metal graphene, we will analyze the reliability of two many-body formalisms, the so-called static COHSEX and constant-screening approximation, for an efficient calculation of electron-phonon interactions in organic systems at the MBPT level. After a short conclusion, the Appendix containing details and derivations of the formalisms presented before closes this work.

Nesta tese obtemos, de um ponto de vista matemáticamente rigoroso, a parte inferior do espectro de energia-momento de dois modelos de teorias quânticas de campo com tempo imaginário em redes de dimensão $d+1$ (resultados explícitos para o caso $d = 3$ e matrizes de Dirac) que contém férmions: um modelo puramente fermiônico com interação quártica nos campos fermiônicos de $N$ componentes (modelo de Quatro-Férmions) e um modelo de cromodinâmica quântica. Para o modelo de Quatro-Férmions, $\\kappa$ é o parâmetro de hopping, $M_0$ é a massa bare dos férmions e $\\lambda$ é o parâmetro de interação. Uma expansão de polímeros garante a existência das funções de correlação no limite termodinâmico, na região onde $|\\frac{\\kappa}|$ é pequeno. A análise do espectro é baseada em representações espectrais para funções de correlação de dois e quatro férmions. A análise das funções de correlação adequadas é simplificada pelo uso de simetrias, em particular, de uma {\\em nova} simetria de Reflexão Temporal que aparece no nível das funções de correlação. A determinação do espectro é executada através de um estudo detalhado das taxas de decaimento das funções de correlação. Até próximo ao limiar de três partículas, o espectro de energia e momento exibe curvas de dispersão isoladas que são identificadas com partículas e estados ligados de duas partículas. No subespaço de uma partícula, o espectro consiste em uma curva de dispersão isolada. A massa da partícula é de ordem $-\\ln \\kappa$. O espectro de duas partículas aparece como soluções de uma equação de Bethe-Salpeter, resolvida primeiro em uma aproximação em escada. O espectro de duas partículas contém uma banda de duas partículas livres de largura finita. A existência de estados ligados acima ou abaixo da banda de duas partículas depende do fato do modelo apresentar ou não dominação gaussiana. Um parâmetro $\\aleph$ é dado para medir a dominação gaussiana. Para $\\aleph=0$, nenhum estado ligado ocorre. Para $\\aleph>0$, o estado ligado ocorre abaixo da banda de duas partículas. Para $\\aleph<0$, o estado ligado aparecem acima desta banda. Os resultados obtidos nesta aproximação em escada podem ser estendidos para o modelo completo através de um controle rigoroso das contribuições que diferenciam essas duas situações. Em uma segunda parte, idéias análogas são aplicadas para analisar o espectro do modelo de cromodinâmica quântica. Em particular, nós mostramos a existência dos pentaquarks no regime de acoplamento forte (acoplamento entre as plaquetas $0 <\\beta= \\frac{g^2_0} \\ll \\kappa $). O modelo possui simetria de calibre $SU(3)_c$ e de sabor $SU(2)_f$. Os pentaquark revelados são superposições de estados ligados de mésons e bárions. Apenas estados com um número ímpar de férmions e abaixo do limiar de energia meson-bárion são considerados. O pentaquark é determinado usando uma aproximação em escada para uma equação Bethe-Salpeter. Na ordem dominante em $\\beta$, a massa deste estado é aproximadamente $-5 \\ln\\kappa$ e sua energia de ligação é de ordem $\\textrm(\\kappa^2)$. O estado mais fortemente ligado tem isospin $I=\\frac$. Para $I=\\frac$ não há estados ligados. Estes resultados mostram uma dependência nos spins dos méson e bárion. Esta análise mostra que um potencial de troca de quark-anti-quark de $\\textrm(\\kappa^2)$ é a interação dominante, mas não há uma interpretação de troca de mésons. / In this thesis, we obtain, from a mathematically rigorous point of view, the low-lying energy-momentum spectrum of two $3+1$ dimensional imaginary time lattice quantum filed theory with fermion fields (we give explicit results for the case $d = 3$ and Dirac matrices): a pure fermionic model with quartic interaction in the $N$-component fermion field and a quantum chromodynamics model. For the Four-Fermion model, $\\kappa$ denotes the hopping parameter, $M_0$ the fermion bare mass and $\\lambda$ the interaction parameter. A polymer expansion show the existence of the model correlation functions in the thermodynamic limit, in the region where $|\\frac{\\kappa}|$ is small enough. The analysis of the spectrum is based on spectral representations of two- and four- point correlation functions. The analysis of such adequate correlation functions is simplified by the help of symmetries, in particular, by a {\\em new} Time Reflection symmetry, which appear in the level of correlation functions. The exact determination of the spectrum is done using a detailed study of the decay rates of the correlations. Up to near the 3 particle threshold, the energy-momentum spectrum exhibits isolated dispersion curves that are identified as particles and bound states. In the one-particle subspace, the spectrum consist in just a isolated dispersion curve. The mass of the associated particle is of order $-\\ln \\kappa$. The two-particle spectrum shows up as solutions of a Bethe-Salpeter equation, which is solved first in a ladder approximation. The two-particle spectrum contains a two free particles band of finite width. The existence of bound states above or below the band depends on wherever the model Gaussian domination holds. A parameter $\\aleph$ is given to measure the Gaussian domination. For $\\aleph=0$, no bound state occurs. For $\\aleph>0$, a bound state appears bellow the two-particles band. For $\\aleph<0$, the bound state appears above this band. The result obtained in this ladder approximation can be extended to the full model by a rigorous control of the contributions that differ these two cases. In a second part, analog ideas are applied to analyze the spectrum of a quantum chromodynamics model. In particular, we show the existence of pentaquarks in the strong coupling regime (plaquette coupling $0 <\\beta= \\frac{g^2_0} \\ll \\kappa $). The model has a $SU(3)_c$ gauge symmetry and a $SU(2)_f$ flavor symmetry. The reveled pentaquarks are superpositions of meson-baryon bound states. Only states with an odd number of fermions and bellow the meson-baryon threshold are considered. The pentaquark are determined using a ladder approximation to the Bethe-Salpeter equation. In the dominant order in $\\beta$, the bound state mass is $\\approx -5 \\ln\\kappa$ and the binding energy is of order $\\textrm(\\kappa^2)$. The most strongly bounded bound state has isospin $I=\\frac$. For $I=\\frac$, there is no bound state. These results shows a dependence in the spins of the meson and baryon. This analysis show that a $\\textrm(\\kappa^2)$ quark-anti-quark exchange potential is the dominant interaction, although there is not a meson exchange interpretation.

Nesta tese usamos o conceito de propagador quântico quadridimensional representado nas coordenadas da frente de luz para definir a função de Green no tempo x+(= t+z). Calculamos a correção perturbativa à função de Green de dois corpos na frente de luz considerando os diagramas "escada"e "escada-cruzado". Construímos um conjunto hierárquico de equações acopladas para a função de Green de dois corpos na frente de luz, que é fisicamente equivalente à equação de Bethe-Salpeter quadridimensional. Aplicamos este formalismo para analisar sistemas de dois bósons e de dois férmions. Mostramos que a corrente eletromagnética de um sistema composto por dois bósons carregados, tem uma estrutura de muitos corpos mesmo na aproximação de impulso, quando descrita com o tempo x+. Em termos da componente de dois corpos do sistema ligado, a corrente contém operadores de dois corpos. Discutimos o processo de criação de par pelo fóton no referencial de Drell-Yan, e interpretamos isto como uma contribuição do modo zero para a corrente. Construímos um conjunto de equações hierárquicas acopladas para a função de Green de dois férmions no modelo de Yakawa na frente de luz, que é equivalente ao propagador de dois férmions na aproximação "escada". Demonstramos que a expansão sistemática do kernal da equação de Bethe-Salpeter na frente de luz elimina naturalmente as divergências nas integrais nos momentos transversos. A renormalização da equação de Bethe-Salpeter na frente de luz e na aproximação "escada" segue da expansão sistemática do seu kernel derivada das equações hierárquicas para as funções de Green.

In this thesis we describe the development of a real-space implementation of the Bethe-Salpeter equation (BSE) and use it in conjunction with a semi-empirical tight-binding model to investigate the optoelectronic properties of colloidal quantum- confined nanostructures. This novel implementation exploits the limited radial extent and small size of the atomic orbital basis to treat finite systems containing up to &Tilde;4000 atoms in a fully many-body framework. In the first part of this thesis our tight-binding model is initially benchmarked on zincblende CdSe nanocrystals, before subsequently being used to investigate the electronic states of zincblende CdSe nanoplatelets as a function of thickness. The band-edge electronic states are found to show minimal variation for a range of thicknesses and the results of our tight-binding model show good agreement with those predicted using a 14-band k·p model for a nanoplatelet of 4 monolayers (ML) in thickness. Optical absorption spectra were also computed in the independent-particle approximation. While the results of the tight-binding model show good agreement with those of the 14-band k·p model in the low-energy region of the spectrum, agreement with experiment was poor. This reflects the need for a many-body treatment of optical absorption in nanoplatelet systems. In the second part of this thesis we apply our tight-binding plus BSE model to study the excitonic properties of CdSe nanocrystals and nanoplatelets. Simulations performed on CdSe nanocrystals examined an approximation of the BSE equivalent to configuration interaction singles (CIS), and found that both the optical gap and the low-energy spectral features were unaffected by the approximation. A comparison of exciton binding energies with those predicted by CIS demonstrates the sensitivity of results to the exact treatment of dielectric screening and the decision of whether or not to screen exchange. Our model predicts optical gaps that are in strong agreement with average experimental data for all but the smallest diameters, but was not able to reproduce low-energy spectral features that were fully consistent with experiment. This was attributed to the absence of the spin-orbit interaction in the model. Simulations performed on CdSe nanoplatelets investigate the optical gaps and exciton binding energies as a function of thickness. Exciton binding energies were found to reach &Tilde;200 meV for the thinnest system, however, optical gaps were slightly overestimated in comparison to experiment. This is attributed to the reduced lateral dimensions used in our simulations and our bulk treatment of dielectric screening. A two-dimensional treatment of dielectric screening is expected to further increase binding energies. Calculations of the excitonic absorption spectrum reproduce the characteristic spectral features observed in experiment, and show strong agreement with the spectra of nanoplatelets, with thicknesses ranging from 3 ML to 5 ML.

La théorie de la fonctionnelle de la densité dépendante du temps (TDDFT) est aujourd'hui une méthode de référence pour le calcul des énergies d'excitation électroniques. Cependant, dans les approximations usuelles, elle n'est pas capable de décrire correctement les excitations de Rydberg, à transfert de charge ou présentant un caractère multiple. La séparation de portée de l'interaction électronique permet de combiner rigoureusement les méthodes fonctionnelles pour décrire la courte portée de l'interaction et les méthodes fonctions d'onde ou fonctions de Green pour la longue portée. Dans cette thèse, les effets de cette séparation de portée sur les énergies d'un système en interaction partielle sont d'abord étudiés le long de la connection adiabatique dans le cas indépendant du temps afin d'aider le développement des méthodes à séparation de portée pour les énergies d'excitation. La séparation de portée est ensuite appliquée dans le cadre de la TDDFT aux noyaux d'échange et de corrélation, où dans le cas d'une approximation monodéterminentale, la longue portée du noyau de corrélation est absente. Afin de prendre en compte l'effet des doubles excitations, un noyau de corrélation de longue portée dépendant de la fréquence est développé en s'inspirant du noyau Bethe-Salpeter. Ce noyau est alors ajouté de façon perturbative au noyau TDDFT à séparation de portée afin de prendre en compte les effets des excitations doubles. / Linear-response time-dependent density-functional theory (TDDFT) is nowadays a method of choice to compute molecular excitation energies. However, within the usual adiabatic semi-local approximations, it is not able to describe properly Rydberg, charge-transfer or multiple excitations. Range separation of the electronic interaction allows one to mix rigorously density-functional methods at short range and wave function or Green’s function methods at long range. When applied to the exchange functional, it already corrects most of these deficiencies but multiple excitations remain absent as they need a frequency-dependent kernel. In this thesis, the effects of range separation are first assessed on the excitation energies of a partially-interacting system in an analytic and numerical study in order to provide guidelines for future developments of range-separated methods for excitation energy calculations. It is then applied on the exchange and correlation TDDFT kernels in a single-determinant approximation in which the long-range part of the correlation kernel vanishes. A long-range frequency-dependent second-order correlation kernel is then derived from the Bethe-Salpeter equation and added perturbatively to the range-separated TDDFT kernel in order to take into account the effects of double excitations.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.