A general exact matrix renormalization group method is developed for solving quantum transmission through networks. Using this method transmission of spinless electrons is calculated for a Hanoi network and a (newly introduced) fully connected Bethe lattice. Plots of the transmission and wavefunctions are obtained through application of the derived Renormalization Group recursion relations. The plots reveal band gaps (which has possible application in nano devices) in HN3 networks while no band gaps are observed in HN5 networks. With the fully connected Bethe lattice a drastic reduction in the transmission (in comparison to the normal Bethe lattice) is observed. This reduction can be found to be a purely quantum mechanical effect.

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Ein präzises Verständnis über die Entstehung der Elemente im Universum stellt ein hoch- relevantes Kernthema der nuklearen Astrophysik dar. Vor diesem Hintergrund wurde die 12C(p,γ)13N-Reaktion untersucht, die als Startreaktion des CNO-Zyklus Einfluss auf das Verhältnis von 12C zu 13C im Universum nimmt. Die analysierten Messdaten wurden in in- verser Kinematik am Tandetron-Beschleuniger des Helmholtz-Zentrum Dresden-Rossendorf aufgenommen. Der resultierende S-Faktor, vermessen im Bereich der 421keV-Resonanz, liegt im Mittel 23% unterhalb etablierter Literaturdaten, deckt sich aber mit den Ergeb- nissen anderer kürzlich veröffentlichter Messdaten. Die in dieser Analyse ebenfalls erschwerte präzise Untersuchung niedriger, aber astrophy- sikalisch relevanter Energien kann durch Untertagelabore und der damit einhergehenden Abschirmung vor kosmischer Strahlung erreicht werden. In der vorliegenden Arbeit werden in diesem Bestreben erste mit dem 5 MV-Tandem-Beschleuniger untersuchte Kernreaktio- nen am Felsenkeller-Untertagelabor in Dresden vorgestellt. Aus Untersuchungen der 14N(α,γ)18F-, der 13C(p,γ)14N- und der 27Al(p,γ)28Si-Reaktion wurden dabei präzise Werte für die Energiekalibration des Ionenbeschleunigers ermittelt. Es wird ein Vergleich mit weiteren Möglichkeiten zur Bestimmung dieses Kalibrationsfaktors präsentiert und aus diesem Vergleich ein Wert von k = 0,9572 ± 0,0004 zur Kalibration der Hochspannung des Beschleunigers abgeleitet. Die vorgestellte Herangehensweise zur Bestimmung dieses Faktors und die dokumentier- ten Erkenntnisse und Analysen zu optimalen Betriebsparametern von Beschleuniger und der zugehörigen Radiofrequenz-Ionenquelle werden auch für zukünftige protonen- und he- liumstrahlinduzierte Untersuchungen im Felsenkeller-Untertagelabor von Relevanz sein.

The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.

O objetivo deste trabalho foi o estudo do Ansatz de Bethe Algébrico (ABA), que é uma técnica utilizada na obtenção dos auto-estados do hamiltoniano de inúmeros modelos da Mecânica Estatística e da Teoria Quântica de Campos. Aplicamos este procedimento na diagonalização de três modelos de spins: o modelo de Heisenberg, o modelo de Heisenberg-Sklyanin e o modelo de Heisenberg-Cherednik. Na diagonalização do primeiro modelo, não foi possível encontrar todos os auto-estados do hamiltoniano através do ABA e, durante o procedimento de obtenção das expressões analíticas, nos deparamos com um conjunto de identidades inédito na literatura. A matriz de borda do modelo de Heisenberg-Sklyanin acopla o último e o primeiro sítios, generalizando o modelo anterior, e permite estabelecer uma relação limite com outros modelos integráveis. Neste caso também não conseguimos obter todos os auto-estados utilizando a técnica do ABA. Diferentemente do que ocorreu para os primeiros modelos, o de Heisenberg-Cherednik, com acoplamentos que alternam a intensidade ao longo da cadeia de spin, apresentou um conjunto completo de auto-estados quando diagonalizado pelo ABA. / The goal of this work was to study the Algebraic Bethe ansatz (ABA), which is a technique used to obtain the eigenstates of Hamiltonian of many models of Statistical Mechanics and Quantum Field Theory. We apply this procedure to diagonalize three types of spin models: the Heisenberg model, the Heisenberg-Sklyanin model and the Heisenberg-Cherednik model. On diagonalization of the …rst model, we could not …nd all the eigenstates of Hamiltonian through ABA, and during the procedure for obtaining the analytical expressions, we face an unprecedented set of identities in literature. The Sklyanin´s boundary matrix couples the fi…rst and last sites, generalizing the previous model, and provides a limit for other integrable models. In this case also did not get all eigenstates using the technique of ABA. Unlike what happened with the …rst models, the Heisenberg-Cherednik model, with alternating couplings the intensity along the spin chain, presented a complete set of eigenstates when diagonalized by ABA.

Nesta tese estudamos as propriedades críticas dos modelos anisotrópicos (isotrópicos) de Heisenberg com spin s arbitrário. O espectro das Hamiltonianas, com condições periódicas de contorno, foi calculado para redes finitas, resolvendo-se as equações do Bethe ansatz associadas. Nossos resultados indicam que a anomalia conforme destes modelos tem o valor c=3s/(1+s), independente da anisotropia, e os expoentes críticos variam continuamente com a anisotropia assim como no modelo de 8-vértices. O conteúdo de operadores destes modelos indica que a teoria de campos que governa a criticalidade destes modelos de spin é descrita por operadores formados pelo produto de um operador Gaussiano por outro com simetria Z(2s). Estudando estes modelos, com certas condições especiais de contorno, mostramos que eles são relacionados com uma nova classe de teorias unitárias recentemente propostas / This thesis is concerned with the critical properties of anisotropic (isotropic) Heisenberg chain,with arbitrary spin-s. The eigenspectrum of these Hamiltoniana, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The results indicate that the conformal anomaly hás the value c=3s/1+s, independently of the anisotropy, and the exponentes vary continuously with the anisotropy like in the 8-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-s models are described by composite fields formed by the product of Gaussian and Z(2s) fields. Studying these models, with some special boundary conditions, we show that they are related with a large class of unitary conformal field theories recntly introduced

Nesta dissertação, mostramos as primeiras aplicações do recém criado Anstz do Produto Matricial [8] na solução exata das matrizes de transferência associadas a modelos de vértices. A integrabilidade dos modelos é obtida diagonalizando-se a matriz de transferência diagonal-para-diagonal. Foram estudados duas classes de modelos. Na primeira delas introduzimos novos modelos de vértices, que denominamos de modelos de 5 vértices interagentes. Nestes modelos os vértices além das interações usuais de vizinhos próximos, dadas pela regra do gelo, possuem também interações de natureza repulsiva ao longo da diagonal. O famoso modelo de 6 vértices é obtido num limite particular deste novo modelo. O espectro da matriz de transferência, analogamente ao que acontece no ansatz de Bethe tradicional é dado em termos de solução de equações não lineares. Um estudo analítico e numérico destas equações foi feito para o modelo de 6 vértices que está contido nesta primeira classes de modelos. Tais resultados, juntamente com as idéias de invariância conforme, nos permitiram estudar o modelo em seu regime crítico. A segunda classe de modelos que estudamos foram os modelos de 10 vértices que satisfazem às regras do gelo. Obtivemos todos os possíveis modelos exatamente integráveis desta classe, reobtendo resultados da literatura bem como novos resultados. / In this dissertation we present the first application of a recent introduces Matrix Product Ansatz [8], in the exact solution of the transfer matrices associated to vertex models. The exact integrability is obtained through the diagonalization of the diagonal-to-diagonal transfer matrix. We studied two classes of models. In the first one we introduced new vertex models, that we call as interacting 5 vertex models. On these models beyond the nearest-neighbor interactions among the vertices, imposed by the ice rule, they also have repulsive interactions along the diagonal. The famous 6-vertex model is just a special case this class of models. The eigenspectrum of this transfer matrix, analogously as in the traditional Bethe ansatz, is obtained in terms of the roots of nonlinear equation. An analytical and numerical study of these equations we done on the first class. These results together with the machinery coming from conformal invariance allow us the study the model on its critical region. The second class of models we considered were the 10 vertex models that satisfy ice rules we obtained all the possible exact integrable models on this class, rederiving earlier results on the literature as were producing new ones.

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.

Made available in DSpace on 2016-06-02T20:15:31Z (GMT). No. of bitstreams: 1 5872.pdf: 509530 bytes, checksum: 496d8919c2fd01cb49ba79b80563de69 (MD5) Previous issue date: 2014-05-26 / Universidade Federal de Minas Gerais / In this work we study vertex models with non-diagonal boundaries, characterized by reection matrices with an upper triangular form. By means of an extension of the algebraic Bethe ansatz, we construct generalized Bethe states as well as the respective eigenvalues for two classes of models: six and nineteen vertex models. As usual, in both cases the exact solution is given in terms of the Bethe equations. / Neste trabalho estudamos modelos de vértices com fronteiras não-diagonais, caracterizadas por matrizes de reflexão com estrutura triangular. Por meio de uma extensão do ansatz de Bethe algébrico usual, construímos estados de Bethe generalizados e os respectivos autovalores para duas classes de modelos: seis e dezenove vértices. Como usual, em ambos os casos a solução exata é dada em termos das equações de Bethe.

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.