The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.

Universiẗat, Diss., 2005--Münster (Westfalen).

Dissertation (Ph.D.) -- Worcester Polytechnic Institute. / Keywords: ladder approximation; inelastic process; Bethe-Salpeter; BSE; nucleon; scalar field; pion. Includes bibliographical references (p.161-163).

Neste trabalho nós desenvolvemos uma abordagem baseada no método do ansatz de Bethe e no princípio variacional para encontrar a energia do estado fundamental para sistemas unidimensionais formados por um número pequeno de particulas interagentes. Particularmente, nós investigamos sistemas de duas e três partículas interagentes aprisionados em uma armadilha harmônica unidimensional. Nossos resultados apresentam uma boa concordância com as soluções analíticas e numéricas existentes na literatura. Também determinamos a densidade de probabilidade e a função de correlação de pares do sistema. Nossa abordagem é bastante genérica e permite o estudo de sistemas de poucas partículas mais complexos, alguns de interesse experimental, que não apresentam solução analítica. / In this work we develop an approach based on the Bethe ansatz method and the variational principle to nd the ground state energy for a one-dimensional few-body system. We investigate a system of two and three interacting particles con ned in a one-dimensional harmonic trap. Our results show a good agreement with existing analytical and numerical results. We also determine the probability density and the pair correlation function of the system. Our approach is very general and enables the study of more complex few-body systems, some of them of experimental interest, where no exact analytical solution is available.

Made available in DSpace on 2016-06-02T20:15:17Z (GMT). No. of bitstreams: 1 1813.pdf: 547388 bytes, checksum: 3c5fafe377da474a62c0ea847b158e20 (MD5) Previous issue date: 2002-03-21 / Financiadora de Estudos e Projetos / The procedure for obtaining integrable open spin chain Hamiltonians via reflection matrices explicitly carried out for some three-state vertex models. We have considered the 19-vertex models of Zamolodchikov-Fateev and Izergin-Korepin and the Z2 graded 19-vertex models with sl(2/1) and osp(1/2) invariances. In each case the eigenspectrum is determined by application of the coordinate Bethe ansatz. / O procedimento para resolução de cadeias quânticas integráveis de spin 1 com termos de superfícies diagonais para os modelos de vértices de três estados é apresentado. Consideramos os modelos de 19-vértices Zamolodchikov-Fateev e Izergin- Korepin e os modelos de 19-vértices com graduação Z2 sl(2/1) e osp(1/2) . Em cada caso os autovalores de energia são determinados pela aplicação do ansatz de Bethe de coordenadas.

Neste trabalho nós desenvolvemos uma abordagem baseada no método do ansatz de Bethe e no princípio variacional para encontrar a energia do estado fundamental para sistemas unidimensionais formados por um número pequeno de particulas interagentes. Particularmente, nós investigamos sistemas de duas e três partículas interagentes aprisionados em uma armadilha harmônica unidimensional. Nossos resultados apresentam uma boa concordância com as soluções analíticas e numéricas existentes na literatura. Também determinamos a densidade de probabilidade e a função de correlação de pares do sistema. Nossa abordagem é bastante genérica e permite o estudo de sistemas de poucas partículas mais complexos, alguns de interesse experimental, que não apresentam solução analítica. / In this work we develop an approach based on the Bethe ansatz method and the variational principle to nd the ground state energy for a one-dimensional few-body system. We investigate a system of two and three interacting particles con ned in a one-dimensional harmonic trap. Our results show a good agreement with existing analytical and numerical results. We also determine the probability density and the pair correlation function of the system. Our approach is very general and enables the study of more complex few-body systems, some of them of experimental interest, where no exact analytical solution is available.

Neste trabalho nós desenvolvemos uma abordagem baseada no método do ansatz de Bethe e no princípio variacional para encontrar a energia do estado fundamental para sistemas unidimensionais formados por um número pequeno de particulas interagentes. Particularmente, nós investigamos sistemas de duas e três partículas interagentes aprisionados em uma armadilha harmônica unidimensional. Nossos resultados apresentam uma boa concordância com as soluções analíticas e numéricas existentes na literatura. Também determinamos a densidade de probabilidade e a função de correlação de pares do sistema. Nossa abordagem é bastante genérica e permite o estudo de sistemas de poucas partículas mais complexos, alguns de interesse experimental, que não apresentam solução analítica. / In this work we develop an approach based on the Bethe ansatz method and the variational principle to nd the ground state energy for a one-dimensional few-body system. We investigate a system of two and three interacting particles con ned in a one-dimensional harmonic trap. Our results show a good agreement with existing analytical and numerical results. We also determine the probability density and the pair correlation function of the system. Our approach is very general and enables the study of more complex few-body systems, some of them of experimental interest, where no exact analytical solution is available.

No description available.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Indiana University-Purdue University Indianapolis (IUPUI) / We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.