Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Critical phenomena study was for many years dominated by analysis of transitions generated by thermal fluctuations. This thermal fluctuations cease at T-0, however, quantum fluctuations does not end at zero temperature. These quantum fluctuations may, under certain conditions, trigger phase transitions. In this work the Hubbard model is used to study quantum state and quantum phase transitions in strongly correlated electron systems, considering the terms of intersite hopping and Coulomb repulsion intrasite. It was developed an algorithm based on Lanczos method to solving the Hubbard model applied in different types of lattices. Analysis of algorithms efficiency were made an was observed that the standard approaches to evaluate the properties of the ground state in the Hubbard model by Lanczos method presents convergence problems when there is a significant difference between hopping parameters and Coulomb interaction. This difference is very important since the energy convergence does not necessarily reflect in a convergence of the ground state. In this work are discussed several algorithms as standard Lanczos method, the Explicit Restarted Lanczos algorithm and the Modified Explicit Restarted Lanczos algorithm. A protocol based on these algorithms using the operator S2 as s stopping criterion was developed, since through this the operator it is possible to assess the error getting from the ground state itself. The algorithm based on the ERL provides better accuracy and it is 5 times faster compared with conventional ones. The MERL-based algorithm keeps the error at the last significant digit, and its processing time is about 2.5 times longer than the ERL-based algorithm, although it is still faster than the standard Lanczos method. These analyzes pave the way for a reliable and practical evaluation of the ground-state properties not only of the Hubbard model, but also for other manybodies quantum systems. The systems analyzed were clusters of polymeric lattice AB2 tipe, one-dimensional lattice considering nears and next nears neighbors hoppings and cluster of fcc lattice. All systems showed quantum state transitions. Through the study of the spin-spin correlations of the AB2 lattices clusters it was possible to analyze in detail the behavior of these spin-spin correlation functions between sublattices of a finite system. The analysis of one-dimensional lattice with next near neighbor made it possible to use an extrapolation method, which has determined that the quantum phase transition critical point, Uc/t = 4.7, from which the system changes from a paramagnetic behavior to a ferromagnetic behavior. In the fcc lattice clusters were examined the ground state energy as a function of the particle density showed a minimum value for all the structural sizes studied. The minimum energy decreases with increasing the interaction parameter U. It was observed that the ground state energy has a minimum at n = 0.6 for U/t = W, where W denotes the non-interacting bandwidth and the face-centered cubic structure is ferromagnetic. These results, when compared to the nickel properties, shown great similarity analysis in literature, made at finite temperature and support the results of Hirsh, which proposes that the interatomic interaction exchange is dominant to driving the system to a ferromagnetic phase. / O estudo dos fenômenos críticos foi, por muitos anos, dominado pela análise das transições geradas por flutuações térmicas. As flutuações térmicas cessam em T-0, porém flutuações quânticas não acabam na temperatura zero. Essas flutuações de caráter quântico podem, sob certas condições, desencadear transições de fase. Neste trabalho o modelo de Hubbard é utilizado para o estudo de transições de estado quântico e de fase quântica em sistemas de elétrons fortemente correlacionados, considerando os termos de hopping intersítios e de repulsão coulombiana intrasítio. Foi desenvolvido um algoritmo com base no método de Lanczos para resolver o modelo de Hubbard aplicado a diferentes tipos de rede. Foram feitas análises da eficiência de algoritmos, nelas foi possível observar que as abordagens padrão para avaliar as propriedades do estado fundamental do modelo de Hubbard através do método de Lanczos apresentam problemas de convergência quando há uma significante diferença entre os parametros de hopping e de interação coulombiana. Esta diferença é muito relevante uma vez que a convergência da energia não reflete necessariamente em uma convergência do estado fundamental. Neste trabalho são discutidos vários algoritmos como o método de Lanczos padrão, o algoritmo Explicit Restarted Lanczos e o algoritmo Modified Explicit Restarted Lanczos. Foi desenvolvido um protocolo baseado nesses algoritmos que utiliza o valor de S2 como critério de parada do método, uma vez que através dessa grandeza é possível avaliar o erro na obtenção do estado fundamental. O algoritmo baseado no ERL proporciona uma melhor precisão é 5 vezes mais rápido quando comparado com o convencional. O algoritmo baseado no MERL mantém o erro no último dígito significativo e seu tempo de processamento é cerca de 2.5 vezes mais longo do que o algoritmo baseado no ERL, embora ainda seja mais rápido do que o método Lanczos padrão. Essas análises abrem caminho para uma avaliação confiável e prática das propriedades do estado fundamental, não só do modelo de Hubbard, mas também para muitos outros sistemas quânticos de muitos corpos. Os sistemas analisados foram clusters de rede polimérica tipo AB2, de rede unidimensional considerando hoppings tanto de primeiros quanto de segundos vizinhos e clusters de rede fcc. Todos os sistemas apresentaram transições de estado quântico. Através do estudo das correlações spin-spin do cluster da rede AB2 foi possível analisar detalhadamente o comportamento das referidas funções de correlação spin-spin entre sub-redes de um sistema finito. A análise da rede unidimensional com hopping entre segundos vizinhos possibilitou utilizar um método de extrapolação, o qual determinou que o ponto crítico de transição de fase quântica, Uc/t = 4.7, a partir do qual o sistema passa de um comportamento paramagnético para um comportamento ferromagnético. Nos clusters de rede fcc foram examinadas as energias do estado fundamental em função da densidade de partícula, observando-se a existência de um valor de mínimo de energia para todas os tamanhos estruturais estudados. Os mínimos de energia diminuem com o aumento do parâmetro de interação U. Foi observado que a energia do estado fundamental tem um mínimo em a densidade eletrônica igual a 0.6 para U/t=W, em que W denota a largura de banda não-interagente e a estrutura cúbica de face centrada mostrou-se ferromagnético. Esses resultados, quando comparados com as propriedades do níquel, mostam grande semelhança com análises na literatura feitas sob temperatura finita e suportam os resultados de Hirsh, o qual propõe que a interação interatômica de exchange é dominante na condução do sistema à uma fase ferromagnética.

Bound state systems consisting of three nonrelativistic particles are numerically studied. Calculations are performed employing two-body and three-body forces as input in the Hamiltonian in order to study the role or contribution of three-body forces to the binding in these systems. The resulting differential Faddeev equations are solved as three-dimensional equations in the two Jacobi coordinates and the angle between them, as opposed to the usual partial wave expansion approach. By expanding the wave function as a sum of the products of spline functions in each of the three coordinates, and using the orthogonal collocation procedure, the equations are transformed into an eigenvalue problem. The matrices in the aforementioned eigenvalue equations are generally of large order. In order to solve these matrix equations with modest and optimal computer memory and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction with the so-called tensor trick method. Furthermore, we incorporate a polynomial accelerator in the algorithm to obtain rapid convergence. We applied the method to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)

Bound state systems consisting of three nonrelativistic particles are numerically studied. Calculations are performed employing two-body and three-body forces as input in the Hamiltonian in order to study the role or contribution of three-body forces to the binding in these systems. The resulting differential Faddeev equations are solved as three-dimensional equations in the two Jacobi coordinates and the angle between them, as opposed to the usual partial wave expansion approach. By expanding the wave function as a sum of the products of spline functions in each of the three coordinates, and using the orthogonal collocation procedure, the equations are transformed into an eigenvalue problem. The matrices in the aforementioned eigenvalue equations are generally of large order. In order to solve these matrix equations with modest and optimal computer memory and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction with the so-called tensor trick method. Furthermore, we incorporate a polynomial accelerator in the algorithm to obtain rapid convergence. We applied the method to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)

In this work we calculate the binding energies and root-mean-square radii for A−body nuclear bound state systems, where A ≥ 3. To study three−body systems, we employ the three−dimensional differential Faddeev equations with nucleon-nucleon semi-realistic potentials. The equations are solved numerically. For this purpose, the equations are transformed into an eigenvalue equation via the orthogonal collocation procedure using triquintic Hermite splines. The resulting eigenvalue equation is solved using the Restarted Arnoldi Algorithm. Ground state binding energies of the 3H nucleus are determined. For A > 3, the Potential Harmonic Expansion Method is employed. Using this method, the Schr¨odinger equation is transformed into coupled Faddeev-like equations. The Faddeevlike amplitudes are expanded on the potential harmonic basis. To transform the resulting coupled differential equations into an eigenvalue equation, we employ again the orthogonal collocation procedure followed by the Gauss-Jacobi quadrature. The corresponding eigenvalue equation is solved using the Renormalized Numerov Method to obtain ground state binding energies and root-mean-square radii of closed shell nuclei 4He, 8Be, 12C, 16O and 40Ca. / Physics / M. Sc. (Physics)

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A..=2b, where A 2 Rnn is a large, sparse symmetric positive definite matrix and b 2 Rn is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LLT is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L..T z, with x = A..1=2z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form n = A..=2b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t..=2 and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A..=2b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

In this work we calculate the binding energies and root-mean-square radii for A−body nuclear bound state systems, where A ≥ 3. To study three−body systems, we employ the three−dimensional differential Faddeev equations with nucleon-nucleon semi-realistic potentials. The equations are solved numerically. For this purpose, the equations are transformed into an eigenvalue equation via the orthogonal collocation procedure using triquintic Hermite splines. The resulting eigenvalue equation is solved using the Restarted Arnoldi Algorithm. Ground state binding energies of the 3H nucleus are determined. For A > 3, the Potential Harmonic Expansion Method is employed. Using this method, the Schr¨odinger equation is transformed into coupled Faddeev-like equations. The Faddeevlike amplitudes are expanded on the potential harmonic basis. To transform the resulting coupled differential equations into an eigenvalue equation, we employ again the orthogonal collocation procedure followed by the Gauss-Jacobi quadrature. The corresponding eigenvalue equation is solved using the Renormalized Numerov Method to obtain ground state binding energies and root-mean-square radii of closed shell nuclei 4He, 8Be, 12C, 16O and 40Ca. / Physics / M. Sc. (Physics)