Este trabalho calcula analítica e numericamente a densidade espectral para o modelo de Anderson de duas impurezas sem correlação eletrônica (U=0). Nossos resultados servem como passo inicial para se entender o modelo com a correlação eletrônica. O modelo estudado descreve a interação entre elétrons de um metal e impurezas magnéticas localizadas, e a simplificação, U = 0, torna o Hamiltoniano quadrático permitindo assim que se divida o mesmo em dois termos: um envolvendo apenas operadores pares (canal par) e outro envolvendo apenas operadores ímpares (canal ímpar). Cada termo encontrado difere pouco do Hamiltoniano de Nível Ressonante. Nossos resultados abrangem tanto a diagonalização analítica como a numérica pelo método do Grupo de Renormalização, adaptado para o caso de duas impurezas. A simplicidade do Hamiltoniano permite que (1) se identifique características do modelo que afetam adversamente a precisão do cálculo numeríco e (2) se encontre uma maneira de circundar tais dificuldades. Os resultados aqui encontrados ajudaram o desenvolvimento do cálculo da densidade espectral do modelo correlacionado, desenvolvido paralelamente em nosso grupo de pesquisa. / This work calculates analytically and numerically the spectral density for the two impurity uncorrelated Anderson model (U = O). Our results serve as an initial step towards understanding models with electronic correlation. The studied model describes the interaction between conduction-band electrons of a metal and localized magnetic impurities. The simplification U = O turns the Hamiltonian quadratic, allowing us to split it into two parts: one involving only even operators (even channel), the other involving odd operators (odd channel). Each term has a form differing a little from that for the Resonant Level Hamiltonian. Our results include analytic diagonalization as well as numerical calculations using the method of the Renormalization Group, adapted for the two impurity case. The traditional tridiagonalization method imposes particle-hole symmetry, while our treatment preserves the energy dependence of the coupling, between the impurities and the conduction-band, and consequently, the natural asymmetry of the model. The simplicity of the Hamiltonian allowed us to (1) identify characteristics of the model that affect adversely the acuracy of the numeric calculation and (2) find a way to surround such difficulties. The results here found helped the development of the calculation of the spectral density of the correlated model, developed simultaneously in our research group.

Thesis (MSc (Physics))--University of Stellenbosch, 2005. / It was Wigner that in the 1950’s first introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of Random Matrix Theory has grown tremendously, with applications ranging from fluctuations on the economic markets to M-theory. It is the purpose of this thesis to discuss the basic concepts of Random Matrix Theory, using the ensembles of random matrices originally introduced by Wigner, the Gaussian ensembles, as a starting point. As Random Matrix Theory is classically concerned with the statistical properties of levels sequences, we start with a brief introduction to the statistical analysis of a level sequence before getting to the introduction of the Gaussian ensembles. With the ensembles defined, we move on to the statistical properties that they predict. In the light of these predictions, a few of the classical applications of Random Matrix Theory are discussed, and as an example of some of the important concepts, the Anderson model of localization is investigated in some detail.

Estudamos o calor específico de um sistema formado por duas impurezas adsorvidas, sem spin, em meio fermiônico (banda de condução do metal) e que contém um buraco (elétron) tunelando entre elas. Modelamos esse sistema por dois níveis acoplados e que sofrem interação Coulombiana com a banda de condução. Através da análise das curvas de calor específico, investigamos a alteração (renormalização) da taxa de tunelamento em função da interação eletrostática entre os elétrons da banda de condução e o buraco tunelante e da separação entre as impurezas. Utilizamos o Hamiltoniano de Kondo de tunelamento para representar esse modelo e usamos o Grupo de Renormalização Numérico para diagonalizá-lo. Analisamos a influência de cada termo do Hamiltoniano na renormalização da taxa de tunelamento e verificamos que a troca de paridade das funções de onda do buraco tunelante e dos elétrons da banda desempenha papel essencial. Encontramos uma expressão que combina a distância entre as impurezas e a interação Coulombiana em um único parâmetro (a), de tal forma que sistemas diferentes mas que apresentam o mesmo a e a mesma taxa de tunelamento livre têm a mesma curva de calor específico. / We calculate the specific heat of the two-spinless impurity coupled to a fermionic bath. The model takes into account the tunneling of a hole between the impurities. The two-level system representing the impurities is coupled electrostatically with the conduction electrons. Through the specific heat curves, we analyse the renormalization of the tunneling rate as a function of the Coulomb interaction and distance between impurities. The Numerical Renormalization Group is used to diagonalize the tunneling Hamiltonian proposed by Kondo. We analyse the role of each term of the Hamiltonian in the renormalization of the bare tunneling rate and we stress the importance of the exchange parity between impurity states and conduction states. Finally, a parameter a, is found which combines the distance between impurities and Coulomb interaction in such a way that every curve is specified only by a and the bare tunneling rate.

Este trabalho se dedica à paralelização de um programa para cálculos de propriedades físicas de ligas magnéticas diluídas. O método do grupo de renormalização aplicado ao modelo de Anderson de duas impurezas se mostrou particularmente adequado ao processamento paralelo visto que grande parte dos cálculos pode ser executada simultaneamente, assim como variações nos conjuntos de dados requeridas pelo método. Para tal reescrevemos o programa seqüencial usado anteriormente pelo Grupo de Física Teórica do IFSC e implementamos três versões paralelas. Essas versões diferem entre si em relação à abordagem dada à paralelização. O uso de clusters de computadores se revelou uma opção conveniente pois verificamos que o limitante no desempenho é o tempo tomado pelos cálculos e não pela comunicação. Os resultados mostram uma grande redução no tempo total de execução, porém deficiências no speedup e escalabilidade devido a problemas de balanceamento de carga. Analisamos esses problemas e sugerimos alternativas para solucioná-los. / This dissertation discuss the parallelization of a program that calculates physical properties of dilute magnetic alloys. The renormalization group method applied to Anderson's two impurities model showed to be specially suitable to parallel processing because a large amount of calculations as well as variations of data entries required by the method can be performed simultaneously. To achieve this we rewrote the sequential program previously used by the Theoretical Physics Group of the IFSC and wrote three parallel versions. These versions differ from each other by the parallelization approach. The use of computer clusters revealed to be an appropriate option because the calculation time is the limiting factor on performance instead of communication time. The results show a good reduction of execution time, but speedup and scalability lack due to load balancing problems. We analyze these problems and suggest possible solutions.

Long range order (LRO) is one of the most important properties of physics systems, especially for a strongly correlated system. In this thesis, the long range order in a few strongly correlated systems is investigated both rigorously and numerically. / Magnetic orders in the two-dimensional periodic Anderson model (PAM) were investigated in the project. Several numerical methods including exact diagonalization, mean field methods and the constrained path Monte Carlo (CPMC) method were used here. We studied the effect of the dispersion of the impurity band on the magnetism and gave estimated phase diagrams on the band filling and impurity chemical potential plane, by comparing the ground state energies and by studying the Fourier transformation of the spin-spin correlations. / The Neel long range order in various quantum spin models was studied. A spatially anisotropy antiferromagnetic Heisenberg system was studied and a critical point Jup⊥S was obtained. When J⊥ is larger than Jup⊥S , the Neel long range order was proved to exist in the ground state of the system. Then an onsite single-ion anisotropy D-term was imposed on the above system and its effect on the existence of the long range order was studied. We obtained a critical line on the DJ⊥ plane such that above which the long range order exists. To get insight into the long range order in the two-dimensional isotropic Heisenberg model, the XXZ model and the Heisenberg system with next nearest neighbors interaction were studied. For the XXZ model, two critical couplings DeltaXY and DeltaIsing were obtained such that when 0 ≤ Delta ≤XY or Delta ≥ DeltaIsing, the Neel order appears. For the antiferromagnetic Heisenberg model with next nearest neighbors interaction both spin S = 1/2 and S = 1 were studied. For S = 1/2, a critical next nearest neighbors ferro-coupling Jc2 was obtained such that when J2 ≤ Jc2 , the Neel long range order appears, while for S = 1 an improved Jc2 was obtained such that when J2 ≤ Jc2 the Neel order still exists even with frustration. / Wang Yongqiang = 强关联系统中的长程序 / 王永强. / "June 2005." / Adviser: Haiqing Lin. / Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3861. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 156-162). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Text in English; abstracts in English and Chinese. / School code: 1307. / Wang Yongqiang = Qiang guan lian xi tong zhong de chang cheng xu / Wang Yongqiang.

Orientador: Antônio Carlos Ferreira Seridônio / Resumo: Por meio da equação de Weyl que descreve o bulk de um semimetal de Weyl, inserimos duas impurezas no interior deste semimetal, com o intuito de medir a condutância desse sistema por meio de uma ponta de um microscópio de corrente de tunelamento (STM: scanning tunneling microscope), assim como exploramos teoricamente a influência que o semimetal de Dirac-Weyl causa na estrutura dos orbitais das impurezas. Verificamos que nessas condições apresentadas: (i) é possível resgatar e obter os mesmos resultados publicados por Phys. Rev. B 96, 041112(R) (2017) para um semimetal de Dirac em três dimensões; (ii) ao quebrarmos a simetria de inversão do sistema, ocorre um alargamento dos picos de ressonância, até o ponto que uma impureza deixa de sentir a presença da outra, o que caracteriza um sistema que segue o modelo de Anderson de uma impureza (SIAM: single impurity Anderson model), ao passo que, caso continuemos aumentando o valor deste parâmetro, a estrutura de bandas do material torna-se um metal; (iii) ao quebrarmos a simetria de reversão temporal nas direções dos eixos x e z, ocorre formação de um par de orbitais de ligações π para os estados ligantes e antiligantes, o que significa que os elétrons são excitados de ondas s para ondas p. / Abstract: Through the Weyl equation that describes the bulk of a Weyl semimetal, we inserted two impurities inside this semimetal, in order to measure the conductance of this system by means of a scanning tunneling microscope (STM), as well as we theoretically explore the influence that Dirac-Weyl semimetal causes on the structure of the impurities orbitals. We verified that under these conditions presented: (i) it is possible to retrieve and obtain the same results published by Phys. Rev. B 96, 041112 (R) (2017) for a three-dimensional Dirac Semimetal; (ii) when we break the inversion symmetry of the system, there is a widening of the resonance peaks, to the point that one impurity cannot feel the presence of the other one, which characterizes a single impurity Anderson model (SIAM) system, whereas, if we continue increasing the value of this parameter, the bands structure of the material becomes a metal; (iii) when we break the time reversal symmetry in the x and z-axis directions, a pair of π-bond orbitals forms for the bonding and antibonding states, which means that electrons are excited from s-waves to p-waves. / Mestre

Este trabalho se dedica à paralelização de um programa para cálculos de propriedades físicas de ligas magnéticas diluídas. O método do grupo de renormalização aplicado ao modelo de Anderson de duas impurezas se mostrou particularmente adequado ao processamento paralelo visto que grande parte dos cálculos pode ser executada simultaneamente, assim como variações nos conjuntos de dados requeridas pelo método. Para tal reescrevemos o programa seqüencial usado anteriormente pelo Grupo de Física Teórica do IFSC e implementamos três versões paralelas. Essas versões diferem entre si em relação à abordagem dada à paralelização. O uso de clusters de computadores se revelou uma opção conveniente pois verificamos que o limitante no desempenho é o tempo tomado pelos cálculos e não pela comunicação. Os resultados mostram uma grande redução no tempo total de execução, porém deficiências no speedup e escalabilidade devido a problemas de balanceamento de carga. Analisamos esses problemas e sugerimos alternativas para solucioná-los. / This dissertation discuss the parallelization of a program that calculates physical properties of dilute magnetic alloys. The renormalization group method applied to Anderson's two impurities model showed to be specially suitable to parallel processing because a large amount of calculations as well as variations of data entries required by the method can be performed simultaneously. To achieve this we rewrote the sequential program previously used by the Theoretical Physics Group of the IFSC and wrote three parallel versions. These versions differ from each other by the parallelization approach. The use of computer clusters revealed to be an appropriate option because the calculation time is the limiting factor on performance instead of communication time. The results show a good reduction of execution time, but speedup and scalability lack due to load balancing problems. We analyze these problems and suggest possible solutions.

Estudamos o calor específico de um sistema formado por duas impurezas adsorvidas, sem spin, em meio fermiônico (banda de condução do metal) e que contém um buraco (elétron) tunelando entre elas. Modelamos esse sistema por dois níveis acoplados e que sofrem interação Coulombiana com a banda de condução. Através da análise das curvas de calor específico, investigamos a alteração (renormalização) da taxa de tunelamento em função da interação eletrostática entre os elétrons da banda de condução e o buraco tunelante e da separação entre as impurezas. Utilizamos o Hamiltoniano de Kondo de tunelamento para representar esse modelo e usamos o Grupo de Renormalização Numérico para diagonalizá-lo. Analisamos a influência de cada termo do Hamiltoniano na renormalização da taxa de tunelamento e verificamos que a troca de paridade das funções de onda do buraco tunelante e dos elétrons da banda desempenha papel essencial. Encontramos uma expressão que combina a distância entre as impurezas e a interação Coulombiana em um único parâmetro (a), de tal forma que sistemas diferentes mas que apresentam o mesmo a e a mesma taxa de tunelamento livre têm a mesma curva de calor específico. / We calculate the specific heat of the two-spinless impurity coupled to a fermionic bath. The model takes into account the tunneling of a hole between the impurities. The two-level system representing the impurities is coupled electrostatically with the conduction electrons. Through the specific heat curves, we analyse the renormalization of the tunneling rate as a function of the Coulomb interaction and distance between impurities. The Numerical Renormalization Group is used to diagonalize the tunneling Hamiltonian proposed by Kondo. We analyse the role of each term of the Hamiltonian in the renormalization of the bare tunneling rate and we stress the importance of the exchange parity between impurity states and conduction states. Finally, a parameter a, is found which combines the distance between impurities and Coulomb interaction in such a way that every curve is specified only by a and the bare tunneling rate.

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.

This research studied the Anderson localization of electrons in two-channel wires with correlated disorder and in DNA molecules. It involved an analytical calculation part where the formula for the inverse localization length for electron states in a two-channel wire is derived. It also involved a computational part where the localization length is calculated for some DNA molecules. Electron localization in two-channel wires with correlated disorder was studied using a single-electron tight-binding model. Calculations were within second-order Born-approximation to second-order in disorder parameters. An analytical expression for localization length as a functional of correlations in potentials was found. Anderson localization in DNA molecules were studied in single-channel wire and two-channel models for electron transport in DNA. In both of the models, some DNA sequences exhibited delocalized electron states in their energy spectrum. Studies with two-channel wire model for DNA yielded important link between electron localization properties and genetic information.