In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

Alors que les approches standards de résolution de la structure électronique présentent un coût de calcul à la puissance 3 par rapport à la complexité du problème, des solutions permettant d’atteindre un régime asymptotique linéaire,O(N), sont maintenant bien connues pour le calcul de l'état fondamental. Ces solutions sont basées sur la "myopie" de la matrice densité et le développement d'un cadre théorique permettant de contourner le problème aux valeurs propres. La théorie des purifications de la matrice densité constitue une branche de ce cadre théorique. Comme pour les approches de type O(N) appliquées à l'état fondamental,la théorie des perturbations nécessaire aux calculs des fonctions de réponse électronique doit être révisée pour contourner l'utilisation des routines coûteuses.L'objectif est de développer une méthode robuste basée uniquement sur la recherche de la matrice densité perturbée, pour laquelle seulement des multiplications de matrices creuses sont nécessaires. Dans une première partie,nous dérivons une méthode de purification canonique qui respecte les conditions de N-representabilité de la matrice densité à une particule. Nous montrons que le polynôme de purification obtenu est auto-cohérent et converge systématiquement vers la bonne solution. Dans une seconde partie, en utilisant une approche de type Hartree-Fock, nous appliquons cette méthode aux calculs des tenseurs de réponses statiques non-linéaires pouvant être déterminés par spectroscopie optique. Au delà des calculs à croissance linéaire réalisés, nous démontrons que les conditions N-representabilité constituent un prérequis pour garantir la fiabilité des résultats. / Whereas standard approaches for solving the electronic structures present acomputer effort scaling with the cube of the number of atoms, solutions to overcomethis cubic wall are now well established for the ground state properties, and allow toreach the asymptotic linear-scaling, O(N). These solutions are based on thenearsightedness of the density matrix and the development of a theoreticalframework allowing bypassing the standard eigenvalue problem to directly solve thedensity matrix. The density matrix purification theory constitutes a branch of such atheoretical framework. Similarly to earlier developments of O(N) methodology appliedto the ground state, the perturbation theory necessary for the calculation of responsefunctions must be revised to circumvent the use of expensive routines, such asmatrix diagonalization and sum-over-states. The key point is to develop a robustmethod based only on the search of the perturbed density matrix, for which, ideally,only sparse matrix multiplications are required. In the first part of this work, we derivea canonical purification, which respects the N-representability conditions of the oneparticledensity matrix for both unperturbed and perturbed electronic structurecalculations. We show that this purification polynomial is self-consistent andconverges systematically to the right solution. As a second part of this work, we applythe method to the computation of static non-linear response tensors as measured inoptical spectroscopy. Beyond the possibility of achieving linear-scaling calculations,we demonstrate that the N-representability conditions are a prerequisite to ensurereliability of the results.

Orientador: Rogerio Custodio / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-05T16:17:29Z (GMT). No. of bitstreams: 1 Politi_JoseRobertodosSantos_D.pdf: 1434250 bytes, checksum: 9907fe783c6f36299444800b1f937bc8 (MD5) Previous issue date: 2005 / Doutorado / Físico-Química / Doutor em Ciências

Orientador: Rogerio Custodio / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-12T23:54:09Z (GMT). No. of bitstreams: 1 Angelotti_WagnerFernandoDelfino_D.pdf: 827045 bytes, checksum: 044a9aecfaec7ac3e14153480c625d9f (MD5) Previous issue date: 2009 / Resumo: Esta tese explorou diferentes objetivos envolvendo o método Monte Carlo Quântico, dos quais se destacam: avaliação do método convencional em propriedades eletrônicas; formalização das relações existentes entre a teoria de matriz densidade e os métodos Monte Carlo Quântico Variacional e de Difusão; estudo da correlação eletrônica com diferentes funções correlacionadas e também através de método misto envolvendo a teoria de perturbação e o Monte Carlo Quântico Variacional; aplicações para átomos do primeiro e segundo período da tabela periódica e moléculas diatômicas. Experimentos computacionais com o método Monte Carlo Quântico e separação de spins foram realizados produzindo excelentes resultados para cálculos de potenciais de ionização sucessivos para átomos, ionização atômica e molecular e construção de curvas de potencial para moléculas simples. Foram ainda obtidas duas formulações analíticas que descrevem exatamente o vínculo formal entre a matriz densidade e o Monte Carlo Quântico. Esta associação proporcionou ótimos resultados para os métodos Variacional e de Difusão, apresentando semelhanças e significativas diferenças quando comparado ao tratamento convencional com respeito à estrutura nodal para cada estado eletrônico estudado. Além disso, a matriz densidade aliada às funções correlacionadas é capaz de recuperar parte da correlação eletrônica e torna possível a correção de funções de onda dentro da associação do Monte Carlo Quântico e teoria de perturbação. / Abstract: This thesis explored different goals involving the quantum Monte Carlo method, of which stand out: assessment of the conventional method in electronic properties; formalization of relations between the density matrix theory and the variational and diffusion quantum Monte Carlo methods; study of the electronic correlation with different correlated functions and also through mixed method involving the perturbation theory and variational quantum Monte Carlo; applications to atoms of the first and second period of the periodic table and diatomic molecules. Computational experiments with the quantum Monte Carlo method and separation of spins were achieved producing excellent results for calculations of successive ionization potentials for atoms, single ionization of atoms and simple molecules and calculation of potential curves for simple molecules. Two analytical formulations were obtained that describes exactly the formal link between the density matrix and quantum Monte Carlo. This association has provided excellent results for variational and diffusion methods, presenting similarities and significant differences when compared to conventional treatment with respect to the nodal structure for each electronic state studied. Furthermore, the density matrix together with correlated wave functions is able to recover part of the electronic correlation and makes possible the correction of the wave functions within the association of quantum Monte Carlo and perturbation theory. / Doutorado / Físico-Química / Doutor em Ciências

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18774号 / 理博第4032号 / 新制||理||1581(附属図書館) / 31725 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 佐々 真一, 教授 高橋 義朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis is part of an effort to enable large-scale Hartree-Fock/Kohn-Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn-Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is σ(n3). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix-matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix-matrix multiplication can be performed with complexity σ(n). A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO-LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method. / <p>QC 20101123</p>

No description available.

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.

Thesis (MSc)--University of Stellenbosch, 2011. / ENGLISH ABSTRACT: The selective excitation of an arbitrary vibrational level of a polyatomic molecule, without passage through an intermediary electronic excited state is demonstrated. This was achieved by simulating the interaction of a shaped, femtosecond pulse with one vibrational mode of the molecule. The carrier frequency of the pulse is chosen near resonant to the ground-to- rst-excited vibrational transition of the mode, and the pulse shape is optimized via closed-loop feedback. The simulation concentrates on the rst few vibrationally excited states since the density of states is still low, thus ensuring that the inter-vibrational decoherence time is relatively long compared to the pulse length. While various molecules were investigated this study focuses onUF6 for which detailed spectroscopic data for the v3 vibrational mode is available in literature. A multilevel model was developed and can be adapted for any number of levels. The model reported here was limited to a vibrational quantum number of four. The spectroscopic data included anharmonic splitting as well as forbidden transitions. The effect of rotational levels was not included. A density matrix approach was followed because this will allow for the introduction of dephasing of the coherent excitation via thermalizing collisions with the reservoir, as well as inter-vibrational relaxation. The time evolution of the density matrix is given by the Von Neumann equations. / AFRIKAANSE OPSOMMING: Die selektiewe opwekking van 'n arbitrêre vibrasionele vlak van 'n poliatomies molekule sonder oorgang na 'n intermediëre elektroniese opgewekte toetstand word gedemonstreer. Dit was bereik deur die interaksie te simuleer van 'n gevormde, femtosekonde pulse met een vibrasionele mode van 'n molekule. Die draer frekwensie van die pulse is so gekies dat dit naby resonansie van die grond-tot-eerste-opgewekte vibrasionele oorgang van die mode is, die puls vorm word geoptimeer deur 'n geslote-lus terugvoer. Die simulasie konsentreer op die eerste paar vibrasionele opgewekte toestande, omdat die digtheid van toestande nog steeds laag is, dus verseker dit dat inter-vibrasionele de-koherensie tyd relatief lank is in vergelyking met die puls se lengte. Verskillende molekules was ondersoek vir die studie. Die fokus is op UF6 waarvoor gedetaileerde spektroskopiese data vir die v3 vibrasionele beskikbaar is in die literatuur. 'n Multivlak model was ontwikkel en kan aangepas word vir enige aantal van vlakke. Die model wat hier aangemeld is, is beperk tot die vibrasionele kwantum getal van vier. Die spektroskopiese data het anharmonies splitting so wel as nie toegelaatbare oorgange bevat. Die effek van rotasionele vlakke was nie in berekening geneem nie. 'n Digtheids matriks benadering was gevolg, omdat dit toelaat vir die dekoherensie. Die tyd evolusie van die digtheids matriks word gegee deur die Von Neumann vergelykings.

No description available.