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Orientador: Fabiano Borges da Silva / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T12:59:08Z (GMT). No. of bitstreams: 1 Fernandes_AristonLopes_M.pdf: 1807643 bytes, checksum: 1b8f6422f12372fd95aefa1b55d12fc7 (MD5) Previous issue date: 2011 / Resumo: Este trabalho tem como propósito estudar trajetórias geradas por sistemas hamiltonianos no plano. Para isso, são analisados os diversos tipos de retratos de fase dos sistemas lineares planares e a classificação destes. Sistemas hamiltonianos surgiram na mecânica clássica e seus pontos de equilíbrio são classificados em selas ou centros, conforme os sinais dos autovalores da matriz do sistema linearizado. Além disso, é apresentada a relação entre campos de vetores hamiltonianos e espaços vetoriais simpléticos / Abstract: This work has the objective of studing trajectories generated by Hamiltonian systems on the plane. For this, we analyse the various types of phase portraits of planar and linear systems. Hamiltonian systems have emerged in the mechanical and their classical equilibrium points are classified into saddles or centers, as the signs of the eigenvalues of linearized system matrix. We have also illustrated the connection between Hamiltonian vector fields and symplectic spaces / Mestrado / Matemática Universitária / Mestre em Matemática Universitária

Electronic structure methods enable first-principles calculations of the properties of molecules and materials. But numerically exact calculations of systems relevant to chemistry are computationally intractable due to the exponentially scaling cost of solving the associated Schrödinger equation. This thesis describes the application of quantum Monte Carlo (QMC) methods that enable the accurate solution of this equation at reduced computational cost. Chapter 2 introduces the fast randomized iteration (FRI) framework for analyzing discrete-space QMC methods for ground-state electronic structure calculations. I analyze the relative advantages of applying different strategies within this framework in terms of statistical error and computational cost. Chapter 3 discusses the incorporation of strategies from related stochastic methods to achieve further reductions in statistical error. Chapter 4 presents a general framework for extending these FRI-based approaches to calculate energies of excited electronic states. Chapter 5 demonstrates that leveraging the best of these ground- and excited-state techniques within the FRI framework enables the calculation of very accurate electronic energies in large molecular systems. In contrast to Chapters 2–5, which describe discrete-space QMC methods, Chapter 6 describes a continuous-space approach, based on diffusion Monte Carlo, for calculating optical properties of materials with a particular layered structure. I apply this approach to calculate exciton, trion, and biexciton binding energies of hybrid organic-inorganic lead-halide perovskite materials using a semiempirical Hamiltonian.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.

We present a method of constructing the phase diagram at low temperatures, using the low temperature expansions. We consider spin Iattice systems described by a Hamiltonian with a d-dimensional perturbation space. We prove that there is a one-one correspondence between subsets of the phase diagram and extremal elements of some family of convex sets. We also solve a linear programming problem of the phase diagram for a set of affine functionals. / Ph. D.

Achieving the quantum control of ever more complex systems has been a driving force of atomic, molecular, and optical physics. This goal has materialized in the harnessing of systems with increasingly rich structures and interactions: the more sophisticated the system, the more faceted and fascinating its application to fields as varied as quantum simulation, quantum information, many body physics, metrology, and quantum chemistry. One of the current frontiers of quantum control is ultracold dipolar molecules. They present rich internal structures and long-range, anisotropic dipole-dipole interactions which promise to revolutionize AMO physics, for example by realizing realistic Hamiltonians in quantum simulation, by providing a new platform for quantum information, and by achieving a novel kind of quantum liquid. Despite its promises, the full quantum control of dipolar molecules has been over a decade in the making. The difficulties in either directly laser cooling molecules or in collisionally stabilizing their bulk samples have been major roadblocks that have hampered the development of this quantum system. The realization of a Bose-Einstein condensate of dipolar molecules has been a particularly elusive milestone. In this thesis, I report on the first observation of this quantum state of matter. The work that brought us to this achievement parallels the historical evolution of AMO physics in the last thirty years. To reach a BEC of molecules, we initially constructed a dual species experiment capable of realizing the simultaneous Bose-Einstein condensation of atomic sodium (Na) and cesium (Cs). Individual BECs of sodium and cesium were first reported in 1995 and 2003 respectively, while our experiment was the first instance of their concurrent condensation. The study of the Na-Cs interatomic scattering properties in an homogeneous magnetic field showed us the path to the Feshbach association of loosely-bound sodium-cesium (NaCs) molecules, a technique first demonstrated in 2006 for heteronuclear molecules but never attempted on our species. Following the Feschbach association, we determined a novel pathway to the molecular electronic, vibrational and rotational ground state using STIRAP. From this point, we found ourselves at the forefront of the field: bulk samples of bosonic molecules such as NaCs had neither been stabilized against collisional losses nor evaporatively cooled. At first, we successfully applied a single-frequency microwave shielding approach to decrease in-bulk losses by a factor of 200 and reach lifetimes on the order of 2 s, allowing us to measure high elastic scattering rates and characterize their dipolar anisotropy. Moreover, we demonstrated the first evaporative cooling of a bosonic molecular gas by increasing its phase-spacedensity by a factor of 20 and reaching a temperature of 36(5) nK. Since this proved insufficient to achieve Bose-Einstein condensation due to unexpected three-body losses, we introduced an enhanced microwave shielding technique, double microwave shielding. This further decreased loss rates enabling efficient evaporative cooling of our sample to a long-lived Bose-Einstein condensate of dipolar molecules. This new double microwave shielding technique also allows the tunability of the strength of dipole-dipole interaction, establishing ultracold bosonic dipolar molecules as a new quantum liquid for the exploration of many body physics. In addition to the experimental work on dipolar NaCs, we have theoretically explored the field of direct molecular laser cooling. Our aim was twofold: we aimed to expand the category of molecules that can be laser cooled and to simplify the identification of laser cycling schemes. For the former goal, we lifted the widespread assumption that only molecules with diagonal Franck-Condon factors could be laser cooled. For the latter, we decided to employ publicly available repositories of molecular transitions. A second consequence of the use of these databases is that they contain data on molecules of interest to other scientific fields, further establishing direct laser cooling as a technique that could be of interest beyond AMO physics. Our work was successful in that we identified laser cycling schemes for C₂ and OH+. To simplify the determination of laser cycling schemes, we developed a graph-based algorithm form their identification starting from spectroscopic data.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-04-19T13:09:32Z No. of bitstreams: 1 Francisco José dos Santos Nascimento.pdf: 743351 bytes, checksum: 997f8a5009a3bbc979a7206041daf583 (MD5) / Made available in DSpace on 2017-04-19T13:09:32Z (GMT). No. of bitstreams: 1 Francisco José dos Santos Nascimento.pdf: 743351 bytes, checksum: 997f8a5009a3bbc979a7206041daf583 (MD5) Previous issue date: 2017-02-17 / CAPES / In this work, we discuss the article The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem due to Hallan and Rana. For this we present some basic definitions and results abut Hamiltonian systems such as equilibrium stability of linear Hamiltonian systems. We set out the restricted problem of the three bodies and show some classic results of the problem. Finally we present the Robe’s problem and discuss the main results using Hamiltonian systems theory. / Nesse trabalho, dissertamos sobre o artigo \The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem" devido a Hallan e Rana. Para isso apresentamos definições e resultados básicos sobre sistemas Hamiltonianos tais como estabilidade de equilíbrios de sistemas Hamiltonianos lineares. Enunciamos o problema restrito dos três corpos e mostramos alguns resultados clássicos do problema. Por fim apresentamos o problema de Robe e discutimos os principais resultados usando a teoria de sistemas Hamiltonianos.

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.

Two well-known types of water waves are shallow water waves and the solitary waves. The former waves are those waves which have larger wavelength than the local water depth and the latter waves are used for the ones which retain their shape and speed after colliding with each other. The most well known of the latter waves are Korteweg de Vries (KdV) equations, which are widely used in many branches of physics and engineering. These equations are nonlinear long waves and mathematically represented by partial differential equations (PDEs). For solving the KdV and KdV-type equations, several numerical methods were developed in the recent years which preserve their geometric structure, i.e. the Hamiltonian form, symplecticity and the integrals. All these methods are classified as symplectic and multisymplectic integrators. They produce stable solutions in long term integration, but they do not preserve the Hamiltonian and the symplectic structure at the same time. This thesis concerns the application of energy preserving average vector field integrator(AVF) to nonlinear Hamiltonian partial differential equations (PDEs) in canonical and non-canonical forms. Among the PDEs, Korteweg de Vries (KdV) equation, modified KdV equation, the Ito&rsquo / s system and the KdV-KdV systems are discetrized in space by preserving the skew-symmetry of the Hamiltonian structure. The resulting ordinary differential equations (ODEs) are solved with the AVF method. Numerical examples confirm that the energy is preserved in long term integration and the other integrals are well preserved too. Soliton and traveling wave solutions for the KdV type equations are accurate as those solved by other methods. The preservation of the dispersive properties of the AVF method is also shown for each PDE.

Various properties of linear infinite-dimensional Hamiltonian systems are studied. The structural stability of the Vlasov-Poisson equation linearized around a homogeneous stable equilibrium [mathematical symbol] is investigated in a Banach space setting. It is found that when perturbations of [mathematical symbols] are allowed to live in the space [mathematical symbols], every equilibrium is structurally unstable. When perturbations are restricted to area preserving rearrangements of [mathematical symbol], structural stability exists if and only if there is negative signature in the continuous spectrum. This analogizes Krein's theorem for linear finite-dimensional Hamiltonian systems. The techniques used to prove this theorem are applied to other aspects of the linearized Vlasov-Poisson equation, in particular the energy of discrete modes which are embedded within the continuous spectrum. In the second part, an integral transformation that exactly diagonalizes the Caldeira-Leggett model is presented. The resulting form of the Hamiltonian, derived using canonical transformations, is shown to be identical to that of the linearized Vlasov-Poisson equation. The damping mechanism in the Caldeira-Leggett model is identified with the Landau damping of a plasma. The correspondence between the two systems suggests the presence of an echo effect in the Caldeira-Leggett model. Generalizations of the Caldeira-Leggett model with negative energy are studied and interpreted in the context of Krein's theorem. / text