Cette thèse traite de la description de la corrélation électronique et de la spectroscopie dans le cadre de la Théorie de la Fonctionnelle de la Matrice Densité Réduite (RDMFT). Dans la RDMFT, les propriétés de l'état fondamental d'un système physique sont des fonctionnelles de la matrice densité à un corps. Plusieurs approximations à la corrélation électronique ont été proposées dans la littérature. Beaucoup d'entre elles peuvent être reliés au travail de Müller, qui en a proposé une similaire à l'approximation Hartree-Fock mais qui peut produire des nombres d'occupation fractionnaires. Cela n'est pas toujours suffisant, notamment dans les matériaux fortement corrélés. Par ailleurs, l'expression des observables du système en terme de la matrice densité n'est pas toujours connue. Tel est le cas, par exemple, pour la fonction spectrale, qui est liée aux spectres de photoémission. Dans ce cas, il y a des annulations d'erreur entre l'approximation à la corrélation électronique et l'approximation à l'observable, ce qui affaiblit la théorie. Dans cette thèse, nous recherchons des approximations plus précises en exploitant le lien entre les matrices densité et les fonctions de Green. Dans la première partie de la thèse, nous nous concentrons sur la fonction spectrale. En utilisant le modèle de Hubbard, qui peut être résolu exactement, nous analysons les approximations existantes à cette observable et nous soulignons leurs points faibles. Ensuite, à partir de sa définition en terme de la fonction de Green à un corps nous dérivons une expression pour la fonction spectrale qui dépend des nombres d'occupation naturels et d'une énergie efficace qui prend en compte toutes les excitations du système. Cette énergie efficace dépend de la matrice densité à un corps ainsi que des ordres supérieurs. Des approximations simples à cette énergie efficace donnent des spectres précis dans des systèmes modèles dans des régimes à la fois de faible et de forte corrélation. Pour illustrer notre méthode sur les matériaux réels, nous calculons le spectre de photoemission du NiO massif: notre méthode donne une image qualitativement correcte dans la phase antiferromagnétique et dans la phase paramagnétique, contrairement aux méthodes de champ moyen utilisés actuellement, qui donnent un métal dans le dernier cas. La deuxième partie de la thèse est plus explorative et traite des phénomènes dépendant du temps dans la RDMFT. En général, l'évolution temporelle des matrices densité est donnée par la hiérarchie des équations de Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY), dans lequel l'équation du mouvement de la matrice densité a n corps est donnée en termes de la matrice densité à n+1 corps. La première équation de la hiérarchie relie la matrice densité à un corps à la matrice densité à deux corps. La tâche difficile est de trouver des approximations à la matrice densité à deux corps. Les approximations existantes sont des extensions adiabatiques des approximations de l'état fondamental. Nous explorons cette question en examinant de nouvelles approximations qui nous tirons de la théorie à plusieurs corps (MBPT) basée sur les fonctions de Green ainsi que de la solution exacte du modèle de Anderson à deux niveaux dans son état fondamental. Nos premiers résultats sur le modèle de Anderson soumis à divers champs externes montrent quelques caractéristiques intéressantes, qui suggèrent d'explorer davantage ces approximations aussi sur des systèmes modèles plus grands. / This thesis addresses the description of electron correlation and spectroscopy within the context of Reduced Density-Matrix Functional Theory (RDMFT). Within RDMFT the ground-state properties of a physical system are functionals of the ground-state reduced density matrix. Various approximations to electron correlation have been proposed in literature. Many of them, however, can be traced back to the work of Müller, who has proposed an approximation to the correlation which is similar to the Hartree-Fock approximation but which can produce fractional occupation numbers. This is not always sufficient. Moreover, the expression of the observables of the system in terms of the reduced density matrix is not always known. This is the case, for example, for the spectral function, which is closely related to photoemission spectra. In this case there are error cancellations between the approximation to correlation and the approximation to the observable, which weakens the theory. In this thesis we look for more accurate approximations by exploiting the link between density matrices and Green's functions. In the first part of the thesis we focus on the spectral function. Using the exactly solvable Hubbard model as illustration, we analyze the existent approximations to this observable and we point out their weak points. Then, starting from its definition in terms of the one-body Green's function, we derive an expression for the spectral function that depends on the natural occupation numbers and on an effective energy which accounts for all the charged excitations. This effective energy depends on the one-body as well as higher-order reduced density matrices. Simple approximations to this effective energy give accurate spectra in model systems in the weak as well as strong-correlation regimes. To illustrate our method on real materials we calculate the photoemission spectrum of bulk NiO: our method yields a qualitatively correct picture both in the antiferromagnetic and in the paramagnetic phases, contrary to currently used mean-field methods, which give a metal in the latter case. The second part of the thesis is more explorative and deals with time-dependent phenomena within RDMFT. In general the time evolution of the reduced density matrices is given by the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations, in which the equation of motion of the n-body reduced density matrix is given in terms of the (n + 1)-body reduced density matrix. The first equation of the hierarchy relates the one-body to the two-body reduced density matrix. The difficult task is to find approximations to the two-body reduced density matrix. Commonly used approximations are adiabatic extension of ground-state approximations. We explore this issue by looking at new approximations derived from Many-Body Perturbation Theory (MBPT) based on Green's functions as well as from the exact solution of the two-level Anderson impurity model in its ground state. Our first results on the two-level Anderson model subjected to various external fields show some interesting and, at the same time, puzzling features, which suggest to explore further these approximations.

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The ability to predict and control laminar-to-turbulent transition in high-speed flow has a substantial effect on heat transfer and skin friction, thus improving the design and operation of hypersonic vehicles. The control of transition on blunt bodies is essential to improve the performance of lifting and control surfaces. The objective of this Ph.D. research is to develop efficient and accurate algorithms for the detection of high-frequency heat flux fluctuations supported by hypersonic flow in transitional boundary layers. The focus of this research is on understanding the mathematical properties of the reconstruction such as regularity, sensitivity to noise, multi-resolution, and accuracy. This research is part of an effort to develop small-footprint heat flux sensors able to measure high-frequency fluctuations on test articles in a hypersonic wind tunnel with a small curvature radius. In the present theoretical/numerical study a multi-resolution formulation for the direct and inverse reconstruction of the heat flux from temperature sensors distributed over a multidimensional solid in a hypersonic flow was developed and validated. The solution method determines the thermal response by approximating the system Green's function with the Galerkin method and optimizes the heat flux distribution by fitting the distributed surface temperature data. Coating and glue layers are treated as separate domains for which the Green's function is obtained independently. Connection conditions for the system Green's function are derived by imposing continuity of heat flux and temperature concurrently at all interfaces. The solution heat flux is decomposed on a space-time basis with the temporal basis a multi-resolution wavelet with arbitrary scaling function. Quadrature formulas for the convolution of wavelets and the Green's function, a reconstruction approach based on isoparametric mapping of three-dimensional geometries, and a boundary wavelet approach for inverse problems were developed and verified. This approach was validated against turbulent conjugate heat transfer simulations at Mach 6 on a blunted wedge at 0 angle of attack and wind tunnel experiments of round impinging jet at Mach 0.7 It was found that multidimensional effects were important near the wedge shoulder in the short time scale, that the L-curve regularization needed to be locally corrected to analyze transitional flows and that proper regularization led to sub-cell resolution of the inverse problem. While the L2 regularization techniques are accurate they are also computationally inefficient and lack mathematical rigor. Optimal non-linear estimators were researched both as means to promote sparsity in the regularization and to pre-threshold the inverse heat conduction problem. A novel class of nonlinear estimators is presented and validated against wind tunnel experiments for a flat-faced cylinder also at Mach 6. The new approach to hypersonic heat flux reconstruction from discrete temperature data developed in this thesis is more efficient and accurate than existing techniques. / Doctor of Philosophy / The harsh environment supported by hypersonic flows is characterized by high-frequency turbulent bursts, acoustic noise, and vibrations that pollute the signals of the sensors that probe at high frequencies the state of the boundary layers developing on the walls. This research describes the search for optimal estimators of the noisy signal, i.e., those that lead to the maximum attenuation of the risk of error pollution by non-coherent scales. This research shows that linear estimators perform poorly at high-frequency and non-linear estimators can be optimized over a sparse projection of the signal in a discrete wavelet basis. Optimal non-linear estimators are developed and validated for wind tunnel experiments conducted at Mach 6 in the Advanced Propulsion and Power Laboratory at Virginia Tech.

Interferometry in geosciences uses mathematical techniques to image subsurface properties. This method turns a receiver in to a virtual source through utilizing either random noises or engineered sources. The method in seismology has been discussed extensively. Electromagnetic interferometry at high frequencies with coupled electromagnetic fields was developed in the past. However, the problem was not addressed for diffusive electromagnetic fields where the quasi-static limit holds. One of the objectives of this dissertation was to theoretically derive the impulse response of the Earth for low-frequency electromagnetic fields. Applying the theory of interferometry in the regions where the wavefields are diffusive requires volumetrically distributed sources in an infinite domain. That precondition imposed by the theory is not practical in experiments. Hence, the aim of this study was to quantify the important areas and distribution of sources that makes it possible to apply the theory in practice through conducting numerical experiments. Results of the numerical analysis in double half-space models revealed that for surface-based exploration scenarios sources are required to reside in a region with higher diffusivity. In contrast, when the receivers straddle an interface, as in borehole experiments, there is no universal rule for which region is more important; it depends on the frequency, receiver separation and also diffusivity contrast between the layers and varies  for different scenarios. Time-series analysis of the sources confirmed previous findings that the accuracy of the Green\'s function retrieval is a function of both source density and its width. Extending previous works in homogenous media into inhomogeneous models, it was found that sources must be distributed asymmetrically in the system, and extend deeper into the high diffusivity region in comparison to the low diffusivity area. The findings were applied in a three-layered example with a reservoir layer between two impermeable layers. Bayesian statistical inversion of the data obtained by interferometry was then used to estimate the fluid diffusivity (and permeability) along with associated uncertainties. The inversion results determined the estimated model parameters in the form of probability distributions. The output demonstrated that the algorithm converges closely to the true model. / Ph. D.

<p>Experiments shows that the conductance of a quantum point contact is quantized in steps of  2e²/h, where e is the charge of the electron and h is Planck’s constant, and thereby Ohm’s law is not valid for nanostructures. By using the approximation method finite difference, the transmission for one-dimensional contacts and one- and two-dimensional potentials are investigated. In the case of two-dimensional contacts and a two-dimensional potential the Green’s function method is used. It turns out that if electrons are treated as waves, the transmission and the conductance just differ by the constant 2e²/h, which in this thesis is interpreted numerically in Matlab by using the Green’s function method.</p>

A three-dimensional parabolic equation (PE) and perturbation approach is used to invert for the depth- and range-dependent geoacoustic characteristics of the seabed. The model assumes that the sound speed profile is the superposition of a known range-independent profile and an unknown depth- and range-dependent perturbation. Using a Green’s function approach, the total measured pressure field in the water column is decomposed into a background field, which is due to the range-independent profile, and a scattered field, which is due to the range-dependent perturbation. When the Born approximation is applied to the resulting integral equation, it can be solved for the range-dependent profile using linear inverse theory. Although the method is focused on inverting for the sound speed profile in the bottom, it can also invert for the sound speed profile in the water column. For simplicity, the sound speed profile in the water column was assumed to be known with a margin of error of ± 5 m/s. The range-dependent perturbation is added to the index of refraction squared n2(r), rather than the sound speed profile c(ro). The method is implemented in both Cartesian (x,y,z) and cylindrical (r,q,z) coordinates with the forward propagation of the field in x and r, respectively. Synthetic data are used to demonstrate the validity of the method [1]. Two inversion methods were combined, a Monte Carlo like algorithm, responsible for a starting approximation of the sound speed profile, and a steepest descent method, that fine-tuned the results. In simulations, the inversion algorithm is capable of inverting for the sound speed profile of a flat bottom. It was tested, for three different frequencies (50 Hz, 75 Hz, and 100 Hz), in a Pekeris waveguide, a range-independent layered medium, and a range-dependent medium, with errors in the inverted sound speed profile of less than 3%. Keywords: Three-dimensional parabolic equation method, geoacoustic inversion, range-dependent sound speed profile, linear inversion, Born approximation, Green’s functions. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection

The growing fields of spintronics and nanotechnology have created increased interest in developing the means to manipulate the spin of electrons. One such method arises from the combination of the spin-orbit interaction and the broken inversion symmetry that arises at surfaces and interfaces, and has prompted many recent investigations on metallic surfaces. A method by which surface states, in the absence of spin orbit effects, have been successfully investigated is the Green function embedding scheme of Inglesfield. This has been integrated into a self consistent FLAPW density functional framework based on the scalar relativistic K¨olling Harmon equation. Since the spin of the electron is a direct effect of special relativity, calculations involving the spin orbit interaction are best performed using solutions of the Dirac equation. This work describes the extension of Green’s function embedding to include the Dirac equation and how fully relativistic FLAPW surface electronic structure calculations are implemented. The general procedure used in performing a surface calculation in the scalar relativistic case is closely followed. A bulk transfer matrix is defined and used to generate the complex band structure and an embedding potential. This embedding potential is then used to produce a self consistent surface potential, leading to a Green’s function from which surface state dispersions and splittings are calculated. The bulk embedding potential can also be employed in defining channel functions and these provide a natural framework in which to explore transport properties. A relativistic version of a well known expression for the ballistic conductance across a device is derived in this context. Differences between the relativistic and nonrelativistic methods are discussed in detail. To test the validity of the scheme, a fully relativistic calculation of the extensively studied spin orbit split L-gap surface state on Au(111) is performed, which agrees well with experiment and previous calculations. Contributions to the splitting from different angular momentum channels are also provided. The main advantages of the relativistic embedding method are the full inclusion of the spin orbit interaction to all orders, the true semi infinite nature of the technique, allowing the full complex bands of the bulk crystal to be represented and the fact that a only small number of surface layers is needed in comparison to other existing methods.

<p>This Licentiate presents the main results of theoretical study of light propagation in photonic structures, namely lasing disk microcavities and photonic crystals. In the first two papers (Paper I and Paper II) we present the developed novel scattering matrix technique dedicated to calculation of resonant states in 2D disk microcavities with the imperfect surface or/and inhomogeneous refractive index. The results demonstrate that the imperfect surface of a cavity has the strongest impact on the quality factor of lasing modes.</p><p>The generalization of the scattering-matrix technique to the quantum-mecha- nical case has been made in Paper III. That generalization has allowed us to treat a realistic potential of quantum-corrals (which can be considered as nanoscale analogues of optical cavities) and to obtain a good agreement with experimental observations.</p><p>Papers IV and V address the novel effective Green's function technique for studying propagation of light in photonic crystals. Using this technique we have analyzed characteristics of surface modes and proposed several novel surface-state-based devices for lasing/sensing, waveguiding and light feeding applications.</p> / Report code: LIU-TEK-LIC 2006:5

Experiments shows that the conductance of a quantum point contact is quantized in steps of  2e²/h, where e is the charge of the electron and h is Planck’s constant, and thereby Ohm’s law is not valid for nanostructures. By using the approximation method finite difference, the transmission for one-dimensional contacts and one- and two-dimensional potentials are investigated. In the case of two-dimensional contacts and a two-dimensional potential the Green’s function method is used. It turns out that if electrons are treated as waves, the transmission and the conductance just differ by the constant 2e²/h, which in this thesis is interpreted numerically in Matlab by using the Green’s function method.

We discuss a relativistic transport theory of condensed matter based on a microscopic system containing bosonic and fermionic degrees of freedom interacting via 3 - and 4-point interactions. We use the Dyson hierarchy as a solution to the underlying field theory and truncate this hierarchy by parametrizing the 2-particle-irreducible kernels of the 4- and 5-point vertex functions. We then perform a complete crossing-symmetric reduction of the 2-particle intermediate states of the theory. We obtain a reduction hierarchy and show how to explore the quality of our truncation scheme using this reduction hierarchy. Finally we discuss the problem of regularization of the theory in the case of hadronic matter by either putting form factors directly in the action or by using dispersion relations to introduce causal form factors into singular diagrams. / Graduation date: 1992