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In this dissertation, we use exact diagonalization to study a few strongly correlated systems, ranging from the Fermi-Hubbard model to the fractional quantum Hall effect (FQHE). The discussion starts with an overview of strongly correlated systems and what is meant by strongly correlated. Then, we extend cluster perturbation theory (CPT), an economic method for computing the momentum and energy resolved Green's function for Hubbard models to higher order correlation functions, specifically the spin susceptibility. We benchmark our results for the one-dimensional Fermi-Hubbard model at half-filling. In addition we study the FQHE at fillings $nu = 5/2$ for fermions and $nu = 1/2$ for bosons. For the $nu = 5/2$ system we investigate a two-body model that effectively captures the three-body model that generates the Moore-Read Pfaffian state. The Moore-Read Pfaffian wave function pairs composite fermions and is believed to cause the FQHE at $nu = 5/2$. For the $nu = 1/2$ system we estimate the entropy needed to observe Laughlin correlations with cold atoms via an ansatz partition function. We find entropies achieved with conventional cooling techniques are adequate. / Doctor of Philosophy / Strongly correlated quantum many-body physics is a rich field that hosts a variety of exotic phenomena. By quantum many-body we mean physics that is concerned with the behavior of interacting particles, such as electrons, where the quantum behavior cannot be ignored. By strongly correlated, we mean when the interactions between particles are sufficiently strong such that they cannot be treated as a small perturbation. In contrast to weakly correlated systems, strongly correlated systems are much more difficult to solve. That is because methods that reduce the many-body problem to a single independent body problem do not work well. In this dissertation we use exact diagonalization, a method to computationally solve quantum many-body systems, to study two strongly correlated systems: the Hubbard model and the fractional quantum Hall effect.The Hubbard model captures the physics of many interesting materials and is the standard toy model. Originally developed with magnetic properties in mind, it has been extended to study superconductivity, topological phases, cold atoms, and much more. The fractional quantum Hall effect is a novel phase of matter that hosts exotic excitations, some of which may have applications to quantum computing.

Cette thèse porte sur l'étude des interactions faibles et de la spectroscopie vibrationnelle en phase condensée à partir d'un développement théorique basé sur la théorie de la perturbation de la fonctionnelle de densité. D'une part des corrections de la fonction d'onde Born-Oppenheimer ont été calculées pour déterminer le moment magnétique induit par les vibrations et ainsi calculer des spectres de dichroïsme circulaire vibrationnel. D'autre part, une modélisation des effets de polarisation est réalisée à l'aide d'une nouvelle représentation de la susceptibilité électronique non-locale. / This thesis deals with the development and application of computational methods for the efficient and accurate calculation of spectroscopic parameters and non-covalent inter-molecular interactions in condensed-phase systems from quantum chemical methods. Specifically, electronic current densities and polarizability effects are computed using density functional perturbation theory. The nuclear velocity perturbation theory is rigorously derived from the exact factorization of the electron-nuclear wave function. Its implementation within a large-scale electronic structure program package is reported and the calculation of dynamical vibrational circular dichroism in the condensed phase is demonstrated. A position-dependent mass of nuclei in molecules is derived, addressing the fundamental questions as to how masses move in a molecule. First steps towards a density-based modeling of inter-molecular interactions using a compact representation of the electronic susceptibility are devised.

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Numerical simulation is becoming increasingly used to support theoretical effort into understanding the turbulence problem. We develop theoretical ideas related to the transfer and dissipation of energy, which clarify long-standing issues with the energy balance in isotropic turbulence. These ideas are supported by results from large scale numerical simulations. Due to the large number of degrees of freedom required to capture all the interacting scales of motion, the increase in computational power available has only recently allowed flows of interest to be realised. A parallel pseudo-spectral code for the direct numerical simulation (DNS) of isotropic turbulence has been developed. Some discussion is given on the challenges and choices involved. The DNS code has been extensively benchmarked by reproducing well established results from literature. The DNS code has been used to conduct a series of runs for freely-decaying turbulence. Decay was performed from a Gaussian random field as well as an evolved velocity field obtained from forced simulation. Since the initial condition does not describe developed turbulence, we are required to determine when the field can be considered to be evolved and measurements are characteristic of decaying turbulence. We explore the use of power-law decay of the total energy and compare with the use of dynamic quantities such as the peak dissipation rate, maximum transport power and velocity derivative skewness. We then show how this choice of evolved time affects the measurement of statistics. In doing so, it is found that the Taylor dissipation surrogate, u^3 / L, is a better surrogate for the maximum inertial flux than dissipation. Stationary turbulence has also been investigated, where we ensure that the energy input rate remains constant for all runs and variation is only introduced by modifying the fluid viscosity (and lattice size). We present results for Reynolds numbers up to Rλ = 335 on a 1024^3 lattice. Using different methods of vortex identification, the persistence of intermittent structure in an ensemble average is considered and shown to be reduced as the ensemble size increases. The longitudinal structure functions are computed for smaller lattices directly from an ensemble of realisations of the real-space velocity field. From these, we consider the generalised structure functions and investigate their scaling exponents using direct analysis and extended self-similarity (ESS), finding results consistent with the literature. An exploitation of the pseudo-spectral technique is used to calculate second- and third-order structure functions from the energy and transfer spectra, with a comparison presented to the real-space calculation. An alternative to ESS is discussed, with the second-order exponent found to approach 2/3. The dissipation anomaly is then considered for both forced and free-decay. Using different choices of the evolved time for a decaying simulation, we show how the behaviour of the dimensionless dissipation coefficient is affected. The Karman-Howarth equation (KHE) is studied and a derivation of a work term presented using a transformation of the Lin equation. The balance of energy represented by the KHE is then investigated using the pseudo-spectral method mentioned above. The consequences of this new input term for the structure functions are discussed. Based on the KHE, we develop a model for the behaviour of the dimensionless dissipation coefficient that predicts Cɛ= Cɛ(∞)+CL/RL. DNS data is used to fit the model. We find Cɛ(∞) = 0.47 and CL = 19.1 for forced turbulence, with excellent agreement to the data. Theoretical methods based on the renormalization group and statistical closures are still being developed to study turbulence. The dynamic RG procedure used by Forster, Nelson and Stephen (FNS) is considered in some detail and a disagreement in the literature over the method and results is resolved here. An additional constraint on the loop momentum is shown to cause a correction to the viscosity increment such that all methods of evaluation lead to the original result found by FNS. The application of statistical closure and renormalized perturbation theory is discussed and a new two-time model probability density functional presented. This has been shown to be self-consistent to second order and to reproduce the two-time covariance equation of the local energy transfer (LET) theory. Future direction of this work is discussed.

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Cette thèse, ayant pour thème les algorithmes de la chimie quantique, s'inscrit dans le cade du changement de paradigme observé depuis une douzaines d'années, dans lequel les méthodes de calcul séquentielles se doivent d'être progressivement remplacées par des méthodes parallèles. En effet, l'augmentation de la fréquences des processeurs se heurtant à des barrières physiques difficilement franchissables, l'augmentation de la puissance de calcul se fait par l'augmentation du nombre d'unités de calcul. Toutefois, là où une augmentation de la fréquence conduisait mécaniquement à une exécution plus rapide d'un code, l'augmentation du nombre de cœurs peut se heurter à des barrières algorithmiques, qui peuvent nécessiter une adaptation ou un changement d'algorithme. Parmi les méthodes développées afin de contourner ce problème, on trouve en particulier celles de type Monte-Carlo (stochastiques), qui sont intrinsèquement "embarrassingly parallel", c'est à dire qu'elles sont par construction constituées d'une multitudes de tâches indépendantes, et de ce fait particulièrement adaptées aux architectures massivement parallèles. Elles ont également l'avantage, dans de nombreux cas, d'être capables de produire un résultat approché pour une fraction du coût calculatoire de l'équivalent déterministe exacte. Lors de cette thèse, des implémentations massivement parallèles de certains algorithmes déterministes de chimie quantique ont été réalisées. Il s'agit des algorithmes suivants : CIPSI, diagonalisation de Davidson, calcul de la perturbation au second ordre, shifted-Bk, et Coupled Cluster Multi Références. Pour certains, une composante stochastique a été introduite en vue d'améliorer leur efficacité. Toutes ces méthodes ont été implémentées sur un modèle de tâches distribuées en TCP, où un processus central distribue des tâches par le réseau et collecte les résultats. En d'autres termes, des nœuds esclaves peuvent être ajoutés au cours du calcul depuis n'importe quelle machine accessible depuis internet. L'efficacité parallèle des algorithmes implémentés dans cette thèse a été étudiée, et le programme a pu donner lieu à de nombreuses applications, notamment pour permettre d'obtenir des énergies de références pour des systèmes moléculaires difficiles. / This thesis, whose topic is quantum chemistry algorithms, is made in the context of the change in paradigm that has been going on for the last decade, in which the usual sequential algorithms are progressively replaced by parallel equivalents. Indeed, the increase in processors' frequency is challenged by physical barriers, so increase in computational power is achieved through increasing the number of cores. However, where an increase of frequency mechanically leads to a faster execution of a code, an increase in number of cores may be challenged by algorithmic barriers, which may require adapting of even changing the algorithm. Among methods developed to circumvent this issue, we find in particular Monte-Carlo methods (stochastic methods), which are intrinsically "embarrassingly parallel", meaning they are by design composed of a large number of independent tasks, and thus, particularly well-adapted to massively parallel architectures. In addition, they often are able to yield an approximate result for just a fraction of the cost of the equivalent deterministic, exact computation. During this thesis, massively parallel implementations of some deterministic quantum chemistry algorithms were realized. Those methods are: CIPSI, Davidson diagonalization, computation of second-order perturbation, shifted-Bk, Multi-Reference Coupled-Cluster. For some of these, a stochastic aspect was introduced in order to improve their efficiency. All of them were implemented on a distributed task model, with a central process distributing tasks and collecting results. In other words, slave nodes can be added during the computation from any location reachable through Internet. The efficiency for the implemented algorithms has been studied, and the code could give way to numerous applications, in particular to obtain reference energies for difficult molecular systems.