La déformation plastique des minéraux des roches mantelliques joue un rôle-clé dans les phénomènes de convection globale à l’intérieur de la Terre. Cependant, les mécanismes de déformation des minéraux sont encore mal compris. Généralement, la déformation plastique est assurée par le mouvement des dislocations qui sont des défauts linéaires du cristal. La mobilité des dislocations est déterminée par la structure de coeur des défauts. Le modèle de Peierls-Nabarro auquel nous avons recours est un concept fondamental qui décrit la résistance qu’oppose le cristal au mouvement des dislocations au travers du calcul de la contrainte de Peierls et qui fournit également l’étalement du coeur de la dislocation dans le cristal. Développé il y a plus d'un demi-siècle, ce modèle connut un regain d’intérêts dans les années 70 avec l’introduction du concept de fautes d’empilement généralisées qui permet une description plus générale des coeurs de dislocations. La détermination de ces fautes consiste à calculer ab initio ou à l’aide de potentiels empiriques des barrières d’énergie associées au cisaillement d’un plan cristallographique donné dans une, deux ou toutes les directions. L’étude menée durant cette thèse s’est portée sur la détermination des structures de coeur des dislocations de la phase post-perovskite présente à la base du manteau mais également de la wadsleyite, de la forsterite et du diopside, trois minéraux majeurs entre la croûte et 520km de profondeur. L’utilisation du modèle de Peierls-Nabarro unidimensionnel dans un premier temps a été étendue aux trois dimensions de l’espace dans le but de modéliser des structures de coeur de dislocations de plus en plus complexes. / Plastic deformation of minerals in rocks plays an important role in the global convection of the Earth’s mantle. However, the deformation mechanisms of minerals are not well-known. In general, plastic deformation is due to the dislocation motion. The dislocations are linear defects of the crystal. The mobility of the dislocation is governed by the dislocation core structure.Here, we used the Peierls-Nabarro model that is a theoretical model to address the issue of dislocation core modelling. This model provides also the lattice friction of the crystal against the dislocation motion through the calculation of the Peierls stress. Developed more than 50 years, the model has a great interest since the 70’s with the introduction of the generalized stacking fault (GSF) concept that provides a general description of the dislocation core. The determination of the GSF consists in the calculations of energy barriers associated with the shearing of the crystal in a crystallographic plane in one or more directions. In this study, we have determined the dislocation core structures of the post-perovskite phase present at the core-mantle boundary and of the wadsleyite, diopside and forsterite, major minerals from the crust to 510Km deep. We used the one-dimension formulation of the Peierls-Nabarro model in a first time that has been extended to three dimensions in order to model much more complex dislocation cores.

This thesis presents studies of the low energy properties of nseveral frustrated spin-1/2 Heisenberg antiferromagnets using various analytic and computational methods. The models studied include the union jack model, the alternating Heisenberg chain, the Heisenberg bilayer model, and the spin-Peierls model. The union jack model is a Heisenberg antiferromagnetic spin model with frustration, and is analyzed using spin-wave theory. For small values of the frustrating coupling $\alpha$, the system is N{\' e}el ordered, while for large $\alpha$ the frustration is found to induce a canted phase. Spin wave theory with second order corrections finds the critical coupling at $\alpha \simeq 0.645$,which agrees quantitatively with series expansion results. No intermediate spin-liquid phase is found to exist between the two phases. The alternating Heisenberg chain is studied using an alternative triplet-wave expansion formalism for dimerized spin systems, modification of the ??bond operator?? formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. Comparisons are made with the results of dimer series expansions and exact diagonalization. The S=1/2 Heisenberg bilayer spin model at zero temperature is studied in the dimerized phase using analytic triplet-wave expansions and dimer series expansions. The occurrence of two-triplon bound states in the S=0 and S=1 channels, and antibound states in the S=2 channel, is predicted with triplet-wave theory and confirmed by series expansions. All bound states are found to vanish at or before the critical coupling separating the dimerized phase from the N{\' e}el phase. The critical behavior of the total and single-particle static transverse structure factors is also studied by series expansion methods and found to conform with theoretical expectations. The Heisenberg spin-Peierls model with dispersive, gapless phonons is studied with Density Matrix Renormalization Group methods. We investigate the zero temperature properties of the model using the crossover method. The calculations were found to converge poorly and no conclusive results could be found using this method. An analysis of the convergence problems and the discovery of an anomalous triplet ground state is presented in this chapter.

This thesis presents studies of the low energy properties of nseveral frustrated spin-1/2 Heisenberg antiferromagnets using various analytic and computational methods. The models studied include the union jack model, the alternating Heisenberg chain, the Heisenberg bilayer model, and the spin-Peierls model. The union jack model is a Heisenberg antiferromagnetic spin model with frustration, and is analyzed using spin-wave theory. For small values of the frustrating coupling $\alpha$, the system is N{\' e}el ordered, while for large $\alpha$ the frustration is found to induce a canted phase. Spin wave theory with second order corrections finds the critical coupling at $\alpha \simeq 0.645$,which agrees quantitatively with series expansion results. No intermediate spin-liquid phase is found to exist between the two phases. The alternating Heisenberg chain is studied using an alternative triplet-wave expansion formalism for dimerized spin systems, modification of the ??bond operator?? formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. Comparisons are made with the results of dimer series expansions and exact diagonalization. The S=1/2 Heisenberg bilayer spin model at zero temperature is studied in the dimerized phase using analytic triplet-wave expansions and dimer series expansions. The occurrence of two-triplon bound states in the S=0 and S=1 channels, and antibound states in the S=2 channel, is predicted with triplet-wave theory and confirmed by series expansions. All bound states are found to vanish at or before the critical coupling separating the dimerized phase from the N{\' e}el phase. The critical behavior of the total and single-particle static transverse structure factors is also studied by series expansion methods and found to conform with theoretical expectations. The Heisenberg spin-Peierls model with dispersive, gapless phonons is studied with Density Matrix Renormalization Group methods. We investigate the zero temperature properties of the model using the crossover method. The calculations were found to converge poorly and no conclusive results could be found using this method. An analysis of the convergence problems and the discovery of an anomalous triplet ground state is presented in this chapter.

University, Diss., 1999--Dortmund. / Dateiformat: PDF.

Th. 3e cycle--Phys. de la matière et du rayonnement--Grenoble 1, 1983. N°: 51.

<p> As the first inorganic spin-Peierls compound, CuGe03 can be described with an AF Heisenberg model. The magnetization process of CuGe03 is studied using the numerical Density Matrix Renormalization Group (DMRG) technique. The M - H curve of CuGe03 and other one dimensional Heisenberg systems are described, and their different nature are analyzed. In particular, the middle field cusp singularity appears in the M- H curve of CuGe03 is described. </p> / Thesis / Master of Science (MSc)

A aproximação da densidade local para a teoria do funcional da densidade tem, no passado, levado a resultados conflitantes para a dimerização de cadeias infinitas do trans-poliacetileno (trans-PA). Estes resultados variaram deste forte dimerização, próximo dos resultados experimentais, até fraca dimerização, ou até mesmo nenhuma dimerização. Desde que a aproximação da densidade local usualmente descreve transições de fase estruturais corretamente, esta situação insatisfatória no caso do trans-PA clama por uma investigação detalhada. Neste trabalho estudamos o problema descrevendo a molécula como um modelo de Hubbard-Peierls unidimensional. Estabelecemos uma teoria do funcional da densidade e construímos uma aproximação da densidade local para este modelo. Em acordo com os cálculos ab initio encontramos que este tipo de aproximação não descreve adequadamente a dimerização. Propomos então um Formalismo alternativo, baseado em funcionais da matriz-densidade. Neste formalismo a energia de troca-correlação é escrita como um funcional da densidade de carga (como no método tradicional) e do parâmetro de ordem para a transição. Desta maneira obtemos um tratamento aperfeiçoado para a fase dimerizada. Nossos resultados sugerem que uma descrição adequada de sistemas espacialmente correlacionados dentro da teoria do funcional da densidade requer uma nova classe de funcionais, que vai além da aproximação da densidade local por levar explicitamente em conta parâmetros de ordem espaciais de longo alcance. / The local-density approximation to density-functional theory has, in the past, lead to conflicting results for the dimerization of infinite trans-polyacetylene (trans-PA) chains. These results range from strong dimerization, close to the one observed experimentally, to weak dimerization, or even no dimerization at all. Since the local-density approximation usually describes structural phase transitions correctly, this unsatisfactory situation in the case of trans-PA calls for a detailed investigation. In this work we study the problem by describing the polyacetylene molecule as a one-dimensional Hubbard-Peierls model. We set up a density-functional theory for this model, and construct a local-density approximation. In agreement with the ab initio calculations we find that this type of approximation does not consistently describe the dimerization. We therefore propose an alternative formalism, based on density-matrix functionals. In this formalism the exchange-correlation energy is written as a functional of the charge density (as in the traditional method) and the order parameter for the dimerization transition. In this way we achieve an improved treatment of the dimerized phase. Our results suggest that a reliable description of spatially correlated systems within density-functional theory requires a new type of functionals, going beyond the local-density approximation by explicitly accounting for long-range spatial order.

A aproximação da densidade local para a teoria do funcional da densidade tem, no passado, levado a resultados conflitantes para a dimerização de cadeias infinitas do trans-poliacetileno (trans-PA). Estes resultados variaram deste forte dimerização, próximo dos resultados experimentais, até fraca dimerização, ou até mesmo nenhuma dimerização. Desde que a aproximação da densidade local usualmente descreve transições de fase estruturais corretamente, esta situação insatisfatória no caso do trans-PA clama por uma investigação detalhada. Neste trabalho estudamos o problema descrevendo a molécula como um modelo de Hubbard-Peierls unidimensional. Estabelecemos uma teoria do funcional da densidade e construímos uma aproximação da densidade local para este modelo. Em acordo com os cálculos ab initio encontramos que este tipo de aproximação não descreve adequadamente a dimerização. Propomos então um Formalismo alternativo, baseado em funcionais da matriz-densidade. Neste formalismo a energia de troca-correlação é escrita como um funcional da densidade de carga (como no método tradicional) e do parâmetro de ordem para a transição. Desta maneira obtemos um tratamento aperfeiçoado para a fase dimerizada. Nossos resultados sugerem que uma descrição adequada de sistemas espacialmente correlacionados dentro da teoria do funcional da densidade requer uma nova classe de funcionais, que vai além da aproximação da densidade local por levar explicitamente em conta parâmetros de ordem espaciais de longo alcance. / The local-density approximation to density-functional theory has, in the past, lead to conflicting results for the dimerization of infinite trans-polyacetylene (trans-PA) chains. These results range from strong dimerization, close to the one observed experimentally, to weak dimerization, or even no dimerization at all. Since the local-density approximation usually describes structural phase transitions correctly, this unsatisfactory situation in the case of trans-PA calls for a detailed investigation. In this work we study the problem by describing the polyacetylene molecule as a one-dimensional Hubbard-Peierls model. We set up a density-functional theory for this model, and construct a local-density approximation. In agreement with the ab initio calculations we find that this type of approximation does not consistently describe the dimerization. We therefore propose an alternative formalism, based on density-matrix functionals. In this formalism the exchange-correlation energy is written as a functional of the charge density (as in the traditional method) and the order parameter for the dimerization transition. In this way we achieve an improved treatment of the dimerized phase. Our results suggest that a reliable description of spatially correlated systems within density-functional theory requires a new type of functionals, going beyond the local-density approximation by explicitly accounting for long-range spatial order.

The determination of the phase boundary between the charge density wave and the Luttinger Liquid phase for the one-dimensional Peierls extended Hubbard model is done using Stochastic Series Expansion and comparison is done with the phase boundary for the Extended Hubbard model. The result of adding an electron-phonon interaction is that the charge density wave phase weakens. par The energy autocorrelation time is reported for the Extended Hubbard Model and the Peierls Hubbard Model. For coupling near and above the critical coupling the autocorrelation time increases exponentially. par Also investigated is the presence of a spin gap and the critical value of phonon coupling with respect to two parameters, (1) the bare phonon frequency and (2) different U, at one-quarter filling.

Stuttgart, Univ., Diss., 2005.