Cette thèse porte principalement sur l'étude de modèles de fermions en interactions contenant un couplage spin-orbite. Ces modèles (i) peuvent décrire une classe de matériaux composés d'iridates sur le réseau en nid d'abeille ou (ii) pourraient être réalisés artificiellement dans des systèmes d’atomes froids. Nous avons étudié, dans un premier temps, le système à demi-remplissage avec l'interaction de Hubbard et un couplage spin-orbite anisotrope. Nous avons trouvé plusieurs phases: la phase isolant topologique pour de faibles corrélations, et deux phases avec des ordres magnétiques frustrés, l'ordre de Néel et l'ordre spiral, dans la limite de très fortes corrélations. La transition entre les régimes de faibles et de fortes corrélations est une transition de Mott dans laquelle les excitations électroniques se fractionnent en excitations de charge et de spin. Les charges sont localisées par l'interaction. Le secteur de spin présente de fortes fluctuations qui sont modélisées par un gaz d’instantons. Nous avons ensuite exploré la physique d'un système régi au demi-remplissage par le modèle de Kitaev-Heisenberg, qui présente une phase magnétique de type zig-zag. En dopant le système, autour du quart remplissage, la structure de bande présente de nouveaux centres de symétrie en plus de la symétrie d'inversion. Le couplage de spin de Kitaev-Heisenberg favorise alors la formation de paires de Cooper dans un état triplet autour de ces centres de symétrie. La condensation de ces paires de Cooper autour de ces vecteurs d'onde non triviaux se manifeste par une modulation spatiale du paramètre d'ordre supraconducteur, comme dans la supraconductivité de Fulde–Ferrell–Larkin–Ovchinnikov (FFLO). La dernière partie de la thèse propose et étudie une implémentation des phases topologiques dite de Haldane et de Kane-Mele dans un système avec deux espèces de fermions sur le réseau en nid d'abeille, stabilisée grâce à l’interaction RKKY médiée par l’espèce rapide et qui agit sur l’espèce lente. / This thesis is dedicated largely to the study of theoretical models describing interacting fermions with a spin-orbit coupling. These models (i) can describe a class of 2D iridate materials on the honeycomb lattice or (ii) could be realized artificially in ultra-cold gases in optical lattices. We have studied, in the first part, the half-filled honeycomb lattice model with on-site Hubbard interaction and anisotropic spin-orbit coupling. We find several different phases: the topological insulator phase at weak coupling, and two frustrated magnetic phases, the Néel order and spiral order, in the limit of strong correlations. The transition between the weak and strong correlation regimes is a Mott transition, through which electrons are fractionalized into spins and charges. Charges are localized by the interactions. The spin sector exhibits strong fluctuations which are modeled by an instanton gas. Then, we have explored a system described by the Kitaev-Heisenberg spin Hamiltonian at half-filling, which exhibits a zig-zag magnetic order. While doping the system around the quarter filling, the band structure presents novel symmetry centers apart from the inversion symmetry point. The Kitaev-Heisenberg coupling favors the formation of triplet Cooper pairs around these new symmetry centers. The condensation of these pairs around these non-trivial wave vectors is manifested by the spatial modulation of the superconducting order parameter, by analogy to the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductivity. The last part of the thesis is dedicated to an implementation of the Haldane and Kane-Mele topological phases in a system composed of two fermionic species on the honeycomb lattice. The driving mechanism is the RKKY interaction induced by the fast fermion species on the slower one.

Motivated by the recent experimental progress in the search for Majorana fermions, we identify signatures of topological superconductivity and propose realistic experiments to observe these signatures. In the first part of this thesis, we study charge transport through a topological superconductor with a pair of Majorana end states, coupled to leads via quantum dots with resonant levels. The nonlocality of the Majorana bound states opens the possibility of Cooper pair splitting with nonlocal shot noise. In the space of quantum dot energy levels, we find a characteristic four-peaked cloverlike pattern for the strength of noise due to Cooper pair splitting, distinct from the single ellipsoidal peak found in the absence of Majorana end states. Semiconductor-superconductor hybrid systems are promising candidates for the realiza- tion Majorana fermions and topological order in solid state devices. In the second part, we show that the topological order is mirrored in the excitation spectra and can be observed in nonlinear Coulomb blockade transport through a ring-shaped nanowire. Especially, the ex- citation spectrum is almost independent of magnetic flux in the topologically trivial phase but acquires a characteristic h/e magnetic flux periodicity in the nontrivial phase. The transition between the trivial and nontrivial phase is reflected in the closing and reopening of an excitation gap. In the third part, we investigate characteristic features in the spin response of doped three-dimensional topological insulators with odd-parity unequal-spin superconducting pairing, which are predicted to have gapless Majorana surface modes. These Majorana modes contribute to the spin response, giving rise to a characteristic temperature behavior of the Knight shift and the spin-lattice relaxation time in magnetic resonance experiments.

[pt] Nesta tese estudamos dois importantes sistemas de Isoladores Topológicos (TIs), onde nos concentramos particularmente no papel das interações e percolação nos estados de borda topológicos. Primeiro, analisamos o papel das interações vizinhas mais próximas em um protótipo de TI unidimensional, o modelo Su-Schrieffer-Heeger (SSH). Com base em um formalismo de função de Green, aplicamos a equação de Dyson em combinação com a aproximação da matriz-T para verificar a correspondência bulk-edge na presença de interações. Os expoentes críticos próximos às transições de fase topológicas são os mesmos do modelo SSH não interagente, indicando que o sistema permanece na mesma classe de universalidade, apesar da presença de interações. O segundo sistema é um TI bidimensional simétrico na inversão de tempo, ou seja, o modelo de Bernevig-Hughes-Zhang (BHZ) em conjunto com um metal ferromagnético com quebra de reversão do tempo (FMM), onde investigamos a percolação do estado Hall de spin quântico do modelo BHZ para o FMM por meio de um modelo de ligações fortes (tight-binding). Demonstramos que dependendo de se o estado de borda do cone de Dirac submerge nas sub-bandas do FMM e da direção da magnetização do FMM, a percolação do estado de borda e seu spin-momentum-locking são afetados de maneiras diferentes. Surpreendentemente, descobrimos que a corrente de spin de borda de equilíbrio no modelo BHZ, naturalmente esperada dos estados de borda de propagação do spin polarizado, está de fato ausente devido ao cancelamento das bandas de valência. No entanto, fluxos laminares de correntes de carga e spin persistente à temperatura ambiente são produzidos perto da interface da junção BHZ / FMM. Usando teoria de resposta linear, investigamos a polarização de spin induzida pela corrente causada pela percolação do estado de borda, que serve como um torque de rotação que é encontrado ser predominantemente field-like. Além disso, a polarização do spin é dramaticamente aumentada perto das impurezas na borda do modelo BHZ. / [en] In this thesis we studied two important Topological Insulators (TIs), where we focused particularly on the role of interactions and percolation on the topological edge states. First, we analyzed the role of nearest-neighbor interactions in a prototype one-dimensional TI, namely the Su-Schrieffer-Heeger (SSH) model. Based on a Green s function formalism, we applied Dyson s equation in combination with T-matrix approximation to verify the bulk-edge correspondence in the presence of interactions. The critical exponents near topological phase transitions are found to be the same as the noninteracting SSH model, indicating that the system stays in the same universality class despite the presence of interactions. The second system is a two-dimensional timereversal symmetric TI, namely the Bernevig-Hughes-Zhang (BHZ) model in conjunction with a time-reversal breaking ferromagnetic metal (FMM), where we investigated the percolation of the quantum spin Hall state from the TI layer to the FMM by means of a tight-binding model. We demonstrated that depending on whether the edge state Dirac cone submerges into the FMM subbands and the direction of the magnetization of the FMM, the percolation of the edge state and its spin-momentum locking are affected in different ways. Surprisingly, we uncover that the equilibrium edge spin current in the BHZ model, naturally expected from the spin polarized propagating edge states, is in fact absent due to the cancellation from the valence bands. Nevertheless, laminar flows of room temperature persistent charge and spin currents are produced near the interface of the BHZ/FMM junction. Using a linear response theory, we investigate the current-induced spin polarization caused by the percolation of the edge state, which serves as a spin torque that is found to be predominantly field-like. Moreover, the spin polarization is dramatically enhanced near the impurities at the edge of the BHZ model.

[pt] A investigação da geometria quântica em semicondutores e isoladores tornou-se significativa devido às suas implicações nas características dos materiais. A noção de geometria quântica surge considerando a métrica quântica do estado de Bloch da banda de valência, que é definido a partir da sobreposição dos estados de Bloch em momentos ligeiramente diferentes. Ao integrar a métrica quântica em toda a zona de Brillouin, introduzimos uma quantidade que chamamos de número de fidelidade, que significa a distância média entre estados de Bloch adjacentes. Além disso, apresentamos um formalismo para expressar o número de fidelidade como um marcador de fidelidade local no espaço real que pode ser definido em qualquer sítio da rede. O marcador pode ser calculado diretamente diagonalizando o hamiltoniano da rede que descreve o comportamento das partículas na rede. Posteriormente, o conceito de número e marcador de fidelidade é estendido para temperatura finita utilizando a teoria de resposta linear, conectando-os a medições experimentais que envolvem analisar o poder de absorção óptica global e local quando o material é exposto à luz linearmente polarizada. Particularmente para materiais bidimensionais, a opacidade do material permite a determinação direta do número de fidelidade espectral, permitindo a detecção experimental do número de fidelidade. Finalmente, um marcador de fidelidade não local é introduzido considerando a divergência da métrica quântica. Este marcador é postulado como um indicador universal de transições de fase quântica, assumindo que o momento cristalino permanece um número quântico válido. Este marcador não local pode ser interpretado como uma função de correlação dos estados de Wannier, que são funções de onda localizadas que descrevem estados eletrônicos em um cristal. A generalidade e aplicabilidade destes conceitos são demonstradas através da investigação de vários isoladores topológicos e transições de fase topológicas em diferentes dimensões. Essas descobertas elaboram o significado dessas quantidades e sua conexão com vários fenômenos fundamentais na física da matéria condensada. / [en] The investigation of quantum geometry in semiconductors and insulators has become significant due to its implications for material characteristics. The notion of quantum geometry arises by considering the quantum metric of the valence-band Bloch state, which is defined from the overlap of the Bloch states at slightly different momenta. By integrating the quantum metric through-out the Brillouin zone, we introduce a quantity that we call fidelity number, which signifies the average distance between adjacent Bloch states. Furthermore, we present a formalism to express the fidelity number as a local fidelity marker in real space that can be defined on every lattice site. The marker can be calculated directly by diagonalizing the lattice Hamiltonian that describes particle behavior on the lattice. Subsequently, the concept of the fidelity number and marker is extended to finite temperature using linear-response theory, connecting them to experimental measurements which involves analyze the global and local optical absorption power when the material is exposed to linearly polarized light. Particularly for two-dimensional materials, the material s opacity enables straightforward determination of the fidelity number spectral, allowing for experimental detection of the fidelity number. Finally, a nonlocal fidelity marker is introduced by considering the divergence of the quantum metric. This marker is postulated as a universal indicator of quantum phase transitions, assuming the crystalline momentum remains a valid quantum number. This nonlocal marker can be interpreted as a correlation function of Wannier states, which are localized wave functions describing electronic states in a crystal. The generality and applicability of these concepts are demonstrated through the investigation of various topological insulators and topological phase transitions across different dimensions. These findings elaborate the significance of these quantities and their connection to various fundamental phenomena in condensed matter physics.

Les signatures de l’existence des phonons sont omniprésentes dans les propriétés des matériaux. En première approximation, on peut scinder l'effet des phonons sur la structure électronique en deux contributions. D’une part, l'interaction électron-phonon capture la réponse électronique aux vibrations des noyaux du cristal, et d’autre, l'énergie libre de la population de phonons modifie le volume cristallin à l’équilibre. En plus d'être responsables de la dépendance en température de la structure électronique, ces deux mécanismes affectent les niveaux d'énergie à température nulle, à travers le mouvement du point zéro et l'énergie du point zéro. Cette thèse analyse l’apport de ces deux mécanismes à la renormalisation du point zéro (ZPR) de l'énergie de la bande interdite des semi-conducteurs. Une généralisation du modèle de Fröhlich prenant en compte l'anisotropie et les dégénérescences présentes dans les matériaux réels révèle que l'interaction non adiabatique entre les électrons et les noyaux domine le ZPR dans les matériaux polaires. La prise en compte de ce mécanisme dans l'évaluation de l'interaction électron-phonon est déterminante pour reproduire adéquatement les données expérimentales. L'approche développée par Grüneisen, qui néglige communément les effets du point zéro, reproduit la dilatation du point zéro du réseau (ZPLE) et sa contribution au ZPR obtenues avec la méthode standard basée sur la minimisation de l'énergie libre à moindre coût numérique, y compris pour les matériaux anisotropes. La contribution du ZPLE au ZPR total, qui a reçu peu d'attention dans la littérature, peut atteindre de 20% à plus de 80% de la contribution de l'interaction électron-phonon, y compris dans des matériaux constitués de noyaux légers. Elle domine même le ZPR du GaAs dans le contexte de la DFT semi-locale. Il est donc essentiel de traiter les deux contributions sur le même pied d'égalité pour modéliser le ZPR avec précision. L'inclusion du couplage spin-orbite (SOC) diminue le ZPR d'un ensemble substantiel de matériaux cubiques de structure zinc-blende, diamant et rock-salt. L'essentiel de cette variation tire son origine de l'effet du SOC sur les énergies électroniques statiques, qui provient en grande partie de la variation des masses effectives des bandes de valence au point \(\Gamma\). La réduction du ZPR peut être estimée à partir d'un modèle de Fröhlich généralisé auquel on a introduit le SOC. Les subtilités numériques liées au traitement de la séparation de Dresselhaus dans les matériaux non centrosymétriques sont discutées. On démontre enfin comment l'effet combiné de l'interaction électron-phonon et de la dilatation thermique affecte le diagramme de phase topologique du BiTeI. L'augmentation de la température repousse l'apparition de la phase d'isolant topologique \(\mathbb{Z}_2\) vers des pressions plus élevées et élargit la plage de pressions correspondant à la phase intermédiaire de type semi-métal de Weyl. Le caractère orbital dominant des extrema de bande influence significativement leur sensibilité à la pression et au changement de topologie. Pour guider la recherche expérimentale de phases topologiquement non triviales dans les matériaux de façon adéquate, les études numériques doivent donc considérer l'effet de la température. / Phonon signatures are ubiquitous in material properties. At first order, the effect of phonons on the electronic structure can be split into two contributions. On the one hand, the electron-phonon interaction captures the electronic response to the vibrations of the nuclei. On the other hand, the free energy of the phonon population modifies the crystalline volume at equilibrium. In addition to driving the temperature dependence of the electronic structure, these two mechanisms affect the energy levels at zero temperature through zero-point motion and zero-point energy. This thesis investigates the contribution of these two mechanisms to the zero point renormalization (ZPR) of the band gap energy of semiconductors. A generalized Fröhlich model taking into account the anisotropy and degeneracies occurring in real materials reveals that the non-adiabatic interaction between electrons and nuclei dominates the ZPR in polar materials. Taking this mechanism into account when evaluating the electron-phonon interaction is crucial to reproduce experimental data adequately. The Grüneisen formalism, which commonly neglects zero-point effects, reproduces the zero-point lattice expansion (ZPLE) and its contribution to the ZPR obtained from the standard method based on free energy minimization at lower numerical cost, including for anisotropic materials. The ZPLE contribution to the total ZPR, which has received little attention in the literature, can reach from 20% to more than 80% of the contribution of the electron-phonon interaction, including in materials containing light atoms. It even dominates the ZPR of GaAs within semilocal DFT. Therefore, both contributions should be treated on an equal footing to model the ZPR accurately. The inclusion of spin-orbit coupling (SOC) decreases the ZPR of a substantial set of cubic materials of zincblende, diamond and rocksalt structure. This variation originates mostly from the effect of SOC on the static electronic eigenvalues, which comes largely from the variation of the effective masses of the valence bands at the \(\Gamma\) point. The reduction of the ZPR can be estimated from a generalized Fröhlich model in which SOC has been introduced. Numerical subtleties related to the treatment of Dresselhaus separation in non-centrosymmetric materials are discussed. We finally show how the combination of electron-phonon interaction and thermal expansion affects the topological phase diagram of BiTeI. An increase in temperature pushes the \(\mathbb{Z}_2\) topological insulator phase towards higher pressures and widens the pressure range corresponding to the Weyl semi-metal intermediate phase. The leading orbital character of the band extrema significantly influences their sensitivity to variations in pressure and topology. To adequately guide the experimental search for topologically non-trivial phases in materials, numerical studies must therefore consider the effect of temperature.

Le développement d’un médicament pour une cible thérapeutique donnée passe par plusieurs étapes qui se résument en une étape de criblage, une phase préclinique et plusieurs phases cliniques. Ces étapes permettent de sélectionner une substance active et de démontrer son efficacité thérapeutique et sa sécurité toxicologique. Ces deux critères définissent la qualité du médicament qui, une fois démontrée, doit être garantie pendant toute sa durée de validité. La qualité est évaluée au moyen d’études de stabilité qui sont réalisées d’abord sur la matière première de la substance active au cours de la phase de pré-développement du médicament, ensuite sur le produit fini. La stabilité intrinsèque de la substance active concerne à la fois ses propriétés chimiques et ses propriétés physiques qui sont liées à la nature de la substance. L’étude de stabilité repose d’abord sur la caractérisation de ces propriétés, et ensuite sur l’étude de la sensibilité de la substance à l’égard des facteurs environnementaux pouvant modifier les propriétés intrinsèques de la substance. L’approche adoptée dans ce travail repose d’une part sur l’évaluation de la stabilité chimique c’est à dire de la réactivité chimique des substances à usage pharmaceutique au travers des études de pureté chimique et des études de dégradation forcée de ces substances en solution, et d’autre part, sur l’évaluation de la stabilité physique. Dans ce cadre, l’étude du polymorphisme cristallin revêt une grande importance, tout comme l’aptitude à la formation d’hydrates ou de solvates. Cette étude, basée sur la thermodynamique, consiste pour l’essentiel à construire un diagramme de phases pression-température permettant de définir les domaines de stabilité relative des différentes formes cristallines. Cinq substances actives, existant à l’état solide et entrant dans la composition de médicaments administrés par voie orale, ont été étudiées dans le cadre de ce travail. L’analyse chimique du tienoxolol, présentant un effet anti-hypertenseur, a montré qu’il est très sensible à l’hydrolyse et à l’oxydation. Sept produits de dégradation ont été identifiés pour ce produit dont un schéma probable de fragmentation a été établi. Des diagrammes de phases pression-température ont été construits pour le bicalutamide et le finastéride, médicaments du cancer de prostate, en utilisant une approche topologique basée simplement sur les données disponibles dans la littérature. Cette étude a montré que la relation thermodynamique (énantiotropie ou monotropie) entre les formes cristallines sous conditions ordinaires peut être modifiée en fonction de la température et de la pression. Ce résultat est important pour la production des médicaments car il montre comment une telle information peut être obtenue par des mesures simples et accessibles aux laboratoires de recherche industrielle, sans que ces derniers soient contraints d’expérimenter sous pression. La méthode topologique de construction de diagramme de phases a été validée ensuite en la comparant à une méthode expérimentale consistant à suivre, par analyse thermique, des transitions de phases en fonction de la pression. La méthode expérimentale a été appliquée à deux composés, la benzocaine, anesthésique local, et le chlorhydrate de cystéamine, médicament utilisé pour les cystinoses. Les deux formes étudiées de benzocaine présentent une relation énantiotrope qui se transforme en relation monotrope à haute pression. Une nouvelle forme cristalline (forme III) du chlorhydrate de cystéamine a été découverte au cours de ce travail. La relation thermodynamique entre cette forme III et la forme I est énantiotrope dans tout le domaine de température et de pression. De plus, le chlorhydrate de cystéamine, classé hygroscopique, a fait l’objet d’une étude quantitative de sa sensibilité à l’eau, montrant qu’il devient déliquescent sans formation préalable d’hydrate (...) / The development of a drug for a given therapeutic target requires several steps, which can be summarized by drug screening, a preclinical phase and a number of clinical phases. These steps allow the selection of an active substance and a verification of its therapeutic efficacy and toxicological safety. The latter two criteria define the quality of the drug, which once demonstrated, must be guaranteed throughout its shelf life. Quality is assessed through stability studies that are carried out with the raw material of the active substance (preformulation phase) and with the final product. The intrinsic stability of the active substance depends on its chemical and physical properties and their characterization is the core of the stability studies, which in addition consists of sensitivity studies of the active pharmaceutical ingredient (API) for environmental factors that can modify the intrinsic properties of the substance. The approach presented in this work is based on the one hand on the assessment of the chemical stability, i.e. the reactivity of APIs through chemical purity studies and forced degradation in solution, and on the other hand on the assessment of the physical stability. For the latter, crystalline polymorphism is of great importance, as is the ability of the API to form hydrates or solvates. The study of crystalline polymorphism is based on the construction of pressure-temperature phase diagrams in accordance with thermodynamic requirements leading to the stability condition domains of the different crystalline forms. The stability behavior of five APIs used or meant for oral applications has been studied as part of this work. The chemical analysis of tienoxolol, an antihypertensive drug, has demonstrated its sensitivity for hydrolysis and oxidation. Seven degradation products were identified and patterns of fragmentation have been established. Pressure-temperature phase diagrams have been constructed for bicalutamide and finasteride, drugs against prostate cancer, using a topological approach based on data available in the literature. The study demonstrates that the thermodynamic relationship (enantiotropy or monotropy) between crystalline forms under ordinary conditions can change depending on the pressure. This is important for drug development as it demonstrates how stability information can be obtained by standard laboratory measurements accessible to industrial research laboratories without the necessity to carry out experiments under pressure. The topological approach for the construction of phase diagrams has subsequently been validated by measuring transition temperatures as a function of pressure. Experiments have been carried out with benzocaine, a local anesthetic, and with cysteamine hydrochloride, a drug used against cystinosis. Two crystalline forms were observed in the case of benzocaine. They exhibit an enantiotropic relationship that becomes monotropic at high pressure. For cysteamine hydrochloride, a new crystalline form (form III) was discovered. The thermodynamic relationship between the new form III and the known form I is enantiotropic for the entire temperature and pressure range. Cysteamine hydrochloride’s sensitivity to water has been studied, as it is hygroscopic. It has been demonstrated that it becomes deliquescent in the presence of water and no trace of a hydrate has been found. Finally, a study combining thermal and chromatographic methods showed that, under the effect of temperature, cysteamine hydrochloride turns into cystamine in the solid as well as in the liquid state, The latter is known to be an important impurity of cysteamine hydrochloride. In conclusion, the approach developed in this work allowed to characterize the stability properties of a number of APIs and to determine the factors that may change these properties and influence the intrinsic stability (...)