In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.

In recent years there has been significant work aimed at understanding what effect the variation of electronic doping has on material properties. In the high transition-temperature (high-T$_c$) cuprate superconductors hole doping has an impact on the superconducting transition temperature. In the underdoped regime, the cuprates exhibit anomalous properties due to a pseudogap which forms and is thought to be related to Mott insulating physics. While there is no general consensus as to the mechanism underlying high temperature superconductivity, the resonating valence bond (RVB) theory proposed by Anderson in 1987 with a Gutzwiller projected d-wave BCS wave function could give a first picture of the high-T$_c$ cuprates. We have calculated properties of the cuprates using the assumption that the pseudogap state acts as a normal state to an otherwise standard BCS mean field theory. We find that the phenomenological RVB spin liquid model proposed by Yang, Rice and Zhang (YRZ) is highly successful at describing the doping dependent features of the cuprates. Through application of the YRZ model and the tools of many-body theory we present results on anomalous properties observed in: electronic specific heat; Raman and angle-resolved photoemission spectroscopy (ARPES) data; effective mass renormalization; and thermal broadening seen in ARPES. We verify that the YRZ ansatz qualitatively describes these anomalies along with their doping dependent variations. We conclude from this work that the physics underlying the pseudogap, while distinct in origin from superconductivity, is likely to arise from an RVB wavefunction that is closely related to the BCS state. In graphene, variation in doping modifies the polarization function which describes a screened electron-electron interaction. This leads to additional features in the spectral function which are due to electron-plasmon coupling. In this work, we calculated the electronic density of states including this interaction along with its doping dependence with and without an electron-phonon interaction. We find clear features of electron-electron interactions in the density of states. These features are related to the energies of plasmaron bands in the spectral function and can be modified through doping so as to be distinct from the phonon energy scales.

Les réactions de type Ugi sont connues depuis une cinquantaine d’années. Ces réactions multicomposants mettent en jeu un aldéhyde, une amine, un isonitrile ainsi qu’un dérivé acide (acide carboxylique ou phénol activé). Dans ces travaux, la structure électronique des isonitriles, composé au coeur de ces réactions a été étudiée. La forme carbénique RN=C s’est révélée majoritaire contrairement à toute attente. La linéarité de la molécule a pu être interprétée grâce aux contributions des formes minoritaires. La seconde partie s’est focalisée sur l’étude théorique et expérimentale des couplages de type Ugi. Contrairement à la réaction de Ugi-Smiles, le réarrangement final de la réaction de Ugi, un réarrangement de Mumm, s’est révélé non cinétiquement déterminant. L’importance du rôle de l’environnement a été étudiée par le biais d’une microsolvatation. La compréhension accrue de la réaction de Ugi-Smiles a fait émerger deux nouveaux partenaires acides : les trichlorophenols et les nitrosophenols. Les premiers ont permis d’isoler pour la première fois desaryl-imidates de phénol. Les seconds ont été utilisés pour développer une nouvelle voie de synthèse debenzimidazoles à partir des adduits de Ugi-Smiles correspondants. Enfin, les adduits de type Ugi ont été mis à profit pour développer une nouvelle réaction palladocatalysée d’ouverture d’aminocyclopropanes. Cette ouverture conduit à l’accès rapide de produits hétérocycliques dont des tétracycles complexes suite à une réaction tandem impliquant un couplagede Heck. / Ugi type reactions are well known for decades. These multicomponent reactions involve an aldehyde,an amine, an isocyanide and an acid compound (carboxylic acid or activated phenol). Herein, electronicstructure of isocyanides, as key compounds of these reactions, was studied. Surprisingly, thecarbenic form RN=C was found to be the major one. Contribution of minor mesomeric structureexplains the linearity of the molecule. Then, Ugi and Ugi-Smiles couplings were studied theoreticallyand experimentally. Contrary to the Ugi-Smiles reaction, the final rearrangement of the Ugi coupling,a Mumm one, was not found to be a rate determining step. Importance of microsolvation was investigated.The comprehension of the mechanism of Ugi-Smiles reaction prompted us to investigatetheoretically new acidic partners. Trichlorophenols permitted to isolate the aryl-imidate for the firsttime with a phenol. Nitrosophenols were also considered. Confirmed experimentally, these new adductswere used to propose a new synthesis of benzimidazole. Finally, Ugi type adducts have been used todevelop a new palladium-catalyzed ring-opening of aminocyclopropanes. Rapid access to heterocycliccompounds, such as complex tetracycle with a tandem Heck coupling, can be obtained by this strategy.

Enolate formation by proton abstraction from an sp3-hybridized carbon atom situated next to a carbonyl or carboxylate group is an abundant process in nature. Since the corresponding nonenzymatic process in water is slow and unfavorable due to high intrinsic free energy barriers and high substrate pKa s, enzymes catalyzing such reaction steps must overcome both kinetic and thermodynamic obstacles. Computer simulations were used to study enolate formation catalyzed by glyoxalase I (GlxI) and 3-oxo-Δ5-steroid isomerase (KSI). The results, which reproduce experimental kinetic data, indicate that for both enzymes the free energy barrier reduction originates mainly from the balancing of substrate and catalytic base pKas. This was found to be accomplished primarily by electrostatic interactions. The results also suggest that the remaining barrier reduction can be explained by the lower reorganization energy in the preorganized enzyme compared to the solution reaction. Moreover, it seems that quantum effects, arising from zero-point vibrations and proton tunnelling, do not contribute significantly to the barrier reduction in GlxI. For KSI, the formation of a low-barrier hydrogen bond between the enzyme and the enolate, which is suggested to stabilize the enolate, was investigated and found unlikely. The low pKa of the catalytic base in the nonpolar active site of KSI may possibly be explained by the presence of a water molecule not detected by experiments. The hemoglobin-degrading aspartic proteases plasmepsinI and plasmepsin II from Plasmodium falciparum have emerged as putative drug targets against malaria. A series of C2- symmetric compounds with a 1,2-dihydroxyethylene scaffold were investigated for plasmepsin affinity, using computer simulations and enzyme inhibition assays. The calculations correctly predicted the stereochemical preferences of the scaffold and the effect of chemical modifications. Calculated absolute binding free energies reproduced experimental data well. As these inhibitors have down to subnanomolar inhibition constants of the plasmepsins and no measurable affinity to human cathepsin D, they constitute promising lead compounds for further drug development.

La recherche de phases magnétiques exotiques de la matière qui fondent même à T=0 uniquement sous l'action des fluctuations quantiques a été long et ardu, à la fois théoriquement et expérimentalement. La percée est venue récemment avec la découverte de l'Herbertsmithite, un composé formant un réseau kagome parfait avec des moments magnétiques de spin-1/2. Des expériences pionnières, mêlant des mesures de NMR, µSR et de diffusion de neutrons, ont montré une absence totale de gel ou d'ordre des moments magnétiques de spin, fournissant ainsi une forte signature d'une phase paramgnétique quantique. Théoriquement, l'Herbertsmithite est extrêmement bien modélisé par le modèle de Heisenberg quantique antiferromagnétique pour des spins-1/2 sur le réseau kagome, problème qui n'a pas été résolu jusqu'à présent. Plusieurs méthodes approximatives numériques et analytiques ont donné différents états fondamentaux, allant des liquides de spins Z2 gappés et un liquide de spins exotique algébrique U(1) de Dirac aux liquides de spins chiraux et les cristaux à liaisons de valence. Dans cette thèse, le problème est traité dans le cadre d'une approche particule-esclave fermionique, à savoir le formalisme des fermions de Schwinger SU(2). Il est conclu qu'un liquide de spins sans gap algébrique de Dirac a l'énergie variationnelle la plus basse et peut en fait constituer un vrai état fondamental physique de liquide de spins. Une implémentation sophistiquée de méthodes numériques de pointes comme le Monte-Carlo variationnel, le Monte-Carlo fonctions de Green et l'application de pas Lanczos dans un schéma variationnel ont été utilisés. Il est montré que contrairement à la croyance habituelle, le liquide de spins de Dirac U(1) projeté en "2+1" dimensions est remarquablement robuste par rapport à une large classe de perturbations, incluant les liquides de spins topologiques Z2 et les cristaux à liaisons de valence. De plus, l'application de deux pas Lanczos sur la fonction d'onde du liquide de spins de Dirac U(1) montre que son énergie est compétitive avec celles proposées pour les liquides de spins topologiques Z2. Ce résultat, combiné avec les indications expérimentales qui pointent vers un liquide de spins sans gap pour l'Herbertsmithite, appuie l'affirmation que le vrai état fondamental de ce modèle est en fait un liquide de spins algébrique de Dirac.