In this work, we develop a many-body theory of electronic transport through single molecule junctions based on nonequilibrium Green’s functions (NEGFs). The central quantity of this theory is the Coulomb self-energy matrix of the junction ∑(C). ∑(C) is evaluated exactly in the sequential-tunneling limit, and the correction due to finite lead-molecule tunneling is evaluated using a conserving approximation based on diagrammatic perturbation theory on the Keldysh contour. In this way, tunneling processes are included to infinite order, meaning that any approximation utilized is a truncation in the physical processes considered rather than in the order of those processes. Our theory reproduces the key features of both the Coulomb blockade and coherent transport regimes simultaneously in a single unified theory. Nonperturbative effects of intramolecular correlations are included, which are necessary to accurately describe the highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gap, essential for a quantitative theory of transport. This work covers four major topics related to transport in single-molecule junctions. First, we use our many-body theory to calculate the nonlinear electrical response of the archetypal Au-1,4-benzenedithiol-Au junction and find irregularly shaped ‘molecular diamonds’ which have been experimentally observed in some larger molecules but which are inaccessible to existing theoretical approaches. Next, we extend our theory to include heat transport and develop an exact expression for the heat current in an interacting nanostructure. Using this result, we discover that quantum coherence can strongly enhance the thermoelectric response of a device, a result with a number of technological applications. We then develop the formalism to include multi-orbital lead-molecule contacts and multi-channel leads, both of which strongly affect the observable transport. Lastly, we include a dynamic screening correction to ∑(C) and investigate the optoelectric response of several molecular junctions.

We investigate properties of quantum mechanical systems in the light of quantum information theory. We put an emphasize on systems with infinite-dimensional Hilbert spaces, so-called continuous-variable systems'', which are needed to describe quantum optics beyond the single photon regime and other Bosonic quantum systems. We present methods to obtain a description of such systems from a series of measurements in an efficient manner and demonstrate the performance in realistic situations by means of numerical simulations. We consider both unconditional quantum state tomography, which is applicable to arbitrary systems, and tomography of matrix product states. The latter allows for the tomography of many-body systems because the necessary number of measurements scales merely polynomially with the particle number, compared to an exponential scaling in the generic case. We also present a method to realize such a tomography scheme for a system of ultra-cold atoms in optical lattices. Furthermore, we discuss in detail the possibilities and limitations of using continuous-variable systems for measurement-based quantum computing. We will see that the distinction between Gaussian and non-Gaussian quantum states and measurements plays an crucial role. We also provide an algorithm to solve the large and interesting class of naturally occurring Hamiltonians, namely frustration free ones, efficiently and use this insight to obtain a simple approximation method for slightly frustrated systems. To achieve this goals, we make use of, among various other techniques, the well developed theory of matrix product states, tensor networks, semi-definite programming, and matrix analysis. / Die stürmische Entwicklung der Quanteninformationstheorie in den letzten Jahren brachte einen neuen Blickwinkel auf quantenmechanische Probleme. Insbesondere die fundamentale Eigenschaft der Verschränkung von Quantenzuständen spielt hierbei eine Schlüsselrolle. Einstein, Podolsky und Rosen haben 1935 versucht die Unvollständigkeit der Quantenmechanik zu demonstrieren, indem sie zeigten, dass sie keine lokale, realistische Therie ist und der Ausgang einer Messung an einem Ort von Messungen abhängen kann, die an beliebig weit entfernten Orten gemacht wurden. John Bell stellte 1964 eine, später nach ihm benannte, Ungleichung auf, die eine Grenze an mögliche Korrelationen von Messergebnissen in lokalen, realistischen Theorien gibt. Die Vorhersagen der Quatenmechanik verletzen diese Ungleichung, eine Tatsache, die 1981 von Alain Aspect und anderen auch experimentell bestätigt wurde. Solche nicht-lokalen Quantenzustände werden verschränkt'' genannt. In neuerer Zeit wurde Verschränkung nicht mehr nur als mysteriöse Eigenschaft der Quantenmechanik sondern auch als Resource für Aufgaben der Informationsverarbeitung gesehen. Ein Computer, der sich diese Eigenschaften der Quantenmechanik zu nutze macht, ein sogenannter Quantencomputer, würde es erlauben gewisse Aufgaben schnell zu lösen für die normale'' Computer zu lange brauchen. Das wichtigste Beispiel hierfür ist die Zerlegung von großen Zahlen in ihre Primfaktoren, für die Shor 1993 einen Quantenalgorithmus präsentierte. In dieser Arbeit haben wir uns mit den Eigenschaften von Quantensystemen, die durch sogenannte kontinuierliche Variablen beschrieben werden, beschäftigt. Diese sind nicht nur theoretisch sonder auch experimentell von besonderem Interesse, da sie quantenoptische Systeme beschreiben, die sich verhältnismäßig leicht im Labor präparieren, manipulieren und messen lassen. Wenn man eine vollständige Beschreibung eines Quantenzustandes erhalten will, braucht man, auf Grund der Heisenberg'schen Unschärferelation, mehrere Kopien von ihm an denen man dann Messungen durchführt. Wir haben eine Methode, compressed-sensing genannt, eingeführt um die Anzahl der nötigen Messungen substantiell zu reduzieren. Wir haben die theoretische Effizienz dieser Methode bewiesen und durch numerische Simulationen auch ihre Praktikabilität demonstriert. Desweiteren haben wir beschrieben, wie man compressed-sensing für die schon erwähnten optischen Systemen sowie für ultrakalte Atome experimentell realisieren kann. Ein zweites Hauptthema dieser Arbeit war messbasiertes Quantenrechnen. Das Standardmodell des Quantenrechnens basiert auf sogenannten Gattern, die eine genaue Kontrolle der Wechselwirkung zwischen den Bestandteilen des Quantencomputers erfordern. Messbasiertes Quantenrechnen hingegen kommt mit der Präparation eines geeigneten Quantenzustands, Resource genannt, gefolgt von einfachen Messungen auf diesem Zustand aus. Wir haben gezeigt, dass Systeme mit kontinuierlichen Variablen eine vorteilhafte Realisierung eines Quantencomputers in diesem Paradigma erlauben, es jedoch auch wichtige Beschränkungen gibt, die kompliziertere Zustandspräparationen und Messungen nötig machen.

Thermoelektrika (TE) sind Materialien die Elektrizität aus Abwärme gewinnen können. Eine wichtige Kenngröße für die Effizienz, und damit die Anwendbarkeit, von TE ist ihre Gitterwärmeleitfähigkeit. In meiner Doktorarbeit habe ich die Invarianz dieser Größe im Kontext der Linear-Response Theorie (LR) bewiesen. Dies ermöglichte es, eine Korrektur der Boltzmann-Transport Gleichung (BTE) für die Gitterwärmeleitfähigkeit in kristallinen Materialien mittels LR herzuleiten. Diese Korrektur ist wichtig um zu beurteilen, wie genau die BTE die Wärmeleitfähigkeit eines Kristalls vorhersagen kann. Um die dafür notwendigen symbolischen Umformungen durchzuführen, habe ich ein Computer-Algebra System (CAS) entwickelt. Die Anzahl an Beiträgen zum finalen Resultat stellte sich als zu groß heraus um Grenzfälle zu analysieren oder prüfbare Approximationen herzuleiten. Aus diesem Grund habe ich alle Beiträge mit so wenigen Approximationen wie möglich ausgewertet. Dafür habe ich eine Software entwickelt, um diese Terme numerisch auszuwerten. Damit habe ich meine Korrektur für altbekannte wie auch vielversprechende TE ausgewertet, nämlich PbTe, Bi2Te3 , SnSe und B4 C. Zusätzlich habe ich MgO und KF untersucht. Das Resultat lässt sich wie folgt zusammenfassen: Die Korrektur zur BTE für die Gitterwärmeleitfähigkeit hat in keinem der untersuchten Materialien und bei keiner der simulierten Temperaturen einen nennenswerten Einfluss. Meine Untersuchung legt nahe, dass die BTE für eine große Bandbreite an Materialien sicher angewandt werden kann, auch besonders stark Anharmonische. Folglich ist diese Arbeit in Übereinstimmung mit der Literatur, dass die am stärksten anharmonischen Materialien genau die mit der niedrigsten Wärmeleitfähigkeit sind. Es scheint daher sinnvoll, dass sich zukünftige Forschung weniger auf die Herleitung solcher Korrekturen zur BTE als vielmehr auf die korrekte Berechnung des Phononpropagators in stark anharmonischen Materialien konzentrieren sollte. / Thermoelectrics (TE) are materials that can be used to generate electricity from waste heat. A key quantity to the efficiency, and therefore the applicability, of TE is the lattice thermal conductivity. In this work, I prove the invariance of the lattice thermal conductivity in the context of linear-response theory (LR). This invariance enables me to derive novel formulas for a correction to the widely used Boltzmann-transport equation (BTE) for lattice thermal transport in crystalline solids using LR. It turned out that these derivations cannot be performed by a human by hand, using the formalism I chose. To perform the necessary symbolic manipulations, I programmed a computer algebra system (CAS), that implements LR, starting from expectation values, over Feynman diagrams to mathematical formulas. The number of resulting terms turned out to be too large for an analysis of all limiting cases. Consequently, I aimed at evaluating all terms, with as few approximations as possible, to generate a simple, numerical result. To do so, I developed a software package to evaluate the formulas numerically without further approximation and applied it to long-serving as well as promising new TE, namely PbTe, Bi2 Te3 , SnSe, and B4C. Additionally I investigated MgO and KF. The result can be summed up as follows: The correction to the BTE for the lattice thermal conductivity has almost no influence in the investigated materials at any simulated temperature. My investigation suggests that the BTE can be used for a wide range of materials, including the most anharmonic ones. Consequently, this work is in agreement with the literature, that the most anharmonic materials are exactly those with the lowest lattice thermal conductivity. It suggests that future theoretic work on lattice thermal conductivity should focus to find the correct phonon-propagator of strongly anharmonic systems.

Diese Dissertation beschäftigt sich mit theoretischen Untersuchung der elektronischen und magnetischen Eigenschaften von 4f-Systemen mit Filmgeometrie. Die vorgestellte Theorie basiert auf dem s-f-Modell, welches durch einen intra-atomaren Austausch zwischen einem System lokaler magnetischer Momente und den Leitungselektronen charakterisiert ist. Das Modell wird für den Fall des leeren Leitungsbandes untersucht. Der untersuchte Spezialfall ist anwendbar auf die Klasse der ferromagnetischen Halbleiter mit den Europiumchalkogeniden EuO und EuS als Prototypen solcher Substanzen. Für den Grenzfall ferromagnetischer Sättigung des Systems lokaler magnetischer Momente existiert eine exakte Lösung für das Problem. Für endliche Temperaturen wird eine Methode vorgestellt, die auf einer momentenerhaltenden Entkopplungsprozedur für passend definierte Green-Funktionen basiert. Die Theorie für endliche Temperaturen leitet sich dabei übergangslos aus dem exakt lösbaren Grenzfall ab. Mit Hilfe der vorgestellten Theorie wird das temperaturabhängige Quasiteilchenspektrum eines ferromagnetischen Modellfilmes berechnet. Die Rechnungen zeigen ein deutliches korrelationsinduziertes Aufspalten der Spektren, das in der Existenz eines neuen Quasiteilchens, des magnetischen Polarons, resultiert. Der zweite Teil der Dissertation beschäftigt sich mit der Berechnung der elektronischen und magnetischen Eigenschaften eines realen ferromagnetischen Halbleiterfilms. Um den vielfachen Leitungsbändern eines realen Systems Rechnung tragen zu können, wird das ursprüngliche s-f-Modell zu einem Mehrbandmodell erweitert. Das so erweiterte s-f-Modell wird dazu benutzt, die temperaturabhängige Bandstruktur von Volumen-EuO und von EuO(100)-Filmen zu berechnen. Die T=0-Bandstrukturen, die als Input für die Modellrechnungen dienen, werden hierbei mittels einer TB-LMTO-ASA-Bandstrukturrechnung berechnet. Die spezielle Struktur der Lösung des s-f-Modells für den exakt lösbaren Grenzfall von T=0 verhindert dabei das Auftreten von Doppelzählungen relevanter Wechselwirkungen bei der Kombination von ab-initio-Rechnungen und s-f-Modellrechnungen. Die erhaltenen temperaturabhängigen Bandstrukturen geben wertvolle Einblicke in das Wechselspiel zwischen elektronischen und magnetischen Eigenschaften in EuO-Systemen und gestatten es, verifizierbare Vorhersagen für künftige Experimente zu machen. Insbesondere wird die Existenz eines EuO(100)-Oberflächenzustandes vorhergesagt, der das Auftreten eines Oberflächen-Metall-Isolator-Übergangs induzieren kann. / This dissertation is concerned with the theoretical investigation of the electronic and magnetic properties of 4f systems with film geometry. The presented theory is based on the s-f model which features an intra-atomic exchange between a system of localized magnetic moments and the conduction electrons. The model is investigated for the special case of zero band occupation of the conduction bands which is applicable to the situation in ferromagnetic semiconductors such as the europium chalcogenides EuO and EuS. For the special case of ferromagnetic saturation of the local-moment system the problem is exactly solvable. For finite temperatures, the presented approach is based on a moment-conserving decoupling approximation for suitably defined Green functions and evolves continuously from the exact limiting case. The theory is used to calculate the temperature-dependent quasiparticle spectrum of a ferromagnetic model film. Within these calculations, one finds a marked correlation-induced splitting of the spectra resulting in the existence of a new quasiparticle, the magnetic polaron. The second part of the thesis is devoted to the calculation of the electronic and magnetic properties of a real ferromagnetic semiconductor film. The original s-f model is extended to a multi-band s-f model to account for the multiple conduction bands in a real system. Based on the resulting model, the temperature-dependent band structures of bulk EuO and EuO(100) films are calculated. Here, the T=0 band structures of the systems, which have to be taken as input for the model calculations, are calculated using the TB-LMTO-ASA band-structure technique. Due to the special form of the solution of the s-f model for the exactly solvable limiting case of T=0 the employed approach for combining the first-principles calculations with the model calculations prevents the problem of double counting of relevant interactions. The calculated temperature-dependent band structures yield a valuable insight into the temperature-dependent interplay between the magnetic and electronic properties in the EuO systems and allow to make verifiable predictions for future experiments. In particular, the existence of a EuO(100) surface state has been predicted and been shown to possibly induce a surface insulator-metal transition.

Uma técnica de projeção é usada para tratar o problema de condição inicial na teoria quântica de campos. Neste formalismo, equações de movimento do tipo cinético são deduzidas para o conjunto de variáveis dinâmicas de um corpo. Estas equações são submetidas a uma expansão não perturbativa. Tratamos esta expansão em ordem mais baixa, correspondente a aproximacão de campo médio, para um sistema uniforme de muitos fermions fora do equilíbrio descrito pelo modelo fermiônico quiral de gross-neveu. Nesta aproximação recuperamos os resultados existentes na literatura, tais como, geração dinâmica de massa, liberdade assintótica e o fenômeno de transmutação dimensional. Estudando ainda nesta aproximção o regime de pequenas oscilações em torno do equilíbrio, obtemos soluções analíticas para a evolução dinâmica de nossas variáveis. Verificamos também as condições para existencias de estados ligados neste regime. / A time-dependent projection technique is used to treat the initial value problem in Quantum Field Theory. On the basis of the general dynamics of the fields, we derive equations of kinetic type for the set of one-body dynamics variables. A non-perturbative expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Mean-Field Approximation, for a non-equilibrium uniform many-fermions system described by Chiral Gross-Neveu Model. Several literature results are obtained such as dynamical mass generation, dimensional transmutation and asymptotic freedom. In this approximation we study the small oscillations regime obtaining analytical solution for one-body dynamical variables. We have also examined the condition for the existence of bound-state in this case.

Uma técnica de projeção é usada para tratar o problema de condição inicial na teoria quântica de campos. Neste formalismo, equações de movimento do tipo cinético são deduzidas para o conjunto de variáveis dinâmicas de um corpo. Estas equações são submetidas a uma expansão não perturbativa. Tratamos esta expansão em ordem mais baixa, correspondente a aproximacão de campo médio, para um sistema uniforme de muitos fermions fora do equilíbrio descrito pelo modelo fermiônico quiral de gross-neveu. Nesta aproximação recuperamos os resultados existentes na literatura, tais como, geração dinâmica de massa, liberdade assintótica e o fenômeno de transmutação dimensional. Estudando ainda nesta aproximção o regime de pequenas oscilações em torno do equilíbrio, obtemos soluções analíticas para a evolução dinâmica de nossas variáveis. Verificamos também as condições para existencias de estados ligados neste regime. / A time-dependent projection technique is used to treat the initial value problem in Quantum Field Theory. On the basis of the general dynamics of the fields, we derive equations of kinetic type for the set of one-body dynamics variables. A non-perturbative expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Mean-Field Approximation, for a non-equilibrium uniform many-fermions system described by Chiral Gross-Neveu Model. Several literature results are obtained such as dynamical mass generation, dimensional transmutation and asymptotic freedom. In this approximation we study the small oscillations regime obtaining analytical solution for one-body dynamical variables. We have also examined the condition for the existence of bound-state in this case.

À l’heure actuelle, l’interaction entre nucléons est décrite par une théorie effective des champs chiraux. Dans ce cadre théorique, les contributions aux observables nucléaires sont organisées en suite d’importance décroissante. En particulier, le calcul de la contribution principale nécessite de résoudre exactement l’équation de Schrödinger pour un certain Hamiltonien. Une description alternative de l’interaction nucléaire, dite théorie effective des champs sans pion, considère uniquement des nucléons comme degrés de liberté et mène à la même nécessité d’une résolution exacte de l’équation de Schrödinger. En pratique, de tels calculs sont irréalistes, même numériquement, pour des observables à N corps dès que N >> 10. Par conséquent, des approximations supplémentaires doivent être développées. Dans cette thèse, des approximations non-perturbatives basées sur des fonctions de Green auto-cohérentes (SCGF) ainsi que des approximations basées sur des théories des perturbations à N corps (MBPT) sont considérées dans le cadre de la théorie effective des champs sans pion. Le but de cette thèse est d’étudier l’invariance par le groupe de renormalisation d’observables à N corps calculées avec ces approximations supplémentaires. L’espoir étant de pouvoir ensuite étendre les conclusions tirées au cas de la théorie effective des champs chiraux. Dans le cas des approximations SCGF, l’analyse des résultats numériques produits avec un code à l’état de l’art révèle une instabilité critique amenant à des observables dépendant de la renormalisation. Un correctif est proposé et devra être implémenté avant tout futur calcul SCGF au sein de la théorie effective des champs sans pion. Cette étude révèle l’importance critique des approximations numériques sur l’invariance par le groupe de renormalisation des observables. Dans le cas des approximations perturbatives basées sur MBPT, une étude formelle ouvre la voie pour dériver, de manière systématique, une renormalisation adéquate pour un large ensemble d’approximation à N corps. / The current paradigm to describe the nuclear interaction is within the frame of Chiral Effective Field Theory (ₓEFT) which organizes contributions to observables in a serie of decreasing importance. It happens that the leading contribution already requires to solve exactly the Schrödinger equation with a particular Hamiltonian. The same requirement is at play in pionless EFT which considers only nucleonic degrees of freedom. Such calculations are numerically intractable for A-body observables with A >> 10. One must design an additional expansion and truncation for many-body observables. In this thesis, non-perturbative approximations on the basis self-consistent Green’s function (SCGF) and on many-body perturbation theory (MBPT) are considered together with a pionless EFT. The goal of the present thesis is to investigate, in such framework, the renormalization invariance of many-body observables computed in A-body sectors with A >> 10. Hopefully the lessons learnt can be extended to ₓEFT. Analysis of numerical calculations realized with a state-of-the-art SCGF code reveals a critical numerical approximation leading to renormalization dependent observables. A necessary fix is proposed and must be implemented before any calculations based on SCGF and EFT in the future. This emphasizes the criticality of numerical approximations for any calculation within a pionless EFT. At the same time, renormalization invariance of observables computed within MBPT is studied formally, opening the path to formulate the renormalization of a wide range of many-body truncation schemes in the future.

Les dernières décennies ont donné lieu à un développement rapide des théories ab initio visant à décrire les propriétés des noyaux à partir de l'interaction nucléonique. Un tel développement a été rendu possible à la fois par la très importante croissance de la puissance de calcul et de nouveaux développements formels. Le présent travail se consacre au développement de la théorie de perturbation à N corps de Bogolioubov récemment proposée, qui repose sur l'usage d'un état de référence brisant la symétrie associée au nombre de particules pour permettre une description des noyaux à simple couche ouverte. Le formalisme est tout d'abord décrit en détails, son lien avec la théorie de perturbation à N corps standard est établi, tout comme sa connexion avec la théorie de cluster couplés de Bogolioubov. L'extension du formalisme à des ordres plus élevés à partir de méthodes de théorie des graphes est ensuite présentée ainsi que le programme ADG qui génère et évalue les diagrammes BMBPT à un ordre quelconque. Les implications de ce développement formel dépassent le cadre du présent travail, les méthodes développées pouvant être appliqués à d’autres méthodes à N corps. Pour terminer, de premiers résultats numériques pour les isotopes de l'oxygène, du calcium et du nickel sont présentés. Ces résultats établissent la théorie de perturbation à N corps de Bogolioubov comme une méthode de premier intérêt pour des calculs à grande échelle sur les chaînes isotopiques et isotoniques de masse moyenne. / The last few decades in nuclear structure theory have seen a rapid expansion of ab initio theories, aiming at describing the properties of nuclei starting from the inter-nucleonic interaction. Such an expansion relied both on the tremendous growth of computing power and novel formal developments. This work focuses on the development of the recently proposed Bogoliubov Many-Body Perturbation Theory that relies on a particle-number-breaking reference state to tackle singly open-shell nuclei. The formalism is first described in details, and diagrammatic and algebraic contributions are derived up to second order. Its link to standard Many-Body Perturbation Theory is made explicit, as well as its connexion to Bogoliubov Coupled-Cluster theory. An automated extension to higher orders based on graph theory methods is then detailed, and the ADG numerical program generating and evaluating BMBPT diagrams at arbitrary order is introduced. Such a formal development carries implications that are not restricted to the present work, as the developed methods can be applied to other many-body methods. Finally, first numerical results obtained for oxygen, calcium and nickel isotopes are presented. They establish BMBPT as a method of interest for large-scale computations of isotopic or isotonic chains in the mid-mass sector of the nuclear chart.

The physics of iron pnictides has been the subject of intense research for half a decade since the discovery of superconductivity in doped LaFeAsO in 2008. By now there exists a large number of different materials that are summarized under the term "pnictides'' with significant differences in their crystal structure, electronic properties, and their phase diagrams. This thesis is concerned with the investigation of the various phase transitions that are observed in the underdoped compounds of the pnictide subgroups RFeAsO, where R is a rare-earth element, and AFe_2As_2, where A is an alkaline-earth element. These compounds display two closely bound transitions from a tetragonal to an orthorhombic phase and from a paramagnetic to an antiferromagnetic metal. Both symmetry-broken phases are suppressed by doping or pressure and close to their disappearance superconductivity sets in. The superconducting state is stabilized until some optimal doping or pressure is reached and gets suppressed thereafter. The central goal of this thesis is to improve our understanding of the interplay between these three phases and to describe the various phase transitions. We start from an itinerant picture that explains the magnetism as a result of an excitonic instability and show how the other phases can be included into this picture. This approach is based on the the observation that the compounds we are interested in have a Fermi surface with multiple nested electron and hole pockets and that they have small to intermediate interaction strengths. The thesis starts with a study of the doping dependence of the antiferromagnetic phase transition in four different five-orbital models. We use the random-phase approximation to determine the transition temperature, the dominant ordering vector, and the contribution of the different orbitals to the ordering. This allows us to identify the more realistic models, which give results that are in good agreement with experimental observations. In addition to the frequently made assumption of orbital-independent interaction potentials we study the effect of a reduction of the interaction strengths that involve the d_{xy} orbital. We find that this tunes the system between two different nesting instabilities. A reduction of the interactions that involve the d_{xy} orbital also enhances the tendency towards incommensurate (IC) order. For a weak reduction this tendency is compensated by the presence of the orthorhombic phase. However, for a reduction of 30%, as it is suggested by constrained random-phase-approximation calculations, we always find large doping ranges, where a state with IC order has the highest transition temperature. We continue the investigation of the magnetic phase transition by studying the competition of different possible types of antiferromagnetic order that arises from the presence of two degenerate nesting instabilities with the ordering vectors (pi,0) and (0,pi). We derive a Ginzburg-Landau free energy from a microscopic two-band model and find that the presence of the experimentally observed stripe phase strongly depends on the number and size of the hole pockets in the system and on the doping. We show that within the picture of a purely magnetically driven nematic phase transition, which breaks the C_4 symmetry and induces the orthorhombic distortion, the nematic phase displays exactly the same dependence on the model parameters as the magnetic stripe phase. We propose that in addition to the purely magnetically driven nematic instability there is a ferro-orbital instability in the system that stabilizes the nematic transition and, thus, explains the experimentally observed robustness of the orthorhombic transition. We argue that including a ferro-orbital instability into the picture may also be necessary to reproduce the transition from simultaneous first-order transitions into an orthorhombic antiferromagnetic state to two separate second-order transitions, which is observed as a function of doping. Finally, a study of the superconducting phase transition inside the antiferromagnetic phase that is observed in some pnictide compounds is presented. We present an approach to calculate the fluctuation-mediated pairing interaction in the spin-density-wave phase of a multiband system, which is based on the random-phase approximation. This approach is applied to a minimal two-band model for the pnictides to study the effect of the various symmetry-allowed bare on-site interactions on the gap symmetry and structure. We find a competition between various even- and odd-parity states and over a limited parameter range a p_x-wave state is the dominant instability. The largest part of the parameter space is dominated by even parity states but the gap structure sensitively depends on the bare interactions. We propose that the experimentally observed transition from a nodeless to a nodal gap can be due to changes in the on-site interaction potentials.

This thesis considers quantum impurity models that exhibit a quantum phase transition (QPT) between a Fermi liquid strong coupling (SC) phase, and a doubly-degenerate non-Fermi liquid local moment (LM) phase. We focus on what can be said from exact analytic arguments about the LM phase of these models, where the system is characterized by an SU(2) spin degree of freedom in the entire system. Conventional perturbation theory about the non-interacting limit does not hold in the non-Fermi liquid LM phase. We circumvent this problem by reformulating the perturbation theory using a so-called `two self-energy' (TSE) description, where the two self-energies may be expressed as functional derivatives of the Luttinger-Ward functional. One particular paradigmatic model that possesses a QPT between SC and LM phases is the pseudogap Anderson impurity model (PAIM). We use infinite-order perturbation theory in the interaction, U, to self-consistently deduce the exact low-energy forms of both the self-energies and propagators in each of the distinct phases of the model. We analyse the behaviour of the model approaching the QPT from each phase, focusing on the scaling of the zero-field single-particle dynamics using both analytical arguments and detailed numerical renormalization group (NRG) calculations. We also apply two `conserving' approximations to the PAIM. First, second-order self-consistent perturbation theory and second, the fluctuation exchange approximation (FLEX). Within the FLEX approximation we develop a numerical algorithm capable of self-consistently and coherently describing the QPT coming from both distinct phases. Finally, we consider a range of static spin susceptibilities that each probe the underlying QPT in response to coupling to a magnetic field.