In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U-J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated self-consistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system's response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the two-field theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential U, coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries.

In this thesis we review the gauge/gravity duality and how it can be used to compute the thermodynamic properties and low-energy excitations of holographic quantum liquids - strongly-interacting field theories with a non-zero density of matter. We then study in detail the charge density excitations of two such liquids, the D3/D7 theory and the RN-AdS₄ theory, by computing the poles of their charge density Green's functions, and their charge density spectral functions. Although it is not a Landau Fermi liquid, the charge density excitations of the D3/D7 theory display many of the same properties as one, including a collisionless/hydrodynamic crossover as the temperature is increased. In contrast to this, the charge density (and energy density) excitations of the RN-AdS₄ theory do not share these properties but behave in a way that cannot be explained by Landau's theory of interacting fermionic quasiparticles. This is consistent with other results which indicate that this is not a Landau Fermi liquid.

Ce mémoire est constitué de 3 parties.La NP-complétude de la satisfaction de combinaisons booléennes de contraintes de sous-arbres est démontrée dans l'article [Ven87] ; la partie I de ce mémoire étudie dans quelle mesure l'ajout de contraintes régulières laisse espérer conserver la complexité NP. Ce modèle étendu définit une nouvelle classe de langages dont l'expressivité est comparée à celle des Rigid Tree Automata [JKV11]. Puis un début de formalisation des t-dags est donné.Les patterns ont été étudiés, principalement du point de vue des contraintes sur les données qu'ils demandent. La partie II de ce mémoire les étudie plus finement, en mettant de côté les données. Les squelettes sont définis en tant qu'intermédiaire de calcul et le fait que leur syntaxe caractérise leur sémantique est démontré. Puis un lemme de pompage est donné dans un cas restreint, un autre dans le cas général est étudié et conjecturé. Ensuite des fragments de combinaisons booléennes de patterns sont comparés en expressivité pour terminer avec l'étude de la complexité des problèmes de model-checking, satisfaisabilité et DTD-satisfaisabilité sur les dits fragments.Le contenu de la partie III constitue l'article [FMS11], c'est la démonstration de la caractérisation des langages des automates fortement déterministes de niveau 2 par des systèmes d'équations récurrentes caténatives. Celle-ci utilise, entre autres, des techniques de réécriture, la notion d'inconnues non-réécrivables et les ordres noethériens. Cette caractérisation constitue le cas de base de la récurrence démontrée dans [Sén07]. / This thesis is in 3 parts.The NP-completeness of satisfiability of boolean combinations of subtree constraints is shown in the article [Ven87] ; in the part I of this thesis, we study whether adding regular contraints lets hope for keeping the same complexity. This extended model defines a new class of languages which is compared in expressivity to the Rigid Tree Automata [JKV11]. Then a begining of formalisation of the t-dags is developped.The patterns have been studied mainly from the point of view of the constraints they demand on the data. The part II of this thesis study them more finely, by putting aside the data. The skeletons are defined as calculus intermediate and the characterisation holding between their syntax and their semantics is shown. Then a pumping lemma is prooved in a restreict case, another one is conjectured in the most general case. Then fragments of boolean combinations of patterns are compared in expressivity, this parts ends with the study of complexity of model-checking, satisfiability and DTD-satisfiability on these fragments.The content of part III constitutes the article [FMS11], it is the demonstration of the characterisation of strongly-deterministic 2-level pushdown automata by recurrent catenative equation systems. This proof uses in particular, some rewriting techniques, unrewritable unknowns and noetherian orders. This characterisation provides the base case of the recurrence shown in [Sén07].

Im Mittelpunkt unserer Untersuchungen steht das mehrkriterielle Optimierungsproblem, in einer beliebigen nichtleeren Menge eines halbgeordneten endlich dimensionalen Raumes. Zu dessen Lösung betrachten wir ein Dialogverfahren, in dem der Entscheidungsträger in jedem Schritt seine Wünsche äußert. Bei der Bestimmung einer Lösung, die den Entscheidungsträger zufriedenstellt, müssen wir ein im Allgemeinen nichtkonvexes und nicht triviales skalares Optimierungsproblem lösen. Zur Lösung dieses Problems haben wir zwei Klassen einparametrischer Optimierungsprobleme (Einbettungen) konstruiert. Mit Hilfe der Projektion auf den konvexen Ordungskegel haben wir gezeigt, dass diese Einbettungen wohldefiniert sind. Im Gegensatz zu der in der Literatur untersuchten Standardeinbettung, sind die in dieser Arbeit betrachteten Einbettungen durch die Skalarisierungen der Vektoroptimierungsprobleme mittels streng monotoner skalarisierender Funktionen motiviert. Diese Untersuchung wird unter dem Gesichtspunkt der Theorie der einparametrischen Optimierungsprobleme für den Fall eines beliebigen spitzen polyedrischen Ordnungskegels durchgeführt. Sie umfasst z.B. Fragestellungen nach der Art der Singularitäten, die für die verschiedenen Einbettungen auftreten können, nach den Bedingungen, unter denen eine Zusammenhangskomponente in der Menge stationärer oder verallgemeinerter kritischer Punkte mit Hilfe von Kurvenverfolgungsmethoden numerisch beschrieben werden kann und nach den hinreichenden Bedingungen für die Existenz einer Lösungskurve. Anschließend haben wir das von Guddat und Jongen eingeführte Konzept der strukturellen Stabilität eines skalaren Optimierungsproblems in der Vektoroptimierung verallgemeinert und einen Zusammenhang zur strukturellen Stabilität eines Minimaxproblems erstellt. Dieses Minimaxproblem steht in starker Beziehung zur Skalarisierungsmethode der Vektoroptimierungsprobleme. / In this work we consider the multiobjective optimization in a subset of a partially orded finite dimensional space. In order to solve this problem we use a dialogue procedure in which the decision maker has to determine in each step the aspiration and reservation level expressing his wishes (goals). This leads to an optimization problem which is not easy to solve in the nonconvex case. We solve it proposing two classes of one-parametric optimization problems (embeddings). Using the projection in the ordering cone, we show that these embeddings are well defined, i.e. the corresponding constraint sets depending on real-valued parameters are not empty. Contrary to the very known standard embedding the proposed embeddings are motivated by the use of strongly monotonically increasing functions, which play an important role by the scalarization of multiobjective optimization problems. The two classes of embeddings are investigated from the point of view of parametric optimization considering a pointed polyhedral cone. This investigation includes the determination of the kind of singularities which can appear, the conditions under which a connected component in the set of stationary or generalized critical point can be numerically described using pathfollowing methods and a solution curve may exist. Finally, we extend the concept of structural stability by Guddat and Jongen to the multiobjective optimization problems and establish a connection to the problem of Minimax type, which is related to the scalarization of multiobjective optimization problems.

Die vorliegende Arbeit befasst sich mit der elektronischen Struktur von Bi-Kupraten vom Normalzustand bis in den supraleitenden Zustand. Der Normalzustand von einschichtigen Bi-Kupraten wurde mittels polarisationsabhängiger XAS untersucht. Es konnte eine deutliche Polarisationsabhängigkeit der CuL3- und OK-Kante innerhalb der Kupferoxidebene beobachtet werden. Insbesondere in den Merkmalen, die den dotierten Löchern zugeordnet werden. Die Winkelabhängigkeit geht über die erwartete Hybridisierung von Cu3dx^2-y^2- und O2px,y-Orbitalen hinaus, und unterstützt somit Theorien, die auch Orbitale ausserhalb der Kupferoxidebene zur Beschreibung der elektronischen Struktur einbeziehen. Desweiteren wurde beobachtet, dass die Ladungs-Transfer-Lücke sich mit steigender Lochkonzentration vergrößert konform zu Theorien zum Zusammenbruch der Zhang-Rice-Singuletts im überdotierten Bereich. Mittels ARPES wurden die Anregungen nahe der Fermikante in antinodaler Richtung an zweischichtigen Bi-Kupraten untersucht. Die komplexe Linienform im zweischichtigen Bi-Kuprat, die aus Interlageneffekten resultiert, wurde durch die gezielte Ausnutzung von Matrixelementeffekten vereinfacht. Dadurch konnten, in Kombination mit der spezifischen Ausrichtung der Polarisation, vorherige, sich scheinbar widersprechende Beobachtungen am einschichtigen und zweischichtigen Bi-Kuprat in Einklang gebracht werden. Es konnte gezeigt werden, dass im zweischichtigen Bi-Kuprat eine Anregung zusätzlich zum bindenden und antibindenden Band existiert, welche mit dem antibindenden Band korreliert zu sein scheint. Außerdem zeigt es Gemeinsamkeiten mit dem scharfen Peak, der im einschichtigen Bi-Kuprat gefunden wurde. So besteht es über die supraleitende Sprungtemperatur Tc hinaus, und verschwindet vermutlich bei oder über der Pseudolücken-Temperatur T*. Die ARPES Messungen lassen sich am Besten innerhalb des Modells elektronischer Inhomogenitäten erklären, welches Hochtemperatursupraleitung aus Streifen ableitet. / The electronic structure of Bi cuprates from the normal state down to the superconducting state has been investigated. The normal state electronic structure is probed by polarization dependent XAS on single layer Bi cuprates. With the x-ray beam being incident normal to the CuO2 plane the azimuthal angle was varied to explore the polarization effects on orbitals within the plane. In the CuL3- as well as the OK-edge spectra, the spectral features related to the doped holes showed a distinct polarization dependence within the CuO2 plane. The revealed polarization dependence is more complex than expected from hybridization of Cu3dx^2-y^2 and O2px,y orbitals only. Thus, the results support the inclusion of out-of-plane orbitals into the description of the electronic structure as has been previously theoretically proposed. Furthermore, the charge transfer gap has been observed to rise with rising hole concentration supporting theories of the instability of Zhang-Rice-singlets in the overdoped regime. By ARPES the excitations close to the Fermi surface in the antinodal region of double layer Bi cuprates have been investigated. The complex lineshape in double layer Bi cuprates that results from interlayer effects has been disentangled by exploiting matrix element effects. In combination with distinct polarization settings this enabled to unify seemingly inconsistent observations made on single and double layer Bi cuprates. The existence of an excitation additional to antibonding and bonding band could be shown in the double layer Bi cuprate. This additional excitation is probably connected to the antibonding band. It furthermore shows similarities to the sharp peak observed in single layer Bi cuprates. It persists to temperatures above the superconducting temperature Tc, and presumably vanishes at or above the pseudogap temperature T*. The ARPES results could be best explained within the model of electronic inhomogeneity which derives superconductivity from stripes.

Matériaux dans lesquels les électrons responsables des propriétés de basse énergie son soumis à fortes corrélations sont aujourd'hui très étudiés à la recherche de nouvelles phases émergentes aux propriétés surprenantes et/ou utiles.Les supraconducteurs à base de fer (IBSC) sont maintenant considérés dans cette classe de composés. En utilissant des techniques multi-corps nécessaires pour le traitement théorique de ces corrélations (théorie du champ moyen de spin esclave - SSMFT et théorie du champ moyen dynamique - DMFT - en conjonction avec la théorie du fonctionnelle de la densité, DFT), dans cette thèse, j'etudie plusieurs propriétés d'IBSC.D’abord, j'analyse les composés très dopés de la famille de IBSC, qui montrent expérimentalement certains comportements typiques des ``fermions lourds'', des composés typiquement des terres rares ou des actinides, où des électrons extrêmement corrélés coexistent avec des électrons moins corrélés. En particulier je me concentre sur la chaleur spécifique et le pouvoir thermoélectrique et je montre comment ces propriétés peuvent être comprises dans le paradigme récemment développé ``métaux de Hund''. En effet, l’échange intra-atomique (le ``couplage de Hund'') est responsable de ces matériaux à éléments métal de transition en montrant la physique des fermions lourds. Je montre aussi que les caractéristiques typiquement fermions-lourds du spectre d’excitation, connues car les singularités de Van Hove sont bien capturées par notre modélisation au sein de DFT+SSMFT. J'utilise ensuite DMFT dans un modèle afin d'étudier l'impact direct des singularités de Van Hove sur la force des corrélations.Dans une seconde partie, je montre comment FeSe, le IBSC actuellement le plus étudié, se trouve également dans une phase métal de Hund, mais il est amené à la frontière de cette phase par la pression. Cette frontière est liée à une augmentation de la compressibilité électronique qui est positivement corrélée à l’augmentation de la supraconductivité trouvée dans les expériences.Je réalise une étude analogue sur le détenteur du record pour la température supraconductrice critique la plus élevée, la monocouche FeSe où je trouve également une compressibilité augmentée. Cela appuie la récente proposition selon laquelle la frontière du métal de Hund favorise la supraconductivité à haute température.Enfin, j'étudie la nature du magnétisme dans une autre famille de IBSC, les germanides de fer. J'explore différents ordres magnétiques possibles avec des simulations DFT et leur concurrence (ce qui peut en principe favoriser la supraconductivité) dans plusieurs composés où différents substitutions sont appliquées au composé parent YFe2Ge2. J'étudie également l'effet de la pression chimique sur ce composé. / Materials where the electrons responsible for the low-energy properties experience strong correlations are today very investigated in search of emerging new phases with surprising and/or useful properties. Iron-based superconductors (IBSC) are now considered in this class of compounds. Using the many-body techniques necessary for the theoretical treatment of these correlations (slave-spin mean field theory - SSMFT- and dynamical mean field theory - DMFT- in conjunction with density functional theory, DFT), in this thesis I address several properties of IBSC.First I analyze the very hole-doped compounds in the IBSC family, that show experimentally some behaviors typical of the so-called “heavy fermions”, compounds typically of rare earth or actinides, where extremely correlated electrons coexist with others less correlated. In particular I focus on the specific heat and the thermoelectric power and show how these properties can be understood in the recently developed paradigm of “Hund’s metals”. Indeed the intra-atomic exchange (the “Hund’s coupling”) is responsible for these materials of transition metal elements showing heavy-fermionic physics. I show also that typical heavy-fermionic features of the excitation spectrum, known as Van Hove singularities are well captured by our modelization within DFT+SSMFT. I then use DMFT in a model in order to study the direct impact of the Van Hove singularities on the strength of correlations.In a second part I show how FeSe, the presently most studied IBSC, is also in a Hund’s metal phase, but it is brought to the frontier of this phase by pressure. This frontier is connected to an enhancement of the electronic compressibility which correlates positively then with the enhancement of superconductivity found in experiments. I perform an analogous study on the record holder for the highest critical superconducting temperature, the monolayer FeSe where I also find an enhanced compressibility. This supports the recent proposal that the frontier of a Hund's metal favors high-temperature superconductivity.Finally I study the nature of magnetism in another family of IBSC, the iron-germanides. I explore different possible magnetic orders with DFT simulations and study their competition (which can in principle favor superconductivity) in several compounds where different chemical substitutions are applied to the parent compound YFe2Ge2. I also study the effect of chemical pressure on this compound.

In der vorliegenden Arbeit wird der Einfluß der reduzierten Translationssymmetrie auf die magnetischen Eigenschaften in dünnen Filmen und an Oberflächen auf der Basis des stark korrelierten Hubbard-Modells untersucht. Zunächst wird die Möglichkeit von spontanem Ferromagnetismus im Hubbard-Modell für translationssymmetrische Systeme diskutiert.Verschiedene Näherungsmethoden zur Lösung des Vielteilchenproblems des Hubbard-Modells werden detailliert beschrieben und mit Ergeb nissen von Quanten-Monte-Carlo-Rechnungen verglichen. Die Konsistenz mit exakten Resultaten über die grobe Struktur der Ein-Teilchen-Spektraldichte im Limes starker Coulomb- Wechselwirkung zwischen den Elektronen erweist sich als essentiell wichtig für eine qualitativ korrekte Beschreibung von spontanem Ferromagnetismus. Das Temperaturverhalten in der ferromagnetischen Phase wird anhand von Magnetisierungs kurven sowie mit Hilfe des spinabhängigen Quasiteilchenspekt rums ausführlich diskutiert. Ein genaues Verständnis der Physik des Volumensystems liefert die Basis für den Übergang zu Systemen mit reduzierter Translationssymmetrie. Es wird eine Methode vorgestellt mit der sich approximative Theorien für das translationssymmetrische Hubbard-Modell auf die Behandlung von Filmsystemen verallgemeinern lassen. Die magnetischen Eigenschaften dünner Hubbard-Filme werden mit Hilfe der lagenabhängigen Magnetisierung als Funktion der Temperatur sowie der Filmdicke diskutiert. Die Abhängigkeit der Curie-Temperatur von der Filmdicke wird untersucht. Insbesondere wird auf die Frage nach der magnetischen Stabilität an der Oberfläche eingegangen. In stark korrelierten Elektronensystemen ist für endliche Temperaturen die magnetische Stabilität an der Oberfläche reduziert im Vergleich zu den inneren Lagen, obwohl auf der Basis des bekannten Stoner-Bildes für Bandmagnetismus genau der gegenteilige Trend zu erwarten wäre. Es wird gezeigt,daß sich dieses Verhalten anhand einfacher Argumente versteh en läßt. Die magnetischen Eigenschaften der Hubbard-Filme lassen sich im Detail mit Hilfe der lokalen Quasiteilchenzus tandsdichte sowie der wellenvektorabhängigen Spektraldichte analysieren. Die elektronische Struktur zeigt eine ausgeprägte Spin-, Lagen- und Temperaturabhängigkeit. In einem weiteren Teil der Arbeit wird der temperaturgetrieb ene Reorientierungsübergang der Magnetisierungsrichtung in dünnen metallischen Filmen untersucht. Dazu müssen die die Hubbard-Filme um anisotrope Beiträge der Dipol-Wechselwir kung und der Spin-Bahn-Wechsel wirkung erweitert werden. Das Wechselspiel von Dipol- und Spin-Bahn-Anisotropie führt unter gewissen Bedingungen zu einem Reorientierungsübergang als Funktion der Temperatur. Im Rahmen des hier vorgestellten Zugangs lassen sich sowohl Reorientierungsüber gänge von einer senkrechten in eine parallele Position ("Fe-artig") als auch Reorientierungsübergänge von einer parallelen in eine senkrechte Position ("Ni-artig") der Magnetisierung qualitativ korrekt beschreiben. / In this work the influence of the reduced translational symmetry on the magnetic properties of thin itinerant-electr on films and surfaces is investigated within the strongly correlated Hubbard model. Firstly, the possibility of spontaneous ferromagnetism in the Hubbard model is discussed for the case of systems with full translational symmetry. Different approximation schemes for the solution of the many -body problem of the Hubbard model are introduced and discussed in detail. It is found that it is vital for a reasonable description of spontaneous ferromagnetism to be consistent with exact results concerning the general shape of the single-electron spectral density in the limit of strong Coulomb interaction between the electrons. The temperature dependence of the ferromagnetic solutions is discussed in detail by use of the magnetization curves as well as the spin-dependent quasiparticle spectrum. For the investigation of thin films and surfaces the approximation schemes for the bulk system have to be generalized to deal with the reduced translational symmetry. The magnetic behavior of thin Hubbard films is investigated by use of the layer-dependent magnetization as a function of temperature as well as the thickness of the film. The Curie-temperature is calculated as a function of the film thickness. Further, the magnetic stability at the surface is discussed in detail. Here it is found that for strong Coulomb interaction the magnetic stability at finite temperatures is reduced at the surface compared to the inner layers. This observation clearly contradicts the well-known Stoner picture of bandmagnetism and can be explained in terms of general arguments which are based on exact results in the limit of strong Coulomb interaction. The magnetic behavior of the Hubbard films can be analyzed in detail by inspecting the local quasiparticle density of states as well as the wave vector dependent spectral density. The electronic structure is found to be strongly spin-, layer-, and temperature- dependent. The last part of this work is concerned about the temperature-driven reorientation transition in thin metallic films. For the description of the magnetic anisotropy in thin films the dipole interaction as well as the spin-orbit interaction have to be included in the model. By calculating the temperature-dependence of the magnetic anisotropy energy it is found that both types of temperature-driven reorientation transitions, from out-of-plane to in-plane ("Fe-type") and from in-plane to out-of-plane ("Ni-type") magnetization are possible within the generalized Hubbard films.

Neste trabalho, consideramos o problema autônomo {(-∆u+V(x)u=f(u) em R^N,@u∈H^1 (R^N)\\{0},)┤ em que N≥3, a função V é não periódica, radialmente simétrica e muda de sinal e a não linearidade f é assintoticamente linear. Além disso, impomos que V possui um limite positivo no infinito e que o espectro do operador L≔-∆+V tem ínfimo negativo. Sob essas condições, baseando-se em interações entre soluções transladadas do problema no infinito associado, é possível mostrar que tal problema satisfaz a geometria do teorema de linking clássico e garantir a existência de uma solução fraca não trivial. Em seguida, estabelecemos a existência de uma solução não trivial para o problema não autônomo {(-∆u+V(x)u=f(x,u) em R^N,@u∈H^1 (R^N)\\{0},)┤ sob hipóteses similares ao problema anterior, admitindo também que f(x,u)=f(|x|,u) dentre outras condições. Aplicamos novamente o teorema de linking para garantir que tal problema possui uma solução não trivial. Por fim, provamos que o problema quasilinear {(-∆u+V(x)u-u∆(u^2)=g(x,u) em R^3,@u∈H^1 (R^3)\\{0},)┤ em que o potencial V muda de sinal, podendo ser não limitado inferiormente, e a não linearidade g(x,u), quando |x|→∞, possui um certo tipo de monotonicidade, possui uma solução não trivial. A existência de tal solução é provada por meio de uma mudança de variável que transforma o problema num problema semilinear, nos permitindo, assim, empregar o teorema do passo da montanha combinado com o lema splitting. / In this work, we consider the autonomous problem {(-∆u+V(x)u=f(u) em R^N,@u∈H^1 (R^N)\\{0},)┤ where N≥3, V is a non-periodic radially symmetric function that changes sign and the nonlinearity f is asymptotically linear. Furthermore, we impose that V has a positive limit at infinity and the spectrum of the operator L≔-∆+V has negative infimum. Under these conditions, employing interaction between translated solutions of the problem at infinity, it is possible to show that such problem satisfies the geometry of the classical linking theorem and garantee the existence of a nontrivial weak solution. After that, we establish the existence of a nontrivial weak solution for the nonautonomous problem {(-∆u+V(x)u=f(x,u) em R^N,@u∈H^1 (R^N)\\{0},)┤ under similar hyphoteses to the previous problem, assuming also that f(x,u)=f(|x|,u) among others conditions. We apply again the classical linking theorem to ensure that such problem possesses a nontrivial weak solution. Finally, we prove that the quasilinear problem {(-∆u+V(x)u-u∆(u^2)=g(x,u) em R^3,@u∈H^1 (R^3)\\{0},)┤ where the potential V changes sign and may be unbounded from below and the nonlinearity g(x,u), as|x|→∞, has a kind of monotonicity, has a nontrivial weak solution. The existence of such solution is proved by means of a change of variables that makes the problem become a semilinear problem and hence allow us apply the mountain pass theorem combined with splitting lemma.

La physique de la matière condensée, et notamment les systèmes fortement corrélés, amènent à des problèmes parmi les plus stimulants et difficiles de la physique moderne. Dans ces systèmes, les interactions à plusieurs corps et les corrélations entre les particules quantiques ne peuvent être négligées, sinon, les modèles échoueraient simplement à capturer les mécanismes physiques en jeu et les phénomènes qui en découlent. En particulier, le travail présenté dans ce manuscrit traite du magnétisme quantique et aborde plusieurs questions distinctes à l'aide d'approches computationnelles et méthodes numériques à l'état de l'art. Les effets conjoints du désordre (i.e. impuretés) et des interactions sont étudiés concernant un matériau magnétique spécifique : plutôt qu'une phase de la matière dite localisée, attendue à fort champ magnétique, une phase ordonnée induite par le désordre lui-même est mise en lumière, avec une réapparition inattendue de la cohérence quantique dans ledit composé. Par ailleurs, la réponse dynamique d'aimants quantiques à une perturbation externe, comme celle mesurée dans des expériences de résonance magnétique nucléaire ou de diffusion inélastique de neutrons est étudiée. / Condensed matter physics, and especially strongly correlated systems provide some of the most challenging problems of modern physics. In these systems, the many-body interactions and correlations between quantum particles cannot be neglected; otherwise, the models would simply fail to capture the relevant physics at play and phenomena ensuing. In particular, the work presented in this manuscript deals with quantum magnetism and addresses several distinct questions through computational approaches and state-of-the-art numerical methods. The interplay between disorder (i.e. impurities) and interactions is studied regarding a specific magnetic compound, where instead of the expected many-body localized phase at high magnetic fields, a novel disorder-induced ordered state of matter is found, with a resurgence of quantum coherence. Furthermore, the dynamical response of quantum magnets to an external perturbation, such as it is accessed and measured in nuclear magnetic resonance and inelastic neutron scattering experiments is investigated.

Sejam $F$ um corpo de característica diferente de $2$ e $G$ um grupo. A partir da involução clássica, que envia cada elemento em seu inverso, e uma orientação do grupo $G$ é possível definir uma involução clássica orientada na álgebra de grupo $FG$. O objetivo desta tese é estudar propriedades de Lie do conjunto dos elementos simétricos $(FG)^+$ e, em alguns casos, do conjunto dos elementos anti-simétricos $(FG)^-$. Primeiro, abordamos o caso quando $G$ não tem elementos de ordem $2$. Aqui, mostramos que se $(FG)^+$ (ou $(FG)^-$) é Lie nilpotente ou Lie $n$-Engel, então $FG$ também é Lie nilpotente ou Lie $m$-Engel, respectivamente. Depois, consideramos o caso quando $G$ contém uma cópia do grupo quatérnio de ordem $8$. Neste caso, caracterizamos completamente as álgebras de grupo tais que $(FG)^+$ é fortemente Lie nilpotente, Lie nilpotente e Lie $n$-Engel. Como consequência, provamos que o conjunto das unidades simétricas deste tipo de grupos é nilpotente. Estudamos também o caso em que quando $G$ não contém uma cópia do grupo quatérnio de ordem $8$. Em particular, apresentamos um exemplo que mostra que os resultados obtidos em pesquisas anteriores, com a involução clássica, não devem ser esperados ao trabalhar com involuções clássicas orientadas. Não entanto, damos alguns casos especiais de grupos nos quais esses resultados são obtidos. Finalmente, estudamos o índice de Lie nilpotência de $(FG)^+$. Estabelecemos uma condição necessária e suficiente, para que o índice de Lie nilpotência de $(FG)^+$ e a classe de nilpotência das unidades simétricas de uma álgebra de grupo Lie nilpotente sejam o maior possível. Além disso, consideramos a situação em que o grupo $G$ contém uma cópia de $Q_8$. / Let $F$ be a field of characteristic different from $2$ and $G$ a group. From the classical involution, which sends each element in its inverse and an orientation of $G$, it is possible to define an oriented classical involution on the group algebra $FG$. The goal of this thesis is to study Lie properties of the set of symmetric elements $(FG)^+$ and, in some cases, of the set of skew-symmetric elements $(FG)^-$. We first deal with the case when $G$ does not have elements of order $2$. In this situation, we show that if $(FG)^+$ (or $(FG)^-$) is Lie nilpotent or Lie $n$-Engel, then the whole group algebra $FG$ satisfies the same property. Later we consider the case when $G$ contains a copy of the quaternion group of order $8$. In this instance, we give a complete description of the group algebras such that $(FG)^+$ is strongly Lie nilpotent, Lie nilpotent and Lie $n$-Engel. As a consequence, we get that the set of symmetric units of this kind of groups is nilpotent. Furthermore, we study the case when $G$ does not contain a copy of the quaternion group of order $8$. Here, we present an example that shows that the previews results obtained in former works, with the classical involution, may not hold with an oriented classical involution. However, we give some kinds of groups for which those results are achieved. Finally, we study the Lie nilpotency index of $(FG)^+$. It is given a necessary and sufficient condition to the Lie nilpotency index of $(FG)^+$ and the nilpotency class of the symmetric units to be maximal, in a Lie nilpotent group algebra. In addition, we consider the situation when $G$ contains a copy of the quaternion group of order $8$.