From the time of the unexpected discovery of the insulating nature of NiO by Verwey half a century ago, Oxide materials have continued to occupy the centre stage of condensed matter physics. The recent discovery of high temperature superconductivity in doped cuprates has given a new impetus to the study of the strongly correlated electron systems. Besides, the occurrence of Colossal Magneto-Resistance (CMR) in doped rare earth manganite has also created renewed interest in these rather old systems. Understanding of the rich and complex phase diagram of these materials and their sensitivity to small perturbations e.g. external magnetic field of a few Tesla, temperature, change in isotope etc. are of great theoretical interest and also these materials have many potential technological applications. A common feature of all these oxide materials is that the transition metal ions have partially filled d-shells. Unlike s and p-electrons which gives rise to hybridized Bloch states, the d-electrons retain their atomic nature in a solid. This gives rise to strong Coulomb interaction among d-electrons which may be comparable or more than its kinetic energy. The strong correlation effects are evident from the experimental fact that the undoped parent compounds are insulators rather than metals as suggested by band theory, which favours a metallic state for systems with one electron per unit cell since this gives rise to partially filled bands (and hence a metallic state). These insulators termed Mott insulators, arise solely due to strong electron-electron correlations as compared to the band insulators which arise due to complete filling of one electron bands thereby giving rise to a gap (band gap)in the excitation spectra. The delicate competition between the kinetic energy and the Coulomb energy for d-electrons is broadly responsible for the wide variety of phenomena like Mott metal-insulator transition (MIT), magnetic transitions, charge ordering, orbital ordering, ferro/antiferroelectricity, and most interestingly the observation of high Tc superconductivity in doped cuprates. In this thesis we will restrict our interest to one such class of oxide materials, namely the doped rare earth manganites. In Chapter 1 we give a brief overview of the structure and basic interactions present in the doped manganites. Also, in the same Chapter we give a brief introduction to the phenomenology of manganites, particularly its phase diagram in the doping and temperature plane and various experimental features, e.g. the wide variety of phase transitions and phenomena particularly the observation of CMR, charge ordering and incipient meso-scale phase separations etc.. Then we briefly introduce a recently proposed microscopic model which is believed to be a minimal model which, for the first time, includes the three most important interactions present in the manganites namely the following -1)coupling of the orbitally degenerate eg electrons to local lattice distortions of Jahn-Teller type which gives rise to two species of electrons. The one denoted by by ℓ is associated with Jahn-Teller effects and hence is localized whereas the other denoted by b is an extended state and propagates through the lattice. 2) The strong Hund’s couplingof ℓ and b electrons to the t2g core spin and 3) the strong Coulomb correlation between the two species of electrons. Additionally, the model includes a new doping dependent ferromagnetic exchange between the t2g core spins which can arise from “virtual double exchange” mechanism which will be discussed in great detail in Chapter 1 . Finally, we give a brief account on Dynamical Mean Field Theory (DMFT) and Numerical Renormalization Group (NRG) as an impurity solver for the single impurity problem arising under single site DMFT approximation. In Chapter 2 we study the effect of inter-site ℓ - b hybridization on the ‘ℓ - b’ model. The single impurity problem arising under DMFT approximation has close connection with the Vigman-Finkelshtein (VF)model. Then we briefly introduce the VF model and bring out its close connection with the impurity problem. We consider both the particle-hole symmetric as well as the U → ∞ particle-hole asymmetric cases. We derive various spectral functions at T = 0K and discuss the nature of fixed points under various circumstances. We explicitly show that for the particle-hole symmetric case the Hamiltonian flows from X-ray edge singularity fixed point to Free Electron fixed point under Renormalization Group transformation. This is evident from the spectral properties of the model. We write down the effective Hamiltonian at the free electron fixed point. For the particle-hole asymmetric case the model flows from X-ray edge singularity fixed point to Free Electron/Strong Coupling fixed point with additional potential scattering terms. We write down the effective Hamiltonian at this fixed point and derive various leading order deviations. We found all of them to be irrelevant in nature also most interestingly the quasi-particles describing the under lying Fermi liquid state are found to be asymptotically non-interacting. We also calculate the Fermi liquid parameter, z, by analyzing the energy level structure of a non-interacting Hamiltonian with effective renormalized parameter. Also, we consider the case of ‘self consistent bath hybridization’ without ℓ - b hybridization for Bethe lattice with infinite coordination. Low energy qualitative features are found to be same but some of the high energy features get qualitatively modified. In Chapter 3 we discuss the transport properties of doped manganites in the insulating phases and also the Hall effect in the metallic phase. In the first part of this chapter we calculate the resistivity based on the ‘ℓ - b’model and try to fit it to the semiconducting form: ρ(T )= ρ0(T /T0)−nexp[Δ(T )/kBT ] and extract the “transport gap”, Δ(T ). This gap can be characterized in terms of the “spectral gap” which can be defined for the ℓ - b model. It is found that the transport gap in the paramagnetic phase can be characterized in terms of the near constant “spectral gap” in this phase whereas the same in the ferromagnetic phase can be characterized in terms of the zero temperature spectral gap. In the last part of this chapter we calculate the Hall resistivity (ρxy) of these materials in the metallic phase. Ρxy is found to be negative and linear in applied field -quite consistent with the experimental findings but this fails to explain the positive linear Hall resistivity at low temperatures and its crossover as a function of field and temperature. We then present a reasonable explanation for this discrepancy and support it by calculating the Hall density of states for a two band “toy model” involving inter species hybridization. In Chapter 4 we calculate the optical conductivity, σ(ω), in ℓ - b model. σ(ω) arises from two independent processes. One of the processes involves ‘b’ electrons only and termed as ‘b - b channel’ and this gives rise to a Drude peak in the low frequency region. another process termed as the ‘ℓ - b channel’ involves hopping of an ℓ-electron to a neighbouring empty site and transforms into a ‘b’like state. This process gives rise to a broad mid-infrared peak. The total conductivity is the sum of contributions from these two incoherent channels. Calculated σ(ω) for metallic systems shows lot of similarities with experimental observations particularly the temperature evolution of the mid-infrared peak and the spectral weight transfer between the two peaks. But for the insulating systems the calculated optical conductivity showed trends similar to more recent experimental observations on some insulating systems (x =0.125) but contradicts with earlier experimental observations on some other insulating system (x =0.1). Finally, in the concluding chapter, we summarize results from all the chapters and also sketch some possible future directions of investigations.

The two-dimensional electronic systems (2DESs) have kept surprising physicists for the last few decades. Examples include the integer and fractional quantum Hall effects, cuprate superconductivity, and graphene. This thesis is intended to develop suitable theoretical tools which can be generalized to study new types of 2DESs with strong correlation feature. The first part of this thesis describes the investigation of heterostructures made by Mott insulators. This work is mostly motivated by the significant improvement of techniques for layer-by-layer growth of transition metal oxides in the last few years. We construct a toy model based on generalized Hubbard model complemented with long-ranged Coulomb interaction, and we study it by Hartree-Fock theory, dynamical mean-field theory, and Thomas-Fermi theory. We argue that interesting 2D strongly correlated electronic systems can be created in such heterostructures under several conditions. Since these 2D systems are formed entirely due to the gap generated by electron-electron interaction, they are not addiabatically connected to a noninteracting electron states. This feature makes these 2D systems distinguish from the ones created in semiconductor heterostructures, and they may be potential systems having non-Fermi liquid behaviors. The second part of this thesis is devoted to the study of collective excitations in high-temperature superconductors. One important achievement in this work is to develop a time-dependent mean-field theory for t-U-J-V model, an effective low energy model for cuprates. The time-dependent mean-field theory is proven to be identical to the generalized random-phase approximation (GRPA) which includes both the bubble and ladder diagrams. We propose that the famous 41 meV magnetic resonance mode observed in the inelastic neutron scattering measurements is a collective mode arising from a conjugation relation, which has been overlooked in previous work, between the antiferromagnetic fluctuation and the phase fluctuation of the d-wave superconducting order parameter near momentum ([pi, pi]). Furthermore, we find that this collective mode signals the strength of the antiferromagnetic fluctuations which are responsible for the suppression of the superfluid density in the underdoped cuprates even at zero temperature. Finally, we perform a complete analysis on an effective model with parameters fitted by experimental data of Bi2212 within the GRPA scheme and conclude that the short-range antiferromagnetic interactions which are a remnant of the parent Mott-insulator are more likely the pairing mechanism of the High-T[subscript c] cuprates. / text

This thesis is a critical examination of phenomenological nuclear mean field theories, focusing on reliable description of levels of individual particles. The approach presented here is new in the sense that it not only allows to predict the numerical values obtained with this formalism, but also yields an estimate of the probability distributions corresponding to the experimental results. We introduce the concept of 'theoretical errors' to estimate uncertainties in theoreticalmodels. We also introduce a subjective notion of 'Predictive Power' of nuclear Hamiltonians, which is analyzed in the context of the energy spectra of individual particles. The mathematical concept of 'Inverse Problem' is applied to a realistic mean-field Hamiltonian. This technique allows to predict the properties of a system from a limited number of data. To deepen our understanding of Inverse Problems, we focus on a simple mathematical problem. A function dependent on four free parameters is introduced in order to reproduce 'experimental' data. We study the behavior of the 'fitted' parameters, their correlation and the associated errors. This study helps us understand the importance of the correct formulation of the problem. It also shows the importance of including theoretical and experimental errors in the solution.

Le cortex cérébral des mammifères est constitué de larges et complexes réseaux de neurones. La tâche de ces assemblées de cellules est d’encoder et de traiter, le plus précisément possible, l'information sensorielle issue de notre environnement extérieur. De façon surprenante, les enregistrements électrophysiologiques effectués sur des animaux en comportement ont montré que l’activité corticale est excessivement irrégulière. Les motifs temporels d’activité ainsi que les taux de décharge moyens des cellules varient considérablement d’une expérience à l’autre, et ce malgré des conditions expérimentales soigneusement maintenues à l’identique. Une hypothèse communément répandue suggère qu'une partie importante de cette variabilité émerge de la connectivité récurrente des réseaux. Cette hypothèse se fonde sur la modélisation des réseaux fortement couplés. Une étude classique [Sompolinsky et al, 1988] a en effet montré qu'un réseau de cellules aux connections aléatoires exhibe une transition de phase : l’activité passe d'un point fixe ou le réseau est inactif, à un régime chaotique, où les taux de décharge des cellules fluctuent au cours du temps et d’une cellule à l’autre. Ces analyses soulèvent néanmoins de nombreuse questions : de telles fluctuations sont-elles encore visibles dans des réseaux corticaux aux architectures plus réalistes? De quelle façon cette variabilité intrinsèque dépend-elle des paramètres biophysiques des cellules et de leurs constantes de temps ? Dans quelle mesure de tels réseaux chaotiques peuvent-ils sous-tendre des computations ? Dans cette thèse, on étudiera la dynamique et les propriétés computationnelles de modèles de circuits de neurones à l’activité hétérogène et variable. Pour ce faire, les outils mathématiques proviendront en grande partie des systèmes dynamiques et des matrices aléatoires. Ces approches seront couplées aux méthodes statistiques des champs moyens développées pour la physique des systèmes désordonnées. Dans la première partie de cette thèse, on étudiera le rôle de nouvelles contraintes biophysiques dans l'apparition d’une activité irrégulière dans des réseaux de neurones aux connections aléatoires. Dans la deuxième et la troisième partie, on analysera les caractéristiques de cette variabilité intrinsèque dans des réseaux partiellement structurées supportant des calculs simples comme la prise de décision ou la création de motifs temporels. Enfin, inspirés des récents progrès dans le domaine de l’apprentissage statistique, nous analyserons l’interaction entre une architecture aléatoire et une structure de basse dimension dans la dynamique des réseaux non-linéaires. Comme nous le verrons, les modèles ainsi obtenus reproduisent naturellement un phénomène communément observé dans des enregistrements électrophysiologiques : une dynamique de population de basse dimension combinée avec représentations neuronales irrégulières, à haute dimension, et mixtes. / The mammalian cortex consists of large and intricate networks of spiking neurons. The task of these complex recurrent assemblies is to encode and process with high precision the sensory information which flows in from the external environment. Perhaps surprisingly, electrophysiological recordings from behaving animals have pointed out a high degree of irregularity in cortical activity. The patterns of spikes and the average firing rates change dramatically when recorded in different trials, even if the experimental conditions and the encoded sensory stimuli are carefully kept fixed. 
One current hypothesis suggests that a substantial fraction of that variability emerges intrinsically because of the recurrent circuitry, as it has been observed in network models of strongly interconnected units. In particular, a classical study [Sompolinsky et al, 1988] has shown that networks of randomly coupled rate units can exhibit a transition from a fixed point, where the network is silent, to chaotic activity, where firing rates fluctuate in time and across units. Such analysis left a large number of questions unsolved: can fluctuating activity be observed in realistic cortical architectures? How does variability depend on the biophysical parameters and time scales? How can reliable information transmission and manipulation be implemented with such a noisy code? 
In this thesis, we study the spontaneous dynamics and the computational properties of realistic models of large neural circuits which intrinsically produce highly variable and heterogeneous activity. The mathematical tools of our analysis are inherited from dynamical systems and random matrix theory, and they are combined with the mean field statistical approaches developed for the study of physical disordered systems. 
In the first part of the dissertation, we study how strong rate irregularities can emerge in random networks of rate units which obey some among the biophysical constraints that real cortical neurons are subject to. In the second and third part of the dissertation, we investigate how variability is characterized in partially structured models which can support simple computations like pattern generation and decision making. To this aim, inspired by recent advances in networks training techniques, we address how random connectivity and low-dimensional structure interact in the non-linear network dynamics. The network models that we derive naturally capture the ubiquitous experimental observations that the population dynamics is low-dimensional, while neural representations are irregular, high-dimensional and mixed.

Les matériaux corrélés offrent une richesse de nouveaux phénomènes, dont beaucoup ne sont pas encore - ou seulement partiellement - compris. Au centre de cette thèse sont des modèles multi-orbitalaires que j'etudie à travers une palette de méthodes, dont la théorie du champ moyen dynamique. Dans le modèle de Hubbard multi-orbitalaire proche de la transition de Mott, je mets en évidence un régime de mauvais métal induit par le couplage de Hund. Les propriétés de la transition de Mott dans ce système sont analysées. Dans un deuxèime temps, je traite un modèle élargi pour inclure des degrés de liberté des ligands dans les oxydes. Plus spécifiquement, cette thèse étudie les effets des interactions inter-couches entre orbitales corrélés d'un atome de métal de transition et les orbitales p des ligands. Une partie du travail est dédiée au développement de nouvelles méthodes dont une approche de rotateurs esclaves à ce problème. Le dernier chapitre concerne le domaine de la spintronique moléculaire, où j'étudie la physique du "spin-state switching" en fonction de l'hybridation d'un ion de métal de transition avec ses ligands dans les molecules organométalliques du type porphyrine de nickel. / The physics of correlated materials offers a wealth of new phenomena, many of which are not yet - or only partially - understood. In this thesis, we focus on multi-orbital models, which we study using various methods, including dynamical mean-field theory. We show that in the multi-orbital Hubbard model close to the Mott transition, Hund's coupling gives rise to a bad metal regime the properties of which we analyze. Furthermore, we consider a more general class of models that include oxygen ligands. More specifically, we study the effect of inter-shell interactions between correlated metal- and ligand p-orbitals. In this context, we develop and test a new slave-rotor approach to treat such interactions in an effective manner. The final chapter constitutes an excursion to the field of molecular spintronics, where we study the physics of the hybridization-induced spin-state switching in organometallic nickel porphyrin molecules.

Le sujet de cette thèse est la théorie à partir des premiers principes de la structure électronique de matériaux présentant de fortes corrélations électroniques. D’importants progrès ont été faits dans ce domaine grâce aux implémentations modernes de Théorie de la Fonctionelle de Densité (DFT). Néanmoins, la méthode DFT a certaines limitations. D’une part, elle est faite pour décrire les propriétés de l’état fondamental mais pas des états excités des matériaux, bien que ces derniers soient également importants. D’autre part, les approximations de la fonctionnelle employées en pratique réduisent la validité de la DFT, conceptuellement exacte : en particulier elles décrivent mal les matériaux aux effets de corrélations les plus importants.Depuis les années 1990, différentes théoriques quantiques à N corps ont été utilisées pour améliorer ou compléter les simulations à base de DFT. Une des plus importantes est la Théorie du Champ Moyen Dynamique (DMFT), dans laquelle un modèle sur réseau est relié de manière auto-cohérente à un modèle plus simple d’impureté, ce qui donne de bons résultats à condition que les corrélations soient principalement locales. Nous présentons brièvement ces théories dans la première partie de cette thèse. Les progrès récents de la DMFT visent, entre autres, à mieux décrire les effets non-locaux, à comprendre les propriétés hors équilibre et à décrire de vrais matériaux plutôt que des modèles.Afin d’utiliser la DMFT pour décrire de vrais matériaux, il faut partir d’un calcul de structure électronique traitant tous les électrons au même niveau, puis appliquer une correction traitant les effets à N corps sur un sous-espace de basse énergie d’orbitales autour niveau de Fermi. La définition cohérente d’un tel sous-espace nécessite de tenir compte de la dynamique des électrons en-dehors de cet espace. Ces derniers, par exemple, réduisent la répulsion de Coulomb entre électrons dans le sous-espace. Néanmoins, combiner la DFT et la DMFT n’est pas aisé car les deux n’agissent pas sur la même observable. Dans la deuxième partie de cette thèse, nous étudions les modèles de basses énergies, comme la technique échange écranté + DMFT récemment proposée. Nous analysons l’importance de l’échange non-local et des interactions de Coulomb retardées, et illustrons cette théorie en l’appliquant aux états semi-cœur dans les métaux d10 Zn et Cd.Dans la dernière partie, nous utilisons ces méthodes pour étudier trois matériaux corrélés importants d’un point de vue technologique. Dans un premier temps, nous nous intéressons à la physique des mono-lacunes dans la phase paramagnétique du fer. De façon surprenante pour un défaut aussi simple, son énergie de formation n’a toujours pas été obtenue de manière cohérente par la théorie et l’expérience. Nous démontrons que cela est dû à de subtils effets de corrélations autour de la lacune dans la phase paramagnétique à haute température : cette phase est plus fortement corrélée que la phase ferromagnétique, où des calculs de DFT ont été faits.Dans un deuxième temps, nous étudions la transition métal-isolant dans la phase métastable VO2 B. Nous montrons que cette transition ressemble à celle entre la phase conventionnelle rutile et la phase M2 de VO2, mettant en jeu à la fois des liaisons covalentes dans les dimères et une transition de Mott sur les atomes V restants. Nous étudions également l’effet de lacunes d’oxygène sur la structure électronique de VO2.Enfin, nous proposons une technique au-delà de la DFT pour calculer le champ cristallin dans les oxydes et alliages de terres rares. Bien que l’amplitude de ce champ soit faible pour les orbitales localisées 4f des lanthanides, il est crucial pour leur caractère d’aimant permanent. En modifiant l’approximation Hubbard I pour résoudre les équations de DMFT, nous évitons une erreur d’auto-interaction faible en valeur absolue mais physiquement importante, démontrant l’importance de modèles de basse énergie correctement définis. / The topic of this thesis is the first-principles theory of the electronic structure of materials with strong electronic correlations. Tremendous progress has been made in this field thanks to modern implementations of Density Functional Theory (DFT). However, the DFT framework has some limits. First, it is designed to predict ground state but not excited state properties of materials, even though the latter may be just as important for many applications. Second, the approximate functionals used in actual calculations have more limited validity than conceptually exact DFT: in particular, they are not able to describe those materials where many-electron effects are most important.Since the 1990's, different many-body theories have been used to improve or complement DFT calculations of materials. One of the most significant non-perturbative methods is Dynamical Mean-Field Theory (DMFT), where a lattice model is self-consistently mapped onto an impurity model, producing good results if correlations are mostly local. We briefly review these methods in the first part of this thesis. Recent developments on DMFT and its extensions were aimed at better describing non-local effects, understanding out-of-equilibrium properties or describing real materials rather than model systems, among others. Here, we focus on the latter aspect.In order to describe real materials with DMFT, one typically needs to start with an electronic structure calculation that treats all the electrons of the system on the same footing, and apply a many-body correction on a well-chosen subspace of orbitals near the Fermi level. Defining such a low-energy subspace consistently requires to integrate out the motion of the electrons outside this subspace. Taking this into account correctly is crucial: it is, for instance, the screening by electrons outside the subspace strongly reduces the Coulomb interaction between electrons within the subspace. Yet it is a complex task, not least because DFT and DMFT are working on different observables. In the second part of this thesis, we discuss low-energy models in the context of the recently proposed Screened Exchange + DMFT scheme. In particular, we study the importance of non-local exchange and dynamically-screened Coulomb interactions. We illustrate this by discussing semi-core states in the d10 metals Zn and Cd.In the third and last part, we use the methods described above to study the electronic structure of three fundamentally and technologically important correlated materials. First, we discuss the physics of point defects in the paramagnetic phase of bcc Fe, more precisely the simplest of them: the monovacancy. Surprisingly for such a simple point defect, its formation energy had not yet been reported consistently from calculations and experiments. We show that this is due to subtle but nevertheless important correlation effects around the vacancy in the high-temperature paramagnetic phase, which is significantly more strongly correlated than the ferromagnetic phase where DFT calculations had been done.Second, we study the metal-insulator phase transition in the metastable VO2 B phase. We show that this transition is similar to that between the conventional rutile and M2 VO2 phases, involving both bonding physics in the dimer and an atom-selective Mott transition on the remaining V atoms. Motivated by recent calculations on SrVO3, we study the possible effect of oxygen vacancies on the electronic structure of VO2.Finally, we propose a scheme beyond DFT for calculating the crystal field splittings in rare earth intermetallics or oxides. While the magnitude of this splitting for the localized 4f shell of lanthanides does not typically exceed a few hundred Kelvin, it is crucial for their hard-magnetic properties. Using a modified Hubbard I approximation as DMFT solver, we avoid a nominally small but important self-interaction error, stressing again the importance of carefully tailored low-energy models.

<p>The pi-striped, <em>d</em>-wave superconducting (SC) state, which is a type of pair density wave wherein the SC order is spatially modulated, has recently been shown to generate the key ingredients for quantum oscillations consistent with experimental observations (Zelli <em>et al.</em>, 2011, 2012). This was accomplished with a phenomenological approach using non-self-consistent Bogoliubov-de Gennes (BdG) theory. The objective of this thesis is to explore two aspects of this approach: the addition of a charge density wave (CDW) order to the previous non-self-consistent calculations, and an attempt at stabilizing the pi-striped state in fully self-consistent BdG theory. It was found that the CDW order had a minimal effect on the Fermi surface characteristics of the pi-striped state, but that a sufficiently strong CDW degrades the Landau levels which are essential for the formation of quantum oscillations. The self-consistent mean-field calculations were unable to stabilize the pi-striped state under a range of modifications to the Hamiltonian. Free energy calculations with the modulated SC order treated as a parameter demonstrate that the pi-striped state is always less energetically favourable than the normal state for the scenarios which were considered. The results of this study constitute a basis for future, more comprehensive studies, using the BdG approach, of the stability of possible pi-striped SC phases.</p> / Master of Science (MSc)

In this doctoral thesis, we theoretically investigate the propagation of sound waves in dilute Bose gases, in both the collisionless and hydrodynamic regimes. The study of sound wave is a topic of high relevance for the understanding of dynamical properties of any fluid, classical or quantum, and further provides insightful information about the equation of state of the system. In our work, we focus in particular on the two-dimensional (2D) Bose gas, in which the sound wave is predicted to give useful information about the nature of the superfluid phase transition. Recently, experimental measurement of sound wave in a uniform 2D Bose gas has become available, and we show that the measured data are quantitatively well explained by our collisionless theory. Finally, we study the mixtures of weakly interacting Bose gases, by developing a beyond mean-field theory, which includes the effects of thermal and quantum fluctuations in both the density and spin channels. Our new theory allows for the investigation of sound dynamics, as well as the fundamental problem of phase- separation.

In this thesis, we study the kinetic disordering of systems interacting with an agent or a walker. Our studies divide naturally into two classes: for the first, the dynamics of the walker conserves the total magnetization of the system, for the second, it does not. These distinct dynamics are investigated in part I and II respectively. In part I, we investigate the disordering of an initially phase-segregated binary alloy due to a highly mobile vacancy which exchanges with the alloy atoms. This dynamics clearly conserves the total magnetization. We distinguish three versions of dynamic rules for the vacancy motion, namely a pure random walk , an "active" and a biased walk. For the random walk case, we review and reproduce earlier work by Z. Toroczkai et. al., [9] which will serve as our base-line. To test the robustness of these findings and to make our model more accessible to experimental studies, we investigated the effects of finite temperatures ("active walks") as well as external fields (biased walks). To monitor the disordering process, we define a suitable disorder parameter, namely the number of broken bonds, which we study as a function of time, system size and vacancy number. Using Monte Carlo simulations and a coarse-grained field theory, we observe that the disordering process exhibits three well separated temporal regimes. We show that the later stages exhibit dynamic scaling, characterized by a set of exponents and scaling functions. For the random and the biased case, these exponents and scaling functions are computed analytically in excellent agreement with the simulation results. The exponents are remarkably universal. We conclude this part with some comments on the early stage, the interfacial roughness and other related features. In part II, we introduce a model of binary data corruption induced by a Brownian agent or random walker. Here, the magnetization is not conserved, being related to the density of corrupted bits ρ. Using both continuum theory and computer simulations, we study the average density of corrupted bits, and the associated density-density correlation function, as well as several other related quantities. In the second half, we extend our investigations in three main directions which allow us to make closer contact with real binary systems. These are i) a detailed analysis of two dimensions, ii) the case of competing agents, and iii) the cases of asymmetric and quenched random couplings. Our analytic results are in good agreement with simulation results. The remarkable finding of this study is the robustness of the phenomenological model which provides us with the tool, continuum theory, to understand the nature of such a simple model. / Ph. D.

High harmonic generation (HHG) is an extreme non-linear phenomenon where strong laser fields interact with a medium to produce coherent and high-frequency harmonics of the incident light. It has emerged as a rapidly growing research area in bulk materials since its first observation in ZnO crystals in 2011. Over the past decade, pioneering studies have already been made in understanding the details of the microscopic mechanism behind this phenomenon, like the role of intra- and inter-band transitions, the contribution of the modulus and the phase of the dipole moment to even and odd harmonic peaks, the role of the oscillating dipoles, effects of broken symmetry, etc. However, the role of electron-electron correlations in the HHG from strongly correlated materials is much less understood. In these materials the interactions between electrons play a significant role, leading to complex and intriguing physical behaviors. In this dissertation, on the example of ZnO, perovskites BaTiO3 and BiFeO3, and transition-metal oxide VO2 I will study the role of electron-electron interaction effects in the HH spectra by using the time-dependent density-functional theory (TDDFT) approach with the exchange-correlation kernel obtained with dynamical mean- field theory (DMFT). In DMFT, one takes into account time-resolved on-site electron-electron interactions (neglected in most of other approaches) that are crucial for a larger part of strongly correlated materials. As I demonstrate, correlation effects significantly modify the HH spectrum, e.g., through the ultrafast modification of the spectrum of the system, as it was found for ZnO. As the next step, I explored the effects of electron-electron correlations in the HH spectrum of BaTiO3 perturbed by intense, few-cycle mid-infrared laser excitations. The correlation effects in this system lead to the emergence of "super-harmonics" - periodic enhancements and suppressions of specific harmonic orders that depend on the correlation strength. I extended my analysis to the case of BiFeO3, where in addition to correlation effects the effects of memory in HHG were analyzed. I have found that both correlation effects and memory lead to an extension of the harmonic cutoff. In my final part, I explored the effect of electron correlations on the HH spectrum of VO2 and compared my findings with the experiment. The obtained results may shed light on the often important role of electron correlations in the HH spectra of solids, providing valuable insights into ultrafast dynamics in complex materials, and contributing to advancements in nonlinear optics and strong-field physics, with the potential for novel photonic devices and imaging techniques in the attosecond and femtosecond regimes.