In a stochastic Lotka-Volterra model on a two-dimensional square lattice with periodic boundary conditions and subject to occupation restrictions, there exists an extinction threshold for the predator population that separates a stable active two-species coexistence phase from an inactive state wherein only prey survive. When investigating the non-equilibrium relaxation of the predator density in the vicinity of the phase transition point, we observe critical slowing-down and algebraic decay of the predator density at the extinction critical point. The numerically determined critical exponents are in accord with the established values of the directed percolation universality class. Following a sudden predation rate change to its critical value, one finds critical aging for the predator density autocorrelation function that is also governed by universal scaling exponents. This aging scaling signature of the active-to-absorbing state phase transition emerges at significantly earlier times than the stationary critical power laws, and could thus serve as an advanced indicator of the (predator) population's proximity to its extinction threshold. In order to study boundary effects, we split the system into two patches: Upon setting the predation rates at two distinct values, one half of the system resides in an absorbing state where only the prey survives, while the other half attains a stable coexistence state wherein both species remain active. At the domain boundary, we observe a marked enhancement of the predator population density, the minimum value of the correlation length, and the maximum attenuation rate. Boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern, with two different reaction rates assigned to neighboring patches. We furthermore add another predator species into the system with the purpose of studying possible origins of biodiversity. Predators are characterized with individual predation efficiencies and death rates, to which "Darwinian" evolutionary adaptation is introduced. We find that direct competition between predator species and character displacement together play an important role in yielding stable communities. We develop another variant of the lattice predator-prey model to help understand the killer- prey relationship of two different types of E. coli in a biological experiment, wherein the prey colonies disperse all over the plate while the killer cell population resides at the center, and a "kill zone" of prey forms immediately surrounding the killer, beyond which the prey population gradually increases outward. / Ph. D.

Ce manuscrit est consacré à la caractérisation de structures cristallines pour des applications de technologie quantique. Il est composé de deux parties.Dans un premier projet, nous étudions la transition d'une particule d'un état étendu à un état localisé dans un cristal artificiel quasipériodique, dont le potentiel dépend de la position de la particule. Nous considérons un atome ultrafroid, confiné par un réseau optique et incorporé dans une cavité optique. Le dipôle atomique est en interaction forte avec le champ électrique dans la cavité, ce qui mène à un deuxième potentiel optique pour l'atome. La position de l'atome dans la cavité influence notamment le champ intracavité~: le mouvement de l'atome a donc un effet rétroactif sur le potentiel dans lequel il est confiné. Pour des longueurs d'onde incommensurables, nous montrons que la compétition entre les deux réseaux optiques donne lieu à un potentiel quasipériodique pour l'atome. Nous déterminons les paramètres pour lesquels nous reproduisons le modèle Aubry-André et nous discutons les effets de la rétroaction de la cavité sur la transition de localisation.Le deuxième projet est une proposition pour générer une radiation THz, en utilisant le couplage entre excitons et phonons dans un cristal semi-conducteur. Nous proposons un schéma de conversion de fréquence, basé sur une chaîne d'interactions naturellement présentes dans une cavité semi-conductrice pompée. La partie cruciale du schéma de conversion de fréquence est l'interaction faible entre des excitons et des phonons transverses optiques. Nous la dérivons en commençant avec l'interaction électron-phonon via le potentiel de déformation et en prenant en compte les propriétés de symétrie du cristal. Nous identifions les conditions nécessaires pour générer une radiation THz, nous estimons la puissance de l'émission et nous montrons que l'interaction entre excitons et phonons transverses optiques fournit une susceptibilité non linéaire d'ordre deux. / This thesis is devoted to the characterization of crystalline structures for quantum technological applications. It is composed of two parts.In a first project we study the localization transition of one particle in an one-dimensional artificial quasiperiodic crystal, whose potential depends on the particle position. We consider an ultracold atom in an optical lattice, embedded in an optical cavity. The atom strongly couples to the cavity, leading to a second optical potential. The position of the atom within the cavity affects the cavity field, thus the atomic motion backacts on the potential it is subjected to. For incommensurate wavelengths, we show that the competition between the two potentials yields a quasiperiodic potential. We determine the parameters for which we reproduce the Aubry-Andr'e model and discuss the effects of the backaction on the localization transition.In the second project we propose a frequency down-conversion scheme to generate THz radiation using the exciton-phonon coupling in a semiconductor crystal. Our idea is based on a chain of interactions that are naturally present in a pumped semiconductor microcavity. We derive the crucial exciton-phonon coupling, starting from the electron-phonon interaction via the deformation potential and taking into account the crystal symmetry properties. We identify conditions necessary for THz emission, estimate the emission power and show that the exciton-phonon interaction provides a second-order susceptibility.

Vertical external-cavity surface-emitting lasers are ideal testbeds for studying the influence of the non-equilibrium many-body dynamics on mode locking. As we will show in this thesis, ultra short pulse generation involves a marked departure from Fermi carrier distributions assumed in prior theoretical studies. A quantitative model of the mode locking dynamics is presented, where the semiconductor Bloch equations with Maxwell’s equation are coupled, in order to study the influences of quantum well carrier scattering on mode locking dynamics. This is the first work where the full model is solved without adiabatically eliminating the microscopic polarizations. In many instances we find that higher order correlation contributions (e.g. polarization dephasing, carrier scattering, and screening) can be represented by rate models, with the effective rates extracted at the level of second Born-Markov approximations. In other circumstances, such as continuous wave multi-wavelength lasing, we are forced to fully include these higher correlation terms. In this thesis we identify the key contributors that control mode locking dynamics, the stability of single pulse mode-locking, and the influence of higher order correlation in sustaining multi-wavelength continuous wave operation.

We investigate the relaxation dynamics of two distinct non-equilibrium processes: relaxation of three-dimensional antiferromagnetic lattice spin models with Heisenberg interaction following a critical quench, and a one-dimensional exclusion process inspired by the gear-like motion of molecular motors. In a system of three-dimensional Heisenberg antiferromagnets the non-conserved staggered magnetization components couple non-trivially to the conserved magnetization densities inducing fully reversible terms that enter the Langevin dynamic equation. We simulate the exact microscopic dynamics of such a system of antiferromagnets by employing a hybrid simulation algorithm that combines the reversible spin precession implemented by the fourth-order Runge-Kutta integration method with the standard relaxational dynamics at finite temperatures using Monte Carlo updates. We characterize the dynamic universality class of this system by probing the early temporal window where the system exhibits aging scaling properties. We also verify an earlier renormalization group prediction that the temporal decay exponent in the two-time spin autocorrelation function exhibits non-universality, specifically it depends on the width of the initial spin orientation distribution. We employ a similar numerical technique to study the critical dynamics of an anisotropic Heisenberg antiferromagnet in the presence of an external field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. We study the aging scaling dynamics for the model C critical line, probe the model F critical line by investigating the system size dependence of the characteristic spin-wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point. We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions, and verify this using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent γ≈ 1.34 for periodic boundary and ballistic growth ( γ≈ 2) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile. / Doctor of Philosophy / Most systems found in nature are out of equilibrium. In this dissertation we investigate the relaxation dynamics of two such non-equilibrium systems: 1. We investigate a three-dimensional antiferromagnetic system relaxing towards equilibrium from an initial state that is driven far away from equilibrium at the point in the parameter space where the system undergoes a second-order phase transition. We devise a novel simulation method that captures emerging dynamic universal features and scaling features at these points of continuous phase transition in the early times of relaxation when the system is still far away from equilibrium. 2. Cytoplasmic dyneins are one of three kinds of motor proteins that move on tubular structures called microtubules carrying and transporting cellular cargo inside the cells. Unlike the other molecular motors that move forward with fixed step sizes, the dyneins have been experimentally observed to vary their step size depending on the amount of cargo they are carrying. We model an exclusion process in a one-dimensional lattice inspired by the motion of the dynein molecular motors where the motors can hop from one to four steps depending on their internal states. We study the effect of this variable step size on the dynamics of a collection of dyneins. We observe intriguing oscillating density profiles and discrete peaks in the distribution of empty sites. Our results suggest self-organization among the motors and the empty sites.

Complex networks are an important class of models used to describe the behaviour of a very broad category of systems which appear in different fields of science ranging from physics, biology and statistics to computer science and other disciplines. This set of models includes spin systems on a graph, neural networks, decision networks, spreading disease, financial trade, social networks and all systems which can be represented as interacting agents on some sort of graph architecture. In this thesis, by using the theoretical framework of statistical mechanics, the equilibrium and the dynamical behaviour of such systems is studied. For the equilibrium case, after presenting the region graph free energy approximation, the Survey Propagation method, previously used to investi- gate the low temperature phase of complex systems on tree-like topologies, is extended to the case of loopy graph architectures. For time-dependent behaviour, both discrete-time and continuous-time dynamics are considered. It is shown how to extend the cavity method ap- proach from a tool used to study equilibrium properties of complex systems to the discrete-time dynamical scenario. A closure scheme of the dynamic message-passing equation based on a Markovian approximations is presented. This allows to estimate non-equilibrium marginals of spin models on a graph with reversible dynamics. As an alternative to this approach, an extension of region graph variational free energy approximations to the non-equilibrium case is also presented. Non-equilibrium functionals that, when minimized with constraints, lead to approximate equations for out-of-equilibrium marginals of general spin models are introduced and discussed. For the continuous-time dynamics a novel approach that extends the cav- ity method also to this case is discussed. The main result of this part is a Cavity Master Equation which, together with an approximate version of the Master Equation, constitutes a closure scheme to estimate non-equilibrium marginals of continuous-time spin models. The investigation of dynamics of spin systems is concluded by applying a quasi-equilibrium approach to a sim- ple case. A way to test self-consistently the assumptions of the method as well as its limits is discussed. In the final part of the thesis, analogies and differences between the graph- ical model approaches discussed in the manuscript and causal analysis in statistics are presented. / <p>QC 20160904</p>

This thesis is devoted to study far-from-equilibrium aspects of quantum systems at strong coupling using the holographic duality as a tool. The duality, originated from string theory and further generalized to broader scenarios, relates certain strongly coupled gauge theories to classical gravity theories in higher dimensions. Over the last years, it has proved itself useful as a calculational tool to map difficult questions of interest in the gauge theory into a dual (i.e., equivalent) problem in a higher-dimensional gravity language where the solution may become feasible. The interest in strongly coupled quantum field theories, in particular non-Abelian gauge theories, is motivated by a number of nuclear and condensed matter physics phenomena which are known to take place at a non-perturbative regime, such as the quark-gluon plasma phase of quantum chromodynamics or high-Tc superconducting materials. While dealing with strong coupling is typically a very hard task even at equilibrium, the situation becomes yet more dramatic when non-equilibrium setups are concerned since the main non-perturbative tool available nowadays lattice field theory suffers from serious problems when it comes to real-time dynamics. This is the reason why unconventional techniques such as the ones provided by holography are welcome. Of particular interest here are the problems of thermalization of strongly coupled plasmas as well as the quench dynamics of quantum systems, both of which admit a dual gravitational description involving time-dependent solutions to the corresponding classical equations of motion in the bulk of Anti de Sitter (AdS) spacetimes, such as collapsing solutions describing AdS black hole formation. Specifically, and always from a holographic point of view, in this thesis we deal with three classes of problems: the thermalization properties of a charged non-Abelian plasma after a sudden injection of energy (such as a heavy ion collision); the dynamics of a symmetry breaking quench process from a relativistic to a non-relativistic setup of the Lifshitz type with dynamical exponent z; and, finally, a new analytical approach to the non- equilibrium properties of conformal field theory plasmas placed in an expanding background. Apart from the specific problems, we also provide a self-contained but concise introduction to the holographic duality with a view towards newcomers with an elementary general relativity and quantum field theory background. / Esta tese designa-se ao estudo de sistemas quânticos fortemente acoplados e fora do equilíbrio utilizando como ferramenta a dualidade holográfica. A dualidade, originária da teoria de cordas e posteriormente generalizada a cenários mais abrangentes, relaciona certas teorias de calibre fortemente acopladas e teorias de gravidade clássica em dimensões mais altas. Nos últimos anos, ela tem se mostrado útil como uma ferramenta de cálculo para mapear questões complicadas na teoria de gauge em um problema \\q{dual} (isto é, equivalente) formulado na linguagem completamente diferente de gravidade em dimensões extras, onde obter uma solução pode ser viável. O interesse em teorias quânticas de campo fortemente acopladas, em particular teorias de calibre não-Abelianas, motiva-se por uma variedade de fenômenos das físicas nuclear e da matéria condensada que, reconhecidamente, ocorrem em um regime não-perturbativo, tais como o plasma de quarks e glúons da cromodinâmica quântica ou certos materiais supercondutores com temperatura crítica alta. Em geral, lidar com acoplamentos fortes é uma tarefa bastante complicada mesmo em configurações de equilíbrio, mas a situação se torna ainda mais dramática quando configurações longe do equilíbrio são tratadas, visto que a principal ferramenta não-perturbativa disponível atualmente (teoria de campos na rede) enfrenta sérios problemas em situações dinâmicas. Esta é a principal razão pela qual técnicas alternativas tais como as fornecidas pela dualidade holográfica são bem vindas. De particular interesse aqui são os problemas da termalização de plasmas fortemente acoplados bem como a dinâmica pós-\\emph{quench} de sistemas quânticos, ambos os quais admitem uma descrição gravitacional dual envolvendo soluções dependentes do tempo às correspondentes equações gravitacionais em espaços-tempo de Anti de Sitter (AdS), tais como soluções de colapso descrevendo a formação de buracos negros assintoticamente AdS. Especificamente, e sempre sob um ponto de vista holográfico, nesta tese lidamos com três tipos diferentes de problemas: a termalização de um plasma não-Abeliano carregado como resultado de uma injeção repentina de energia (tal como uma colisão de íons pesados); a dinâmica durante um processo de quebra da simetria relativística para uma simetria não-relativística do tipo Lifshitz com expoente dinâmico $z$; e, finalmente, uma nova abordagem analítica para tratar propriedades fora do equílibrio de plasmas conformes colocados em um fundo que se expande. Além de tais problemas específicos, este texto fornece também uma introdução sucinta e auto-contida à dualidade holográfica direcionada a um leitor com conhecimento elementar de relatividade geral e teoria quântica de campos.

Nature changes continuously and is only seemingly at equilibrium. Environmental parameters like temperature, humidity or insolation may strongly fluctuate on scales ranging from seconds to millions of years. Being part of an ecosystem, species have to cope with these environmental changes. For ecologists, it is of special interest how individual responses to environmental changes affect the dynamics of an entire population – and, if this behavior is predictable. In this context, the demographic structure of a population plays a decisive role since it originates from processes of growth and mortality. These processes are fundamentally influenced by the environment. But, how exactly does the environment influence the behavior of populations? And what does the transient behavior look like? As a result from environmental influences on demography, so called cohorts form. They are age or size classes that are disproportionally represented in the demographic distribution of a population. For instance, if most old and young individuals die due to a cold spell, the population finally consists of mainly middle-aged individuals. Hence, the population got synchronized. Such a population tends to show regular fluctuations in numbers (denoted as oscillations) since the alternating phases of individual growth and population growth (due to reproduction) are now performed synchronously by the majority of the population.That is, one time the population growths, and the other time it declines due to mortality. Synchronous behavior is one of the most pervasive phenomena in nature. Gravitational synchrony in the solar system; fireflies flashing in unison; coordinate firing of pacemaker cells in the heart; electrons in a superconductor marching in lockstep. Whatever scale one looks at, in animate as well as inanimate systems, one is likely to encounter synchrony. In experiments with phytoplankton populations, I could show that this principle of synchrony (as used by physicists) could well-explain the oscillations observed in the experiments, too. The size of the fluctuations depended on the strength by which environmental parameters changed as well as on the demographic state of a population prior to this change. That is, two population living in different habitats can be equally influenced by an environmental change, however, the resulting population dynamics may be significantly different when both populations differed in their demographic state before. Moreover, specific mechanisms relevant for the dynamic behavior of populations, appear only when the environmental conditions change. In my experiments, the population density declined by 50% after ressource supply was doubled. This counter-intuitive behavior can be explained by increasing ressource consumption. The phytoplankton cells grew larger and enhanced their individual constitution. But at the same time, reproduction was delayed and the population density declined due to the losses by mortality. Environmental influences can also synchronize two or more populations over large distances, which is denoted as Moran effect. Assume two populations living on two distant islands. Although there is no exchange of individuals between them, both populations show a high similarity when comparing their time series. This is because the globally acting climate synchronizes the regionally acting weather on both island. Since the weather fluctuations influence the population dynamics, the Moran effect states that the synchrony between the environment equals the one between the populations. My experiments support this theory and also explain deviations arising when accounting for differences in the populations and the habitats they are living in. Moreover, model simulations and experiments astonishingly show that the synchrony between the populations can be higher than between the environment, when accounting for differences in the environmental fluctuations (“noise color”). / Die Natur unterliegt ständigen Veränderungen und befindet sich nur vermeintlich in einem Gleichgewicht. Umweltparameter wie Temperatur, Luftfeuchtigkeit oder Sonneneinstrahlung schwanken auf einer Zeitskala von Sekunden bis Jahrmillionen und beinhalten teils beträchtliche Unterschiede. Mit diesen Umweltveränderungen müssen sich Arten als Teil eines Ökosystems auseinandersetzen. Für Ökologen ist interessant, wie sich individuelle Reaktionen auf die Umweltveränderungen im dynamischen Verhalten einer ganzen Population bemerkbar machen und ob deren Verhalten vorhersagbar ist. Der Demografie einer Population kommt hierbei eine entscheidende Rolle zu, da sie das Resultat von Wachstums- und Sterbeprozessen darstellt. Eben jene Prozesse werden von der Umwelt maßgeblich beeinflusst. Doch wie genau beeinflussen Umweltveränderungen das Verhalten ganzer Populationen? Wie sieht das vorübergehende, transiente Verhalten aus? Als Resultat von Umwelteinflüssen bilden sich in Populationen sogenannte Kohorten, hinsichtlich der Zahl an Individuen überproportional stark vertretene Alters- oder Größenklassen. Sterben z.B. aufgrund eines außergewöhnlich harten Winters, die alten und jungen Individuen einer Population, so besteht diese anschließend hauptsächlich aus Individuen mittleren Alters. Sie wurde sozusagen synchronisiert. Eine solche Populationen neigt zu regelmäßigen Schwankungen (Oszillationen) in ihrer Dichte, da die sich abwechselnden Phasen der individuellen Entwicklung und der Reproduktion nun von einem Großteil der Individuen synchron durchschritten werden. D.h., mal wächst die Population und mal nimmt sie entsprechend der Sterblichkeit ab. In Experimenten mit Phytoplankton-Populationen konnte ich zeigen, dass dieses oszillierende Verhalten mit dem in der Physik gebräuchlichen Konzept der Synchronisation beschrieben werden kann. Synchrones Verhalten ist eines der verbreitetsten Phänomene in der Natur und kann z.B. in synchron schwingenden Brücken, als auch bei der Erzeugung von Lasern oder in Form von rhythmischem Applaus auf einem Konzert beobachtet werden. Wie stark die Schwankungen sind, hängt dabei sowohl von der Stärke der Umweltveränderung als auch vom demografischen Zustand der Population vor der Veränderung ab. Zwei Populationen, die sich in verschiedenen Habitaten aufhalten, können zwar gleich stark von einer Umweltveränderung beeinflusst werden. Die Reaktionen im anschließenden Verhalten können jedoch äußerst unterschiedlich ausfallen, wenn sich die Populationen zuvor in stark unterschiedlichen demografischen Zuständen befanden. Darüber hinaus treten bestimmte, für das Verhalten einer Population relevante Mechanismen überhaupt erst in Erscheinung, wenn sich die Umweltbedingungen ändern. So fiel in Experimenten beispielsweise die Populationsdichte um rund 50 Prozent ab nachdem sich die Ressourcenverfügbarkeit verdoppelte. Der Grund für dieses gegenintuitive Verhalten konnte mit der erhöhten Aufnahme von Ressourcen erklärt werden. Damit verbessert eine Algenzelle zwar die eigene Konstitution, jedoch verzögert sich dadurch die auch die Reproduktion und die Populationsdichte nimmt gemäß ihrer Verluste bzw. Sterblichkeit ab. Zwei oder mehr räumlich getrennte Populationen können darüber hinaus durch Umwelteinflüsse synchronisiert werden. Dies wird als Moran-Effekt bezeichnet. Angenommen auf zwei weit voneinander entfernten Inseln lebt jeweils eine Population. Zwischen beiden findet kein Austausch statt – und doch zeigt sich beim Vergleich ihrer Zeitreihen eine große Ähnlichkeit. Das überregionale Klima synchronisiert hierbei die lokalen Umwelteinflüsse. Diese wiederum bestimmen das Verhalten der jeweiligen Population. Der Moran-Effekt besagt nun, dass die Ähnlichkeit zwischen den Populationen jener zwischen den Umwelteinflüssen entspricht, oder geringer ist. Meine Ergebnisse bestätigen dies und zeigen darüber hinaus, dass sich die Populationen sogar ähnlicher sein können als die Umwelteinflüsse, wenn man von unterschiedlich stark schwankenden Einflüssen ausgeht.

This thesis is devoted to study far-from-equilibrium aspects of quantum systems at strong coupling using the holographic duality as a tool. The duality, originated from string theory and further generalized to broader scenarios, relates certain strongly coupled gauge theories to classical gravity theories in higher dimensions. Over the last years, it has proved itself useful as a calculational tool to map difficult questions of interest in the gauge theory into a dual (i.e., equivalent) problem in a higher-dimensional gravity language where the solution may become feasible. The interest in strongly coupled quantum field theories, in particular non-Abelian gauge theories, is motivated by a number of nuclear and condensed matter physics phenomena which are known to take place at a non-perturbative regime, such as the quark-gluon plasma phase of quantum chromodynamics or high-Tc superconducting materials. While dealing with strong coupling is typically a very hard task even at equilibrium, the situation becomes yet more dramatic when non-equilibrium setups are concerned since the main non-perturbative tool available nowadays lattice field theory suffers from serious problems when it comes to real-time dynamics. This is the reason why unconventional techniques such as the ones provided by holography are welcome. Of particular interest here are the problems of thermalization of strongly coupled plasmas as well as the quench dynamics of quantum systems, both of which admit a dual gravitational description involving time-dependent solutions to the corresponding classical equations of motion in the bulk of Anti de Sitter (AdS) spacetimes, such as collapsing solutions describing AdS black hole formation. Specifically, and always from a holographic point of view, in this thesis we deal with three classes of problems: the thermalization properties of a charged non-Abelian plasma after a sudden injection of energy (such as a heavy ion collision); the dynamics of a symmetry breaking quench process from a relativistic to a non-relativistic setup of the Lifshitz type with dynamical exponent z; and, finally, a new analytical approach to the non- equilibrium properties of conformal field theory plasmas placed in an expanding background. Apart from the specific problems, we also provide a self-contained but concise introduction to the holographic duality with a view towards newcomers with an elementary general relativity and quantum field theory background. / Esta tese designa-se ao estudo de sistemas quânticos fortemente acoplados e fora do equilíbrio utilizando como ferramenta a dualidade holográfica. A dualidade, originária da teoria de cordas e posteriormente generalizada a cenários mais abrangentes, relaciona certas teorias de calibre fortemente acopladas e teorias de gravidade clássica em dimensões mais altas. Nos últimos anos, ela tem se mostrado útil como uma ferramenta de cálculo para mapear questões complicadas na teoria de gauge em um problema \\q{dual} (isto é, equivalente) formulado na linguagem completamente diferente de gravidade em dimensões extras, onde obter uma solução pode ser viável. O interesse em teorias quânticas de campo fortemente acopladas, em particular teorias de calibre não-Abelianas, motiva-se por uma variedade de fenômenos das físicas nuclear e da matéria condensada que, reconhecidamente, ocorrem em um regime não-perturbativo, tais como o plasma de quarks e glúons da cromodinâmica quântica ou certos materiais supercondutores com temperatura crítica alta. Em geral, lidar com acoplamentos fortes é uma tarefa bastante complicada mesmo em configurações de equilíbrio, mas a situação se torna ainda mais dramática quando configurações longe do equilíbrio são tratadas, visto que a principal ferramenta não-perturbativa disponível atualmente (teoria de campos na rede) enfrenta sérios problemas em situações dinâmicas. Esta é a principal razão pela qual técnicas alternativas tais como as fornecidas pela dualidade holográfica são bem vindas. De particular interesse aqui são os problemas da termalização de plasmas fortemente acoplados bem como a dinâmica pós-\\emph{quench} de sistemas quânticos, ambos os quais admitem uma descrição gravitacional dual envolvendo soluções dependentes do tempo às correspondentes equações gravitacionais em espaços-tempo de Anti de Sitter (AdS), tais como soluções de colapso descrevendo a formação de buracos negros assintoticamente AdS. Especificamente, e sempre sob um ponto de vista holográfico, nesta tese lidamos com três tipos diferentes de problemas: a termalização de um plasma não-Abeliano carregado como resultado de uma injeção repentina de energia (tal como uma colisão de íons pesados); a dinâmica durante um processo de quebra da simetria relativística para uma simetria não-relativística do tipo Lifshitz com expoente dinâmico $z$; e, finalmente, uma nova abordagem analítica para tratar propriedades fora do equílibrio de plasmas conformes colocados em um fundo que se expande. Além de tais problemas específicos, este texto fornece também uma introdução sucinta e auto-contida à dualidade holográfica direcionada a um leitor com conhecimento elementar de relatividade geral e teoria quântica de campos.

The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.

Cette thèse présente une étude des propagations des corrélations dans les systèmes avec interaction de longue portée. La dynamique des observables locales ne peut pas être décrite avec les méthodes utilisées pour la physique statistique à l’équilibre et les approches complètement nouvelles doivent être développées. Différentes bornes sur l’évolution temporelle des corrélations ont été dérivées, mais la dynamique réelle trouvée dans des données expérimentales et numériques est beaucoup plus compliquée avec différents régimes de propagation. Une approche plus spécifique est donc nécessaire pour comprendre ces phénomènes. Nous présentons une méthode analytique pour décrire l’évolution temporelle d’observables génériques dans des systèmes décrits par des hamiltoniens quadratiques avec interactions de courte et longue portée. Grâce ces expressions, la propagation des observables peut être interprétée comme la propagation des excitations du système. Nous appliquons cette méthode générique à un modèle de spins et on obtient trois régimes différents. Ils peuvent être directement expliqués qualitativement et quantitativement par les divergences du spectre des excitations. Le résultat le plus important est le fait que la propagation, là où elle n’est pas instantanée, est au plus balistique, voir plus lente, alors les bornes permettent une propagation significativement plus rapide. On applique les mêmes expressions analytiques à un système de bosons sur un réseau avec interaction de longue et courte portée. Nous étudions les corrélations à deux corps qui ont un comportement toujours balistique et les corrélations à un corps qui ont un comportement plus riche. Cet effet peut être expliqué en calculant la contribution aux deux observables des différentes excitations qui déterminent les parties du spectre contribuant à l’observable. Ces résultats démontrent que la propagation des observables n’est pas déterminée uniquement par le spectre des excitations mais également par des quantités qui dépendent de l’observable et qui peuvent changer complètement le régime de propagation. / In this thesis we present our results on the propagation of correlations in long-range interacting quantum systems. The dynamics of local observables in these systems cannot be described with the standard methods used in equilibrium statistical physics and completely new methods have to be developed. Several bounds on the time evolution of correlations have been derived for these systems. However the propagation found in experimental and numerical results is completely different and several regimes are present depending on the long-range character of the interactions. Here we present analytical expressions to describe the time evolution of generic observables in systems where the Hamiltonian takes a quadratic form with long- and short-range interactions. These expressions describe the spreading of local observables as the spreading of the fundamental excitations of the system. We apply these expressions to a spin model finding three different propagation regimes. They can be described qualitatively et quantitatively by the divergences in the energy spectrum. The most important result is that the propagation is at most ballistic, but it can be also significantly slower, where the general bounds predict a propagation faster than ballistic. This points out that the bounds are not able to describe properly the propagation, but a more specific approach is needed. We then move to a system of lattice bosons interacting via long-range interactions. In this case we study two different observables finding completely different results for the same interactions: the spreading of two-body correlations is always ballistic while the one of the one-body correlations ranges from faster-than-ballistic to ballistic. Using our general analytic expressions we find that different parts of the spectrum contribute differently to different observables determining the previous differences. This points out that an observable-dependent notion of locality, missing in the general bounds, have to be developed to correctly describe the time evolution.