In this dissertation, a low energy theory approach is applied to the studies of Dark Matter direct detection experiments and two-dimensional Quantum Chromodynamics (QCD) spectra. We build a general framework of non-relativistic effective field theory of Dark Matter direct detection using non-relativistic operators. Any Dark Matter particle theory can be translated into the coefficients of an effective operator and any effective operator can be related to a most general description of the nuclear response. Response functions are evaluated for common Dark Matter targets. Based on the effective field theory we perform an analysis of the experimental constraints on the full parameter space of elastically scattering Dark Matter. We also formulate an analytic approach to solving two-dimensional gauge theories. We find that in theories with confinement, in a conformal operator basis, the decoupling of high scaling-dimension operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. Consequently the low-energy spectrum of a strongly coupled system like QCD can be calculated using a truncated conformal basis, to an accuracy parametrized exponentially by the cutoff dimension. We apply the conformal basis approach in two models, a two-dimensional QCD with an adjoint fermion at large N, and a two-dimensional QCD with a fundamental fermion at finite N. It is shown that the low energy spectrum converges efficiently in both cases.

Quantum Field Theory (QFT) is the language that describes a wide spectrum of physics. However, it is notoriously hard at strong coupling regime. We approach this problem in an old Quantum Mechanical method - keep a finite number of states and diagonalize the Hamiltonian as a finite-size matrix. To study a QFT, we take the Hamiltonian to be the Conformal Field Theory as the Ultraviolet fixed point of the theory's Renormalization Group Flow, deformed by a relevant operator. We use a recent framework known as the Lightcone Conformal Truncation (LCT), where we use conformal basis and lightcone quantization. As an application of the method, we study the two dimensional Supersymmetric (SUSY) Gross-Neveu-Yukawa Model. The model is expected to have a critical point in the universality class of tri-critical Ising model, a massive phase and a massless SUSY-breaking phase. We use the LCT to compute the spectrum and the spectral density of the theory at all couplings and map the entire phase diagram.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

In this thesis, we extend the numerical S-matrix bootstrap program to 1+1d theories with a boundary, where we bootstrap the 1-to-1 reflection matrix (R-matrix). We review the constraints that a physical R-matrix must obey, namely unitarity, analyticiy and crossing symmetry. We then carve out the allowed space of 2d R-matrices with the O(N) nonlinear sigma model and the periodic Yang Baxter solution in the bulk. We find a variety of integrable R-matrices along the boundary of the allowed space both with and without free parameters. The integrable models without a free parameter appear at vertices of the allowed space, while those with a free parameter occupy the whole boundary. We also introduce the extended analyticity constraint where we increase the domain of analyticity beyond the physical region. In some cases, the allowed space of R-matrices shrinks drastically and we observe new vertices which correspond to integrable theories. We also find a new integrable R-matrix through our numerics, which we later obtained by solving the boundary Yang--Baxter equation. Finally, we derive the dual to the extended analyticity problem and find that the formalism allows for R-matrices which do not saturate unitarity to lie on the boundary of the allowed region.

La cosmologie moderne amène à étudier la théorie quantique des champs en espace-temps courbe. Les champs scalaires légers, notamment, génèrent un mécanisme simple pour l'inflation et les fluctuations primordiales. Cependant, les calculs de boucles de ces modèles contiennent des divergences infrarouges et séculaires qui requièrent des techniques de resommation. Dans ce but, on implémente le groupe de renormalisation non perturbatif pour des champs scalaires en espace-temps de De Sitter. Dans un premier temps, on applique l'Approximation de Potentiel Local (APL). On démontre que les effets infrarouges sont responsables d'une restauration de la symétrie, et qu'une masse est générée en accord avec l'approche stochastique. On étudie ensuite la limite d'espace-temps plat de notre formalisme en prenant la courbure $H\to 0$, ce qui reproduit un certain nombre de résultats connus. Enfin, on s'intéresse à l'expansion dérivative, qui va au-delà de l'APL. Son implémentation semble trop complexe dans le cas général d'un espace-temps courbe, mais les symétries de De Sitter permettent de trouver une représentation simple. On définit une prescription pour tous les ordres de l'expansion, puis on implémente le flot du terme de premier ordre dans le cas simple où la dépendance en champ est négligée / The nonperturbative renormalization group for quantum field theory in de Sitter space.The study of cosmology draws us to the topic of quantum fields in curved space-time. In particular, light scalar fields offer a simple mechanism for inflation and primordial fluctuations. When computing loop corrections to these models however, infrared and secular divergences appear which call for resummation techniques. To this end, we implement the nonperturbative renormalization group for quantum scalar fields on a fixed de Sitter background. First, the Local Potential Approximation (LPA) is applied. We show that there is always symmetry restoration due to infrared effects, and that mass is generated in agreement with the stochastic approach. Next, we study the flat space limit of our formalism by taking the curvature $H\to0$, and we check that it reproduces a number of known results. Finally, we discuss the derivative expansion, which goes beyond the LPA. Its implementation seems too complex in general curved space-times, but de Sitter symmetries allow for a simpler representation. We define a prescription for all orders of the expansion, and discuss the flow of the first order term in the simple case where we neglect the field dependency (LPA')

Orientador: Prof. Dr. Alysson Fábio Ferrari / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017. / Na literatura encontramos argumentos tanto fenomenológicos quanto teóricos que favorecem o congelamento da constante de acoplamento da QCD a valores moderados no regime infravermelho. O acoplamento pode ser parametrizado em termos de uma massa efetiva para o gluon (mg) obtida dinamicamente através das equações de Schwinger- Dyson, cuja soluções são compatíveis com as simulações da QCD na rede. Primeiro nós consideramos o processo de aniquilação elétron-pósitron em hádrons Re+e- até O(3s) e adotamos o método de smearing sugerido por Poggio, Quinn e Weinberg para confrontar os dados experimentais com a teoria. Nós vamos usar como modelo teórico a QCD com uma constante de acoplamento finita no regime de baixas energias. Para encontrar o melhor fit entre os dados experimentais e teóricos, nós realizamos um test de 2, que dentro das incertezas do modelo , tem um valor mínimo quando mg=QCD está entre 1.2 - 1.4. Esses valores concordam com outras determinações fenomenológicas da razão mg=QCD e levam a uma carga efetiva s(0) 0.7. Nós comentamos como essas cargas efetivas poderiam afetar a escala de massa da dualidade global, a qual indica a fronteira entre a física perturbativa e não perturbativa. Calculamos tanto o potencial efetivo aprimorado no caso da QED escalar e da QCD com um escalar sem cor, como também a evolução do acoplamento escalar do Higgs () no Modelo Padrão. Em ambos os casos consideramos pontos fixos. No caso da QCD com o escalar sem carga de cor tanto a barreira associada ao polo de Landau quanto o mínimo do potencial mudam. Por outro lado, encontramos que a existência dos pontos fixos não perturbativos no infravermelho movem a evolução do acoplamento escalar na direcção da estabilidade. Para certos valores da constante de acoplamento da QCD no infravermelho, o potencial do Modelo Padrão pode ficar estável até a escala de Planck. / Several phenomenological and theoretical arguments favor a freezing of the Quantum Chromodynamics (QCD) coupling constant in the infrared region at one moderate value. This coupling can be parameterized in terms of an effective dynamical gluon mass (mg) which is determined through Schwinger-Dyson equations, whose solutions are compatible with QCD lattice simulations. First we consider the electron-positron annihilation process into hadrons Re+e- up to O(3s) and we adopt the smearing method suggest by Poggio, Quinn and Weinberg to confront the experimental data with theory. As a theoretical model we use the aforementioned QCD coupling constant frozen in the low energy regime. In order to find the best fit between experimental data and theory we perform a 2 study, that, within the uncertainties of the approach, has a minimum value when mg=QCD is in the range 1.2 - 1.4. These values are in agreement with other phenomenological determinations of this ratio and lead to an infrared effective charge s(0) 0.7. We comment how this effective charge may affect the global duality mass scale that indicates the frontier between perturbative and nonperturbative physics. We also compute the improved effective potential in the case of scalar QED and QCD with a colorless scalar and compute the Standard Model scalar boson Higgs coupling () evolution. In both cases we consider fixed points. In the case of QCD with a colorless scalar not only the barrier associated to the Landau pole is changed but the local minimum of the potential is also changed. On the other hand we find that the existence of such nonperturbative infrared fixed point moves the evolution towards stability. For the phenomenological preferred IR value of the QCD coupling constant the standard model Higgs potential may be stable up to the Planck scale.

Die vorliegende Dissertationarbeit ist dem Problem der analytischen Beschreibung des Confinement-Mechanismus in der QCD und in anderen Eichtheorien gewidment. Als Leitprinzip der Arbeit wurde das sogenannte Wilsonsche-Confinement-Kriterium gewählt, gemäss welchem diese Erscheinung durch eine effektive Stringtheorie beschrieben werden kann. Die entstehenden Strings des Eichfeldes verbinden farbige-Objekte (Quarks, Gluonen) miteinander und hindern ihr Auseinandergehen auf makroskopische Abstände. Es werden verschiedene Verfahren der Ableitung dieser Stringstheorien aus unterschiedlichen Eichtheorien, einschliesslich der QCD, vorgestellt. Kapitel 2 enthält die Untersuchung der nichtlokalen effektiven Stringwirkung, die im Rahmen des sogenannten stochastischen Vakuum-Modells der QCD entsteht, wobei die Wechselwirkung zwischen den Elementen der String-Weltfläche durch den phänomenologischen Background-Gluon-Propagator vermittelt wird. Durch Entwicklung dieser Wirkung nach Ableitungen wurden die ersten Terme niedrigster Ordnung bestimmt. Die ersten beiden Terme dieser Entwicklung sind die Nambu-Goto- und Rigidity-Terme mit Kopplungskonstanten, die sich durch das Gluon-Kondensat und die Korrelationlänge des QCD-Vakuums ausdrücken lassen. Die Vorzeichen dieser Konstanten zeigen, dass die durch dieses Verfahren erhaltenen Strings stabil sind. Danach wurde eine mögliche Lösung des ``Crumpling'' Problems auf der Basis eines zusätzlichen topologischen Stringtermes im Instantongas-Modell des QCD-Vakuums vorgestellt. Mittels Störungstheorie im nicht-störungstheoretischen QCD-Background berechneten wir zusätzliche-Korrekturen zur ursprünglichen nicht-störungstheoretischen Stringwirkung. Diese Korrekturen führen zu neuen Formen der nichtlokalen effektiven Stringwirkung, die den störungstheoretischen Gluon-Propagator im Backgroundfeld zwischen den Elementen der Weltfläche enthalten. Durch Ableitungsentwicklung dieser Wirkung bekommen wir eine Korrektur zur Kopplungskonstante des Rigidity-Terms; die Stringsspannung des Nambu-Goto-Terms jedoch bleibt unverändert. Am Ende dieses Kapitels wurde der Hamilton-Operator des QCD-Strings mit spinlosen Quarks hergeleitet, der der effektiven Stringwirkung mit Rigidity-Term entspricht. Dieser Hamilton-Operator liefert einen Korrekturterm zur Wechselwirkung im relativistischen Quarkmodell-Operator. Im Kapitel 3 untersuchten wir das Problem der Stringdarstellung von Abelsch-projezierten Eichtheorien. Als erstes wurde die Herleitung der Stringdarstellung der erzeugenden Funktion für das einfachste Modell dieser Art, d.h. die Abelsch-projezierte SU(2)-QCD gegeben, die einem dualen Abelschen Higgs-Modell mit äusseren elektrisch geladendenen Teilchen äquivalent ist. Der Vorteil dieses Stringszuganges im Vergleich zum Zugang des stochastischen Vakuum-Modells der QCD besteht in der Berücksichtigung der Integration über String-Weltflächen, die auf Grund der Integration über den Singulärteil der Higgsfeld-Phase entsteht. Zusätzlich zur Stringdarstellung der erzeugenden Funktion wurde im London-Limes die Stringdarstellung für die erzeugenden Funktionale der Feldstärke- und Monopolstromkorrelatoren hergeleitet. Dies gab uns die Möglichkeit, die entsprechenden bilokalen Kumulanten zu finden und zu zeigen, dass die bilokalen Kumulanten der Feldstärke für grosse Abstände das gleiche Verhalten wie die entsprechenden eichinvarianten Kumulanten der QCD zeigen. Das Letztere wurde durch das stochastische Vakuum-Modell vorhergesagt und durch Gitterexperimente berechnet. Dieses Ergebnis unterstützt einerseits die Methode der Abelschen Projektion und gibt anderseits dem stochastischen Vakuum-Modell der QCD einen neuen feldtheoretischen Status. Danach erweiterten wir unsere Analyse über den Rahmen des London-Limes hinaus untersuchten den Zusammenhang von quartischen Kumulanten und bilokalen Kumulanten. Anschliessend wurde die Stringdarstellung der SU(3)-Gluodynamik hergeleitet. Dabei wurde die Stringdarstellung für ein entsprechendes duales Modell formuliert, das drei Arten des magnetischen Higgs-Feldes enthält. Infolgedessen liefert das Modell drei Strings, von denen nur zwei wirklich unabhängig sind. Alle diese Strings wechselwirken untereinander durch Austausch zweier massiver dualer Eichbosonen. Ausserdem erhielten wir die bilokalen Kumulanten des effektiven dualen Modells der SU(3)-Gluodynamik. Die entsprechenden bilokalen Kumulanten zeigen für grosse Abstände ein Verhalten wie es durch das stochastische Vakuum-Modell vorhergesagt wurde. Zum Schluss dieses Kapitels geben wir eine nützliche Darstellung für erzeugende Funktionen von Abelsch-projezierten Theorien in Form von Integralen über Monopolströme an. Im Kapitel 4 wurde ein weiteres Modell untersucht, das eine analytische Beschreibung des Confinement-Mechanismus zulässt, nämlich die 3D kompakte QED. Für den Wilson-Loop der entsprechenden Theorie mit Monopoldichten wurde die Äquivalenz zur sogenannten Confining-Stringtheorie demonstriert. Ausserdem wurde das Verhalten der bilokalen Kumulante der Feldstärke im Limes schwacher Felder untersucht. Dieses Verhalten befindet sich ebenfalls in Übereinstimmung mit den Voraussagen des stochastischen Vakuum-Modells. Erwartungsgemäss sind die Stringdarstellungen der erzeugenden Funktionen der 3D kompakten QED im Limes schwacher Felder und der dualen Abelschen Higgs-Modelle sehr ähnlich. Wir zeigten ausserdem, dass diese Entsprechung nicht zufällig ist. Die 3D kompakte QED ergibt sich nämlich im Limes verschwindender Eichbosonmasse aus dem 3D Abelschen Higgs-Modell mit äusseren Monopolen. Zum Schluss wurde ein allgemeines Verfahren der Beschreibung der Anregungen der Stringweltfläche in Abelsch-projezierten Theorien (kompakte QED und QCD) ausgearbeitet. Es ist auf der Methode der nicht-linearen Sigma-Modelle gegründet und gibt eine Möglichkeit, die in diesen Fluktuationen quadratische Effektive Wirkung zu erhalten. In der Dissertation wurden analytische nicht-störungstheoretische Verfahren ausgearbeitet, die neue Informationen über den Confinement-Mechanismus in der QCD und anderen Eichtheorien liefern und zum besseren Verständnis der Vakuumstruktur dieser Theorien beitragen können. Sie sind insbesondere relevant für die Herleitung effektiver Stringtheorien aus Eichtheorien. / The main problem addressed in the present Dissertation was an attempt of an analytical description of confinement in QCD and other gauge theories. As a guiding principle for our investigations served the so-called Wilson's picture of confinement, according to which this phenomenon can be described in terms of some effective theory of strings, joining coloured objects to each other and preventing them from moving apart to macroscopic distances. In this Dissertation, we have proceeded with a derivation of such string theories corresponding to various gauge ones, including QCD, i.e. with the solution of the problem of string representation of gauge theories. We have started our analysis with the nonlocal string effective action, arising within the so-called Stochastic Vacuum Model of QCD, where the interaction between the string world-sheet elements is mediated by the phenomenological background gluon propagator. By performing the derivative expansion of this action, we have derived the first few terms of a string Lagrangian. The first two nontrivial of them turned out to be the Nambu-Goto and rigidity terms with the coupling constants expressed completely via the gluonic condensate and correlation length of the QCD vacuum. The signs of these constants ensure the stability of strings in the so-obtained effective string theory. After that, we have investigated the problem of crumpling for the string world-sheets by derivation of the topological string term in the instanton gas model of the gluodynamics vacuum. Next, by making use of perturbation theory in the nonperturbative QCD vacuum, we have calculated perturbative corrections to the obtained string effective action. Those lead to a new form of the nonlocal string effective action with the propagator between the elements of the world-sheet being the one of a perturbative gluon in the confining background. By the derivative expansion of this action, we got a correction to the rigidity term coupling constant, whereas the string tension of the Nambu-Goto term occurs to get no corrections due to perturbative gluonic exchanges. Finally, we have derived the Hamiltonian of QCD string with spinless quarks at the ends, associated with the obtained string effective action including the rigidity term. In the particular case of vanishing orbital momentum of the system, this Hamiltonian reduces to that of the so-called relativistic quark model, albeit with some modifications due to the rigidity term, which might have some influence on the dynamics of the QCD string with quarks. All these topics have been elaborated on in Section 2, and form the essence of the string representation of QCD within the Stochastic Vacuum Model. In Section 3, we have addressed the problem of string representation of Abelian-projected theories. In this way, we have started with the string representation for the partition function of the simplest model of this kind, namely the Abelian-projected SU(2)-QCD, which is argued to be the dual Abelian Higgs Model with external electrically charged particles. The advantage of this approach to the string representation of QCD w.r.t. the one based on the Stochastic Vacuum Model is a possibility to get an integration over the string world-sheets, resulting from the integration over the singular part of the phase of the Higgs field. After the string representation of the partition function in the London limit, we have proceeded with the string representation for the generating functionals of the field strength and monopole current correlators. Those enabled us to find the corresponding bilocal cumulants and demonstrate that the large-distance asymptotic behaviour of the bilocal field strength cumulant matches the one of the corresponding gauge-invariant cumulant in QCD, predicted by the Stochastic Vacuum Model and measured in the lattice experiments. This result supports the method of Abelian projection on the one hand and gives a new field-theoretical status to the Stochastic Vacuum Model on the other hand. After that, we have extended our analysis beyond the London limit, and studied the relation of the quartic cumulant, which appears there, with the bilocal one in the London limit. Next, by making use of the Abelian projection method, we have addressed the problem of string representation of the SU(3)-gluodynamics. Namely, we have casted the related dual model, containing three types of magnetic Higgs fields, into the string form. Consequently, the latter one turned out to contain three types of strings, among which only two ones were actually independent. As a result, we have found, that both the ensemble of strings as a whole and individual strings display confining properties in a sense that all types of strings (self)interact via the exchanges of the massive dual gauge bosons. We have also derived bilocal cumulants in the effective dual model of confinement, corresponding to the SU(3)-gluodynamics, and they turned out to be also in line with the ones predicted by the Stochastic Vacuum Model. In conclusion of this topic, we have obtained another useful representation for the partition functions of the Abelian-projected theories in the form of an integral over the monopole currents. In Section 4, we have studied another model, allowing for an analytical description of confinement, which is 3D compact QED. In this way, by virtue of the integral over the monopole densities, we have derived string representation for the Wilson loop in this theory and demonstrated the correspondence of this representation to another recently found one, the so-called confining string theory. After that, we have calculated the bilocal cumulant of the field strength tensors in the weak-field limit of the model under study. It also turned out to be in line with the general concepts of the Stochastic Vacuum Model and therefore matches the corresponding results known from the lattice measurements in QCD and found analytically for the effective Abelian-projected theories in the previous Section. Besides that, string representations of the partition functions of the weak-field limit of 3D compact QED and of the dual Abelian Higgs Model turned out to be also quite similar. We have illustrated later on that this correspondence is not accidental. Namely, we have shown that 3D compact QED is nothing else, but the limiting case of 3D Abelian Higgs Model with external monopoles, corresponding to the vanishing gauge boson mass. Finally, we have elaborated on a unified method of description of the string world-sheet excitations in the Abelian-projected theories, compact QED, and QCD, based on the techniques of nonlinear sigma-models, and obtained the effective action, quadratic in the world-sheet fluctuations. In conclusion, the proposed nonperturbative techniques provide us with some new information on the mechanisms of confinement in QCD and other gauge theories and shed some light on the vacuum structure of these theories. They also show the relevance of string theory to the description of this phenomenon and yield several prescriptions for the construction of the adequate string theories from the corresponding gauge ones.

Nous étudions le modèle O (N) relativiste, une généralisation quantique de la théorie φ⁴ utilisée en physique statistique pour étudier des transitions de phase. Ce modèle décrit certaines transitions de phase quantiques telles que la transition isolant de Mott-superfluide dans un gaz de bosons piégés dans un réseau optique ou la transition paramagnétique-antiferromagnétique dans un aimant. En deux dimensions d’espace, ces systèmes sont fortement corrélés près de la transition. Nous les étudions à l’aide du groupe de renormalisation non perturbatif, une formulation du groupe de renormalisation de Wilson. Nous nous intéressons aux propriétés universelles au voisinage de la transition de phase quantique à température nulle. Ainsi, nous déterminons les fonctions d’échelle universelles qui définissent la thermodynamique et démontrons que ces fonctions sont reliées à celles décrivant la force de Casimir critique dans un système classique tridimensionel. Ensuite, nous étudions le spectre d’excitation dans la phase ordonnée à température nulle. Pour N = 2 et 3, nous établissons l’existence d’un mode d’amplitude aussi appelé « mode de Higgs » par analogie avec le mécanisme de Higgs en physique des hautes énergies. Pour N = 1, nous montrons l’existence d’un état lié pour des dimensions proches de trois. Enfin, nous calculons la dépendance en fréquence de la conductivité à température nulle et confirmons son universalité, en particulier à la transition. Nous établissons que l’une des composantes du tenseur de conductivité dans la phase ordonnée est une quantité « superuniverselle », ne dépendant ni de la distance au point critique ni de N. / We study the relativistic O (N) model, a quantum generalization of the φ⁴ theory used in statistical physics to study some phase transitions. This model describes quantum phase transitions such as the Mott insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson’s renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which define the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. Then, we study the excitation spectrum in the zero-temperature ordered phase. For N = 2 and 3, we establish the existence of an amplitude mode, also called “Higgs mode” by analogy with the Higgs mechanism in high-energy physics. For N = 1, we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and confirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a “superuniversal” quantity depending neither on the distance to the critical point nor on N.

L'objet de cette thèse est l'étude de phénomènes critiques hors-équilibre. Pour décrire ces systèmes, l'utilisation d'équations de Langevin est souvent incontournable car elles permettent une description heuristique relativement simple du phénomène, construite en ajoutant un terme de bruit à la dynamique macroscopique. J'ai montré qu'il est toutefois possible, dans le cas des processus de réaction-diffusion, d'aller au delà de cette approche et de dériver une équation de Langevin exacte qui décrit la dynamique au niveau microscopique. Une seconde partie de ma thèse est consacrée à l'étude de modèles spécifiques de phénomènes critiques hors-équilibre à l'aide du groupe de renormalisation non-perturbatif (NPRG), une version moderne des blocs de spins de Wilson et Kadanoff. À l'équilibre, cet outil tire son succès de sa capacité à contrôler les fluctuations au voisinage de la transition grâce à un régulateur. Hors équilibre, les fluctuations temporelles doivent être traitées de la même façon, et j'ai donc conçu un régulateur qui contrôle à la fois les fluctuations spatiales et temporelles. Enfin, j'ai appliqué le NPRG à un modèle d'érosion. En effet, l'apparition générique de lois d'échelles dans les paysages suggère l'existence d'un mécanisme sous-jacent qui conduit ces systèmes à leur point critique. L'équation de Kardar-Parisi-Zhang modélise l'érosion à grande échelle (>2 km), mais ne s'accorde pas aux observations à plus petite échelle. Un modèle différent, tenant compte de l'anisotropie (la pente d'une montagne), fut donc suggéré. À l'aide du NPRG, je montre que ce modèle possède une ligne de points fixes qui correspond à un domaine continu d'exposants d'échelle. / This manuscript is focused on the study of critical phenomena taking place out-of-equilibrium. In the description of such phenomena, Langevin equations are ubiquitous and are usually derived in a phenomenological way by adding a noise term to a deterministic mean-field equation. However, I show that for reaction-diffusion processes it is in fact possible to derive an exact Langevin equation from the microscopic process. A second part of my thesis work has been devoted to the study of specific nonequilibrium critical phenomena using the nonperturbative renormalization group (NPRG), which is a modern implementation of Wilson and Kadanoff's block-spin idea. This tool, very powerful in an equilibrium context, takes care of the growing spatial fluctuations that arise near criticality through the use of a regulator. In a nonequilibrium context, the temporal fluctuations also have to be controlled. I have therefore designed a regulator that tackles both spatial and temporal fluctuations. Finally, I have applied the NPRG techniques to a model of landscape erosion: indeed, the generic scaling behaviour that appear in erosional landscapes suggests the existence of an underlying mechanism naturally fine-tuned to be critical. The Kardar-Parisi-Zhang equation seems to give a correct model for landscape erosion at large length scale (>2 km), but fails to predict the scaling observed at smaller scale. A different model was thus suggested which takes into account the intrinsic anisotropy at smaller length scale (the slope of the mountain). Using NPRG techniques, I show that this model possesses a line of fixed points associated with a continuous range of scaling exponents.