Fractional Chern insulators (FCIs) are strongly correlated, topological phases of matter that may exist on a lattice in the presence of broken time-reversal symmetry. This thesis explores the link between FCI states and the quantum Hall effect of the continuum in the context of the Hofstadter model, using a combination of nonperturbative, perturbative and numerical methods. We draw links to experimental realisations of topological phases, and go on to consider a novel way of generating general FCI states using strong interactions on a lattice. We begin by considering the Hofstadter model at weak field, where we use a semiclassical analysis to obtain nonperturbative expressions for the band structure and Berry curvature of the single-particle eigenstates. We use this calculation to justify a perturbative approximation, an approach that we extend to the case when the amount of flux per plaquette is close to a rational fraction with a small denominator. We find that eigenstates of the system are single- or multicomponent wavefunctions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalised Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions, and magnetic field strengths. We present numerical simulations of the Hofstadter model relevant to its recent experimental realisation using optical lattices, noting the additional complications that arise in the presence of an external trap. Finally, we show that even if a noninteracting system is topologically trivial, it is possible to stabilise an FCI state by introducing strong interactions that break time-reversal symmetry. We show that this method may also be used to create a (time-reversal symmetric) fractional topological insulator, and provide numerical evidence to support our argument.

Cette thèse porte sur le modèle de Hofstadter i.e., un électron qui se déplace sur un réseau carré couplé à un champ magnétique homogène et perpendiculaire au réseau. Son spectre en énergie est l'un des célèbres fractals de la physique quantique, connu sous le nom "le papillon de Hofstadter". Cette thèse consiste en deux parties principales: la première est l'étude du lien profond entre le modèle de Hofstadter et la distribution de l’aire algébrique entourée par les marches aléatoires sur un réseau carré bidimensionnel. La seconde partie se concentre sur les caractéristiques spécifiques du papillon de Hofstadter et l'étude de la largeur de bande du spectre. On a trouvé une formule exacte pour la trace de l'Hamiltonien de Hofstadter en termes des coefficients de Kreft, et également pour les moments supérieurs de la largeur de bande.Cette thèse est organisée comme suit. Dans le chapitre 1, on commence par la motivation de notre travail. Une introduction générale du modèle de Hofstadter ainsi que des marches aléatoires sera présentée. Dans le chapitre 2, on va montrer comment utiliser le lien entre les marches aléatoires et le modèle de Hofstadter. Une méthode de calcul de la fonction génératrice de l'aire algébrique entourée par les marches aléatoires planaires sera expliquée en détail. Dans le chapitre 3, on va présenter une autre méthode pour étudier ces questions en utilisant le point de vue "point spectrum traces" et retrouver la trace de Hofstadter complète. De plus, l'avantage de cette construction est qu'elle peut être généralisée au cas de "l'amost Mathieu opérateur". Dans le chapitre 4, on va introduire la méthode développée par D.J.Thouless pour le calcul de la largeur de bande du spectre de Hofstadter. En suivant la même logique, on va montrer comment généraliser la formule de la largeur de bande de Thouless à son n-ième moment, à définir plus précisément ultérieurement. / This thesis is about the Hofstadter model, i.e., a single electron moving on a two-dimensional lattice coupled to a perpendicular homogeneous magnetic field. Its spectrum is one of the famous fractals in quantum mechanics, known as the Hofstadter's butterfly. There are two main subjects in this thesis: the first is the study of the deep connection between the Hofstadter model and the distribution of the algebraic area enclosed by two-dimensional random walks. The second focuses on the distinctive features of the Hofstadter's butterfly and the study of the bandwidth of the spectrum. We found an exact expression for the trace of the Hofstadter Hamiltonian in terms of the Kreft coefficients, and for the higher moments of the bandwidth.This thesis is organized as follows. In chapter 1, we begin with the motivation of our work and a general introduction to the Hofstadter model as well as to random walks will be presented. In chapter 2, we will show how to use the connection between random walks and the Hofstadter model. A method to calculate the generating function of the algebraic area distribution enclosed by planar random walks will be explained in details. In chapter 3, we will present another method to study these issues, by using the point spectrum traces to recover the full Hofstadter trace. Moreover, the advantage of this construction is that it can be generalized to the almost Mathieu operator. In chapter 4, we will introduce the method which was initially developed by D.J.Thouless to calculate the bandwidth of the Hofstadter spectrum. By following the same logic, I will show how to generalize the Thouless bandwidth formula to its n-th moment, to be more precisely defined later.

The study of topological phases in condensed matter physics has seen remarkable advancements, primarily focusing on systems with a well-defined bulk and boundary. However, the emergence of topological phenomena on self-similar systems, characterized by the absence of a clear distinction between bulk and boundary, presents a fascinating challenge. This thesis focuses on the topological phases on self-similar systems, shedding light on their unique properties through the lens of adiabatic charge pumping. We observe that the spectral flow in these systems exhibits striking qualitative distinctions from that of translationally invariant non-interacting systems subjected to a perpendicular magnetic field. We show that the instantaneous eigenspectra can be used to understand the quantization of the charge pumped over a cycle, and hence to understand the topological character of the system. Furthermore, we establish a correspondence between the local contributions to the Hall conductivity and the spectral flow of edge-like states. We also find that the edge-like states can be approximated as eigenstates of the discrete angular-momentum operator, with their chiral characteristics stemming from this unique perspective. We also investigate the effect of local structure on the topological phases on self-similar structures embedded in two dimensions. We study a geometry dependent model on two self-similar structures having different coordination numbers, constructed from the Sierpinski gasket. For different non-spatial symmetries present in the system, we numerically study and compare the phases on both structures. We characterize these phases by the localization properties of the single-particle states, their robustness to disorder, and by using a real-space topological index. We find that both structures host topologically nontrivial phases and the phase diagrams are different on the two structures, emphasizing the interplay between non-spatial symmetries and the local structure of the self-similar unit in determining topological phases. Furthermore, we demonstrate the presence of topologically ordered chiral spin liquid on fractals by extending the Kitaev model to the Sierpinski Gasket. We show a way to perform the Jordan-Wigner transformation to make this model exactly solvable on the Sierpinski Gasket. This system exhibits a fractal density of states for Majorana modes and showcases a transition from a gapped to a gapless phase. Notably, the gapped phase features symmetry-protected Majorana corner modes, while the gapless phase harbors robust zero-energy and low-energy self-similar Majorana edge-like modes. We also study the vortex excitations, characterized by remarkable localization properties even in small fractal generations. These localized excitations exhibit anyonic behavior, with preliminary calculations hinting at their fundamental differences from Ising anyons observed in the Kitaev model on a honeycomb lattice.

This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.