In this dissertation, we use exact diagonalization to study a few strongly correlated systems, ranging from the Fermi-Hubbard model to the fractional quantum Hall effect (FQHE). The discussion starts with an overview of strongly correlated systems and what is meant by strongly correlated. Then, we extend cluster perturbation theory (CPT), an economic method for computing the momentum and energy resolved Green's function for Hubbard models to higher order correlation functions, specifically the spin susceptibility. We benchmark our results for the one-dimensional Fermi-Hubbard model at half-filling. In addition we study the FQHE at fillings $nu = 5/2$ for fermions and $nu = 1/2$ for bosons. For the $nu = 5/2$ system we investigate a two-body model that effectively captures the three-body model that generates the Moore-Read Pfaffian state. The Moore-Read Pfaffian wave function pairs composite fermions and is believed to cause the FQHE at $nu = 5/2$. For the $nu = 1/2$ system we estimate the entropy needed to observe Laughlin correlations with cold atoms via an ansatz partition function. We find entropies achieved with conventional cooling techniques are adequate. / Doctor of Philosophy / Strongly correlated quantum many-body physics is a rich field that hosts a variety of exotic phenomena. By quantum many-body we mean physics that is concerned with the behavior of interacting particles, such as electrons, where the quantum behavior cannot be ignored. By strongly correlated, we mean when the interactions between particles are sufficiently strong such that they cannot be treated as a small perturbation. In contrast to weakly correlated systems, strongly correlated systems are much more difficult to solve. That is because methods that reduce the many-body problem to a single independent body problem do not work well. In this dissertation we use exact diagonalization, a method to computationally solve quantum many-body systems, to study two strongly correlated systems: the Hubbard model and the fractional quantum Hall effect.The Hubbard model captures the physics of many interesting materials and is the standard toy model. Originally developed with magnetic properties in mind, it has been extended to study superconductivity, topological phases, cold atoms, and much more. The fractional quantum Hall effect is a novel phase of matter that hosts exotic excitations, some of which may have applications to quantum computing.

Topologically ordered phases flamboyance a cornucopia of intriguing phenomena that cannot be perceived in the conventional phases including the most striking property of hosting anyon quasiparticles having fractional charges and fractional statistics. Such phases were discovered with the remarkable experiment of the fractional quantum Hall effect and are drawing a lot of recognition. Realization of these phases on lattice systems and study of the anyon quasiparticles there are important and interesting avenue to research in unraveling new physics, which can not be found in the continuum, and this thesis is an important contribution in that direction. Also such lattice models hosting anyons are particularly important to control the movement of anyons while experimentally implemented with ultra-cold atoms in optical lattices. We construct lattice models by implementing analytical states and parent Hamiltonians on two-dimensional plane hosting non-Abelian anyons, which are proposed candidates for quantum computations. Such lattice models are suitable to create both quasiholes and quasielectrons in the similar way and thereby avoiding the singularity problem for the quasielectrons in continuum. Anyons in these models are found to be well-screened with proper charges and right statistics. Going beyond two dimensions, we unravel the intriguing physics of topologically ordered phases of matter in fractional dimensions such as in the fractal lattices by employing our model constructions of analytical states and parent Hamiltonians there. We find the anyons to be well-screened with right charges and statistics for all dimensions. Our work takes the first step in bridging the gap between two dimensions and one dimension in addressing topological phases which reveal new physics. Our constructions are particularly important in this context since such lattices lack translational symmetry and hence become unsuitable for the fractional Chern insulator implementations. The special features of topologically ordered phases make these difficult to probe and hence the detection of topological quantum phase transitions becomes challenging. The existing probes suffer from shortcomings uo-to a large extent and therefore construction of new type of probes become important and are on high demand. The robustness of anyon properties draw our attention to propose these as detector of topological quantum phase transitions with significant advantages including the facts that these are numerically cheaper probes and are independent of the boundary conditions. We test our probe in three different examples and find that simple properties like anyon charges detect the transitions. / Topologisch geordnete Phasen extravagieren ein Füllhorn faszinierender Phänomene, die in den herkömmlichen Phasen nicht wahrgenommen werden können, einschließlich der auffälligsten Eigenschaft, Quasiteilchen mit fraktionierten Ladungen und fraktion- ierten Statistiken aufzunehmen. Solche Phasen wurden mit dem bemerkenswerten Exper- iment des fraktionierten Quanten-Hall-Effekts entdeckt und finden viel Anerkennung. Die Realisierung dieser Phasen auf Gittersystemen und die Untersuchung der Anyon- Quasiteilchen sind wichtige und interessante Wege zur Erforschung der Entschlüsselung neuer Physik, die im Kontinuum nicht zu finden sind, und diese These ist ein wichtiger Beitrag in diese Richtung. Auch solche Gittermodelle, die Anyons enthalten, sind beson- ders wichtig, um die Bewegung von Anyons zu steuern, während sie experimentell mit ultrakalten Atomen in optischen Gittern implementiert werden. Wir konstruieren Gittermodelle, indem wir analytische Zustände und Eltern-Hamiltonianer auf einer zwei- dimensionalen Ebene implementieren, die nicht-abelsche Anyons enthält, die als Kan- didaten für Quantenberechnungen vorgeschlagen werden. Solche Gittermodelle sind geeignet, sowohl Quasi-Löcher als auch Quasielektronen auf ähnliche Weise zu erzeu- gen und dadurch das Singularitätsproblem für die Quasielektronen im Kontinuum zu vermeiden. Jeder in diesen Modellen wird mit angemessenen Gebühren und richtigen Statistiken gut überprüft. Über zwei Dimensionen hinaus enträtseln wir die faszinierende Physik topologisch geordneter Phasen der Materie in fraktionierten Dimensionen wie in den fraktalen Gittern, indem wir dort unsere Modellkonstruktionen von analytischen Zuständen und Eltern-Hamiltonianern verwenden. Wir finden, dass die Anyons mit den richtigen Gebühren und Statistiken für alle Dimensionen gut überprüft werden. Unsere Arbeit macht den ersten Schritt, um die Lücke zwischen zwei Dimensionen und einer Dimension zu schließen und topologische Phasen anzugehen, die neue Physik enthüllen. Unsere Konstruktionen sind in diesem Zusammenhang besonders wichtig, da solche Gitter keine Translationssymmetrie aufweisen und daher für die fraktionierten Chern- Isolatorimplementierungen ungeeignet werden. Die besonderen Merkmale topologisch geordneter Phasen machen es schwierig, diese zu untersuchen, und daher wird die Detek- tion topologischer Quantenphasenübergänge schwierig. Die vorhandenen Sonden leiden in hohem Maße unter Mängeln, weshalb die Konstruktion neuer Sondenarten wichtig wird und eine hohe Nachfrage besteht. Die Robustheit der Anyon-Eigenschaften lenkt unsere Aufmerksamkeit darauf, diese als Detektor für topologische Quantenphasenübergänge mit signifikanten Vorteilen vorzuschlagen, einschließlich der Tatsache, dass dies numerisch billigere Sonden sind und von den Randbedingungen unabhängig sind. Wir testen unsere Sonde in drei verschiedenen Beispielen und stellen fest, dass einfache Eigenschaften wie Ladungen die Übergänge erfassen.

Non-Abelian excitations, including Majorana fermions, parafermions, and Fibonacci anyons, provide potential new settings for realizations of topological quantum computation operations. Topological quantum systems have the advantage of being protected against some types of entanglement with the surrounding environment, but their elusive nature has inspired many to pursue rare systems in which they may be physically realized. In this work we present a new platform for production of parafermions in the ν=1/3 fractional quantum hall effect regime in a two-dimensional hole gas in a Gallium Arsenide quantum well, where spin transitions in the rich Γ₈ Luttinger ground state can be manipulated by gate-controlled electric fields. When numerical and analytical calculations of many-particle interactions combine with a proximity-induced superconducting pairing potential in this system, the spin transition we observe gives rise to a superconducting gap with an onset of six-fold degenerate ground state which disappears at critical values of the gap parameter Δ_k, the energetic signature associated with parafermion production.<br>

Dans certains états quantique de la matière, le courant peut être transporté par des porteurs de charges ayant une fraction e* de la charge élementaire. C'est notamment le cas de l'Effet Hall quantique fractionnaire (EHQF) qui se produit pour des systèmes électroniques bidimensionels à basse température et soumis à un fort champ magnetique perpendiculaire. Quand le nombre de quantum de flux en unité h/e est une fraction du nombre d'électrons, le courant se propage le long des bords de l'échantillon sans dissipation. Les porteurs de charges impliqués dans le transport portent une charge fractionnaire. La mise en évidence de ces charges peut être faite via les faibles fluctuations de courant dûes à la granularité de la charge. Nous présentons ici une méthode fiable de mesure de la charge fractionnaire basée sur des correlations croisées de fluctuations de courant. La dynamique de ces charges fractionnaires lorsque l'échantillon est irradié avec des photons GHz est étudiée, permettant la mesure de la fréquence Josephson des charges fractionnaires. Ces mesures valident les processus photo-assisté en régime d'EHQF et permettent une manipulation résolue en temps des charges fractionnaires, dans le but de réaliser une source d'anyon sur le principe du léviton afin de réaliser des tests de la statistique anyonique de ces charges fractionnaires. / In some quantum matter states, the current may remarkably be transported by carriers that bear a fraction e* of the elementary electron charge. This is the case for the Fractional quantum Hall effect (FQHE) that happens in two-dimensional systems at low temperature under a high perpendicular magnetic field. When the number of magnetic flux in units of h/e is a fraction of the number of electron, a dissipationless current flows along the edges of the sample and is carried by anyons with fractional charge. The observation of the fractional charge is realized through small current fluctuations produced by the granularity of the charge. Here is presented a reliable method to measure the fractional charge by the mean of cross-correlation of current fluctuations. Moreover, the dynamical properties of those charges is probed when the sample is irradiated with photos at GHz frequency. The long predicted Josephson frequency of the fractional charge is measured. Those measurements validate Photoassisted processes in the FQHE and enable timedomain manipulation of fractional charges in order to realize a single anyon source based on levitons to perform tests of the anyonic statistics of fractional charge.

Dans cette thèse, je présente une étude que j'ai réalisée à l'université Paris-sud sous la direction de Thierry Jolicœur sur les phases des Hall Quantiques Fractionnaire (HQF) dans la géométrie du cylindre.Après une rapide introduction dans le premier chapitre, je présente dans le second quelques concepts de base de l'effet HQF et j'introduit certains aspects de la géométrie cylindrique.Le chapitre 3 est consacré à l'étude de la limite du cylindre fin, c'est à dire lorsque la circonférence du cylindre est de l'ordre de quelques longueurs magnétiques. Dans cette limite, on sait que la fonction d'onde de Laughlin au remplissage 1/q se réduit à un cristal unidimensionnel, où une orbitale sur q est occupée. Dans le but d'étudier un limite intermédiaire, nous conservons les quatre premiers termes du développement de l’Hamiltonien lorsque la circonférence est petite devant la longueur magnétique. On trouve alors une expression exacte de l'état fondamental au moyen d'opérateurs de "squeezing" ou de produits de matrices. Nous trouvons également une écriture similaire pour les quasi- trous, les quasi-électron et la branche magnétoroton.Dans les chapitres 4 et 5, je me concentre sur l'étude des excitations de bord chirales des phases de HQF. Je présente une étude microscopique de ces états de bord dans la géométrie du cylindre, lorsque les quasi-particules peuvent passer d'un bord à l'autre par effet tunnel. J'étudie d'abord dans le chapitre 4 la phase de HQF principale dont l'état fondamental est bien décrit par la fonction d'onde de Laughlin. Pour un échelle d'énergie plus faible que le gap du volume, le théorie effective est donnée par un fluide d'électrons unidimensionnel bien particulier : un liquide de Luttinger chiral. À l'aide de diagonalisations numériques exactes, nous étudions le spectre des états de bord formé de le combinaison des deux bord contre-propageant sur chacun des cotés du cylindre. Nous montrons que les deux bords se combinent pour former un liquide de Luttinger non-chiral, où le terme de courant reflète le transfert de quasi-particules entre les bords. Cela nous permet d'estimer numériquement les paramètre de Luttinger pour un faible nombre de particules, et nous trouvons une valeur cohérente avec la théorie de X. G. Wen.J'analyse ensuite dans le chapitre 5 les modes de bord des phases de HQF au remplissage 5/2. À partir une construction basée sur la Théorie des Champs Conformes (TCC), Moore et Read (Nucl. Phys. B, 1991) ont proposé que la physique essentielle de cette phase soit décrite par un état apparié de fermion composites. Une propriété importante de cet état est que ses excitations émergentes permutent sous une statistique non-abéliène. Lorsqu'elles sont localisées sur les bords, ces excitations sont décrites par un boson chiral et un fermion de Majorana. Dans la géométrie du cylindre, nous montrons que le spectre des excitations de bord est fomé des tours conformes du modèle IsingxU(1). De plus, par une méthode Monte-Carlo, nous estimons les différentes dimensions d'échelle sur des grands systèmes (environ 50 électrons), et nous trouvons des valeurs en accord avec les prédictions de la TCC.Dans le dernier chapitre de ce manuscrit, je présente un travail que j'ai réalisé à UBC (Vancouver) en collaboration avec Marcel Franz sur les phase de Hall quantiques de spin induites dans le graphène par des adatomes. Dans ce système, les adatomes induisent un couplage spin-orbite sur les électrons des la feuille de graphène et introduisent du désordre qui est susceptible de détruire le gap spectral. Nous montrons dans ce chapitre que le gap spectral est préservé lorsque des valeurs réalistes de paramètres sont usités. De plus, au moyen de calculs analytiques à base énergie et de diagonalisations numériques exactes, nous identifions un signal caractéristique dans la densité d'états locale mettant en évidence la présence d'un gap topologique. Ce signal pourrait être observé au moyen d'un microscope à effet tunnel. / I present in this thesis a study that I did in the university Paris-sud under the supervision of Thierry Jolicœur onto Fractional Quantum Hall (FQH) phases in the cylinder geometry. After a short introduction in the first chapter, I present some basic concept relative to the FQH effect in the second one and introduce some essential features relative to the cylinder geometry, useful for the chapters 3, 4, and 5. The chapter 3 is dedicated to the study of the thin cylinder limit, i.e. when the circumference of the cylinder is of the order of a few magnetic length. In this limit, it is known that the Laughlin wave function at the filling factor 1/q is reduced to a one dimensional crystal in the lowest Landau level orbitals where one every q orbitals is occupied. We Taylor expand the Hamiltonian when the circumference is small compare to the magnetic length in order to study an intermediate limit. When only the first four terms of the development are kept, it is possible to find exact representations of the ground state with "squeezing" operators or matrix products. We also find similar representations for quasiholes, quasielectrons and the magnetorton branch. These results have been published in the article Phys. Rev. B 85, 155116 (2012). In the chapter 4 and 5 I focus onto the gapless chiral edge excitations of FQH phases. I present a microscopic study of those edges states in the cylindrical geometry where quasiparticles are able to tunnel between edges. I first study the principal FQH phase at the filling fraction 1/3 whose ground state is well described by the Laughlin wave function in the chapter 4. For an energy scale lower than the bulk gap, the effective theory is given by a very peculiar one dimensional electron fluid localized at the edge: a chiral Luttinger liquid. Using numerical exact diagonalizations, we study the spectrum of edge modes formed by the two counter-propagating edges on each side of the cylinder. We show that the two edges combine to form a non-chiral Luttinger liquid, where the current term reflects the transfer of quasiparticles between edges. This allows us to estimate numerically the Luttinger parameter for a small number of particles and find it coherent with the one predicted by X. G. Wen theory. We published this work in Phys. Rev. B 86, 115214 (2012). I then analyze edge modes of the FQH phase at filling fraction 5/2 in the chapter 5. From a Conformal Field Theory (CFT) based construction, Moore and Read (Nucl. Phys. B, 1991) proposed that the essential physics of this phase is described by a paired state of composite fermions. A striking property of this state is that emergent excitations braid with non-Abelian statistics. When localized along the edge, those excitations are described through a chiral boson and a Majorana fermion. In the cylinder geometry, we show that the spectrum of edge excitations is composed of all conformal towers of the IsingxU(1) model. In addition, with a Monte Carlo method, we estimate the various scaling dimensions for large systems (about 50 electrons), and find them consistent with the CFT predictions.In the last chapter of my manuscript, I present a work that I did in UBC (Vancouver) in collaboration with Marcel Franz onto quantum spin Hall phases in graphene induced by adatoms. In this system, adatoms induce a spin orbit coupling for electrons in the graphene sheet and create some disorder which might be responsible for destruction the spectral gap. We show in this chapter and in the article [Phys. Rev. B 89, 201410(R) (2014)] that the spectral gap remains open for a realistic range of parameters. In addition, with analytical computations in the low energy approximation and numerical exact diagonalizations, we find characteristic signal in the local density of states highlighting the presence of topological gap. This signal might be observed in scanning tunneling spectroscopy experiments.

We study three types of theories, using supersymmetry and ideas from string theory as tools to gain understanding of systems of more general interest. Firstly, we introduce non-relativistic Chern-Simons-matter field theories in three dimensions and study their anyonic spectrum in a conformal phase. These theories have supersymmetric completions, which in the non-relativistic case suffices to protect certain would-be BPS quantities from corrections. This allows us to compute one-loop exact anomalous dimensions of various bound states of non-Abelian anyons, analyse some interesting unitarity bound violations, and test some recently proposed bosonization dualities. Secondly, we turn on a chemical potential and break conformal invariance, putting the theory into the regime of the Fractional Quantum Hall Effect (FQHE). This is illustrated in detail: the theory supports would-be BPS vortices which model the electrons of the FQHE, and they form bag-like states with the appropriate filling fractions, Hall conductivities, and anyonic excitations. This formalism makes possible some novel explicit computations: an analytic calculation of the anyonic phases experienced by Abelian quasiholes; analytic relationships to the boundary Wess-Zumino-Witten model; and derivations of a wide class of QHE wavefunctions from a bulk field theory. We also further test the three-dimensional bosonization dualities in this new setting. Along the way, we accumulate new descriptions of the QHE. Finally, we turn away from flat space and investigate a problem in (3+1)-dimensional quantum gravity. We find that even as an effective theory, the theory has enough structure to suggest the inclusion of certain gravitational instantons in the path integral. An explicit computation in a minimally supersymmetric case illustrates the principles at work, and highlights the role of a hitherto unidentified scale in quantum gravity. It also is an interesting result in itself: a non-perturbative quantum instability of a flat supersymmetric Kaluza-Klein compactification.

The quantum Hall (QH) system---cold electrons in two dimensions in a perpendicular magnetic field---is a striking example of a system where unexpected phenomena emerge at low energies. The low-energy physics of this system is effectively one-dimensional due to the magnetic field. We identify an exactly solvable limit of this interacting many-body problem, and provide strong evidence that its solutions are adiabatically connected to the observed QH states in a similar manner as the free electron gas is related to real interacting fermions in a metal according to Landau's Fermi liquid theory. The solvable limit corresponds to the electron gas on a thin torus. Here the ground states are gapped periodic crystals and the fractionally charged excitations appear as domain walls between degenerate ground states. The fractal structure of the abelian Haldane-Halperin hierarchy is manifest for generic two-body interactions. By minimizing a local k+1-body interaction we obtain a representation of the non-abelian Read-Rezayi states, where the domain wall patterns encode the fusion rules of the underlying conformal field theory. We provide extensive analytical and numerical evidence that the Laughlin/Jain states are continuously connected to the exact solutions. For more general hierarchical states we exploit the intriguing connection to conformal field theory and construct wave functions that coincide with the exact ones in the solvable limit. If correct, this construction implies the adiabatic continuation of the pertinent states. We provide some numerical support for this scenario at the recently observed fraction 4/11. Non-QH phases are separated from the thin torus by a phase transition. At half-filling, this leads to a Luttinger liquid of neutral dipoles which provides an explicit microscopic example of how weakly interacting quasiparticles in a reduced (zero) magnetic field emerge at low energies. We argue that this is also smoothly connected to the bulk state.

La spectroscopie d'intrication, initialement introduite par Li et Haldane dans le contexte de l'effet Hall quantique fractionnaire, a suscité un large éventail de travaux. Le spectre d'intrication est le spectre de la matrice de densité réduite, quand on partitionne le système en deux. Pour de nombreux systèmes quantiques, il révèle une caractéristique unique : calculé uniquement à partir de la fonction d'onde de l'état fondamental, le spectre d'intrication donne accès à la physique des excitations de bord. Dans ce manuscrit, nous donnons un apercu de la spectroscopie d'intrication. Nous introduisons les concepts de base dans le cas des chaînes de spins quantiques. Nous présentons une étude approfondie des spectres d'intrication appliqués aux phases de l'effet Hall quantique fractionnaire, montrant quel type d'information est encodé dans l'état fondamental et comment les différentes facons de partitionner le système permettent de sonder différents types d'excitation. Comme application pratique de cette technique, nous discutons de la manière dont cette technique peut aider à faire la distinction entre les différentes phases qui émergent dans les isolants de Chern en interaction forte.

Strong quantum correlations between many particles in low dimensions lead to emergence of interesting phases of matter. These phases are often studied through the properties of the many-body eigenstates of an interacting quantum many-body system. The folklore example of topological order in the ground states is the fractional quantum Hall (FQH) effect. With the current developments in the field of ultracold atoms in optical lattices, realizing FQH physics on a lattice and being able to create and braid anyons is much awaited from the view point of fault tolerant quantum computing. This thesis contributes to the field of FQH effect and anyons in a lattice setting. Conformal field theory has been useful to build interesting lattice FQH models which are few-body and non-local. We provide a general scheme of truncation to arrive at tractable local models whose ground states have the desired topological properties. FQH models are known to host anyons, but, it is a hard task when it comes to braiding them on small sized lattices with edges. To get around this problem, we demonstrate that one can squeeze the anyons and braid them successfully within a smaller area by crawling them like snakes on modest sized open lattices. As a numerically cheap approach to detect topological quantum phase transitions, we again resort to anyons that are only well defined in a topological phase. We create defects and study a simple quantity such as the charge of the defect to test whether the phase supports anyons or not. On the other hand, with the advent of many-body localization (MBL) and quantum many-body scars, interesting eigenstate phases which were otherwise only known to occur in ground states have been identified even at finite energy densities in the many-body spectra of generic systems. This thesis also contributes to the field of non-equilibrium physics by portraying models that display interesting non-ergodic phases and also quantum many-body scars. For instance, we show that an emergent symmetry in a disordered model can be used as a tool to escape MBL in a single eigenstate while not preventing the rest of the states from localizing. This can lead to an interesting situation of weakly broken MBL phase where a non-MBL state lives in the spectrum of MBL like states. We also demonstrate the emergence of a non-ergodic, but also a non-mbl phase in a non-local model with SU(2) symmetry. We provide two constructions of rather different models with quantum many-body scars with chiral and non-chiral topological order.

Les progrès technologiques dans la fabrication des semi-conducteurs permettent, depuis le début des années 80, de réaliser des dispositifs dans lesquels les électrons sont fortement confinés dans un plan, on parle de système d'électrons bidimensionnels. L'application d'un champ magnétique perpendiculaire intense à ce système permit l'observation des effets Hall quantiques (EHQ), entier en 1980 puis fractionnaire en 1982. En présence du champ magnétique et aux températures extrêmement faibles qui sont concernées, le spectre énergétique des électrons bidimensionnels est quantifié en niveaux de Landau macroscopiquement dégénérés. Le comportement du système est alors déterminé par le facteur de remplissage des niveaux de Landau. L'EHQ entier apparaît autour des valeurs de champ magnétiques qui correspondent à un remplissage entier des niveaux Landau, tandis que son pendant fractionnaire est obtenu autour de certaines fractions du facteur de remplissage ν (ν =1/3, 2/5, 5/2, …) . Alors qu'à remplissage ν entier c'est le comportement individuel des électrons qui gouverne le comportement du système, aux facteurs de remplissage fractionnaires les corrélations électroniques dominent. En raison de ce caractère fortement corrélé, l'EHQ fractionnaire sous-tend un effort de recherche expérimental et théorique important depuis sa découverte. En effet, dans le régime fractionnaire les corrélations fortes induisent des propriétés inédites telles l'existence de quasi-particules de charge fractionnaire, mais elles rendent également la description théorique du système ardue. En 1983, Robert Laughlin proposa une fonction d'onde variationnelle modèle pour la description de l'EHQ fractionnaire observé à remplissage ν=1/3, dont il discuta la validité au regard d'une étude numérique approfondie des interactions entre les électrons. Le succès de cette méthode l'éleva au rang de paradigme, et de nombreuses fonctions d'onde d'essai ont depuis été proposées pour l'explication des effets Hall quantiques observés aux autres facteurs de remplissages. Notamment, la fonction d'onde de Moore et Read s'avère pertinente pour la description de l'EHQ observé à demi-remplissage du second niveau de Landau. Celle-ci suggère l'existence de quasi-particules non-abéliennes qui génère des espoirs importants de par ses applications potentielles en informatique quantique protégée topologiquement. Bien que l'EHQ ait également été observé à demi-remplissage du plus bas niveau de Landau, la nature de l'état sous-jacent est encore débatue. Celui-ci n'est observé que dans les systèmes bicouches et dans les puits larges qui sont au centre de ce travail de thèse. Les puits larges désignent les systèmes dans lesquels l'épaisseur du système d'électrons bidimensionnel ne peut plus être négligée, typiquement à des épaisseurs de l'ordre de 100 nm. En raison du potentiel de confinement ressenti par les électrons, leurs niveaux d'énergies dans la direction du confinement sont quantifiés en sous-bandes. Dans un puits extrêmement fin seule la plus basse sous-bande est peuplée et le degré de liberté correspondant est alors gelé, mais dans les puits large les sous-bandes excitées sont pertinentes. Dans ces conditions l'EHQ fractionnaire à demi-remplissage peut également résulter de la stabilisation d'un état à deux composantes qui peuple les sous-bande excitées. Cet état proposé par Bertrand Halperin en 1983 entre en compétition avec l'état de Moore et Read. En plus de ces deux états, un état métallique de fermions composite est possible, ainsi qu'un cristal électronique de Wigner au comportement isolant. La compétition entre ces différents états est arbitrée par une étude de Monte-Carlo variationnel combinée à des calculs de diagonalisation exacte. La nature de l'état qui est stabilisé dépend de la nature du potentiel de confinement. Dans ce manuscrit de thèse sont discutés les dispositifs de la bicouche, du puits large, ainsi que du puits large en présence d'un biais externe. / Due to technological advances in the manufacture of semiconductors enable, in it possible since the early 80s to create devices in which electrons are strongly confined in a plane, thus effectively realizing a two-dimensional electron system. The application of a strong perpendicular magnetic field to this system led to the observation of the integer quantum Hall effect (QHE) in 1980 and fractional QHE in 1982. Under a strong magnetic field the energy spectrum of the two-dimensional electrons is quantified in Landau levels that are macroscopically degenerate, and the behavior of the system is governed by the filling factor of Landau levels. The integer QHE appears around magnetic field values ​​which correspond to an integer filling of the Landau levels, while the fractional equivalent is obtained around certain fractions of the filling factor ν (ν = 1/3, 2/5, 5 / 2, ...). Although for integers values of ν is the individual behavior of electrons dictates the behavior of the system, the fractional filling factors the electronic correlations dominate. Because of those strong correlations, the underlying fractional QHE motivates an important experimental and theoretical research effort since its discovery. Indeed, in the fractional regime the strong correlations induce novel properties such as the existence fractionally-charged quasiparticles, but they also make the theoretical description of the system laborious. In 1983 Robert Laughlin proposed a variational wave function model for the description of the QHE observed at fractional filling ν = 1/3. He discussed the validity of this trial wave function in a comprehensive numerical study of interactions between electrons. The success of this method made it a paradigm, and many test wave functions have been proposed since then for the explanation of quantum Hall effects observed with other fillings factors. In particular, the wave function of Moore and Read is relevant for the description of the QHE observed at half-filling the second Landau level. This suggests the existence of non-Abelian quasiparticles with potential applications in topologically-protected quantum computing. QHE has also been observed at half filling the lowest Landau level, but the nature of the underlying quantum state is still debated; it is observed that in bilayer systems and wells wide. The large wells, which are the focus of this thesis, refer to systems in which the thickness of the two-dimensional electron system cannot be trivially neglected and usually corresponds to a thickness of about 100 nm. Due to the confinement potential felt by the electrons, their energy levels in the direction of confinement are quantized in sub-bands. In a narrow well only the lowest subband is populated and the corresponding degree of freedom is thus frozen, but in a wide well the excited sub-bands are relevant. Under these conditions fractional QHE at half-filling can also result from the stabilization of a two-state components that also populates the excited sub-band. The corresponding trial state, proposed by Bertrand Halperin in 1983, competes with the state of Moore and Read. In addition to these two states, a metal composite fermion state is a relevant trial state as well as an electronic Wigner crystal, the latter behaving as an insulator. The competition between these states is refered by a variational Monte-Carlo study combined with exact diagonalization calculations. The nature of the state that is stabilized depends on the nature of the confinement potential. In this PhD thesis three confinement potentials are studied: the bilayer, the wide well, and the wide well in the presence of an external bias.