No description available.

The prediction and the experimental discovery of topological insulators has set the stage for a novel type of electronic devices. In contrast to conventional metals or semiconductors, this new class of materials exhibits peculiar transport properties at the sample surface, as conduction channels emerge at the topological boundaries of the system. In specific materials with strong spin-orbit coupling, a particular form of a two-dimensional topological insulator, the quantum spin Hall state, can be observed. Here, the respective one-dimensional edge channels are helical in nature, meaning that there is a locking of the spin orientation of an electron and its direction of motion. Due to the symmetry of time-reversal, elastic backscattering off interspersed impurities is suppressed in such a helical system, and transport is approximately ballistic. This allows in principle for the realization of novel energy-efficient devices, ``spintronic`` applications, or the formation of exotic bound states with non-Abelian statistics, which could be used for quantum computing. The present work is concerned with the general transport properties of one-dimensional helical states. Beyond the topological protection mentioned above, inelastic backscattering can arise from various microscopic sources, of which the most prominent ones will be discussed in this Thesis. As it is characteristic for one-dimensional systems, the role of electron-electron interactions can be of major importance in this context. First, we review well-established techniques of many-body physics in one dimension such as perturbative renormalization group analysis, (Abelian) bosonization, and Luttinger liquid theory. The latter allow us to treat electron interactions in an exact way. Those methods then are employed to derive the corrections to the conductance in a helical transport channel, that arise from various types of perturbations. Particularly, we focus on the interplay of Rashba spin-orbit coupling and electron interactions as a source of inelastic single-particle and two-particle backscattering. It is demonstrated, that microscopic details of the system, such as the existence of a momentum cutoff, that restricts the energy spectrum, or the presence of non-interacting leads attached to the system, can fundamentally alter the transport signature. By comparison of the predicted corrections to the conductance to a transport experiment, one can gain insight about the microscopic processes and the structure of a quantum spin Hall sample. Another important mechanism we analyze is backscattering induced by magnetic moments. Those findings provide an alternative interpretation of recent transport measurements in InAs/GaSb quantum wells. / Mit der Vorhersage und der experimentellen Entdeckung von topologischen Isolatoren wurde die Grundlage für eine vollkommen neue Art von elektronischen Bauelementen geschaffen. Diese neue Klasse von Materialien zeichnet sich gegenüber herkömmlichen Metallen und Halbleitern durch besondere Transporteigenschaften der Probenoberfläche aus, wobei elektrische Leitung in Randkanälen an den topologischen Grenzflächen des Systems stattfindet. Eine spezielle Form des zweidimensionalen topologischen Isolators stellt der Quanten-Spin-Hall-Zustand dar, welcher in bestimmten Materialien mit starker Spin-Bahn-Kopplung beobachtet werden kann. Die hier auftretenden eindimensionalen Leitungskanäle sind von helikaler Natur, was bedeutet, dass die Orientierung des Spins eines Elektrons und seine Bewegungsrichtung fest miteinander gekoppelt sind. Aufgrund von Symmetrien wie Zeitumkehr ist elastische Rückstreuung an eventuell vorhandenen Störstellen in solchen helikalen Kanälen verboten, sodass elektrische Leitung als nahezu ballistisch betrachtet werden kann. Prinzipiell bieten sich dadurch neue Möglichkeiten zur Konstruktion von energieeffizienten Transistoren, “Spintronik“-Bauelementen, oder zur Erzeugung von speziellen Zuständen, die für den Betrieb eines Quantencomputers benutzt werden könnten. Die vorliegende Arbeit beschäftigt sich mit den allgemeinen Transporteigenschaften von eindimensionalen, helikalen Randzuständen. Neben dem oben erwähnten topologischen Schutz gibt es zahlreiche Störquellen, die inelastische Rückstreuprozesse induzieren. Die wichtigsten davon werden im Rahmen dieser Dissertation beleuchtet. Entscheidend wirkt hierbei oft die Rolle von Elektron-Elektron-Wechselwirkungen, welche in eindimensionalen Systemen generell von großer Bedeutung ist. Zunächst werden bewährte Techniken der Festkörperphysik wie etwa Abelsche Bosonisierung (mithilfe derer Wechselwirkungen in einer Raumdimension exakt berücksichtigt werden können), die Theorie von Luttinger Flüssigkeiten, oder die störungstheoretische Renormierungsgruppenanalyse rekapituliert. Diese Methoden werden im Weiteren benutzt, um die Korrekturen zum Leitwert eines helikalen Transportkanals zu berechnen, welche aufgrund von ausgewählten Störungen auftreten können. Ein Fokus liegt hierbei auf dem Zusammenspiel vonWechselwirkungen und Rashba Spin-Bahn-Kopplung als Quelle inelastischer Ein-Teilchen- oder Zwei-Teilchen-Rückstreuung. Mikroskopische Details wie etwa die Existenz einer Impulsobergrenze, welche das Energiespektrum beschränkt, oder die Anwesenheit von wechselwirkungsfreien Spannungskontakten, sind dabei von grundsätzlicher Bedeutung. Die charakteristische Form der vorhergesagten Korrekturen kann dazu dienen, die Struktur und die mikroskopischen Vorgänge im Inneren einer Quanten-Spin- Hall-Probe besser zu verstehen. Ein weiterer grundlegender Mechanismus ist Rückstreuung verursacht durch magnetische Momente. Aus der entsprechenden Analyse der Korrekturen zur Leitfähigkeit ergeben sich interessante Übereinstimmungen mit aktuellen Experimenten in InAs/GaSb Quantentrögen.

Cette thèse est consacrée à l'étude du transport dans les nanostructures quantiques unidimensionnelles dont les propriétés sont étudiées en s'appuyant notamment sur le bruit en excès à fréquence finie. La première partie de cette thèse est consacrée à l'étude du transport à travers une impureté dans un liquide de Luttinger couplée à un environnement électromagnétique arbitraire. L'impureté est traitée dans deux limites de transmission, la limite tunnel et la limite de faible rétrodiffusion. Les calculs sont menés dans le formalisme de Keldysh. Nous montrons ainsi que la théorie du blocage de Coulomb dynamique, établie pour une jonction tunnel couplée à un environnement à l'équilibre, demeure valide pour un liquide de Luttinger. Par ailleurs nous montrons que les relations de fluctuation dissipation reliant le bruit à fréquence finie au courant reste valide. Nous montrons que cette théorie peut également s'étendre dans la limite de faible rétrodiffusion à condition de prendre en compte la rétro-action du liquide électronique sur l'environnement. En revanche, les relations de fluctuation dissipation ne sont respectées que pour le bruit en émission. Dans une seconde partie nous intéressons effets d'une modulation radiofréquence sur les propriétés de transport des mêmes systèmes. Nous montrons notamment que ces effets peuvent être décrit par une théorie du blocage de Coulomb dynamique effective en convoluant la statistique d'absorption de photon avec la statistique de Tien-Gordon. Notons cependant que les relations de fluctuation dissipation ne sont plus vérifiées. Ces prédictions théoriques sont comparées aux résultats expériments obtenus par une équipe du SPEC au CEA de Saclay. Enfin nous étudions les propriétés de transport lorsque l'environnement, ici un oscillateur harmonique, est maintenu dans un état excité. Nous montrons que la présence de photons autorise d'une part le processus photo-assistés mais favorise également l'absorption de photons par des processus de bunching. Nous montrons finalement que les propriétés du transport s'obtiennent en convoluant la loi de Poisson du blocage de Coulomb avec la fonction caractéristique de Glauber de l'état peuplant l'oscillateur, menant à des statistiques exotiques. Ce dernier point nous permet d'utiliser ce système comme un détecteur d'état quantique. / This thesis discusses electronic transport in uni-dimensional quantum systems whose properties are studied with an extensive use of the finite-frequency non symmetrised excess noise. The first part focuses on transport through an impurity embedded in a Luttinger liquid coupled to an arbitrary electromagnetic environment. The impurity is treated in two paradigmatic situations : The tunneling and the weak backscattering regime. The out-of-equilibrium situation is dealt with the Keldysh Formalism. We show that the dynamical Coulomb blockade theory, extends to the case of a a tunnel junction between Luttinger liquids. Besides, fluctuations dissipation relations that link noise noise and current remain valid. In the transparent regime, we show that the dynamical Coulomb blockade theory applies to the backscattering current albeit back-action effects of the electronic liquid on the electromagnetic environment that have to be taken into account. Fluctuation-dissipation relations remain valid only for the emission noise. The second part focuses on the effects of a micro-wave modulation on the transport properties of the transport properties of these systems. An effective dynamical Coulomb blockade can be obtained by convolving the statistic of absorption of the environment with the Tien-Gordon statistic. Yet, the fluctuation dissipation relations are not verified in this case. These predictions are compared to the experimental results obtained by a team of the SPEC at the CEA Saclay. Last, we study the transport properties of a tunnel junction coupled to a harmonic oscillator maintained in an excited state. We show that the photons within the cavity lead to two distinct processes: photo-assisted transport that enhance the conductance, and bunching effects that enhance the probability to absorbe a large number of photons. An effective dynamical Coulomb blockade theory can also be derived by convolving the Poisson distribution with Glauber characteristic function leading to exotic statistics. These can be probed by excess noise which can thus be used as a quantum state detector.

No description available.

Dans les systèmes quantiques unidimensionnels, le rôle des fluctuations et des interactions est plus important et les théories utilisées à plus haute dimension ne peuvent plus être employées. Le point de départ pour décrire la plupart des systèmes unidimensionnels est la théorie du liquide de Luttinger. Bien que cette théorie décrive de nombreux phénomènes avec succès, elle a aussi ses limites. Par exemple, elle ne peut décrire que la limite de basse énergie d'un system unidimensionnel, elle échoue aussi lorsqu'il s'agit de décrire la désintégration des excitations du système. Dans cette thèse, nous étudions principalement deux types de problème en une dimension. Le premier est l'interaction effective entre des impuretés dans un liquide de bosons tandis que le deuxième est la désintégrations des quasi-particules dans un mélange bosons-fermions. Dans les deux cas, décrire le système comme un liquide de Luttinger n'est pas suffisant. Afin de pallier à cela, nous développons plusieurs approches pour ces systèmes unidimensionnels qui prennent en compte les différentes anharmonicités nécessaires afin de capturer les mécanismes importants en jeu dans ces problèmes. / In one-dimensional quantum systems, the role of fluctuations and interactions is enhanced and theories used in higher- dimensional systems cannot be employed anymore to describe such strongly-correlated systems. The starting point to describe most one-dimensional systems is the Luttinger liquid theory. Even though this theory is successful to describe many phenomena, it has its shortcomings. For example, it can only treat the low-energy limit of one-dimensional systems and fails to describe the decay of excitations. In this thesis, we mainly study two kinds of problems in one dimension. The first one is the effective interaction between impurities in a Bose liquid whereas the second one is the decay of quasiparticles in a Bose-Fermi mixture. In both cases, the description of the system in terms of a Luttinger liquid is not sufficient. To overcome this, we develop different approaches for these one-dimensional systems to account for the various anharmonicities which are necessary to capture the relevant physics of these problems.

This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.

Single-wall carbon nanotubes (SWCNTs) are a fascinating material because they exhibit many outstanding properties. Due to their unique geometric structure, they are a paradigm for one-dimensional systems. Furthermore, depending on their chirality, they can be either metallic or semiconducting. The SWCNT are arranged in bundles of some ten nanotubes with a random distribution of semiconducting and metallic tubes. They are thus one-dimensional objects embedded in a three-dimensional arrangement, the bundles. In this thesis, the metallic ground state of one-dimensional (1D) and three-dimensional (3D) systems is investigated on the basis of SWCNTs, using angle-integrated photoemission spectroscopy. In particular, a transition from a 1D to a 3D metallic system, induced by a charge transfer, is studied on SWCNTs and C60 peapods. In general, the metallic ground state of materials is greatly influenced by correlation effects. In classical three-dimensional metals, electron-electron interaction mainly leads to a renormalization of the charge carrier properties (e.g. effective mass), as described in Landau's Fermi liquid theory. One-dimensional metals are influenced to a greater extent by interactions. In fact, the Landau-quasiparticle picture breaks down due to the Peierls instability. Instead, one-dimensional metals are described by Tomonaga-Luttinger liquid (TLL) theory which predicts unusual properties such as spin-charge separation and non-universal power laws in some physical properties such as the electronic density of states (DOS). Angle-integrated photoemission spectroscopy provides direct access to the DOS and as such directly addresses the power law renormalization of a TLL. It is first shown, that the bundles of single-wall carbon nanotubes indeed exhibit a power law scaling of the electronic density of states is observed as it is expected from TLL theory. The main part of the thesis is devoted to the investigation of the metallic ground state of SWCNTs upon functionalization. In general, functionalization is a controlled modification of the structural and/or electronic properties of SWCNT. It can be carried out e.g. by doping with electron donors or acceptors, by filling the nanospace inside the tubes with molecules or by substituting carbon atoms. First, the behavior of the SWCNT upon chemical doping was probed. The overall modification of the electronic band structure can be explained well by a rigid band shift model. The one-dimensional character of the metallic tubes in the bundle is retained at low doping, but when the semiconducting tubes in the sample are also rendered metallic by the charge transfer, a Fermi edge emerges out of the power law renormalization of the spectral weight, signifying a transition to a three-dimensional metallic behavior. This can be explained by an increased interaction between the tubes in the bundle. A crossover from a Tomonaga-Luttinger liquid to a Fermi liquid is observed. The filling of SWCNTs with C60 molecules leads to the formation of so-called peapods. It raises questions concerning the role of the additional bands originating from the C60 filling in the one-dimensional system. In the pristine state, the states of the C60 filling were found to have no influence on the metallic ground state. The TLL power law scaling of the density of states is observed. The overall interaction between the SWCNT host and the C60 filling is small. Upon doping however, the modified band structure leads to a qualitative change in the crossover from a TLL to a Fermi liquid. Upon doping, also states in the conduction band of the C60 are filled. The evolution of the power law scaling at intermediate doping can be interpreted as an opening of an additional conduction channel of one-dimensional metallic chains of C60 inside the tubes. This is in good agreement with transport experiments. Upon further doping, a Fermi edge similar to the highly doped SWCNTs is observed.

Made available in DSpace on 2014-12-17T15:14:53Z (GMT). No. of bitstreams: 1 RonivonLE_DISSERT.pdf: 2244591 bytes, checksum: e86660db1b3ec37036777503cc5e8a0d (MD5) Previous issue date: 2010-12-20 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / In this work we investigate the effect of a BCS-type pairing term for free spinless fermions, with a propensity to form a condensate of pairs in a 1+1 dimension. Using the of bosonization technique we explore the possible condition of existence of quasiparticles in a superconducting state. Although there is no spontaneous breaking of chiral symmetry the propagator of one-particle fermion is massive and, in fact, resembles the one-particle Green s function of conventional quasiparticles / Neste trabalho investigamos o efeito de um termo de emparelhamento do tipo BCS para f?rmions livres sem spins, com propens?o a formar um condensado de pares em uma dimens?o 1+1. Utilizando a t?cnica de bosoniza??o vamos explorar a poss?vel condi??o de exist?ncia de quasipart?culas em um estado supercondutor. Embora n?o haja nenhuma quebra espont?nea de simetria quiral o propagador de 1-part?cula fermi?nica ? massivo e de fato assemelha-se a fun??o de Green de 1-part?cula de uma quasipart?cula convencional

Single-wall carbon nanotubes (SWCNTs) are a fascinating material because they exhibit many outstanding properties. Due to their unique geometric structure, they are a paradigm for one-dimensional systems. Furthermore, depending on their chirality, they can be either metallic or semiconducting. The SWCNT are arranged in bundles of some ten nanotubes with a random distribution of semiconducting and metallic tubes. They are thus one-dimensional objects embedded in a three-dimensional arrangement, the bundles. In this thesis, the metallic ground state of one-dimensional (1D) and three-dimensional (3D) systems is investigated on the basis of SWCNTs, using angle-integrated photoemission spectroscopy. In particular, a transition from a 1D to a 3D metallic system, induced by a charge transfer, is studied on SWCNTs and C60 peapods. In general, the metallic ground state of materials is greatly influenced by correlation effects. In classical three-dimensional metals, electron-electron interaction mainly leads to a renormalization of the charge carrier properties (e.g. effective mass), as described in Landau's Fermi liquid theory. One-dimensional metals are influenced to a greater extent by interactions. In fact, the Landau-quasiparticle picture breaks down due to the Peierls instability. Instead, one-dimensional metals are described by Tomonaga-Luttinger liquid (TLL) theory which predicts unusual properties such as spin-charge separation and non-universal power laws in some physical properties such as the electronic density of states (DOS). Angle-integrated photoemission spectroscopy provides direct access to the DOS and as such directly addresses the power law renormalization of a TLL. It is first shown, that the bundles of single-wall carbon nanotubes indeed exhibit a power law scaling of the electronic density of states is observed as it is expected from TLL theory. The main part of the thesis is devoted to the investigation of the metallic ground state of SWCNTs upon functionalization. In general, functionalization is a controlled modification of the structural and/or electronic properties of SWCNT. It can be carried out e.g. by doping with electron donors or acceptors, by filling the nanospace inside the tubes with molecules or by substituting carbon atoms. First, the behavior of the SWCNT upon chemical doping was probed. The overall modification of the electronic band structure can be explained well by a rigid band shift model. The one-dimensional character of the metallic tubes in the bundle is retained at low doping, but when the semiconducting tubes in the sample are also rendered metallic by the charge transfer, a Fermi edge emerges out of the power law renormalization of the spectral weight, signifying a transition to a three-dimensional metallic behavior. This can be explained by an increased interaction between the tubes in the bundle. A crossover from a Tomonaga-Luttinger liquid to a Fermi liquid is observed. The filling of SWCNTs with C60 molecules leads to the formation of so-called peapods. It raises questions concerning the role of the additional bands originating from the C60 filling in the one-dimensional system. In the pristine state, the states of the C60 filling were found to have no influence on the metallic ground state. The TLL power law scaling of the density of states is observed. The overall interaction between the SWCNT host and the C60 filling is small. Upon doping however, the modified band structure leads to a qualitative change in the crossover from a TLL to a Fermi liquid. Upon doping, also states in the conduction band of the C60 are filled. The evolution of the power law scaling at intermediate doping can be interpreted as an opening of an additional conduction channel of one-dimensional metallic chains of C60 inside the tubes. This is in good agreement with transport experiments. Upon further doping, a Fermi edge similar to the highly doped SWCNTs is observed.

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.