We consider the purity-based entanglement detection scheme introduced in [C. Moura Alves and D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004)]. We describe how it could be implemented in an optical lattice using two-atom loss, and prove that in this form it detects all pure entangled states even without any spatial resolution. We then prove that correcting for certain reasonable types of experimental error is possible, and practical for error rates up to the order of one over the number of lattice sites considered. Limited spatial resolution similarly becomes a significant improvement over no spatial resolution only at nearly single site level. We also show how to use this process for state parameter estimation and collapse-revival evidence of entanglement, for which it remains useful even when the error rate is too high to permit unambiguous entanglement detection. We also consider an optical lattice bosonic analogue of the fractional quantum Hall (FQH) effect. This system can reach high “magnetic fields” very difficult to attain in the solid state FQH system, where the discrete nature of the lattice becomes important. Near simple rational numbers l/n of flux quanta per lattice cell, we find that the single particle states become nearly periodic with period n lattice sites, and have an n fold degeneracy which leads to FQH states resembling those of n-internal-state particles. Standard time of flight expansion would reveal this periodicity and be able to distinguish FQH states from vortex lattice or Mott insulator states. Shot noise correlation would provide further information on the nature of the FQH states.

This thesis scrutinizes the condensed matter physics phenomenon known as the fractional quantum Hall effect (FQHE), in particular fractional quantum Hall effects occurring in multicomponent systems. Broadly speaking, the FQHE can be defined as a many-electron quantum phenomenon, driven by strong interactions, that occurs in two-dimensional electron gasses in the presence of a perpendicular external magnetic field (and it is also predicted to occur for any two-dimensional particles, such as confined cold atoms, in an external gauge field). Multicomponent systems are systems where the constituent particles (such as electrons or cold atoms) possess internal degrees of freedom, for instance a spin or valley index. These internal degrees of freedom are often overlooked when modeling the FQHE. Taking into account the multicomponent degree of freedom yields an abundance of possibilities for the intellection of new types of so-called “topological phases of matter”, which are ubiquitously associated with the FQHE. In this thesis several different cases are considered. The first topic discussed herein is a study of phase transitions that can take place between FQHE phases with different net values of their multicomponent degrees of freedom. Examples are phase transitions between phases of different uniform net spin polarization, tunable as a function of certain system parameters. Some significant technical refinements are made to a previous model and comparisons are made with a variety of different experiments. The results are relevant for multicomponent FQHEs occurring in GaAs,AlAs and SiGe semiconductor systems where the electronic structure is confined to two dimensions, as well as in two-dimensional materials such as graphene. The second topic discussed herein is the introduction of the multiparticle multicomponent pseudopotential formalism. This methodology is oriented towards considerably expanding an existing framework for the construction of exactly solvable FQHE models by parameterizing multicomponent interactions. The final topic is the first example application of this new formalism to the construction of an exactly solvable FQHE model.

We consider a series of problems regarding quantum Hall edges, focusing on both dynamics and the mathematical structure of edge states. We begin in Chapter 3 with a limiting case of the Laughlin state placed in a very steep confining potential, but which is weak compared to the interactions. We find that the eigenstates have a Jack polynomial structure and an energy spectrum which is extremely different from the well-known Luttinger liquid edge. In Chapter 5 we analyse the inner products of edge state wavefunctions, using an effective description given by a large-N expansion ansatz proposed by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check the conjecture by calculating overlaps exactly for small system sizes and comparing the numerics with our high-order expansion to find excellent agreement. Finally, Chapter 6 considers the behaviour of quantum Hall edges close to the Luttinger liquid fixed point that occurs in the low energy, large system limit. We construct effective Hamiltonians using a local field theory description and then consider the effect of bulk symmetries on this edge. The symmetry analysis produces remarkable simplifications which allow for very accurate descriptions of the low-energy edge physics even relatively far away from the Luttinger liquid fixed point.

Electron-electron (e-e) interactions in 2-dimensional electron gases (2DEGs) can lead to many-body correlated states such as the the fractional quantum Hall effect (FQHE), where the Hall conductance quantization appears at fractional filling factors. The experimental discovery of an anomalous integer quantum Hall effect in graphene has faciliated the study of the interacting electrons which behave like massless chiral fermions. However, the observation of correlated electron physics in graphene is mostly hindered by strong electron scattering caused by charge impurities. We fabricate devices, in which, electrically contacted and electrostatically gated graphene samples are either suspended over a SiO₂ substrate or deposited on a hexagonal boron nitride layer, so that a drastic suppression of disorder is achieved. The mobility of our graphene samples exceeds 100,000 cm²/Vs. This very high mobility allows us to observe previously inaccessible quantum limited transport phenomena. In this thesis, we first present the transport measurements of ultraclean, suspended two-terminal graphene (chapter 3), where we observe the Fractional quantum Hall effect (FQHE) corresponding to filling fraction ν=1/3 FQHE state, hereby supporting the existence of interaction induced correlated electron states. In addition, we show that at low carrier densities graphene becomes an insulator with a magnetic-field-tunable energy gap. These newly discovered quantum states offer the opportunity to study correlated Dirac fermions in graphene in the presence of large magnetic fields. Since the quantitative characterization of the observed FQHE states such as the FQHE energy gap is not straight-forward in a two-terminal measurement, we have employed the four-probe measuremt in chapter 4. We report on the multi-terminal measurement of integer quantum Hall effect(IQHE) and fractional quantum Hall effect (FQHE) states in ultraclean suspended graphene samples in low density regime. Filling factors corresponding to fully developed IQHE states, including the ν±1 broken-symmetry states and the ν=1/3 FQHE state are observed. The energy gap of the 1/3 FQHE, measured by its temperature-dependent activation, is found to be much larger than the corresponding state found in the 2DEGs of high-quality GaAs heterostructures, indicating that stronger e-e interactions are present in graphene relative to 2DEGs. In chapter 5, we investigate the e-e correlations in graphene deposited on hexagonal boron nitride using the thermopower measurements. Our results show that at high temperatures the measured thermopower deviates from the generally accepted Mott's formula and that this deviation increases for samples with higher mobility. We quantify this deviation using the Boltzmann transport theory. We consider different scattering mechanisms in the system, including the electron-electron scattering. In the last chapter, we present the magnetothermopower measurements of high quality graphene on hexagonal boron nitride, where we observe the quantized thermopower at intermediate fields. We also see deviations from the Mott's formula for samples with low disorder, where the interaction effects come into play . In addition, the symmetry broken quantum Hall states due to strong electron-electron interactions appear at higher fields, whose effect are clearly observed in the measured in mangeto-thermopower. We discuss the predicted peak values of the thermopower corresponding to these states by thermodynamic arguments and compare it with our experimental results. We also present the sample fabrication methods in chapter 2. Here, we first explain the fabrication of the two-terminal and multi-terminal suspended graphene and the current annealing technique used to clean these samples. Then, we illustrate the fabrication of graphene on hexagonal boron nitride as well as encapsulated graphene samples with edge contacts. In addition, the thermopower measurement technique is presented in Appendix A, in which, we explain the temperature calibration, DC and AC measurement techniques.

L’effet Hall quantique (EHQ), observé par exemple dans des gaz bidimensionnels d’électrons (2DEGS) à basse température et sous fort champ magnétique, a révolutionné la métrologie des résistances car il permet d’obtenir un étalon quantique de résistance qui ne dépend que de e et h (respectivement la charge de l’électron et la constante de Planck). Une des missions des métrologues est de développer les étalons en améliorant leurs performances ou en les rendant plus facile à mettre en oeuvre (travaillant à plus haute température ou plus faible champ magnétique). Dans ce contexte, la physique du graphène suscite l’intérêt pour une application en métrologie. Une monocouche de graphène est une feuille d’un seul atome d’épaisseur constituée d’atomes de carbone disposés en nid d’abeille. Une bicouche de graphène est formée par empilement de deux monocouches. Les écarts en énergie entre les premiers niveaux de Landau dans la monocouche et dans la bicouche sont supérieurs par rapport à ceux dans GaAs ce qui rend l’EHQ dans le graphène plus robuste et laisse envisager le développement d’un étalon plus pratique. Durant ma thèse, nous avons mis en place un protocole de fabrication de barres de Hall en graphène exfolié comprenant un repérage optique, des lithographies électroniques, la métallisation, la gravure plasma… L’utilisation de substrat de silicium oxydé en surface rend possible l’utilisation d’une grille en face arrière. En outre la géométrie des échantillons répond au mieux aux critères métrologiques (canal central large, prises de tension bien définies, …). A basse température, le dopage résiduel obtenu après le recuit in situ est de l’ordre de 3-4x1011 cm-2. Les mobilités sont proches de 3000 cm2/(V.s) et 4000 cm2/(V.s) respectivement pour les échantillons monocouche et bicouche à la fois pour les électrons et les trous. Le transport mésoscopique a été caractérisé à basse température par des mesures de localisation faible et de fluctuations universelles de conductance. La longueur de cohérence que nous avons extraite est de l’ordre de 0.5 µm à 1.5 K. La résistance des contacts mesurée en régime d’EHQ est plutôt faible (typiquement quelques ohms). L’EHQ a été étudié en détail à basse température (300 mK < T <1.5 K) et sous fort champ magnétique (jusqu’à 18.5T) à la fois dans la monocouche et la bicouche en mesurant de manière précise la résistance de Hall (RH) et la résistance longitudinale (Rxx). Les mesures fines de RH sont réalisées à l’aide d’un pont de comparaison basé sur un Comparateur Cryogénique de Courant ; elles consistent à comparer indirectement l’EHQ dans l’échantillon de graphène à l’EHQ obtenu dans une barre de Hall en GaAs/AlGaAs qui est supposée fournir la valeur exacte RH/2. Nos mesures révèlent un accord entre la résistance de Hall dans le graphène et la valeur attendue avec une incertitude de quelques 10-7. Au plus faible courant et dans l’état de dissipation minimale (Rxx→0), nous avons obtenu un accord avec une incertitude relative de 3.10-7. Ce niveau de précision est principalement limité par la petite taille de nos échantillons et par les inhomogénéités de la densité qui y sont présents, ces deux caractéristiques amenant de faibles courants de rupture de l’EHQ (1-2 µA). Toutefois, nos résultats sont à ce jour les tests les plus précis concernant l’EHQ dans du graphène exfolié et les premiers tests sur une bicouche. Ils confirment le potentiel de l’EHQ dans le graphène pour une application en métrologie. / The quantum Hall effect (QHE) observed in two dimensional electron gases (2DEGs) at low temperature and under high magnetic induction, has revolutionized the resistance metrology because it leads to a universal and very reproducible quantum resistance standard only dependent on e and h (respectively the electron charge and Planck's constant). One of the metrologists' missions is to develop standards with improved performances and to notably make them more practical, working for example at higher temperature or lower magnetic induction. In this context, graphene physics could be very interesting for metrological applications. Monolayer graphene is a one atom thick layer of carbon atoms condensed in a honeycomb lattice. A bilayer graphene consists in two stacked monolayers. Larger energy spacings between the first Landau Levels in monolayer and in bilayer than in GaAs make the QHE in graphene more robust and give hope that more practical standards could be developed. During the PhD, we have set a protocol up in order to fabricate exfoliated graphene based Hall bars, including location with an optical microscope, e-beam lithography, metallization, plasma etching… Backgated using oxidized silicon wafers the devices were designed to fulfill at best the metrological requirements (large conduction channel, well defined voltage probes…). At low temperature, the typical charge carrier residual doping obtained after the annealing process was 3-4x1011 cm-2. Mobilities were close to 3000 cm2/(V.s) and 4000 cm2/(V.s) respectively for the monolayer and the bilayer based device both for holes and electrons. Mesoscopic transport was characterized at low temperature by weak localization and universal conductance fluctuations (UCF) measurements. The phase coherence length deduced was about 0.5 µm below 1.5 K. The resistance of the contacts, measured in the QHE regime, appeared to be rather low (typically few ohms). The QHE was investigated in details at low temperature (300 mK < T <1.5 K) and high magnetic field (up to 18.5 T) in both monolayer and bilayer graphene by refined measurements of the Hall resistance (RH) and also of the longitudinal resistance (Rxx). The accurate measurements of RH were performed using a Cryogenic Current Comparator based resistance bridge. They consist in an indirect comparison between the QHE in graphene and the QHE obtained in a GaAs based Hall bar, supposed to deliver the expected value RH/2. Our measurements showed an agreement of the Hall resistance in graphene with the expected value within some parts in 107. At the lowest biasing current and in the lowest dissipation state (where Rxx→0) it is possible to demonstrate an agreement within an uncertainty of 3 parts in 107. That accuracy is essentially limited by the small size, and the poor homogeneity of the carrier density of the graphene electronic systems, both acting for a very reduced breakdown current of the QHE (1-2 µA). Nevertheless these results are the most accurate tests of the QHE performed in exfoliated graphene and the first universality test of the QHE with bilayer graphene. They confirm the potential of the QHE in graphene for the metrological application.

No description available.

Using the method of breaking circular symmetry and the subsequent symmetry restoration via projection techniques, we calculate the ground-state energies and excitation spectra of N-electrons confined in parabolic quantum dots in strong magnetic fields in the medium-size range 10<=N <=30. The physical picture is that of finite rotating electron molecules (REMs) comprising multiple rings, with the rings rotating independently of each other. A derived analytic expression for the energetics is applicable to arbitrary sizes given the corresponding ring configuration of classical point charges. Also by exact diagonalization (EXD) method, we show the spectrum and structure of few electrons, 2<=N<=3, confined in elliptical dots at finite magnetic field. The results suggest the formation of a state of Wigner-molecular properties with spin associated, which has great instructions for the development of quantum register in quantum computing.

This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional. / text

In this thesis we discuss quantum Hall effects in bilayer graphene and other novel two-dimensional electron systems, focusing on the interplay between nontrivial Fermi surface topology and electron-electron interactions. In the first chapter I will give a brief introduction to some aspects of the quantum Hall effects. The second chapter discusses the physics in bilayer graphene in the absence of external magnetic fields. The first half discusses the band gap opening and trigonal warping effects in its bandstructure, and the second half focuses on the insulating ground state that results from electron-electron interactions. The third chapter discusses the single-particle Landau level structure in bilayer graphene. We will see that when both the band gap and trigonal warping effects are present, the highest Landau level in the valence band is three-fold degenerate at small magnetic fields. As the field increases, the three fold degeneracy is lifted and the Landau level structure gradually reduces to that in the absence of trigonal warping effects. At the end of the chapter we will demonstrate a formalism to map the momentum distribution of the single-particle Landau level structure. Such a mapping will give valuable information about the single-particle bandstructure. The fourth chapter deals with electron-electron interactions in the integer quantum Hall regime, where there is no fractional filling of the orbital degrees of freedom. In such a regime, the effect of electron-electron interactions often leads to spontaneous ordering of the internal degrees of freedom, such as spin, layer and valley. The first part of the chapter will establish the general formalism of Hartree-Fock theory in the quantum Hall regime, and then a specific theory for gapped bilayer graphene with trigonal warping effects is constructed. The resulting ground states are analyzed in the last part of the chapter. / text

When electrons are confined to a two dimensional layer with a perpendicular applied magnetic field, such that the ratio of electrons to flux quanta \((\nu)\) is a small integer or simple rational value, these electrons condense into remarkable new phases of matter that are strikingly different from the metallic electron gas that exists in the absence of a magnetic field. These phases, called integer or fractional quantum Hall (IQH or FQH) states, appear to be conventional insulators in their bulk, but behave as a dissipationless metal along their edge. Furthermore, electrical measurements of such a system are largely insensitive to the detailed geometry of how the system is contacted or even how large the system is... only the order in which contacts are made appears to matter. This insensitivity to local geometry has since appeared in a number of other two and three dimensional systems, earning them the classification of "topological insulators" and prompting an enormous experimental and theoretical effort to understand their properties and perhaps manipulate these properties to create robust quantum information processors. The focus of this thesis will be two experiments designed to elucidate remarkable properties of the metallic edge and insulating bulk of certain FQH systems. To study such systems, we can use mesoscopic devices known as single electron transistors (SETs). These devices operate by watching single electrons hop into and out of a confining box and into a nearby wire (for measurement). If it is initially unfavorable for an electron to leave the box, it can be made favorable by bringing another charge nearby, modifying the energy of the confined electron and pushing it out of the box and into the nearby wire. In this way, the SET can measure nearby charges. Alternatively, we can heat up the nearby wire to make it easier for electrons to enter and leave the box. In this way, the SET is a sensitive thermometer. First, by operating the SET as an electrometer, we measure the local charge of the \(\nu = 5/2\) FQH state. An immediate consequence of measuring fractionally quantized conductance plateaus is that the charge of local excitations should be a fraction of \(e\), the charge of an electron. The simplest charge that would be expected at \(\nu = 5/2\) would \(e/2\). However, if the charged particles that condense into the \(\nu = 5/2\) FQH state are paired, the expected local charge becomes \(e/4\). By watching these local charges being added to compressible puddles at \(\nu = 5/2\) and \(\nu = 5/7\), we find that the local charge at \(\nu = 5/2\) is indeed \(e/4\), indicating that objects of charge \(e\) are pairing to form the ground state of the system. This has implications for the future possibility of detecting non-Abelian braiding statistics in this state, and is described in detail in Chapter 2. By further monitoring how eagerly these \(e/4\) particles enter puddles as we increase the temperature, we can attempt to identify the presence of some excess entropy related to an unconventional degeneracy of their ground state. Such an entropy would be expected if the \(\nu = 5/2\) state exhibited non-Abelian braiding statistics. Progress on these experiments and prospects for building a quantum computer are presented in Chapter 3. Next, by operating the SET as a thermometer, we monitor heat flow along the compressible edge and through the bulk of IQH and FQH states. As an edge is heated and charge on that edge is swept downstream by the external magnetic field, we expect that charge to carry the injected energy in the same downstream direction. However, for certain FQH states, this is not the case. By heating an edge with a quantum point contact (QPC) and monitoring the heat transported upstream and downstream, we find that heat can be transported upstream when the edge contains structure related to \(\nu = 2/3\) FQH physics. Surprisingly, this can be present even when the bulk is in a conventional insulating (IQH) state. Additionally, we unexpectedly find that the \(\nu = 1\) bulk is capable of transporting heat, while the \(\nu = 2\) and \(\nu = 3\) bulk are not. These experiments are presented in Chapter 4. Finally, in Chapter 5, we describe preliminary work on a very different type of topological material, the quantum spin Hall (QSH) insulator. Here, the spin of electrons takes the place of the external magnetic field, creating edge states that propagate in both directions. Each of these edges behaves as an ideal one-dimensional mode, with predicted resistance \(h/e^2\). By creating well-defined regions where these modes can exist, we identify and characterize the conductance associated with topological edges. / Physics