Electronic Fabry-Perot Interferometry of Quantum Hall Edge States

In the pattern-of-zeros approach to quantum Hall states, a set of data ${n;m;{S}_{a}|a=1,\dots{},n;n,m,{S}_{a}∊\mathbb{N}}$ (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data ${n;m;{S}_{a}}$ corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the pattern of zeros alone does not completely characterize the quantum Hall states. In this paper, we find that the following expanded set of data ${n;m;{S}_{a};c|a=1,\dots{},n;n,m,{S}_{a}∊\mathbb{N};c∊\mathbb{R}}$ provides a more complete characterization of quantum Hall states. Each expanded set of data completely characterizes a unique quantum Hall state, at least for the examples discussed in this paper. The result is obtained by combining the pattern of zeros and ${Z}_{n}$ simple-current vertex algebra which describes a large class of Abelian and non-Abelian quantum Hall states ${\ensuremath{\Phi}}_{{Z}_{n}}^{\mathrm{sc}}$. The more complete characterization in terms of ${n;m;{S}_{a};c}$ allows us to obtain more topological properties of those states, which include the central charge $c$ of edge states, the scaling dimensions and the statistics of quasiparticle excitations.

The two dimensional electron gas subjected to a perpendicular magnetic field is a model system that supports a variety of electronic phases. Perhaps the most well-known are the fractional quantum Hall states, but in recent years there has been an upsurge of interest in the charge ordered phases commonly referred to as electron solids. These solids are a consequence of electron-electron interactions in a magnetic field. While some solid phases form in the lowest Landau level, the charged ordered phases are most abundant in the higher Landau levels. Examples of such phases include the Wigner solids, electronic bubble phases and stripe or nematic phases. Open questions surround the exact role of disorder, confinement potential, temperature and the Landau level index in determining the stability and competition of these phases with other ground states. The interface of GaAs/AlGaAs remains the cleanest host for the two-dimensional electron gas due to the extremely high quality of materials available and the advancement in molecular beam epitaxy growth techniques. As a result, exceptionally high electron mobilities in this system have been instrumental in the discovery of numerous electron solids. In this Thesis, I discuss the discovery and properties of several electron solids that develop in such state-of-the-art two dimensional electron gases. These electron solids often develop at ultra low temperatures, in the milliKelvin temperature range. After an introduction to the physics of the quantum Hall effect in two dimensions, in chapter 3, I discuss electron solids developing in the N=1 Landau level. While these solids have been known for some time, details of the competition of these phases xiii with the nearby fractional quantum Hall states remains elusive. A number of reports observe new fractional quantum Hall states at filling factors where electron solids are found in other experiments. We undertook a systematic study to answer some of these unsettled questions. We see evidence for incipient fractional quantum Hall states at 2+2/7 and 2+5/7 at intermediate temperatures which are overtaken by the electronic bubble phases at lower temperatures. Several missing fractional states including those at filling factors 2+3/5, 2+3/7, 2+4/9 highlight the relative stability of the electronic solids called the bubble phases in the vicinity in our sample. In chapter 4, I discuss a newly seen electron crystal which manifests itself in transport measurements as a reentrant integer quantum Hall state. Reentrant integer behavior is common in high Landau levels, but so far it was not observed in the lowest Landau level in narrow quantum well samples. In contrast to high Landau levels, where such reentrant integer behavior was associated with electronic bubbles, we believe that the same signature in the N=0 Landau level is due to an electronic Wigner crystal. The filling factors at which we observe such reentrance reveal that it is a crystal of holes, rather than electrons. The discovery of this reentrant integer state paints a complex picture of the interplay of the Wigner crystal and fractional quantum Hall states. Finally, in chapter 5, I discuss the observation of a novel phenomenon, that of reentrant fractional quantum Hall effect. In the lowest Landau level, we observe a fractional quantum Hall state, but as the field is increased, we see a deviation and then a return to quantization in the Hall resistance. Such a behavior indicates a novel electron solid. In contrast to the collective localization of electrons evidenced by the reentrant integer quantum Hall effect, such reentrance to a fractional Hall resistance clearly points to the involvement of composite fermion quasiparticles. This property thus distinguishes the ground state we observed as a solid formed of composite fermions. Such a solid phase is evidence for exotic electron-electron correlations at play which are clearly different from those in the traditional Wigner solid of electrons.

The Hall resistance of a homogeneous electron system is well known to be antisymmetric with respect to the magnetic field and the sign of charge carriers. We have observed that such symmetries no longer hold in planar hybrid structures consisting of partly single layer graphene (SLG) and partly bilayer graphene (BLG) in the quantum Hall (QH) regime. In particular, the Hall resistance across the SLG and BLG interface is observed to exhibit quantized plateaus that switch between those characteristic of SLG QH states and BLG QH states when either the sign of the charge carriers (controlled by a back gate) or the direction of the magnetic field is reversed. Simultaneously reversing both the carrier type and the magnetic field gives rise to the same quantized Hall resistances. The observed SLG-BLG interface QH states, with characteristic asymmetries with respect to the signs of carriers and magnetic field, are determined only by the chirality of the QH edge states and can be explained by a Landauer-B\"uttiker analysis applied to such graphene hybrid structures involving two regions of different Landau level structures.

The single-particle energy spectrum of a two-dimensional electron gas in a perpendicular magnetic field consists of equally spaced energy states, known as Landau levels. Each level is split owing to spin interactions, and its degeneracy is proportional to the magnetic field strength. When the ratio, ν (or ‘filling factor’), of the number of electrons and the degeneracy of a Landau level takes an integer or particular fractional values, quantum Hall effects1 occur, characterized by a vanishingly small longitudinal resistance and a quantized (transverse) Hall voltage2. The quantum Hall regime may be used for the controlled study of many-particle cooperative phenomena, such as order–disorder phase transitions (analogous to those observed in conventional magnets). Both isotropic and anisotropic ferromagnetic ground states have been predicted3,4,5,6,7,8 to occur in the quantum Hall regime, some of which have been investigated experimentally9,10,11,12,13 in samples with different geometries and filling factors. Here we report evidence for first-order phase transitions in quantum Hall states (ν = 2,4) confined to a wide gallium arsenide quantum well. We observe hysteresis and an anomalous temperature dependence in the longitudinal resistivity, indicative of a transition between two distinct ground states of an Ising quantum Hall ferromagnet. The microscopic origin of the anisotropy field is identified using detailed many-body calculations.

We study a class of Abelian quantum Hall (QH) states which are topologically unstable (T-unstable). We find that the T-unstable QH states can have a phase transition on the edge which causes a binding between electrons and reduces the number of gapless edge branches. After the binding transition, the single-electron tunneling into the edge gains a finite energy gap, and only certain multi-electron co-tunneling (such as three-electron co-tunneling for $\nu=9/5$ edges) can be gapless. Similar phenomenon also appear for edge state on the boundary between certain QH states. For example edge on the boundary between $\nu=2$ and $\nu=1/5$ states only allow three-electron co-tunneling at low energies after the binding transition.

We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors \nu = 2/(2m-p) for odd integers $m$ and $p$, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge $c=1$ at radius $R=\sqrt{p/2}$. When $p=1$ the state is Abelian. The state with $p=3$ is the $Z_4$ Read Rezayi state, and the series of $p\ge 3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge $e$ electrons coexist with charge $4e$ bosons and a strong clustered phase where the electrons are strongly bound in groups of 4. The transition between these two phases on a wire is mapped to the critical point of the 4 state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge $4e$ bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.

The frequency dependence of the capacitance between two-dimensional electron systems (2DESs) and the gate has been studied at the quantum Hall (QH) plateaus. The capacitance minima at the QH plateaus are highly dependent on the area of edge channels at high frequencies. The residual contribution of bulk channel to the capacitance has been estimated, and the widths of edge channels have been evaluated. The diagonal conductivity σxx of less than 10-11 S has been obtained accurately by capacitance measurements at the QH plateaus. The relationship between σxx and capacitance has been discussed.

The quantum Hall (QH) effect represents a unique playground where quantum coherence of electrons can be exploited for various applications, from metrology to quantum computation. In the fractional regime it also hosts anyons, emergent quasiparticles that are neither bosons nor fermions and possess fractional statistics. Their detection and manipulation represent key milestones in view of topologically protected quantum computation schemes. Exploiting the high degree of phase coherence, edge states in the QH regime have been investigated by designing and constructing electronic interferometers, able to reveal the coherence and statistical properties of the interfering constituents. Here, we review the two main geometries developed in the QH regime, the Mach-Zehnder and the Fabry-Perot interferometers. We present their basic working principles, fabrication methods, and the main results obtained both in the integer and fractional QH regime. We will also show how recent technological advances led to the direct experimental demonstration of fractional statistics for Laughlin quasiparticles in a Fabry-Perot interferometric setup.

In this paper, we examine the viscoelastic properties of integer quantum Hall\n(IQH) states in a tilted magnetic field. In particular, we explore to what\nextent the tilted-field system behaves like a two-dimensional electron gas with\nanisotropic mass in the presence of strain deformations. We first review the\nKubo formalism for viscosity in an external magnetic field, paying particular\nattention to the role of rotational symmetry and contact terms. Next, we\ncompute the conductivity, stress, and viscosity tensors for IQH states in the\npresence of a tilted field and vertical confining potential. By comparing our\nresults with the recently developed bimetric formalism, we show that, at the\nlevel of the contracted Hall viscosity tensor, the mapping between tilted field\nand effective mass anisotropy holds only if we simultaneously modify the\nbackground perpendicular magnetic field; in other words, a simultaneous\nmeasurement of the density, contracted Hall viscosity, and Hall conductivity at\nfixed particle number can distinguish between tilted field and effective mass\nanisotropy. Additionally, we show that in the presence of a tilted magnetic\nfield, the stress tensor acquires an unusual anisotropic ground state average,\nleading to anomalous elastic response functions. We develop a formalism for\nprojecting a three-dimensional Hamiltonian with confining potential and\nmagnetic field to a two-dimensional Hamiltonian in order to further address the\nphenomenology of the tilted-field IQH fluid. We find that the projected fluid\ncouples non-minimally to geometric deformations, indicating the presence of\ninternal geometric degrees of freedom.\n

We report the observation of a metal-insulator transition (MIT) in a two- dimensional electron gas (2DEG) in a Si/SiGe heterostructure at zero magnetic field. On going through the MIT we observe the corresponding evolution of the magnetic field induced transition between the insulating phase and the quantum Hall (QH) liquid state in the QH regime. Similar to the previous reports for a GaAs sample, we find that the critical magnetic field needed to produce the transition becomes zero at the critical electron density corresponding to the zero field MIT. The temperature dependence of the conductivity in a metallic-like state at zero field is compared with the theory of the interaction corrections at intermediate and ballistic regimes $k_{B}T\tau/\hbar\geq1$. The theory yields a good fit for the linear part of the curve. However the slope of that part of $\sigma_{xx}(T)$ is about two times smaller than that reported in other 2D systems with similar values of $r_s$. At the same time, the recent theory of magnetoresistance due to electron-electron interaction in the case of arbitrary $k_{B}T\tau/\hbar$, smooth disorder and classically strong fields does not seem to be quite adequate for the description of the parabolic magnetoresistance observed in our samples. We attribute these results to the fact that neither of these theories deals with the whole scattering potential in a sample but leaves either its long range or its short range component out of consideration.

We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.

In the fractional quantum Hall effect, quasiparticles are collective excitations that have a fractional charge and show fractional statistics as they interchange positions. While the fractional charge affects semi-classical characteristics such as shot noise and charging energies, fractional statistics is most notable through quantum interference. Here we study fractional statistics in a bilayer graphene Fabry-Pérot interferometer. We tune the interferometer from the Coulomb-dominated regime to the Aharonov-Bohm regime, both for integer and fractional quantum Hall states. Focusing on the fractional quantum Hall state with a filling factor ν = 1/3, we follow the evolution of the Aharonov-Bohm interference of quasiparticles while varying the magnetic flux through an interference loop and the charge density within the loop independently. When their combined variation is such that the Landau filling remains 1/3, the charge density in the loop varies continuously. We then observe pristine Aharonov-Bohm oscillations with a period of three flux quanta, as expected for quasiparticles of one-third of the electron charge. Yet, when the combined variation leads to discrete events of quasiparticle addition or removal, phase jumps emerge and alter the phase evolution. Notably, across all cases with discrete and continuous charge variation, the average phase consistently increases by 2π with each addition of one electron to the loop, as expected for quasiparticles, obeying fractional statistics.

We consider a hybrid quantum Hall-superconductor system, where a superconducting finger with oblique profile is wedged into a two-dimensional electron gas in the presence of a perpendicular magnetic field, as considered by Lee et al., Nat. Phys. 13, 693 (2017). The electron gas is in the quantum Hall regime at filling factor $ν=1$. Due to the Meissner effect, the perpendicular magnetic field close to the quantum Hall-superconductor boundary is distorted and gives rise to an in-plane component of the magnetic field. This component enables nonlocal crossed Andreev reflection between the spin-polarized chiral edge states running on opposite sides of the superconducting finger, thus opening a gap in the spectrum of the edge states without the need of spin-orbit interaction or nontrivial magnetic textures. We compute numerically the transport properties of this setup and show that a negative resistance exists as a consequence of nonlocal Andreev processes. We also obtain numerically the zero-energy local density of states, which systematically shows peaks stable to disorder. The latter result is compatible with the emergence of Majorana bound states.

We calculate the dynamical structure factor of the \nu=5/2 non-abelian quantum Hall state in the dipole approximation, valid for large momenta. Due to the fact that both quasi-particles (qps) and quasi-holes (qhs) have an internal Majorana degree of freedom, a qp-qh pair has a fermionic degree of freedom which can be either empty or occupied, and leads to a splitting of the roton mode. Observation of this splitting by means of finite wavelength optical spectroscopy could provide evidence for Majorana modes in the \nu=5/2 quantum Hall state.

Selective control of edge-channel trajectories by scanning gate microscopy