Edge states in the fractional-quantum-Hall-effect regime

Unusual Quasiparticles and Tunneling Conductance in Quantum Point Contacts in ν=2/3 Fractional Quantum Hall Systems

Understanding topological matter in the fractional quantum Hall (FQH) effect requires identifying the nature of edge state quasiparticles. FQH edge state at the filling factor ν=2/3 in the spin-polarized and unpolarized phases is represented by the two modes of composite fermions (CF) with the parallel or opposite spins described by the chiral Luttinger liquids. Tunneling through a quantum point contact (QPC) between different or similar spin phases is solved exactly. With the increase of the applied voltage, the QPC conductance grows from zero and saturates at e^{2}/2h while a weak electron tunneling between the edge modes with the same spin transforms into a backscattering carried by the charge q=e/2 quasiparticles. These unusual quasiparticles and conductance plateau emerge when one or two CF spin-polarized modes in the QPC tunnel into a single mode. We propose experiments on the applied voltage and temperature dependence of the QPC conductance and noise that can shed light on the nature of edge states and FQH transport.

Fingerprints of Anti-Pfaffian Topological Order in Quantum Point Contact Transport

Despite recent experimental developments, the topological order of the fractional quantum Hall state at filling ν=5/2 remains an outstanding question. We study conductance and shot noise in a quantum point contact device in the charge-equilibrated regime and show that, among Pfaffian, particle-hole Praffian, and anti-Pfaffian (aPf) candidate states, the hole-conjugate aPf state is unique in that it can produce a conductance plateau at G=(7/3)e^{2}/h by two fundamentally distinct mechanisms. We demonstrate that these mechanisms can be distinguished by shot noise measurements on the plateaus. We also determine distinct features of the conductance of the aPf state in the coherent regime. Our results can be used to experimentally single out the aPf order.

Higher-Order Topological Peierls Insulator in a Two-Dimensional Atom-Cavity System

In this work, we investigate a two-dimensional system of ultracold bosonic atoms inside an optical cavity, and show how photon-mediated interactions give rise to a plaquette-ordered bond pattern in the atomic ground state. The latter corresponds to a 2D Peierls transition, generalizing the spontaneous bond dimerization driven by phonon-electron interactions in the 1D Su-Schrieffer-Heeger (SSH) model. Here the bosonic nature of the atoms plays a crucial role to generate the phase, as similar generalizations with fermionic matter do not lead to a plaquette structure. Similar to the SSH model, we show how this pattern opens a nontrivial topological gap in 2D, resulting in a higher-order topological phase hosting corner states, that we characterize by means of a many-body topological invariant and through its entanglement structure. Finally, we demonstrate how this higher-order topological Peierls insulator can be readily prepared in atomic experiments through adiabatic protocols. Our work thus shows how atomic quantum simulators can be harnessed to investigate novel strongly correlated topological phenomena beyond those observed in natural materials.

Anomalous quantized plateaus in two-dimensional electron gas with gate confinement

Quantum information can be coded by the topologically protected edges of fractional quantum Hall (FQH) states. Investigation on FQH edges in the hope of searching and utilizing non-Abelian statistics has been a focused challenge for years. Manipulating the edges, e.g. to bring edges close to each other or to separate edges spatially, is a common and essential step for such studies. The FQH edge structures in a confined region are typically presupposed to be the same as that in the open region in analysis of experimental results, but whether they remain unchanged with extra confinement is obscure. In this work, we present a series of unexpected plateaus in a confined single-layer two-dimensional electron gas (2DEG), which are quantized at anomalous fractions such as 9/4, 17/11, 16/13 and the reported 3/2. We explain all the plateaus by assuming surprisingly larger filling factors in the confined region. Our findings enrich the understanding of edge states in the confined region and in the applications of gate manipulation, which is crucial for the experiments with quantum point contact and interferometer.

Half-Integer Conductance Plateau at the ν=2/3 Fractional Quantum Hall State in a Quantum Point Contact

The ν=2/3 fractional quantum Hall state is the hole-conjugate state to the primary Laughlin ν=1/3 state. We investigate transmission of edge states through quantum point contacts fabricated on a GaAs/AlGaAs heterostructure designed to have a sharp confining potential. When a small but finite bias is applied, we observe an intermediate conductance plateau with G=0.5(e^{2}/h). This plateau is observed in multiple QPCs, and persists over a significant range of magnetic field, gate voltage, and source-drain bias, making it a robust feature. Using a simple model that considers scattering and equilibration between counterflowing charged edge modes, we find this half-integer quantized plateau to be consistent with full reflection of an inner counterpropagating -1/3 edge mode while the outer integer mode is fully transmitted. In a QPC fabricated on a different heterostructure which has a softer confining potential, we instead observe an intermediate conductance plateau at G=(1/3)(e^{2}/h). These results provide support for a model at ν=2/3 in which the edge transitions from a structure having an inner upstream -1/3 charge mode and outer downstream integer mode to a structure with two downstream 1/3 charge modes when the confining potential is tuned from sharp to soft and disorder prevails.

Emergence of Neutral Modes in Laughlin-like Fractional Quantum Hall Phases

Chiral gapless boundary modes are characteristic of quantum Hall (QH) states. For hole-conjugate fractional QH phases counterpropagating edge modes (upstream and downstream) are expected. In the presence of electrostatic interactions and disorder these modes may renormalize into charge and upstream neutral modes. Orthodox models of Laughlin phases anticipate only a downstream charge mode. Here we show that in the latter case, in the presence of a smooth confining potential, edge reconstruction leads to the emergence of pairs of counterpropagating modes, which, by way of mode renormalization, may give rise to nontopological upstream neutral modes, possessing nontrivial statistics. This may explain the experimental observation of ubiquitous neutral modes, and the overwhelming suppression of anyonic interference in Mach-Zehnder interferometry platforms. We also point out other signatures of such edge reconstruction.

Heat Equilibration of Integer and Fractional Quantum Hall Edge Modes in Graphene

Hole-conjugate states of the fractional quantum Hall effect host counterpropagating edge channels which are thought to exchange charge and energy. These exchanges have been the subject of extensive theoretical and experimental works; in particular, it is yet unclear if the presence of integer quantum Hall edge channels stemming from fully filled Landau levels affects heat equilibration along the edge. In this Letter, we present heat transport measurements in quantum Hall states of graphene demonstrating that the integer channels can strongly equilibrate with the fractional ones, leading to markedly different regimes of quantized heat transport that depend on edge electrostatics. Our results allow for a better comprehension of the complex edge physics in the fractional quantum Hall regime.

Determination of topological edge quantum numbers of fractional quantum Hall phases by thermal conductance measurements

To determine the topological quantum numbers of fractional quantum Hall (FQH) states hosting counter-propagating (CP) downstream (N_d) and upstream (N_u) edge modes, it is pivotal to study quantized transport both in the presence and absence of edge mode equilibration. While reaching the non-equilibrated regime is challenging for charge transport, we target here the thermal Hall conductance G_Q, which is purely governed by edge quantum numbers N_d and N_u. Our experimental setup is realized with a hexagonal boron nitride (hBN) encapsulated graphite gated single layer graphene device. For temperatures up to 35 mK, our measured G_Q at ν = 2/3 and 3/5 (with CP modes) match the quantized values of non-equilibrated regime (N_d + N_u)κ₀T, where κ₀T is a quanta of G_Q. With increasing temperature, G_Q decreases and eventually takes the value of the equilibrated regime ∣N_d − N_u∣κ₀T. By contrast, at ν = 1/3 and 2/5 (without CP modes), G_Q remains robustly quantized at N_dκ₀T independent of the temperature. Thus, measuring the quantized values of G_Q in two regimes, we determine the edge quantum numbers, which opens a new route for finding the topological order of exotic non-Abelian FQH states.

The knowledge of quantum numbers of the edge modes is essential for understanding fractional Hall states containing counter-propagating downstream and upstream modes. Here the authors identify the edge quantum numbers by probing a crossover from non-equilibrated to equilibrated edge mode regime in thermal conductance.

Counterpropagating topological and quantum Hall edge channels

The survival of the quantum spin Hall edge channels in presence of an external magnetic field has been a subject of experimental and theoretical research. The inversion of Landau levels that accommodates the quantum spin Hall effect is destroyed at a critical magnetic field, and a trivial insulating gap appears in the spectrum for stronger fields. In this work, we report the absence of this transport gap in disordered two dimensional topological insulators in perpendicular magnetic fields of up to 16 T. Instead, we observe that a topological edge channel (from band inversion) coexists with a counterpropagating quantum Hall edge channel for magnetic fields at which the transition to the insulating regime is expected. For larger fields, we observe only the quantum Hall edge channel with transverse resistance close to h/e². By tuning the disorder using different fabrication processes, we find evidence that this unexpected ν = 1 plateau originates from extended quantum Hall edge channels along a continuous network of charge puddles at the edges of the device.

Edge Reconstruction of a Time-Reversal Invariant Insulator: Compressible-Incompressible Stripes

Two-dimensional (2D) topological electronic insulators are known to give rise to gapless edge modes, which underlie low energy dynamics, including electrical and thermal transport. This has been thoroughly investigated in the context of quantum Hall phases, and time-reversal invariant topological insulators. Here we study the edge of a 2D, topologically trivial insulating phase, as a function of the strength of the electronic interactions and the steepness of the confining potential. For sufficiently smooth confining potentials, alternating compressible and incompressible stripes appear at the edge. Our findings signal the emergence of gapless edge modes which may give rise to finite conductance at the edge. This would suggest a novel scenario of a nontopological metal-insulator transition in clean 2D systems. The incompressible stripes appear at commensurate fillings and may exhibit broken translational invariance along the edge in the form of charge density wave ordering. These are separated by structureless compressible stripes.

Observation of ballistic upstream modes at fractional quantum Hall edges of graphene

The presence of "upstream" modes, moving against the direction of charge current flow in the fractional quantum Hall (FQH) phases, is critical for the emergence of renormalized modes with exotic quantum statistics. Detection of excess noise at the edge is a smoking gun for the presence of upstream modes. Here, we report noise measurements at the edges of FQH states realized in dual graphite-gated bilayer graphene devices. A noiseless dc current is injected at one of the edge contacts, and the noise generated at contacts at length, L = 4 μm and 10 μm away along the upstream direction is studied. For integer and particle-like FQH states, no detectable noise is measured. By contrast, for "hole-conjugate" FQH states, we detect a strong noise proportional to the injected current, unambiguously proving the existence of upstream modes. The noise magnitude remains independent of length, which matches our theoretical analysis demonstrating the ballistic nature of upstream energy transport, quite distinct from the diffusive propagation reported earlier in GaAs-based systems.

Distinguishing between non-abelian topological orders in a Quantum Hall system

[Figure: see text].

Vanishing Thermal Equilibration for Hole-Conjugate Fractional Quantum Hall States in Graphene

Transport through edge channels is responsible for conduction in quantum Hall (QH) phases. Robust quantized values of charge and thermal conductances dictated by bulk topology appear when equilibration processes become dominant. We report on measurements of electrical and thermal conductances of integer and fractional QH phases, realized in hexagonal boron nitride encapsulated graphite-gated bilayer graphene devices for both electron and hole doped sides with different valley and orbital symmetries. Remarkably, for complex edges at filling factors ν=5/3 and 8/3, closely related to the paradigmatic hole-conjugate ν=2/3 phase, we find quantized thermal conductance whose values (3κ_{0}T and 4κ_{0}T, respectively where κ_{0}T is the thermal conductance quantum) are markedly inconsistent with the values dictated by topology (1κ_{0}T and 2κ_{0}T, respectively). The measured thermal conductance values remain insensitive to different symmetries, suggesting its universal nature. Our findings are supported by a theoretical analysis, which indicates that, whereas electrical equilibration at the edge is established over a finite length scale, the thermal equilibration length diverges for strong electrostatic interaction. Our results elucidate the subtle nature of crossover from coherent, mesoscopic to topology-dominated transport.

Andreev reflection of fractional quantum Hall quasiparticles

Electron correlation in a quantum many-body state appears as peculiar scattering behaviour at its boundary, symbolic of which is Andreev reflection at a metal-superconductor interface. Despite being fundamental in nature, dictated by the charge conservation law, however, the process has had no analogues outside the realm of superconductivity so far. Here, we report the observation of an Andreev-like process originating from a topological quantum many-body effect instead of superconductivity. A narrow junction between fractional and integer quantum Hall states shows a two-terminal conductance exceeding that of the constituent fractional state. This remarkable behaviour, while theoretically predicted more than two decades ago but not detected to date, can be interpreted as Andreev reflection of fractionally charged quasiparticles. The observed fractional quantum Hall Andreev reflection provides a fundamental picture that captures microscopic charge dynamics at the boundaries of topological quantum many-body states.

Fractional Coulomb blockade for quasi-particle tunneling between edge channels

In the fractional quantum Hall effect, the elementary excitations are quasi-particles with fractional charges as predicted by theory and demonstrated by noise and interference experiments. We observe Coulomb blockade of fractional charges in the measured magneto-conductance of a 1.4-micron-wide quantum dot. Interaction-driven edge reconstruction separates the dot into concentric compressible regions with fractionally charged excitations and incompressible regions acting as tunnel barriers for quasi-particles. Our data show the formation of incompressible regions of filling factors 2/3 and 1/3. Comparing data at fractional filling factors to filling factor 2, we extract the fractional quasi-particle charge e */e = 0.32 ± 0.03 and 0.35 ± 0.05. Our investigations extend and complement quantum Hall Fabry-Pérot interference experiments investigating the nature of anyonic fractional quasi-particles.

Quantized charge fractionalization at quantum Hall Y junctions in the disorder dominated regime

Fractionalization is a phenomenon where an elementary excitation partitions into several pieces. This picture explains non-trivial transport through a junction of one-dimensional edge channels defined by topologically distinct quantum Hall states, for example, a hole-conjugate state at Landau-level filling factor ν = 2/3. Here we employ a time-resolved scheme to identify an elementary fractionalization process; injection of charge q from a non-interaction region into an interacting and scattering region of one-dimensional channels results in the formation of a collective excitation with charge (1-r)q by reflecting fractionalized charge rq. The fractionalization factors, r = 0.34 ± 0.03 for ν = 2/3 and r = 0.49 ± 0.03 for ν = 2, are consistent with the quantized values of 1/3 and 1/2, respectively, which are expected in the disorder dominated regime. The scheme can be used for generating and transporting fractionalized charges with a well-defined time course along a well-defined path.

Magnetic-Field-Dependent Equilibration of Fractional Quantum Hall Edge Modes

Fractional conductance is measured by partitioning a ν=1 edge state using gate-tunable fractional quantum Hall (FQH) liquids of filling 1/3 or 2/3 for current injection and detection. We observe two sets of FQH plateaus 1/9, 2/9, 4/9 and 1/6, 1/3, 2/3 at low and high magnetic field ends of the ν=1 plateau, respectively. The findings are explained by magnetic field dependent equilibration of three FQH edge modes with conductance e^{2}/3h arising from edge reconstruction. The results reveal a remarkable enhancement of the equilibration lengths of the FQH edge modes with increasing magnetic field.

Topological Classification of Shot Noise on Fractional Quantum Hall Edges

Electrical and thermal transport on a fractional quantum Hall edge are determined by topological quantities inherited from the corresponding bulk state. While electrical transport is the standard method for studying edges, thermal transport appears more challenging. Here, we show that the shot noise generated on the edge provides a fully electrical method to probe the edge structure. In the incoherent regime, the noise falls into three topologically distinct universality classes: charge transport is always ballistic while thermal transport is either ballistic, diffusive, or "antiballistic." Correspondingly, the noise either vanishes, decays algebraically, or is constant up to exponentially small corrections in the edge length.

Superconducting Correlations Out of Repulsive Interactions on a Fractional Quantum Hall Edge

We consider a fractional quantum Hall bilayer system with an interface between quantum Hall states of filling fractions (ν_{top},ν_{bottom})=(1,1) and (1/3,2), motivated by a recent approach to engineering artificial edges [Y. Ronen et al., Nat. Phys. 14, 411 (2018)NPAHAX1745-247310.1038/s41567-017-0035-2]. We show that random tunneling and strong repulsive interactions within one of the layers will drive the system to a stable fixed point with two counterpropagating charge modes which have attractive interactions. As a result, slowly decaying correlations on the edge become predominantly superconducting. We discuss the resulting observable effects and derive general requirements for electron attraction in Abelian quantum Hall states. The broader interest in fractional quantum Hall edge with quasi-long-range superconducting order lies in the prospects of hosting exotic anyonic boundary excitations, which may serve as a platform for topological quantum computation.

Synthesizing a ν=2/3 fractional quantum Hall effect edge state from counter-propagating ν=1 and ν=1/3 states

Topological edge-reconstruction occurs in hole-conjugate states of the fractional quantum Hall effect. The frequently studied filling factor, ν = 2/3, was originally proposed to harbor two counter-propagating modes: a downstream v = 1 and an upstream v = 1/3. However, charge equilibration between these two modes always led to an observed downstream v = 2/3 charge mode accompanied by an upstream neutral mode. Here, we present an approach to synthetize a v = 2/3 edge mode from its basic counter-propagating charged constituents, allowing a controlled equilibration between the two counter-propagating charge modes. This platform is based on a carefully designed double-quantum-well, which hosts two populated electronic sub-bands (lower and upper), with corresponding filling factors, v_l and v_u. By separating the 2D plane to two gated intersecting halves, each with different fillings, counter-propagating chiral modes can be formed along the intersection line. Equilibration between these modes can be controlled with the top gates’ voltage and the magnetic field.

The boundaries of fractional quantum Hall states can host multiple, interacting one-dimensional edge modes, which test our understanding of strongly interacting systems. Here the authors observe the edge-mode equilibration transition that was predicted for the ν=2/3 fractional quantum Hall state.

Counter-propagating charge transport in the quantum Hall effect regime

The quantum Hall effect, observed in a two-dimensional (2D) electron gas subjected to a perpendicular magnetic field, imposes a 1D-like chiral, downstream, transport of charge carriers along the sample edges. Although this picture remains valid for electrons and Laughlin's fractional quasiparticles, it no longer holds for quasiparticles in the so-called hole-conjugate states. These states are expected, when disorder and interactions are weak, to harbor upstream charge modes. However, so far, charge currents were observed to flow exclusively downstream in the quantum Hall regime. Studying the canonical spin-polarized and spin-unpolarized v = 2/3 hole-like states in GaAs-AlGaAs heterostructures, we observed a significant upstream charge current at short propagation distances in the spin unpolarized state.

Transmission of heat modes across a potential barrier

Controlling the transmission of electrical current using a quantum point contact constriction paved a way to a large variety of experiments in mesoscopic physics. The increasing interest in heat transfer in such systems fosters questions about possible manipulations of quantum heat modes that do not carry net charge (neutral modes). Here we study the transmission of upstream neutral modes through a quantum point contact in fractional hole-conjugate quantum Hall states. Employing two different measurement techniques, we were able to render the relative spatial distribution of these chargeless modes with their charged counterparts. In these states, which were found to harbor more than one downstream charge mode, the upstream neutral modes are found to flow with the inner charge mode-as theoretically predicted. These results unveil a universal upstream heat current structure and open the path for more complex engineering of heat flows and cooling mechanisms in quantum nano-electronic devices.

Spin Mode Switching at the Edge of a Quantum Hall System

Quantum Hall states can be characterized by their chiral edge modes. Upon softening the edge potential, the edge has long been known to undergo spontaneous reconstruction driven by charging effects. In this Letter we demonstrate a qualitatively distinct phenomenon driven by exchange effects, in which the ordering of the edge modes at ν=3 switches abruptly as the edge potential is made softer, while the ordering in the bulk remains intact. We demonstrate that this phenomenon is robust, and has many verifiable experimental signatures in transport.

Charge Density Waves in Graphite: Towards the Magnetic Ultraquantum Limit

Graphite is a model system for the study of three-dimensional electrons and holes in the magnetic quantum limit, in which the charges are confined to the lowest Landau levels. We report magneto-transport measurements in pulsed magnetic fields up to 60 T, which resolve the collapse of two charge density wave states in two, electron and hole, Landau levels at 52.3 and 54.2 T, respectively. We report evidence for a commensurate charge density wave at 47.1 T in the electron Landau level, and discuss the likely nature of the density wave instabilities over the full field range. The theoretical modeling of our results predicts that the ultraquantum limit is entered above 73.5 T. This state is an insulator, and we discuss its correspondence to the "metallic" state reported earlier. We propose that this (interaction-induced) insulating phase supports surface states that carry no charge or spin within the planes, but does, however, support charge transport out of plane.

Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices

Charge carriers in the quantum Hall regime propagate via one-dimensional conducting channels that form along the edges of a two-dimensional electron gas. Controlling their transmission through a gate-tunable constriction, also called quantum point contact, is fundamental for many coherent transport experiments. However, in graphene, tailoring a constriction with electrostatic gates remains challenging due to the formation of p–n junctions below gate electrodes along which electron and hole edge channels co-propagate and mix, short circuiting the constriction. Here we show that this electron–hole mixing is drastically reduced in high-mobility graphene van der Waals heterostructures thanks to the full degeneracy lifting of the Landau levels, enabling quantum point contact operation with full channel pinch-off. We demonstrate gate-tunable selective transmission of integer and fractional quantum Hall edge channels through the quantum point contact. This gate control of edge channels opens the door to quantum Hall interferometry and electron quantum optics experiments in the integer and fractional quantum Hall regimes of graphene.

Quantum point contacts are gate-tunable constrictions allowing for control of charge carrier transmission in 2D electron gases. Here, the authors fabricate a hBN/graphene/hBN van der Waals heterojunction to enable quantum point contact devices in the integer and fractional quantum Hall regimes.

Suppression of Interference in Quantum Hall Mach-Zehnder Geometry by Upstream Neutral Modes

Mach-Zehnder interferometry has been proposed as a probe for detecting the statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states. Here, we focus on interferometers made of multimode edge states with upstream modes. We find that the interference visibility is suppressed due to downstream-upstream mode entanglement; the latter serves as a "which path" detector to the downstream interfering trajectories. Our analysis tackles a concrete realization of a filling factor of ν=2/3, but its applicability goes beyond that specific case, and encompasses the recent observation of the ubiquitous emergence of upstream neutral modes in FQH states. The latter, according to our analysis, goes hand in hand with the failure to observe Mach-Zehnder anyonic interference in fractional states. We point out how charge-neutral mode disentanglement will resuscitate the interference signal.

Coulomb blockade with neutral modes

We study transport through a quantum dot in the fractional quantum Hall regime with filling factors ν=2/3 and ν=5/2, weakly coupled to the leads. We account for both injection of electrons to or from the leads, and quasiparticle rearrangement processes between the edge and the bulk of the quantum dot. The presence of neutral modes introduces topological constraints that modify qualitatively the features of the Coulomb blockade (CB). The periodicity of CB peak spacings doubles and the ratio of spacing between adjacent peaks approaches (in the low temperature and large dot limit) a universal value: 2∶1 for ν=2/3 and 3∶1 for ν=5/2. The corresponding CB diamonds alternate their width in the direction of the bias voltage and allow for the determination of the neutral mode velocity, and of the topological numbers associated with it.

Nonequilibrated counterpropagating edge modes in the fractional quantum Hall regime

It is well established that density reconstruction at the edge of a two-dimensional electron gas takes place for hole-conjugate states in the fractional quantum Hall effect (such as v=2/3, 3/5, etc.). Such reconstruction leads, after equilibration between counterpropagating edge channels, to a downstream chiral current edge mode accompanied by upstream chiral neutral modes (carrying energy without net charge). Short equilibration length prevented thus far observation of the counterpropagating current channels-the hallmark of density reconstruction. Here, we provide evidence for such nonequilibrated counterpropagating current channels, in short regions (l=4  μm and l=0.4  μm) of fractional filling v=2/3 and, unexpectedly, v=1/3, sandwiched between two regions of integer filling v=1. Rather than a two-terminal fractional conductance, the conductance exhibited a significant ascension towards unity quantum conductance (GQ=e(2)/h) at or near the fractional plateaus. We attribute this conductance rise to the presence of a nonequilibrated channel in the fractional short regions.

Recent experimental progress of fractional quantum Hall effect: 5/2 filling state and graphene

The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally, FQHE has been observed in high-mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene.

Transmission phase lapses through a quantum dot in a strong magnetic field

The phase of the transmission amplitude through a mesoscopic system contains information about the system's quantum mechanical state and excitations thereof. In the absence of an external magnetic field, abrupt phase lapses occur between transmission resonances of quantum dots and can be related to the signs of tunneling matrix elements. They are smeared at finite temperatures. By contrast, we show here that in the presence of a strong magnetic field, phase lapses represent a genuine interaction effect and may occur also on resonance. We identify a relevant physical regime where these phase lapses are robust against finite temperature broadening.

Edge reconstruction in the ν=2/3 fractional quantum Hall state

The edge structure of the ν=2/3 fractional quantum Hall state has been studied for several decades, but recent experiments, exhibiting upstream neutral mode(s), a plateau at a Hall conductance of 1/3(e2/h) through a quantum point contact, and a crossover of the effective charge, from e/3 at high temperature to 2e/3 at low temperature, could not be explained by a single theory. Here we develop such a theory, based on edge reconstruction due to a confining potential with finite slope, that admits an additional ν=1/3 incompressible strip near the edge. Renormalization group analysis of the effective edge theory due to disorder and interactions explains the experimental observations.

Topological Order: From Long-Range Entangled Quantum Matter to a Unified Origin of Light and Electrons

We review the progress in the last 20–30 years, during which we discovered that there are many new phases of matter that are beyond the traditional Landau symmetry breaking theory. We discuss new “topological” phenomena, such as topological degeneracy that reveals the existence of those new phases—topologically ordered phases. Just like zero viscosity defines the superfluid order, the new “topological” phenomena define the topological order at macroscopic level. More recently, we found that at the microscopical level, topological order is due to long-range quantum entanglements. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional charges and fractional statistics. Long-range quantum entanglements can even provide a unified origin of light and electrons; light is a fluctuation of long-range entanglements, and electrons are defects in long-range entanglements.

Chargeless heat transport in the fractional quantum Hall regime

We demonstrate a direct approach to investigate heat transport in the fractional quantum Hall regime. At a filling factor of ν=4/3, we inject power at quantum point contacts and detect the related heating from the activated current through a quantum dot. The experiment reveals a chargeless heat transport from a significant heating that occurs upstream of the power injection point, in the absence of a concomitant electrical current. By tuning in situ the edge path, we show that the chargeless heat transport does not follow the reverse direction of the electrical current path along the edge. This unexpected heat conduction, whose mechanism remains to be elucidated, may play an important role in the physics of the fractional quantum Hall regime.

Upstream neutral modes in the fractional quantum Hall effect regime: heat waves or coherent dipoles

Counterpropagating (upstream) chiral neutral edge modes, which were predicted to be present in hole-conjugate states, were observed recently in a variety of fractional quantum Hall states (ν=2/3, ν=3/5, ν=8/3, and ν=5/2), by measuring the charge noise that resulted after partitioning the neutral mode by a constriction (denoted, as N→C). Particularly noticeable was the observation of such modes in the ν=5/2 fractional state--as it sheds light on the non-Abelian nature of the state's wave function. Yet, the nature of these unique, upstream, chargeless modes and the microscopic process in which they generate shot noise, are not understood. Here, we study the ubiquitous ν=2/3 state and report of two main observations: First, the nature of the neutral modes was tested by "colliding" two modes, emanating from two opposing sources, in a narrow constriction. The resultant charge noise was consistent with local heating of the partitioned quasiparticles. Second, partitioning of a downstream charge mode by a constriction gave birth to a dual process, namely, the appearance of an upstream neutral mode (C→N). In other words, splitting "hole conjugated" type quasiparticles will lead to an energy loss and decoherence, with energy carried away by neutral modes.

Shot noise and charge at the 2/3 composite fractional quantum Hall state

The exact structure of edge modes in "hole conjugate" fractional quantum Hall states remains an unsolved issue despite significant experimental and theoretical efforts devoted to their understanding. Recently, there has been a surge of interest in such studies led by the search for neutral modes, which in some cases may lead to exotic statistical properties of the excitations. In this Letter, we report on detailed measurements of shot noise, produced by partitioning of the more familiar 2/3 state. We find a fractional charge of (2/3)e at the lowest temperature, decreasing to e/3 at an elevated temperature. Surprisingly, strong shot noise had been measured on a clear 1/3 plateau upon partitioning the 2/3 state. This behavior suggests an uncommon picture of the composite edge channels quite different from the accepted one.

Observation of chiral heat transport in the quantum Hall regime

Heat transport in the quantum Hall regime is investigated using micron-scale heaters and thermometers positioned along the edge of a millimeter-scale two dimensional electron system (2DES). The heaters rely on localized current injection into the 2DES, while the thermometers are based on the thermoelectric effect. In the nu=1 integer quantized Hall state, a thermoelectric signal appears at an edge thermometer only when it is "downstream," in the sense of electronic edge transport, from the heater. When the distance between the heater and the thermometer is increased, the thermoelectric signal is reduced, showing that the electrons cool as they propagate along the edge.

Universal periods in quantum Hall droplets

Using the hierarchy picture of the fractional quantum Hall effect, we study the ground-state periodicity of a finite size quantum Hall droplet in a quantum Hall fluid of a different filling factor. The droplet edge charge is periodically modulated with flux through the droplet and will lead to a periodic variation in the conductance of a nearby point contact, such as occurs in some quantum Hall interferometers. Our model is consistent with experiment and predicts that superperiods can be observed in geometries where no interfering trajectories occur. The model may also provide an experimentally feasible method of detecting elusive neutral modes and otherwise obtaining information about the microscopic edge structure in fractional quantum Hall states.

Universality of the edge-tunneling exponent of fractional quantum Hall liquids

In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for nu=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "external" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.

Interactions suppress quasiparticle tunneling at Hall bar constrictions

Tunneling of fractionally charged quasiparticles across a two-dimensional electron system on a fractional quantum Hall plateau is expected to be strongly enhanced at low temperatures. This theoretical prediction is at odds with recent experimental studies of samples with weakly pinched quantum-point-contact constrictions in which the opposite behavior is observed. We argue here that this unexpected finding is a consequence of electron-electron interactions near the point contact.

Strongly correlated fractional quantum Hall line junctions

We have studied a clean finite-length line junction between interacting counterpropagating single-branch fractional quantum Hall edge channels. Exact solutions for low-lying excitations and transport properties are obtained when the two edges belong to quantum Hall systems with different filling factors and interact via the long-range Coulomb interaction. Charging effects due to the coupling to external edge-channel leads are fully taken into account. Conductances and power laws in the current-voltage characteristics of tunneling are strongly affected by interedge correlations.

Nonuniversal behavior of scattering between fractional quantum Hall edges

In a fractional quantum Hall system with a narrow constriction, tunneling of quasiparticles between states at different edges can lead to resistance and to shot noise. The ratio of the shot noise to the backscattered current, in the weak scattering regime, measures the fractional charge of the quasiparticle, which has been confirmed in several experiments. However, the predicted nonlinearity of the resistance was apparently not observed in some of these cases. As a possible explanation, we consider a model where coupling between the current carrying edge mode and additional phononlike edge modes can lead to nonuniversal exponents in the current-voltage characteristic.